MULTIPLE DIMENSIONS INVOLVED IN THE DESIGN OF

TEACHING LEARNING SITUATIONS TAKING ADVANTAGE OF

TECHNOLOGY

EXAMPLES IN DYNAMIC MATHEMATICS TECHNOLOGY

Colette Laborde, Colette Laborde, IUFM & University Joseph Fourier,IUFM & University Joseph Fourier,Grenoble, FranceGrenoble, FranceColette.Laborde@imag.frColette.Laborde@imag.fr

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

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

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)

WHAT ARE DYNAMIC WHAT ARE DYNAMIC MATHEMATICS ENVIRONMENTS?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.

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

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

ORDINARY USE OF DYNAMIC GEOMETRY IN CLASSROOMSFirst instrumentation of Dynamic Geometry by math teachers

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 spectator of 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

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

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 taughtNo need of instrumentation by students Interaction between students and software

avoided or strongly controlled

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)

DESIGN OR CHOICE OF TASKS BASED ON DYNAMIC GEOMETRY BY TEACHERS

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?

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?

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

KINDS OF DG USE IN EXERCISESIn 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

AN EXERCISE OF A TEXTBOOK GRADE 61) Mark two points A and B. Use the tools : “Point” “Segment”,“Perpendicular line”, “Polygon” and“Hide/Show” to construct a rectangle ABCD.

2) Drag point A or point B. Write down what you observe.

A B

CD

Dragging only A and B provides an incomplete invalidationof the constructions by eye.

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 ?

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

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

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

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

DIMENSIONS (2/2)

DidacticOpen 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

INTERVENTIONS IN THE CLASSROOM

Even if the teacher uses ready made dynamic activities, his/her role is critical in the solving process

THE MISSING WHEEL EXAMPLETHE 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

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”

BRUNER’S SIX SCAFFOLDING BRUNER’S SIX SCAFFOLDING FUNCTIONSFUNCTIONS

(1) Recruiting the learner’s interest (2) Reducing degrees of freedom (3) Maintaining direction (4) Marking critical features (5) Controlling frustration (6) Demonstrating

IN TEACHER PREPARATION

To enlarge the scope of possible tasks making use of DG, teacher preparation is needed

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 perspectiveThey must solve situations proposed by the

teacher educatorThey must analyze them from a pedagogical

perspective: what is the contribution of DG to learning in the task? And learning what?

Session 2 Pedagogical use of DG

Session 1 Initiation to Cabri

StudentTeachers adapt or create tasks - 2

Student Teachers adapt or create tasks - 3

Session 3 Didactique

Schedule of observations of Schedule of observations of preservice teachers preservice teachers

designing tasksdesigning tasks

Student Teachers adapt or create tasks - 1

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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

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

AN EXAMPLE OF ADAPTATION OF TASK (TAPAN 2006)

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?

Preservice teachers were proposed the following task

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’. 

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)

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.

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

“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

FROM THE SAME DIFFICULTY EMERGES A NEW TASK - EXAMPLE OF ANOTHER PAIR

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 moveG but it moves, what are you saying?E the real axis does not moveG precisely no no … I don’t think that… no noE 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)

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.

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

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

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