colette laborde, iufm & university joseph fourier, grenoble, france [email protected]
DESCRIPTION
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 [email protected]. TOOLS IN TEACHING. - PowerPoint PPT PresentationTRANSCRIPT
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, [email protected]@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