The design of a mixed media learning arrangement for the concept of function in grade 8

Paul Drijvers

PPT FIsme

OutlineThe project

The concept

The tools

The educational arrangement

The data and analysis The conclusions

PPT FIsme

OutlineThe project

The concept

The tools

The educational arrangement

The data and analysis

The conclusions

PPT FIsme

1 The research project: fact sheetTool Use in Innovative Learning Arrangements for Mathematics Granted by the Netherlands Organisation for Scientific ResearchTimeline: 2006 2008Research team:Peter Boon, programmer / researcher0.2 fteMichiel Doorman, researcher 0.4 ftePaul Drijvers, PI / researcher 0.4 fteSjef van Gisbergen, teacher / researcher0.2 fteKoeno Gravemeijer, supervisorPMwww.fi.uu.nl/nl/tooluse

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Research Questions1. How can applets be integrated in an instructional sequence for algebra, so that their use fosters the learning?

2. How can teachers orchestrate tool use in the classroom community?

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AppletsFor the design products see:

www.fi.uu.nl/wisweb/en/ (primary)www.fi.uu.nl/rekenweb/en/ (secondary) So far: a lot of design / development of games / applets than research on their use

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Theoretical frameworkRealistic Mathematics Education

Vygotsky and others on tool use

Instrumental approach to tool use

Cultural-historical activity theory

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Research setup and methodologyDesign research (Gravemeijer): design of an educational arrangement includingStudent worksheets (paper)Applet with tasks embedded in an electronic learning environmentTeacher guide with scenarios of use

Three cycles:PilotOne class, 2006First cycleThree classes, 2007Second cycleScaling up, 2008

Data: both qualitative (video, screenvideo) and quantitive (student work saved in ELE)

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The project: ConcretisationMathematical subject: the notion of fonction Tools: an applet embedded in an electronic learning environment

Target group: mid high achieving students in grade 8 (14 year olds)

Teaching sequence: 7 lessons of 50 minutes

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OutlineThe project

The concept

The tools

The educational arrangement

The data and analysis The conclusions

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2 The concept of functionThe function: Lower secondary level (13 15 year olds): a way to describe a calculation process, a relation, a machineUpper secondary level (16 18 year olds): a mathematical object, with several faces, which one can consider as membre of a family, or that can be submitted to a higher level procedure such as differentiation.

Intention: bridge the gap between the two, facilitate the transition and promote a rich conception of the notion of function.

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2 The concept of function

The function is a special kind of dependence, that is, between variables which are distinguished as dependent and independent. (...) This - old fashioned - definition stresses the phenomenologically important element: the directedness from something that varies freely to something that varies under constraint. (Freudenthal, 1983, p. 496).

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2 The concept of functionThree aspects of the notion of function:

A directed relation from input to outputA dynamical process of co-variationA mathematical object with several representations

Mathematical phenomenology, but how about the didactical phenomenology?

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OutlineThe project

The concept

The tools

The educational arrangement

The data and analysis The conclusions

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3 The toolsThe applet AlgebraPijlen (AlgebraArrows): chains of operations, connected by arrows, with tables, graphs and formulas.

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3 The toolsThe Digital Mathematics Environment (EDM) : Author: design tasks and activities, Digital textbookStudent: work, look back, improve, continue, Digital worksheetTeacher: prepare, comment, assess, Collection of digital worksheetsResearcher: observe, analyse the digital results, Digital database

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The tools and the function conceptThe function as a directed relation from input to output: construct and use chains

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The tools and the function conceptThe function as a dynamical process of co-variation: change input values to study the effect, use trace (graph) and scroll (input/table)

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The tools and the function conceptThe function as a mathematical object with several representations: compose chains, construct inverse chains, link representations and study families of functions

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OutlineThe project

The concept

The tools

The educational arrangement

The data and analysis The conclusions

PPT FIsme

4 The educational arrangementMain ideas:

Mixture of working formats: group work, individual work, work in pairs with the computer, plenary teaching and discussion

Mixture of tools: paper pencil, posters, cards, applet, DME, both in school and at home First step: a hypothetical learning trajectory

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The educational arrangement: lesson 1Group work on three central problems

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The educational arrangement: lesson 2Posters, presentations and living chains

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The educational arrangement: lesson 3First work in pairs with the applet after introduction

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The educational arrangement: lesson 4Second work in pairs with the applet after plenary homework review

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The educational arrangement: lesson 5Group work on the matching of representations

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The educational arrangement: lesson 6Third applet session in pairs after plenary discussion

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The educational arrangement: lesson 7Final work with the applet and reflections on the concept of function and its notation

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OutlineThe project

The concept

The tools

The educational arrangement

The data and analysis

The conclusions

PPT FIsme

5 The data and analysisData on the use of applets and learning:Videos of the group work du travail en groupeScreen video of pairs of studentsStudents work (digital, paper, posters)

Data on the use of applets and teaching:Videos of plenaries in the classroomVideos of student - teacher interactions

Data analysis ongoing

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Digital student work

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Coding in Atlas tiCODES 13 15 16 17----------------------------------------docent-huishoudelijk 10 11 11 7 39docent-inhoudelijkeh 5 10 8 8 31docent-proceshulp 17 8 3 3 31docent-reflectiehulp 4 10 10 6 30docent-technischehul 8 4 3 4 19functie-dynamiek-cov 5 1 2 4 12functie-globale-in-u 0 6 7 7 20functie-locale-in-ui 3 2 2 1 8functie-object-repre 1 3 7 2 13functie-variabele 2 4 4 2 12techniek-in-uitzoome 3 5 5 1 14techniek-ketting-mak 3 5 2 0 10techniek-ketting-rec 1 0 0 0 1techniek-scrollen-ta 1 3 1 3 8techniek-variatie-in 2 5 3 4 14techniek-verbinden 1 1 0 1 3----------------------------------------Totals 68 93 87 62 310

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Some prelimenary resultsDifficulties to phrase the reasoning

Interaction paper-pencil / applet fruitful

Some lasting conceptual difficulties Interaction teacher student crucial

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A Difficulties to phrase the reasoningGoes up sidewardsStraigt lineFurther and further away from 0All equally steepWith the same jumpsThe point is always movingIt goes up steeper and steeperIt gets higher and higher

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B Interaction paper-pencil / applet fruitful

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C Some lasting conceptual difficultiesThe work of Juliette, who insists on representing a function with several chains.

Atlas

Online

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D Interaction teacher student crucialEither in individual contact and in plenary discussions

The teacher plays an important role while orchestrating the discussion on, the reflection on, and the convergence of techniques and thinking within the class

PPT FIsme

OutlineThe project

The concept

The tools

The educational arrangement

The data and analysis

The conclusions

PPT FIsme

6 Provisional conclusionsHow can applets be integrated in an instructional sequence for algebra, so that their use fosters the learning?

Global learning trajectory OK, but which problem does the function concept solve for the students?Dynamic did not work -> change appletMore insight needed into the relation between applet technique and concept development

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6 Provisional conclusionsHow can teachers orchestrate tool use in the classroom community?Technical class management time consuming and not self-evident!Mixture of scenarios and media (ICT and P&P) led to a fruitful arrangement that fosters transfer between mediaDemonstration/presentation/class discussion important for reflection and collective instrumental genesisDME offers means to monitor the learningTeacher interactions that focus both on concept and technique are effective.

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6 Provisional conclusionsTheoretical questions:

Is the framework of instrumental genesis, with its stress on the relation between technical and conceptual development, is useful in case the tool is as simple as an applet?

How to observe and to analyse the orchestration by the teacher?

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I rather am a design researcher than a designerJuliette 1-2, 1-3, 1-7 en 2-3 voor dubbele kettingenJuliette 1-5, 1-6, 1-8 voor de truuk van het aftrekken van gratis belminuten bij on-line inloggen als michiel.