the design of a mixed media learning arrangement for the concept of function in grade 8 paul...
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The design of a mixed media learning arrangement for the concept of function in grade 8
Paul Drijvers
Outline1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
Outline1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
1 The research project: fact sheet “Tool Use in Innovative Learning Arrangements
for Mathematics“ Granted by the Netherlands Organisation for
Scientific Research Timeline: 2006 – 2008 Research team:• Peter Boon, programmer / researcher 0.2 fte• Michiel Doorman, researcher 0.4 fte• Paul Drijvers, PI / researcher 0.4 fte• Sjef van Gisbergen, teacher / researcher 0.2 fte• Koeno Gravemeijer, supervisor PM
www.fi.uu.nl/nl/tooluse
Research Questions
1. 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?
Applets
For 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
Theoretical framework
Realistic Mathematics Education
Vygotsky and others on tool use
Instrumental approach to tool use
Cultural-historical activity theory
Research setup and methodology Design research (Gravemeijer): design of an
educational arrangement including• Student worksheets (paper)• Applet with tasks embedded in an electronic learning
environment• Teacher guide with scenario’s of use
Three cycles:• Pilot One class, 2006• First cycle Three classes, 2007• Second cycle Scaling up, 2008
Data: both qualitative (video, screenvideo) and quantitive (student work saved in ELE)
The project: Concretisation
Mathematical 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
Outline1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
2 The concept of function
The function: • Lower secondary level (13 – 15 year olds): a way to
describe a calculation process, a relation, a ‘machine’• Upper 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.
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).
2 The concept of function
Three aspects of the notion of function:
A directed relation from input to output A dynamical process of co-variation A mathematical object with several
representations
Mathematical phenomenology, but how about the didactical phenomenology?
Outline1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
3 The tools
The applet AlgebraPijlen (“AlgebraArrows”): chains of operations, connected by arrows, with tables, graphs and formulas.
3 The tools The Digital Mathematics
Environment (EDM) : • Author: design tasks and
activities, ‘Digital textbook’• Student: work, look back,
improve, continue, ‘Digital worksheet’
• Teacher: prepare, comment, assess, ‘Collection of digital worksheets’
• Researcher: observe, analyse the digital results, ‘Digital database’
The tools and the function concept
The function as a directed relation from input to output: construct and use chains
The tools and the function concept
The function as a dynamical process of co-variation: change input values to study the effect, use trace (graph) and scroll (input/table)
The tools and the function concept
• The function as a mathematical object with several representations: compose chains, construct inverse chains, link representations and study families of functions
Outline1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
4 The educational arrangement
Main 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
The educational arrangement: lesson 1
Group work on three central problems
The educational arrangement: lesson 2
Posters, presentations and ‘living chains’
The educational arrangement: lesson 3 First work in pairs with the applet after introduction
The educational arrangement: lesson 4 Second work in pairs with the applet after plenary
homework review
The educational arrangement: lesson 5 Group work on the ‘matching’ of representations
The educational arrangement: lesson 6 Third applet session in pairs after plenary discussion
The educational arrangement: lesson 7
Final work with the applet and reflections on the concept of function and its notation
Outline1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
5 The data and analysis
1. Data on the use of applets and learning:• Videos of the group work du travail en groupe• Screen video of pairs of students• Students’ work (digital, paper, posters)
2. Data on the use of applets and teaching:• Videos of plenaries in the classroom• Videos of student - teacher interactions
Data analysis ongoing
Digital student work
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
Some prelimenary results
A. Difficulties to phrase the reasoning
B. Interaction paper-pencil / applet fruitful
C. Some lasting conceptual difficulties
D. Interaction teacher – student crucial
A Difficulties to phrase the reasoning
Goes up sidewards Straigt line Further and further away
from 0 All equally steep With the same jumps The point is always
moving It goes up steeper and
steeper It gets higher and higher
B Interaction paper-pencil / applet fruitful
C Some lasting conceptual difficulties
• The work of Juliette, who insists on representing a function with several chains.
Atlas
Online
D Interaction teacher – student crucial
Either 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
Outline1. The project
2. The concept
3. The tools
4. The educational arrangement
5. The data and analysis
6. The conclusions
6 Provisional conclusions
1. How 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 applet More insight needed into the relation
between applet technique and concept development
6 Provisional conclusions2. How can teachers orchestrate tool
use in the classroom community? Technical class management time
consuming and not self-evident! Mixture of scenario’s and media
(ICT and P&P) led to a fruitful arrangement that fosters transfer between media
Demonstration/presentation/class discussion important for reflection and collective instrumental genesis
DME offers means to monitor the learning Teacher interactions that focus both on
concept and technique are effective.
6 Provisional conclusions
Theoretical 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?