An ICT-rich learning arrangement for the concept of function in grade 8: student perspective and teacher perspective Paul Drijvers Freudenthal Institute for Science and Mathematics Education Utrecht University Universität Köln,

Outline 1. 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

Outline 1. 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 

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?

Applets For collections of applets see:  (primary)  (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

Outline 1. 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 concept Two 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?

Outline 1. 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.AlgebraPijlen

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 concept a.The function as a dependency relation from input to output: construct and use chains

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

Outline 1. 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 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 trajectoryhypothetical 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

Outline 1. 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)Atlas  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 independent input values Graphic representation Table of dependent output values Chain of operations Formula

Outline 1. 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 orchestrations Main orchestrations observed: 1. Technical demo 2. Explain the screen 3. Link screen board 4. Discuss the screen 5. Spot and show: exampleexample 6. Sherpa at work: exampleexample

Orchestrations by teacher Orchestration typeTeacherA cycle1 TeacherA cycle2 TeacherB cycle2 TeacherC cycle3 TeacherA cycle3 Total Technical-demo Explain-the-screen Link-screen-board Discuss-the-screen Spot-and-show Sherpa-at-work Total

BRelations 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 – student Different types of interactions:  Content of interaction: Mathematical meaning Technical meaning Situational meaning Interaction-meaning-technical  Form of interaction: Revoicing Questioning Answering

Outline 1. 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 learning 1. 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 teaching 2. 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 theory Theoretical 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?