Colette Laborde, IUFM & University Joseph Fourier, Grenoble, France Colette.Laborde@imag.fr

1. MULTIPLE DIMENSIONS INVOLVED IN THE DESIGN OF TEACHING LEARNING SITUATIONS TAKING ADVANTAGE OF TECHNOLOGY EXAMPLES IN DYNAMIC MATHEMATICS TECHNOLOGY Colette Laborde, IUFM & University Joseph Fourier, Grenoble, France Colette.Laborde@imag.fr

2. TOOLS IN TEACHING • Human activity resorts to tools • In mathematics, • a long tradition of paper and pencil • more recently, ICT: calculators and computers • Tools in teaching activity have a dual role (Two levels of instrumentation, cf instrumental approach Rabardel) • they can be used to carry out a mathematical activity • they can be used as pedagogical tools • ICT has been recently introduced and as such may cause a perturbation in the habits of the teacher

3. FOCUS OF THE TALK • Adaptation of the teacher to the use of ICT as a pedagogical tool • In the design of teaching/learning situations • Window on • Teachers’ conceptions of learning • Types of required knowledge

4. CONTENT OF THE TALK • In what part of teaching activity is the new tool used? • Design or choice of tasks by the teacher • Interventions of the teacher in the classroom • Different kinds of knowledge required from the teacher: illustration with preservice teachers • Chosen ICT: Dynamic mathematics, Cabris (II Plus, 3D, Cabri Elem)

5. WHAT ARE DYNAMIC MATHEMATICS ENVIRONMENTS? • Computer environments where the user (student, teacher…) can interact directly with mathematical objects on a computer screen • Objects are not inert but “behave” mathematically to the actions of the user (co-action, Hegedus & Moreno) • Variation, variables, co-variation are reified in this kind of environment, they become central and provide new ways of approaching mathematics focusing on variation even outside calculus.

6. WHAT IS THE FOCUS WHEN DRAGGING? • Variable x A, Variations of the Object O(x) by looking • at each state when x vary in A (robust constructions) • Ex: angle inscribed in a semi-circle depends on x • Focus: on the invariants among all individuals

7. WHAT IS THE FOCUS WHEN DRAGGING? • at the set of all states of O(x) obtained as a trajectory (Trace) • Ex: trajectory of the vertex of A of triangle ABC with A being a right angle • Focus: on the family of individuals as an object and on the way O(x) varies • at changes of O(x) when x vary in A • more global view of relationships among elements of O(x) : what changes, what does not change? Why? • at a specific state obtained in the change of O(x) (soft constructions) • Ephemeral state

8. ORDINARY USE OF DYNAMIC GEOMETRY IN CLASSROOMS First instrumentation of Dynamic Geometry by math teachers

9. PREVALENT USE OF DG IN CLASSROOMS: DEMONSTRATION USE • Convergence of research studies from various countries. • The most immediate use by teachers is just “showing” geometrical theorems: teachers manipulate themselves or the students are allowed to have a restricted manipulation (dragging a point on a limited part of line) • It would take a long time in order for them to master the package and I think the cost benefit does not pay there… And there is a huge scope for them making mistakes and errors, especially at this level of student… and the content of geometry at foundation and intermediate level does’nt require that degree of investigation » (quoted by Ruthven et al.) • The student is a spectatorof beautiful figures (showing the power of the software) or of properties part of the content of the curriculum (Belfort and Guimaraes in a study of resouces written by teachers in inservice sessions) • Focus on invariants

10. RESOURCES AVAILABLE FOR THE TEACHERS • In France, institutional request to use DG from the first year of secondary school since 1996 • More than 1/3 of the schoolbooks propose a CDROM for the teacher (Caliskan 2006) • with mainly the files of the figures of the book that can be animated by dragging or ready made constructions that can be replayed step by step • Demonstration use prevails in these CDROMs • Internet resources: same observation

11. MINIMAL PERTURBATION • This demonstration use offers a minimal perturbation in the teaching system with regard to the state of the system without technology • It meets two constraints of the didactic system: time and content to be taught • No need of instrumentation by students • Interaction between students and software avoided or strongly controlled

12. UNDERLYING CONCEPTION ABOUT LEARNING • No awareness of the fact that what is apparent and visible for teachers may be not visible for students • Avoiding mistakes from students • Facing directly the student to the correct and official formulation of the theorem • No view of a construction of a partially correct knowledge by the learner developing over time • Learning time does not differ from teaching time • Technology is only an amplifier and not a conceptual reorganizer (Pea)

13. DESIGN OR CHOICE OF TASKS BASED ON DYNAMIC GEOMETRY BY TEACHERS

14. IMPORTANCE OF TASKS stressed by research in maths education: “importance of tasks in mediating the construction of students’ scientific knowledge” (Monaghan) • Central role in several theoretical frameworks about teaching and learning processes • even if they do not use the word “task” itself • Constructivist and socio-constructivist approach: problematic tasks for the learners • Problem is the source and criterion for knowledge (Vergnaud) • Learning comes from adapting to a new situation creating a perturbation (Brousseau) • Mathematical knowledge as providing an economical solving process of a problem • For each piece of knowledge, what are the problems for which it provides an efficient solution?

15. A PROFESSIONAL ACTIVITY • Designing or choosing tasks is a teacher professional activity (Robert & Rogalski 2005) • It is a complex activity involving several dimensions • Epistemological dimension: choosing • features of mathematical knowledge • how to use them • Cognitive dimension: what kind of learning does promote the task? • Didactic dimensions: • How does the task fit the constraints and needs • of the teaching system, • of the curriculum, • of the specific class and of its didactic past?

16. HOW DOES A TEACHER USUALLY DESIGN TASKS IN PAPER AND PENCIL ENVIRONMENT? • Resources are usually available in textbooks for tasks in paper and pencil environment • In France, the choice of a textbook by teachers is essentially driven by the number of exercises • “Bricolage” (Perrenoud) from the available resources • Very few teachers design tasks from scratch

17. KINDS OF DG USE IN EXERCISES In textbooks (Caliskan) • A figure has to be constructed by students • construction steps are given • Question: drag an element and • observe that… • or tell what you observe • is this property always satisfied? • Possible additional question: Justify • Sometimes only a construction task without mention of dragging or incomplete control by dragging • Again the role of the dragging is to focus on robust constructions and invariants

18. AN EXERCISE OF A TEXTBOOK GRADE 6 Mark two points A and B. Use the tools : “Point” “Segment”,“Perpendicular line”, “Polygon” and “Hide/Show” to construct a rectangle ABCD. D C B A 2) Drag point A or point B. Write down what you observe. Dragging only A and B provides an incomplete invalidation of the constructions by eye.

19. ANALYSIS OF A TASK • Choosing a task for supporting learning requires • Considering all possible solving ways depending • on prior knowledge of students • on available tools • on available feedback (possibly to be interpreted by students) • “Milieu” (Brousseau) • A task may be completely changed when moving from an environment to another one • Compare “Calculate 117 + 29” • In a paper and pencil technology • In the purely mental “technology” • On a calculator • Knowlege required in each case ? Learning ?

20. DIFFICULTY IN CHOOSING OR ADAPTING A TASK • It is difficult for teachers to carry out an • A priori analysis of the possible solving strategies of students • With restricted experience and knowledge of usual behaviors of students in a technology environment

21. A GAP BETWEEN RESEARCH AND USUAL PRACTICE • In usual practice • no tasks such as those mentioned in research, in which • Dragging is a means of exploring the problem • Dragging is the source of the problem • no other uses of dragging • Objects as trajectories • Observations of changes

22. THREE CATEGORIES OF TASKS IN CABRI • Cabri as facilitating the task while not changing it conceptually (visual amplifier, provider of numerical data) • Similar problem • Same available tools • Cabri modifies the ways of solving the task • The task itself is grounded in Cabri and could not be given outside of the environment • Black boxes tasks • Tasks requiring a dynamic linking between different registers

23. DIMENSIONS (1/2) • Epistemological: • Geometry is permeated with paper and pencil (discrete use) • Some teachers have difficulties in accepting the drag mode: “this point” should refer to a fixed point • DG software is often called “geometric construction software” (as in the French syllabus) and not DG software • Proof is only related to formal proof and not to mathematical experiments or exploration • Cognitive: • Implicit assumptions about learning are not necessarily constructivist

24. DIMENSIONS (2/2) • Didactic • Open ended tasks as used in research • are too long, favour a larger scope of students strategies • increase the possibilities of instrumental problems • Instrumental is seen as independent of mathematics • Incomplete instrumentation by teachers • in particular of dragging

25. INTERVENTIONS IN THE CLASSROOM Even if the teacher uses ready made dynamic activities, his/her role is critical in the solving process

26. THE MISSING WHEEL EXAMPLE • Experiment performed at grades 4, 5 and 6 within project MAGI (Better Learning Geometry with ICT) • A priori analysis of possible solving strategies: 3 possible strategies, • Two visual strategies • The successful strategy may not emerge • The a priori analysis leads to plan an intervention of the teacher after some trials by students

27. TEACHER’S INTERVENTIONS • Two possible theoretical interpretations of the role of the teacher • Analysis in terms of scaffolding (Bruner) • Analysis in terms of didactic situations • Feedback coming from the “milieu” invalidates visual or semi visual strategies • A geometrical interpretation of the feedback may be difficult for students • The teacher provides an interpretation of the feedback • without giving the solution: change of the “milieu” by the teacher • In order to avoid effects of the “didactic contract”

28. BRUNER’S SIX SCAFFOLDING FUNCTIONS • (1) Recruiting the learner’s interest • (2) Reducing degrees of freedom • (3) Maintaining direction • (4) Marking critical features • (5) Controlling frustration • (6) Demonstrating

29. IN TEACHER PREPARATION To enlarge the scope of possible tasks making use of DG, teacher preparation is needed

30. PRESERVICE TEACHERS AS A WINDOW ON THE COMPLEXITY OF THE DESIGN OF TASKS (TAPAN 2006) • Three preparation sessions • First session • introduction to the use of Cabri • and presentation of many examples of situations commented by the teacher educator • Second session: How to use Cabri from a pedagogical perspective • They must solve situations proposed by the teacher educator • They must analyze them from a pedagogical perspective: what is the contribution of DG to learning in the task? And learning what?

31. 13 Schedule of observations of preservice teachers designing tasks Session 1Initiation to Cabri Session 2Pedagogical use of DG Session 3Didactique [ ] [ ] [ ] StudentTeachers adapt or create tasks - 2 Student Teachers adapt or create tasks - 3 Student Teachers adapt or create tasks - 1

32. ADAPTION OR CREATION OF TASKS • Involving the notion of reflection and axial symmetry • After the first session, student teachers • proposed in Cabri exactly the same task as proposed in paper and pencil • Very little use of dragging in their tasks • After the second session • Dragging for validating/invalidating and for conjecturing planned in their tasks • Contribution of DG contrasted with paper and pencil tasks • They started from reference tasks (given either in paper and pencil or Cabri) to create new tasks: very seldom new tasks

33. PRESERVICE TEACHERS - CONCLUSIONS • Presenting a variety of tasks to them is not enough for preparing teachers • They must themselves manipulate and analyze the tasks to take advantage of them When they design tasks • The full pedagogical use of dragging requires time • The creation of really new tasks is very rare • Evolution over time and after the second session in the design of tasks

34. AN EXAMPLE OF ADAPTATION OF TASK (TAPAN 2006) Preservice teachers were proposed the following task Below is presented a task for learning reflection in paper and pencil environment at grade 6. 1) Would you propose it as such to your pupils ? 2) Analyse the different ways of using Cabri for such a situation. What could be the contribution of dynamic geometry?

35. THE INITIAL PAPER AND PENCIL TASK TO BE MODIFIED In the figure below, triangle T’ is reflected from triangle T. Construct the line of the reflection transforming T into T’. 

36. INITIAL TASK • Very common in textbooks and in classroom • Not demanding in terms of knowledge • The reflection line is vertical • The line joining one point and its image is drawn • It is enough to know that the midpoint of two symmetrical points is on the axis • The midpoints are on an already drawn line • The task can even be solved using the only perception (feeling of balance)

37. REACTIONS OF PRESERVICE TEACHERS • The task was chosen because of the unusual behavior of the reflection line with respect to the “theorem in action” in DG : “a constructed object depending on a moving object must move when this latter moves” • But the reconstructed reflection line does not move when moving the triangle • Contrasting two pairs of students when faced with this difficulty • Method: Facing a teacher or a student with a difficulty or with a situation outside the routine situations is a good method to know more about them.

38. SPECIFIC DG SOLUTION BASED ON A DIFFERENT USE OF DRAGGING • Soft construction • Coincidence of a point and its reflected image by dragging • The reflection line can be determined by two such coinciding points

39. “IT’S TOO HARD IN CABRI” • A pair did not criticize the paper and pencil task and then in Cabri: “It is terribly harder in Cabri without squared paper” • They used the grid in Cabri • They found that it is possible to find the axis by counting the squares just as in paper and pencil • They decided to keep the same task but did not see any contribution of Cabri • Little exploration of the figure • Lack of didactical knowledge and belief that a task must not be too hard

40. FROM THE SAME DIFFICULTY EMERGES A NEW TASK - EXAMPLE OF ANOTHER PAIR • They first thought that in a robust construction, the reflection line should move when the triangle moves and set value on Cabri for this feedback through dragging. But student E discovered that even in a correct and robust construction, the reflection line does not move. E there is still something that worries me, honestly there is something that worries me it does not move… cause if she (a student) manages, she will move the axis and say yes “it is almost like that” (E makes a line by eye and adjusts it visually) G why? E because the axis does not move G but it moves, what are you saying? E the real axis does not move G precisely no no … I don’t think that… no no E or some of them will become aware that … when moving this triangle they will think « it is passing there » (when one vertex and its image are superimposed) and no matter how much the teacher does, no matter how much the teacher moves, the axis is right (he drags the vertices of the triangle)

41. A FIRST MODIFICATION • They soon found a value to this solution making use of invariant points (mathematical interpretation) • They discussed whether this solution could be found by students of this age (cognitive and didactic analysis) • G: yes but which property do they use ? If the kid knows how to justify • E: yes but there • G: why when they really touch, it means that the axis is passing through A, if it is really justified • E: yes but at grade 6 don’t dream • G: precisely at grade 6 they won’t do what you did. Frankly I would be surprised • E: at the beginning of grade 6, on the other hand when folding they see also that every point on the axis remains on the axis

42. A SECOND MODIFICATION • Then they thought that students must know how to find the axis by using the classical method making use of the perpendicular bisector (reference to a demand of the curriculum) • Finally they decided to ask students to provide two methods for finding the hidden line (didactical decision not part of the usual didactic contract) • For them, contribution of Cabri: existence of two possible methods • Constructing the line as perpendicular bisector • Method of invariant points made possible

43. RESORTING TO SEVERAL TYPES OF KNOWLEDGE • Three types of knowledge strongly intertwined in the design of this task • Mathematical knowledge • Knowledge of Cabri (the axis does not move) • Didactical knowledge • about students’ knowledge • about the curriculum • and about the ways of using Cabri for fostering learning • Therefore teacher education must establish relationships between all these components