Paul Drijvers Freudenthal Institute for Science and Mathematics Education Utrecht University

An ICT-rich learning arrangement for the concept of function in grade 8: student perspective and teacher perspective

Paul DrijversFreudenthal Institute for Science and Mathematics EducationUtrecht University

Universität Köln, 20.01.09www.fi.uu.nl

Outline1. The project

2. The function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

Outline1. The project

2. The function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

1 The project Project name: Tool Use in Innovative Learning

Arrangements for Mathematics Granted by the Netherlands Organisation for Scientific

Research NWO Timeline: 2006 – 2008 Research team:

• Peter Boon, programmer / researcher• Michiel Doorman, researcher • Paul Drijvers, PI / researcher • Sjef van Gisbergen, teacher / researcher• Koeno Gravemeijer, supervisor• Helen Reed, master student

www.fi.uu.nl/tooluse/en

Project theme: math & technology Integrating technology in mathematics education seems

promising

But optimistic claims are not always realized!

Technology for ‘drill & practice’ or also for conceptual development?

If yes, how to achieve this?

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?

AppletsFor collections of applets see:

www.fi.uu.nl/wisweb/en/ (primary)

www.fi.uu.nl/rekenweb/en/ (secondary)

So far: rather much design / development of games / applets than research on their use in the classroom

Project concretisation Mathematical subject: the concept 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: 7-8 lessons of 50 minutes

Outline1. The project

2. The function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

2 The function conceptTwo quotes:

“The very origin of function is stating and producing dependence (or connection) between variables occurring in the physical, social, mental world (i.e. in and between these worlds).”(Freudenthal, 1982)

“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)

Function definitions

"a quantity composed in any of [a] variable and constant" (Bernoulli, 1718)

an "analytic expression" (Euler, 1747)

f is a function from a set A to a set B if f is a subset of the Cartesian product of A (the domain) and B (the range), so that for each a in A there exists exactly one b in B with (a, b) in f. (Dirichlet-Bourbaki, 1934)

How useful are these definitions for lower secondary mathematics education?

The ‘function gap’

Lower secondary level (SI, 13 – 15 year olds): a way to describe a calculation process, an input-output ‘machine’ for numerical values.

Upper secondary level (SII, 16 – 18 year olds): a mathematical object, with several representational faces, which one can consider as membre of a family, or that can be submitted to a higher level procedure such as differentiation.

Intentions and didactical ideas

Intentions: To bridge the gap between the two, facilitate the

transition and promote a rich conception of the notion of function including both the process and the object view.

Relevant ideas from mathematics didactics: Vinner (1983), Vinner & Dreyfus (1989):

Concept definition and concept image Janvier (1987):

Multiple representations – formula, graph, table Sfard (1991): Process – object duality Malle (2000): Function as assignment and as co-variation

Proces-object duality (Sfard, 1991):

Operational conception: processes

Structural conception: objects

In the process of concept formation, operational conceptions precede the structural

Three aspects of the notion of function:

a. Dependency relation from input to output

b. Dynamical process of co-variation

c. Mathematical object with several representations

Mathematical phenomenology or didactical phenomenology?

Outline1. The project

2. The function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

3 The ICT tools (1) Freudenthal (1983) mentions activities with arrow chains

as one means to approach the function concept ICT tool: The applet AlgebraPijlen (“AlgebraArrows”):

chains of operations, connected by arrows, with tables, graphs and formulas.

3 The ICT tools (2)The Digital Mathematics Environment (DME) :

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 concepta. The function as a dependency relation from input to

output: construct and use chains

The tools and the function conceptb. 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 conceptc. 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 function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

4 Learning 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

Learning arrangement: lesson 1 Group work on three central problems

Learning arrangement: lesson 2 Posters, presentations and ‘living chains’

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

Learning arrangement: lesson 4 Second work in pairs with the applet after plenary

homework review

Learning arrangement: lesson 5 Group work on the ‘matching’ of representations

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

Learning arrangement: lesson 7 (+8) Final work with the applet and reflections on the concept

of function and its notation

Outline1. The project

2. The function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

5 Some results on learning

A. Difficulties to express the reasoning

B. Mixed media approach fruitful (paper-pencil <-> applet)

C. Form-function shift as a model for describing conceptual change in ICT-rich learning

A Difficulties to express the reasoning

Students explaining dynamic co-variation:

“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 Mixed media approach fruitful

C Form-function shift

Form-function shift as a model for describing conceptual change in ICT-rich learning

Example: task 1.6

The work of two girls Their work ‘real time’: Atlas (clip 59:9) Their final product:

Hypothesis: form-function shift (1)

A form-function shift (Saxe, 1991) takes place concerning the functions that arrow chains have for the student:

Initially, the arrow chain represents a calculation process, and is a means to calculate the output value once the input value is given. The arrow chain helps to organize the calculation process.

Evidence: students make new chains for the same calculation:

Hypothesis: form-function shift (2) Later, the arrow chains become object-like entities that

represent functional relationships and can be compared and reasoned with.

Verfication of the hypothesis: Task 4.1

The work of the two girls

Results of three classes

Theoretical interest Form-function shift here might be a suitable construct to

explain conceptual change when there is little technical development in the use of the ICT tool.

Instrumental genesis, which was one of the points of departure of this study, seems to be more appropriate for more versatile technological tools.

Table of independentinput values

Graphic representation

Table of dependent output values

Chain of operationsFormula

Outline1. The project

2. The function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

6 Some results on teaching

A. Different whole-class orchestrations

B. Relations with teachers’ views on teaching and learning

C. Interaction teacher – student

A Different whole-class orchestrationsMain orchestrations observed:

1. Technical demo2. Explain the screen3. Link screen board4. Discuss the screen5. Spot and show: example6. Sherpa at work: example

Orchestrations by teacher

Orchestration type TeacherA cycle1

TeacherAcycle2

TeacherB

cycle2

TeacherCcycle3

TeacherA cycle3

Total

Technical-demo 5 3 2 7 5 21

Explain-the-screen 0 0 0 7 1 8

Link-screen-board 3 0 6 0 3 12

Discuss-the-screen 4 4 3 1 2 13

Spot-and-show 0 1 12 2 2 19

Sherpa-at-work 2 7 0 0 1 10

Total 14 15 23 17 14 83

B Relations with teachers’ views on teaching and learning

Teacher A: “…so you could discuss it with the students using the images that you say on the screen, […] it makes it more lively…”

Teacher B:“I use the board to take distance from the specific ICT-environment, otherwise the experience remains too much linked to the ICT”

Teacher C:“I am a typical teacher for mid-ability students, and these students need clear demonstrations and explanations”

C Interaction teacher – studentDifferent types of interactions: Content of interaction:

• Mathematical meaning• Technical meaning• Situational meaning• Interaction-meaning-technical

Form of interaction:• Revoicing• Questioning• Answering

Outline1. The project

2. The function concept

3. The ICT tools

4. Learning arrangement

5. Some results on learning

6. Some results on teaching

7. Conclusion

7 Conclusion on learning1. How can applets be integrated in

an instructional sequence for algebra, so that their use fosters the learning?

Global learning trajectory works, but which problem does the function concept solve for the students?

Mixed media approach fruitful Subtle relation between applet technique and concept development (instrumentation, FFS)

Form-function shift as a model for describing conceptual change in ICT-rich learning

7 Conclusion on teaching2. How can teachers orchestrate tool

use in the classroom community?

Technical class management not self-evident! Mixture of whole-class orchestrations, related to teachers’

views Demonstration/presentation/class discussion important for

reflection and collective instrumental genesis DME offers means to monitor the learning The teacher important for orchestrating discussion /

reflection / convergence of techniques and thinking

7 Conclusion on theoryTheoretical questions:

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