The design of tasks taking full advantage of dynamic geometry:

what kinds of knowledge does it

require from teachers?

Colette LabordeUniversity Joseph Fourier

Grenoble, FranceColette.Laborde@imag.fr

Shared research claims

• There are several technological environments very promising in terms of learning

• The usual teaching practice does not take full advantage of these possibilities

• A critical element in the integration of technology into usual teaching is the teacher (Artigue, Bottino and Furinghetti, Guin and Trouche, Monaghan, Ruthven, Sutherland…)

Focus of the talk

• Technology: Dynamic geometry• Design of tasks by teachers• Based on research literature• and on two research and development projects in Grenoble: writing scenarios with Cabri geometry meant for teachers of

• high school• of elementary and middle school (MAGI)

– within the French curriculum

Importance of tasks (1/2)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)

Importance of tasks (2/2)

In the praxeological approach (Chevallard)

• Knowledge used in an institution is characterized by a system of – tasks, – techniques to solve the tasks,– justifications of the techniques, – and theories from which justifications may come

A professional activity

• Designing tasks is a teacher professional activity (Robert)

• 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 and institutional 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?

• 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

Designing technology based tasks is problematic

• Designing technology based tasks is out of the range of the ordinary activities of teachers– Limited number of such tasks in textbooks– Limited number of resources

• Including the new element “technology” is not just adding it but affecting all dimensions of the design activity

• And introducing a hidden dimension: the instrumental dimension

Instrumental dimension• A tool affects the way of solving a task• A tool is not transparent but must be appropriated by the user

• The user constructs schemes of utilization of the tool to perform tasks with the tool

• Construction process of these schemes: instrumental genesis (Rabardel)

• Using a tool shapes the way to do mathematics and consequently may affect mathematical knowledge constructed by the user

Three possibilities

• Three possibilities for the design of tasks – using ready made tasks for technology

– adapting tasks designed for paper and pencil

– designing his/her own tasks

In French schoolbooks (Caliskan, PhD thesis

Paris)

French Schoolbooks

Without DG

With DG

Edition 11996-1999

19 13

Edition 22000-2004

5 32

There is an institutional request in the French syllabus for using DG environments at each level since 1996

DG is present in French schoolbooks for middle school

But a “weak” use of DG• 5% of proposed activities have recourse to DG

• DG is mainly present in exercises– 11% in presented activities– 5% in the exposition of the content– 84% in exercises

• More than 1/3 of the schoolbooks propose a CDROM for the teacher– 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

Demonstration use• Prevailing in the resources given to teachers with textbooks

• Also mentioned by other research studies 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)

• Sometimes the students must formulate the theorems

Reasons invoked by teachers

• Benefit for teachers – Facing the students– More comfort (no pain in arms and back)– Clean, precise and beautiful figures– Saving construction and time

• Benefit for students– Saving construction and time– Multiplying cases– Amplified Visualization

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– The tasks given to students remain the same as in paper and pencil environment

Kinds of DG use in exercisesIn schoolbooks or tasks proposed by teachers

(Caliskan)• A figure has to be constructed by students• Question: drag an element and

• tell what you observe• is this property always satisfied?• make a conjecture

• sometimes discrete use of DG: – construct several points, are they on a line? – measure for several cases and by calculation find a numerical constant

• Possible additional question: Justify• Construction tasks are in smaller number• No tasks such as those mentioned in research

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)

• 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– of dragging

Modes of instrumental integration of DG

(Assude)• Instrumental integration refers to the way instrumental and mathematical dimensions are organized and related to each other in teaching

• Assude distinguishes between different modes– Instrumental and mathematical are treated independently by the teacher

– Mathematical tasks are given calling for• either the construction of new instrumentation• or both new mathematical knowledge and new instrumentation

• Such coordination requires several types of knowledge from teachers

• It was observed in research and development projects or after inservice or preservice education

R&D project of writing teaching scenarios with Cabri for high school• Various profiles in the team of each project– Teachers

• Experienced teachers familiar with the use of technology

• Experienced teachers novice with the use of technology

• Novice teachers with different degree of familiarity with technology

– Teacher educators– Researchers

Several phases in the design of the scenario

• First phase: one member wrote a scenario project and experimented it in a class

• Second and further phases: discussion in the team, possibly re-experimentation, and modification

• We assumed that the two kinds of feedback to the design of scenarios was critical for their evolution– Experimenting the tasks in classroom– Working in team and discussing the tasks in a team

An example: First version of scenario “Enlargement” 1/2• In Cabri mark a point I; create by means of

the tool Polygon any quadrilateral ABCD.• 1) Select the tool Number and type 3

– Construct the image of ABCD by using the tool Enlargement in the following way: point out successively the quadrilateral, point I and number 3. Label A’, B’, C’ and D’ the corresponding vertices of the new quadrilateral.

– Compare vectors • IA and IA’, IB and IB’, • AB and AB’, BC and BC’, • area (ABCD) and area (A’B’C’D’)

– Which equality is valid for vectors IA and IA’? For vectors IB and IB’?

An example: First version of scenario

“Enlargement”•Modify number 3 into -0.5 and answer again questions of activity 1

•Do several trials by changing the position of I and then point A

First version: Cabri provider of data

• Provider of static diagrams and data• No use of continuous drag of points and updating a computation on displayed measurements

• Questions about numerical relationships• No qualitative questions• Strong guidance of pupils

– Elements to be compared and to be changed were given

• Task as such is possible in paper and pencil environment

• Minimizing perturbation and uncertainty for the teacher

Scenario “Enlargement”: Second

version• In the toolbox Transformation of Cabri, in addition to reflection and point symmetry, there is the tool “Enlargement”. You will study this transformation.

• Create a point I, edit a number k by using as starting value 2.5, create by means of the tool Polygon a quadrilateral ABCD and construct its image through the dilation with centre I and ratio k (tool Enlargement, point the quadrilateral, then the centre I and number k). Characterize the obtained image.

Scenario “Enlargement”: Second

version•Characterize the obtained image•Do not hesitate to drag polygon ABCD, points A, B, C and D, centre I and to modify number k; do not forget that you can display measures with tool “Distance and length” and that a calculator is available in Cabri.

• Give to k a negative value (choose in a first step -0.5) and complete the previous characterisation. Do not hesitate to vary k.

Changes from version 1 to 2

• Central place to dragging including for numbers

• More open ended questions• Qualitative exploration made possible• Reference to a larger number of tools• Task as such impossible in paper and pencil environment

• Larger variety of possible answers, more potential questions encountered by pupils when manipulating

• More uncertainty for the teacher

Three categories of tasks

• Cabri as facilitating the task while not changing it conceptually (visual amplifier, provider of numerical data)

• Cabri modifies the ways of solving the task

• The task takes its meaning from Cabri

The story of tasks design

with DG by teachers reflects the difficult situation of teachers– when trying to give more autonomy to students they increase the uncertainty in their classroom management

– when reducing the autonomy of students, they decrease the learning potential

• Professional development is critical for contributing to increase the confidence of teachers (accompanying strategy Grugeon)

• The role of research is also crucial– Time for investigating different kinds of tasks– A better knowledge of students faced with different kinds of tasks

– Informing professional development