metacognition in higher education - core

Metacognition in Higher Education

Henk Vos

2001

Ph.D. thesisUniversity of Twente

Also available in print:http://www.tup.utwente.nl/

T w e n t e U n i v e r s i t y P r e s s

http://www.tup.utwente.nl/

Metacognition in Higher Education

Promotiecommissie

Voorzitter Prof.dr. J.M. Pieters Universiteit Twente, Toegepaste Onderwijskunde

Secretaris Prof.dr. J.M. Pieters Universiteit Twente, Toegepaste Onderwijskunde

Promotoren Prof.dr. S. Dijkstra Universiteit Twente, Toegepaste Onderwijskunde

Prof.dr. A. Lesgold University of Pittsburgh, School of Education

Leden Prof.dr.ir. J. van Amerongen Universiteit Twente, Elektrotechniek

Prof.dr. B. Collis Universiteit Twente, Toegepaste Onderwijskunde

Prof.dr. J. Greve Universiteit Twente, Technische Natuurkunde

Prof.dr. A. Pilot Universiteit Utrecht, Scheikunde; Natuur- en Sterrenkunde;

Lerarenopleiding

Publisher: Twente University Press,

P.O. Box 217, 7500 AE Enschede, the Netherlands, www.tup.utwente.nl

Cover design: Jo Molenaar, [deel 4] ontwerpers, Enschede

Print: Grafisch Centrum Twente, Enschede

©No part of this work may be reproduced by print, photocopy or any other means

without the permission in writing from the publisher.

ISBN 9036516658

T w e n t e U n i v e r s i t y P r e s s

H.Vos, Ed.D. thesis, Enschede, 2001

METACOGNITION IN HIGHER EDUCATION METACOGNITIE IN HET HOGER ONDERWIJS

PROEFSCHRIFT

ter verkrijging vande graad van doctor aan de Universiteit Twente,

op gezag van de rector magnificus,prof.dr. F.A. van Vught,

volgens besluit van het College voor Promotiesin het openbaar te verdedigen

door

Hendrik Vosgeboren op 10 oktober 1943

te Huizen (NH)

op woensdag 10 oktober 2001 te 15.00 uur

Dit proefschrift is goedgekeurd door de promotoren

prof. dr. S. Dijkstraprof. dr. A. Lesgold

Thiss sentence contains threee errors.

Deeze zin bevat drie fauten.

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Voorwoord (Preface) ...........................................................................................5Samenvatting (Summary) ...................................................................................7Summary (Samenvatting) .................................................................................11

Chapter 1. General Introduction: Metacognition in Engineering andInstruction................................................................................................15Abstract.........................................................................................................................15

1.1 Cognition and Engineering...........................................................................................181.1.1 Knowledge, Skills, and Information.............................................................................191.2 Information and Its Exchange.......................................................................................21

References ....................................................................................................................22

Chapter 2. Metacognition and its Development .............................................25Abstract.........................................................................................................................25

2.1 Metacognition and Cognition .......................................................................................262.1.1 Models of Metacognition..............................................................................................272.2 Development of Metacognition ....................................................................................292.3 Structures in Metacognition and Cognition..................................................................302.3.1 Metacognitive Knowledge, Strategies, and Schemata..................................................302.3.2 Three Views on Concepts.............................................................................................322.3.3 Restrictions of Working Memory and Metacognition..................................................342.3.4 Metacognitive Skills and Metacognitive Information ..................................................342.3.5 Combinations of Metacognition and Cognition ...........................................................352.4 Instruction in Metacognition ........................................................................................35

References ....................................................................................................................36

Chapter 3. Metacognition and Reasoning Schema Representations............39Abstract.........................................................................................................................39

3.1 Structure of and Access to Three-value Reasoning......................................................413.1.1 Structure of Reasoning .................................................................................................413.1.2 Three-value Reasoning .................................................................................................423.1.3 Access to the Structure of Reasoning ...........................................................................433.1.4 Hypothesis and Expectations........................................................................................443.2 Method..........................................................................................................................463.2.1 Participants ...................................................................................................................463.2.2 Material.........................................................................................................................463.2.3 Procedures ....................................................................................................................463.2.4 Data and Analysis.........................................................................................................473.3 Results ..........................................................................................................................483.3.1 The Weather Case.........................................................................................................493.3.2 The Oscilloscope Case..................................................................................................503.3.3 Comparison of the Weather and the Oscilloscope Cases .............................................513.4 Discussion.....................................................................................................................513.4.1 The Structures of Reasoning.........................................................................................513.4.2 The Dependence on the Content...................................................................................523.4.3 The Difference between Well-developed and Poorly-developed Metacognition ........533.4.4 Hints for Development of a (Metacognitive) Schema..................................................54

References ....................................................................................................................55

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Chapter 4. Development of Metacognition in an Instructional Modelwith Double Sequencing .........................................................................57Abstract.........................................................................................................................57

4.1 Instructional Design and Design of Research: Theoretical Considerations .................594.1.1 Strategy Variant ............................................................................................................594.1.2 Task Variant .................................................................................................................614.1.3 Knowledge Variant.......................................................................................................624.1.4 Design of the Study ......................................................................................................644.2 Method..........................................................................................................................654.2.1 Situation and Participants .............................................................................................654.2.2 Instructional Design......................................................................................................654.2.3 Analysis of the Data .....................................................................................................674.3 Results ..........................................................................................................................694.3.1 The (Metacognitive) Performance during the Course ..................................................694.3.2 Interactive Behavior in Class........................................................................................714.4 Discussion.....................................................................................................................714.4.1 Development of Metacognitive Skills ..........................................................................714.4.2 Interactions in Class......................................................................................................734.4.3 General Remarks ..........................................................................................................73

References ....................................................................................................................74

Chapter 5. Metacognition in Teachers’ Knowledge of a Course ..................77Abstract.........................................................................................................................77

5.1 Dimensions in Knowledge and Design of Research ....................................................795.1.1 Design of Research.......................................................................................................815.2 Method..........................................................................................................................825.2.1 Participants ...................................................................................................................825.2.2 Material and Observation Instrument ...........................................................................825.2.3 Procedures ....................................................................................................................855.2.4 Data...............................................................................................................................865.3 Results ..........................................................................................................................875.3.1 Metacognitive Characteristics of the Categorization Task...........................................875.3.2 The Categories Formed ................................................................................................885.3.3 The Role of Representations.........................................................................................885.3.4 Relations among the Categories ...................................................................................905.3.5 Hierarchical Relations: Abstract, Ideal, General ..........................................................915.4 Discussion.....................................................................................................................925.4.1 Metacognitive Characteristics of the Categorization Task...........................................925.4.2 The Categories Formed ................................................................................................925.4.3 The Role of Different Representations .........................................................................935.4.4 Relations among the Categories ...................................................................................935.4.5 Hierarchical Dimensions ..............................................................................................945.4.6 Critical Remarks ...........................................................................................................945.4.7 Consequences for Instruction .......................................................................................945.4.8 Concluding Remarks ....................................................................................................95

References ....................................................................................................................97

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Chapter 6. The Development of and Reflection on Metacognition ..............99Abstract.........................................................................................................................99

6.1 Access to Metacognition and the Role of Information.................................................996.2 Triple Coding and a Model of the Concept ‘Concept’ ...............................................1016.3 Metacognition and Instruction....................................................................................1026.4 Future Research ..........................................................................................................104

References ..................................................................................................................105

APPENDICES..................................................................................................107A The content of the lab course: lab guide; assignments; introduction .....................107A.1 The Three Parts of the Course ................................................................................107A.2 Four Types of Assignments....................................................................................107A.3 The Introduction to the Lab Guide: Its Construction .............................................110A.4 Presentation of the Content.....................................................................................110B The metacognitive strategy: a framework for investigation...................................119C Contradictory and comparative assignments ..........................................................120C.1 Contradictions Between Thoughts of the Students and

Observed Behavior of Circuits ...............................................................................120C.2 Comparative assignments .......................................................................................121D The data collection in the lab course and their analysis .........................................123D.1 Diagnostic entrance test and remedial exercise......................................................123D.2 Example of a grading sheet ....................................................................................126D.3 Example of a questionnaire ....................................................................................128D.4 Analysis of the metacognitive performance ...........................................................130E Overall performance before and after the change in the lab course .......................135E.1 Procedure ................................................................................................................135E.2 Results ....................................................................................................................135E.3 Discussion...............................................................................................................136F The influence of the composition of the class on performance..............................137F.1 Procedure ................................................................................................................137F.2 Results ....................................................................................................................137F.3 Discussion...............................................................................................................137G Characteristics of the categories made by the participants.....................................138H Methods to solve problems in NA..........................................................................144I Hierarchical dimensions seen by the teachers ........................................................145J Metacognitive aspects during execution of the tasks .............................................147K Analysis of the concepts and methods of NA.........................................................149K.1 Basic and Prerequisite Concepts.............................................................................149K.2 Systematic Problem Solving Approach..................................................................149L Salient features of metacognition in the experts verbal protocols..........................152L.1 Some Characteristics of E8.....................................................................................152L.2 Phased Recognition of Nonstandard Formulae. .....................................................152L.3 Category of Nodes or Connections.........................................................................152L.4 Special Categories ..................................................................................................152L.5 Types of Representation as Aids in Categorization................................................153L.6 Views on Instruction...............................................................................................153

Biographical note.............................................................................................154

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Voorwoord (Preface)

Mijn promotie in de natuurkunde in 1972 was de afronding van een proces waarin hetbegrijpen van de natuur met technische middelen op de voorgrond stond. Dit proefschrift ishet resultaat van het traject dat daarna is ingeslagen, waarin het leren begrijpen, de overdrachtvan de leerstof en de inrichting van het onderwijs aan de orde komen.

De aanleiding voor dit onderzoek was dat grote aantallen studenten niet slaagden voor heteerstejaars practicum Netwerkanalyse. Na een ontwikkelingsonderzoek werd het practicum-onderwijs gewijzigd en verbeterd. Later rees de behoefte om beter te begrijpen hoe hetonderwijs in de techniek ‘werkt’ en de ervaringen uit de praktijk te voorzien van eentheoretisch kader. Immers, er is niets zo praktisch als een goede theorie, mits de theorieaansluit op de praktijk. Naast het werk in de faculteit heb ik daarom mijn ‘vrije’ tijd, ruimaanwezig in de vorm van ADV-dagen, avonden en weekends, vrij gemaakt voor ditonderzoek naar metacognitie.

Het gebied van de onderwijskunde en de psychologie had voor mij vele onbekendevalleien in petto. Sanne Dijkstra, dank voor je begeleiding: soms geduldig elementaire zakenuitleggend, nooit voorschrijvend, altijd ruimte gevend voor het meest eigene. Het oplossenvan problemen staat centraal in je denken, de elektrotechniek als een vak van problemenoplossers moest daar wel goed bij aansluiten. Misschien is er gelegenheid om die aansluitingnog nauwer te maken. Alan Lesgold, I first met you in Pittsburgh right after the ‘blizzard ofthe century’ with 90 cm of snow in 24 hour. Thanks for your stimulating comments anddiscussions from such a large distance.

Op deze plaats wil ik alle studenten, docenten, en bestuurders van de faculteit bedanken,die mij zonder enige terughoudendheid van hun inzichten in het onderwijs lieten weten. Ikheb daarvan veel geleerd over elektrotechniek, en ook over goed onderwijs. Verder wil ikmijn collegae van het Onderwijskundig Centrum (OC) bedanken. Zij hebben mij geleerd ophoeveel verschillende wijzen je als onderwijskundige in de praktijk kunt bijdragen aanonderwijsontwikkeling.

Een aantal mensen hebben bijgedragen aan het onderzoek. Wiebe van der Veen en Henkvan den Hengel wil ik bedanken voor het zorgvuldig analyseren van de gegevens vanhoofdstuk 3. Het vak Netwerkanalyse was een product van de staf van de voormaligevakgroep “Theorie van (Netwerken,) Informatie, Communicatie en Systemen”. Ik wil in hetbijzonder Willem Gröneveld, Reint Brink, Carl Naber, Henk Tattje, en Leo Veelenturfbedanken voor de vakinhoudelijke basis waarvan ik gebruik heb gemaakt in hoofdstukken 4en 5. Ook wil ik Joke Oosterhuis-Geers van het Onderwijskundig Centrum bedanken voorhaar bijdragen aan het onderzoek van hoofdstuk 4. Dank gaat uit naar het College van Bestuurvan de Universiteit Twente, en de Decaan van de faculteit der Elektrotechniek, die het bezoekaan de docenten in het buitenland ten behoeve van het onderzoek in hoofdstuk 5 mogelijkmaakten, door het toestaan van educatief verlof en het verschaffen van financiëleondersteuning. Thanks are due to all participants to the study in Chapter 5 for their time andattention.

Velen hebben bijgedragen tot de gedachtevorming en ontwikkeling die aan dit proefschriftte grondslag liggen. Ik wil hier noemen de discussies in de Bolhaarkring (Henk Wagenaar,Cees Terlouw, Peter van der Sijde, Jan Gulmans), vaak gegrondvest in de leerpsychologievolgens de traditie van Vygotskij, Gal’perin en Davydov, net als de werkgroep van het OCdie gericht was op het oplossen van problemen (o.a. Albert Pilot, Frank Pothof, CeesTerlouw). Carel van Parreren heeft me geleerd wat abstraheren is. John Cowan, still workingfor the Open University in Schotland as a teacher of reflection, you renewed my interest inmetacognition. Otto Herrmann, Jan Gulmans, Henk Tattje, en het AIO-netwerk van TO(ProIST), bedankt voor jullie commentaar op eerdere concepten van hoofdstukken. Thyra

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Kamphuis wil ik bedanken voor haar bijdragen aan de lay-out, en Kevin McKenney voor hetbijschaven van het Engels.

Hier wil ik ook aan mijn ouders denken die op hun eigen wijze en met eindeloosvertrouwen de basis hebben gelegd voor mijn ontwikkeling. Tenslotte veel dank aan mijnzonen Albert, Gerard, Herman, Maarten, Willem-Jan en mijn vrouw Dineke. Jullie hebben mevaak gemist. Jullie hebben me soms ontzien als mijn gedachten elders waren, soms ook niet.Jullie hielpen me om met beide benen op de grond te blijven. Dank zij jullie heb ik nietkunnen vergeten dat de wereld van ideeën waarin een onderzoek zich afspeelt een beperktgedeelte is van de realiteit.

Henk Vos, mei 2001

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Samenvatting (Summary)

Dit proefschrift onderzoekt (1) het redeneren van studenten, (2) het leren in een practicumen (3) de ideeën van docenten over de structuur van een theoretisch vak. Dezeonderzoekingen worden verenigd door het begrip metacognitie dat globaal opgevat kanworden als cognitie over cognitie. Het redeneren over de inhoud van een vak wordt opgevatals een metacognitieve vaardigheid, het practicum onderwijs bevat metacognitieveaanwijzingen en ontwikkelt de metacognitie van de studenten, en docenten hebben meta-cognitieve kennis over de vakinhoud. Het proefschrift legt verbanden tussen theoretischebegrippen uit onderwijskunde en psychologie aan de ene kant en de praktijk van het onderwijsaan de andere kant. Aan de hand van de praktijkgevallen worden respectievelijk onderzocht:(1) toestanden van ontwikkeling van metacognitie, (2) de componenten van metacognitie inhet practicumonderwijs en hun effecten, en (3) de metacognitieve structuur van de vakinhoudbij docenten. De onderzoeksvragen kunnen kortweg geformuleerd worden als: hoe kanmetacognitie zich voordoen, waar is het te vinden in het (hoger) onderwijs, en welkemetacognitieve aspecten zijn belangrijk om te ontwikkelen in een vak. Voordat deze vragenonderzocht kunnen worden, is meer kennis over metacognitie nodig, en een geschikteonderzoeksplek.

In hoofdstuk 1 worden argumenten aangevoerd voor de aanname dat cognitie zowelkennis, vaardigheden (beide in personen aanwezig) als informatie (publiek toegankelijk)omvat. Hierdoor wordt metacognitie aan informatie gekoppeld. Dit is van belang omdat in hethoger onderwijs grote hoeveelheden informatie worden aangeboden aan studenten. Dezeinformatie bevat metacognitieve componenten, en speelt een rol bij de ontwikkeling vanmetacognitie. Bovendien kan in informatie gemakkelijker dan in kennis en vaardigheden, eenstructuur worden gevonden. Het kennisdomein van de elektrotechniek is geschikt ommetacognitieve componenten in het hoger onderwijs te bestuderen omdat het zelf eengestructureerd vakgebied is, sterk hiërarchisch van aard, met abstracte concepten, waarbij derealiteit een rol speelt in de relatie tussen theorie en technologische toepassingen. Verderwordt in dit hoofdstuk een algemene omschrijving van domeinkennis gegeven, wordt hettechnische begrip ‘informatie’ in relatie gebracht met onzekerheid en tekens (signs), wordt hetbelang van de vorm van tekens als metacognitief aspect van informatie benadrukt en wordt derol van uitwisseling van informatie besproken.

De componenten van metacognitie en het verschil tussen cognitie en metacognitie warenniet duidelijk genoeg gedefinieerd voor dit onderzoek. Om deze begrippen te kunnentoepassen op informatie was er een operationeel onderscheid nodig. Daarom werd een analysevan de literatuur uitgevoerd dat een model van metacognitie en cognitie moest opleveren. Hetonderzoek en de resultaten worden beschreven in hoofdstuk 2. Op basis van de analysesworden definities van cognitie en metacognitie voorgesteld. Tevens worden hiërarchischemodellen van metacognitie gepresenteerd, die instructie-ontwerpregels voor de ontwikkelingvan metacognitie leveren: maak de toegang tot metacognitie gemakkelijk; vestig de aandachtop karakteristieke trekken van metacognitie; koppel metacognitieve componenten aan elkaar;dwing mensen hun eigen cognitieve toestanden waar te nemen via contradicties in kennis,door het vergelijken van vaardigheden of het checken van de voortgang van hun activiteiten.

In metacognitie worden vier varianten onderscheiden: kennis, strategieën (of acties), taken(of doelen), en ervaringen. Een analyse van mentale schemata voor het oplossen vanproblemen, die probleemkennis en probleemaanpak verenigen, leidt tot een model vanschemata voor metacognitieve strategische kennis, die onafhankelijk zijn van de (vak)inhoud.Deze schemata zijn leeg (inhoudsvrij). Een integratie van de drie klassieke opvattingen overconcepten (als een klasse, een definitie, of een prototype) wordt voorgesteld als raamwerkvoor de abstracte begrippen van de elektrotechniek. Metacognitieve kennis over de capaciteit

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van het werkgeheugen (ongeveer zeven ‘brokken’) en de competitie tussen cognitie enmetacognitie om die ruimte, leidt tot de instructie-ontwerpregel dat het verstandig is om hetleren van de structuur van metacognitie te scheiden van het leren van de toegang tot diestructuur. Omdat metacognitie vier varianten omvat, cognitie drie componenten bevat, enmetacognitie over cognitie gaat, kan een matrix gepresenteerd worden van twaalfcombinaties. Tenslotte worden de ontwikkelde instructieontwerpregels vergeleken met die uitde literatuur.

In hoofdstuk 3 wordt een onderzoek naar toestanden van ontwikkeling van metacognitiebeschreven. De invloed van het kennisdomein op de structuur van en toegang tot hetredeneren werd bestudeerd, alsmede de vaardigheid in het hanteren van onzekerheid in hetredeneren. De deelnemers aan het onderzoek waren eerstejaars elektrotechniek studenten dieredeneertaken kregen over het weer en over de oscilloscoop. Er werd aangenomen dat demetacognitieve componenten van het redeneren waren verworven via een implicietleerproces. Een theoretisch model van redeneren werd ontworpen waarin de toegang tot destructuur van het redeneren werd verworven door het vormen van waarheidswaarden, en destructuur van het redeneren zelf werd voorgesteld door een waarheidstabel. De structuur vanhet redeneren betrof hier het samenstel van de redeneerprocedures volgens een conjunctie(zoals: de zon schijnt als het dag is en er geen wolk voor de zon zit). Onzekerheid wasopgenomen door een driewaardige logica te gebruiken, waarin de waarheidswaarden ‘waar’,‘onwaar’ en ‘onzeker’ waren. In de redeneertaken werd een vaststaande bewering alsuitgangspunt genomen om effecten van een tekort aan domeinkennis te minimaliseren.Vervolgens werd een uitspraak gedaan waarvan de correctheid bepaald moest worden opbasis van de gegeven bewering, met als mogelijkheden: juist, onjuist, het is niet te zeggen(bijvoorbeeld: het is niet te zeggen of de zon schijnt om 8 uur, dat hangt van het jaargetijdeaf). De antwoorden en de argumenten daarvoor werden verzameld.

In het redeneren over het weer konden drie structuren van redeneren geïdentificeerdworden: een ‘verborgen’ structuur waarin de deelnemers meestal maar niet consistent volgenseen gemeenschappelijk schema redeneerden (gebruikt door ongeveer 75 % van dedeelnemers), een ‘empirische’ die 25 % van de deelnemers consistent gebruikten, en een‘theoretische’ structuur die in overeenstemming was met de theoretische waarheidstabel (1van de 42 deelnemers). De deelnemers waren meestal in staat onzekere factoren correct tehanteren in hun redeneringen over het weer. In het redeneren over de oscilloscoop werdenveel minder correcte redeneringen aangetroffen, en werd alleen een verborgen structuuraangetroffen, die verschilde van die bij het redeneren over het weer. Nu waren de deelnemersniet in staat alle onzekere aspecten te hanteren in overigens correcte redeneringen. Hieruitwordt geconcludeerd dat zowel de structuur van het redeneren verandert als de toegang ertoevermindert bij de overgang naar een nieuw kennisdomein. Met name de vaardigheid in hethanteren van onzekere aspecten vermindert. De implicaties van dit unieke resultaat voor deverwerving van metacognitieve (redeneer)vaardigheden zijn: maak de verborgen structuurmeer consistent, maak de empirische structuur meer expliciet, en benadruk de theoretischestructuur door het gebruik van waarheidswaarden en waarheidstabellen. De relaties metonderzoek op het gebied van expertise worden eveneens besproken.

De resultaten worden in hoofdstuk 4 gebruikt om metacognitieve aspecten inpracticumonderwijs op te sporen en de ontwikkeling van de metacognitieve prestaties vanstudenten te bestuderen. Het practicumonderwijs in kwestie, waarin een systematische aanpakvan experimenteel onderzoek (een metacognitieve strategie) werd onderwezen, was grondigherzien wat leidde tot een stabiele verbetering van de slaagpercentages. De beschikbareonderzoeksgegevens werden geanalyseerd om de vier metacognitieve varianten in hetonderwijs te identificeren en te beschrijven, de wijze van ordening (sequentiëring) van deleertaken te expliciteren en de ontwikkeling van metacognitieve prestaties te bestuderen.

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De strategie variant van metacognitie was aanwezig in de vorm van een explicietgeformuleerde strategie voor experimenteel onderzoek, en omvatte o.a. de aanwijzing om deuitkomsten van metingen te vergelijken met die van berekeningen. Deze strategie werd in heteerste, bekende, sub-domein (deel van de het practicum) geïntroduceerd (de structuurontwikkelen in een bekend domein waar de toegang gemakkelijk is), in het tweede, nieuwe,sub-domein geoefend (toegang tot de structuur oefenen in een nieuw domein) en tenslottetoegepast in het derde, nieuwe, sub-domein. De metacognitieve taak variant omvatte de eisom de meetprocedures (het ‘kookboek’) zelf samen te stellen en beïnvloeding van demeetresultaten door de meetinstrumenten te overwegen. De metacognitieve kennis variant vielte onderscheiden in de structuur van de kennis, de vereiste relaties tussen woorden, formulesen schema’s, en het verschil tussen waarneming en mentale voorstelling. Om metacognitieveervaringen te stimuleren, bevatten sommige taken contradicties in kennis, andere eenvergelijking van verschillende methoden, en waren sommige opdrachten open.

De karakteristieken van de volgorde van de taken (sequentiëring) met betrekking totmetacognitie in het instructiesysteem over de drie sub-domeinen waren ‘fading’(aanwijzingen in het eerste sub-domein, later niet), ‘boosting’ (in de eerste twee sub-domeinen opdrachten van een hoger niveau dan het beoogde niveau, dat in het laatste sub-domein werd gevraagd), ‘just-in-time feedback’ (de resultaten van het eerste sub-domein snelnakijken en bespreken met de studenten), en ‘early marking’ (vanaf het eerste sub-domeincijfers toekennen). De taken werden in elk sub-domein geordend naar toenemende mate vancognitieve complexiteit. Op deze wijze kon het instructie-ontwerp van het practicumgekarakteriseerd worden als een systeem van dubbele sequentiëring van taken, namelijk opmetacognitief en cognitief niveau.

De ontwikkeling van de metacognitieve prestaties gedurende het practicum werdbestudeerd door de studenten in drie categorieën te verdelen op basis van hun cijfers voorandere vakken (goed, matig, zwak) en hun activiteiten en prestaties waar te nemen. Hiertoewerden scoringslijsten voor de logboeken van elk sub-domein en vragenlijsten voor elkezitting gebruikt. Er werden significante verschillen in prestaties voor metacognitievevaardigheden gevonden. Het leren van de strategie was geen lineair proces. De meestevoortgang werd geboekt in het tweede sub-domein, nadat feedback was gegeven op deprestaties in het eerste deel. In het derde deel kwamen de prestaties van de goede en de matigestudenten dichter bij elkaar, een uniek resultaat. De conclusie hieruit is dat in het nieuwepracticum minder studenten faalden doordat meer matige studenten dan in het verleden hetgeval was, metacognitieve prestaties leverden.

In hoofdstuk 5 wordt het onderzoek naar de metacognitieve kennis van docenten over deinhoud van hun vak gepresenteerd. De vraag was hoe de kennis van docenten van het vakNetwerkanalyse was georganiseerd. Mogelijke dimensies van de organisatie van kenniswerden geïdentificeerd en toegepast op de cursusinhoud om een meetinstrument teontwikkelen bestaande uit ongeveer 350 kaarten waarin deze dimensies gerepresenteerdwaren. Aan acht professoren werd gevraagd deze kaarten te sorteren volgens hun eigeninzicht. Op deze wijze konden in ongeveer anderhalf uur gegevens over de organisatie vanhun kennis verzameld worden.

De deelnemers werden verdeeld in twee groepen: zij die benoemd waren voor onderzoekop het vakgebied Netwerktheorie, en zij die onderzoek deden in andere onderwerpen. Dezegroepen categoriseerden de kaarten op een verschillende wijze, en gaven verschillenderelaties aan tussen de categorieën. De eerste groep had een meer coherente en theoretischerorganisatie van kennis dan de tweede. De woorden die elementen aanduidden, kwamen in deeerste groep vaker in dezelfde categorieën als de bijbehorende elementvergelijkingen enschematische symbolen (triple coding: woorden, schema’s en algebraïsche vergelijkingenhoren bij elkaar). Dit wijst op gedachteeenheden die bestaat uit vier componenten:

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woordvorm, algebraïsche vorm, schematische vorm, en betekenis. In deze groep was eenorganisatie van kennis toegankelijk die correspondeerde met een systematische probleemaanpak (een metacognitieve strategie) voor problemen in de Netwerkanalyse. Dezecorrespondentie was een uniek resultaat. Verder werden drie categorieën van basisconceptengeïdentificeerd: praktijk, voorkennis en basiskennis, naast meer complexe en afgeleidegeavanceerde concepten. Er werden aanwijzingen gevonden voor drie typen van hiërarchischedimensies die een rol speelden in de organisatie van kennis, die aangeduid worden als:abstract- specifiek; algemeen- gedetailleerd; ideaal- reëel. De implicaties van de resultatenworden besproken.

De resultaten van de onderzoekingen tonen aan dat het begrijpen van een nieuwkennisdomein door studenten niet alleen moeilijk is door de onbekende cognitieve inhoud,maar ook doordat de structuur van de noodzakelijke metacognitie en de toegang daartoeontwikkeld moeten worden. Dit kan gemakkelijk tot overbelasting van het korte-termijngeheugen leiden. In het systeem van dubbele sequentiëring werd gevonden dat matigestudenten in de ontwikkeling van hun metacognitie dichter bij de goede studenten komen. Ditbetekent dat metacognitieve componenten in korte tijd aan grote groepen studenten geleerdkunnen worden met speciaal ontworpen onderwijs, of, in termen van opbrengst, dat deslaagpercentages hoog kunnen zijn zonder de eisen aan de prestaties van de studenten teverminderen. Verder is gebleken dat metacognitieve strategieën eerder gevonden worden bijde (theoretische georiënteerde) docenten die onderzoek doen in het vak waarin ze les geven.

In de algemene discussie in hoofdstuk 6 wordt geconcludeerd dat toegang tot metacognitieplaats vindt door het modelleren van de resultaten van cognitieve acties in speciale concepten,zoals waarheidswaarden en fysische grootheden. De theoretische ontwikkeling van dezeconcepten vindt blijkbaar plaats via een symbolisering in algebraïsche en schematische vormen een verdere coherente integratie. Op basis van de bevindingen wordt een recursievedefinitie van het begrip ‘begrip’ voorgesteld. Tenslotte worden enige algemene aanbevelingengeformuleerd voor de ontwikkeling van metacognitieve strategieën bij studenten en voorverder onderzoek.

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Summary (Samenvatting)

This thesis studies (a) the reasoning of students, (b) teaching and learning in a laboratorycourse, and (c) the ideas of teachers about the structure of a theoretical course. These studieshave been united by the concept of metacognition that can be understood globally ascognition on cognition. Reasoning about the content of a course is considered to be ametacognitive skill, the instruction in the lab course contains metacognitive hints anddevelops the metacognition of the students, and teachers have metacognitive knowledge aboutthe domain. The thesis relates theoretical concepts from instructional science and psychologyon the one hand with the practice of education on the other hand. The studies of the threepractical cases are focused on: (a) the states of development of metacognition (Chapter 3), (b)the components of metacognition in (lab) education (Chapter 4), and (c) the metacognitivestructure of knowledge about a theoretical domain (Chapter 5), respectively. The researchquestions can be formulated shortly as: what forms of metacognition can be distinguished,where can metacognition be found, and what aspects of metacognition are important to bedeveloped in a course. Before these questions can be studied, more knowledge onmetacognition is required (see Chapter 2) and a suitable research environment.

In the introductory Chapter 1 it is argued that cognition comprises knowledge, skills(both present in persons), and information (publicly accessible). The vague concept ofmetacognition is understood here as cognition about cognition. Metacognition is related toinformation in this way. This is of importance because in higher education large quantities ofinformation are presented to the students. This information comprises metacognitivecomponents, and plays a role in the development of metacognition. Moreover, it is easier tofind a structure in information than it is in knowledge and skills. The domain of electricalengineering is appropriate to study metacognitive components in higher education, because itis highly structured, hierarchical of character, using abstract concepts, but even so realityplays a role in the relation between theory and technological applications. Further a generaldescription of knowledge of a domain is presented as a framework in this chapter. Thetechnical concept ‘information’ is discussed and related to uncertainty and signs. Theimportance of the form of signs as a metacognitive aspect of information is emphasized, andthe role of exchange of information is discussed.

The components involved in metacognition and the difference between cognition andmetacognition were not defined clearly enough to apply these concepts to information. Ananalysis of the literature was therefore carried out in order to construct a model ofmetacognition. This research and its results are described in Chapter 2. Definitions of bothcognition and metacognition have been formulated on the basis of the analyses. Hierarchicalmodels of metacognition have been constructed that represent instructional-design rules forthe development of metacognition: facilitate the access to metacognition; draw attention tocharacteristic features of metacognition; relate metacognitive components to each other;compel people to observe their own cognitive states by contradictions in knowledge orcomparison of skills.

Four variants of metacognition were distinguished: knowledge, strategies (or actions),tasks (or goals), and experiences. An analysis of problem schemata, that unite problemknowledge and a problem approach, led to a model of content-independent schemata formetacognitive strategic knowledge. These schemata are empty (content-free). An integrationof the three classical views on concepts (as a class, a definition, and a prototype) wasproposed as a framework for the abstract concepts of electrical engineering. Themetacognitive knowledge on the capacity of working memory (about seven chunks) and thecompetition between cognition and metacognition for this space, leads to the instructionaldesign rule of separating the learning of the structure of metacognition from learning to get

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access to this structure. Because four variants are distinguished in metacognition, and threecomponents in cognition, and metacognition is about cognition, a matrix of twelvecombinations can be distinguished. Finally, the consequences of the model for instruction inmetacognition are compared to design prescriptions from literature.

In Chapter 3 a study of states of the development of metacognition is described. Theinfluence that the content of a subject had on the structure of and access to reasoning wasstudied, as well as the ability to handle uncertainty in reasoning. The participants were first-year electrical engineering students who were presented with reasoning tasks about theweather and about the oscilloscope. It was assumed that the metacognitive components ofreasoning were acquired by an implicit learning process. Theoretically, access to the structureof reasoning was acquired by the formation of truth values, and the structure itself wasrepresented by a truth table. The structure of reasoning comprised here the system of thereasoning procedures according to a conjunction like: the sun shines if it is daytime and ifthere is no cloud in front of the sun). Uncertainty was included by using a three-value logic inwhich the truth-values were ‘true’, ‘false’ and ‘uncertain’. In the reasoning tasks, an assertionwas given as a starting point to diminish effects from a lack of domain knowledge. Astatement was then presented, the correctness of which had to be determined on the basis ofthe assertion, with the possible answers: correct, wrong, one cannot tell (of example: onecannot tell whether the sun shines at eight o’clock, that depends on the season). The answersand the arguments supporting them were collected.

In reasoning about the weather, three structures of reasoning could be distinguished: ahidden structure in which the participants mostly but not consistently reasoned according to acommon schema (used by 75 % of the participants), an empirical structure consistently usedby 25 % of the participants, and a theoretical structure in agreement with the theoretical truthtable (used by 1 of the 40 participants). The participants were mostly able to handle alluncertain components in their reasoning about the weather. In reasoning about theoscilloscope, less correct reasoning procedures were found. A hidden structure could bedistinguished that differed from the one found in reasoning about the weather. Now, theparticipants were not able to handle all uncertain components in otherwise correct reasoning.From these findings it is concluded that both the structure of and access to reasoning changein a new domain. Especially the ability to handle uncertain components diminishes. Theimplications of this unique result for the acquisition of metacognitive (reasoning) skills are:make the hidden structure more consistent, make the empirical structure more explicit, andemphasize the theoretical structure by using truth values and truth tables. The relations withresearch on expertise are discussed.

The results were used in Chapter 4 to identify metacognitive components in a laboratorycourse and to study the development of the metacognitive performance of the students. Thelab course, in which a systematic approach to experimental investigation (a metacognitivestrategy) was taught, was thoroughly revised, which revision led to a stable improvement inthe passing rates. The data and observations on the course were reanalyzed in order to identifyand describe the four metacognitive variants in the instructional format, to explicate thesequencing of the learning tasks, and to study the development of metacognitive performance.The lab course consisted of three parts, considered as three separate subdomains ofknowledge.

The metacognitive strategy variant was explicitly formulated and included among others,the prescription to compare the results of measurements with the outcome of calculations.This strategy was introduced in the first, known, subdomain (developing the structure in awell-known domain in which access is easy), practiced in the second, new, subdomain(practice access to the structure in a new domain), and applied in the third, new, subdomain.The metacognitive task variant included the requirement to the students to construct the

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measurement procedures (the ‘cookbook’) themselves and to consider the possible influenceof the measurement instruments on the results. The metacognitive knowledge variant includedthe structure of knowledge, the relation between words, formulae, and schematics, and thedifference between observation and mental image. To stimulate metacognitive experiences,some of the tasks included contradictions in knowledge, others a comparison of differentmethods, and some tasks were open-ended.

The characteristics of the sequencing of the tasks across the three subdomains with respectto metacognition were ‘fading’ (hints in the first subdomain only), ‘boosting’ (tasks in thefirst and second subdomain of a higher level than the final level, that was asked in the thirdsubdomain), ‘just-in-time feedback’ (fast correction of the results of the first subdomain, andfast discussion of the results with the students), and ‘early marking’ (starting in the firstsubdomain according to the final requirements). The tasks were sequenced according toincreasing cognitive complexity in each subdomain. Thus the instructional design of thecourse was characterized by a system of double sequencing of the tasks, namely both on ametacognitive and a cognitive level.

The development of the metacognitive performance of the students during the course wasstudied by dividing them, based on their results for other courses, into three categories (good,moderate, and weak students) and observing their activities and performance. The usedinstruments were scoring lists for assessing the logbooks in each subdomain, andquestionnaires for each session. Significant differences in performance on severalmetacognitive skills were found. Learning the strategy was not a linear process. It was foundthat the students made most progress in learning the strategy in part two of the course, afterfeedback had been given on their performance in the first part. In the third part of the course,the performance of the good students and that of the moderate ones on the metacognitiveskills approached each other, a unique result. This provided the explanation that the lower rateof failure in the new situation could be attributed to the improved learning of themetacognitive strategy by the moderate students.

The study of the metacognitive knowledge of teachers on the content of their course ispresented in Chapter 5. The question explored empirically was how the knowledge of teachersof a course on Network Analysis (NA) was organized. Possible dimensions of organization ofknowledge have been identified and applied to the content of this course to develop anobservation instrument. An unstructured, unsupervised, unconstrained categorization task ofabout 350 cards containing these dimensions was presented to eight professors. Data about theorganization of the knowledge of the teachers could be collected in about an hour and a half.

The participants were divided into two groups: those appointed for research on the subjectmatter of the NA, and those doing research in other domains. It turned out that the groups ofparticipants categorized the cards differently, and gave different relations among thecategories. The first group had more coherent and theoretical metacognitive knowledge thanthe second group. For the appointed group, the words indicating examples of elements comemore often in the same categories as the physical equations and the schematics associatedwith the examples (triple coding: words, schematic symbols, and algebra belonged together).This indicates units of thought consisting of four components: meaning, word-form, algebraicequation, and schematic. An organization of knowledge corresponding to a metacognitivestrategy for solving problems (a general, systematic problem-solving approach) in networkanalysis was accessible in this group, a unique result. Three categories of basic concepts wereidentified, called practical, prerequisite, and basic respectively, along with more complex orderived advanced concepts. Indications were found that three types of hierarchical dimensionsplayed a role in the organization of knowledge. The implications of the findings are discussed.

The results show that understanding a new domain of knowledge is difficult not onlybecause of new unknown content, but also because the structure of the necessary

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metacognition and access to it must first be developed. This might easily lead to an overloadof short-term memory.In the system of double sequencing it was found that moderate studentsapproached the level of the good students with respect to the development of theirmetacognition. This means that a metacognitive strategy can be taught to large groups ofstudents with specially designed instruction in a rather short time. In terms of output it meansthat the passing rates can be high without a concomitant lowering of the requirements for theperformance of the students. Further it is found that metacognitive strategies can be identifiedmore easily with the (more theoretically oriented) teachers who are appointed for research inthe subject in which they teach.

In the general discussion (Chapter 6) it is concluded that access to metacognition takesplace by modeling the results of cognitive actions in special concepts like truth values orphysical quantities. The theoretical development of these concepts apparently takes place viaa representation in algebraic and schematic form and a further coherent integration. On thebasis of the findings, a recursive definition of the concept ‘concept’ is proposed. Finally,some general instructional-design rules for stimulating the development of metacognitivestrategies with students are formulated and several lines for future research are proposed.

General introduction 15

Chapter 1

General Introduction: Metacognition in Engineering andInstruction

AbstractThis thesis studies (a) the reasoning of students, (b) teaching and learning in a laboratory

course, and (c) the ideas of teachers about the structure of a theoretical course. These studiesare united by the concept of metacognition that can be understood globally as cognition oncognition. Reasoning about the content of a course is considered to be a metacognitive skill,the instruction in the lab course contains metacognitive hints and develops the metacognitionof the students, and teachers have metacognitive knowledge about the domain. The thesisrelates theoretical concepts from instructional science and psychology on the one hand withthe practice of education on the other hand. The studies of the three practical cases arefocused on: (a) the states of development of metacognition (Chapter 3), (b) the components ofmetacognition in (lab) education (Chapter 4), and (c) the metacognitive structure ofknowledge about a theoretical domain (Chapter 5), respectively. The research questions canbe formulated shortly as: what forms of metacognition can be distinguished, where canmetacognition be found, and what metacognition is important to be developed in a course.Before these questions can be studied, more knowledge on metacognition is required (seeChapter 2). In this introductory chapter it is argued that cognition can be considered tocomprise knowledge and skills (present in persons), and information (publicly accessible).The components of metacognition, their relations, and the states of development ofmetacognition will be studied, i.e. structural elements of metacognition for identification ofmetacognition in instruction. It is assumed that the large quantities of information presentedto the students in higher education contain metacognitive components, while information alsoplays a role in the development of metacognition. Electrical engineering is an appropriatedomain to identify and study metacognitive components in higher education because of itsstructured, hierarchical nature, the abstract character of its concepts, and the role realityplays in the relation between theory and technology. A general characterization of domainknowledge is presented as a framework. The technical concept of information is discussedand related to metacognition and signs. The importance of the form of the signs as ametacognitive aspect of information is emphasized. The role of the exchange of information inthe process of learning is discussed.

When a child sees a painting for the first time, he (or, equivalently, she) might call it a‘calendar’. This solution to a classification problem can be viewed (a) as a lack of knowledgeabout paintings (in this case, the mother will say that this is not a calendar but a painting), or(b) as faulty reasoning (the mother will say that it is not a calendar because it has no dates onit). The child himself would not view the problem in either of these ways: his knowledgeabout his own behavior - his metacognition - has not yet fully developed.

Adults have strategies to solve classification problems. They know that in order toconclude that something is a calendar, it is first necessary to know the attributes of a calendar.If adults were not sure about these attributes, they would ask for these. This strategy is notalways used, however. Adults say for instance ‘electrical current is expensive’ withoutreference to the attribute of electricity that can be sold, i.e. the electrical energy. Now, this

16 General introduction

might be viewed as either a lack of knowledge about electricity or a lack of reasoning skills,because adults have developed metacognition. It will be argued that reasoning itself can alsobe a metacognitive skill.

Basic to the concept of metacognition is thinking about one’s own thoughts (e.g. Hacker,1998, p. 3). Both in knowledge and in reasoning about some topic, and also in other strategiesof checking the solution to a problem, cognitive and metacognitive aspects go together.People know what they know and what not. People know when to reason with the things theyknow and when to ask the things they do not know. In practice, no one bothers about thedifference between cognition and metacognition. In higher education, however, manymetacognitive skills and knowledge are of importance (e.g. Hacker, Dunlosky, & Graesser,1998; Weinert, 1987). Therefore it is important for instruction to be able to distinguishbetween them.

It seems that metacognition is well-developed in the case of calendars, but ill-developed inthe case of electricity. It is not clear, however, what the difference is and whether thisoriginates from differences in knowledge about the domain or lies on a level that isindependent of domain knowledge. It seems often that all that matters are the answer to aproblem (‘This is a painting’) or how to get to the answer (‘This is not a calendar because thepicture has no dates on it’). In many cases, however, it is important to know how you have toget to know the attributes of an object, or some other way to check progress towards theanswer. It seems often that information and lecturing are just to read and hear, respectively.Some parts of the information can contain metacognitive hints, however, that are easilyoverlooked or misunderstood, and are needed later to guide your cognitive enterprise.

The studies in this thesis focus on the identification and characterization of metacognitivecomponents in information and instruction. Not the thinking itself is the subject of research,but the structure of thinking about thinking: its components, their relations, and thedifferences in states of development. Weinert & Kluwe (1987) argued that motivation playsan important role in the development of metacognition, but this aspect was not studied andneither was the aspect of development itself. Where a development is mentioned, this shouldbe read as the state of being developed, independent of the way the process of developmenttook place. Perkins and Salomon (1989) stated that a general skill does not developindependent of the content of a knowledge domain, but that in this process bothcontextualization and decontextualization with respect to the content play a role. The presentstudy focuses on the metacognitive components in the way such a process is guided and in theresults of it.

Cognitive development can take place without much reference to metacognition. Thereason for studying these questions is that in higher education students have to handleknowledge in a way different from what they were used to. Earlier, in high school, studentshad already learned to construct knowledge; they formed concepts, acquired skills and learnedto process information. At university the knowledge to be acquired is more complex than inhigh school, the subject matter is to be studied with less guidance and help, and information isintroduced at a much higher rate. Also, the problems posed are often ill-defined and moredifficult to solve.

In order to meet this situation, the students have to be able to handle more and morecomplex concepts, skills and information simultaneously. Knowledge has to be betterorganized. Metacognitive skills like content-free reasoning, and solving realistic problems ina systematic way become more important. Also metacognitive knowledge, like having acoherent overview over the domain of knowledge, have to get more attention. The studentsneed to be more aware of their own cognitive processes and have to be better able to monitor,control and evaluate their strategies in learning. They need more metacognitive strategies foracquiring knowledge. These developments have to be guided by the information and


instruction they get.Sometimes the difference between metacognition and cognition was highlighted in

instruction. For instance, in solving problems of a domain, a cognitive skill like calculatingwas used. This skill was distinguished from the systematic use of that skill (at what time, inwhat circumstances) and from checking the progress and the outcome of the problem-solvingprocess (Schoenfeld, 1982). Thus knowledge of the difference between metacognition andcognition (see Chapter 2) could help to distinguish these aspects in education.

But such a systematic approach was not learned in the same way as calculating. Ifcalculating was considered to be a cognitive part of problem solving, solving problems in amethodical way included metacognitive aspects that were learned in a different way. It shouldbecome clear why metacognition could not be learned as cognition could. Probably this had todo with the difference between the content of metacognition when applied to a domain andthe content of cognition for that domain.

In short, metacognition and metacognitive skills should get more attention from instructorsand curriculum designers. Exploiting knowledge about metacognition to an advantage forlearning effectiveness should be possible ( Weinert,1987). Therefore the following questionswere addressed. What is metacognition, and how does it differ from cognition? What states ofdevelopment of metacognition could be traced? What components of metacognition could beidentified in information and instruction and how did metacognitive performance develop?What types of metacognition were relevant for a course as a whole? After the answers havebeen found in some case studies, it becomes possible to discuss how the findings can begeneralized and used for further development of metacognition. Each of these questions hasbeen approached in a separate study that is presented in a corresponding chapter.

In order to answer the questions about metacognition in higher education courses it wasnecessary to know what aspects of metacognition could be distinguished (Chapter 2). Afterthis the questions could be addressed as to which features of metacognition were different indifferent states of development, which features of development could be fostered byinstruction, and which features of metacognition were of enough importance to warrantattention and to be developed.

Human beings have some general metacognitive knowledge available. Metacognition oftendevelops spontaneously. For instance, human beings can reason without training in formallogic. They can solve many problems they meet. However, problem solving techniques werelearned by solving problems, reasoning was learned by reasoning about things, but whatprecisely developed in the learning process was not clear. It was found that more preciseinformation was needed on the states of development of a specific metacognitive skill inrelation to the cognitive domain knowledge (see Chapter 3).

In cases where teachers wanted metacognitive development, instruction was used tofacilitate or accelerate the learning process. It was found that some examples of instructioncould be identified that stimulated metacognitive development but either for special classes ofchildren (e.g. Brown & Campione, 1977) or just as an extra stimulus in education. Thereforemore knowledge was needed on how the development of metacognition was facilitated in aregular course, for example a course focused on the development of a metacognitive skill (seeChapter 4).

Also the question arose which metacognitive aspects could be distinguished in thearguments that lead to the structure of a course program. If several teachers presented thesame course, it turned out that they had their own ideas about the way the content should betaught, or was treated in the available books. Teachers often made their own lecture notes andwrote a book of their own. It was not clear why this was the case, and how their ideasdiffered. Therefore more knowledge was needed about the way different teachers structuredthe content of a course and what the metacognitive features in this structure were (see


Chapter 5).On the one hand, it could be supposed that metacognition was so general that many

components of it had developed, could be identified and were used both inside and outsideschools. On the other hand, it was supposed that the metacognition involved was connected tothe content of the domain. The study of the structure of metacognitive components ininformation and instruction was carried out in the domain of knowledge of electricalengineering. For this reason, the characteristics of this domain of knowledge that areimportant for this study will be highlighted first.

1.1 Cognition and EngineeringThe studies focused on the domain of Network Analysis (NA), also called Electrical

Circuit Theory. An electrical network is considered as an abstraction from an electricalsystem (circuit) and consists of ideal electrical elements, the electrical connections amongthese elements, and the electrical currents through and voltages across the elements in thenetwork. Interestingly, the results of NA can be generalized to many other domains in whichsystems can be defined, like mechanics and thermodynamics (e.g. Breedveld, 1984).

Some general characteristics of electrical engineering were of importance here.1. The basis of engineering is thinking in layers of systems and their parts, or, equivalently,

in a hierarchical structure of systems and subsystems. Solving a design problem starts with aprogram of requirements negotiated with the principal. In top-down design, on the mostabstract level the sketch of the object to be designed is a functional model, called a blockdiagram or block schematic. This describes the global way of functioning of the system as anartifact of engineering composed of subsystems that are first considered as black boxes. Theseblocks become more detailed in the designing process, until available components are met.Mathematical analysis and computer simulations are used to check whether the artifact willwork as designed. Next, a concrete, material product is produced following the detailedschematics. Top-down design can be summarized as representing ideas in more and moredetail until material components are met with which a product can be realized that meets theprogram of requirements.

Bottom-up design starts from the technical systems or material elements (calledcomponents) that are available. Both practical and theoretical knowledge about how theyfunction and interact with each other guide the combination of these components in serial,parallel, or more complicated ways. The product has a certain structure: components areconnected and electrically interact through these connections. A mathematical model of theproduct is made, and a mathematical analysis is carried out to ascertain that the product willfunction as supposed. Bottom-up design can be summarized as constructing (literally: puttingtogether) a complex new product from available products in a certain structure, and making amodel to check and explain its functioning. In practice, top-down and bottom-up design arecombined and applied alternatively. This means that ideas and reality meet each other in themathematical analysis of a design.

2. Another characteristic of electrical engineering is that the concepts and methods areabstract and require abstract logical thinking. For example, a basic concept like electrical‘resistance’ was in name borrowed from other domains by analogy and defined in an abstractway as a relation between electrical current and voltage (U = I.R, Ohm’s law). The word‘abstract’ here means ‘not directly observable’, but also means ‘applicable to a broad range ofobjects’. Thus abstract and broad are equivalent here, meaning that the concept can be appliedto every material object if necessary to describe its electrical properties. For example, thedetermination of the position of a probe, brought into the heart of a living patient via the veinsto repair holes in a partition membrane, can be done by using the electrical resistance of theheart.


3. Also the use of terms is precise, e.g. in distinguishing related terms. Definitions areformulated in such a way that they can be taken literally and used for generalization. Thesefeatures not only facilitate application, but also aid teaching because the basic instructions canbe precise and detailed, providing a reliable basis for the required knowledge.

It was assumed that the characteristics of structure in networks, hierarchical systems,practical and theoretical objects of thinking, abstract concepts, and precise languageembedded in this domain of knowledge, were favorable to study structure and hierarchy incognition and metacognition. Since both schematics and mathematical formulae next to wordswere used in information and instruction and were intended to guide thinking, it was alsoassumed that the role of metacognition in information could be studied here better than inother domains where words only were used.

1.1.1 Knowledge, Skills, and InformationInformation in instruction is considered to be outside there, in the form of books, lecture

notes, results of measurements, observable characteristics of objects, etc. Information has gotform in signs. In communication about and understanding of the domain, the role of signs(including symbols, tokens, etc.) is essential. It is however not generally accepted thatcognition, let alone metacognition, can be situated in information or signs. Therefore a shortdiscussion on this point is necessary.

Seel & Winn (1997) pointed out that sharing of cognition takes place between anindividual and a medium in which the signs are represented (distributed cognition). Before thedevelopment of cognitive science, cognition was considered to be a characteristic of humans,and so was metacognition. Cognitive science described cognition in such a way that byconsequence it included intelligent machines. The idea of distributed cognition extendedcognition to include signs. The signs were thought to be embedded in a culture, and to carryinformation.

Therefore, three types of cognition were both distinguished and related in this work.Knowledge or concepts, available in the brain of people. Skills or potential actions, to becarried out in thinking and in external actions. And information, with observablecharacteristics like other objects.

Knowledge and skills can be used to solve problems. The ability to use the knowledge isitself a skill. In instruction, problems are designed in such a way that the student can constructthe required knowledge and train problem-solving procedures at the same time (Dijkstra,1990). The information in the signs, in the data that come from the manipulation of objects,and in the communication that takes place in direct social interaction with teachers and peersplay a crucial role here. Information can also be stored as declarative or procedural knowledgefor later use.

Posner (1989) described cognitive science as the study of intelligent systems like humans.The results of this science yielded information on the way the memory works and on the wayhuman beings solve problems. If metacognition is loosely defined as cognition on cognition,then the results of cognitive science are metacognitive of character. Even a step further can bemade: if cognitive science is to describe cognition, it must include components that refer toitself (reflexive components).

Much of the knowledge on cognition resulted from cognitive science. Cognitive sciencestudied the cognitive processes of human beings by giving them tasks and information. But inthe instruction for these tasks, and in general in learning material, there were often clues thatintended to let persons think about the way in which the tasks should be fulfilled or thelearning should take place, part of which was metacognitive information. Thus part of theinstructional information addressed metacognition. It was supposed that education in generalmade use of information that included metacognitive components in many places. Which


places these were had to be clarified, and also the role this information played.The basis of cognition can be considered as follows. Posner (1973) considered

(conceptual) knowledge to be a static component of cognition, consisting of all kinds ofconcepts, facts, and literally remembered information. Skills were capacities to apply andgenerate knowledge, and were considered to be dynamic components of cognition (Posner,1973). It is, however, not satisfactory to call a concept static because keeping a concept inmind and in the brain requires activity and no indications have been found that a static storageof knowledge is possible in the brain. To open the possibility of cognition in information too,a different classification is proposed here.

A skill as a dynamic component of cognition -and certainly the related action- can beviewed as a dynamic cognitive state (of the brain), a state that changes during the action andhas to be controlled. Knowledge can be viewed as a stationary cognitive state (of the brain), astate that is maintained for a certain amount of time, as long as attention is drawn to aconcept, and that can be monitored. Information can be considered to be a static component ofcognition, because it is represented in signs in some medium. As long as these signs areobserved by a person, a stationary state is maintained by external stimuli.

Knowledge and skills, can thus be distinguished from information. This is in accordancewith the view of Dijkstra (1997) that information is personal knowledge that is made explicit,sometimes validated by scientific experiments, made public, agreed upon by the scientificcommunity, and stored. In this work, information is considered to be a mental representationthat has taken form, and thus can be called formalized knowledge. Information can be put intopapers and books, and can be made available in libraries. If it is read aloud, it can still beconsidered (quasi-) static cognition.

1.1.2 Knowledge of a DomainThe components of (formalized) domain knowledge can be found from the information

about a domain. These include laws, problem-solving methods, theories, and hypotheses.They are considered as combinations of the three basic types of cognition. For example,Ohm’s law includes three concepts, a multiplication, and an equivalence relation. If Ohm’slaw is considered as knowledge, this would mean that multiplication (a skill) and equivalence(a concept) should have become conceptual components (see Chapter 2). For application ofthe law (a skill) it is necessary to know how electrical current and voltage can be generatedand measured. The law could be externally represented as information in at least two forms:in words, and as a mathematical formula. The law could also be used as a method to calculatethe current from voltage and resistance, and thus develop into a skill.

For complicated problems more advanced methods are available, based on theory andrequiring the mastery of more skills (see Chapter 5). In general a theory is a system ofvalidated relations among knowledge, skills, and signs, in the form of statements that explainphenomena. It is usually externally represented as information in books. When someonestudies the book, its information can develop into knowledge, skills, and the reminiscence ofinformation in the brain of the student.

Two types of knowledge will be distinguished from the viewpoint of their foundation:empirical and theoretical. Empirical knowledge concerns knowledge that has been developedby manipulating the reality outside us, and reflecting on that reality. An example is Ohm’slaw: this relation has been validated experimentally, but can not yet be founded otherwise.Theoretical knowledge concerns knowledge that is founded in its own system of knowledgecomponents. It should be in accordance with, and explain empirical cognition, but its scope isdeeper and more general, and its foundation is in the axioms (a kind of hypotheses) about theexistence of basic components and basic procedures. Examples of theoretical cognition aremathematical theories of logic, and Network Theory. Theoretical knowledge is considered a


reality inside us that can be applied to outside reality in accordance with the theoreticalinformation available.

A hypothesis is a statement that, if found true, helps to solve a problem. It is knowledgethat seems highly probable, is made explicit, but is still subject to validation. In a personalproblem to be solved, a hypothesis is a purely subjective cognitive construct. In problems ofthe scientific community, a hypothesis is based on empirical and theoretical knowledge andcan become a proposal for an axiom, such as the invariance of the speed of light. Suchhypotheses are often called principles, like the conservation of electric charge is.

The concept of information was mentioned several times. It was included in cognition, andwill also be included in metacognition (see Chapter 2). This concept will be elaborated in thenext section and related to uncertainty. Since uncertainty in connection with knowledge is ametacognitive component (knowing that one is not certain about one’s knowledge), this willrelate information, uncertainty and metacognition (to be used in Chapter 3).

1.2 Information and Its ExchangeHuman forms of expression and communication take place on the basis of the use of signs.

The interactions of humans with other humans via signs lead to an exchange of informationthat can be distinguished from a communication of signs. Information is a concept that is usedin electrical engineering in a technical sense. The amount of information transmitted bycertain signs (signals) from a sender to a receiver via a medium can be defined if the range ofsigns that the sender could produce and the probability that each sign is produced, are known(Shannon, 1947). This is a property that cannot be attributed to the signs alone, but to thesystem as a whole only including the production of the signs by the sender and theirobservation by the receiver.

The amount of information in the transmission of certain signs is not stored in the signsalone, but is also dependent on the expectations of the receiver. The more possibilities thereceiver knows of, the larger the amount of information. One could say that the amount ofinformation corresponds to the amount of unexpectedness (for signs that are on the list ofpossibilities). Information has to do with a reduction of the uncertainty of the receiver withrespect to some state of affairs. With the foregoing in mind, Sheridan and Ferrell (1974)defined information as that property of messages, data, or other evidence (signs) that reducesone’s uncertainty about the true state of affairs. Interestingly, this definition could be used tomake a first try at calculating the study effort, expressed in hours, needed to study the contentof a course (Bosman, 1993).

The technical definition of information implies that those persons who are not uncertain,i.e. have no questions about what is the case or how to do something, and have no hypothesesthat show their uncertainty, can not receive any information in the technical sense from amessage or data. This also implies that if it is to be effective, instruction has to arousepersonal interest and questions. Otherwise the learner can at most store the signs for later use.Storing signs has to be distinguished from symbol (sign) processing as a form of thinking (cf.the discussion between Vera & Simon, 1993, and Greeno & Moore, 1993). With this in mind,information and signs are used as equivalent terms in the following chapters. But first therelation between metacognition and information is further elaborated.

Information can come to humans in all possible forms: books, lecture notes, audio-tapedexplanations of a teacher, computer-generated hints, video-taped role models, demonstrationsof skills, and computer programs, to name but a few. In instruction information is exchangedin a two-way communication of a beginner with an expert about objects and phenomena andhow to interact with them. The medium by which this communication takes place hasdeveloped in the past from the use of the voice and gestures to computers and the internet.Seel and Winn (1997) argued that the medium had an influence on knowledge construction,


and thus on instruction.It is argued here that not only the medium but also the form of the signs in one medium,

influence knowledge construction. Vygotskij (1962) found in his research the unit of verbalthought to be word meaning, so a unit of external form and internal representation. Heconsidered thinking to be a differentiation of social speech into inner speech and beyond, aprocess nearly without signs, in which the concepts got sense from the contexts in which theywere used and the associations with other memories. To form the concepts and to express theconcepts the form played an essential role. In this tradition Davydov (1977) stated that thebasis of theoretical thinking is finding the general form. Thus form and medium influencethinking. The step further is that also metacognition is influenced by the form of signs.

An example from mathematics might be appropriate. Today numbers are presented in thedecimal positional system, but the Roman way of presenting numbers was equally as valid(cf. Reif and Heller, 1982). However, multiplication in the Roman system was different,tedious and required much more training than our positional system requires. This is anexample of how the form of the information regulates cognitive skills. A person who knowsthe Arab digits and the use of the base-ten number system can multiply or learn to multiplymuch easier than someone who does not. Knowledge can be expressed in signs, and signsinfluence the construction of knowledge. The form and media of the signs will be integratedin the resulting knowledge.

From a psychological point of view, information is an aid in functioning and an object ofcognition, but from a sociological viewpoint it is a type of shared or distributed cognition, acultural property of cognition. In this view, cognition is a property of the signs of the culturewith the initiated people sharing this cognition. They possess the common codebook todecipher the meaning of the signs. And this codebook can also contain codes on how tohandle the form of the signs (meta-information).

Not all interactions or exchanges of information lead to learning. Learning is considered tobe a more or less long lasting change in the behavior of human beings. For instance, solving aproblem is based on processing information, but does not necessarily lead to learning. Onlysome special categories of problems lead to the construction of knowledge (and thus changebehavior). Solving problems is considered to lead to learning if the expectation of the studentsis that some general traits in the solution process lead to the capability to solve a category ofsimilar problems better, faster, etc. in the future. Some of these metacognitive aspects oflearning will be met in Chapter 4.

In the next chapter metacognition is studied in more detail in order be able to identify it ininformation and instruction.

ReferencesBosman, D. (1993). Indirect measurement of study effort. In Proceedings of the IMEKO

colloquium on the state and advances of measurement and instrumentation science (pp.152-157). London.

Breedveld, P.C. (1984). Physical system theory in terms of bond graphs. Unpublisheddoctoral dissertation, Twente University, Enschede, The Netherlands.

Brown, A.L., & Campione, J.C. (1977). Training strategic study time apportionment ineducable retarded children. Intelligence, 1, 94-107.

Davydov, V.V. (1977). Arten der Verallgemeinerung im Unterricht [Forms of generalizationin instruction]. Berlin: Volk und Wissen.

Dijkstra, S. (1990). The description of knowledge and skills for the purpose of instruction. InS. Dijkstra, B.H.A.M. van Hout Wolters & P.C. van der Sijde (Eds.), Research oninstruction: design and effects. Englewood Cliffs, NJ: Educational TechnologyPublications.


Dijkstra, S. (1997). The integration of instructional system design models and constructivisticdesign principles. Instructional Science, 25, 1-13.

Greeno, J.G., & Moore, J.L. (1993). Situativity and symbols: response to Vera and Simon.Cognitive Science, 17(1), 49-60.

Hacker, D.J., Dunlosky, J., & Graesser, A.C. (Eds.) (1998). Metacognition in educationaltheory and practice. London: Lawrence Erlbaum.

Perkins, D.N., & Salomon, G. (1989, January). Are cognitive skills context-bound?Educational Researcher, 16-25.

Posner, M.I. (1973). Cognition: an introduction. Glenview, IL: Scott, Foresman andCompany

Posner, M.I. (Ed.) (1989). Foundations of Cognitive Science. Cambridge, MA: Bradford/ MITPress.

Reif, F., & Heller, J.I. (1982). Knowledge structure and problem solving in physics.Educational Psychologist, 17(2), 102-127.

Schoenfeld, A.H. (1982). Measures of problem-solving performance and of problem-solvinginstruction. Journal for Research in Mathematics Education, 13(1), 31-49.

Seel, N.M., & Winn, W.D. (1997). Research on media and learning: distributed cognition andsemiotics. In R.D. Tennyson, F. Schott, N.M. Seel & S. Dijkstra (Eds.), InstructionalDesign: International Perspectives. Vol. 1: Theory, Research, and Models. Mahwah, NJ:Lawrence Erlbaum Associates.

Shannon, C.E. (1947). A mathematical theory of communication. The Bell System TechnicalJournal 27, 379-623.

Sheridan, T.B., & W.R. Ferrell (1974). Man-machine systems: Information, control, anddecision models of human performance. Cambridge: MIT Press.

Vera, A.H., & Simon, H.A. (1993). Situated action: A symbolic interpretation. CognitiveScience, 17(1), 7-48.

Vygotskij, L.S. (1962). Thought and Language. Cambridge, MA: MIT Press.Weinert, F.E. (1987). Introduction and overview: Metacognition and motivation as

determinants of effective learning and understanding. In F.E. Weinert & R.H. Kluwe(Eds.), Metacognition, motivation and understanding (pp. 1-16). Hillsdale, NJ: LawrenceErlbaum.

Weinert, F.E., & Kluwe, R.H. (1987). Metacognition, motivation and understanding.Hillsdale, NJ: Lawrence Erlbaum.

Metacognition and its development 25

Chapter 2

Metacognition and its Development

AbstractThe components involved in metacognition and the difference between cognition and

metacognition were not defined clearly enough to distinguish cognition from metacognition inhigher education. An analysis of the literature was therefore carried out in order to constructa model of metacognition. Definitions of both cognition and metacognition have beenformulated. Hierarchical models of metacognition have been constructed that giveinstructional-design rules for the development of metacognition: facilitating access to theorganization of metacognition; drawing attention to characteristic features of metacognition;relating metacognitive components to each other; compelling people to observe their owncognitive states by contradictions in knowledge or comparison of skills. Four variants ofmetacognition were distinguished: knowledge, strategies or actions, tasks or goals, andexperiences. An analysis of problem schemata, uniting problem knowledge and problemapproach, led to a model of content-free schemata for metacognitive strategic knowledge. Anintegration of the three classical views on concepts (as a class, a definition, and a prototype)was proposed as a framework for the abstract concepts of electrical engineering. Themetacognitive knowledge on the capacity of working memory restricts the use ofmetacognition in competition with cognition to about seven chunks. This leads to theinstructional design rule of separating the learning of the structure of metacognition fromgetting access to this structure. Because four variants are distinguished in metacognition,three components in cognition (knowledge, skills, and information), and metacognition isabout cognition, twelve combinations can be distinguished. Finally, the consequences of themodel for instruction in metacognition were compared to design prescriptions from literature.

The skill needed to read a text differs from the skill of human beings to monitor theirunderstanding of the text. The first is an example of a cognitive skill, the second of ametacognitive skill. The knowledge of Ohm’s law is cognitive, the knowledge that you arebetter in reading than in executing calculations is of a metacognitive character. Feeling anelectric shock from the 220 V mains is a cognitive experience, the belief that you are near thesolution of a problem you are working on, is a metacognitive experience. These examplesmay seem clear, but some questions remain.

In higher education, the students get information and have to solve problems in whichcognition and metacognition are not clearly distinguished. The characteristics ofmetacognition have to be formulated in such a way that it is possible to distinguish it fromcognition in information and in problem solving. Does the understanding and application of aconcept, the meaning of a mathematical symbol, etc. belong to cognition or to metacognition?What components are involved in metacognition? And what role does information play inmetacognition?

In the following chapters differences in the development of metacognition will be studied.In Chapter 3, differences between metacognition in two domains of knowledge. In Chapter 4,differences between metacognitive skills of students and the development of those skills. InChapter 5, differences between metacognitive knowledge of teachers about the content oftheir courses. In all studies problems were give to the participants in which metacognitivecomponents were represented.

26 Metacognition and its development

In the present chapter the following questions will be addressed: (a) What is the differencebetween cognition and metacognition? (b) How can metacognition be structured in a modelthat will enable the description of differences in its development? (c) What features ofmetacognition are relevant for solving problems? To be clear, those aspects of metacognitionwill be emphasized, that enable identification of metacognitive components in informationand in instruction. The psychological aspects of metacognition will be subordinate to thisgoal. The data to be used came from the literature and from an analysis of metacognition.

2.1 Metacognition and CognitionKnowledge of metacognition was first developed in research on memory (e.g. Flavell &

Wellman, 1977). Flavell (1971) first used the term metamemory, and later the termmetacognition. Metacognition comprised metacognitive skills like monitoring and regulatingcognitive processes (Flavell, 1987). This concept has been used and studied in many domains.Where the terms self-appraisal, self-management, higher order skills, metaskills, etc. are used,different aspects of metacognition are, in fact, being studied in different contexts.

Metacognition involved “active monitoring and consequent regulation and orchestration”of cognitive processes to achieve cognitive goals (Flavell, 1976). Monitoring, regulation, andorchestration could take the form of checking, planning, selecting, and inferring (Brown &Campione, 1977); self-interrogation and introspection (Brown, 1978); interpretation ofongoing experience (Flavell & Wellman, 1977); or simply making judgments about what aperson knows or does not know about how to accomplish a task (Nelson, 1996; Metcalfe &Shimamura, 1994). Kluwe (1982) stresses that human beings can understand themselves asagents of their own thinking, and can also assess themselves as such, as self-regulatoryorganisms.

The research on cognitive skills in general included different tasks, such as memory tasks,reading text, writing, language acquisition, problem solving, social cognition, but alsoperforming calculations, measurements, mathematical modeling, constructing, drawing,reading schematics and diagrams, etc. Cognition not only included the observation andmanipulation of objects, entities, reality, but also the processing of information, i.e. of signslike words or figures, often coupled to previously learned skills.

The description and meaning of the concept of metacognition differed among domains ofapplication. The concept itself was fuzzy as Garner (1987, chap. 2) stated. The precisemeaning of metacognition was not clear (cf. Weinert, 1987; Posner, 1989; Forrest-Presley,1985; Hacker, 1998). Different authors in different fields used different terms, often withoverlapping concepts. Metacognition was described simply as knowledge of knowledge,thinking about thinking, cognition about cognitive processes, or “knowledge and cognitionabout cognitive phenomena” (Flavell, 1979, p. 906). The basic description of metacognitionwas that of cognition about cognition.

Simon and Kaplan (1989) described cognition as the capacity to use intelligence inexecuting tasks, or the capacity to execute cognitive tasks. De Groot and Van Peet (1997)understood cognition as equivalent to cognitive functioning. By this definition, allcombinations of psychological functions could play a role, like observing, memory, thinking,making a sound choice and deciding, but also processing emotions and intuition. Cognitionwas the act or process of knowing, including both awareness and judgement, and could alsobe a product of this act (Wellman, 1985). It could be a product of this act as well. But thedistinction between cognition and metacognition was not made here.

In order to separate cognition from metacognition a more precise description of both wasrequired. The descriptions found showed that it was possible to separate these concepts inseveral ways. The contents differed: metacognition was about cognition (part of the mentalworld), while cognition was about things in both the real world and mental images thereof.


The content of cognition included objects, persons, events, physical phenomena, signs, etc.,skills to handle these entities, and information on the tasks. The contents of metacognitionwere the knowledge, skills, and information about cognition.

The function of cognition and metacognition also differed. The function of cognition wasto solve problems, to bring cognitive enterprises to a good end. The function of metacognitionwas to regulate a person’s cognitive functioning in solving a problem or executing a task.

Flavell (1979) assumed in his model on metacognition that metacognition and cognitiondiffer in their content and function, and are similar in their form and quality. Both cognitionand metacognition can be acquired, be forgotten, be correct or incorrect, etc. Metacognitioncan be expressed in external formulations, with said information being either correct or not,subjective, shared, or validated, just like cognition.

Brown (1978) distinguished between two clusters of metacognitive activities: knowledgeof cognition (e.g. that one’s memory for a new phone number may only be short-term), andactivities used to regulate and have an overview over cognition (e.g., that a person needs toactively rehearse information in order to retain it in working memory). Kluwe (1982) linkedthe two general attributes already mentioned by Flavell as person and task variables, todeclarative and procedural knowledge. Chi (1987) also distinguished declarative andprocedural knowledge in metacognition.

Thus it became clear that cognition and metacognition were also supposed to be equivalentin that knowledge, skills, and information were distinguished. In metacognition, therefore,knowledge was identified with metacognitive knowledge, and skills or potential actions wereidentified with metacognitive strategies. On a cognitive level tasks are strongly related withinformation (assignment, explanation, supporting material). On the metacognitive levelinformation included concepts and skills, providing material for the goal of knowing aboutcognition.

From the aforementioned characteristics ‘definitions’ of metacognition (see later) andcognition will be constructed. Cognition can be defined as involving (a) signs, objects, andphenomena in the real world, (b) mental images thereof, and also (c) knowledge of andoperations on (a) and (b), including mental operations. Signs include the formulation ofcognitive tasks, goals, and any other information. Knowledge of signs includes the meaningof the sign. Operations on signs without reference to their meaning were also part ofcognition. So understanding a text was cognition, while ‘turning on the meaning’ wasmetacognitive (Garner, 1987, p. 121).

2.1.1 Models of MetacognitionFlavell (1979, p. 906) stated: “I believe that the monitoring of a wide variety of cognitive

enterprises occurs through the actions and interactions among four classes of phenomena: (a)metacognitive knowledge, (b) metacognitive experiences, (c) goals (or tasks), (d) actions (orstrategies).” Knowledge, tasks, strategies and experiences can all be both cognitive andmetacognitive in character (Flavell, 1987). The four variants of metacognition will be used inthe consecutive studies. A metacognitive strategy will be studied in Chapter 3, all variantswill be used in Chapter 4 to analyze how development of metacognition can be stimulated,and metacognitive, structural knowledge of knowledge in a domain will be studied in Chapter5. Therefore the distinction between cognition and metacognition in each of the four variantswill first be elaborated.

The information that is available for executing the task is a cognitive variable, whereas thecharacterization of this information as sufficient in quantity and quality to warrant confidencein solving the problem is a metacognitive one. Qualification of a task as easy or difficult isalso a metacognitive variable. A cognitive strategy is a strategy for making progress towards agoal or subgoals of a task, a metacognitive one is a strategy to monitor and regulate progress.


Human beings have a cognitive experience when they feel that they invest effort to attain thegoal, and metacognitive experience when they have a feeling that they are far away from thegoal without knowing why. What they do when faced with such a situation would have to dowith their metacognitive knowledge. Metacognitive knowledge consists of knowledge andbeliefs about personal cognition like ‘You can fail to understand something or someone intwo different ways: (a) by not achieving any coherent representation at all, or (b) byunderstanding incorrectly, i.e. misunderstanding’.

Metacognitive experiences influence the structuring and execution of cognitive tasks. Bothmetacognitive knowledge and experiences can lead human beings to select, evaluate, revise,and abandon cognitive tasks, goals and strategies. This suggests that metacognitive thoughtsare deliberate, planned, intentional, goal-directed, and future-oriented mental actions that areused to accomplish cognitive tasks. They are similar to mental actions in cognitive tasks. Thusmany researchers use the label metacognition for conscious and deliberate thoughts that haveother thoughts as their object, but some include nonconscious processes.

When metacognition about a task is disturbed, even nonconsciously, performance in thistask can drop dramatically. Cornoldi (1998) told an experimental group of technical schoolstudents that remembering concrete concepts was a sign of intelligence, whereas they knewand had shown that concrete concepts were easier to remember than abstract ones. Bydisturbing their metacognition in this way, the number of concrete concepts remembered bythe group decreased to half of that of the control group, about equal to the number of abstractconcepts remembered. A skilled typist who starts thinking about where the keys are will alsotype much more slowly. So highlighting metacognition does not seem profitable in automaticfunctioning, unless this functioning has to be relearned because of errors.

Flavell (1979) stated that most metacognitive knowledge contains interactions betweenmetacognitive knowledge, task variables, strategy variables, and experiences. Sternberg(1998) pointed out that metacognition interacts with many other aspects of the student:abilities, personality, learning styles and so on. In the abilities domain many variablescorrelate and “it is easy to slip into causal inferences from these correlations, despiteadmonitions to the contrary from elementary statistics teachers”. He argues the view thatmetacognition converges with other abilities necessary for academic success, in a construct ofdeveloping expertise. Research has revealed that cognitive knowledge can be activated andcan influence cognition -even without becoming conscious (Nelson, 1996). Flavell (1979)argues that the same applies to metacognitive knowledge. So metacognition can influence notonly cognition, but also (reflexively) develop itself into increasing metacognitive expertise.

The main points that arose from these descriptions are that metacognition has a controllingposition with respect to cognition and to itself, and that it has four variants (knowledge, goalsor tasks, strategies or actions, and experiences) that influence each other. The interactions inthis model are bidirectional: monitoring in one direction and regulating in the other. Theseconsiderations lead to two alternative models of metacognition each of which differs fromearlier models in their reflexive character (Metcalfe & Shimamura, 1994; Hacker, 1998;Garner, 1987). The first model is recursive, because metacognition controls itself here: thismodel can be repeated infinitely. The second model is reflexive, because metacognitioncontrols a mirror image of itself. The last model looks like a separation of the brain in twoidentical halves (see Figure 2.1).

A corresponding ‘definition’ of metacognition will now be formulated, that helps todistinguish cognition and metacognition. Metacognition is supposed to involve (A) signsdenoting knowledge and operations of cognition, (B) mental images thereof, and (C)knowledge, monitoring and regulation of, and experiences on (A), (B), (C), and theknowledge and operations denoted in the signs of (A) and (B). This implies that if signs areprocessed without reference to their denotation, a cognitive operation takes place, whereas a


metacognitive operation takes place if these operations include the reflection on the meaningof the signs.

The foregoing description has implications for the aspects of metacognition that can bestudied. If metacognition has generally developed in humans, and has a controlling relation tocognition, its study can focus on the organization of metacognition and the access to it inperforming tasks (see Chapter 3; and e.g. Nelson, 1996). This controlling relation does notnecessarily mean that cognitive science is automatically metacognitive with respect to domaincontent. Processing of signs in cognitive science can be of a cognitive level, while regulatingcomponents in domain knowledge can be of metacognitive level. The foregoing statementsalso have consequences for the development of metacognition.

2.2 Development of MetacognitionIf cognition and metacognition differ in content and function, but were alike in form and

quality, this would have consequences for development of metacognition. The contentdifference would be that the objects of metacognition would be internal cognitive states andtheir representations, whereas the objects of cognition would be the objects and phenomena inthe outside, real world and their representations. The functional difference would be thatcognition should lead to observable results in tasks and problem solving, while metacognitionshould regulate the performance in cases where functioning without metacognition is notsufficient.

B.

Metacognition Metacognition

Cognition Cognition

Figure 2.1. Two models of metacognition. The double pointed arrows represent control: one point forregulation (from metacognition) and the other for monitoring (towards metacognition). Figure Arepresents a recursive model: the outer model can be imagined to repeat itself as the inside model,leading to an infinite series of models. Figure B represents a reflexive one: metacognition can monitorand regulate a mirror image of itself.

Thus metacognitive development is especially important in automated (nonconscious)functioning that contains errors and needs an ‘adaptation’ to the environment (Piaget, 1926;

A. Metacognition

Cogn

Metacognition

Cognition

MMeettaaccooggnniittiioonn

CCooggnniittiioonn


Ginsburg & Opper, 1969) and in cases where a human being is inquisitive and wants toexplore his own further autonomous development (Vygotskij, 1962).

The model presented leads to the consequence that metacognitive development isstimulated in four ways.

1. Facilitating access to the organization of metacognition will stimulate the developmentof metacognition. The attention is, for instance, easily drawn to familiar features ofmetacognition that have to be used in a task in a well-known cognitive domain.

2. Drawing the attention to some characteristic features of metacognition will stimulate itsdevelopment. This will be the case if familiar features of metacognition are emphasized asimportant to reach a certain cognitive goal that otherwise would be difficult to attain.

3. Relating metacognition to itself by relating metacognitive variants to each other willstimulate metacognitive development. This can be attained, for instance, by inducingmetacognitive experiences of success while the students use metacognitive information in atask in order to stimulate the development of a metacognitive strategy.

4. Metacognitive development will be stimulated when humans are compelled to observetheir cognitive states. If two stationary cognitive states (concepts) are in conflict, thecontradiction can only be reconciled by mentally observing the states and taking a highermental position. If two dynamic cognitive states (actions or skills) have to be compared, thesestates have to be observed and the observations turned into stationary conceptual states, whichcan be analyzed to find similarities and differences.

These principles were applied in the study of Chapter 4. In the next section, theconsequences of the controlling hierarchy in metacognition and cognition are analyzed.

2.3 Structures in Metacognition and CognitionFlavell (1979) already mentioned some clear combinations of metacognition and cognition:

metacognitive knowledge about cognitive knowledge (interpersonal, intrapersonal, andgeneral knowledge), about cognitive strategies (how to make progress towards a goal), andabout cognitive tasks or task-information (e.g. ‘this is difficult to do’). This metacognitiveknowledge has to be distinguished from a metacognitive strategy (the skill to check progresstowards a goal) or a metacognitive goal (e.g. to find out whether your knowledge is sufficientto pass tomorrow’s exam). First the status and use of schemata will be analyzed, in whichanalysis attention will be given to what happens when metacognitive knowledge representedin schemata is made independent of the cognitive content, because such content free schemataare certainly metacognitive of character.

2.3.1 Metacognitive Knowledge, Strategies, and SchemataMetacognitive knowledge and schemata

Conceptual metacognitive knowledge may concern cognitive knowledge; e.g. thatknowledge is organized in schemata. It may also be about cognitive skills; e.g. that withoutspecial means it is not possible to have much more than seven chunks of information in shortterm memory and pay attention to these (Miller, 1956). It may also concern information (e.g.the knowledge that anything written about cognition, even if it is scientifically validated, maynot apply to me because of statistical variations).

It is now possible to present some consequences of the foregoing using the example of aschema (several types of schemata will be discussed in the following research). Research onthe mental representation of the concept ‘bird’ (Smith, 1989, p. 511) led to a possible externalrepresentation in the form of the schema in Table 2.1. Such a schema represents the wayhuman beings use the concept ‘bird’ to categorize animals into birds and nonbirds, or toreason about birds. Thus the schema represents both the concept and skills of application.


The first column numbers the attributes of the concept. In the second column, the attributesare presented together with their importance in the diagnosis of membership, i.e. theirusefulness in discriminating concept instances from noninstances. The third column gives therelations among the attributes using the numbers from the first column. The relation betweenlocomotion and dwelling in trees can be enabling, or causal, or otherwise. The supersetindicates the more abstract concept that comprises the concept ‘bird’. In the fourth column,possible values for each attribute are presented that can be assumed at the same time byinstances of the concept (e.g. flies and walks). The defaults are the values most often met. Inthe first row there is an indication of the type or superset, to which the concept belongs,providing connections with other schemata in a semantic network.

Table 2.1A possible schema for bird

Attributes (# = diagnosticity) Relations Possible values (*=default)1 Type Superset Animal2 Locomotion (4) 2-3, 2-4

2-5, 2-6Flies*, walks, swims

3 Communication (3) 3-2 Sings*, squawks4 Size (1) 4-2 Small*, medium, large5 Habitat (3) 5-2 Trees*, land, water6 Food (2) 6-2 Insects*, seeds, fish

A schema like that for bird has metacognitive aspects. It is itself of a metacognitivecharacter because it provides information about how you yourself and other people thinkabout birds. It describes the structure of your knowledge about birds and thus stimulatesdevelopment of metacognitive knowledge. It fosters the kinds of mental processes that humanbeings presumably execute implicitly. For instance, the schema guides a person to first look atthe size of the animal, then to the food it eats, etc., and in each case to compare theobservations with the possible values. Thus the schema represents a process, in that thefootprints of the steps are presented, while it is supposed that the steps are basic and need nofurther explanation. It is supposed that this is a general feature of the description of mentalprocesses: these processes can be described as far as they can be split into partial processes,with the intermediate products of each partial process being used to describe the process as aprocedure.

A second reason for the importance of schemata is that they can form quite complexstructures: schemata can form part of other schemata and, conversely, schemata can containsubschemata and can thus form branched hierarchies of trees in chunks. They may beproposition-like, picture-like or both (Simon and Kaplan, 1989). These schemarepresentations differ among individual persons but generally exist in relation to natural-typeconcepts like ‘bird’, and artifact concepts like ‘furniture’. They all comprise metacognitivecomponents next to cognitive ones.

Schema representations can relate to concepts, situations or problem solving. Differentnames have been used in reference to different applications. A schema representation is astable mental structure that organizes a set of related concepts, propositions, and procedures(Rumelhart & Ortony, 1977; Bartlett, 1932). It has been proposed that schemas comprisegeneric descriptions of situations and play a role as active recognition devices. A frame(Minsky, 1975) and a script (Schank & Abelson, 1977) are types of schemas that emphasizerepresenting a - developing - situation in which a procedure is applied. A plan is a type ofschema that is activated to perform a particular task: ‘a plan is a set of sequenced cognitiveoperations that we apply to information to complete a task’ (Martin, 1984).


A problem schema representation organizes knowledge components on the basis of theirrelevance to solving a type of problem (VanLehn, 1989, p. 545). The problem schemaconsists of information about the class of problems the schema applies to, and informationabout their solutions. Problem schemas have two main parts: one for describing problems andthe other for describing solutions. Such problem schemata are used in routine problemsolving. Here the solutions consist of three processes: selecting a schema, adapting(instantiating) it to the problem, and executing its solution procedure. Schema selectionfollows a poorly-understood triggering process that seems to take place early in the perceptionof the problem, before the whole problem presentation has been read.

Metacognitive strategiesUp till now the discussion has concerned the metacognitive knowledge of schemata that

comprise cognitive knowledge and strategies to solve problems. Persons who haveinternalized and integrated a schema like the one of Table 2.1 into their knowledge, can alsouse the schema as a metacognitive strategy. When talking about or classifying other things,like mammals or electrical resistors, they may ask, for example, for the attributes and thepossible values of the concept, the default values, for the importance of the attribute, for therelations among the values or the attributes, for the superset and some subsets. The schemaprovides a check on the types of questions that have to be asked.

When used in this way, the schema can be seen as a kind of empty template, a content freestrategy, or a structural example for the categorization of all entities. It is much more generalthan for birds only, and can be filled in any specific cognitive case. The variables are nolonger the values for birds, but the number of attributes to be asked for, the values and defaultvalues to be filled in, etc. Thus, if the necessary metacognitive skills have been learned, theempty schema representation may develop into another metacognitive variable, a strategyvariable as described by Flavell (1979). Such an empty template gives access to a structure ofmetacognition. An example will be presented in Chapter 3 in connection with reasoning.

Based on the foregoing analysis of schemata, an integration of the 3 classical views onconcepts is proposed in order to present a framework for the abstract concepts of electricalengineering.

2.3.2 Three Views on ConceptsConcepts serve to partition the world into classes of objects (including persons, processes,

events, symbols, relations as special types of objects). In the classical view (Smith and Medin,1981) the classes were sharp and distinct. For fuzzy concepts, the classes are less sharplydefined. It is supposed that the extension of any concept refers to a class of objects of somekind, whereas the intension of a concept specifies the type of concept. Thus it is possible todefine the concept of multiplication as referring to all possible types of multiplication,multiplication itself being a skill. The possible views on concepts are of importance inChapter 5, where the organization of knowledge is studied. The views correspond to threebasic types of problems as distinguished by Dijkstra (1997): classification, explanation, anddesign problems.

Smith (1989) distinguished three views on concepts. These views were taken as basicviews that could be generalized to any kind of concept, either based on direct observation ofobjects and symbols, or based on internalized observation, i.e. imagination.

1. First, concepts can be viewed as exemplar-based proposals where on-line computationof prototypes takes place, like in the assertion “An ostrich, a penguin, a sparrow and a duck,are all birds”.

2. Secondly as definitions like the scientific definition or the cognitive schema of a bird(see Table 2.1).


3. Third, as prototypes giving an image of the concept, like “the eagle is a typical bird”.It is supposed that from these three types of cognitive constructs, objects are assigned to

categories along different rules. It is supposed that a concept is reliably formed if learning inthe three views leads to the same successful categorization of objects into members andnonmembers of the class. It is further assumed that the different rules are integrated in thatcase. These principles will be applied in the design of instruction (see Chapter 4).

The first view starts from exemplars presented as members of the class (a set). The conceptin this view is a concept superordinate to the exemplars presented, like ‘bird’ is superordinateto ‘robin’ and element is superordinate to resistor. Membership of a class can be determinedhere by checking correspondence of an object with one of the exemplars. The exemplars canbe considered as examples of the concept. The dimension of abstraction involves the conceptand examples of the concept, a class and its members. It is the dimension ranging fromconcrete to abstract or backward. The processes involved are categorization (formingcategories) and instantiation (giving examples) respectively. They are important to partitionthe world into classes of objects.

The second view on concepts is based on a description or a definition of the concept. Thescientific definition of a bird assigns objects to the category ‘bird’ in a theoretical way, whilethe cognitive schema of ‘bird’ (see Table 2.1) is derived from the way human beingscategorize in practice, and represents an empirical approach. The definition specifies thecharacteristic properties (attributes) of members of the class and the way to combine thesecharacteristics in deciding on membership. In this view, membership in the class isdetermined by analyzing the properties of an object, comparing attributes and deciding onmembership in a logical way. The dimension of abstraction here involves a more and lessdetailed description of the concept, from a definition like ‘animal that lays eggs and haswings’ to a name like ‘bird’. It is the dimension ranging from detailed to global, fromprecisely described to a general description, or backwards. The processes involved aregeneralization (finding a general description, giving a name or a symbol) and specification(finding a more detailed description) respectively. These processes are important inexplaining things.

Third, concepts can be viewed as prototypes. A prototype is a kind of idealized instance ofthe concept that represents all members of the class in a perceptually salient way. Theprototype represents the general form of the concept, like an eagle is a prototype of ‘bird’ anda schematic is a prototype of the circuits produced. In this view, membership in the class isdetermined by similarity in form with the prototype. The dimension of abstraction involves anideal instance and the real forms of the members of the class. It is the dimension ranging fromreal to ideal or backwards. The processes involved are modeling (finding a general form forthe members) and realization (finding specific forms) respectively. These processes areimportant in the design of artifacts.

Categorization of many concepts takes place along a conjunctive reasoning process:members of the class (examples of the concept) should have some characteristics in common(like in: ‘a grandmother is female and her child has a child’). Such common properties cannotbe found for concepts like ‘multiplication’ involving skills. Multiplication processes are allabout repeatedly adding a number to a given number (or constant), but these are the objectsrather than a property of the skills.

In science courses, a scientific definition is given for many concepts that is confusinglyabstract in all three senses, e.g. a resistor is a device that obeys Ohm’s law. It is idealized inthe sense that in reality it can be only approximated. It is sufficiently general to be applicableto all material objects to describe one of their electrical properties. It is sufficiently abstract todescribe all types of subordinate concepts, and at the same time sufficiently concrete todiscriminate between coordinate concepts (Vos, 1987a; Vos, 1987b; see Chapter 5). Such a


theoretical concept differs widely from the notion of what a concept is (a class, prototype,categorization rule, or an empirical schema).

2.3.3 Restrictions of Working Memory and MetacognitionThe metacognitive knowledge about the amount of knowledge a person can handle in the

working or short term memory (STM) is of importance, too. This implies that in order toreach an overview of the subject matter of a course, it should be possible to integrate thisknowledge in about seven basic chunks. This is only possible when the chunks representconcepts that are so abstract and general that they can cover the whole field. Such aknowledge base would probably be so general that it can be instantiated in terms and domainsthat the students can understand. This would make it possible to construct a true ‘advanceorganizer’ for the course (Ausubel, 1978). Such knowledge bases have been constructed forseveral courses (Vos, 1995; Vos, 1996; Reif and Heller, 1982; Ferguson-Hessler, 1989). Theone developed earlier by Vos (1991) has been used in Chapter 4.

The metacognitive knowledge about a course will also be constrained by the space inSTM. It is expected that a structure of metacognitive knowledge will contain about sevenchunks if access is made easy and does not use space in STM (see Chapter 5).

Another consequence is that metacognitive and cognitive skills will compete for the spacein working memory. It is supposed that especially the weak students, who need much space inSTM for cognitive goals, will not easily develop metacognition at the same time. A strategyin developing metacognition is to separate learning the structure of it from getting access tothis structure (see Chapter 4). Another possible strategy is to separate the cognitive tasks frommetacognitive development by doing the work on metacognition as a reflection just after thetask has been completed.

2.3.4 Metacognitive Skills and Metacognitive InformationIn addition to conceptual metacognitive knowledge, metacognitive skills or strategies are

also distinguished such as monitoring and regulating cognitive processes. Regulatoryfunctions always include a check -a comparison of a desired situation with the actualsituation. Metacognitive skills are therefore most easily identified by looking for possibilitiesto check the progress of cognitive tasks.

These metacognitive skills can monitor cognitive knowledge (e.g. in monitoring what ismentally represented inside one’s head in such a way that it can be expressed externally in anequivalent way). They can also control cognitive skills (e.g. using heuristics or otherstrategies not only to aid, but also to monitor the progress of a solution procedure of aproblem towards the goal). Or they may monitor information (e.g. checking that thetranslation of a written problem into a ‘foreign’ language like mathematical symbols ordiagrams has the same content, structure and meaning as the original).

Third, metacognition can include metacognitive information. Metacognitive informationmay concern cognitive knowledge. An example is the sentence above: “anything writtenabout cognition, even when scientifically validated to be generally the case, may not apply tome because of statistical variations”. Metacognitive information can also relate to cognitiveskills (e.g. by presenting a strategy to monitor progress towards subgoals in problem solving).Such metacognitive information is not automatically transformed into metacognitiveknowledge. Finally, metacognitive information can be about cognitive information.

The sentence: ‘A sentence like “I am telling the truth” is useless’, provides themetacognitive information that reflexive information about oneself is useless from a cognitiveviewpoint, because liars could also say the same. Nelson (1996) discussed such sentences asdifferent level meanings in the same form. The sentence: ‘A sentence like “I am a liar”provides an example of a paradox’, provides the metacognitive information that information


about oneself can be a paradox, because neither a liar nor a truth-teller will say they are liars.Metacognitive information can also be included into a message and lead to contradictions,like in the painting of a smoking pipe by Magritte, with the subtitle: this is not a pipe. Suchsentences are relevant in studying metacognition.

Since metacognition can be seen as cognition about cognition, reflexive statements canturn up. The sentence ‘I know that I know it’ gives no information because metacognitiveknowledge can be wrong. Some teachers would say ‘you have to learn independently’(implying: so listen to your teacher). This sentence contains a paradox because in listening toyour teacher you are not learning independently. Contradictions lead to cognitive conflictswhose solutions can promote metacognitive development. An example is: ‘Determine thenon-linearity of the given circuit’ in a case where the circuit seems to be linear (see Chapter4). The adage “the best teachers are those who learn most from their students” providesanother example of a contradiction: a teacher cannot be a teacher and a student at the sametime. If teachers succeed in reconciling this contradiction, they can be a role model for theirstudents.

2.3.5 Combinations of Metacognition and CognitionSince metacognition has four variants and cognition three components, twelve

combinations are possible, as represented in the cells of Table 2.2. The four variants ofmetacognition (first column) are combined with each of three components of cognition (rows2 to 4). An example of each cell is already presented in the foregoing text. The table bothdistinguishes and relates cognition and metacognition.

Table 2.2Basic components of metacognition

CognitionMetacognition Knowledge

(stationary, personal)Skills/ strategies

(dynamic, personal)Information

(static, public)MC knowledge(stationary, personal)

Internalized schemarepresentations.Knowledge of theorganization ofknowledge.

Internalized conditionsof application.Procedural knowledge.

The internalizedmeaning of schemata.Knowing thedistinction betweenform and meaning ofsigns.

MC skills/ strategies(dynamic, personal)

Monitoring of theexternal representationof one’s knowledge.Consciously learning aconcept. Reasoning.

Regulating cognitiveprocesses. Monitoringprogress. Categorizingskills. Learning a skill

Checking correctnessof information againstpersonal experiences.

MC information(static, public)

Explicit conceptualschema (Table 2.1).Results of research onknowledge.

Explicit strategy formonitoring. Conditionsof application.Limitations of memory.

Explicit informationthat information isparadoxical,contradictory, useless.

MC experience Feeling of (not)knowing. Uncertainty.

Judgement of learning. Confidence in answers.Ease of learning.

2.4 Instruction in MetacognitionGenerally speaking, metacognition develops slowly and is difficult to teach. Brown and her

group developed successful teaching methods for metacognition (e.g. Brown & Campione,1977; Brown, 1978; Brown, Campione & Barclay, 1979). Teaching metacognition often ledto disappointing results, as in courses on systematic problem solving approaches, but e.g.Schoenfeld (1982) and Mettes and Pilot (1980) reported successes. Elshout-Mohr (1992)


stated that expectations on metacognitive learning results were often unreasonably high.Much effort was needed to teach metacognition, leading to high costs in terms of effort andsalaries. A clear understanding of instruction aimed at metacognitive development wastherefore important.

Instruction can be described as organizing a learning environment for the students in whicha coomunication between a beginner and an expert takes place, or, as Cowan (1998) stated, as‘the purposeful creation of situations from which motivated learners should not be able toescape without learning or developing’. The environment should provide motivating tasks tobe carried out, means to carry out these tasks, information to the students with respect to thecognitive and metacognitive goals and the product required. Information should be providedabout the evaluation and its criteria, and the execution of the tasks, and of course,opportunities to fill gaps in prior knowledge.

Elshout-Mohr (1992) summarized several examples of instruction in metacognition. It waspossible to train self-regulatory skills in the way Brown, Campione and Barclay (1978) haddone for retarded children. The basics of this training consisted of repeating core questionslike “Do I know this, or do I not?” This should take place in a pedagogical setting in whichquestions like this have been integrated, for example by letting the students pose them to eachother (reciprocal teaching). Teaching was most successful if strategies where trained with acognitive executive function and a metacognitive regulating function (e.g. “loudly repeatdefinitions” both as a means to learn and to check the results); if the students gained insightinto what they had to learn (goal) and the reason why the strategy could help them overcometheir difficulties (use); and if the students became experienced in independent control andregulation of their behavior during application of the strategy. In such education, coordinationof control and execution should be appropriate and quality requirements should be high.

In that way a strategy of regulation could be taught that provided the amount of controlneeded for progress in learning, and that was maintained after the training. The strategy ofregulation did not change however. Resnick (1984) thought it should be better to let thestudents experience the difference between comprehension of text after ‘having readinformation’ and the type of comprehension after ‘having summarized the information’, thusdeveloping their metacognitive awareness. Elshout-Mohr (1992) proposed separatinginstructional principles for students with low metacognitive awareness (developingmetacognitive awareness, i.e. access) and for students with high metacognitive awareness(developing metacognitive strategies, i.e. structure). In Chapter 4 it was found advantageousto apply both these principles.

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Metacognition and reasoning 39

Chapter 3

Metacognition and Reasoning Schema Representations

AbstractThe influence that the content of a subject had on the structure of and access to reasoning

was studied, as well as the ability to handle uncertainty. It is assumed that thesemetacognitive components were acquired by an implicit learning process. Theoretically,access to the structure of reasoning was acquired by the formation of truth values, and thestructure itself was represented by a truth table. The participants were first-year electricalengineering students who were presented with reasoning tasks about the weather and aboutthe oscilloscope. The structure of reasoning comprised several reasoning procedures to reacha conjunctive conclusion with two premises; uncertainty was involved by using a three-valuelogic in which the truth-values were ‘true’, ‘false’ and ‘uncertain’. In the reasoning tasks, anassertion was given as a starting point. A statement was then presented, the correctness ofwhich had to be determined on the basis of the assertion (correct, wrong, one cannot tell).The answers and the arguments supporting them were collected. In reasoning about theweather, three structures of reasoning could be distinguished: a hidden structure in which theparticipants mostly but not consistently used a common schema (used by 75 % of theparticipants), an empirical structure consistently used by 25 % of the participants, and atheoretical structure in agreement with the theoretical truth table (used by 1 of the 40participants). The participants were mostly able to handle all uncertain components in theirreasoning. In reasoning about the oscilloscope, less correct reasoning procedures werefound. A hidden structure could be distinguished that differed from the one found in theweather case because the participants were not able to handle all uncertain components inotherwise correct reasoning here. From these findings it is concluded that both the structureof and access to reasoning change in a new domain, and that the ability to handle uncertaincomponents is diminished. The implications of this unique finding for the acquisition ofmetacognitive (reasoning) skills are to make a hidden structure more consistent, to make anempirical structure more explicit, and to emphasize the theoretical structure by using truthvalues and truth tables. The relations with research on expertise are discussed.

The structure and use of metacognition have been studied with the aid of reasoning.Halpern (1984) asserted that reasoning is an inferential process by which a person, beginningwith some given information or premises, makes an inference which enables that individual toreach a conclusion or provide some new (inferred) information that was not given. This newinformation was not known in the beginning, but was needed to determine some state ofaffairs. Thus reasoning is also considered as a way to solve categorization problems in whichthe outcome is uncertain (see Chapter1). This way could be called a metacognitive strategy,because it is about knowledge in a certain domain, it was felt that information was needed,and usually involved a check along other means. For instance, if a person calls an olderwoman ‘granny’, it can be concluded that she is a grandmother, but that is not certain and hasto be checked by asking her whether she is female and has a grandchild.

In this example, the content of the reasoning is family relations, but content, structure andaccess are mixed up. Voss and Means (1991) stated that human thinking resembles theprocesses emphasized by classical rhetoric (cf. Aristotle, 1960) more than processessuggested by logic (e.g. Lemmon, 1965; Schoenfeld, 1967). They made a distinction between

40 Metacognition and reasoning

formal reasoning, according to the laws of rhetoric or logic, and informal reasoning, as ispracticed in argumentation in school classes and elsewhere (see also Means & Voss, 1996).This distinction was not specified in the former studies; in the present study the possibility hasbeen left open that people reasoned in either of these ways.

From earlier research, e.g. in selection tasks (Wason, 1977, 1983), it was known that thecontent has an effect on reasoning. In that research it was impossible to discriminate betweenan effect on access to reasoning and an effect on the structure of reasoning. In the presentstudy the influence of the content on reasoning was diminished, and the structure of reasoningwas separated from access to reasoning.

In this study, first the structure of reasoning will be addressed, followed by the distinctionbetween the structure of and access to reasoning. Then the influence of the content of thesubject on the structure and the access to it will be studied, followed by the differencebetween well-developed and poorly developed reasoning.

The motivation for this study was the complaint of teachers that in some cases theirstudents could not ‘reason’, while in other cases they reasoned rather well. This was supposedto be a content effect. It was supposed that in a familiar or well-known subject like theweather, students could reason correctly, whereas in a new topic, handling the contentinterfered with their reasoning skills.

The goal of this study was both to understand the origins of this effect and to apply thisknowledge to accelerate the process of learning to reason in a new domain, and in general, tofoster the development of metacognition. It was supposed that from the results of this study,indications could be found about how to stimulate the development of metacognitiveknowledge and skills.

The participants were first year electrical engineering students, who were given reasoningtasks. The tasks were of the following form:

“Given that the sun shines if there is no cloud in front of the sun and it is daytime”, whatcan you tell about the statement ‘The sun shines because it rains a little bit and it is in themonth of July? Is it correct, or not, or can’t you tell what is the case? These reasoning taskswere about the weather, a well-known domain (the weather case). Other tasks were about theoscilloscope, a complicated measuring instrument of electrical engineering (the oscilloscopecase).

The structure of reasoning was supposed to consist of related types of reasoningprocedures. Access to reasoning was described as the ability to make use of reasoning. Todetermine the structure of reasoning, the reasoning procedures and their relations had to befound from the reasoning tasks.

From mathematical logic theory (e.g. Schoenfeld, 1967) it was known that the(metacognitive) truth-values of propositions like ‘this person is a grandmother’ were the basisof reasoning procedures. The truth-values were values of logical variables, comparable to thevalues of mathematical variables. In mathematical problems, the value of a variable in anequation was often called an unknown, which had to be solved. Analogous, the unknowns inreasoning problems were thus the truth-values of the logical variables that had to bedetermined.

As long as the unknown was not determined, the problem solvers had to handle theuncertainty connected to the mental state of not knowing. Handling uncertainty with respectto the outcome of the process of reasoning was introduced by using three-value reasoning, inwhich the third truth-value was ‘uncertain’ (in addition to ‘true’ and ‘false’, see later).

Handling uncertainty was of practical interest in instruction. Chi, Bassok, Lewis, Reimannand Glaser (1989) found that good students in physics concluded more often than beginnersthat they did not understand some point in the problem-solving process and thereforegenerated self-explanations and self-monitoring statements more frequently than did poor


students. This was interpreted as a better metacognitive awareness of uncertainty ofconditions, actions, or goals that required elaboration and justification.

The type of reasoning chosen was a conjunctive conclusion based on two premises,because such reasoning is often used. The effects of content on reasoning were studied bychanging from reasoning about the weather to reasoning about the oscilloscope. The contentof the (three-value) reasoning was the shining of the sun and the functioning of theoscilloscope, respectively.

The students had no formal training or instruction in reasoning, and certainly not in three-value reasoning because this logic is rather unknown. Thus, their mental reasoning schemarepresentation, the structure and access of which were determined, must have developedspontaneously, either wired in the human organism or by previous, implicit learningprocesses. Such processes produced tacit knowledge (Reber, 1989; cf. learning withoutawareness, Hartman, Knopman & Nisson, 1989), in the domain of reasoning. Suchknowledge can be nonconsciously used (Gleitman, Fridlund, & Resiberg, 1999).

According to Reber (1989), this knowledge could be characterized as deep, abstract, andrepresentative of the structure inherent in the underlying invariance patterns of the stimulusenvironment. Such knowledge, called intuitive knowledge by Lewicki, Czyzewska, andHoffman (1987), and implicit knowledge by Broadbent, FitzGerarld, and Broadbent (1986),had been acquired by an implicit, tacit process. Thus it was assumed that without awareness, amental schema representation had been acquired from previous experience with reasoning.

The schema must have been accumulated from experience with practical tasks. It wasstored in memory, but not separated from the context, feelings, etc. Whether the participantshad become aware of this schema or not was not relevant for this research because thestructure of their reasoning was deduced from the reasoning tasks, not from what they couldtell about this structure (cf. Stadler, 1989; Hartman, Knopman, and Nissen, 1989).

Below, the relation of reasoning tasks to the structure of reasoning and its access will beclarified. Also, three-value reasoning will to be introduced. Finally, the expectations will beformulated.

3.1 Structure of and Access to Three-value ReasoningReasoning consists of reasoning procedures. A procedure is a series of steps that are

described by their (intermediate) results. A procedure can often be depicted by ademonstration, but in this case be described by the footprints of only the smallest steps to betaken. Reasoning procedures are based on the truth-values of their propositions. Thus, areasoning procedure can be described by presenting the truth-values involved, the way theseare combined, and the results of these combinations. First, the reasoning procedures in a two-value conjunctive conclusion based on two premises are described and related to the structureof conjunctive reasoning.

3.1.1 Structure of ReasoningA conjunctive conclusion is defined as a complex logical sentence, which is true if and

only if each of its premises is true. The assertion that the sun shines if there is no cloud infront of the sun and it is daytime, is a conjunction. To make the connection with mathematicallogic, two logical variables A and B are introduced, representing the truth-value of premises 1and 2, respectively. With the truth-values of these logical variables, the reasoning procedurescan be represented without reference to their content. Four combinations of the truth-values‘true’ (symbol: 1) and ‘false’ (0) are possible for the two logical variables. The logicalconjunction A^B is a third logical variable, whose truth-values are given by the truth table ofTable 3.1.

Each row of the truth-table represents a valid reasoning procedure. For instance, row three


represents the reasoning procedure ‘if premise 1 is true (there isn’t a cloud in front of the sun)and premise 2 is false (it is not daytime) then the conclusion is false (the sun is not shining)’.The content knowledge, the truth-value of a logical variable, and the reasoning procedurediffered here from the aforementioned example, but it was still a reasoning procedure fromthe same structure of reasoning, i.e. a conjunction.

Table 3.1The theoretical reasoning procedures of conjunctive reasoning in two-value logic with two logicalvariables A and B (truth table for a conjunction)

A B Conjunction (A^B)1010

1100

1000

Note. Each row represents a valid theoretical reasoning procedure. 1 means: true. 0 means: false.

The structure of reasoning according to a conjunction is defined as the relations among thereasoning procedures as represented in the truth table for a conjunction. The truth table for adisjunction represents a different structure of reasoning from that of a conjunction, althoughsome of the reasoning procedures are identical. Truth tables do not tell people what toconclude, but how to conclude in each particular case, by means of the truth-values of thepremises. The content of the reasoning is separated from the procedure of reasoning by theuse of logic variables.

Such reasoning procedures are called theoretical, because they are independent of thecontent and in accordance with the truth-table of mathematical logic theory. In theoreticalreasoning (it was supposed that other types of reasoning were also possible), a truth tableregulates the reasoning processes in any case in which it is applied.

3.1.2 Three-value ReasoningIn accordance with mathematical logic theory, reasoning can be based on more than two

truth-values. Rescher (1969) studied several multiple-value logics, including three-valuereasoning as used here. In the usual two-value logic schema, a statement could be either trueor false. The words right and wrong, or correct and incorrect, are used in everyday language.In the three-value logic schema of Lukasiewicz (Rescher, 1969), used in this study, the threetruth-values: true, false, and uncertain were possible for all propositions. In this three-valuelogic, ‘not true’ does not necessarily infer ‘false’, as opposed to the usual two-value logicwhere ‘not true’ does imply ‘false’.

It should be emphasized here that in logic, the value ‘uncertain’ (symbol: ?) has the samestatus as ‘true’ or ‘false’. Interpretations of ‘uncertain’ as ‘it cannot be known whether it istrue or not’ (universals of personal knowledge) or ‘I do not know whether it is true or not’(intrapersonal knowledge) place the third value above the first two ones. This is apsychological interpretation of logical values, and does not change the logic used.

The third truth-value is related to metacognition. Being able to handle uncertainty is ametacognitive aspect of cognitive tasks. This uncertainty was included in the tasks. In three ofthe five types of reasoning tasks of the experiment, it was impossible to say whether a givenanswer was correct or not. In total, six uncertain premises were involved in these five types.

A theoretical three-value type of conjunctive reasoning, including the supposedlymetacognitive component ‘uncertain’, was used in the reasoning tasks. The conjunction oftwo logical variables in this three-value logic was schematized in the three-value truth table of


Table 3.2 (see Appendix A for more properties of this truth table). This table represents thetheoretical structure of three-value reasoning studied.

3.1.3 Access to the Structure of ReasoningTo make the skill of reasoning independent from domain knowledge (content) in the best

possible way, it was decided to present all information needed in an assertion. The assertiongave the students enough information on the content to be able to draw a conclusion in acorrect way. The assertion, common to all reasoning tasks in the weather case, was: “The sunis shining if there isn’t a cloud in front of the sun and it is daytime”. It was explicitly statedthat this assertion was correct. The assertion was equivalent to a ‘definition’ of thephenomenon that the sun is shining. Two propositions could be distinguished in this assertion,(1) ‘there isn’t a cloud in front of the sun’ and (2) ‘it is daytime’. If these propositions weretrue, the valid conclusion (the third proposition) could be drawn: ‘the sun is shining’.

Table 3.2The theoretical reasoning procedures of conjunctive reasoning in three-value logic with two logicalvariables A and B (truth table for a conjunction)

A B Conjunction (A^B)11000?1??

1010?0?1?

100000???

Note. Each row represents a theoretical reasoning procedure. 1 means: true. 0 means: false. ? means:uncertain, one cannot tell.

Access to reasoning was supposed to be the determination of the truth-values needed toapply the truth table in each concrete reasoning task. The earlier example will be analyzed toillustrate this. The problem was whether the statement ‘The sun shines because it rains a littlebit and it is in the month of July’ was correct, or not, or it was impossible to tell. To solve thestated problem, the procedure theoretically started with an evaluation of both premises (Table3.3). Premise 1 was uncertain because if it rains the cloud need not be in front of the sun.Premise 2 was uncertain because the time of the day was not given. In these steps, calledintermediate inferences, the given propositions were compared with the premises, and theoutcomes were logic truth-values. To find the truth-value of the conjunction, the truth table(Table 3.2) was used. From the table it was found that the truth-value of the conjunction wasuncertain (last row). Translating this back to the problem, it was found that the conclusion ofthe assertion was uncertain, so it was not certain whether the sun was shining (which is thecorrect answer). The complete and explicit theoretical way of reasoning is represented inTable 3.3.

The reasoning procedure discussed above is represented by ? ? → ? In some cases theorder of the givens is reversed with respect to the premises in the assertion. Such a reasoningprocedure is labeled by the truth-values of its propositions as: rev ? ? → ?.

In this model, reasoning followed a procedure with intermediate inferences in which thetruth-values of the logical variables involved had to be determined. These intermediateinferences gave access to the reasoning procedures. The truth-values of the logical variablesmust - whether the intermediate inferences were correct or not - lead to a truth-value for the


conjunction along the reasoning procedures represented in Table 3.2. Therefore, thecorrectness of the reasoning procedure had to be distinguished from the correctness of theaccess.

Table 3.3A reasoning procedure in a conjunctive conclusion based on two premises and the access to it

Logic Premise 1 Premise 2 ConclusionProblem: (it rains a bit

↓AND it is July)

↓What about the sun?

Assertion: (no cloud in front of sun AND daytime) The sun is shiningLogic variables:Present access:Structure of reasoning:Reasoning procedure:

A?A?

B?

^ B^ ?

Conjunction? (result)

Answer: It is uncertainwhether the sun isshining or not

3.1.4 Hypothesis and ExpectationsThe intention of the study was to measure the structure of the participants’ reasoning, and

how it was accessed, but not to determine the exact processes in their reasoning procedures.Therefore, the participants were asked to state their reasons for the answers they gave. Fromthe arguments given in each of the reasoning procedures, the structure of reasoning could bededuced. It was assumed that the reasons given for the answer were the same reasons used inthe reasoning procedures to infer the answer. Thus, it was supposed that the number and typeof reasons presented were prescribed by the reasoning procedure involved. This follows fromthe assumption that reasoning is a metacognitive skill whose structure regulated both the wayconclusions were drawn and the way arguments were given. The entire set of reasons wouldrepresent the structure of reasoning.

To find the structure of reasoning, correct reasoning procedures had to be identified.However, only the answer and the reasons for this answer were available from the reasoningtasks. From neither a correct answer nor correct reasons alone could it be concluded that thereasoning procedure was correct. If both the answer and the reasons were right, the researchercould indeed assume that the reasoning procedure was correct, too. But in other cases, it couldhave been that the answer was wrong because of incorrect reasons, while the reasoning wascorrect. It was also possible that the answer was correct but the reasoning was not, because ofan incorrect reasoning based on wrong reasons. Thus the correctness of answers, reasons, andreasoning procedures had to be checked separately.

All information that is needed to draw a conclusion from the givens in the statement waspresent in the assertion. However, the information needed to evaluate the correctness of thepremises (the intermediate inferences) was not presented. This information, that was contentdependent and involved logical reasoning (especially in the oscilloscope case), wasconsidered basic. It was supposed that lack of knowledge or mistakes in the intermediateinferences would be clear from the reasons given by the students. It was expected that the waythey interpreted the given statements and their propositions would be reflected in the reasonspresented.

The results should be related to expertise. The students, when questioned about theweather, could be seen as experts in reasoning because they had at least ten years experiencewith this type of question. When questioned about the oscilloscope, they were somewhere in


between a pre-novice (who has no knowledge about the subject) and a novice (who hassuccessfully passed a course on the subject). Chi, Feltovich and Glaser (1981) found thatexperts in solving problems could work better with “unknowns” than novices do. Thus it wasexpected that the participants could handle logic variables in reasoning about the weatherbetter than when reasoning about the oscilloscope.

It was expected that in the case of the oscilloscope, the reasoning of students would bemore confused than in the weather case, and that if reasoning was really confused, theanswers would be guided by chance. Confused reasoning was supposed to show up inincorrect reasoning procedures. It was expected that in incorrect reasoning procedures, thenumber of correct answers would be in accordance with a random frequency distribution overall possible answers. It was also expected that the number of incorrect reasons would belarger, and the access less, than in the weather case.

It was supposed that the tacit knowledge used in correct reasoning processes could be ofdifferent types, and that different valid structures of reasoning (i.e. combinations of reasoningprocedures) were possible for the same type of reasoning problems. Any valid reasoningschema was supposed to generate correct reasoning procedures and correct answers. Since thesame applies to a theoretical schema, theoretical and other reasoning schemata cannot be incontradiction from a theoretical viewpoint.

First, it might be that the underlying structure couldn’t readily be detected from thebehavior of the participants. Such a schema was labeled hidden. In reasoning according to ahidden structure the participants give now this, and than that reason for their answers.Reasoning according to a hidden structure was supposed to be the case if the participants gavemore correct answers than were possible by chance, but showed no consistent pattern ofreasons in their reasoning procedures (no structure). In cases where a structure could bedetected, such a structure was called either empirical or theoretical.

Secondly, if participants’ reasoning procedures consistently showed the structure ofreasoning in accordance with the aforementioned theoretical truth table for a conjunction,their reasoning was labeled theoretical. Such reasoning was supposed to be the case if aparticipant consistently gave a correct answer and mentioned two reasons in correct reasoningprocedures. In this case consistent meant that an individual participant used correct reasoningprocedures and gave reasons following the same pattern in each reasoning task.

A third structure of reasoning was labeled empirical. In reasoning according to anempirical structure there is a structure in the reasons for the answers, but not always that tworeasons are given. Such reasoning was supposed to be the case when a rather large number ofparticipants gave reasons according to an identical pattern differing from a theoretical one.Consistent here meant that in a group of participants, all individuals gave reasons in theirreasoning procedures following the same pattern as the others did.

Summarizing, it was supposed that at least three types of a nonconscious structure ofsuccessful reasoning were possible:

1. In theoretical reasoning, reasons must be present in a stereotype manner, in which aparticipant gives two reasons every time.

2. In hidden reasoning, more answers are correct than would be expected by chance alone,but the reasons in otherwise correct reasoning show no consistent pattern.

3. Empirical reasoning is reasoning according to a structure that is identical among a groupof participants. The structure of empirical reasoning might differ from that of theoreticalreasoning, but should not be in contradiction with it.

The questions to be answered in this research are the following: How do students answerquestions about the weather involving a conjunctive conclusion based on two premises in thecontext of a three-value logic? How do these students give arguments for their answers ineach reasoning procedure? What schema representations of their reasoning can be deduced


from their answers and arguments? How do these answers, arguments and schemarepresentations change when students answer questions and give arguments in the case of theoscilloscope? And what can be concluded about the access to and the structure of reasoning,and about the handling of uncertainty in reasoning?

3.2 Method3.2.1 Participants

Two groups of electrical engineering students (in total 44 students) were randomly drawnfrom a total of 160 first-year students and were asked to volunteer as participants in thisstudy. The students had to pass a pre-test at the start of a laboratory course on electricalcircuits. The test was intended to determine whether they had acquired sufficient knowledgeabout the oscilloscope, a kind of monitor on which electrical signals can be made visible andmeasured, that was used to investigate the circuits. The students had already attended a priorlaboratory course in measuring instruments, including the oscilloscope. In this laboratorycourse, the students had to become acquainted with the principles and functions of theoscilloscope and the way to use it. Because the students had been working in pairs, some ofthem had not touched the oscilloscope and still could be considered pre-novices in thisdomain. Others could be regarded as novices because they had mastered the oscilloscope tosome extent.

The seven reasoning tasks about the oscilloscope were given to all 160 students, togetherwith some other questions and a measurement task, as part of the entrance test of the labcourse on electrical network analysis. Three weeks later, five reasoning tasks about theweather were presented to the two randomly drawn groups of students (44 students in total) inthe lab course, at the same scheduled time of the week. The tasks were presented as anecessary continuation of the first test in order to clear up some confusion about the reasoningcapabilities of the students. The data included all reasoning tasks about the weather and theoscilloscope given to these 44 participants.

3.2.2 MaterialIn the weather case, five reasoning tasks that met the conditions discussed in the theoretical

section were presented to the participants. The five tasks included a common assertion andfive different statements (see Table 3.4), each characterized by the truth values of thecorresponding theoretical reasoning procedure.

In the oscilloscope case, seven reasoning tasks were given to the participants,corresponding to the same theoretical reasoning procedures as in the weather case. With thefirst assertion five statements were given, with the second assertion two, similar to the firsttwo of the series of five (see Table 3.4).

The reasoning was about a subsystem of the oscilloscope. The task of the internaltriggering was to keep the picture on the monitoring screen constant in place, otherwise thepicture would be moving or look like lightning. This was a complex situation because bothpremises - although basic to the reasoning procedures - were complex in the sense that eachwas composed of several content components: ‘sufficiently large’ (with respect to what?) and‘the controls’ (which ones?) respectively.

3.2.3 ProceduresThe tasks were fulfilled during lab hours. First the assertion was presented to the

participants. It was explicitly stated that this assertion was correct. Then the statements werepresented. The participants were asked whether each statement was correct, with threepossibilities for an answer (yes, no, one cannot tell), and to give reasons for this answer(answer: ……because: ….).


In post-experimental interviews the participants were asked whether they knew of a three-value truth table for a conclusion based on two premises. It was not asked how much theparticipants knew about the oscilloscope because all necessary data to determine the structureof reasoning could be derived from the reasons and answers given.

Table 3.4Statements and corresponding theoretical reasoning procedures as related to the assertion

The weather case ProcedureAssertion:The sun is shining when there isn’t a cloud in front of the sunand it is daytime.Statements:

1. The sun is shining because there is one cloud in the sky and itis a quarter of an hour before sunrise.

? 0→0

2. The sun is shining because the weather is clear and it is sixo’clock in the evening.

1 ?→?

3. The sun is shining because it is eight o’clock and the weatherforecast says it is cloudy.

rev? ?→?

4. The sun is shining because there are no clouds and the sunhas just set.

10→0

5. The sun is shining because it is raining a little and it is in themonth of July.

? ?→?

The oscilloscope caseAssertion:For an internal triggering to work well the input signal needsto be sufficiently large and the controls of the oscilloscopeneed to be correctly adjusted.Statements:

1. The trigger works well because the trigger level is 1 V andthe oscilloscope is off.

? 0→0

2. The trigger works well because the input signal is sufficientlylarge and the electron-beam is badly focused.

1 ?→?

3. The trigger works well because the oscilloscope is on and theinput signal is 1 V.

rev? ?→?

4. The trigger works well because the input signal is sufficientlylarge and the time base control is on the position x via yA.

1 0→0

5. The trigger works well because the input signal is on yA andthe triggering is on yA.

Assertion: For a correct image dot both the focusing and theintensity have been adjusted correctly.

? ?→?

6. The image dot is correct because the focusing works well andso does the illumination.

1 ?→?

7. The image dot is correct because the focusing control isturned a little bit to the right and the intensity control isturned to the left as far as possible.

? 0→0

Note. Each statement is labeled by the corresponding theoretical reasoning procedure.

3.2.4 Data and AnalysisThe data consisted of the answers given and their reasons. After the data had been

collected1. the answers were classified according to correctness,2. the number of reasons per answer were counted,3. the reasons presented were classified according to both correctness and relevance, and


4. the correctness of the reasoning procedures as based on the reasons presented, waschecked. In case of doubt, the reasoning procedure was labeled as correct.

Below are a few examples showing how the analysis worked:In the weather case on statement 1 the answer ‘one cannot tell, because: since the sun has

not risen, it is not observable whether the sun actually shines’. Categorization will be asfollows. Answer: wrong; premise 1: absent; premise 2: irrelevant; reasoning procedure:correct (because when it is not observable whether the sun actually shines, it is possible todeduce that a person cannot tell whether the sun shines). Thus in this case the participant usesone premise, an irrelevant argument and a reasoning procedure taken as correct.

Or ‘two conditions have to be fulfilled and one of the two – because the sun has not risen -- is not true, thus the statement is not true’. Categorization: answer: correct; premise 1:mentioned but not present, assumed correct; premise 2: correct; reasoning procedure: correct.

The following considerations were used in these classifications.Some participants might give reasons that were not based on the assertion, like “The sun is

not shining because it is raining a little”. Such reasons were classified as not correct. Thegivens were also sometimes misunderstood or misinterpreted. A participant might think, forexample, that cloudy meant a hundred percent clouded over, labeled as incorrect. If a reasonpresented was incorrect, the way the conclusion was drawn might, nevertheless, be a correctreasoning procedure, leading to a wrong answer. An example might be a participant statingconclusion 3 of Table 3.4 (weather case) as wrong: ‘the sun is not shining because in cloudyweather there are clouds in front of the sun’.

If a reason did not relate a given to a premise, it was called irrelevant (e.g. “the weatherforecast can be right or wrong”). In this case the reasoning procedure was considered to beincorrect. As incorrect reasoning procedures were in general labeled: a procedure based on acorrect reason but leading to a wrong answer; or, a procedure based on an incorrect reasonand leading to an answer not permitted by the theoretical truth table.

There was a possibility that in some of these cases a subjective, not correct judgement wasmade. Therefore three persons classified the reasons and reasoning procedures in the weathercase, and two in the oscilloscope case. It turned out that the differences between theclassifications were small. There was difference of opinion in less than 4 % of the data.Differences could be reduced to differences in interpretation with respect to the supposedreasoning procedure of a participant that sometimes could not be reconstructed with certainty.

After the classifications the frequency distributions of the answers, the number of reasonsand the reasoning procedures in each category were analyzed. The consistency of theelaboration of the participants’ reasons for each participant and for groups of participants wasalso analyzed. From this, information was deduced about the reasoning schemas of theparticipants. Finally, it was checked whether there was a significant change in the way ofreasoning from the case of the weather to that of the oscilloscope.

3.3 ResultsIn the weather case it was found that 31 participants reasoned in a way in which the

structure was hidden, 10 showed an empirical structure of reasoning and one participantreasoned according to a theoretical structure. For empirical reasoning, a schema for thereasoning procedures was constructed. In the oscilloscope case no structure in reasoning wasidentified. However, more answers were correct than was possible by chance. The hiddenstructure of reasoning was different from the weather case because the use of themetacognitive component ‘uncertain’ changed significantly. A total number of reasoningprocedures (and answers) for the weather case and the oscilloscope of 210 and 308respectively were analyzed.


3.3.1 The Weather CaseThe absolute frequencies of both correct and incorrect reasoning procedures and

answers are presented in Table 3.5. A total of 191 correct answers were given, (91 % of all210 answers), with 70 % of the participants having all five answers correct. Chance wouldhave produced only 1 correct answer in 3, or 70 correct answers, much less than was the case.

Table 3.5Absolute frequencies of correct and incorrect answers and reasoning procedures

Classes Weather OscilloscopeCorrect reasoning procedures

with correct answerswith incorrect answers

Incorrect reasoning procedureswith correct answerswith incorrect answers

196

14

18313

86

150

158

9951

57101

correct answers, subtotals:incorrect answers, subtotals:

19119

156152

Totals 210 210 308 308

From all reasoning procedures, 93 % were correct. The frequency distribution of types ofreasons within the class of correct reasoning procedures is presented in Table 3.6. It is clearfrom the table that most participants gave a number of reasons that were dependent on thereasoning procedure involved. When both premises were uncertain, most of the participantswho reasoned correctly gave two reasons, otherwise one reason sufficed. But also in about 14% of the last cases, two reasons were presented (as in theoretical reasoning).

Table 3.6Frequency distribution of reasons in correct reasoning procedures (%, rounded off)

Procedure Weather OscilloscopeCorrect reasons Wrong/ Correct reasons Wrong/1 2 irrelevant 1 2 irrelevant

? 0→01 ?→?rev ? ?→?1 0→0? ?→?

8383228326

1315691262

538513

8350643345

2014010

1547236745

The reasons given by the majority of participants reasoning correctly were identified. Forinstance, in the first reasoning task the reason used most often was the false premise (‘it is notdaytime’). In Table 3.7 the most often used reasons are presented by the truth-values of thecorresponding premises. In correct reasoning procedures most of the participants used thereasons as shown in this table. When the individual participants’ data were analyzed, it wasfound that 10 participants consistently used the reasoning procedures as presented in Table3.7.

Further, it was found that the Pearson correlation coefficient between the truth values ofthe answers in the reasoning procedures with a false premise in the weather case showed acorrelation of r = 1.00, p < 0.001.

In post-experimental interviews the participants reported they did not know of a three-


value truth table for a conclusion based on two premises.When the individual participants’ data were examined, it was found that one participant

consistently used two reasons as required by theoretical reasoning, thus using a theoreticalreasoning schema representation.

Table 3.7Reasons most often given in correct reasoning procedures (weather case)

Procedure Reason %? 0→01 ?→?rev? ?→?1 0→0? ?→?

false premise (0)uncertain premise (?)both premises (? ?)false premise (0)both premises (? ?)

8383698362

Three participants consistently used one reason. The choice of the reason from among thetwo possible reasons did not show any pattern: it varied among these participants.

The other 28 participants did not give reasons consistent with respect to number or choice,although the majority of reasoning procedures was in accordance either with the empirical orwith the theoretical structure. Therefore at least 28 participants reasoned according to ahidden structure, mostly providing correct answers, but not being able to give reasons in aconsistent manner. Their pattern in reasoning looked like that of Table 3.7.

3.3.2 The Oscilloscope CaseTable 3.5 gives the absolute frequencies of the distribution of the results in classes of

correct and incorrect reasoning procedures and answers. Here 49 % of the answers werewrong, and 51 % of the reasoning procedures were wrong.

About 50 % of the answers, i.e. 156, were correct. If participants had chosen their answersat random, one third of the answers would have been correct, i.e. 103. Thus more answerswere correct than was possible by chance alone (p (x ≥ 156; M=103) <0.001), the differencebeing smaller than in the weather case.

It could also be observed that half of the group of participants (158 out of 308) failed to usecorrect logical reasoning when answering questions about the oscilloscope. The ratio of thenumber of correct answers to the wrong ones in case of incorrect reasoning procedures was 57to 101, and did not differ from the ratio that would be expected purely by chance (1:2) (χ2 (1,N = 158) = 0.5, a non significant result).

Correct answers could be based on wrong reasoning procedures, and wrong answers oncorrect reasoning. Table 3.5 gives some data that are characteristics for performance ofparticipants in a complex reasoning situation. About 1/3 of the correct answers (57 out of 156,37 %) were based on wrong reasoning procedures. About 1/3 of the wrong answers (51 out of152, 34 %) were based on correct reasoning procedures.

The relative distribution of the number of reasons in correct reasoning procedures ispresented in Table 3.6. No equivalent pattern to the weather case could be identified, thus noequivalent of Table 3.7 could be constructed. On examining the individual participants’ data,five persons were found who used only correct reasoning procedures, but no persons whoused either two reasons consistently (theoretical reasoning) or otherwise gave reasonsconsistently (empirical reasoning).

Since the total number of right answers was more than could have been obtained bychance, it was concluded that correct reasoning in the oscilloscope case, as represented inTable 3.6, made use of a hidden structure. The pattern of two uncertain premises as reasons


could not be found. However, the two tasks involving a false premise were answered correctlymore often than the other tasks (cf. step 1 of Table 3.9). The data of the weather and theoscilloscope cases will be compared in the next section.

3.3.3 Comparison of the Weather and the Oscilloscope CasesIn the weather case, 90 % of all answers were correct, as were 94 % of all reasoning

procedures. In the oscilloscope case, 51 % of all answers were correct and 49 % of allreasoning procedures (see Table 3.5). Both the number of correct answers and the number ofcorrect reasoning procedures decreased.

In the weather case, one participant out of 42 used theoretical reasoning, 10 used empiricalreasoning and 28 reasoned along a hidden structure. In the oscilloscope case, no consistentreasoning procedures, either empirical or theoretical, could be found.

The difference in the reasons presented was examined. The percentage of incorrect andirrelevant reasons given by participants in otherwise correct reasoning procedures increasedfrom M = 6 (SD = 4) percent in the weather case to M = 34 (SD = 20) percent in theoscilloscope case (cf. Table 3.6). In 1/3 of the correct reasoning procedures, the reasonspresented were incorrect, thus leading to a wrong answer.

The number of correct reasons as presented in correct reasoning procedures differed. (cf.Table 3.8). The hypothesis, that in correct reasoning procedures the number of correct reasonsgiven by the participants for their answers is independent of the domain of application of thereasoning procedure is tested by the value of chi-squared for each reasoning procedure aspresented in Table 3.8.

Table 3.8The number of correct reasons in correct reasoning procedures for the weather and the oscilloscope

WeatherCorrect reasons

OscilloscopeCorrect reasons

χ2

Procedures 1 2 1 2 (1 df)

? 0→01 ?→?rev ? ?→?1 0→0? ?→?

333383510

5625524

45191469

10302

2.6 (n.s.)1.8 (n.s.)13.4**

0.07.4*

Totals: 119 65 93 6* p < 0.01, one-tailed. **p < 0.001, one-tailed

For three reasoning procedures the shift was nil or not significant. For the reasoningprocedures ? ?→? and rev? ?→? the hypothesis was rejected with a significance value smallerthan 0.01 (see Table 3.8). In correct reasoning procedures based on correct reasons with twouncertain premises, the number of reasons given changed significantly when the logicproblems had a new and unknown content, whereas in reasoning with one uncertain premiseno change was observed.

3.4 Discussion3.4.1 The Structures of Reasoning

The main results of this study confirm that, indeed, three structures of reasoning: hidden,empirical and theoretical structures, are identified. In theoretical reasoning all reasoningprocedures are in accordance with the logic truth table as a mental reasoning schemarepresentation. This reasoning is used by one percent of the participants.


Since 25 % of the participants consistently use reasons as shown in Table 3.7, it isconcluded that this table represents the structure of an empirical reasoning schema. The maincharacteristic of this schema is that in case of two uncertain premises, both are used, but in theother cases only one premise is used. The answers in the empirical schema do not contradictthe theoretical one, so it can be assumed that the other premise is used nonconsciously in suchreasoning, and is not made conscious when reasons are given for the answer.

It is possible to construct a reasoning schema of sequential steps from these data (see Table3.9). The sequence must be that first the false premises are identified and used to draw aconclusion. The strength of this first step was confirmed by the strong correlation between theanswers in reasoning procedures with a false premise. Then the cases in which both premisesare uncertain are handled. At last the case in which one uncertain premise and one truepremise are present is handled. In the last two steps the uncertain premises are presented asreasons.

The order of the steps presented is consistent with all data.

Table 3.9The steps that are used in the empirical reasoning schema (weather case)

1. If one of the premises is incorrect (0), the conclusion is incorrect (0) because of onereason, the incorrect premise. The other premises (? or 1) are neglected.

2. In the remaining cases, if both premises are uncertain (?) the conclusion is uncertain (?)because of two reasons (both premises).

3. In the remaining cases, if one of the premises is uncertain (?) the conclusion is uncertain(?) because of one reason (the uncertain premise). The other, true premise, is neglected.

In three-value reasoning, people first reason with the aid of the false premise (start withwhat can be excluded), while in definitions such as the one for the conclusion in two-valuereasoning, the true premises were used (use what remains).

Table 3.7 gives an empirical reasoning schema representation consisting of five proceduresto give reasons; Table 3.9 gives an empirical reasoning schema consisting of three sequentialsteps in accordance with Table 3.7. From a viewpoint of parsimony Table 3.9 is preferred asrepresenting empirical reasoning.

Most participants reasoned in a way in which the majority of the reasoning procedureswere in accordance with the empirical schema, although no consistent structure among theseprocedures could be found. About 15 % of the reasoning procedures were in accordance withtheoretical reasoning.

It follows from the post-experimental interview that no metacognitive awareness withrespect to the structure of reasoning is involved. It is also highly unlikely that participants hadpreviously encountered this three-value structure of reasoning. This means that they show ametacognitive regulation of their tasks, although they are not aware of it.

Most participants were aware that if two premises are uncertain, both should be involved inreasoning. Thus it is clear that all participants were able to handle uncertainty rather well inreasoning in a well-known domain, though some more consistently than others. This handlingof uncertainty also shows access to reasoning by using two premises, as is shown incidentallyin other cases.

3.4.2 The Dependence on the ContentIn the case of more difficult - new and unknown - content, the situation is different.

Although many correct reasoning procedures can be observed, no structure of reasoning canbe deduced. The structure of reasoning changes. The hidden structure is different from thehidden structure in the well-known domain.

The number of wrong or irrelevant reasons in correct reasoning increased, showing a


decrease in access to reasoning. However, a decrease in the access to reasoning does notnecessarily influence the correctness of the reasoning procedures, but does influence thestructure of reasoning.

In most cases of correct reasoning with correct reasons only one premise is used, also incases where two premises are uncertain. In the oscilloscope case the capacity to handle twouncertain premises disappears. Handling of uncertain premises is an indication of access toreasoning (cf. Tables 3.1 and 3.3). Thus it is found that both the access to reasoning and thestructure of reasoning change because of the influence of the unknown content on reasoning;in a new domain they both have to be developed anew.

If the participants meet a case with two uncertain premises in which they can not tellwhether the answer given is correct, they give one reason only, in otherwise correct reasoningprocedures. This means that they can no longer reason backward clearly from an uncertainanswer to two uncertain premises, which they were able to do in the weather case. Why is thisso?

To handle two givens, two premises, the conclusion, and a logical variable at the sametime, requires at least 6 chunks in working memory. In the weather case this raises noproblems: making the two intermediate inferences (comparing the givens and premises) andhandling the two resulting truth-values is automated and can be done well. In the oscilloscopecase, where the content is new, and the premises and givens are more complex, additionalchunks have to be activated because of the extra difficulty in handling the complex premises.Then one of the premises probably is not retained or is forgotten.

The objection can be raised that the participants simply did not have the knowledge of agiven bit of information and thus were unable to evaluate the premise needed as true, false oruncertain. Given this, their reasoning should be inaccurate. However, such cases were putaside as incorrect reasons, irrelevant reasons, and/or incorrect reasoning. The shift can beobserved in cases of correct reasons and correct reasoning procedures based on them.

Another objection might be that participants had a different interpretation of the task. Aparticipant stated conclusion 3 as wrong because “there can be a cloud in front of the sun”.Here the answer was wrong and the reason was correct. Comparing this to the sameparticipant’s performance in the other tasks, it turned out that he reasoned that each statementas a whole was incorrect, instead of the conclusions only. This was the only participant whohad a different interpretation of the end of the task.

3.4.3 The Difference between Well-developed and Poorly-developed MetacognitionWhen participants reason about the weather they can be regarded as experts, when

reasoning about the oscilloscope as (pre-) novices. Chi, Bassok, Lewis, Reimann, and Glaser(1989) hypothesized from earlier research on problem solving and expertise that expertspossess problem schemata that novices lack. The results of Chi et al. and the present resultscorrespond, while in the last case it is more obvious what aspects of reasoning schemata arelacking: the nonconscious features of both access to and structure of metacognition have notyet developed.

An interesting observation is the finding that in complex subject matter, the reasoning iscorrect in 1 out of 3 wrong answers, and that in 1 out of 3 correct answers the reasoning iswrong. Teachers who value both correct reasoning and correct answers know this very well:in an exam situation they check not only the answers, but also the way participants came tothese answers. Since conjunctive conclusions are often used in practical reasoning, thenumbers found here might be indicative for the number of cases in which a teacher has tocorrect the mark.


3.4.4 Hints for Development of a (Metacognitive) SchemaThe structure of reasoning was either supposed to be “wired in” or acquired by an earlier,

implicit, learning process. This ‘tacit’ knowledge can be used to implicitly solve problems, aswas the case in the reasoning problems (Reber, 1989). The present findings extend this viewto the assumption that the participants’ implicit learning can lead to different structures intheir reasoning: a hidden one, an empirical one and a theoretical type.

Reber’s conclusion, however, is too general. “Tacit knowledge” in reasoning can be usedto solve reasoning problems only in a well-known domain like the weather. In a new domainlike the oscilloscope this is not the case. Thus implicit learning leads to tacit knowledge of aspecific domain. This raises the question of transfer to a new domain.

It follows that either the required knowledge has to be developed in a new domain in animplicit learning process, or the required knowledge is taught explicitly in such a way thattransfer will be stimulated. Empirical reasoning and hidden reasoning of the “experts” in awell-known domain are rather closely in accordance with each other, and used by nearly allparticipants. It seems advantageous, therefore, to make the empirical reasoning schemaexplicit in order to improve the reasoning capabilities of participants. That would lead to anexplicit schema: a set of empirical ‘rules’ that can be advantageous in transfer to otherdomains. It might be possible to explicitly stimulate metacognitive development in thissituation (see also Chapter 4). This conclusion might be related to transfer from priorexperience or prior capabilities by advance organization (Gagné, 1977; Ausubel, Novak, &Hanesian, 1978). For the case of reasoning, some other directions for instruction that can begeneralized can be given.

The hidden structure in reasoning should be made consistent (e.g. by discussions amongthe participants). The empirical reasoning procedures should be made explicit (e.g. byrequiring an explicit description from groups of participants). The theoretical reasoningshould be emphasized (e.g. by introducing truth-values and the ways to handle them). Thismeans that both highlighting the use of truth-values in reasoning, and exchanging informationon the way participants reason, can help. Presenting problems to the participants in whichforgetting one uncertain premise prevents them from solving the problem can also stimulatemetacognitive development (cognitive conflict).

Reasoning-problem schema representations that are really content-free have to rely ontruth-values. For transfer of reasoning truth tables as empty templates, filled with symbols,can be internalized as mental reasoning schema representations. The three-value conjunctionused in this study has been successfully used in teaching. With the aid of the theoretical truthtable, Veklerova (1980; also under her maiden name Tjoplenkaja, 1977) was able to teachabstract concepts with two attributes to children of kindergarten age. The truth table was usedthere for classification of objects, since the table can be used to draw a conclusion about classmembership (Vos, 1988).

The truth-value ‘true’ was learned in the interpretation ‘matches the attribute’, therepresentational symbol for which had to be learned by a separate process. The children notonly gradually internalize the truth table as a schema during the teaching experiments, butwhen new concepts were introduced, also started to ask for the attributes of these newconcepts, thus showing metacognitive skills. This can also be interpreted as backwardreasoning, an important characteristic of problem solving.

It has to be investigated to what other capabilities besides reasoning these findings can begeneralized, and what hinders application. For the generalization of using truth variables formanipulating unknowns in equations, the present findings suggest that 25 % of theparticipants can handle unknowns consistently in a well-known domain, and about 75 % cannot (cf. Chi et al., 1981). The same numbers could apply to the participants who either do ordo not show metacognitive awareness of understanding the content of the subject (cf. Chi et


al., 1989).It is found that logical capacities are potentialities that are present in reasoning within a

well-known domain, but these knowledge and skills are not available in a new domain ofknowledge. This prompts teachers to remark that participants “cannot reason”. It means thatwhen faced with new tasks in a new domain, participants do not handle the content logically.Flexibility of set (Gagné, 1988) has not yet been acquired.

Bassok and Holyoak (1989) argue that context, content and structure are the mostimportant factors involved in transfer of skills. This is in accordance with the present findings.The choice between empirical schemata such as shown in Table 3.7 (parallel structure) and inTable 3.9 (sequential structure) introduces a choice of structure in instruction. Tjoplenkaja(1977) stated that the theoretical schema has a more general form as compared to an empiricalschema. This introduces a fourth factor: form (cf. Chapter 5). The form she used was that ofTable 3.2.

A general (theoretical) structure has the advantage that it can be applied more consistentlyin more cases, and that the structure itself can be generalized. However, the data make clearthat few participants have a theoretical schema representation, so the schema is more difficultto find. An explicit empirical schema might be used to improve metacognitive skills. Thedisadvantage is that most participants already use such a schema in a well-known domain, anddon’t see the benefits of doing explicitly what they already can do implicitly. A solution tothis dilemma might be to encourage the participants to make their reasoning explicit (seeChapter 6). The advantage of an empirical schema is the participants’ familiarity with it.

An empirical metacognitive schema that is made explicit consists of a set of proceduresthat can be used consistently. This empirical schema controls the choice and number ofelements to be used. It is supposed that instruction for and learning of such an empiricalschema can help participants to further develop metacognition.

ReferencesAristotle. (1960). Rhetoric (Cooper, Trans.). New York: Appleton-Century-Crofts.Ausubel, D.P., Novak, J.D., & Hanesian, H. (1978). Educational Psychology: A Cognitive

View (2nd ed.). New York: Holt.Bassok, M., & Holyoak, K.J. (1989). Interdomain transfer between isomorphic topics in

algebra and physics. Journal of Experimental Psychology: Learning, Memory, andCognition, 15, 153-166.

Broadbent, D.E., FitzGerarld, P., & Broadbent, M.P.H. (1986). Implicit and explicitknowledge in the control of complex systems. British Journal of Psychology, 77, 33-50.

Chi, M.T.H., Bassok, M., Lewis, M., Reimann, P., & Glaser, R. (1989). Self-explanations:How participants study and use examples in learning to solve problems. Cognitive Science,13, 145-182.

Chi, M.T.H., Feltovich, P.J., & Glaser, R. (1981). Categorization and representation ofphysics problems by experts and novices. Cognitive Science 5(2), 121-152.

Gagné, R.M. (1977). The Conditions of Learning (3rd ed.). New York: Holt.Gagné, R.M. (1988). Some reflections on thinking skills. Instructional Science, 17, 387-390.Gleitman, H., Fridlund, A.J., & Resiberg, D. (1999). Psychology (fifth ed.). New York:

Norton.Halpern, D.F. (1984). Thought and Knowledge. Hillsdale, NJ: Lawrence Erlbaum Associates.Hartman, M., Knopman, D.S., & Nisson, M.J. (1989). Implicit learning of new verbal

associations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15,1070-1069.

Lewicki, P., Czyzewska, M., & Hoffman, H. (1987). Unconscious acquisition of complexprocedural knowledge. Journal of Experimental Psychology: Learning, Memory, and


Cognition, 13, 523-530.Lemmon, E.J., (1965). Beginning Logic. London: Nelson.Means, M.L., & Voss, J.F. (1996). Who reasons well? Two studies of informal reasoning

among children of different grade, ability, and knowledge levels. Cognition andInstruction, 14(2), 139-178.

Reber, A.S. (1989). Implicit learning and tacit knowledge. Journal of ExperimentalPsychology: General, 118, 219.

Rescher, N. (1969). Many-valued Logic. New York: McGraw-Hill.Schoenfeld, J.R. (1967). Mathematical Logic. Reading, MA: Addison-Wesley.Stadler, M.A. (1989). On learning complex procedural knowledge. Journal of Experimental

Psychology: Learning, Memory, and Cognition, 15, 1061-1069.Tjoplenkaja, Ch.M. (1977). On the problem of concept formation by children of kindergarten

age. (In German. Tjoplenkaja is the maiden name of Veklerova). In J. Lompscher,Probleme der Ausbildung geistiger Handlungen [Problems in the formation of mentalactions] (pp. 41-70). Berlin: Volk und Wissen.

Veklerova, H.M. (1980). Vorming van logische structuren bij vijf- en zesjarigen [Formationof logical structures with five- and six years old children] (Unpublished doctoraldissertation). In C.F. van Parreren and J.A.M. Carpay (Eds.), Sovjetpsychologen overonderwijs en cognitieve ontwikkeling [Sovjet psychologists on education and cognitivedevelopment] (2nd. ed., pp. 60-68). Groningen, the Netherlands: Wolters-Noordhoff.

Vos, H. (1988). Hoge abstractieniveaus in het onderwijs van Veklerova [High levels ofabstraction in the teaching of Veklerova] (EL-OC doc 88-29). Enschede, the Netherlands:University of Twente, Educational Center.

Voss, J.F., & Means, M.L. (1991). Learning to reason via instruction in argumentation.Learning and Instruction, 1, 337-350.

Wason, P.C. (1977). Self-contradictions. In P.N. Johnson-Laird and P.C. Wason (Eds.),Thinking : Readings in cognitive science. Cambridge: Cambridge University Press.

Wason, P.C. (1983). Realism and rationality in the selection task. In J.St.B.T. Evans (Ed.),Thinking and reasoning: Psychological approaches. London: Routledge & Kegan Paul.

Metacognition in instruction 57

Chapter 4

Development of Metacognition in an Instructional Model withDouble Sequencing

AbstractThe thorough revision of a laboratory course, in which a systematic approach to

experimental investigation (a metacognitive strategy) was taught, led to a stable improvementin the passing rates. The data and observations on the course were reanalyzed in order todescribe four metacognitive variants in the instructional format, to explicate the sequencingof tasks, and to study the development of metacognition. The metacognitive strategy variantwas explicitly formulated and included the capability of the students to find the informationthey needed for themselves. In the three domains (parts) of the course the strategy wasintroduced (developing the structure in a well-known domain in which access was easy),practiced (access to the structure in a new domain), and applied respectively. Thecharacteristics of this system of instruction with respect to metacognition were fading,boosting, just-in-time feedback, and early marking in three separate subdomains. Themetacognitive task variant included the requirement to construct measurement proceduresand to judge the influence of the measurement instruments. The tasks were sequencedaccording to increasing complexity on a cognitive level in each subdomain. Thus theinstructional design of the course was characterized by a system of double sequencing of thetasks on a metacognitive and a cognitive level. The metacognitive knowledge variant includedthe structure of knowledge, the relation between words, formulae, and schematics, and thedifference between observation and mental image. To stimulate metacognitive experiences,some of the tasks included contradictions, others a comparison of methods, and some wereopen-ended. The development of metacognition during the course was studied by dividing thestudents in three categories (good, moderate, and weak) and analyzing observations of theiractivities and performance from scoring lists used for assessing the logbooks in eachsubdomain, and from the questionnaires for each session. Significant differences inperformance on several metacognitive skills were found. Learning the strategy was not alinear process. It was found that the students made most progress in learning the strategy inpart two of the course, after feedback had been given on their performance in the first part. Inthe third part of the course, the performance of the good students and that of the moderateones on the metacognitive skills approached each other, a unique finding. This provided theexplanation that the lower rate of failure in the new situation could be attributed to theimproved learning of the metacognitive strategy by the moderate students.

A laboratory course was thoroughly revised because the passing rates were too low. Themain objective of the course was that the students should learn a systematic approach toexperimental investigation. The students should learn to do experiments in a systematic way,i.e. to carry out an investigation along an explicit methodology by comparing theoretical andexperimental results and acquiring the necessary information to fulfil the tasks. Thesystematic approach consisted of many constituent skills, like being able to formulate ahypothesis (a metacognitive skill) and the skill to operate the measuring instruments (acognitive skill).

58 Metacognition in instruction

The systematic or methodical approach was considered to be a metacognitive strategy. Theobjective of the laboratory course was comparable to teaching thinking skills as investigatedby Reif and St. John (1979). When added to the cognitive content of the lab course to bemastered, the high requirements led to passing rates below 50 % of the participating students.Most students did not show a systematic approach in their work, and had no criteria toevaluate the results of their work. For example, the students asked the assistants questionslike: “Is this the correct outcome of the measurement, sir?” This situation was consideredunacceptable.

The lab course was redesigned. It turned out that it was possible to maintain the highrequirements (the metacognitive level) and at the same time to raise the passing rates to asignificantly higher and stable level (see Appendix E for the data). Both the educationalefficacy and efficiency improved: more students learned it, in less time, needing lessassistance. The results were published as a backward reasoning solution to a practicalinstructional problem (Tattje and Vos, 1995). Understanding these results required furtheranalysis.

At the time of the publication cited, it was thought that the improvement was simply due toa good instructional design of the lab course. Afterwards, the idea surfaced that themetacognitive components in the instructional design might have been responsible for thegood results. Therefore a case study was done, the goal of which was to find out whether themetacognitive components in the instruction could be responsible for the development ofmetacognition, by which the (metacognitive) objectives of the course were met.

The results of the study were considered important because they could help to find someinstructional design rules from good practice in a real life educational setting. It could add toother research on metacognition in educational practice as was, for example, presented inHacker, Dunlosky, and Graesser (1998). The rules found would enable a better design ofinstruction focused on the development of metacognition in other situations. The study couldalso help to better investigate such development.

The lab course was a first year’s course consisting of nine laboratory sessions on NetworkAnalysis running in parallel with lectures on theory and tutoring hours on problem solving.The course presumed prior knowledge of current, voltage, and elementary calculus from thestudents’ high school curriculum. The course was split in three parts of three sessions each. Atask of the students was to write a logbook on each part of the course. Each logbook wasgraded.

The three parts included: (a) the introductory measurements: about circuits with familiarcomponents like resistors and capacitors, and familiar calculations; (b) the time domain: aboutnetwork properties like linearity and calculations involving the solution of differentialequations or convolution integrals; and (c) the frequency domain: about circuits with aninductor as a component and calculations by the complex transfer function. The three partswere considered as three (sub)domains of knowledge.

The content of the course was not changed, but the instructional design was. Some labassignments (tasks) were replaced by new ones, some information was added to the tasks, andnew sections were added to the lab guide. Some of these changes were related to the cognitivelevel, like the sequencing of the instructional tasks from simple to complex, and the inclusionof the concepts and skills from the earlier, simpler ones in the more complex, later ones.Others related to the metacognitive level, like the information on the systematic approach.

To teach students to solve problems in a new course in a systematic way required thedevelopment of a metacognitive strategy. This was comparable to learning a metacognitivestrategy like reasoning in a new domain. In such a case, any previously acquired structure andaccess were not used consistently (see Chapter 3). It was assumed that metacognitivedevelopment was stimulated by the methods described in Chapters 2 and 3.


The questions to be answered were therefore as follows: Which metacognitive variantswere present in the instructional design of the new course? Where in the assignments,information, or assistance were these variants located? What was the expected effect on thecognitive results of the course? How could the sequencing with respect to the metacognitivecomponents in the tasks be described? How did the sequence relate to the cognitivesequencing of tasks? How did the metacognition of the students develop in relation to thesequencing of the tasks?

The study was an exploratory type of action research on the development of themetacognitive knowledge and skills of the new course, on the factors that influenced thedevelopment, and the explanations for that. It must be stressed that no detailed data on thestudents’ performance from the old course were available. Instead of a pretest- posttestdesign, this study included a differential-development analysis design by distinguishing threecategories of students that followed the new course. However, to highlight the changes in theinstructional design, a comparison was made between the instructional design of the oldcourse and the new course in some places where this is explicitly stated. In other cases, thesituation in the new course is described.

4.1 Instructional Design and Design of Research: TheoreticalConsiderations

Four variants were distinguished in metacognition: strategy, task, knowledge, andexperience (see Chapter 2). Each variant will be described shortly, its presence in the labcourse will be exposed, as well as the instructional material and the instructional procedureapplied for its development. This will be presented together with its supposed effect on thedevelopment of metacognition, and on the cognitive performance. Since the development ofeach variant includes metacognitive experiences, these are not treated separately but togetherwith the other variants. An overview is presented in Table 4.1B in the method section.

4.1.1 Strategy VariantThe metacognitive strategy variant regulated cognition and usually involved a check on the

cognitive skills involved. The systematic approach to experimentation to be learned had as abasic principle that the outcome of each measurement should be compared to the outcome ofa calculation, and vice versa. The regulatory function of this approach showed up if theoutcomes did not fit because in such a case, the measurements, the calculations, and themodels made of the circuits all had to be monitored for correctness. So this systematicapproach could indeed be considered as a metacognitive strategy.

MaterialThe schema of the metacognitive strategy (see Table B.1 in Appendix B) consisted of six

phases: (a) problem formulation; (b) information acquisition; (c) formulation of hypotheses;(d) test experiment; (e) conclusion and discussion; (f) reporting. The results of thecalculations and the experimental results had to be compared in the conclusion. The outcomeof an observation and the outcome of a theoretical description should match. The model thusincluded a kind of validation of knowledge in an empirical way (cf. De Groot, 1961). In anycase it should diminish uncertainty as to whether the outcome of either a measurement or acalculation was correct or not.

The model also contained a phase in which (theoretical) information should be gatheredand interpreted. This phase was to stress that information acquisition should be theresponsibility of the student, not of the teacher: a metacognitive communication.


Instructional procedureThis model was presented in the lab guide (Tattje and Vos, 1986; see Appendix A for the

content of the guide) as the way that engineers and scientists work. The model, presented inthe form of a list-like framework for investigation, was explained by an example of a simpleinvestigation in which all phases were present. Some assignments were specially formulatedto let the students practice the application of the model and thus develop their ownmetacognitive schema (see later). An example of an assignments in which the result of acalculation and the result of a measurement had to be compared is given in Assignment 4(Appendix C).

The application of the strategy was stimulated by the assessment criteria for the logbooks.The criteria were not only that the results were correctly represented in tables, etc., but alsothat the students showed that they knew what they were doing and applied the strategy. Afeedback procedure in which feedback on the performance in a part of the course was given atthe beginning of the next part (‘just in time’ feedback) also stimulated the application of thestrategy.

Explanation of the effectsSuch a model of a strategy could be considered as an empirical schema of a metacognitive

skill (see Chapter 3). It was supposed that all students had previously acquired components ofthe model, in solving experimental problems in high school or elsewhere, but mostly did notuse them in a consistent way. The model presented should help them to use the constituentcomponents more consistently in the new subject of NA, and thus to further develop theirhidden metacognitive skill.

It was assumed that this development was stimulated by contradictions in which tworesults did not fit exactly. The differences arose because: (a) the results differed qualitatively,some being numbers and others readings of a pointer along a scale (different representations);(b) every measurement outcome possesses a statistical variation and an inaccuracy, and everycalculation an inaccuracy, causing a scatter in the results; and (c) a measurement ofteninvolved a - somewhat distorted and always limited - picture of a signal on an oscilloscopescreen, whereas a calculation of such a signal resulted in an algebraic expression. So in allcases the students had to accept that things were declared equal from a metacognitive point ofview, even when they were found to be unequal from a cognitive viewpoint.

The tasks of the students required critical consideration of possible causes of anydifferences and elimination of these differences. In order to conclude to a fit between theresults, both measuring and calculation procedures had to be monitored with respect topossible errors and allowed variations after the task was done. Mental images of formulae andobserved signals also had to be monitored. Thus different cognitive stationary and dynamicstates had to be compared, stimulating a metacognitive awareness of these states, theirsimilarities and differences. This method promoted the transformation of cognitive actionsinto objects of thinking.

Access to the regulatory skills was stimulated by the marks for the logbooks. Marks wereconsidered to lead to metacognitive experiences especially if the students had workedintensively - as is usually the case in lab work - and had expected a higher mark than wasgiven. Subsequent feedback on the use of the strategy in the logbook, together with theawarding of the marks, in the next domain just at the beginning of the next logbook, shouldstimulate the student to improve his metacognitive performance.

The development of the schema should lead to the regulation of cognitive actions. It wasexpected that the effect of this strategy would be that students would no longer ask the TAswhether the outcome of a measurement was the correct result, but that they should comparethe outcome of the measurement with some calculated or otherwise deduced theoretical result


by themselves. It was also expected that students would gain more certainty about theoutcomes when their own measurements and calculation results did fit.

The effect of the development of the metacognitive strategy was also assumed to be thatthe students should become more aware of the difference between the cognitive state of notknowing whether an outcome or answer was correct, and knowing that an answer wasvalidated and thus could be considered correct. In the first case, the need for furtherinformation was strengthened by the hint, formulated within the model of the strategy, that thestudents were themselves responsible for the detection and acquisition of missing information.However, this hint was itself a part of the available information, and for instance lacking forthe students who did not read all material carefully from the beginning. So in the beginningthe students needed help on this point (fading).

It was expected that the students would learn to independently acquire information needed,become less dependent on the exact moment the theory was treated in the parallel lectures andtutoring classes, and felt more responsible for finding the theory needed themselves, e.g. bybetter preparing the sessions at home.

4.1.2 Task VariantThe metacognitive task variant comprised not only knowledge of the difficulty,

complexity, and other characteristics of a task, but also metacognitive goals. A metacognitivegoal (a part of the strategy) was that the students should construct the detailed procedures (thecookbook) by themselves. No foolproof description like an algorithm (cookbook), noprocedural description of problem solving methods by steps (partial skills), footprints (theirsubgoals), or their sequence (synthesis of skills) were given. Here a conflict was inducedbecause most assignments required that a measurement or calculation procedure to be carriedout, but these were not described in detail.

Each measurement and calculation procedure had to be constructed from the availableknowledge and skills, or based on the available information and on the experiences in earliertasks. The latter could be done for instance by analyzing the earlier actions into sequences ofsteps, and using some of the steps in a different order. This required that subsequent tasksincluded procedures of earlier tasks, or parts thereof.

Another one of the difficulties the students should meet was the fact that the measuringinstruments could have a substantial influence on the quantity (i.e. the quantitative physicalvariable) to be measured. In such cases, the measurement result was the value of the quantitymeasured in a circuit connected with the instruments, but this result differed from the value ofthe quantity in the circuit without the instruments connected to it. This was analogous to aninteraction of the test and the treatment results. Thus in the case of a difference between tworesults, it was often the case that the origin of this difference could not be found in errors inthe measurement or calculation procedures, but was due to the influence of the instrument. Inthese cases a conflict was induced in the students between a ‘wrong’ or ‘not fitting’ outcomeand the awareness of not having done anything wrong.

Material and instructional procedureInformation was available to help the students solve such conflicts in the relevant

assignments, not as just in time information on the task together with the assignments, but inother sections in the lab guide. The concepts and the principles described were assumed to besufficient for students to construct the cookbook themselves. Information was also presentedabout the different ways to connect measuring instruments to a circuit, the effects in eachparticular case, the ways to diminish the influence of the instrument on the quantity to bemeasured, and the ways to calculate and correct for the influence on the readings.


Homework assignments were introduced to read this information and elaborate on it in thefirst part of the course only (fading). Some assignments were specially designed to induce theconflict caused by the influence of the measuring instruments. It was expected that by lettingthe students work in pairs, together they could handle the conflicts and carry out the tasks.

In order to allow the students to build on earlier experiences, the tasks were sequencedwith respect to complexity in each domain. For instance, measuring a given quantity was asimpler task than, and prerequisite for, the task of comparing two measurements of the samequantity with respect to the accuracy (e.g. Assignment 3, Appendix C). This comparison, inturn, was simpler than carrying out an open ended assignment in which the quantity to bemeasured was not given, and a choice had to be made for the best measurement method. Inthe later, more complex tasks, the students should be able to build on partial skills alreadyacquired in earlier, simpler ones.

Therefore, within each part of the lab course the tasks were sequenced according toincreasing cognitive complexity: (a) homework assignments to process the given information;(b) preparative assignments to illustrate how to measure a new phenomenon or how to use anew instrument; (c) comparative measurement assignments to compare either twomeasurements or a measurement with a calculation (the last one being the final level aimedat); and (d) open assignments, in which neither the quantity to be measured, nor themeasurement or calculation method to be used were given (cf. Dijkstra, 1997).

Furthermore, open assignments were introduced in the first and second domain, while theassignments in the last part were of the lower, final level only (see Table 4.2). This system iscalled ‘boosting’. It is similar to the principle of ‘added difficulty’ that gives poorerperformance during the acquisition phase in learning but better retention (Schmidt & Bjork,1992).

Explanation of the effectsIt was assumed that the students were forced to take a metacognitive position and to find

the necessary information in order to carry out the tasks. The information needed wasavailable in written form, as concepts, explanations, and principles, and separated from thetask. It was expected that those students who would be able to detect the missing knowledgeand find the necessary information, would also be able to carry out the assignments and wouldtherefore develop a metacognitive experience of success, and a trust in their own capacities.

The effect of this metacognitive development on the acquisition of cognitive knowledgewas expected to be that the students would be able to construct and carry out the calculationand measurement procedures largely by themselves, without having to rely on the TAs. Theyshould be able to detect the cases in which the ‘wrong’ measurement result was due to theinfluence of the measurement instrument, and would adjust their instrument or choose themethod of measurement in such a way that influence of the instrument was minimal. It wasassumed that in this way they became critical with respect to the results and execution of theirtasks.

For the system of ‘boosting’, the explanation was as follows. More complex tasks were notonly more difficult, but also were expected to require more strategic regulation. Byintroducing an open-ended assignment in part one, the students were exposed to a level ofhigh strategic regulation. In part three, with comparative assignments only (the intended finallevel), the requirements were lower. It was supposed that this system would raise the chancethat the students would end up at the required level of performance.

4.1.3 Knowledge VariantThe metacognitive knowledge variants in the lab course included oversight over concepts

and skills, and a certain coherence to facilitate the application of the knowledge. An


introduction to the lab course in NA was designed in which the relations among the concepts,skills, schematics, and formulae were presented forming a (metacognitive) structure ofknowledge.

Further, attention was drawn to the possibility of representing the same circuit or circuit-element by different schematics, and alternatively, to build geometrically different circuitsfrom the same schematic. Here different mental representations of the same object wereunited on a metacognitive level of imagination. These representations were important torealize circuits from schematics and to make a schematic model of a circuit.

Material and instructional procedureThe introduction to the lab course is presented in Appendix A. Examples of different

schematics for the same circuit were included in the introduction, and were included in otherparts of the lab guide. Hints for the construction of a circuit from a schematic and for makinga schematic from a circuit were also included. To help students, several stages for making aschematic from a given circuit were distinguished. These stages included: (a) a sketch of thereal circuit with drawings of the elements; (b) a draft schematic in which the leads weresketched but the elements represented by symbols; and (c) a formal schematic representingideal elements, thus describing a development along the dimension from real to ideal. Withrespect to the existence of invisible objects like voltages and signals, these were bothrepresented by diagrams and by mathematical expressions, accompanied by the way theywere measured.

The introduction was presented in the lab guide as a first chapter. A reading assignmentwas given in the beginning of the course. The development of coherence of knowledge bystudents was enhanced by asking them to write in their logbooks not only the results ofmeasurements, but also the schematics, formulae, calculations, and reasoning they used.

Reading assignments for the parts of the lab guide in which different schematics for thesame circuits were presented, were given in the first part of the course. Homeworkassignments to elaborate on the information were added. And as already stated, the studentsworked in pairs.

Explanation of the effectThe introduction was constructed as an advance organizer in the sense described by

Ausubel (Ausubel, Novak & Hanesian, 1978; Vos, 1991; Vos, 1992). Reading theintroduction should contribute to the access of a coherent structure of knowledge and thefurther development of its coherence.

The different schematics for the same circuit provided a contradiction. One type ofrepresentation involved a mental image of the circuit; the other mental images were based ondifferent schematic representations. Here again, the conflicts between the pictorial mentalrepresentations of the circuit and the schematics had to be resolved, the contradiction beingreconciled by declaring things equal that were not. Such a process was assumed to stimulatethe development of imagination. The effects were assumed to be important for the studentswho were weak in geometrical and schematic insight.

Contradictions were used to raise metacognitive awareness of ‘invisible’ objects like avoltage or a property of a circuit like linearity. The conflict arose between the thoughts of thestudents about the object and its measured characteristics here. Examples could be found inthe nonlinearity of a seemingly linear circuit (Assignment 1, Appendix C) and the differentways in which a signal that was ‘constructed’ by a signal generator could be presented on thescreen of the measurement instrument (Assignment 2, Appendix C). The reconciliation of thecontradiction was supposed to induce a conceptual change with respect to the earlier,incomplete or even incorrect, understanding of the student.


Finally, by letting the students work in pairs, discussion, mutual support, andmetacognitive interpersonal experiences were promoted. A confrontation with a difference inknowledge, skills and approach, both on a cognitive and a metacognitive level, was intended.The differences were also expected to help develop metacognition because accuratelyformulating and expressing one’s own knowledge into information and interpreting others’information was required.

Here the exposition of the metacognitive variants in the lab course ends. The followingsection describes the way to study the development of metacognition and the expectations.

4.1.4 Design of the StudyFrom observations in the past it was generally known that large individual differences

existed among the students. It was generally assumed that students could be split into twocategories, weak students and good students. Weak students were supposed to be a categorythat in general would fail courses, while good students could be expected to pass. In this studyit was thought useful to distinguish an intermediate category of moderate students. Thus threecategories of students were distinguished based on their marks for courses in general. It wassupposed that in the past a large number of these moderate students failed, while in the newsituation they would pass the course.

It was also assumed that differences existed among the students with respect to their abilityto observe (differences in) cognitive states. Since learning is dependent on the ability toobserve whether one knows something or not, it was hypothesized that good students havebetter learning results than other students through this metacognitive skill.

The development of different metacognitive variants was assumed to be strongly related.Therefore, the observation of the development of a limited number of metacognitive skillscould be seen as representing the development of the metacognitive variants in general. Theacquisition of metacognitive skills could be tracked in two ways: (a) by assessing thelogbooks with respect to several of these skills; and (b) by assessing the logbooks with respectto specific assignments in which metacognitive skills were necessary. These variables werethe basis for giving marks. Thus the marks were strongly dependent on demonstration ofmetacognitive skills (in general the criterion for assessment was “Do the students understandwhat they are doing?”). The development of the students’ metacognition during the coursewas also supposed to be reflected by the marks on their logbooks.

After categorizing the students into three groups: weak, moderate, and good, thedevelopment of metacognitive knowledge and skills could be monitored during the coursebecause the three logbooks provided observations of the performance of the students in thethree parts of the course. An increase in performance was expected. The study could bedenoted as a time-series treatment design, of the form X1-O-X2-O-X3-O. After each phase inthe treatment X, the metacognitive performance was observed (O).

It was expected that good students would also be more successful in learning the requiredmetacognitive strategy and other metacognitive skills and knowledge: the quality of thestudent (the independent variable) was expected to be positively related with the mastery anddevelopment of metacognitive skills in new domains (the dependent variables). It wasexpected that students with low metacognitive skills would perform especially poorly whenthe external regulation was diminished because of fading. Further, it was expected thatbecause of timely feedback on the first logbook, metacognitive performance would improvein the second part of the course. Finally, because good students better know what they do notknow (Chi, Feltovich, & Glaser, 1981), it was expected that they would better prepare theirwork at home. Elshout (1983) showed that good beginners differ from less good beginners inthe way they handle their being a beginner. Good beginners will work in such a way that there


is a great chance to learn from their experiences, even if this means that they are not so readilyfinished.

The acquisition and mastery of the metacognitive skills as a function of the general qualityof the students was observed in the three parts of the lab course. Thus the progress of themetacognitive development of the students could be studied.

4.2 Method4.2.1 Situation and Participants

Each year about 150 students participated in the lab course, which ran in parallel tolecturing and tutoring hours on NA in the second trimester of the first year. Each of thelectures (28 h in all) was presented to the entire group of students at one time, tutoring hours(18 h) were attended in classes of about 25 students, as were the laboratory classes.Attendance for the lab classes was obligatory, whereas attendance for the lectures and tutoringhours was voluntary. The lab course consisted of 9 sessions of 3½ h each (half days). Thestudents were expected to spend a total of about 55 hours of study at home, includingpreparation for the labs.

A general outline of the course programs for electrical engineering in the Dutch system ofhigher education is found in Vandamme (1990). Basic concepts such as electrical current,voltage, resistor, capacitor, voltage source, electrical lead (connection), Ohm’s law, andmeasurement instruments like ammeter and voltmeter had been treated and examined in theDutch high school system. These concepts have also been investigated in laboratory work inhigh school. In the lectures the more abstract central concepts of NA and the coherence of thetheory were highlighted. In tutoring hours the theory was applied by solving calculationproblems (cf. Vos & De Bruin, 1995; De Bruin & Vos, 1995). In the lab guide someinformation was given about the measuring instruments, the oscilloscope, signal generator,and multimeter (see Appendix A). The use of measuring instruments had ostensibly alreadybeen learned in a prior lab course on measuring instruments, although the students in thiscourse worked in pairs, and mastery was not tested individually.

The students’ results in the labs were graded separately from the theoretical part ofNetwork Analysis. The teaching assistants (TAs) gave a mark for each logbook. The finalmark was the average of the marks for all logbooks, the last two marks counting with moreweight than the first one.

In the labs the students worked together in pairs. The TAs were available to help thestudents in fulfilling their tasks. Each class of 25 students was helped and assessed by severalassistants. The TAs were instructed not to demonstrate the skills students were required toconstruct from knowledge and principles, nor to tell them exactly what to do. They had tohelp students to take the next step in the problems met by asking questions about what wasthe problem, where the problem could be, etc. They were to give hints, to give short answersto questions, or to refer to available information probably overlooked or not understood by thestudent. The marks thus were based on ‘assessment in the zone of proximal development’: theteaching assistants were both helping (teaching) and assessing. The participants in the studywere the first-year students of the academic year 1990/91 (N = 137).

4.2.2 Instructional DesignChanges made to the assignments of the lab course are presented in Table A.2 (see

Appendix A). The first and second parts’ assignments were expanded with the introduction ofcontradictions and addition of open-ended assignments. No changes were made to the thirdpart. The changes to the lab guide can be found in Table A.1 (see Appendix A). The totalnumber of pages increased from 22 to 61. The statements of the assignments in the lab guidewere separated from the information that was needed to fulfill the tasks and from the


information about the strategic schema. Part of the information needed was distributed overdifferent sections of the lab guide and part was presented in the theoretical lecture notes.

The changes in the instructional material and its presentation in order to support thedevelopment of metacognition are summarized in Table 4.1A. The change with respect to themetacognitive strategy is described as follows. In the first part of the lab course, for the fivelab assignments of the second and third session (see Table 4.2), the methodical componentsfrom the strategy to be applied in the particular assignment were listed. Detailed hints werealso given on the places where needed information could be found. In part two, only generalhints were given on the places where information could be found. Such hints were absent inpart three. The principle applied here was fading with respect to the prompting for applicationof the metacognitive strategy.

The cognitive load in part one was supposed to be low because some new concepts wereintroduced here, but no new calculation methods. This situation, where prior knowledge wasextended only, was supposed to be favorable to the introduction of the new strategy. In theother two parts, where new theoretical knowledge was introduced, access to and use of themetacognitive strategy was practiced and tested respectively.

The final aim was to compare measurement with calculation results in the assignments. Anassignment in which measurements had to be compared with other measurements wasincluded in part 1 (session 2), in which part the calculations to be done were rather simple(orientation on the strategy). Assignments in which calculations and measurements had to becompared were included both in the changed part 2 and the unchanged part 3 (practice andapplication of the strategy respectively).

Table 4.1AThe instructional design principles implemented before and after the change

Instructional variants Before AfterCognitive Oral information, concise

written information.More extensive writteninformation.

Information if needed. References in part one and fading.Metacognitive strategy Orally if needed. Written framework/ schema

Oral hints Written hints in part one and fadingacross the other parts.

Metacognition on tasks Oral Written information aboutinfluences of measurementinstruments on quantity.

- Hint: Find information needed.- Hint: Construct a ‘cookbook’.

Metacognitive knowledge - Introduction presenting thestructure of NA knowledge.

In Table 4.1B an overview is presented of the metacognitive variants, the relatedinstructional material, and the instructional procedures as described earlier.

Within each part (domain), the assignments were sequenced according to the ascendingcomplexity of the cognitive tasks involved: activation of prior knowledge, informationprocessing, learning new skills, analyzing and synthesizing skills in comparative assignments,and choosing goals and skills in open ended assignments. Moreover, in the first part adiagnostic test and a remedial task were added. The sequence of the cognitive tasks iselaborated in Appendix A.

In all the assignments the students were required to express their thinking in words,formulae, and schematics in the logbooks. In that way they developed not only metacognitiveskills and thinking skills (Katz and Warner, 1988), but also provided information to the


teachers, enabling them to give feedback on the form and the content of this information andto assess the work. Students were required to record in their logbook not only the results theyachieved, but also the things they would rather omit (e.g. not knowing or not understandingsomething). By expressing their thoughts and understanding in words, schematics andformulae, development of thinking was promoted in units that not only included the word-meanings (Vygotsky, 1962), but also geometrical structure and physical quantity (see Chapter5).

The double sequencing in the instructional design of the new lab course is represented inTable 4.2. The number of homework tasks was increased from 40 to a total of 46, and thenumber of other assignments was diminished from a total of 36 to 23. The distribution of thetypes of assignments has to be read from the bottom of the table upwards for each sessionconsecutively.

Table 4.1BThe metacognitive variants, the instruction material and the instruction procedures

Metacognitive variantsVariables or tasks

Instruction material(instructional components)

Instructional procedures(interactions)

StrategyCompare resultsCompare methods Comparative assignments

Open assignmentsStrategic information

Reading assignment +hintsMonitoring of skills

Fading across partsTask

Get information yourself Sources of information Fading of referencesacross partsInformation separate fromassignments

Construct a cookbook Homework + other student Collaboration in pairDetect influence of measuring

instrumentHomework + other student Discussion in pair

KnowledgeOverview, coherence Advance organizer Reading assignmentRelations schematic - circuitRelations signal - oscilloscope pictureInterpersonal

Contradictions in assignmentsContradiction in assignmentOther student

Discussion in pairDiscussion in pairDiscussion, help

ExperienceCompare performance and

requirementsMark logbook Product evaluation

Compare process vs. result Feedback logbook Process evaluationMost complex tasks Open assignments Boosting across parts

Task, complexityInformation processingAcquire new knowledgeCompare methodsDetermine whether, if, the relation

Homework assignmentsPreparative assignmentsComparative assignmentsOpen assignments

Increasing complexitywithin parts (domains),building upon each other

The instructional design has now been elaborated. The next step will be to describe thedata and the way they were acquired.

4.2.3 Analysis of the DataThe data on the performance of the students were of six general types: (a) the final marks forthe lab course and all other courses available from a large database at the unversity; (b) the


marks for the entrance test and the logbooks recorded by the staff; (c) the scores on gradingsheets to be filled in by the teaching assistants while assessing the logbooks; (d) the answersto the questionnaires given by the students; (f) observation notes on the behavior of thestudents during the labs; and (g) structured interviews held with the TAs.

Table 4.2The number of tasks in each category for the sessions of the course

Part of the course (content)Metacognitive strategy

Introductorystructure

Time domainaccess

Frequency domainapplication

Sessions 1 2 3 4 5 6 7 8 9AssignmentsComparative open-endedComparative measurementPreparative assignmentsHomework assignments

--2*-

-2210*

1*--5

-224

1*-16

-1--

-2-14

-113

-144

Note: The sequence has to be read from bottom to top for all sessions. In homework assignments thestudents read and combine information. Preparative assignments introduce new concepts, newfunctions of instruments, or a new method to determine a variable. In comparative assignments (thefinal level), the students compare and improve the fit of results acquired by two separate methods(metacognitive strategy). In open-ended assignments the variables to be determined and the methodsare not specified.*An assignment containing a major contradiction.

The classification of the participants (in good, moderate, weak students) was done on thebasis of the results for the four theory courses and three lab courses in the first trimester. Thestudents with a mean score of 7.5 or higher on a 1 to 10 scale were defined as good students.The students who scored from 5.5 to 7.4 were called moderate students, while the category ofweak students had a failing mean score of 5.4 or lower. The classification of the students wascombined with the data of the grading sheets and the questionnaires.

The performance of the students with respect to metacognition was assessed from thegrading sheets. The grading sheets were filled in by the teaching assistants while assessing thelogbooks. The grading sheets drew the TAs’ attention to metacognitive skills during grading.The marks for the logbooks also were recorded by the TAs on the grading sheets, which couldbe read automatically. In these detailed grading sheets, relevant performances were scored ona five-point scale (see Appendix D2). The mean scores for the questions on the grading sheetsof the TAs were calculated for each logbook and for each category of students.

The mean scores for some questions were combined as described in Appendix D4 in orderto get a measure of the metacognitive performance of the students with respect to: (a) openassignments; (b) measuring assignments. Further, the performance in metacognition wasmonitored on the basis of six metacognitive skills: (a) the capability to reason; (b) to study thelaboratory manual; (c) to study the lecture notes on Network Analysis; (d) to understand thefunctions of the measurement instruments; (e) to compare several (measuring) methods witheach other; and (f) to work methodically according to the strategic framework presented.

Comparisons were made between the performances of the categories of students: good vs.moderate, moderate vs. weak, and weak vs. good. It was supposed that there were differencesin metacognitive performance between these categories. The null hypothesis was in each casethat the performance variables came from the same distribution. If this hypothesis had to berejected, it was concluded that there were indeed differences between the categories involved.The experimental hypothesis was tested against the null hypothesis by a t-test for the resultsfrom each logbook.


The activities of the students and details of their work, both on the labs and at home, weremonitored by questionnaires (see Appendix D3; see also Oosterhuis, Tattje, & Vos, 1991).The students were required to fill in a questionnaire after each lab session to estimate thefrequency of the activities during the lab classes and the time spent on these activities, andalso to estimate the time spent on each homework assignment and the degree of success. A 5-point scale was used. The questionnaires could be read automatically. The mean scores for theanswers on the questionnaires have been calculated for each category of students for each ofthe sessions. The mean time spent on homework assignments has also been calculated foreach category of students.

The observation notes of the behavior of the students and the TAs during the labs weremade by three faculty members who focused on a few pairs of students. Together with theresults of the structured interviews with the TAs, the notes from the observations wereanalyzed qualitatively by discussions among the 3 staff members involved.

Graphic representations were made of most data.

4.3 ResultsThe boundaries chosen for the categorization of the students into three groups resulted in

45 good students (about 30 % of the population), 84 moderate students (about 60 %), and 15weak students (about 10 %).

4.3.1 The (Metacognitive) Performance during the CourseThe progress of performance over time could be observed from the grading data of the

logbooks.The average performance with respect to open-ended assignments and comparative

measurement assignments for each category of students in the three parts of the lab course ispresented in Figure 4.1.

Figure 4.1. The performance in the application of the metacognitive strategy in NA for the threecategories of students in the three parts of the course. The means of the scores of good, moderate andweak students on measurement and open-ended assignments in the three assessments of the logbooksare presented here. Assessments 1, 2 and 3 relate to the logbooks from the three parts of the lab course,respectively. Where the weak students differ significantly from the others, this is indicated by a dot.Significant increases in performance of the good and moderate students are indicated by circles andcrosses respectively.

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Although all differences seemed quite large, only the largest differences between thescores of the weak students and the others were found to be statistically significant at p < 0.05or 0.01, due to the large spread in the data (see Table D.1 in Appendix D4). The increases inperformance of the good and moderate students on the open-ended assignments from logbook1 to 2 are significant ( t (88) = 5.6 and t (166) = 5.8 respectively, p < 0.001 for both).

The performance for the six metacognitive skills monitored is presented in Figure 4.2. T-tests showed that in most cases the differences between the weak students and the others weresignificant at p < 0.01(see Appendix D4, Table D.2). The significant differences between thegood and moderate students are indicated in Figure 4.2. They occur in logbook 2, and forworking methodically also in logbook 1, but not in logbook 3.

Figure 4.2. The performance in four of the six metacognitive skills for the three categories of studentsin the three parts of the course (mean scores). Significant differences between mean scores of the goodand the moderate students are indicated by two circles. Where the weak students differ significantlyfrom at least one of the other categories, this is indicated by a dot.

The amount of time that the students spent on homework is shown in Figure 4.3 (see alsoAppendix D4, Table D.3). The data show that the good students spent significantly more timeon homework than the weak students for session 2 (t (28) = 2.56, p < 0.01). For session 7,

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both the moderate and good students spent significantly more time on preparations (t (59) =2.22 and t (20) = 2.17 respectively, both p < 0.05).

Figure 4.3. The time spent by the students for the preparation of the lab tasks at each session. The firstand sixth sessions require no preparation and are omitted from the graph. Where the mean scores ofthe weak students differ significantly from the others, this is indicated by a dot.

4.3.2 Interactive Behavior in ClassThe students asked for assistance from the TAs an average of two times per session, with

these questions taking a mean of five minutes to answer. Next to writing in the logbook, mostof the time was spent on deliberations with the partner and handling instruments (both 50minutes). Consulting other pairs of students amounted to 1.5 times taking 3 minutes in thebeginning of the lab course, and 3 times taking 16 minutes at the end of the course (e.g. formoderate students the mean number of interactions were 1.8 and 2.8 on sessions one and sixrespectively, with t (79) = 3.92, p < 0.01). Some differences among the categories of studentswere observed.

In the first session the weak students asked more questions to the TAs (M = 2.5) than thegood students (2.1), and both categories more than the moderate ones (1.8), although thedifferences were not significant. The time used for the answers differed significantly betweenthe weak and the moderate students, M = 9.2 and 3.6 minutes respectively, t (58) = 4.7, p <0.01. This difference disappeared in the next session. The good students used significantlymore time for answers on questions in the sessions seven and nine then the moderate students,7.5 against 5.0 minutes with t (45) = 2.27 (p < 0.05) and 5.5 against 2.2 minutes with t (70) =1.78 (p < 0.05) respectively. The weak students used 13 minutes to read the lab guide in thebeginning and 30 minutes at the end.Table 4.3 shows the help the students needed from the teaching assistants during the labs.

4.4 Discussion4.4.1 Development of Metacognitive Skills

The results show that students who have good marks for courses in general, have a higherscore on the metacognitive skills in the lab course from the beginning on, and weak students alower one. In all significant cases and in nearly all other cases, the performance of the good


students is better than that of the moderate ones, and the performance of the moderate ones isbetter than the performance of the weak ones, with respect to the observed metacognitiveskills. So being a good student (i.e. having good marks) goes together with having bettermetacognitive capabilities and staying better, in accordance with the expectations.

Table 4.3The behavior of the teaching assistants in the old and the new situation

Source Old situation New situationObservations and interviews Much oral information.

Long explanations.Solve problems with handlinginstruments in measurements.Demonstrate skills.Push the students to finishtheir labs.

No oral information.Usually short explanations.Occasional trouble-shooting.

Occasional.Occasional pushing

The performance on the metacognitive strategy shows itself most directly in theperformance on the comparative measurement assignments and the open-ended assignments.These data show the development of the strategy. All categories of students improved theirmetacognitive performance in the second part of the course, except the weak ones. Thereforethe development of the metacognitive strategy primarily takes place in the second part of thecourse.

In the first domain the content is prior knowledge, the structure of the strategy is presented,and therefore access to the structure is easy. After the grading and the fast feedback providedby the teaching assistants on the first logbook, just in time to practice the strategy in thesecond part of the course, access is practiced in a new domain. This has effect.

The performance on the metacognitive skills monitored differs between the good andmoderate students in the second domain. This difference disappears in the third domain whereexternal regulation is diminished (fading) and a new domain is entered. In all cases theperformance of the moderate students approaches that of the good students. This result isquite unique and contrary to the expectations that students with less metacognitive skills arealso slower to develop new metacognitive skills.

From these results it is concluded that more students than before successfully passed thecourse because more of the moderate students raised themselves to a level comparable to thatof the good students with respect to the metacognitive requirements of the course.

The results in this research are based on a post hoc analysis. The instruction had alreadybeen redesigned and the data gathered. There were a large number of design principlesincluded in the course. Therefore it is hard to state which changes produced the effects.Nevertheless, because the course as a whole was successful, it has been assumed that allchanges contributed. The relations proposed in the following are tentative and require furtherresearch.

A decrease of the performance of the good students on the metacognitive strategy in thethird part of the course is ostensibly present, but not shown to be significant. Such a decreasemight be due to (a) fading of the external regulation; (b) being content with the acquired levelof performance and the acquired marks, especially as the examination time came near; (c) thenew and difficult frequency domain, and therefore a drawback in the structure of and accessto the metacognitive strategy (see Chapter 3); or (d) a combinations of these factors. It couldbe that while the good students were already near their top performance in part 2, themoderate students continued to develop metacognitive skills stimulated by the system ofboosting.

Thus learning a metacognitive skill during a lab course does not mean that the performance


of the students in that skill gradually increases. It looks as if learning a metacognitive strategyis not learning a structure once and for all and learning the access to it again, but constructingsomething anew in each new domain.

In accordance with the expectations, good students spend the most time of all onpreparation at home if new information has to be gathered, especially at the start of the firstand third part of the lab course. Moderate students start out spending less time, but spend anequal amount of time at the first session of the third domain. Weak students start low and donot change their behavior much. Moreover, there were some indications that good andmoderate students were better able to select the difficult and most relevant homeworkassignments than weak students. It seems that weak students are not driven by ametacognitive strategy in their choices of the questions, but just select the questions that haveto do with lab work (schematics and measurement instruments), and thus show a superficialapproach to their task.

Concerning the capability to reason, weak students improve their performance between thefirst and second part, but their performance drops in the third part. It looks as if the weakstudents do not know what to ask and how to ask it (cf. Chi, Feltovitch, & Glaser, 1981: theydo not know what they do not know). The weak students need longer explanations, as in thefirst session, but apparently with a reverse metacognitive effect: in the subsequent sessionsthey ask less and use less time than in the beginning. It turns out that the weak students failespecially in the last part where the content is new and difficult, and access to the strategy isnot prompted.

4.4.2 Interactions in ClassFrom analysis of the questionnaire data and from the observations, the staff members had

the impression that during lab hours good students used the least amount of time forinformation gathering and knowledge acquisition, and the most time for writing in thelogbooks. For weak students the reverse was the case. Weak students turned over pages in themanual more often than other students did, probably looking for information, and werefumbling with the instruments. Good students asked the assistants better questions (more tothe point) and thus in general needed shorter answers than the other students did. The sameapplied to their contacts with other students.

The independent acquisition of the information needed, which is a part of the strategy,works out well. Very few questions are asked to the teaching assistants. A great deal ofinteraction takes place between the paired students. The consultation of other pairs of studentsincreases during the course.

Frequent interactions among the students have to be combined with enough students ofgood quality. Some indications were found (see Appendix F) that most of the students whofailed were from a single class. This class performed the worst of all on the entrance test, andso contained fewer good and more weak students than the others. It might have been that abetter distribution of the weak and the good students over the classes would have furtherraised the passing rates.

4.4.3 General RemarksSome critical remarks should be made about the way the data have been collected and

analyzed. Verbal reports about the performance and the way of thinking of the students can befound in the logbooks, and are part of the tasks of the students. This is a favorable situationcompared to think aloud protocols because now the test cannot interfere with the treatment.The measurement of the performance, however, is done by a TA without a check by anotherTA, thus strongly depending on an individual interpretation of the logbook. This dependenceon a subjective interpretation of the meaning of the logbook is somewhat diminished by the


use of grading sheets, probably leading to a more uniform assessment than otherwise wouldhave been the case.

The results show that in well-designed instruction nearly all students can acquire ametacognitive strategy and related metacognitive skills, and also that the metacognitiveperformance of moderate students at the end of a course can approach that of good students.In such instruction the sequencing of tasks with respect to complexity is separated fromsequencing the tasks for metacognitive development (double sequencing). The strategy ofboosting can help also.

The relations between the metacognitive variants and the content of the course have beenelucidated. These relations are used to develop metacognition by the induction ofmetacognitive experiences. A special aspect of the latter is the use of contradictions to inducea cognitive conflict that has to be solved on a metacognitive level, e.g. between theobservation of a circuit and the drawing of its schematic, and between a constructed signaland the observation of that signal.

The foregoing also explains the use of laboratory courses. Laboratories are of generaleducational importance because they provide a concrete basis for the formal and very abstractconcepts used in electrical engineering (Cinquepalmi et al., 1985). Labs are often consideredto be merely an illustration of theoretical concepts. In the terms of Chapter 3 however, accessto the metacognitive theoretical conceptual structure is supposed to be attained by comparingthe results of observation and measurement (or mental images thereof) with the explanationsprovided by the lecture notes or lectures (that could be taken as assertions). The structureitself is represented by the coherence of quantitative variables in the formulae and schematicspresented, just like the coherence of the logical variables in the truth table of Chapter 3. Intutoring hours, the syntax of the formula language is trained, in labs the semantics of theformulae and schematics.

In the present view the labs are even more important because they provide opportunities toattain metacognitive goals. Thus the effort of staff members put into lab courses is not wasted,as every scientist and engineer knows from experience (empirical metacognitive knowledge).Attempts in the past to show effects of lab education by traditional exams have not beensuccessful, however. This is understandable now. Such results cannot be measured by testsother than by actually doing experiments, because in those tests both the measurement skillsand the metacognitive skills and knowledge as developed in a lab course like the present onecan be tested.

Summarizing, a schema of double sequencing has been designed using the three domainsof the course, combined with a system of boosting. The relation between metacognitivevariants and the content of the course has been established - another unique result of thisstudy. In the first part the metacognitive strategy is structured (orientation), in the second partaccess is practiced (exercise), while in the third part unprompted access is tested in a newdomain (application). Metacognitive awareness is stimulated by the introduction ofcontradictions, and by comparison of skills, thus providing metacognitive experiences.Writing down and formulating the activities and results in the logbooks is another phase inmetacognitive development. The next chapter studies the metacognitive knowledge structurethat staff members teaching the same theoretical course have developed during their careers.

ReferencesAusubel, D.P., Novak, J.D., & Hanesian, H. (1978). Educational psychology: a cognitive view

(2nd ed.). New York: Holt.Chi, M.T.H., Feltovich, P.J. & Glaser, R. (1981). Categorization and representation of physics

problems by experts and novices. Cognitive Science, 5, 121-152.


Cinquepalmi, R., Dell’Aquila, C., Fogli-Mu-Ciacca, M.T., Picciarelli, V., Stella, R., &Verrone, G. (1985). The relationship between Piaget-Type questionnaire scores andacademic achievements of engineering freshmen, IEEE Transactions on Education, E-28(2), 111-114.

De Bruin, F.F.G., & Vos, H. (1995). A basic course in network analysis: Part I-Content,Results, Instruction. IEEE Transactions on Education, 38(1), 1-6 .

De Groot, A.D. (1961). Methodologie [Methodology]. Den Haag, The Netherlands: Mouton.Dijkstra, S. (1997). The integration of instructional system design models and constructivistic

design principles. Instructional Science, 25, 1-13.Elshout, J.J. (1983). Een beginner is meer dan iemand die het nog niet kan [A beginner is

more than someone who is not able yet]. In Drenth et al. (Eds.), Psychologie in Nederland[Psychology in the Netherlands]. Amsterdam: Swets & Zeitlinger.

Hacker, D.J., Dunlosky, J., & Graesser, A.C. (Eds.). (1998). Metacognition in EducationalTheory and Practice. London: Lawrence Erlbaum Associates.

Katz, P.S., & Warner, T.E. (1988). Writing as a tool for learning. IEEE Transactions onEducation, 31(3), 214-216.

Oosterhuis-Geers, J.A., Tattje, H.E.P., & Vos, H. (1991). Leerkiem toegepast in practicumNetwerkanalyse: Leerprocessen [Germ of learning applied to a laboratory course inNetwork Analysis: Learning processes]. Doc. 91-85, Onderwijskundig Centrum/ Faculteitder Elektrotechniek, Twente University, Enschede, The Netherlands.

Reif, F., & St.John, M. (1979). Teaching physicists’ thinking skills in the laboratory.American Journal of Physics, 47(11), 950-957

Schmidt, R.A., & Bjork, R.A. (1992). New conceptualizations of practice: Commonprinciples in three paradigms suggest new concepts for training. Psychological Science,3(4), 207-217.

Tattje, H.E.P., & Vos, H. (1995). Improvement of a laboratory course in network analysis:Learning to validate knowledge in an experimental way. IEEE Transactions on Education,38(1), 17-26.

Tattje, H.E.P., & Vos, H. (1986). Praktikumhandleiding Netwerkanalyse, eerste studiejaarElektrotechniek UT, cursus 1989/90, [Lab guide for Network Analysis, first year ElectricalEngineering]. Enschede, The Netherlands: University of Twente, faculty of ElectricalEngineering.

Vandamme, L.K.J. (1990). Electrical Engineering in the New Netherlands EducationalSystem. European Journal of Engineering Education, 15, 45-90.

Ruijter, C.T.A. (1979). Van fysisch meetpraktikum naar praktikum fysische meetmethoden[From a physics measurement laboratory course to a lab course on physics measuringmethods]. Report 40, Onderwijskundig Centrum CDO/AVC, Twente University,Enschede, The Netherlands.

Vos, H. (1991). Leren en transfer: Het gebruik van een leerkiem [Learning and Transfer: theuse of germ learning]. Tijdschrift voor Onderwijsresearch, 16(5), 261-278.

Vos, H. (1992). Two levels of sequencing in germ learning: course and task levels. In Tj.Plomp, J.M. Pieters, A. Feteris (Eds.), European Conference on Educational Research,Book of summaries, Vol.1. (pp. 228-321). Dept. of Education, University of Twente,Enschede.

Vos, H., & De Bruin, F.F.G. (1995). A basic course in network analysis: Part II-Types ofproblems and sequencing. IEEE Transactions on Education, 38(1), 7-12.

Vygotksy, L.S. (1962). Thought and Language. Cambridge, MA: MIT Press.

Metacognition in teacher’s knowledge 77

Chapter 5

Metacognition in Teachers’ Knowledge of a Course

AbstractThe question explored empirically was how the knowledge of teachers of a course on NA

was organised. Possible dimensions of organization of knowledge have been identified andapplied to the content of this course to develop an observation instrument. An unstructured,unsupervised, unconstrained categorisation task of about 350 cards containing thesedimensions was presented to eight professors. Data about the organization of the knowledgeof teachers could be collected in about an hour and a half. The participants were divided intotwo groups: those appointed for research on the subject matter of the course, and those doingresearch in other domains. It turned out that the groups of participants categorised the cardsdifferently, and gave different relations among the categories. The first group had morecoherent and theoretical metacognitive knowledge than the second group. For the appointedgroup, the words indicating examples of elements come in the same categories as the physicalequations and the schematics associated with the examples (triple coding: words, schematicsymbols, and algebra belonged together), indicating a unit of thought consisting of fourcomponents: meaning, word-form, algebraic equation, and schematic. An organization ofknowledge corresponding to a metacognitive strategy for solving problems (a general,systematic problem-solving approach) in network analysis was accessible in this group, aunique finding. Three categories of basic concepts were identified, called practical,prerequisite, and basic respectively, along with more complex or derived advanced concepts.Indications were found that three types of hierarchical dimensions played a role in theorganization of knowledge. The implications of the findings are discussed.

As experts in the subject matter, the teachers in a domain know when: (a) to switch fromtheory to practice and back; (b) to switch from a graph to a formula; and (c) to systematicallyapproach problems. It was supposed that these metacognitive skills should go together with aspecial organization of the knowledge of the domain in which they work and teach. Theorganization of knowledge studied here was its structure, as it had developed naturally bystudying the domain, by teaching the subject matter, and by solving research problems.

Most knowledge of the structure of knowledge has come from research on the approach ofexperts and novices in solving problems (e.g. Chi, Feltovich, & Glaser, 1981; Reif & Heller,1982; De Jong & Ferguson-Hessler, 1996; Savelsbergh, 1998). It was generally supposed thatthe problem representations used were related to the structure of knowledge, but these studiesprovided only indirect information. Therefore, in the present study a more direct observationof the structure of knowledge was intended, from the viewpoint of metacognition.

The difference between cognition and metacognition was supposed to correspond to thefollowing descriptions (see Chapter 2). Cognition involves (a) signs and objects andphenomena in the real world, (b) mental images thereof, and also (c) knowledge of andoperations on (a) and (b), including mental operations. Cognition includes the statement oftasks, goals, and any other information. Knowledge of signs includes the meaning of the sign.Operations on signs, without reference to their meaning, are also part of cognition.

Metacognition involves (A) signs denoting knowledge and operations of cognition, (B)mental images thereof, and (C) knowledge of, operations on, and experiences on (A), (B) and

78 Metacognition in teacher’s knowledge

their denotations. Three metacognitive components in the structure of knowledge have beendistinguished in this study: the way cognitive knowledge components were categorized intogroups (metacognitive conceptual structure of cognitive components), the names anddescriptions assigned to the groups (metacognitive concepts), and the metacognitive relationsamong the groups (conceptual structure of metacognitive knowledge). In order to know thestructure of knowledge, the following questions had to be answered.

What components of cognition belonged together in metacognitive categories? Whatmetacognitive categories could be distinguished? What were the relations between thosecategories? These questions relate to the (metacognitive) structure of cognition, tometacognitive concepts, and to the structure of metacognitive knowledge, respectively.Further, the question was whether the structure of a domain was a constant, an entity fixed bythe domain knowledge, or differed from teacher to teacher, and if so, for what purposes werethe different structures most useful.

When teachers presented a course to students, they imparted information not only todevelop cognition, but also the structure of it. This was done by (a) the choice of the subjectmatter, (b) the sequence and (c) the way of treatment and representation of the subjects.Teachers further facilitated this development by (d) giving hints on how to check progress inthe solution of problems, (e) referring to important literature, and by (f) reference to concreteexamples of concepts and objects of reality, etc.

The motive for this study was the observation that teachers in higher education oftendisagreed about the way the subject matter of a course should be taught, even when theyagreed on the content (personal observation). This divergence of opinion among teachers wasexpressed as criticism of each other’s way of treatment, and by the inclination to write one’sown book or lecture notes, or at least a study guide supplemental to the book used. In thisresearch, it was assumed that these differences originated mostly from differences in themetacognitive structure of knowledge.

More knowledge of these differences seemed required. The three most important aspectswere considered to be more knowledge: (a) of the possible ways of structuring knowledge ona cognitive and a metacognitive level; (b) of the differences among teachers with respect tothe structure of their knowledge; and (c) of the factors that influence the structure and thesedifferences. Such knowledge would be useful in selecting and implementing a structure ofknowledge in instructional design.

The study has been carried out in the domain of Network Analysis (NA, also calledelectrical circuit theory), a basic course in electrical and electronic engineering. A previousanalysis of the content of this domain (Vos and De Bruin, 1995) was used as a basis. De Bruinand Vos (1995) also constructed an integrated representation of about 10 problem-solvingmethods in Network Analysis to calculate the output signal for a given circuit when given theinput signal. The structure of knowledge in Network Analysis serves as an example forsimilar courses in higher education.

In Chapter 3 it was found that among people with much experience in a certain domain ofknowledge, many have developed empirically structured metacognitive skills in that domain.In order to increase the chances of finding clear metacognitive structures, therefore, teacherswere selected with much experience in the given domain. However, while conductingproblem-solving research in Mechanics, Reif (1987, p. 410) found differences among teachersin their demonstrated problem-solving expertise.

For the present research it was assumed that such differences were caused by a differencein the type of experience of the teachers. Teachers with much experience in solvingprofessional research problems in the field of NA could have developed a different structureof knowledge than teachers doing research in other areas. Different types of research expertisewere presumed to lead to different problem-solving skills and also to different structures of


knowledge.In order to study the structures and the differences, it was first necessary to develop an

instrument that would help determine which types of structures of knowledge in the domainof NA could be distinguished. If such an instrument could be constructed, the structures couldbe observed, and it could be determined how these differed among the teachers doing researchin NA and those doing research in a different domain.

First it was determined which dimensions and hierarchies could possibly exist in thestructure of knowledge in general from the point of view of cognitive science. From theseconsiderations, an instrument was constructed to measure the structure of knowledge. Toanswer the questions, this instrument was used to explore the structures in the knowledge ofthe two groups of teachers in field research. The observed differences were then related to theresearch expertise of the groups. Finally the results were evaluated and their consequencesdiscussed.

5.1 Dimensions in Knowledge and Design of ResearchDimensions in knowledge were considered to be of a metacognitive character. A

dimension was considered an aspect of an object of thought. The features of an entity werethe dimensions and values that made up that entity and characterized it. Both the dimensionand its values could be used to organize knowledge in categories or other relations. Thefollowing dimensions could be distinguished in the structure of knowledge, each with itspossible values of importance here (see Table 5.1 for an overview).

The structure of knowledge comprises the relations between the cognitive components:knowledge, skills, and information. Knowledge and skills were supposed to be stationary anddynamic states of the mind respectively (see Chapter 2). To analyze knowledge and skillswith respect to structural characteristics, they first had to be expressed. If expressed, theywere part of information and referred to as concepts and methods, respectively.

Since skills usually referred to problem-solving skills, methods related to problems.Methods included procedures, detailed sequences of steps, but also less specified methods(heuristics). Often, a method was indicated by a name, like Ohm’s law, or a word, like(electrical) power, the principle from which the method was derived being the conservation ofpower. It was assumed that the word or name could represent the associated skills. Aprinciple, a law, and a method were all assumed to be the external expression of skills thatwere used to solve problems.

A method was considered as either a heuristic or an algorithmic way of transforming theinitial representation of a problem into a final representation. For instance, the initialrepresentation might be a known input signal and circuit with an unknown output signal, andthe final representation would be the specific output signal resulting from the given circuitand input signal. A problem representation thus consisted of the representations of two signalsand a circuit representation. If one of these were left out, the problem representation could becalled incomplete.

The following metacognitive components were distinguished.1. MC strategic components were components found useful in monitoring the progress of

(problem-solving) tasks and by preference including a check on this progress. The(independent) conditions, under which a certain method could be applied, fulfilled theserequirements and could therefore be considered as metacognitive strategic information. The(theoretical or empirical) principles, like conservation of power, could be used either as acheck on results of other derivations or on intermediate states of problem-solving (and thus beconsidered as strategic components regulating cognitive behavior), or as methods.

2. A MC task component was, for example, the knowledge that a certain form ofrepresentation (literal, logical, geometrical, or algebraic) was better than another form for a


certain task. Thus the form of representation could be used as a dimension of knowledge.Each type of representation had its elementary units or signs: letters (indices), mathematical-physical symbols (symbols), and schematic symbols (icons, cf. Seel & Winn, 1997). Morecomplex signs (words, formulae, schematics of basic circuits) could be composed intosentences (definitions, equations, schematics of complex circuits). Each type of representationwas also supposed to have its own ‘grammar’ and ‘syntax’, and considered as a language. Ifthis was true, it should be possible to formulate new, nonstandard equations that would berecognized and accepted by teacher-experts.

Table 5.1Metacognitive dimensions of the structure of knowledge and their possible values

Dimensions ValuesMC knowledge components

Categories of knowledge Categories of conceptsCategories of methods

Complexity of concepts Basic - derived - composite conceptsRelations between concepts Similarity relations (formation of categories)

Hierarchical relations (subordinate to, attribute of,member of, part of)

Coherence of knowledgeHierarchical dimensions of concepts From abstract to specific (Categorizational dimension)

From general to detailed (Descriptive dimension)From ideal to concrete or real (Imaginative dimension)

MC task componentsThe form of representation as adimension

The dimension of wordsThe dimension of algebraic formulaeThe dimension of schematics

Language with its syntax and semantics The dimension of definitionsThe dimension of analytical equationsThe dimension of equivalent schematics

Relations between methods The transformation from one problem representation toanother

MC strategic componentsConditions for methods The conditions for the application of methods

3. Metacognitive knowledge components were for instance the categories ‘concepts’,‘methods’, and relations between these categories. For each concept a word was available, butfor many concepts a related algebraic or schematic symbol was also available, or additionallyan expression of its electrical behavior in physical-mathematical symbols (called algebraicequation, elementary equation, or law). These representations were used in combinations inteaching and in problem solving. It was therefore possible that in the process ofinternalization, these categories of concepts became mentally represented by an interaction ofthe three types of external representations, in which case the units of thought would includethese external representations.

Different hierarchical dimensions of knowledge were distinguished in Chapter 2: ‘abstract- specific’, ‘ideal - real’, and ‘general - detailed’. The dimension ‘general - detailed’ was e.g.represented by the sequence letter, word, definition and could lead to related categories (forexample, all letters in one category). The dimension ‘ideal - real’ could lead to categories ofideal concepts and of real things. The hierarchical dimension ‘abstract - specific’ could relatesuperordinate and subordinate concepts in series, as in: living being - animal - bird - robin.

Similarity relations related concepts on the same hierarchical level. These required special


attention because contradictions of two types could be distinguished (see Chapter 4): (a)different meanings in the same form and (b) similar meanings in different forms. Similaritiesand dissimilarities formed the basis of the structure of knowledge. Medin (1989) alreadystated that similarity is a central aspect of conceptual structure. This structural viewpoint hasbeen applied in the present section to analyze conceptual relations in the field of NA becausesuch relations could lead to categories of knowledge.

Coherence of knowledge was another metacognitive variable because, according to Reifand Larkin (1991), coherence of knowledge facilitated selecting important items, predictingwhat will have to be included, and avoiding inconsistencies and contradictions. It wassupposed that coherence of knowledge meant that knowledge components were connected(connectiveness), that no components were lacking (completeness), that the whole contentwas covered (inclusiveness), and that it was clear which components could not be missed(irreducibility).

5.1.1 Design of ResearchAn observation instrument was developed to investigate whether the dimensions described

above were present in the participants’ organization of knowledge. Verbal reports onmetacognitive tasks had serious drawbacks as Weinert (1987) pointed out. Therefore acategorization task was devised using cards, followed by a verbal report on the results of thetask. These verbal reports were not required and served only as an extra check. Nohierarchical sorting was required (cf. Reitman-Olson & Biolsi, 1991).

The participants were asked to categorize the cards along self-chosen categories. Nostandard categories were given. It was assumed that the categories were formed freely sincethe process was unstructured, unsupervised, and unconstrained. Once the categories wereformed, the name for, the description of, and the relations between the categories could beasked for without disturbing the process of categorization.

It was hypothesized that stable structural relations were present in the knowledge of theparticipating experts. It was further hypothesized that when the components of the contentknowledge expressed on cards (cognition) were categorized, some of these structural relationswere accessed and guided the choices made about putting certain cards together.

The relations among the categories were supposed to represent the metacognitive relationsin the knowledge of the participants. However, the meaning of the content of the cards to becategorized could differ for the participants. The choices in interpretation of the cards weresupposed to be governed by activated metacognitive knowledge.

It could be assumed that those dimensions that were most clearly represented in theorganization of knowledge of the participants would be represented in the categories formed.Therefore, many different dimensions had to be represented in the cards. The categoriesformed with the cards were supposed to express the dimensions and values in the structure ofknowledge that governed the choice of the participants, were most easily activated, andtherefore most predominantly present.

The material consisted of a set of cards on which the dimensions were represented by asmany examples of the possible values as possible, taken from the domain of NA (see Table5.3 for some examples). The intent was to include in the cards at least all the values of thepossible dimensions of Table 5.1. Varying representations needed not be included for allconcepts, but for some types of concept only. The number of included concepts was so largethat it could be assumed that the whole domain was covered.

The name the participant gave to a category of cards (e.g. ‘time domain’ or ‘circuittheorems’) and its description (e.g. ‘manipulation of networks’) were considered to denote theMC knowledge of its content. The relations between the categories that the participantsexpressed (e.g. “this is more advanced subject matter than that”) could be considered as the


structure of the MC knowledge of the participants. Any MC interaction between thecategories might be included in this structure.

An observation of the categorization process and a recording of any thinking-aloud couldprovide additional data about the metacognitive experiences and variables in the execution ofthe categorization task by the participants. It was supposed that a consistent pattern ofcategorization, an accurate positioning of cards, and a correct interpretation of nonstandardmathematical expressions would indicate some of the qualities of the metacognition of theparticipants.

The three hierarchical dimensions of concepts were investigated by designing small sets ofcards in which several dimensions were represented. By letting the participants sequence thecards in each set, and afterwards explain its dimension, it could be determined whichdimensions were present in the structure of knowledge of the participants. Thus two taskswere designed: a categorization task involving one large set of cards, and a sequencing task ofseveral smaller sets of cards.

The tasks were presented to two groups of participants, one group with research expertisein NA (condition RNA), the other with research expertise in other fields of electricalengineering (condition NONRNA). It was expected that the knowledge of the participants inthe RNA condition was more integrated and more coherently structured than in the NONRNAcondition. It was also expected that the participants in the RNA condition had betterdeveloped metacognitive empirical rules about problem solving (cf. Chapter 3), and thuscould also more easily access and explain these rules and their relations with the categories ofthe cards (a MC task variable).

5.2 Method5.2.1 Participants

The teachers invited to participate in this investigation were 8 male professors at 7different universities in 5 different countries, each with at least 5 years of experience as ateacher of a course on NA or an equivalent course. Three of the teacher-experts were retiredsenior professors especially appointed in Network Theory, and with around 20 years ofteaching experience in Network or Circuit Analysis (condition RNA). They were denoted byE6 - E8 in all data. The others (E1 - E5) had about 5 years of experience as a teacher of NA orcircuit theory, but their research-related expertise was in fields like control systems,modelling and simulation of electric power systems, or high frequency electromagnetic-fieldtheory (condition NONRNA).

As a check (control group OTHERS), also a bachelor’s student who passed the course withan A (a novice, E0) and a doctoral student with 3 years of experience as a senior tutor forgroups of about 25 students and as an examiner (semi-expert, E½ ), participated in the studyin the test phase.

5.2.2 Material and Observation InstrumentA set of 352 cards was designed, covering the content of the domain and all dimensions, as

an instrument for this research. The cards contained as many categories and dimensions aspossible. Eighteen elements were included, each in three different representations, in total 18literal representations (words), 23 algebraic representations (formulae or symbols), and 22schematic representations (see Table 5.2). Examples of what is meant by a concept involvingthree representations are given in Table 5.3.

The dimensions included three knowledge types, three types of external representation,three hierarchical dimensions, similarity and specification relations, and representations ofsome particular concepts (ideal elements, small networks) in several forms. The methodscomprised 31 names or words that could be considered to represent a method to solve


problems. Some names included the word “method” explicitly, like ‘mesh-current method’.Others did not, like ‘power’, but could be used as a method, like the conservation of power.Conditions for the application of methods were also added. As an example, Thévénin’stheorem was applicable under the condition ‘one-ports, independent sources’ only to replace acomplex circuit by an equivalent element.

Table 5.2Distribution of the 352 cards over the dimensions and values involved

Dimension of organizationValue

N Category N Entities

RepresentationLiteral 217 Names

Definitions in wordsConditions

205

57

14 methods191 others

Application of methodsAlgebraic 84 Symbols

ExpressionsEquations anddefinitions

26

652

Variables, quantities

Similarity/ specification

Pictorial 52 Schematic symbolsSchematics

3616

Elementary conceptsRepresentation

Words 18/ 3 18 12 specific idealelements

Algebra formulae/symbols

20/ 3 and 6 basic networks

Schematics/ symbols 4/ 18Hierarchical

Superordinate- Superordinateconcepts of whichseveral examples wereincluded

E.g. elements, activeelements, powersupplies, voltagesources, current sources

Hierarchical 13 elementsReal 15 2 measuring

instrumentsIdeal 19 17 elements, 2 ideal

metersTasks

Dimension of taskConsistency Identical signs / words 2 / 4Accuracy Algebraic expressions

with errors, incorrectequations

6

Accuracy &interpretation

Nonstandard algebraicequations / expressions

9 / 1

The three independent hierarchical dimensions introduced in Chapter 3 have been includedby representing examples along these dimensions. For example, the dimension ‘abstract -specific’ was represented by the word ‘element’ (the category) and the words ‘resistor’,‘capacitor’, etc. (members). The dimension ‘ideal - real’ was represented by words like


‘voltage source’ versus ‘battery’, or ‘network’ against ‘this actual system’, or ‘voltagedivider’ versus ‘potentiometer’. The dimension ‘general - detailed’ was represented byconcepts and their definitions, e.g. ‘current’ and ‘the net amount of charge flowing through asurface’, or by a mathematical symbol and the defining expression of this symbol.

For monitoring consistency in the task, identical cards were included. To monitor accuracyin the tasks, cards with errors or incorrect equations were devised. Cards with nonstandardalgebraic equations and expressions, that were allowed and meaningful according to thesyntax and semantics of the mathematical language of NA, were designed and included tomonitor accuracy in the task, and interpretation of the equations.

Levels in the complexity of knowledge were elaborated as follows: the basic concepts ofNA were the categories elements (e.g. resistors, capacitors, batteries (sources)), connectionsor nodes (e.g. electrical leads), and interactions among the elements (electrical current andvoltage). Prerequisite concepts were the concepts of time -- making a change in a battery, in aconnection or in a current possible; and charge, magnetic fields and electric force fields --making measurement of current and voltage possible. These form the physical basis of NA,but do not belong to the basic concepts of NA itself (Table 5.3 and Table K.1, Appendix K).These concepts were needed to define the concepts used in NA. Reif (1982, 1987) called theprerequisite knowledge ‘background knowledge’.

Some basic concepts to input-output systems in networks - that are necessary in order tounderstand the full development of NA - are represented in Figure 5.1. Some of the cells ofthe figure were used on cards.

Table 5.3Some basic concepts of NA in three different representations. Each cell contains the content of one ofthe cards. Between brackets the types of concepts (not mentioned on the cards)

Verbal Algebraic Schematic Schematic+algebraic

Word(type)

Symbol Defining equation,elementary equation

Symbol Relation ofrepresentations

Voltage(interaction)

Uu

�E.ds - . . + - . u . +

Current(interaction)

Ii

Q/tdq/dt��J.dS

→ → i _

Resistance(element)

R u = R x iR = u / i

Capacitance(element)

C i = C du/dt || C i•||→•+ u -

Node(connection)

u = 0R = 0

•

DC voltage source(element)

u(i,t) = E0 ▌|

Note. The equations for a node and the equation for a DC voltage source are examples of nonstandardequations.


Figure 5.1. The basic concepts port, oriented structure, signal, complete, and incomplete networks.

Next to this large set of cards for the categorization task, six small sets were prepared forthe sequencing tasks. Each of the small sets contained examples of one type of concept thatcould be sequenced along some dimensions (see Table 5.4). For instance, set 5 includedexamples of voltage sources: (a) active element; (b) voltage source; (c) DC voltage source; (d)battery; (e) 12V accumulator; and (f) the accumulator in my car. The first was the most idealconcept, the last the most real one, if the labels were taken as indicating imaginary prototypes.However, the labels could alternatively be viewed as descriptions of a certain voltage source(dimension general - detailed), or as class concepts (dimension abstract - specific).

Table 5.4The characteristics of the six sequential sets of cards. For the complete sets of cards, see Appendix I

Set # Entity Numberof cards

Some of the labels

1. Circuit 7 Photograph of circuit. Sketch of circuit. Idealschematic.

2. Input-output system 8 Network. Input-output system. 2nd order IO system.3. Resistor 5 This specific resistor. A resistor of 100 ohm. Parasitic

capacitance. Simulation model of resistor.4. Capacitor 8 Element. One-port. Passive element. Capacitor.

Electrolytic capacitor.5. Voltage source 8 Active element. Voltage source. Battery. 12 V

accumulator.6. Signal 8 Port. Signal. Voltage. Digital. Periodic. A frequency of

1 kHz.

5.2.3 ProceduresThe control group (E½ and E0) was used to develop and test the instrument. Subsequently,


the tasks were offered to the eight teacher-experts selected (E1-E8). The two tasks(categorization and sequencing) were presented as taking about 1 ½ to 2 hours time in total. Itwas stated that the tasks were part of an educational research on the ‘image’ of the content ofNA that the participants, selected as teachers of NA, had developed. The experience inteaching and the research interest of the teachers were noted down or collected from theInternet.

In the first task, the participants were asked to put the cards that belong together incategories. The cards were handed out in five piles of 70 cards each, numbered on the back ofthe cards. After putting all 350 cards into categories, the categories were given a label (anumber). The participants were asked to name, describe or otherwise characterize thecategories. Then they were asked to relate the categories to each other, if possible. Timepermitting, they were also asked how they liked the task, and how they went about it(retrospective interview). In the second, sequencing, task the participants were asked to putthe cards of each of the six sets in some order they thought appropriate. Afterwards they wereasked to describe that sequence.

In both tasks the participants were asked to put unknown terms or signs aside on a separatepile, and to think aloud while categorizing the cards. In order not to influence the results, noclasses of cards were suggested. No hints were given. No further time limit was set. Thinkingaloud was suggested but not required.

The remarks of the participants were recorded on tape. The researcher noted the time takenby the participants, characteristics of groupings worded by the expert, remarkable details, etc.

After each session, the cards in each category were registered in the order of the pile, bynumber and content. The 352 cards were sorted back into the original five piles and each pileof 70 cards was shuffled randomly before the next session. The total number of notations waschecked against the number of cards. So for each participant a hand-written listing of hiscategories was available.

The preceding procedure was repeated for each participant. Protocols of the taperecordings were typed out. The notes taken were compared with the protocol of the tape-recording. The protocol was completed from the notes where necessary. A protocol, includingthe notes taken, is thus available for each participant.

5.2.4 DataThe terms used to denote a category were taken from the words the participants used while

thinking aloud, and from the naming and description of the piles afterwards. A name wasselected from these terms in accordance with the content of the categories and the relationsamong them as explained by the participant. A review was made of the categories formed bythe participants containing the label, the name, other terms or descriptions, the number ofelements, and examples from the content of each category (see Appendix G).

An overview was made of the methods and the way the participants categorized them (seeAppendix H). For each participant, an overview was made of all the cards with an indicationof their place in one of the categories indicated by the label of the category. An overview wasmade of the hierarchical order of the small piles for all participants, and their description ofthis order (Appendix I).

First, the categories were analyzed. Each category of cards was expected to represent acommon metacognitive conceptual dimension. The common dimensions found from the nameand description given by the participant to each of his categories were checked by an analysisof the cards in the category. The relations among the categories were also checked against thecontent of the cards in the categories.

Then an analysis was carried out (a) on the characteristics of the categorization tasks withrespect to its MC aspects during the execution, like the recognition of characteristics of


mathematical language and of conditions, and (b) on the categories formed with respect to thedimensions that were distinguished above. The analysis further considered (c) the role playedby different representations, (d) the relations among the categories, including coherence of theknowledge, and (e) the hierarchical relations in the sequencing task.

5.3 ResultsThe main result was that the organization of the knowledge of the group RNA of

participants differed significantly in several aspects from the group of participants incondition NONRNA. The results can be specified as follows.

Six of the teachers carried out the first task (categorization of the large set), five did thesecond task, too. Two of the teachers, because of time constraints, chose instead for a shorterstructured interview about the organization of their knowledge. The interview of participantE3 was about solving circuit problems and student laboratory applications. The interview ofparticipant E5 was about teaching.

In the next sections the literal text on a card is presented ‘between quotes’. A literalcitation of the words of participants, the name they used for a category, or a translation ofthese words into English are presented “between double quotes”.

5.3.1 Metacognitive Characteristics of the Categorization TaskThe participants categorized the 352 cards for a first round in a mean time of 50 min (SD =

33 min). The participants RNA used 38, 35, and 115 min, the others (NONRNA) 35, 26, and51 min, whereas the controls used 100 and 80 min. Participant E8 took significantly moretime (115 min) than the others (the average time spent by the others on the first round was M= 37, SD = 9, p(x ≥ 115) < 0.001). It was observed that most categories had already beenformed after categorizing the first pile of 70 cards.

Identical cards were put together in the same category in most cases, and if categorized in adifferent category (because different interpretations were allowed) usually signaled asidentical (see Table 5.5). The participants showed that they remembered the position ofindividual cards.

Table 5.5The number of cards categorized in a special way as indicated.

Condition: NONRNA RNA OTHERSParticipant: E1 E2 E4 E6 E7 E8 E½ E0

Identical cards put together in thesame category (6)

3 6 5 6 5 1 2 1

Identical cards signaled but putelsewhere

0 0 1 0 1 3 4 5

Incorrect formulae signaled (from6)

0 0 0 3 1 3 0 0

Nonstandard formulae recognizedor signaled (from 10)

0 0 0 4 0 9 0 0

Conditions recognized (7) 0 0 0 0 0 0 0 0Note. The number of cards involved is shown between parentheses.

Table 5.5 shows that only those in the condition RNA signaled some of the cards withincorrect formulae as wrong, in contrast to the other participants. They also recognized thenonstandard formulae for some source elements, either by explicitly stating it in thinkingaloud or by putting them in a category that contained related elements (cf. Table 5.3). From


the protocols, it became clear that this recognition for E6 did not happen at once, but in stages(see Appendix L), whereas E8 recognized the cards immediately. Table 5.5 also shows thatthe conditions were not recognized.

The participants often made remarks about the tasks while carrying them out. The mosttypical examples of these metacognitive remarks are shown in Table J.1 (Appendix J). Othersalient features of metacognition in the experts’ verbal protocols are shown in Appendix L.

5.3.2 The Categories FormedThe characteristics of the categories formed by each participant are described in Appendix

G. The number of the categories made by the participants, are shown in Table 5.6A. Thesehad a mean of 10 in condition RNA and 12 in condition NONRNA. The intermediate expert(E½) made a much more detailed and hierarchically ordered categorization of 48 categories.Correspondingly, the mean number of items per category was 32, but for E½ only 7.

The sizes of some categories are shown in Table 5.6B. A category labeled “waste” (alsocharacterized by “garbage”, “rest”, “aside”, or “wrong”, and including the category“forbidden” and “not correct”) was made by several participants. This category was thesmallest in condition NONRNA and differed significantly from that in condition RNA (M =6, SD = 7 and M = 78, SD = 29 respectively, t(4) = 4.2, p < 0.01). E½ and E0 did not make acategory like “waste”.

Table 5.6AThe number of categories made by participants and the number of items in the categories

NONRNA RNA OTHERSParticipant: E1 E2 E4 E6 E7 E8 E½ E0The number of categories 9 15 12 9 11 11 48 13The number of items percategory

39 23 29 39 32 32 7 27

Table 5.6BThe size of some categories (number of items)

NONRNA RNA OTHERSParticipant: E1 E2 E4 E6 E7 E8 E½ E0“Waste” - 14 3 80 106 49 - -“Practical things” 25 14 10 “waste” “waste” 36 30 38“Algebra, math. symbols” 14 31 69 - - - - -

All participants had a category “practical things” (characterized by “lab work”,“instrumentation”, “metering”, and similar words). The participants E6 and E7 (conditionRNA) included this category in “waste”. If the categories “waste” and “practical things” werecombined, the difference between the two conditions was still significant ( t(4) = 7.3, p <0.01).

The three participants in condition NONRNA made categories containing mathematicalsymbols only, the participants in condition RNA did not, as can be seen in Table 5.6B.

5.3.3 The Role of RepresentationsRepresentations can play different roles, both as a dimension of categorization

(metacognitive variable), and as a dimension of concepts (cognitive variable).


Representation as a dimension for categorizationThe values iconic and algebraic representation of the dimension ‘representation’ were used

by several participants as categories, and thus as metacognitive concepts. The overview of thelabels of the categories and the number or cards assigned to the categories are shown inTable 5.7.

Table 5.7The labels of the categories of pictorial and algebraic representations made by the experts and thenumber of cards in the categories. The category “Circuit diagrams” was temporarily only and laterredistributed into the other categories

Expert Pictorial representation* N Algebraic representation** NE1 “Circuit diagrams” Redistributed “Mathematical symbols” 14

“Circuit equations” 33E2 “Algebraic symbols” 31

“Complex variables” 9E4 “Graphical

representation”52 “Mathematical and analytical

representation” (69 cards)68

“Letters to indicate physicalquantities”

16

E7 “Circuits, networks” 34* The total number of cards is 52** The total number of cards is 84

Representation as an attribute of a conceptThe categories, in which the words for the 18 elements of NA were placed, were identified.

The number of categories involved for each expert is shown in Table 5.8. The words were putinto slightly different numbers of categories, as can be seen in the upper part of Table 5.8.

For each element, one or more corresponding cards in a algebraic or schematicrepresentation were available (see Table 5.2). The lower part of Table 5.8 shows the number

Table 5.8The number of cards of different representations of elements (algebraic, schematic) put in the samecategory as the words for the elements. Also the number of categories into which the words wereplaced is shown. The number of the cards used differed slightly for participants E8, E½, and E0.

NONRNA RNA OTHERSParticipant: N* E1 E2 E4 E6 E7 N2** E8 E½ E0The number of categories inwhich the words forelements were placed

185 5 4 4 5 7 14 3

The number of algebraicexpressions/ symbols placedin the same category as thewords

23

7 7 “algebra” 10 8

21

17 11 3

The number of schematicsigns in the same categoryas the words

2211 9 “schem.” 19 “schem.”

2012 16 18

Totals (algebraic +schematic)

18 16 - 29 - 29

Note. Small hyphenated words indicate the category in which the cards are placed.* N indicates the total number of cards involved.** N2 indicates the different numbers of cards used for E8, E½, and E0.

of cards with a different representation of an element, that each participant placed in the samecategory as the word for the corresponding element. The total number of algebraic


expressions and schematic signs for elements, placed in the same category as the relatedwords, differed significantly between the pairs of participants in both conditions (t (2) = 12,p < 0.01).

5.3.4 Relations among the CategoriesIn the design of the material, a distinction was made between prerequisite knowledge and

basic knowledge. Basic knowledge of NA like the categories “elements” (or components, i.e.practical elements) and “connections” were not distinguished. The connection amongcomponents had a paradoxical place. The connection among components was considered as acomponent itself (see Appendix L). It was not found that the word “node” better distinguisheditself from “element”, than “connection” did from “component”. The category “signals” couldbe distinguished (E1, E7, E8, Appendix G).

Other basic categories could be distinguished (see Appendix G for the categories, theirnames and content). Categories like “basic physics”, “basic definitions”, “physical properties”and “fields” and their contents (like ‘energy’, ‘magnetic field’, and ‘charge’) containedprerequisite knowledge from outside NA. The categories “background” and “basics”, bothcontaining basic concepts of network theory, could also distinguished (participant E6).

Basic knowledge of cognition could turn up in categories like “concepts”, “methods” and“conditions”. No categories of concepts or conditions were made. Categories of methodscould be identified: participant E2 put nearly all methods (27 methods out of 31) in thecategory “circuit solution techniques” (see Appendix H). E4 put 19 methods in his category“methods of analysis” of 25 items, but 11 methods in “concepts”. A further analysis of thecategorization of methods follows later.

Some participants showed categories of instructional-design knowledge. E4 (NONRNA)made a category of words that have the same meaning in the normal language and inengineering, as distinct from words with a specific meaning in NA (cf. Reif and Larkin,1991). E6 (RNA) denoted the category of “basics” as an introduction into NA that should betaught in the beginning. His categories “advanced methods”, “very advanced methods” and“diverse advanced methods” corresponded to this line of categorization.

Categorization of methodsIn Appendix H a survey is given of all 31 methods involved, and the distribution over the

categories for each participant. The methods were placed into several categories, each alsocontaining other cards in most cases. The participants in condition NONRNA, on average,used four categories (SD=2), whereas in the condition RNA nine categories were used(SD=0), which yielded a significant difference (t (4)=5.3, p<0.01).

In the survey of methods, the distribution of E8 over seven categories is used to representthe results in groups of methods. The names of the groups, i.e. the headings in Appendix H,were taken from the descriptions given by the participants. Characteristics that are not takenfrom the data have been added between brackets to clarify the categories.

1. The first group of methods, named “modelling elements”, included elementary equationslike Ohm’s law and ‘background’ variables like energy.

2. The second group, “modeling network structures”, included structural equations likeKirchhoff’s laws, both in word and algebraic form.

3. The third group “operations on signals” / “transforms” included “signal analysis” and“transforms of representations”. This group related to the choice of the independent variable,called domain in NA, with which the interactions in the network should be described: thetime-domain, the steady state domain (real-frequency domain), or the complex-frequencydomain. The transforms were used to make a change of representation of the signal (modeling


the signal).4. Then followed the group “mathematical operations” or “systematic analysis”, a category

for solving the mathematical equations.5. Fifth was the group “network transformations”, theorems for the replacement of an

(incomplete) network by an equivalent one, either simpler or otherwise easier to use.6. Next the group “principles from network theory”, principles like symmetry by which a

network can be related to other networks in order to solve problems.7. Finally the group “measurement and instrumentation”, containing a single method to

solve a problem with the aid of a measurement instrument (by observation).

Coherence of the knowledgeSome direct comments were made by the participants with respect to coherence (see

Table 5.9).

Table 5.9Comments of the experts on coherence

Expert CommentE8 “The task contained surprisingly few puzzles for me.”E1 “You get a textbook if you join them all together.”E6 “Where did you get these cards, have you screened a book?”E2 “Some of the cards, I did not understand what they were, but I can see where... all of

it was intended to go somewhere. So, nothing seeming kind of it was not appropriatefor circuits.”

E8 “Now I am in a split. ‘Linearity’ is left, ‘superposition’ is right, but in this formal,mathematical definition of linearity (on one of the cards) both are present!”

E7 “Sometimes you are not content with what others have written on a certain subject inone part of a book, and in fact therefore you write the whole book anew. And that isunderstandable, because always you will at some moment arrive at a place wherethings have to be known, where some conditions have to be fulfilled, andunfortunately those always come at some time together. You always meet with aproblem.”

No missing knowledge components were signaled, when the participants were asked. Allcomponents spontaneously signaled as missing during the task, showed up later in the task.

5.3.5 Hierarchical Relations: Abstract, Ideal, GeneralIn Appendix I the data about the sequencing of the small sets of cards (cf. Table 5.4) are

presented. In Table 5.10A, a summary of the dimensions according to the sequencing andexplanation by the participants is shown.

From the protocols, some other data about the use of the word ‘ideal’ were observed (seeTable 5.10B). Here a distinction was made between a ‘formal schematic’, having a meaning,and an ‘ideal schematic’, being the meaning denoted.

The intermediate expert E½ made a detailed hierarchical categorization, partiallyrepresented in Appendix G. Some of the hierarchical relations were membership relationsalong the dimension ‘abstract - specific’ (e.g. components, passive components, resistor andcapacitor; domain, time domain). Others involved a description along the dimension general -detailed (e.g. ‘x(t)’, ‘E.sin(ωt)’).


Table 5.10ADimensions in the sequencing of the small sets of cards as denoted by the participants. The numberrefers to the small set of the cards, and is followed by the topic the cards were taken from

# Content of the set. Denotation by the participants1 Circuit and

schematic.FROM real (actual, physical, outward appearance, practice, construction,realization)TO ideal (formal representation, theory, conception)VIA abstraction.

2 Network and(input/output) system.

FROM general (general description, general concept, desire, conception)TO detailed (complex example, increasing complexity, finish, specific one,realization)VIA specification.

3 Describing resistors. FROM general (component, constructive element)TO detailed (description in more detail, more complicated)VIA abstraction.

4 Passive element andcapacitor.

FROM general (general description, global description, theoretical network)TO detailed (particular element, examples, specific models, specificdescription)VIA increasing information.

5 Active element andvoltage source.

FROM abstract (least specific, general concept, general component, theory )TO specific (most specific, technical things, example, specific case,practice).

6 Signal and frequency. FROM abstract (theoretical description )TO specific (specifying, specification, is ..., is a, has a ..., an example)VIA specification.

Table 5.10BComments about the use of the label ‘ideal’

Expert CommentE1 “The ‘formal schematic’ is a formal representation of the ideal schematic”E8 “A ‘formal schematic’, I think an ideal network is meant by that.”

“ ‘Ideal schematic’, I think this is another word for network.”

5.4 Discussion5.4.1 Metacognitive Characteristics of the Categorization Task

All teachers can categorize a large amount of concepts, methods, and signs on cards in aconsistent way showing a metacognitive view on a task like the categorization task. Theviews differ. The data show that some teachers are able to understand and interpret new,nonstandard formulae. This means that formulae constitute a formal language in which new‘sentences’ can be produced according to its grammar and syntax and understood on the basisof its semantics.

5.4.2 The Categories FormedTeachers in the condition RNA put a larger category ‘waste’ of cards aside than the others.

This means that they have a more restricted ‘image’ of what belongs to the other categories inNA. They also put more ‘practical things’ in either a separate category or in ‘waste’. Thismeans that along the dimension ‘ideal - real’ more cards belong to the side “real” and less tothe side “ideal”. They have a way of thinking that is more theoretical, more separated frompractice and the laboratory, than the others.

In general it can be stated that separate categories of prerequisite and basic knowledge (seeTable K.1, Appendix K), separate categories for elements and signals, and a category formethods are possible, and thus also the mental states in which the corresponding MC


dimensions are used.

5.4.3 The Role of Different RepresentationsThe teachers in condition RNA have no categories for mathematical symbols as the others

do. Thus their thinking depends less on the form of the mathematical representation. Theirconcept of elements includes the three different forms of representation together in onecategory, more than the others’ do. This must entail that they can switch more easily amongdifferent representations.

Each of these representations can be taken as a sign in the sense Seel and Winn (1997)describe, a word corresponding to an index, a schematic to an icon, and an algebraic symbolto a symbol. Each can be viewed as a dimension of thought. External representations likeconcepts do not fully represent the richness of knowledge, however; for this, meaning has tobe added. Therefore the units of thought (cf. Vygotskij, 1962) here include four ‘components’(dimensions): the meaning, the verbal form, the algebraic form, and the schematic form, in anintegrated manner.

In general, the three representations in verbal, mathematical and schematic form can bepresent separately as dimensions in the organization of knowledge, sometimes as temporarymental states.

5.4.4 Relations among the CategoriesTeachers in condition RNA have the methods of NA related to more categories than the

others do. This means that the relation of their problem-solving methods with the categoriesof their knowledge is more detailed, and their knowledge is more coherent. This coherence isstrengthened because they have a clearer, more restricted view on what belongs to thetheoretical part of NA. This all corresponds to a more coherent, theoretical view on NA.

In this group of teachers, a person can be present (one out of three here) who is able toverbalize a metacognitive strategy (a systematic problem-solving approach) that is explicitlyrelated to his categories of knowledge. He is also the one that is most definite about thestability of his categorization. Learning to solve problems involves learning to reorganizeones knowledge in a way appropriate for the problems. A systematic problem-solvingapproach seems to entail a specific knowledge organization, appropriate for problems in acertain domain.

A systematic approach for solving problems is identified among teachers who do researchin the specific domain of knowledge involved and have a great deal of experience in problem-solving in this area. This is in accordance with the expectation based on Chapter 3 that moreexperience in a domain leads to better structured metacognition. This structure not onlyincludes the metacognitive problem-solving strategy, but also the (metacognitive) structure ofcognition: a unique result, contrary to the expectation that experts generally do not knowexplicitly how their conceptual understanding is linked to their practical, rule-basedknowledge (Chi, Glaser & Farr, 1988).

An interesting future study might be to see if the coupling of a problem-solving strategy toa structure of cognition could answer the question of how one can recognize an incompleteproblem representation.

The strategy is not a product of the type of task. None of the participants had done acategorization task like this before. Therefore the characteristics that showed up in the wayparticipants categorized cards in this field of knowledge could be attributed to the knowledgethey had of the domain and their metacognitive skills in structuring the content.

It is salient that conditions to the application of methods are not recognized. This might bedue to either less accurate categorization, or the coherence of the knowledge of the teachers.Reif (1987) stated earlier that the coherence of experts’ knowledge means that they are able to


recognize and repair inconsistencies. Supposedly, this applies to unfulfilled conditions aswell, providing a reason why experts do not need conditions as an external check on theapplicability of methods.

The remark of an expert in the group RNA that a linear, logical introduction into thedomain of knowledge without meeting a problem is not possible, can be explained by thestrong coherence of the knowledge of the domain. This can explain a cognitive conflict of aparticipant in the group RNA (see Table J.1, Appendix J) who tried to arrange the cards in theform of a teaching sequence, but did not succeed and blamed that on the special constructionof the card set.

Another view on the relations between the categories is an instructional one. Theknowledge in the domain can be organized in such a way that the categories of knowledgeform a succession of increasing complexity. Such an order would mean a sequencing of tasks(teaching) on a cognitive level, while teaching a systematic problem-solving approach wouldbe teaching on a metacognitive level and require another sequencing. Both ways ofsequencing should be combined in instruction to develop metacognition, (see Chapter 4).

5.4.5 Hierarchical DimensionsThe three hierarchical dimensions (‘abstract - specific’, ‘general - detailed’, and ‘ideal -

real’) are distinguished by the teachers in the domain. The label abstraction is used for thedimension ‘real - ideal’. The words ‘practice’, ‘technical thing’, ‘construction’, ‘realization’are used for the value “real”. Detailed and complex or complicated are used together. Thelabel theoretical is used as abstract, general, and ideal.

Ambiguities have to do with the dimension ‘general - detailed’. The most generaldescriptions are in words or signs, and these are often not unique. However, the dimension‘general - detailed’ also has an important use. Making general descriptions from specificexamples is taken as the basis of generalization. According to Davydov (1983), finding thegeneral form is the basis of theoretical concepts.

The mathematical relations with equal signs (similarity relations) are distributed among theother categories according to their meaning from the viewpoint of the content. This might bean indication that it is not the cognitive logic of concepts that determines metacognition, butthe content (cf. Chapter 3).

5.4.6 Critical RemarksThe instrument is rather easy to handle and demands little time of the teacher/ expert. Since

the instrument does not rely on verbal protocols, the analysis by one researcher suffices.However, the instrument is rather tedious for some participants, and requires a tediousanalysis. It is not possible to implement the instrument on a computer yet. Contemporarycomputer screens cannot show enough detail to facilitate the positioning of 352 cards, whileat the same time maintaining an easy overview of the categories that enables the participantsto reshuffle the cards if they wish to.

This investigation was exploratory in character, because (a) the dimensions could bedescribed tentatively and qualitatively only, (b) it was not known what the influence of thecondition variable (the type of expertise) was. The instrument can be improved because nowit is better known which cards are important and which are not, allowing some cards to beadded and others deleted.

5.4.7 Consequences for InstructionSome characteristics of expert’s knowledge organization have been identified. These can

help to select and implement a structure of knowledge in educational practice. For instance,different representations of objects belong together. In order to form a common concept that


unites these representations, ‘translations’ from schematics into algebraic expressions andwords and vice versa can be helpful. This is equivalent to saying that the students must beable to switch from one representation to another in order to solve problems. In general, suchmetacognitive structures develop in problem solving together with cognition. However,special problems like these ‘translation’ problems can accelerate specific metacognitivedevelopment.

Some of the basic concepts are more advanced but can be derived from other basicconcepts. A conceptual change is needed for this transition. This is a transition not fromprerequisite or outside world concepts to the domain, but an internal conceptual changeneeded to come to a full development of the domain. It thus requires attention.

What about the necessary instruction to develop a coherent structure of knowledge? Theproblem seems to be that a linear approach is not possible because one will always encountercontradictions or things that should already be known. Introducing a complete set of basisconcepts, which cover the whole field of the domain, can solve this problem. Such a nucleusof knowledge can be so small, that a linear approach is no longer necessary because the fullpicture can be treated in one lecture hour.

The smallest nucleus of knowledge can be called an irreducible set of concepts. It shouldcontain the most abstract, general, and ideal concepts. This may be interpreted as aprescriptive principle for the construction of Ausubel’s advance organizer in the field ofscience and engineering (cf. Vos & De Bruin, 1995; Tattje & Vos, 1995; Vos, 1995). If thisnucleus of knowledge is small enough, its use in developing networks of concepts and solvingproblems in these networks can be combined with the development of a metacognitivestrategy, e.g. according to the method of Van Merriënboer (1997).

5.4.8 Concluding RemarksStructure of cognition

The metacognitive structure of cognitive knowledge of a course in higher education can beviewed as follows.

Three types of hierarchies of concepts play a role. The labels abstract, general, and idealare used as equivalent to theoretical. The dimension ‘real - ideal’ corresponds to a separationbetween practice and theory.

The three types of representations verbal (index), algebraic (symbol), and schematic(icon), correspond to three types of languages, integrated in units of thought (triple-coding).These are equivalent to e.g. in chemistry: words, reaction equations, and diagrams ofmolecules; and in physics: words, laws, and diagrams of forces, etc.

A formulation of all the relations among the components in a network allows for acomplete specification of all variables (a solution of a problem), if all influences from and tothe outer world are included and specified (complete network, cf. Acciani, Cafaro, Dilecce &Vacca, 1989).

Incomplete networks can be handled, too, solving for relations among variables thatdescribe the behavior of the incomplete network (the “law” for it). Incomplete networks canbe inserted as components in other more complicated networks by network transformations(hierarchical system viewpoint, corresponding to the dimension ‘general - detailed’).

Descriptive independent domain variables (time and frequency) can be interchanged, andall other variables and relations can be correspondingly transformed into the other domainvariable.

Metacognitive conceptsMetacognitive knowledge of cognitive knowledge is represented in the categories

‘practical things’ and ‘waste’. This distinction separates the theoretical core of the course


from other things.

The structure of metacognitive knowledgeRelations among the MC categories represent a structure of MC knowledge. Such a

structure is integrated in a systematic problem-solving approach if each of its stepscorresponds to a metacognitive conceptual category. Each category contains concepts,methods, and data on a cognitive level. The steps that have been found in this study are nowgeneralized to all networks involving components, connections among them, and interactionsvia these connections. They are formulated without reference to specific concepts from NA.The problems to be solved concern classification problems in which an unknown interactionhas to be determined for a given complete network. The four to seven steps that are needed tosolve a problem are:

1. Knowledge of the ideal components and their ideal properties is used to describe thecomponents of the network with respect to the interactions (model the components);

2. Knowledge of the structure of the connections among the components is used todescribe the structure of the interactions (model the structure);

3. Knowledge of the interactions is used to describe the interactions in as much detail aspossible (model the interactions);

4. Knowledge of reasoning, mathematics, or other formal methods is applied to combinethe descriptions, and to derive the solution (solve the problem);

5. Knowledge of the equivalence of the external behavior of complicated networks to thatof corresponding elements is used to replace a network or a part of it by an element in order tosimplify steps 1 through 4 in a hierarchical way (hierarchical solution);

6. Knowledge that relates networks to others by general properties like symmetry is used toshorten steps 1 through 4 (insightful solution);

7. Knowledge of observation, measurement, and instruments can be used to directlyobserve solutions to problems in reality (observed solution).

Organizations of knowledgeTeachers do not have a uniform organization of knowledge of the content of NA. The

organization of their knowledge is related to the type of problems they think the knowledgeshould be used for. Three main types of knowledge organization can be distinguished. Astructural type (units of thought), in which the relations among the knowledge componentspredominate. A representational type in which the way knowledge is represented prevails.And third, a utilitarian type (metacognitive strategy for problem solving, sequence forteaching), focused on the way the knowledge is used. The strategic type of organization canbe connected to a knowledge representation in which knowledge is organized in four to sevenchunks that can be used in separate steps of a systematic problem-solving approach. Each ofsuch chunks contains combinations of knowledge and methods.

These types of knowledge organizations can be distinguished, but exist together andinfluence each other. A more theoretical view goes together with a larger coherence of thesubject matter, by which the teachers are more easily able to decide that many things do notbelong to the categories of the content. They are also aware of that. These features go togetherwith less orientation on the external representation of the elementary concepts, and more onan elementary concept as a concept in which a meaning and all representations are united.

The categories of knowledge differ among teachers, as do the relations among thecategories. The differences are related to the view of the participants on the purpose of thetask, next to their area of expertise in research. This means that the organization of knowledgeis not a given, but depends on the tasks presented and the ways these tasks are viewed andexecuted. Thus the organization of knowledge is not a fixed entity, but depends on the


problem to be solved.The goals of this study were to get more metacognitive information on the possible ways

of organizing knowledge, the differences among teachers with respect to the structure ofknowledge, and the factors that influence the structures and the differences. These goals werereached. In the next chapter a discussion is presented about metacognition in general.

ReferencesAcciani, G., Cafaro, G., Dilecce, B., & Vacca, F. (1989). A unifying approach to electrical

network model analysis based on a complete model. IEEE Transaction on Education,32(3), 305-313.

Chi, M., Glaser, R., & Farr, M. (Eds.) (1988). The nature of expertise. Hillsdale, NJ:Lawrence Erlbaum Associates.

Chi, M.T.H., Feltovich, P.J., & Glaser, R. (1981). Categorization and representation ofphysics problems by experts and novices. Cognitive Science, 5, 121-152.

Davydov, V.V. (1983). In: J. Haenen & B. van Oers. Begrippen in het onderwijs: De theorievan Davydov [Concepts in education: The theory of Davydov]. Amsterdam: Pegasus.

De Bruin, F.F.G., & Vos, H. (1995). A Basic Course in Network Analysis: Part I - Content,Results, Instruction. IEEE Transactions on Education, 38, 1-6.

De Jong, T., & Ferguson-Hessler, M.G.M. (1996). Types and qualities of knowledge.Educational Psychologist, 31(2), 105-113.

Medin, D.L. (1989). Concepts and conceptual structure. American Psychologist, 44D(12),1469-1481.


Reif, F. (1987). Interpretation of scientific or mathematical concepts: cognitive issues andinstructional implications. Cognitive Science, 11, 395-416.

Reif, F., & Larkin, J.H. (1991). Cognition in Scientific and everyday domains: Comparisonand Learning Implications. Journal of Research on Science Teaching, 28, 733-760.

Reitman Olson, J., & Biolsi, K.J. (1991). Techniques for representing expert knowledge. In K.Anders Ericsson & J. Smith (Eds.), Toward a general theory of expertise: Prospects andlimits (pp. 240-285). Cambridge: Cambridge University Press.

Savelsbergh, E. (1998). Improving mental representations in physics problem-solving.Unpublished doctoral dissertation, Twente University, Enschede, The Netherlands.

Seel, N.M., & Winn, W.D. (1997). Research on media and learning: Distributed cognition andsemiotics. In S. Dijkstra, N.M. Seel, F. Schott, & R.D. Tennyson (Eds.), Instructionaldesign: International perspectives: Vol. 2. Solving instructional design problems (pp. 293-326). Mahwah, NJ: Lawrence Erlbaum Associates.

Tattje, H.E.P., & Vos, H. (1995). Improvement of a Laboratory Course in Network Analysis:Learning to Validate Knowledge in an Experimental Way. IEEE Transactions onEducation, 38(1), 17-26.

Van Merriënboer, J.J.G. (1997). Training complex cognitive skills. Englewood Cliffs, NJ:Educational Technology Publications.

Vos, H., & De Bruin, F.F.G. (1995). A Basic Course in Network Analysis: Part II - Types ofProblems and Sequencing. IEEE Transactions on Education, 38, 7-12.

Vos, H. (1995). Advance organizers for technical university courses. In: C. Aarnoutse, F. DeJong, H. Lodewijks, R.J. Simons & D. Van Der Aalsvoort (Eds). 6th European Conferencefor Research on Learning and Instruction: Book of Abstracts. Tilburg, The Netherlands:MesoConsult.

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The development of and reflection on metacognition 99

Chapter 6

The Development of and Reflection on Metacognition

AbstractFirst it is concluded that access to metacognition takes place by modeling the results of

cognitive actions in special concepts like truth values or physical quantities. The theoreticaldevelopment of these concepts apparently takes place together with a representation inalgebraic and schematic form and a further coherent integration. Based on the findings, areflexive definition of the concept ‘concept’ is proposed. Thirdly, some general instructional-design rules for stimulating development of metacognitive strategies are formulated and,finally, several lines for future research are proposed.

The results show that understanding a new domain of knowledge is difficult not onlybecause of new unknown content, but also because the structure of the necessarymetacognition and access to it must first be developed. This might easily lead to overloadingshort-term memory. In the system of double sequencing, such an overload is diminished and itis found that the metacognitive performances of the moderate students approaches that of thegood students. This means that a metacognitive strategy can be taught to large groups ofstudents with specially designed instruction. Metacognitive strategies can be found moreeasily with the (more theoretically oriented) teachers who are appointed for research in thesubject in which they teach. In the present research, a metacognitive structure of knowledgeof a course was identified that was related to such a metacognitive strategy (a systematicproblem-solving approach). Some other general conclusions can be drawn.

6.1 Access to Metacognition and the Role of InformationFrom Chapter 5 it is clear that a way of access to the theoretical content of a course takes

place by modeling the components of a practical problem. The problem is expressed inequations that fit into the theoretical framework of the course and can be solved. Theseequations contain the equal sign, signifying that the left and right hand sides correspond bothqualitatively and quantitatively, which is of a metacognitive character. The equal sign allowsthe problem solver to transform the problem into a form appropriate to find the solution.

The use of algebraic symbols for physical quantities is essential here. It can be held thatconscious access to the metacognitive structure that is appropriate to a theoretical course takesplace via mathematical symbols. At the moment that the modeling has taken place and theproblem has been represented in symbols, metacognitive information has been generated.Because the mathematical operations are also represented by symbols, the operations can becarried out without reference to their meaning, in a cognitive way. Then to all appearance theoperations seem cognitive.

However, the final result is not just a symbol, but also has a meaning that must betranslated back to the problem situation. Thus using metacognitive actions the problem istransformed into symbols, cognitive operations are carried out on the symbols, and the final(new) result is transformed by a metacognitive action back to its meaning. In this processmetacognition plays a role at the beginning (modeling) and at the end (interpretation), whilecognition alone can handle the rest.

It has to be emphasized that the symbols (or signs) are external here. Greeno and Moore

100 The development of and reflection on metacognition

(1993) and Vera & Simon (1993) discussed whether or not the internal operations could beconsidered as purely symbolic processing. In the present view metacognition plays a role assoon as meaningful processing of symbols and social interaction are involved. As long asmetacognition has not been made explicit and symbolized itself, it escapes research on thispoint.

This procedure is analogous to that for theoretical reasoning as presented in Chapter 3. Intheoretical reasoning procedures, the propositions are turned into truth-values, each with itsown symbol. These form the access to the structure of the theoretical reasoning that isrepresented in the truth-table, one component of which is used in each reasoning procedure.Operating with the truth-table on logical symbols to get the truth-value of the conclusionseems a cognitive operation, but is in fact a metacognitive theoretical operation on the truth-values of the propositions according to a theoretical schema represented in the truth-table.Through a system of symbols, access is acquired to a system of operations that represent themetacognitive theoretical strategy (schema).

The access can also be acquired in the lab. In laboratory practice, observations andmeasurements on material objects are made. The observations concern the connectionsbetween the elements - the geometrical structure of the network - that are represented inschematic symbols. The measurements concern the properties of the elements, the interactionsvia the connections, and the properties of the network as a whole. The direct measuringresults are observations of the position of a pointer or of an oscilloscope trace. These can beconverted into numbers or algebraic expressions. In this manner, access to the theory can beacquired from reality.

It is supposed that the function of observations and measurements on material objects is tobring the notion of metacognition to the awareness: the symbols generated have a meaning,they are the results of and represent actions. Once this notion of metacognitive meaning hasbeen developed, everything related to the laboratory and practical concerns can be releasedand a pure theoretical thinking can develop like that found in Chapter 5 for the group ofteachers appointed for research in the domain of instruction. For a theoretical teacher, thecontact with reality such as is found in a laboratory, is either an illustration of theory or‘waste’. For a student it is a condition for the development of metacognitive notions in thedomain of knowledge.

Other forms of metacognitive information were presented in Chapter 4. Next to theframework for investigation (information about a schema of a metacognitive strategy),metacognitive information was also presented with respect to the task, such as drawingattention to the influence of the measuring instrument. Other metacognitive informationconcerned the coherence of knowledge and the relation between the different representationsof elements. Finally, some tasks were designed such that cognitive conflicts were introducedby the information given.

The role of information in all these cases was assumed to support the reflection onmetacognition. The information referred to and thus drew attention to split, scattered, andoften non-conscious metacognitive knowledge that was developed earlier, and that wasneeded now to fulfill a task or to reconcile contradictions. Basically, tasks to reconcilecontradictions are equivalent to the conservation tests of Piaget (Davydov, 1977, chap. 7;Ginsburg & Opper, 1969, p. 163; Vygotskij, 1962, p. 116). The role of the (metacognitive)information was to bring the metacognitive knowledge more to the level of awareness and tomake it more coherent. In this way it was better accessible and could be more consistentlyapplied in the new situation.

In general, if signs representing concepts or skills are available, these have become objectsof thought. On the other hand, access to one’s own cognitive network and reflection on itsstructure is stimulated by discussions, for which signs are necessary. Access to thinking is


acquired by representing it in signs, by designing a model for it, thus making it an object ofthought. To think about objects of thought requires a reflexive model, as was the case withmetacognition.

6.2 Triple Coding and a Model of the Concept ‘Concept’A reflexive model of a concept is possible from the findings in Chapter 5 about a triple

coding of concepts. This means that representations in words, schematic symbols, andalgebraic formulae belong together - a process equivalent to knowledge encapsulation(Boshuizen & Schmidt, 1992). This phenomenon can be used to present a reflexive schemafor the concept ‘concept’. Such a schema can be compared to other reflexive schemata likethe one for modeling in engineering developed by De Vries and Breedveld (1992), thereflexive discussion on reflection by Von Wright (1992), or the reflexive definition ofnumbers (Conway, 1976).

A word like ‘resistor’ designates a concept including a class of objects, all called resistors.The schematic symbol represents the structure of the designated objects, for instance that allthe objects possess two connection terminals. The algebraic equation describes thecharacteristic behavior of the objects. This behavior forms a rule like Ohm’s law with whichall resistors can be identified. Thus the equation provides a classification rule, and also gives adescription of the behavior of the ideal resistors under electrical conditions.

The concept involves a class of objects (a category of entities), a classification rule for theobjects (a description), and a mental representation of the objects (a prototype of the concept).The entities are material objects here. A category, a description, and a prototype are also thethree viewpoints on a concept (see Chapter 2). This means that triple coding can in fact beidentified as a form in which the three views emerge. The three views thus form attributes ofthe concept ‘concept’. A possible schema for a concept is represented in Table 6.1. Thisschema for ‘concept’ can be compared to the schema for ‘bird’ in Chapter 2.

Table 6.1The components of the proposed SCHEMA for the concept ‘concept’. The word SCHEMA designatesthe meaning of the concept and is in capital to highlight its reflexive character

Attributes Possible values Examples from NA, fromthe concept ‘resistor’

Type Cognitive component Element of networkThe category of subordinateconcepts

Abstract - specific concept Class (R) - subclass ofresistors (100 ohm)

The description of the concept General - detailed description Name - definition,Symbol - algebraic equation,orSchematic symbol - detailedschematic

The prototype of the concept Ideal - real concept Ideal resistor (e.g. Ohm’slaw) - real resistor

The SCHEMA of the concept Mental representation -external representation

Integrated schema by triplecoding

The possible values of the attributes lie along three dimensions: (a) abstract (e.g. resistors)- specific (resistors of 10 ohm); (b) general (‘resistor’) - detailed (‘an element with twoconnections, a cylindrical body, and constructed from carbon or metal film’); (c) ideal (aresistor obeying Ohm’s law) - real (this specific resistor in my hand). The concept as acategory is more abstract than its subclasses or members. The subclasses or members arespecific examples of the concept. The designation of the concept by its name is a more


general description than the concept as formulated in a (classification) rule. The rule is a moredetailed description of the concept. The concept represented in the prototype is more idealthan each of the real examples of the concept is. A material model is more realistic than theideal prototype.

It is not necessary that all attributes be present with the same intensity. Concepts can befuzzy, relational, denote actions, etc. in contrast to classical concepts that consist of aconjunction of a couple of attributes. Some concepts allow no prototypes. In the abovedefinition of ‘concept’ the disjunction applies: a concept is either a class or a prototype or arule or a combination, but it should have a meaning. The type of the concept is of course theuniverse the concept comes from. Finally, a concept can have a name (an index) or not.

In words, a concept is a unit of thought composed of four elements that consists of (a) acategory of entities that belong to the concept (instances), (b) a rule that gives a description ofthe ‘behavior’ of the concept (definition or categorization rule), and/or (c) a prototype of theconcept that serves to recognize concepts; and (d) the meaning of the concept, designated bythe word schema. The name of the concept can refer to each of the components of the unit.The description can be used for categorization - and is called a categorization rule in that case- but also for other purposes like analysis of the concept or the prototype.

The three attributes or dimensions can be seen as the three independent forms ofabstraction from reality that have been distinguished. In order to be able to think about one’sconcepts and make metacognition possible, the fourth dimension is added with its range ofvalues between mental representation and external representation. This dimension representsthe meaning of the concept ‘concept’ and is denoted by the word SCHEMA in Table 6.1.

According to Vygotsky (1962) the meaning of words has developed as the unit of thought‘word-meaning’ from a differentiation of the social speech. In this development the wordsbecame internalized and were enriched by the sense they gained from the context, whilespeech shortened and turned into inner speech. This inner speech developed into thinkingnearly without words. In an analogous way, the meaning of a concept is assumed to developas a differentiation of the use of the three representations of the concept in the socialinteractions of the students with their teachers and peers.

The development of the fourth component of the unit of thought goes together with aninteraction of the three other components, but forms an independent dimension. Each thoughtof this concept has developed by abstraction from the three representations. In other words,the meaning of the concept develops when the concept becomes free from the forms of therepresentations.

In the above paragraphs four attributes of the concept ‘concept’ are presented. These fourattributes have values along dimensions, each of which includes an abstraction from reality. Itwas demonstrated that the representation of the concept ‘concept’ could only consist of a de-representation, i.e. being released from the external representational forms in a movementfrom the concrete reality to the internal plane of mental representations related to ‘innerspeech’ and thinking. In this way the concept ‘concept’ is represented in a schema that isapplicable to itself (reflexive). The consequences of this schema have to be studied yet.

The results of the present study point to some recommendations with respect to the designof the instruction in laboratory courses.

6.3 Metacognition and InstructionBy distinguishing metacognition from cognition, the role of access to metacognition can be

distinguished from the structure of metacognition. This leads to instruction specially designedto foster the development of metacognition in a lab course. It is supposed that suchcharacteristics in general develop metacognition. The following recommendations can beformulated.


It is suggested that the domain of a course program be divided into three subdomains, thefirst of which contains well-known subject matter only (see Chapter 4). Perhaps this can be inthe form of prior knowledge, generalized and translated into a form appropriate for the presentdomain. A metacognitive schema is explicitly formulated, agreed upon by the staff involved,and presented to the students. In the first subdomain, the relation between the schema and thetasks is explained and illustrated, while in the subsequent subdomains it is not (metacognitivefading). Further, the maximum number of elements stimulating metacognitive experiencesshould be included in the instruction (see below). The development that takes place can beexplained as follows.

It can be supposed that the students already have some notions of the elements of theschema (see Chapter 3). It is assumed that the explanation of the strategy in the firstsubdomain makes its structure clear. Thus in the first subdomain the structure of themetacognitive strategy is developed. In the second subdomain, the access to this structure in anew domain of knowledge is practiced, and the learning of the structure is continued. In thethird domain, the metacognitive performance of the good students might decrease becausethey have already reached good results, but moderate students will continue to develop theirmetacognition. Assigning a mark for the students’ performance in each subdomain and just-in-time feedback on their work at the beginning of the next subdomain will stimulatemetacognitive experiences and foster strategic regulation. Essential aspects are collaborativelearning and the availability of feedback on the work in the second subdomain (seeChapter 4).

To get a model for a metacognitive strategy several roads can be taken. It is possible tostudy the way people execute tasks in a well-known domain and derive the strategy from thedata like in Chapter 3. Alternatively, it could be asked to the students how they do it. It is notsure yet that one gets consistent answers in the last case. It can be developed by a Delphimethod like in Chapter 4, by recording thinking aloud protocols and other means to find outhow experts approach and solve problems. One can search for a first draft of such a model inthe literature or books. If the objective of the course is to develop theoretical thinking, themethod of Chapter 5 is appropriate for finding the approach to problem solving and itsrelation with a structure of the theoretical knowledge. The computer can offer some assistancehere (Paquette, Aubin & Crevier, 1999).

It is assumed that to develop a metacognitive strategy, the development of othermetacognitive variants has positive influence. To include other variants in the instruction(next to fading, early marking, just-in-time feedback), an analysis of the subject matter andthe tasks is needed. This analysis is needed in order to (a) find a possible metacognitivestructure of the content used, (b) design tasks that contain contradictions, (c) sequence thetasks with respect to complexity in each subdomain (snowballing), and (d) include a systemof boosting. The last means to include tasks of a slightly higher level of complexity than therequired final level in all subdomains except the last one.

The metacognitive task elements are specific for each course as the coherence ofknowledge is. Coherence is developed by abstraction from reality to theory and bytranslations from each representation into all others and vice-versa (triple-coding).Abstraction can be stimulated by letting the students make schematics and formulae fromconcrete objects and phenomena for themselves. Triple-coding can be fostered by letting thestudents translate schematics into equations (modeling), but also by translating formulae intoschematics (implementing), or translating verbal problems into schematics and formulae(designing).

What is the best metacognitive strategy to be developed by instruction? In order to developa metacognitive strategy as mentioned above, it is first necessary to develop a model of thestrategy and to design the instruction in such a way that metacognitive knowledge and skills


can develop. An alternative is to offer so much practice to the students that the domainbecomes well known, at which time metacognition will have developed spontaneously (cf.Chapter 3). The second alternative has as a consequence that the strategy has to be madeaware and explicit in a collaborative environment. But, from Chapter 5 it became clear that anexplicit strategy seems unnecessary unless you become an expert in the domain (independentof whether you become a teacher in the domain or not).

How to reconcile these conflicting requirements? In Chapter 4 a solution was chosen inwhich the teachers decided on the strategy to be learned, and chose the contradictions thestudents should encounter. From Chapter 5 it can be concluded that students should developthe strategy of the domain they want to become an expert in. This is a domain in which theylike to solve problems, especially new problems that have not been solved yet. Thus it isimportant that the students find the domain in which they are interested as soon as possible,and that the teachers help them to develop the strategies of that domain. This domain candiffer for each student.

6.4 Future ResearchA possible way to solve this problem is to offer much variation in problem situations and

in the form of the problems, and to let the students approach the problems from manydifferent ways. In this way metacognition and cognition will develop at the same time. Tosupport metacognitive development, a system of individual reflection (J. Cowan, 1998; F.A.J.Korthagen and T. Wubbels, 1995; J.A. Moon, 1999) seems appropriate. In reflective papersmetacognition is expressed in signs and made addressable. This makes it possible to supportreflection on metacognition. The teachers should help the students to focus their attention andto become aware of their interests, of the way they approach problems, of the quality of theirapproach, and of the way they learn.

If metacognitive development is intended, the teachers should not tell their students what isthe case, but help them to discover it for themselves by metaphorically ‘holding a mirror up tothem’. The teachers should help the students to find others with the same interest tocollaboratively learn, to discuss and formulate their strategies, and to find out which oneworks best in what situations. In such a system, the students decide for themselves whichcontradictions they want to reconcile, because in a rich environment such as is found atuniversities, contradictions and new problems always arise that wait for a solution.

At the same time, the students also have to prepare for their jobs in the best possibly ways.Developing metacognitive strategies on learning is recommended for their jobs as well(learning to learn, life-long learning).

The study on the structure of students’ reasoning has been carried out with electricalengineering students at university level. It would be interesting to investigate whether thesame results can be found with students from other faculties, or from polytechnics. It could bepossible that electrical engineering attracts students who already reason well and are well ableto handle uncertainty, or that other students are better in these respects. If metacognitive skillsare developed differently, this could also have an effect on the instruction for metacognition.

It would also be interesting to study the capacity of the students to explain the way theyreason in a three-value context. This has not been done here, because it is known that whatpeople say they do does not always match with what they in fact do. Nevertheless, it would beinteresting to find out whether the students can explicitly formulate the structure of ametacognitive strategy that they have not learned explicitly and thus must be implicit(nonconscious).

Mathematics, that has the same status as theoretical reasoning, can also be viewed asmetacognitive knowledge and strategies to solve practical problems. The strategies have aspecific access via the symbols for quantities, and a specific structure represented by the


properties of the mathematical operations. Mathematical skills acquired in some domain oftenare not used in other domains. This lack of transfer could be based on both a diminishedaccess to and a disappearance of the structure of mathematics in the other domain. It would beinteresting to study this phenomenon from the viewpoint of metacognition.

In high school physics, the use of mathematics is often viewed as cumbersome and notnecessary to understand the concepts of physics. This view denies the metacognitive role ofmathematics in physics. In physics, the mathematical symbols no longer represent numbers,but physical quantities including the dimension of the physics as a unit. The meaning of thisunit can only be acquired from measurements. It is therefore recommended that a largeamount of laboratory work and measurements remain in the physics curricula. It would beinteresting to study the lack of transfer of physics lessons in high school to other domainsfrom the viewpoint of metacognition.

Finally, the question of the relation between problem solving and the construction ofknowledge has to be addressed. It is possible to solve a problem without learning anything buthow to solve the problem. Certain types of problems involving contradictions force thestudents to detect the conflicting cognitive states and reconcile them by reconstructing theirknowledge. Problems in which the students have to compare different activities force them toobserve dynamic cognitive states and to turn this observation into a stationary cognitive statethat can be used as a concept. So an important study for instructional design is: which types ofproblems lead to what types of reconstruction of knowledge, and why?

ReferencesBoshuizen, H.P.A., & Schmidt, H.G. (1992). The role of biomedical knowledge in clinical

reasoning by experts, intermediates and novices. Cognitive Science, 16, 153-184.Conway, J.H. (1976). On numbers and games. London: Academic Press.Cowan, J. (1998). On becoming an Innovative University Teacher: Reflection in Action.

Buckingham: The Society for Research into Higher Education and Open University Press.Davydov, V.V. (1977). Arten der Verallgemeinerung im Unterricht (Forms of generalization

in teaching). Berlin: Volk und Wissen.De Vries, T.J.A., & Breedveld, P.C. (1992). A model of the modelling process. In P.C.

Breedveld & G. Dauphin-Tanguy (Eds.), Bond graphs for engineers (pp. 291-302).Amsterdam, The Netherlands: Elsevier.

Ginsburg, H., & Opper, S. (1969). Piaget’s theory of intellectual development: anintroduction. Englewood Cliffs, NJ: Prentice Hall.

Greeno, J.G., & Moore, J.L. (1993). Situativity and symbols: Response to Vera and Simon.Cognitive Science, 17(1), 49-59.

Korthagen, F.A.J., & Wubbels, T. (1995). Characteristics of reflective practitioners: towardsan operationalization of the concept of reflection. Teachers and Teaching: theory andpractice, 1(1), 51-72.

Moon, J.A. (1999). Reflection in Learning and Professional Development. Theory andPractice. London: Kogan Page.

Paquette, G., Aubin, C., & Crevier, F. (1999, August). MISA, A knowledge-based method forthe engineering of learning systems. Journal of Courseware Engineering, 2.

Vera, A.H., & Simon, H.A. (1993). Situated action: A symbolic interpretation. CognitiveScience, 17(1), 7-49.

Von Wright, J. (1992). Reflections on reflection. Learning and instruction, 2, 59-68.Vygotsky, L.S. (1962). Thought and Language. Cambridge, MA: MIT Press.

106 Appendices

Appendices 107

APPENDICES

Appendix A The content of the lab course: lab guide; assignments;introduction

The lab course consisted of nine sessions, split into three parts of three sessions each. First a shortdescription of the content of each part (domain) is given. After that the content of the lab guide isdescribed compared to the earlier lab guide (see Table A.1). Next the changes in the assignments aredescribed (Table A.2). Then the types of assignments as used in the sequencing are described (thesequencing itself is presented in Table 2 of Chapter 4). Finally, an outline of the introduction to thelaboratory course on Network Analysis is presented and the introduction itself (in Dutch).

A.1 The Three Parts of the CourseThe content of the first three sessions, the introductory part of the lab course, was input-output

systems in simple networks, their elements, and the methods to measure characteristics of theseelements like the values of their electrical resistance and capacitance (RC-circuits). Also thecharacteristics of input signals, from sources, and output signals (responses), into loads, like their formand frequency, the phase difference between the input and the response, and the power transmitted bythe signals were treated. The forms of the input signals used were square wave and harmonic.

In the next three sessions the signals were described as functions of time, the reason why this partof the course was called the time domain. After definitions of network properties like linearity andtime-invariance were given, the students had to detect linearity in circuits and to construct responsesby superposition. Differential equations were used to calculate step- and impulse responses, whichwere also determined experimentally. With the aid of the response to an impulse as an input signal themathematical concept of convolution could be used to calculate responses to other signals.

Finally in the last three sessions the signals were described as a superposition of harmonics, inwhich way the signals became a - for the students difficult to understand - function of the frequenciesof the constituent harmonics, called the transfer function. This part therefore was called the frequencydomain. The transfer function was expressed either by a Bode plot (consisting of two related graphs)or by a polar diagram (one diagrammatic representation). The circuits included an inductor (L) here(RLC-circuits).

A.2 Four Types of AssignmentsThe first session of the lab course was used to remedy any shortcomings in the prior measurement

skills needed. The remedial task after the entrance test included the metacognitive experience based onAssignment 2 (appendix C). Further in the course the following types of assignments weredistinguished in relation to their instructional function.

Homework assignments were intended to activate prior knowledge and to study information neededlater on. Questions to check understanding of this information were included. The informationpresented was intended to be sufficient to fulfill the homework tasks. Not more than reading,understanding and combining the information presented was required. No new developments wereintended, and no assistance required. Here learning of facts was the case, and also literal learning ofschematic symbols, basic formulae and definitions to be used later on.

Special preparative assignments introduced a new concept or a new phenomenon in anexperimental, operational way. For instance the concept of step response was introduced as the limit ofresponses to ever longer square signals. Separate assignments introduced the concept of electricalground. Students found out, for example, that it is not possible to measure current and voltage in acircuit at the same time without thinking, because signal generator (source) and oscilloscope(measuring the response) are both grounded (a contradiction to be solved).

Other preparative assignments introduced new types of measurements or calculations. Successiveassignments were designed in such a way that measurement and calculation procedures had to beadapted to each task by taking a different choice of steps in a different order, and could not be repeatedby plugging different numbers in the same procedure. For instance, if the students had to measurephase differences at several, deliberately chosen frequencies, they first got an assignment to measure aphase difference at one frequency in three different ways.

108 Appendices

Table A.1The content of the lab guide in the new and the old course: sections and number of pages

Contents New OldGeneral information 37 pp. 6 pp.

1. Introduction 12 41.1 The functions of the lab course on Network Analysis. (7) (1)1.2 Goals of the lab course:To have operational knowledge of the basis concepts; to be able to carry outmeasurements given some theoretical models; to have a systematic approach inexperimenting.

(1) (1/3)

1.3 Organization of the lab-work:Time, place, working in pairs, logbooks, attendance, preparation, entry test.

(1) (2/3)

1.4 Preparation for assignments:Information and hints how to prepare, tasks of teaching assistants (TA’s).

(1) (1/3)

1.5 Execution of assignments:Information and hints how to do assignments, task of TA’s.

(1) (1/6)

1.6 Assessment: criteria for marking (1) (1/3)Schedule, logbook, report - (1)

2. General information on experimentation 25 22.1 Method of experimentation (framework for investigation): (4)2.2 Writing a logbook: Why, how, what to write. (3)2.3 Measuring and constructing circuits:

Port connections; electrical ground; how to construct a circuit from aschematic; how to make a schematic from a circuit; exercise.

(3)

2.4 Measuring and the influence of measurement instruments (7)Symbols and schematics for measuring circuits: a drawing, draught schematic,ideal -, equivalent -, measuring schematic.Ways of connecting the metersCorrections for influence of meters on resultsInfluences of meters on responsesMeasuring the RC-time constant2.5 Responses of simple circuits (combinations of resistor, capacitor, inductor) (3)2.6 Functions of signal generator and oscilloscope (what can they do?) (5)Specifications of measuring instruments (signal generator, oscilloscope,multimeter): how well are they constructed?

(2)

Assignments 24 pp. 16 pp.1. Introductory part 13 4First session: The instruments

Diagnostic entrance test and remedial exercises. Homework.Second session: Possible and impossible connections. Measuring connections.

Homework assignments.Four assignments, each with the activities from the framework to be used

Third session: Influence of source and measuring instrument

(1)

(2)(8)(2)

2. Time domain 5 6Superposition, time-invariance, causality, impulse response, step response (3 pp.)Fourth session: Homework assignments. Lab assignments.Fifth and sixth session: Homework assignments. Lab assignments.

(3)(1)(1)

3. Frequency domain 6 6(the same instructions as before the change of the course).Session 7: Introduction, homework questions, complex numbers and complexharmonic functions, transfer function, lab assignments for RC-networkSession 8: idem for LRC-networkSession 9: Polar diagram for RC- and LRC-network

(3)(1)(2)

(3)(1)(2)

Appendices 109

Table A.2The assignments of the new and the old lab course

Session Old course New courseIntroductory measurements: RC-circuits and harmonic signals1 Measuring R, measuring C, measuring

and drawing square-wave responses.Diagnostic entrance test (practice andtheory). Remedial exercises*.

2 Measuring phase differences betweeninput and response for harmonic signals

Connecting measuring instruments tocircuits: possible and impossible ways.Observation of changes in responses fromchanges in input signals.Compare two different ways ofdetermination of the product RxC.

3 Harmonic signals: Determination of thepower transfer.

The influence of voltage source andmeasuring instruments on the quantity tobe measured:Investigation of elements* (open ended).

Time domain: network properties, differential equations and convolution4 Network properties: Detecting linearity.

Constructing responses by superposition.Network properties and differentialequations: Determination of step- andimpulse response. Responses bysuperposition.

5 Differential equations: Determination ofstep- and impulse response.

Detecting non-linearity by experimentalinvestigation* (open ended).

6 Convolution: Determination of a saw-tooth response.

Convolution: Determination of a saw-tooth response.

Frequency domain: transfer function7 Determination of a Bode plot for a RC-

circuit.Determination of a Bode plot for a RC-circuit.

8 Determination of a Bode plot for a RLC-circuit.

Determination of a Bode plot for a RLC-circuit.

9 Construction and verification of polardiagrams.

Construction and verification of polardiagrams.

Note. Determination was meant to include both a calculation and a measurement, and a comparisonof the results.* These exercises induce a cognitive conflict because they contain a contradiction

Preparative assignments were exercises in measurement procedures and calculations on a cognitivelevel, in which students had to analyze the skills practiced in preceding assignments and to construct(synthesize) new measurement or calculation procedures from the procedural components embeddedin preceding assignments. Mastery learning i.e. exercising prescribed procedures until the skills weremastered was not intended.

As already pointed out, to stimulate metacognitive experiences comparative measurementassignments were introduced in which the students had to compare two ways to determine a givenquantity or a relation. In comparative tasks the quantity to be determined was given, but the methodsand criterion still had to be generated. Usually the criterion for comparison was accuracy. The studentshad e.g. to measure the product RC (a time characteristic of certain circuits) in different ways. Theycould use a choice from two AC-methods or three step response methods (see Assignment 3 inAppendix C for two of the methods).

Some open-ended assignments were introduced in which neither the quantity nor the relation to bemeasured was given (no goal), nor the way in which that quantity could be determined (no method). Inorder to perform open-ended experiments the students had to be aware of the different ways ofmeasurement or calculation they could make a choice from, and of the criteria for such a choice(metacognitive strategy variables and executive control). Examples were Assignment 1 in Appendix C

110 Appendices

and the assignment in the time domain ‘Investigate which of the given components (materialelements)’ electrical behavior is most in accordance with the equation i = C.du/dt’.

A.3 The Introduction to the Lab Guide: Its ConstructionThe introduction to the lab course was constructed by a Delphi method. Repeatedly discussions

were held by the author with the staff members of both the lab course and the accompanyingtheoretical course. The discussions started from an analysis of the most abstract concepts in the course.These were ordered in such a way that they could be introduced one by one, starting from priorknowledge of electrical circuits and measuring instruments, and measurement and calculation skillsalready available from high school. The discussions lead to adaptations and improvements and a newround of discussions until agreement was reached.

The introduction was concise, complete, and understandable for students already before theystudied the subject matter (private communication from another faculty that used the lab guide withnon electrical engineering students). The relations among the concepts provided information on thestructure of knowledge (see Chapter 5), and could lead to an integration of the concepts and skillsinvolved. It could be considered as an advance organizer in the sense Ausubel described (Ausubel,Novak, & Hanesian, 1978) because it organized the prior knowledge of students to provide anchoringpoints for new knowledge, rather than giving a - not yet understandable - outline of the course. It alsomade that knowledge more coherent and fit for use in the course.

Alternatively, the introduction could be considered a germ from which learning could develop(Vos, 1991; Vos 1992) if practical and theoretical assignments were focused on extension of theknowledge from the simple examples given. The introduction could be viewed as the presentation of alimited number of chunks with which the subject matter can be held in working memory withoutoverloading it. Using this chunk in writing the logbooks should add to their development. Reading theintroduction, and using the chunks, should contribute to the access of a coherent structure and thefurther development of its coherence.

The final result had the following characteristics.

A.4 Presentation of the ContentIn the presentation of this introduction, observation and measurement were considered to be the

basis to verify any statement about the concepts to be introduced, especially the invisible electricinteractions and properties. The starting point was from real, concrete, elements (called components),that were known and could be observed. Components were chosen that could be connected while theobservable effect of this construction could be used to introduce the next concept. For instance, currentand voltage were introduced by the way they were measured.

To measure the current the lead was to be cut and the ammeter put in between the open ends, tomeasure the voltage the voltmeter was to be connected in two places to the uncut leads. This alsoexplained the essential difference between current and voltage. It was thought that an explanation interms of moving charges (current) or line-integral of the electric field strength (voltage) could nevercompletely replace the above definition. The working of the ammeter was explained by magneticeffects, that of the voltmeter by the effect of the voltage on free moving charges, other explanationsthat played an introductory role only.

The introduction was thus that the description of concrete examples of circuits, components,connections, and interactions, could be generalized to other objects, and that the properties of thecircuits and their components could be idealized to conceptual networks and their elements(e.g. U = I x R, Ohm’s law, as the behavior of an ideal resistor). Then an abstraction was madefrom the concrete elements to classes of elements. Also the ways to construct new, more complexcircuits, from given simple ones, including measurement circuits, were presented as a way to extendthe knowledge from the simple examples presented.

A.5 Content of the IntroductionIn this introduction examples of the most abstract, general, and ideal central concepts of NA were

put forward and introduced one at a time. There were three categories of concepts found: the elements,the connections among them (the leads), and the interactions between the elements (electrical current

Appendices 111

and voltage). Interactions, elements, and connections, and their properties (or ‘behavior’) wererepresented both by schematic symbols and mathematical symbols and units.

The central concepts involved the concept element both as an idealized circuit element and as anabstract, superordinate concept for idealized resistors, capacitors, voltage sources, etc. The conceptnetwork referred to an idealized circuit, the simplest one being a loop. The concept node denoted theconnection among a number of elements. The concept structure stood for the way elements wereconnected: in series, in parallel, two, three, or four at a node, etc. The concept port meant a twoterminal connection through which power is flowing (see Appendix A).

The concept of port was important because it provided a way to view the network with elements,voltages across them, and currents through them and their leads (the physical network viewpoint)alternatively as subsystems connected by ports through which signals and power were flowing (theinput-output system viewpoint). This concept was needed to describe measurements.

Then followed a further specification in: (a) passive and active elements; (b) the schematic symbolsused for them; (c) different schematic representations of the same node or circuit; (d) the mathematicalequations for the electrical behavior of the elements (elementary equations, like Ohm’s law for theresistor); (e) the equations for the properties of current and voltage in circuits (structural equations likeKirchhoff’s laws); (f) representation of nonideal elements by equivalent schematics of ideal elements;(g) and influence of meters on measurement results.

The introduction read as follows.

112 Appendices

Appendices 113

114 Appendices

Appendices 115

116 Appendices

Appendices 117

118 Appendices

Appendices 119

Appendix B The metacognitive strategy: a framework for investigation

A model of the strategy was constructed by a Delphi method. Repeatedly discussions were held bythe author with the staff members of the lab course and other lab courses based on a framework usedin chemical engineering (Ruijter, 1979). The discussions lead to adaptations and improvements untilagreement was reached and the schema was fit for electrical engineering (see Table B.1).

Table B.1A framework for investigation

Phase ActivityTHE PROBLEM

Analyze

INFORMATION

Gathering

Interpreting

HYPOTHESES

Formulate

Select

TEST EXPERIMENTDesign

ExecuteEvaluate

CONCLUSIONSDraw

REPORTING

1. What is exactly asked?2. Try to focus and confine the problem3. Separate (partial) problems4. Represent the connections between the partial problems in a scheme5. Which variables play a role?6. Which relations contain these variables?7. Which relations contain quantities that relate to the variables?8. What are the criteria for a satisfactory solution to the problem?11. What information is needed in each phase/ activity? (about variables,

relations, signals, networks, measurement procedures, etc.)12. Gather information from literature, lecture notes, lab guides, etc.13. Gather information by preliminary measurements.14. Analyze and interpret information, calculate signals and system behavior.15. Focus information onto the problem, select consequences.If an hypothesis (idea) turns out to be correct, it helps solving the problem.21. Hypotheses about signals or network models if useful relations are found22. Hypotheses if relations need further investigation23. Select hypotheses that you want to test24. Consider all kinds of boundary conditions (available time, instruments, etc.)25. What are the criteria for a test experiment?

31. Which variables can be adjusted in a measurable way, can be keptconstant, cannot be manipulated?

32. Find other quantities that are related to these variables and that can beadjusted in a measurable way.

33. Write down how these quantities can be measured34. Compare different measuring methods and make a choice35. Consider the required accuracy, precision, and available instruments and

time36. Write down a global plan of work37. Write down a step by step detailed measuring procedure (cookbook)41. Execute the procedure42. Analyze how precise and accurate your observations and results are43. Check any suppositions and limitations in connection with the criteria.44. Possible improvements of the investigation

51. Draw conclusions about the hypotheses; formulate new ones if necessary.52. Check whether the problem has been solved53. Formulate a final conclusion

61. Write down all your thoughts, observations, do’s and don’ts in yourlogbook

62. Make a report of your investigation

120 Appendices

Appendix C Contradictory and comparative assignments

C.1 Contradictions Between Thoughts of the Students and Observed Behavior ofCircuits

Assignment 1. Investigation of Nonlinearity.The assignment reads “Show the non-linearity of the circuit on your desk using the harmonic

signals and the DC-offset of the Krohn-Hite signal generator”.Note. Students are provided with the circuit of Figure C.1 enclosed in a box. In order to determine

the non-linearity of a circuit one has to choose a method, e.g. an input-output ratio method (harmonicsor DC), a phase method or superposition. Using ratio methods one finds the circuit to be linear. Noexplanation of the circuit’s behavior is required (the assignment is to be considered as a categorizationproblem).

Here most students thought at first of linearity as a relation between the amplitudes of input andoutput signals. From measuring the ratio of the input and output signals they had to conclude that thecircuit was linear while it was implicitly given in the assignment that the circuit was non-linear. Byreturning to the mathematical theory they had to become aware of the implications of the definition oflinearity including superposition. Thus they were forced to correct their idea about linearity, ametacognitive experience.

Figure C.1. A seemingly linear circuit.The circuit, made according to the schematic, is enclosed in a box. The students see a black box withtwo input terminals and two output ones. The circuit behavior to be measured (see Assignment 1) islinear when either harmonics or DC are used, but nonlinear for a superposition of a harmonic and DC.

Another example containing three contradictions in one task is presented in Assignment 2

Assignment 2. Contradiction between real signal and oscilloscope picture.The assignment reads: “Realize the signal in Figure C.2 and have it checked by the teaching

assistant”.Note. Usually the students do get the picture on the oscilloscope screen. The divisions are usually

correct but the amplitude and/or the period are not. The signal generator has no adjustment stops at 4V amplitude, at 0.4 ms period or at a symmetry of 1 to 3. This means that first the oscilloscope has tobe used as a measuring instruments with calibrated stops. Then the continuous adjustment of the signalgenerator is used to get 4 V and 0.4 ms. Third, the oscilloscope has to be used in continuousadjustment to get the divisions right while the signal generator adjustment should not be changed anymore in this phase. Both the co-ordinations of various functions of two instruments and the differencebetween the electrical signal and the picture on the screen are stressed in this assignment. Thisassignment can be considered as a problem in which a signal is designed.

Appendices 121

Figure C.2. A square wave to be realized.

C.2 Comparative assignments

Assignment 3. Comparison of measuring methods.The assignment reads: “The product R*C is an important quantity of this (resistor-capacitor)

circuit. Choose two different methods to measure this quantity. Compare the accuracy of themethods”.

Note. The possible methods are in this case the use of square wave or sinusoidal signals. In the firstcase there are several ways to determine RC (tangent method, ratio method (see Figure C.3); use ofstep, square or impulse signals). In the second case it is possible to measure either amplitude or phasedifferences. Again several procedures are appropriate (for the phase difference three procedures areavailable: direct, relative and Lissajous).

Figure C.3. Methods for the determination of the product R*C.The ratio methods use an approximation of e (1/e ≈ 3/8).

122 Appendices

Assignment 4. Comparison of measurement and calculation (time domain).The assignment reads: “Calculate and measure the response of the RC-circuit to a triangular signal

with height A (arbitrary) and length 5*RC (circuit value). Draw the signal and response in one graph(see Figure C.4).

Figure C.4. A triangular signal and the response of a RC circuit to this signal (see Assignment 4).

Appendices 123

Appendix D The data collection in the lab course and their analysis

Appendix D.1 Diagnostic entrance test and remedial exercise

The diagnostic entrance test and the remedial exercise follow on the next page (in Dutch)

Note. The initial settings of the oscilloscope were disarranged in a standard way before each test(oscilloscope defocused, the focus-spot turned away from the screen, the sensitivity adjustedextremely high, etc.).

124 Appendices

Appendices 125

126 Appendices

Appendix D.2 Example of a grading sheet

The grading sheet for the first logbook follows on the next page (in Dutch).The first block of questions identifies the student and the teaching assistant.The second block contains questions specific to assignments of the three sessions relevant to this

logbook.The third block contains questions about relevant general metacognitive skills.The next block contains a general open question about the places where student was stuck.The last block contains a space for the mark.

Appendices 127

128 Appendices

Appendix D.3 Example of a questionnaire

The questionnaire for the third session follows on the next page (in Dutch).These questionnaires had to be answered by the students at the end of each session.It was asked to score the frequencies and amount of time for several activities during the session:

(a) asking help from the TA; (b) getting unasked help from the TA; (c) consulting the partner; (d)consulting another pair of students; (e) read the lab guide or lecture notes; (f) handling the measuringinstruments; (g) chatter with other students; (h) other activities. Also the preparatory activities at homehad to be scored for each homework assignment in the categories: (a) not tried; (b) stuck during thesolution; (c) finished; and the time spent on the assignment.

Appendices 129

130 Appendices

Appendix D.4 Analysis of the metacognitive performance

The metacognitive performance of the students was monitored by calculating an indicator of specificperformances for each logbook. The activities of the students were monitored by calculating an indicatorof the activities for each of the sessions 2 to 9.

The indicators of the specific types of performance were calculated from the grading sheets by takingthe following combinations of scores.

1. The performance in the comparative open assignments.Logbook 1: the mean of the 4 scores for assignment O:3.1 of the grading sheet for the first logbook(see Appendix D2).Logbook 2: the score for assignment O:5.2 of the grading sheet for the second logbook

2. The performance in the comparative measuring assignments.Logbook 1: the mean of the score for assignment O:2.3 and both scores for assignment O:2.4Logbook 2: the mean of the scores for assignments O:4.2, O:4.3 and O:5.3.Logbook 3: the mean of the three scores for assignment O:6.9.1 and the scores for assignments O:6.11.1,O:6.11.2, O:6.11.3.

3. The performance on the metacognitive skills.For each of six metacognitive skills a general question was posed in the grading sheets for the logbooks(see Appendix D2). The score on the following questions was used for:Reasoning: question 1.Studying the lab manual: question 2.Studying the lecture notes: question 3.Understanding the functions of the measurement instrument: question 4.Comparative measuring methods: question 6.Working methodically according to the strategic framework: question 5.

4. The time spent on preparation at home.For each session: the mean time estimated by the students for each homework assignment (seeAppendix D3) summed over all assignments for that session.

The results from the analysis under 1 to 4 are presented hereafter. Significant differences between thescores of the categories of students are indicated. In all cases a t-test has been used, if necessary with acontinuity correction.

Appendices 131

Table D.1The performance of the three categories of students on the measurement and open assignments for thethree parts of the course, and t-tests with continuity correction.

Logbook # M SD n (M1-M2) SDpool d.f. (M1-M2)c tcorr

MEASUREMENT ASSIGNMENTS

Weak students Good-weak students1 2.21 0.67 15 1.03 0.77 58 0.53 2.33 *2 2.56 0.53 15 1.34 0.68 58 0.84 4.17 **3 2.62 0.51 15 0.60 0.87 58 0.10 0.37Moderate students Weak-moderate1 2.74 0.71 84 -0.54 0.70 97 -0.04 -0.202 3.53 0.72 84 -0.97 0.69 97 -0.47 -2.41 **3 3.21 0.89 84 -0.59 0.84 97 -0.09 -0.37Good students Moderate-good1 3.24 0.79 45 -0.49 0.74 127 0.00 0.002 3.90 0.72 45 -0.37 0.72 127 0.00 0.003 3.21 0.96 45 -0.01 0.91 127 0.00 0.00OPEN ASSIGNMENTS

Weak students Good-weak1 1.89 0.75 15 0.81 0.69 58 0.31 1.492 2.63 0.74 15 1.56 0.93 58 1.06 3.84 **Moderate students Weak-moderate1 2.21 0.77 84 -0.32 0.77 97 0.00 0.002 3.52 1.01 84 -0.89 0.98 97 -0.39 -1.43Good students Moderate-good1 2.70 0.67 45 -0.49 0.74 127 0.00 0.002 4.19 0.98 45 -0.67 1.00 127 -0.17 -0.92* p < 0.05 two-tailed** p < 0.01 two-tailed

132 Appendices

Table D.2The performance of the three categories of students on the six metacognitive skills for the three partsof the course, and t-tests with continuity correction

Logbook # M SD n M1-M2 SDpool d.f. (M1-M2)c t-corrREASONING

Weak students Good-weak students1 2.69 1.03 15 1.13 0.91 58 0.63 2.31 *2 3.22 0.44 15 1.28 0.67 58 0.78 3.89 **3 2.93 0.73 15 1.24 0.65 58 0.74 3.82 **Moderate students Weak-moderate1 3.20 0.89 84 -0.51 0.91 97 -0.01 -0.042 3.53 0.86 84 -0.31 0.81 97 0.00 0.003 3.65 0.67 84 -0.72 0.68 97 -0.22 -1.16Good students Moderate-good1 3.82 0.87 45 -0.62 0.88 127 -0.12 -0.742 4.50 0.73 45 -0.97 0.82 127 -0.47 -3.12 **3 4.17 0.62 45 -0.52 0.65 127 -0.02 -0.14STUDYING THE LABGUIDEWeak students Good-weak students1 2.82 0.90 15 0.91 1.06 58 0.41 1.302 2.89 0.80 15 1.61 0.73 58 1.11 5.13 **3 3.00 1.00 15 1.50 0.72 58 1.00 4.68 **Moderate students Weak-moderate1 3.00 0.90 84 -0.18 0.90 97 0.00 0.002 3.61 0.80 84 -0.72 0.80 97 -0.22 -0.983 4.10 0.70 84 -1.10 0.75 97 -0.60 -2.85 **Good students Moderate-good1 3.73 1.10 45 -0.73 0.97 127 -0.23 -1.282 4.50 0.70 45 -0.89 0.77 127 -0.39 -2.75 **3 4.50 0.60 45 -0.40 0.67 127 0.00 0.00STUDYING THE LECTURE NOTESWeak students Good-weak students1 2.73 0.90 15 0.91 0.83 58 0.41 1.672 2.89 0.90 15 1.67 0.75 58 1.17 5.21 **3 2.75 1.00 15 1.64 0.72 58 1.14 5.33 **Moderate students Weak-moderate1 3.06 0.70 84 -0.33 0.73 97 0.00 0.002 3.69 0.90 84 -0.80 0.90 97 -0.30 -1.193 3.83 0.80 84 -1.08 0.83 97 -0.58 -2.49 **Good students Moderate-good1 3.64 0.80 45 -0.58 0.74 127 -0.08 -0.592 4.56 0.70 45 -0.87 0.84 127 -0.37 -2.40 **3 4.39 0.60 45 -0.56 0.74 127 -0.06 -0.44* p < 0.05 two-tailed** p < 0.01 two-tailed

Appendices 133

Table D.2part 2.

Logbook # M SD n M1-M2 SDpool d.f. (M1-M2)c t-corrFUNCTIONS OF INSTRUMENTSWeak students Good-weak1 3.79 0.70 15 0.21 0.78 58 0.00 0.002 3.85 0.80 15 0.59 0.73 58 0.09 0.423 3.33 1.10 15 1.34 0.81 58 0.84 3.46 **Moderate students Weak-moderate1 3.73 0.80 84 0.06 0.79 97 0.00 0.002 3.89 0.80 84 -0.04 0.80 97 0.00 0.003 4.02 0.90 84 -0.69 0.93 97 -0.19 -0.73Good students Moderate-good1 4.00 0.80 45 -0.27 0.80 127 0.00 0.002 4.44 0.70 45 -0.55 0.77 127 -0.05 -0.353 4.67 0.70 45 -0.65 0.84 127 -0.15 -0.97COMPARING METHODSWeak students Good-weak1 2.27 0.90 15 1.00 0.90 58 0.50 1.86 *2 2.44 0.70 15 1.69 0.86 58 1.19 4.66 **3 2.67 0.90 15 1.05 0.83 58 0.55 2.24 *Moderate students Weak-moderate1 2.57 0.90 84 -0.30 0.90 97 0.00 0.002 2.93 1.00 84 -0.49 0.96 97 0.00 0.003 3.20 0.80 84 -0.53 0.82 97 -0.03 -0.13Good students Moderate-good1 3.27 0.90 45 -0.70 0.90 127 -0.20 -1.202 4.13 0.90 45 -1.20 0.97 127 -0.70 -3.92 **3 3.72 0.80 45 -0.52 0.80 127 -0.02 -0.14METHODICAL STEPSWeak students Good-weak1 2.82 1.40 15 1.00 1.32 58 0.50 1.272 2.78 1.60 15 1.35 1.17 58 0.85 2.43 **3 2.58 0.80 15 1.59 0.80 58 1.09 4.57 **Moderate students Weak-moderate1 2.88 1.30 84 -0.06 1.31 97 0.00 0.002 3.13 1.10 84 -0.35 1.19 97 0.00 0.003 3.47 1.00 84 -0.89 0.97 97 -0.39 -1.43Good students Moderate-good1 3.82 1.30 45 -0.94 1.30 127 -0.44 -1.83 *2 4.13 1.00 45 -1.00 1.07 127 -0.50 -2.54 **3 4.17 0.80 45 -0.70 0.94 127 -0.20 -1.16* p < 0.05 two-tailed** p < 0.01 two-tailed

134 Appendices

Table D.3The time spent on preparation at home for the sessions, in minutes, for the three categories of studentsand t-tests with continuity correction

Session # M SD n (M1-M2) SDpool d.f. t-corrWeak students Good-weak students2 26.7 37.1 12 45.0 41.9 28 2.56 **3 25.0 20.6 12 13.1 20.2 28 1.074 19.3 13.7 7 9.1 18.3 21 0.495 25.9 19.1 11 7.1 15.5 24 0.347 14.4 27.4 8 49.5 46.2 20 2.17 *8 30.0 30.0 1 16.4 30.5 6 0.359 30.0 14.4 2 20.5 17.1 10 1.17Moderate students Weak-moderate students2 46.2 50.4 54 -19.5 48.4 64 -0.943 32.2 22.4 49 -7.2 22.1 59 -0.314 24.0 17.3 51 -4.7 17.0 56 0.005 29.5 19.2 55 -3.6 19.2 64 0.007 70.8 64.1 53 -56.4 60.9 59 -2.22 *8 27.9 18.2 22 2.1 18.2 21 0.009 54.4 43.8 16 -24.4 42.6 16 -0.61Good students Moderate-good students2 71.7 44.7 18 -25.5 49.1 70 -1.533 38.1 20.0 18 -5.9 21.8 65 -0.154 28.4 19.9 16 -4.4 17.9 65 0.125 33.0 12.4 15 -3.5 18.0 68 0.007 63.9 53.7 14 6.9 62.2 65 0.108 46.4 30.5 7 -18.5 21.5 27 -1.449 50.5 17.4 10 3.9 36.2 24 0.00* p < 0.05 two-tailed** p < 0.01 two-tailed

Appendices 135

Appendix E Overall performance before and after the change in the labcourse

The earlier analysis for the redesign of the course was focused on the main objective of the course(the systematic approach), the performance of the students, and the information flow. It was carriedout by interviews with the two staff members responsible for the labs and three experienced teachingassistants.

Based on the analysis (Tattje & Vos, 1995) the instructional design was changed. From theperformance of the students before and after the change the effect of this change could be measured.

E.1 ProcedureFrom the database of the university the passing rates for the lab course were extracted from 1982

on as a function of the year of entrance. The passing rates during four years before and four years afterthe introduction of the new lab course were compared. These data provided observations in a posttest-only design with non-equivalent groups, of the form X1-O and X2-O.

Also an analysis of the performance of the students in other courses was carried out in order tomonitor any changes in the environment that could have had an influence on the performance in thelab course.

E.2 ResultsThe passing rates for the course during four years before and four years after the change are

presented in Figure E.1. Whereas before the change the mean passing rates during the course were 48% (with another 13 % passing by additional effort after the course), after the change the immediate

Figure E.1. Passing rates of students in the lab course NA during eight consecutive years.Percentages of students of the entrance lists passing the lab course are shown. Immediate results referto grading obtained during the course. Additional results are obtained during the next trimester,holidays or next year.

136 Appendices

passing rates were at the level of 79 %, a significant increase (t(6)=6.66, p<0.01). These resultspertained until 1993 in which year the curriculum was completely revised. Also the total passing rate(after additional tasks in the following trimester) increased significantly from 61 % to 83 % (t(6)=6.62,p<0.01).

E.3 DiscussionThe increase in the passing rates is significant and stable. The explanation must be found in the

special traits of the instruction. In Tattje and Vos (1995) it was reported which data were used toexclude alternative explanations of the increase in the passing rates: (a) the students after 1987 werenot better than before, thus they belong to the same population; (b) the students were not trained in thesame cognitive skill, thus no mastery learning was the case; (c) the time spend on the coursediminished slightly with respect to earlier measurements; (d) the lab course did slightly better thanbefore discriminate between good en weak students, so the students were not dragged through thecourse by assistance or otherwise; (e) no other changes in the curriculum could attribute to the increase(no environmental effects).

Appendices 137

Appendix F The influence of the composition of the class onperformance

F.1 ProcedureThe marks for the entrance test and each of the three logbooks provided data about the performance

of the students during the course. The percentage of the students performing sufficiently (above 5.4 onthe 10-point scale) was calculated. This analysis was carried out for the classes of about 25 students.

F.2 ResultsIn Table F.1 the relative rates at a sufficient level of performance (5.5 and larger on a 10-point

scale) are presented. The data are shown for all classes of students.

Table F.1The percentage of each class with a sufficient mark on the entrance test and the logbooks (percentagesof the number N of students enlisted in the beginning of the class)

Part of the course: Introductory Time domain Frequency domainMC strategy: Orientation Practice ApplicationSessions: 1 2 3 4 5 6 7 8 9

Class # N DropoutsClass 1Class 2Class 3Class 4Class 5Class 6

212224251811

115405

144442522291

474541604428

766858807246

439062808955

Totals 121 16 41 47 69 71Note. Marks on the entrance test were obtained in session 1, marks on the 3 logbooks shortly aftersessions 3, 6 and 9.

The results on the entrance test and the first logbook (handed in after session 3) showed that 41 %respectively 47 % were performing at a sufficient level. About 13 % of the students dropped out afterthe entrance test. The results of the second and third logbooks showed an increase in general.

Class 1 performed lowest on the entrance test, and less than 50 % of this class performedsufficiently in the third part of the course and passed the course, the same percentage as in the past forthe whole population. The class had good marks for logbook 2, but bad marks for logbook 3.

F.3 DiscussionIn general the results of each of the classes are good. Class 1 shows a divergent behavior (see Table

F.1) indicating that these students did learn during two thirds of the course but failed during the lastpart of the course, the difficult frequency domain, although the assignments in part 3 were of a lowerlevel than those of parts 1 and 2. The foregoing does not explain why the effect of failure in the lastpart of the course is so strong in one class only. Effects of the assistant grading the logbooks areunlikely because the assistants used explicit scoring criteria in the form of the grading sheets andseveral assistants graded each class. It is supposed that the explanation can be found in thecomposition of the class. According to the results on the entrance test (see Table F.1) this is theweakest class of students. Probably the number of good students in this class is too low to attain andprocess the information necessary for understanding the metacognitive requirements and forgenerating the knowledge necessary to solve the cognitive conflicts.

138 Appendices

Appendix G Characteristics of the categories made by the participants

A review of the categories formed by the experts: names, descriptions, examples of contents,together with the expertise of the expert and his experience in teaching. The number of cards in eachcategory are given (between brackets). The names finally chosen are underlined. The mean number ofcards per category are calculated for each expert. Comments of the researcher are given in italics. Thenumbers refer to the labels of the categories.

Participant E1

Expertise: Control systems & theory, sampled data systems, image processing for robotics, sonarand vision for underwater vehicles.

Experience in teaching: 29 years ago: design and circuit theory; last 3 years: Circuit Analysis andA.C. Theory (2nd year), Systems and Signal Analysis (3rd year).

1. Transform methods (introductory ideas) (70)pulses, signals, signal characteristics, frequency, signal analysis, systematic analysis or circuit

analysis through Fourier (description of signals) and Laplace (descriptions of systems) transformtechniques, convolution,

2. Fields (5)magnetic field, electric field, relating to the inductors, capacitors in 3.3. Components and basic physics (70)Charge, inductance, field effects, energy, circuit configurations with Q-factor and resonance,

inductors, resistors, capacitors, passive elements, properties of capacitors, energy, power, differentialequations to describe circuits, solution of differential equations, and the solution for sinusoidal inputsand sinusoidal supplies to that circuits, transients on circuits and sinusoids on circuits, frequencydependence. Modeling type descriptions.

Components and basic physics of inductance, capacitance, resistance through into mutualinductance through to circuits and resonance. Current.

4. Sources (41)current sources, connections, voltages, power supplies, properties, sources, batteries, supplies

Circuit theory (5 and 10 relate)5. Circuit theorems (74)circuit analysis theorems, circuit analysis, Norton’s theorem, two-port networks, series connected

pass wires, star-delta, delta-wye configurations, components, nodes, active circuits, transients,voltages, sources

10. Circuit equations (33)equations, mesh equations, equations going with circuit theorems,6. General instrumentation (25)instruments, calibration, meters, connections, ground connections, devices, transformers, gauges,

bridges7. Circuit diagrams: pictures, diagrams, loops (later distributed into the other columns)8. General terms (21)9. Mathematical symbols (14)If I did another pass then I would distribute 8, 9 and 10 back into the other columns because I think

they are related to those. And also 7.Mean 353/9 = 39 cards per category

Participant E2

Expertise: modelling and simulation of electric power systems; modelling of production andpropagation of harmonics in power networks.

Experience in teaching: 5 - 10 years Circuit Theory (2nd year). About 15 years professor.

Appendices 139

1. Special types of networks (12) specific types of networks, cube, RLC.2. Basic definitions (11) like frequency, current, power.3. Operating characteristics of circuits (8): continuous, symmetrical, time dependent, phase, steady

state. The way the network is operating,4. Inputs (8) or waves, square wave, triangle wave, step wave, ramp, form of the wave, form factor.5. Responses (3) of networks: step response, cut-off frequency, charge, charge impulse, key

characteristics of networks, response of the circuit.6. Characteristics of networks (9), like stable, passive, not a response7. Instrumentation (14), metering8. Circuit solution techniques (127): basic laws, all other aspects combined, things you use to solve.9. Circuit elements (70) like inductors, voltage sources, transformers, ground, potentiometer,

physical elements, capacitor, source, physical devices10. Port networks (25), ports, in and out. [word: port; symbol: square; word: in-out]11. Physical connections (6). [word: connection]12. Electromagnetics (6): E-field, magnetic field, force.13. Complex variables (9)14. Algebraic symbols (31).15. Aside (14): things that I do not know where else to putMean: 353/ 15 = 24 cards per category

Participant E4

Expertise: High frequency electrical engineering and field theory. Microwaves, satellitecommunications, remote sensing.

Experience in teaching: 5 years1. Typical sentences (25), statementsGroup 1 are statements I can imagine myself using through the a lecture.2. Graphical representation (52)3. Mathematical and analytical representation. (69) equations, symbols, expressions4. Methods of analysis (25) (structural equations, mesh current method)5. Notation or expression (123) Ausdrücke, Begriffe, fundamental concepts; electric concepts like

(elements, connections, network, characteristic quantities like Q-factor, pulse, port, one-port, two-port). Voltage, current.

The biggest group, 5, has a main name, title, notation or expression.Within this title you can divide it into:- fundamental expressions, like input, output, current, one-port, equivalent resistor, impedance and

so on, which are belonging to circuits, circuit analysis, and- mathematical expressions, like eigenvalue, algebraic elimination, frequency domain, algebraic

elimination, differential operator.The groups 2, 3, 4, 5 are the main road sections.6. Energy sources (10)7. Circuit elements (4)8. Verifications, verifying measurements (10)measurements, measurement device, instrumentation.9. Letters (16)used to indicate some physical quantities.10. Numerical examples (7)numerical verification, numerical values11. Words with the same meaning in the normal language and engineering (8). Stable, complete,

etc. This should be distinguished from the notation, because the notations are expressions of wordswhich have a special meaning related to the subject.

12. Rest group (3)Mean 352 /12 = 29 cards per category

140 Appendices

Participant E6

Expertise: Professor in Network Theory. Research: Quantum electronics and microwave networktheory. Digital and analog signal processing, with emphasis on adaptive filtering.

Experience in teaching: 18 years of teaching Network Theory. Modern Optics. Wrote a book onNetwork Analysis.

1. Background (20)Introduction. “Ideal model laws”, “structure”. Connection. Voltage. Fundamentals, that is the

fundament.2. Basics (104)Rather fundamental. The first basics, the introduction in network theory. Elements, Kirchhoff’s’

laws, simple circuits, simple networks, resistor networks, general concepts. First true network termsand network concepts.

3. Transients (40). Time domain. Differential equation4. Steady-state (64). Complex calculations. Sine, sine-waves, eigenfunction, frequency, frequency

domain. Periodical, ground, sinusoidal.5. Advanced methods (18). Port, two-port, Tellegen.6. Very advanced methods (7). Fourier transform, Laplace transform.7. Diversity, advanced (19). Graph theory, symmetric, dual, signal components, output, input-

output8. Waste-paper basket (72)9. Forbidden (8)Mean 352 / 9 = 39 cards per category.

Participant E7

Expertise: Network theorem’s, approximation. Professor in Network Theory. Design of timediscrete systems. VLSI-design.

Experience in teaching: taught Network Analysis at least 20 times.1. Frequency domain (38)3. Definitions, concepts (28): definitions of important concepts. Voltage, current. including

concepts [2]4. Signals (32)5. Structure, systematic analysis (53): the elements come in this structure. Connection. Including

formula manipulation.6. Two-ports (16)7. Basics, fysical properties (21)Passive8. Time domain (22)9. Circuits, networks (34): schematics of circuits and parts of circuits10. General (2) : can be placed in all domains.Parameters, Fourier-synthesis11. The rest (106)Practical things, measurements, everything connected with Measuring Techniques, or Electronic.

Algebraic symbols. Things that are not correct.Mean: 352 / 11 = 32 cards per category.

Participant E8

Expertise: Professor in Network TheoryExperience in teaching: taught Network Analysis about 20 times.1. Garbage, English, I do not understand. (49).2. Network theory, network properties: concerns other networks related to the given one (8).

Appendices 141

3. Lab work, experimental analysis, experiments, experimental things, connection, and: electricitysupplies, laboratory, practical work, practical instructions, energy supplies (36).

4. Harmonic analysis, fundamentals of the analysis, mathematical analysis, signal, signal analysis,signal descriptions, frequency dependent, frequency domain, properties of transfer functions andimpulse responses, also: spectrum, eigenfunctions, analysis, frequency domain (92).

5. Ports, port, two-port, external description, external references (29).6. Circuits. Bridge circuit, ladder network (4).7. Circuits. RC-, RLC-, R/L-circuit (3).8. Ideal description of elements, components, elements, fundamentals of a network, forming

elementary equations, description of the network, properties of components, (physical) modelling,discussions of components, descriptions of components, elementary description of network properties,physical description of the elements, idealisation of elements, physical description of elements,physical properties, analysis, time dependent, ideal schematic. Connection wire. Voltage. (77).

9. Systematically forming the equations, fundamentals in the analysis, idealisation, belongs to theanalysis, laws, structural laws, discussion about structures, topological, forming structural equations(of a given network). Current. (23).

10. Systematic analysis, working out of equations, algebraic analysis, manipulation of formulae,solving equations, methods, mathematical description, analysis (15).

11. Tricks, seeing that it is possible to replace something, manipulations of networks, funnypuzzles, flavor adding problem, also: parallel circuit (pitfall) (16).

Mean 352/11 = 32 cards in each category.

Control E½: Intermediate expert

The hierarchy in categories can not be fully represented, but has been indicated by indentation1. basics, terms from the beginning, start, ideal/ non-ideal, concepts, what you have to explain in

the beginning. word: characteristics of ideal elements (1)1.1. node (3)1.2. ideal (5)1.3. voltage (2)1.4. current (4)

2. physical background (7)words: field, charge conservation

3. theory, mathematics, algebraic operators (13)4. leads and components

4.2. leads (3), words: connection4.3. parts, components

4.3a. voltage source, non-ideal voltage source (11)word: battery4.3b. current source (3)4.3c. passive element (1)

4.3c1. resistor (13)4.3c2. capacitor (6)4.3c3. inductor (7)4.3c4. transformer (7)4.3c5. dependent source (2)4.3c6. zero-port (1)

5. network manipulations (8)Kirchhoff (2), reference direction (4)

ground (2)mesh-current-, node-voltage-, branch method (7)

6. superpositionsuperposition is a basic concept (used in transients), linear (4)7. special networks (ladder, voltage divider),

7.1. equivalent resistor (1)

142 Appendices

parallel connection (3), series connection (2)7.2. voltage divider (3), current divider (1) (next to each other)7.3. bridge circuits (6), circuits: a gauge is an application, equilibrium belongs to a bridge circuit7.4. special networks, rest (3)7.5. means (3)7.6. continuous (2)

8. power, energy, passive/ active (11)9. domain (2)

time domain, time signal, variation in time (1)10. differential equation (12)words: state, RC-time

11. convolution-integral, impulse response, time dependent, current signal above response (16)13. group (u(i,t),i(u,t)) (4)14. literal: this capacitor, this voltage, this system (3)15. input/output relation (7)

incomplete, (not)complete (3)16. steady-state, link between time and complex domain, harmonic (5)16.1. DC (3) next to AC (4) “belongs to harmonic”

16.2. resonance and time description, in both domains, resonance-situations,RC-type network (9)

17. frequency domain (1)complex calculations, complex, impedance (15)18. Fourier transform (16)19. Laplace transform (9)20. Fourier analysis (10)words: period, component of a signal, form, approximation

21. port [under time and frequency domain!] (1)21.1. one-port, Thévénin and component, Norton “Norton belongs to voltage divider, tricks”

(11)21.2. two-port, “has 4 parameters”, input signal (18)

22. laboratory work, ideal/ non-ideal “lab work or beginning. Lab work is how it works in reality,parallel capacitors for example (30)

23. rest, not categorized (7 + 3 loose cards)24. unknown (6)25. signaled as double: 4 out of 6, the other 2 consistently categorized.Mean (351-7)/48= 7,2 cards per category.

Control E0: Novice

1. Mathematics, ‘mathematical things’ (32)20 algebraic symbols and formulae:all symbols for mathematical operations (10) and relations (3);among which 6 formulae for: continuous (limit), impulse (limit), time invariance,time dependence of x and impulse-delta, period T1 schematic symbol: for the definition of polarity of a port;11 words: for mathematical operators and operation: formula manipulation, algebraic elimination,equivalence method, differential operator, differential equation, convolution-integral[formula: in category unknown], means, pure imaginary, complex (2x), reciprocal

2. Spatial geometry, mathematics, symmetrical (3)3 words: symmetrical, structure, triangle

mathematical descriptions, ways to describe a circuit as a whole, has also to do with components, e.g.u=Ri, elementary equations, descriptions of networks in the form of a formula, sum of i=0 for a nodee.g., descriptions of a network in a mathematical manner. Divided into:3. Descriptions of a network (abstract) “mathematical descriptions of networks” (33)

Appendices 143

words: transfer function (method), input-output-relation25 formulae: 1 elementary equations, linear

4. Elementary equations, (mathematical) description(s) of components (11)words: ideal model laws, characteristics of ideal elements8 formulae: elementary equation for L, R, G, transformer (2)

5. Signals, frequency, time, what one presents to a circuit, everything that has to do with frequencyetc. Maybe frequency could be split off. Time invariant. (52)words: current signal, time domain, frequency domain, periodic, harmonic [without a hierarchy]1 formula: for the value of the frequency2 graphical symbols: drawings of signals

6. Lab work, “pieces of lab work”, measurement circuit, sketch (of a circuit), real things, non-ideal;‘this system’, ‘this capacitor’ belong to practical things, values also;real components (conductance of 0,01 S, equalizing? capacitor of 0,1 mF) go from parts tolab work, that is practice, reality. (38)1 formula: specification of values4 equivalent schematics of measuring instruments

7. Approximations of reality, linear is a kind of approximation too (3)word: linearnetworks, divided into

8. Parts , parts of networks, elements, I would have liked to divided these intoideal/ non-ideal elements, components,maybe components had better be put with practice, connections (54)words: resistor, capacitor, transformer, inductance3 algebraic. symbols: R, L, C20 schema’s/symbols

9. Networks, “network-like things”, complete and not complete networks,manners of connection (in series and in parallel),could have been divided into complete/not complete (41)words: port, zero-port, one-port, two-port, RR, RLC16 schematic symbols: one-port, two-port, etc., reference direction (2x)

10.Methods, theorems, laws, ideal (26), mesh, loop, node, orientation, reference direction11.Quantities, current, description of current (19), dual

6 algebraic symbols and formulae: E U I u i Q/t12.Divers, rest, left-overs (19).

words: passive, active1 formula

13.Unknown, laid aside (18)12 formulae/ symbols

Signaled as double: complex; spectrum; reciprocity theorem; passive; times or x; (5 out of 6)Mean 351/13 = 27 cards per category.

144 Appendices

Appendix H Methods to solve problems in NA

Legends to Table H.1: Experts, their categories and code numbersE8 E6 E7

894

11

10

25

forming elementary equationsforming structural equationsharmonic signal analysis(t, ω, ϕ, s - domain)

network manipulations,tricks

working out equations,syst.an.

network theoryports

1234567

/

fundamentalsbasic network termstransients, time domainsteady state, time domainadvanced methods, 2-portsvery adv. methods, transformsdiverse advanced, graph

theorywaste paper basket

754183

610, /

basics, physical propertiesstructure, systematic analysissignalsfrequency domaintime domaindefinitions of important

concepts2-portsgeneral, rest

E1 E2 E4354198

components, basic physicscircuit theoremssourcestransform methodsmathematical symbolsgeneral

289

basic definitionscircuit solution techniquescircuit elements

54

conceptsmethods of analysis

DC-, AC(ω, ϕ )-,transient(t)-,frequency domain(s)-

Table H.1Methods

E8 E1 E2 E4 E6 E7Modeling elements

elementary equations 8 3 8 5 2 7substitution method 8 5 8 4 / 5

Use of background variablespower [conservation] 8 3 2 5 2 1energy [conservation] 8 3 2 5 2 3charge conservation 8 3 8 5 1 3current injection 8 5 8 5 / /

Modeling network structuresstructural equations 9 4 8 4 3 5Kirchhoff’s first law; Σi=0 for a node 9 5 8 4 2 5Kirchhoff’s second law; Σu=0 for a loop 9 5 8 4 2 5

Operations on signals, transformsFourier synthesis 4 1 8 4 3 10Fourier analysis 4 1 8 4 6 4superposition 4 5 8 5 2 3Fourier transform 4 1 8 4 6 1Laplace transform; X(s)=�x(t)exp(-st)dt 4 1 8 4 6 1transfer function; Y(ω)=H(jω) X(ω); Y(s) 4 1 8 5 4 1convolution integral: initial state, impulse response 4 1 8 4 7 8

Mathematical operations, systematic analysisformulae manipulations 10 9 8 4 / 5algebraic elimination 10 5 8 5 / 5branch method 10 5 8 4 / 5mesh-current method 10 5 8 4 5 5node-voltage method 10 5 8 4 5 5differential equation 10 4 8 5 3 8characteristic equation 10 1 8 5 3 8

Network transformationsequivalence method 11 8 8 4 / 5Thévénin’s theorem 11 5 8 4 2 5

Norton’s theorem 11 5 8 4 4 5delta-wye transformation 11 5 9 4 4 5

Principles from network theorydual 2 5 8 5 7 5reciprocity theorem 2 8 8 4 5 6Tellegen’s theorem 2 8 8 4 5 5

Measurement and instrumentationmeasurement circuit 3 6 7 8 1 /

Appendices 145

Appendix I Hierarchical dimensions seen by the teachers

Table I.1The dimensions seen by the teachers in the six small sets of cards

[1] Circuit and schematicE1: From actual circuit to ideal schematic and a formal representation of thatE6: From physical, outward appearance via abstraction to formal treatment.E7: From circuit to schematic. From practice to theory.E2: [reversed] From conception to constructionE4: [reversed] From ideal configuration (start from a certain aim, a certain object) till

realization.[2] Network and (input-output-)system

E1: Input-output system through to a complex exampleE2: Desired input and output, increasing order of complexity, until I got a system that will

do what I want it to do and then finalize it. From conception to finish.E4: Specification. First a general description of the system, and then to a specific system.E6: From the most general concept (input-output) to realization.E7: Specification, but it is wrong! no clear dimension, and moreover incorrect.

[3]Describing resistorsE1: Get other effectsE2: ??E4: More complicated modelE6: A resistor of a 100 ohm, describe it further, analyze in more detail, description in more

detail.E7: From constructive element (component) abstracting to a description in terms of network

elements for simulation.[4] Passive element and capacitor

E1: General description, a particular element, examples of that element, specific models.E2: Order of increasing information. From global description of an element to the most

specific descriptionE6: To particular element, specific elementE4: [reversed] Examples of ...E7: [reversed] From constructive element (component) to [theoretical] network. Wish for

another component.[5] Active element and voltage source

E2: From least specific to most specific.E6: From the general concept two-pole to technical things. Three do not belong to the row.E1: [reversed] This is a ... [interpreted as: this is an example of a ...]E4: [reversed] ... is a special case. General component to a special case. Some examples.E7: [reversed] From practice to theory. One of these does not fit.

[6] Signal and frequencyE1: It could be a ... , an example of ....E2: Specifying a signal.E4: ??E6: Signal, is periodical, has a frequency of 1 kHz, it is a voltage: specification.E7: [reversed] From signal to theory, to theoretical description.Legends:[#]: number of the set of cards.In italics: the content of a set of cards.[between brackets]: comments

In Table I.2 the cards in the small sets and the sequences as seen by the participants are presented.For each of the sets and each teacher the sequences are presented. The sequence of the presentation of

146 Appendices

the set (R) is the one thought most appropriate by the researcher. The second column (B) gives thenumber labels on the back of the cards. The sequences made by the participants are represented by thenumbers on the back of the cards. Numbers in bold indicate cards in the same position as the presentedsequence.

Table I.2The cards and the sequences in the small sets

R B E1 E2 E4 E6 E7[1] Circuit and schematic

actual circuit 2 4 4 4 4 2photograph of circuit 4 2 2 2 2 4sketch of circuit 5 5 7 5 6 5rough draught schematic 6 6 6 6 5 6formal schematic 7 3 3 3 7 7electrical schematic 3 1 5 7 3 3ideal schematic 1 7 1 1 1 1

[2] Network and systemstructure of network 3 5 8 8 5 7input-output system 5 1 1 3 8 5relation (in, out) 8 8 3 1 1 8relation (uin, uout) 1 3 2 5 2 1IO system, 1 pole 2 7 6 2 6 22nd order IO system 4 2 4 4 7 61 pole at 1 MHz 6 6 5 6 3 4this specific system 7 4 7 7 4 3

[3] Resistorsthis specific resistor 1 2 2 1 1 1a resistor of 100 ohm 2 1 5 2 2 2parasitic capacitance 3 3 3 5 5 3high frequency equivalent schematic ofresistor

4 4 1 3 4 4

simulation model of resistor 5 5 4 4 3 5[4] Passive element, capacitor

an element 2 6 2 4 4 6a one-port 4 4 4 2 2 4a passive element 6 2 6 6 6 2a capacitor 7 7 7 7 7 7a polarized capacitor 8 1 8 8 5 8an electrolytic capacitor 1 3 1 1 3 1a coupling capacitor of 30 V, 0.1 µF 3 8 5 5 1 3this specific capacitor of 10 nF 5 5 3 3 8 5

[5] Active element, voltage sourcecomponent 5 2 7 5 7 7two-pole 7 3 5 2 2 2active element 2 7 2 7 5 5voltage source 3 5 3 3 3 4DC voltage source 4 4 4 4 4 3battery 6 6 6 6 6 812V accumulator 8 8 8 8 1 1the accumulator in my car 1 1 1 1 8 6

[6] Signal and frequencyport connection 2 6 2 2 6 1reference 4 4 6 8 7 4signal 6 3 3 7 3 2voltage 8 1 5 5 5 3digital 1 7 8 4 8 6periodic 3 5 1 1 1 5a frequency of 1 kHz 5 8 4 3 2 8this specific signal 7 2 7 6 4 7

Appendices 147

Appendix J Metacognitive aspects during execution of the tasks

Table J.1Metacognitive remarks by the participants on the tasks

Aspects of the task Participant Remarks or questionsMetacognitiveexperiences

E4 “How can you transfer this records of my test to make apicture of the logical sequence with which I teach?”

(usefullness) E6 “I have my doubts, if I can do something to it. Whetheryou can use it, draw conclusions from it. This is a coursemethod, you must not draw too many conclusions fromit.”

Metacognitiveknowledge of goals

E4 “I know the objective is not related to a specific person.It is actually to have an overview of how the differentpeople, or just examples or samples, are teaching.”

E6 “Probably you want to check whether I am consistent ornot. You want a validation.” (repeated several times)

E8 “This card I saw earlier, I think. This is of course a testwhether I am consistent or not.”

E8 “I have not seen any differential equations up till now.”“I think ‘dual’ is a separate category. This card will stayalone.”

Metacognitiveknowledge ofstrategies

E1E2

E4

“Are the cards in random order?”“On many times I have to organize things in some wayand nobody gives me rules, so I do it the best I can, forexample here. I just start going.”“First I would like to distribute them randomly, for afirst orientation, in order to have a global picture, arough draught classification.”

E6 Are there much more of these things coming? Where didyou get these cards, have you screened a book? Am I thefirst one who does this task? Who were the others? Howdid they find it?”..

Metacognitiveexperiences(results)

E4 “I will try to arrange the cards in the sequence by whichI would like to speak to the students, in the order bywhich I would like to introduce the theory to thestudents”…“If I would have made this classification myself, itwould have had another form, it is not corresponding tomy logic... These cards are belonging to a certainimagination. They are belonging to the system,according to which they have been made... So I havethere to see the aim of the classification. These cardsdefinitely should be classified in a certain way. Thisclassification is more or less corresponding to the natureof these cards.”

E6 “This pile is very high, what could lead to dividing it,but it can of course be in the choice of the cards, whythis pile is so high. The meaning of it is not clearenough.”

E8 “The result is a little bit unbalanced, but I do not know ifthat is my fault or that it goes back to the one whodesigned the cards.”“Teachers who do not make a pile “forbidden” or“incorrect”, are not teachers appointed in the subject

148 Appendices

area.”Metacognitiveexperiences

E1 “Tomorrow you might get a different sequence of allthese”

(process) E8 “I believe that if I had to repeat it, I would do it againlike this”.

E2 “I should probably organize my teaching a little bitmore, organize my thoughts better.”

E2 “The task is similar to writing a report. One has todecide what it is the people want to see, what categoriesdo you want to include, and then how divide it and youwork up on at that line. It is similar to writing a paper.”

Feelings about thetask

E4 “The task is lengthy, longwise, but not necessarilymentally tiring.”

E6 “I became tired. This whole research does not muchappeal to me. I have the feeling that you cannot do muchwith it.”

E7 “Strange, queer, I never did something like this.”E8 “I had a little bit of examination tension.”

Appendices 149

Appendix K Analysis of the concepts and methods of NA

K.1 Basic and Prerequisite ConceptsFrom an analysis of the content of NA before the study, and from an analysis of the data gathered,

the following overview of prerequisite and basic concepts of NA could be constructed.The category of “prerequisite knowledge” is distinguished from the basic concepts of NA.

Prerequisite knowledge contains independent, external concepts, like time and frequency, energy andcharge, electric field and magnetic field, which are necessary prior knowledge to understand NA. Ofthese, time and frequency are used as independent domain variables in NA. Some concepts that arebasic to NA also cannot be defined within the content of Network Analysis, like current, voltage,element, and connection (see Table K.1).

Table K.1Prerequisite concepts, basic concepts, their representations, and their relations

Status ofconcept

Name of concept, literalrepresentation

Use of concept Otherrepresentations

Used in

Prerequisite conceptsTime Domain variable t I = C du/dtAngular frequency Domain variable ω U = j ω L ICharge Conserved quantity q, Q I = dq/dtEnergy Conserved quantity E E = �p(t)dtElectric field Definition of voltage,

measurement of voltageE �E.ds

Magnetic field Measurement of current HBasic concepts

Current, voltage Interaction i, uElements, nodes Network structure (Schematic)Port System structure : (Schematic)Power System interaction p p = u.i

(Definition)Signal Information transport x(t)Complete network Specifiable interactions (Schematic)N-port, incompletenetwork

Nonspecifiable interactions (Schematic)

K.2 Systematic Problem Solving ApproachParticipant E8 grouped the methods and coupled the groups to categories of his organization of

knowledge (see H). From this grouping the following metacognitive strategy to solve networkproblems in a systematic way could be derived. This strategy couples types of concepts to groups ofmethods. His grouping relates to that of E7.

Participant E7 used a wider category for systematic analysis than E8 did. His category “structure,systematic analysis” contained 12 methods, including “systematic analysis” and “networktransformations” and methods from two other categories of E8. He used the separate categories “timedomain” and “frequency domain”. These categories were mentioned by E8 as candidates to split thecategory ‘Systematic analysis’. The categories of E8 also had several characteristics in common withthe others. E.g., the methods categorized by E8 in “harmonic signal analysis” were also by E1 put intoone category, called “transform methods”.

The following survey is more complete than from the data alone could be derived, relates to thedistinguished prerequisite and basic knowledge, and includes a few results of interviews not recorded.

The category “forming elementary equations” was a category of concepts and methods containingfundamental knowledge of physics and electricity necessary to model the behavior of circuit elementsinto algebraic equations. The category “forming structural equations” was the equivalent for

150 Appendices

systematically writing down the necessary relations between all the currents and voltages in thenetwork in algebraic equations. A check on the number of equations and variables was available. Bothtypes of equations could be represented either in the frequency domain or the time domain.

With the content of the category “harmonic signal analysis” the dependence of the signals on theindependent (prerequisite) variables could be analyzed. The currents and voltages could be constant(direct current, DC), or be harmonic (alternating current, AC, with frequency ω and phase ϕ) in thetime domain. Transients could be described either in the time domain or in the complex-frequencydomain denoted by s. A transform from time domain to frequency domain and back was included.With the aid of these three categories the domain to be used should be chosen and the algebraicequations be written down.

The next category “systematic analysis, working out equations” should be used to solve thecoupled mathematical equations. Another category dealt with general external properties ofcomplicated circuits: sometimes equivalent circuits could be defined that were much simpler but gavethe same external behavior, i.e. the same input-output solution. A simple example was the replacementof two parallel resistors by one equivalent resistor. These “tricks”, “network manipulations”, or“network transformations” could be used to simplify the networks before the equations were writtendown. They allowed complicated networks to be replaced by an equivalent simpler one.

The first five steps formed a metacognitive strategy because in the step checks on the progress tothe goal were available and the results of step four could often be included in step five.

Next to this strategy for problem solving some other methods were available. There was a category“network theory” treating networks different from the given one. The principle involved was thatsometimes a fast solution to the original problem could be found by considering related networks, e.g.because of symmetry or reciprocity relations.

For a survey of elements and methods along these lines, see Table K.2. In order to present thissurvey, a conceptual change from elements, nodes, voltages, currents, and complete networks, toincomplete networks, ports and signals was necessary. In incomplete networks some sources or loadswere missing, leaving a not specified, open port. Complete networks include all sources and loads andtherefore held a unique solution in terms of the values of the variables voltage and current (Acciani etal., 1989). Incomplete networks can be solved for relations among variables only, but in terms of theirvalues.

The problem approach for Network Analysis found consists of steps, each with an own category ofconcepts and methods. The first five steps can be summarized as:

1. Model the elements in the given network,2. Model the structure of the connections between the elements,3. Model the interactions between the elements through the connections. The models can be

expressed in one of two subdomains (time-domain and frequency-domain) and form coherentexpressions.

4. Solve the resulting coupled mathematical equations. For complicated networks this can be donein a hierarchical way by simplification of the networks:

5. Take parts of the network together as one element.

Appendices 151

Table K.2The basic concepts, types of methods, and methods in a strategy for problem solving in 5 steps, andsome other methods.

Basic concepts Types of methods Names of methods

Steps in a strategy for problem solving

1. Elements Modeling elements Elementary equations2. Structure Modeling structures Structural equations

Kirchhoff’s first & second laws3. Signals Modeling signals in Fourier analysis,

time or frequency domain Fourier synthesis(signal analysis, Superpositiontransforms of representations) Laplace transform

Fourier transform4. Complete networks Mathematical solution methods Formula manipulations

Algebraic eliminationSubstitution methodBranch methodMesh-current methodNode-voltage method

in time domain: Differential equationsCharacteristic equationsConvolution integral

in frequency domain: Transfer function5. Incomplete networks Network transformations Equivalence method

Thevenin’s theoremNorton’s theoremDelta-wye transformationCurrent injection

Other methodsDifferent networks Principles from network theory Symmetry

Reciprocity theoremTellegen’s theoremDuality

Prerequisite concepts Conservation laws Power, energy, chargeReal circuits Measurements Compensation, .... etc.

152 Appendices

Appendix L Salient features of metacognition in the experts verbalprotocols

L.1 Some Characteristics of E8Since the identification of the metacognitive strategy depended heavily on the categorization of

participant E8, some characteristics of E8 are reviewed here.Categorization:The observations revealed that he usually did it just like the others. However, nearly all his cards

were categorized in one round, that took significantly more time than the others used. He used muchtime for the identification and categorization of particular cards. E8 took sometimes a long time toread and think.

PerformanceHe recognized the 9 out of 10 nonstandard formulae, from the first instance on (see Table 5.5).

He characterized his category ‘waste’ as: “I do not understand it, I do not want it, I do not see anysense in it.”

Memory of position of cardsHe remembered the position of individual cards:“ ‘passive’, I saw that before, o sorry, it was ‘active’ I saw before”.CorrectionsHe sometimes corrected himself. First he put ‘Σ un in’ (the expression for the power developed in a

circuit) wrongly together with ‘Tellegen’s theorem’, but 50 minutes later he found ‘ u1′i1″ + u2′i2″ =u1″i1′ + u2″i2′ ‘, identified it as Tellegen’s Theorem, and corrected the former one.

AmbiguitiesAmbiguities with respect to the interpretation of the cards were recorded in the protocols.He said: “ ‘j’ might be a current from a source or the complex square root of -1.” and “ ‘Limited’,

relates to network? current? time?” or “ ‘Soldering connection’, then you can have an ideal solderingconnection, but I will put it in the experimental pile.” or “ ‘Connection’? an ideal connection belongsto network descriptions, but a connection can also belong to experimental work. Today inexperimental work.”

GeneralizationHe mentions variables of a generalized type: “x(t), the signal as a function of time, is in my

thoughts a variable”.

L.2 Phased Recognition of Nonstandard Formulae.For E6 the process of recognition of nonstandard formulae went as follows.The cards in the first pile of seventy cards (like ‘u(i,t)=e0(t)’ ) were put aside in “waste”.The reactions to the card ‘i(u,t)=I0’ in the second pile of seventy , were: “what is that ‘u’? Is that

the unit step? Yes, current as a function of ‘u’? …But I would like to know what this means …. Yes,“waste”, isn’t it? But I would like to know where this is from … ‘u’, that is sometimes the unit step, isit not, but then you wouldn’t put that next to ‘t’. And then you think of ‘i’, then you think of a variableresistor or something like that ….”.

In the fourth pile of cards, about a similar card, he remarked “Now I slowly begin to understandwhat you mean. I would say that it is nót a function of the voltage, so it is a current source”.

And still later, in the fifth pile of cards: “ ‘u(i,t)=E0cos(ω.t + ϕ)’, this is a real source, where thevoltage does not depend on the current. That it is!”

L.3 Category of Nodes or ConnectionsA category of nodes could not be distinguished. Instead, Participant E4 stated “ ‘Connections’

belong to the elements”. E8: “A ‘short’ probably is a component, a strange component”. Theconnection among the components was put into the category of components, as a special component.

L.4 Special CategoriesSome participants made categories not made by others.E1, an expert in under-water vehicles had “sources” (41 items) as a category.

Appendices 153

A characteristic of E2, an expert in electric power networks, was his categories “inputs”,“responses”, “operating characteristics of circuits” and “characteristics of networks, other thanresponses”. Participant E4, an expert in satellite communications and remote sensing, constructed acategory “energy sources” of 10 items.

E4 also had a category “notation or expression” containing words denoting fundamental concepts,including words for mathematical expressions like ‘algebraic elimination’, ‘differential operator’.

L.5 Types of Representation as Aids in Categorization.Sometimes the type of representation was used as an intermediate phase in categorization.For instance, E1 said: “I have taken an easy way in putting all diagrams together rather than trying

to put the diagrams with the various topics... and this pile started there because I did not know quiteother what doing, and then, you see lot of words and circuits to match... I decided not to try to matchthem now, and well, I guessed it might be done later, if need be. Optimizing time.” He said this after30 minutes, and distributed the diagrams afterwards back into the piles.

E2: “Pictures ... put them back in piles, I guess.”Also the other way around could be observed: after 48 minutes time this participant put a lot of

symbols together in a category “algebraic symbols”.

L.6 Views on InstructionThe view of E6 on teaching was: “Teachers have to bring order and logic and fascinating things in

the course.”Another participant (E5) had a different instructional viewpoint on methods. He considered

research as a model for learning. Therefore problems and assignments had to include somecharacteristics of research, some unusual, realistic traits. For instance, like in research, questions hadto be stated with too much information (so the student had to select the information needed), or withmissing information (the student had to fill in information, or to state that the problem could not besolved without some specified information). Such problems could be based on standard types ofproblems, just applications of simple methods, and did not require much calculus. He did not specifyhowever the methods available to solve such problems.

154 Biographical note

Biographical note

Henk Vos werkt sinds 1985 aan de Faculteit der Elektrotechniek van de UniversiteitTwente in Enschede als faculteitsonderwijskundige (tegenwoordig opleidingsonderwijs-kundige) mee aan de daar gebruikelijke voortdurende ontwikkeling van het onderwijs, waarstudenten, docenten, bestuurders en de facultaire gremia nauw in samenwerken. In die functieheeft hij bijgedragen aan vernieuwingen in het onderwijs door het ontwikkelen vanvoorbeelden, door zelf onderwijs te geven, docenten te ondersteunen in het verbeteren vanhun onderwijs, deel te nemen aan facultaire commissies en adviezen te verstrekken aanbesturen. Met name heeft hij als coördinator bijgedragen aan de ontwikkeling van een rodedraad door projecten en practica. In een eerdere functie heeft hij docententrainingen verzorgdaan de Universitas Gadjah Mada te Yogyakarta in Indonesië, eerst in opdracht van de faculteitder Technische Natuurkunde van de Universiteit Twente, later van de faculteit der Wis- enNatuurkunde van de Vrije Universiteit te Amsterdam. Van 1972 tot 1979 heeft hij als docentbijgedragen aan het opzetten en uitvoeren van de opleiding tot natuurkundeleraar aan deNieuwe Leraren Opleiding “de Vrije Leergangen - Vrije Universiteit” te Amsterdam. Hij is in1972 gepromoveerd in de natuurkunde, aan de Vrije Universiteit te Amsterdam, na een studiewis- en natuurkunde aldaar vanaf 1960. Hij heeft een eerstegraads bevoegdheid voor hetonderwijs behaald in 1972. De motivatie tot zijn promotie was dat hij op middelbare-schoolleeftijd -aan de afdeling Gymnasium van het Christelijk Lyceum te Hilversum opLindenheuvel- tot de slotsom was gekomen dat hij wilde begrijpen hoe de natuur in elkaar zat.Later heeft hij begrepen dat ook het begrijpen zelf tot de natuur behoort en bestudeerd kanworden, wat leidde tot deze studie in de onderwijskunde en psychologie.

metacognition in higher education - core

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