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

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?