Teachers writing resources integrating dynamic geometry a

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					The design of tasks taking
      full advantage
  of dynamic geometry:
 what kinds of knowledge
   does it require from

        Colette Laborde
    University Joseph Fourier
        Grenoble, France
   Shared research claims
• There are several technological
  environments very promising in terms of
• 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,
           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
       • 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
  – even if they do not use the word “task” itself
• Constructivist and socio-constructivist
  approach: problematic tasks for the
  – Problem is the source and criterion for
    knowledge (Vergnaud)
  – Learning comes from adapting to a new
 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
   A professional activity
• Designing tasks is a teacher professional
  activity (Robert)
• It is a complex activity involving several
  – 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
  – 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
Designing technology based
   tasks is problematic
• Designing technology based tasks is out
  of the range of the ordinary activities of
  – 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
  – using ready made tasks for technology
  – adapting tasks designed for paper and
  – designing his/her own tasks
      In French schoolbooks
   (Caliskan, PhD thesis Paris)
 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
         French      Without With
         Schoolbooks DG      DG

         Edition 1      19       13
         Edition 2      5        32
   But a “weak” use of DG
• 5% of proposed activities have recourse to
• 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
         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
      Reasons invoked by
• 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
Kinds of DG use in exercises
In schoolbooks or tasks proposed by teachers
• 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
         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
• 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
  – Instrumental is seen as independent of
• 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
• Such coordination requires several types of
  knowledge from teachers
• It was observed in research and development
  projects or after inservice or preservice
  R&D project of writing
  teaching scenarios with
   Cabri for high school
• Various profiles in the team of each
  – Teachers
     • Experienced teachers familiar with the use of
     • Experienced teachers novice with the use of
     • 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
 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
 • Provider of static diagrams and data
 • No use of continuous drag of points and
   updating a computation on displayed
 • 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
   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
• 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
• Cabri modifies the ways of solving
  the task
• The task takes its meaning from
 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
• 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
s also crucial
   – Time for investigating different kinds of tasks
   – A better knowledge of students faced with different
     kinds of tasks
   – Informing professional development

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