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Research results about the contribution of

dynamic geometry to the teaching of mathematics

Their possible use in implementing the new senior

secondary curriculum in Hong Kong

- Colette Laborde
- University Joseph Fourier, Grenoble, France
- Colette.Laborde_at_imag.fr

Dynamic geometry (DG)

- DG offers tools for constructing dynamic diagrams

on the screen of computers or calculators - If a construction has been done by using

geometric properties, these properties and all

the deriving properties are preserved when

dragging an element of the diagram

Dragging

- Essential feature of dynamic geometry
- Offers visualization of phenomena ruled by

mathematical laws - Representations of mathematical objects behaving

mathematically - Mathematics becomes tangible

The potential of dragging for learning maths

- Those properties remaining invariant in the drag

mode emerge from the contrast with the changing

properties of the figure in the drag mode - Being able to move screen objects around in

space (and so over time) can add significantly to

the users sense of the underlying concept as an

object not just in itself but a something

invariant amidst change Mason Heal (1995,

p.301) - Conceptualization of invariant structures amidst

changing phenomena is often regarded as a key

sign of knowledge acquisition (A. Leung 2008) - Externalisation
- of the duality invariant/variable in a spatial

form - of quantifiers on variables
- Ex HK curriculum framework Learning objective

10.2 The angle in a semi circle is a right

angle

It is not a new idea

- Sollen unsere Schüler in die heutige Form der

Wissenschaft und zwar gelegentlich in deren

Anwendung eingeführt werden, so müssen sie

beizeiten daran gewöhnt werden, die Figuren als

jeden Augenblick veränderlich zu denken und dabei

auf die gegenseitige Abhängigkeit ihrer Stücke zu

achten, diese zu erfassen und beweisen zu können - (Treutlein 1911)

Brief history

- Prehistory 1963 Ian Sutherland IBM
- Before 1990
- Enthusiastic Pioneers
- Some uses in teaching (ex Switzerland,

Catalunya) - Starting in 1990
- Research in mathematics education at the

international level on the use of dynamic

geometry - Second half of the 90s
- First application of dynamic geometry in an

advanced calculator (1995) - Development of dynamic geometry systems in the

world around 70, less than 10 are original - Integration in the curricula in some countries
- Beginning of XXI st century
- Dynamic geometry application in popular graphing

calculators (2002) - Beginning of integration in the teaching practices

Content of the talk

- Evolution of research
- Objects of study and questions
- Findings
- Two main phases
- Study of the interaction between the students and

DG - Study of teaching sequences making use of DG

Students using DG

First tasks construction tasks

- At the beginning of the 90s
- Construction tasks similar to those done in paper

and pencil were given to students who were

observed when solving the tasks (Sträßer, Hoyles,

Noss, Healy, Jones) - Students were asked to do robust geometric

constructions, i.e. preserved by dragging - Dynamic geometry revealed students conceptions

in geometry

Findings (1/2)

- 1. Importance of constructions done by eye by

students supposed to be familiar with geometry - Dragging to adjust by means of a visual control

Construct a point C collinear with A and B so

that AB AC.

Construct a point Q on d2 such that OQ OP

Findings (2/2)

- 2. Change of students construction processes

through dragging - Intertwining of visual and theoretical controls
- Dragging leads to change in the construction

processes (Jones 1998, Hoyles Noss 1996)

Second type of tasks in research studies

- Open ended problems Identifying conditions for

the existence of objects (Hölzl, Arzarello,

Olivero, Healy) - Ex Relationship between a quadrilateral and the

quadrilateral built by the perpendicular

bisectors of its sides (Olivero) - These tasks could exist in paper pencil but they

would be much harder - The solving processes differs in DG
- Dragging makes possible extensive exploring and

experimenting

Findings

- Again intertwining of empirical and theoretical

processes but students go further in the problem

than in paper and pencil and use in different

ways the drag mode to experiment - Link between the type of dragging and the type of

reasoning (Olivero) - Wandering dragging
- Guided dragging (induction)
- Lieu muet dragging (abduction)

Soft/Robust Constructions

- Distinction robust constructions/soft

constructions (Healy 2000) - Robust Construction preserved by dragging
- Soft Construction displays a property just at a

specific moment when dragging , ephemeral, not

permanent - Are two congruent sides and one congruent angle

enough to make two triangles congruent?

Robust/soft An example

Robust Construction of Tim and Richard First they

did not notice the second intersection

point Focused on the fact that the construction

is preserved by dragging Soft Construction of

Karen and Abby who moved P and immediately found

a counter example

A unexpected use of dragging by students

- In common with other researchers when we began,

it was our intention to encourage students to

build robust constructions. In practice, we

found that some students preferred to investigate

a second type of Cabri-object, soft

constructions, in which one of the chosen

properties is purposely constructed by eye,

allowing the locus of permissible figures to be

built up in an empirical manner under the control

of the student. (Healy 2000)

All these studies show

- A diversity of uses of dragging
- Interaction between tools given to students and

their solving process - Tools affect the actions but also forge the

conceptualisation (Vygotskian perspective) (Noss

Hoyles) notion of situated abstraction - The nature of mathematics that we study will be

inevitably changed (F. Leung 2004)

Second phase

- Design of teaching learning situations

Teaching/learning situations

- DG no longer just a window on students solutions

but a tool in the classroom to promote learning - In geometry (soft constructions to introduce

properties) - And beyond the only geometry, as dragging can be

related to variation and variable - Extension to coordinate geometry,

functions,differential equations - Multiple linked representations

Soft constructions to help understand properties

- Circumscribed circle of a triangle (HK curriculum

10.1) - A tangent to a circle is perpendicular to the

radius through the point of contact (HK

curriculum 10.5)

Mediation of the notion of implication through DG

- Difficulty of students to understand the notion

of necessity - They do not distinguish between hypothesis and

conclusion - Introduction of an asymmetry between hypotheses

and conclusion through action - Hypotheses are not fulfilled at the beginning
- They are met through dragging by student
- The student action is oriented towards obtaining

given conditions (hypotheses) - The conclusion is the obtained effect
- until the hypothesis is not satisfied, the effect

is not visible, - it differs from the focus of the action
- One year long teaching of geometry at grade 7

(Coutat, 2006) to introduce students to the world

of deductive geometry

Introduction of causality

- Causality is more likely to be observed with

able students who seek to understand phenomena in

depth than by weak students who usually are

satisfied with what ever the teacher presents.

(Harel, 1999, Linear Algebra and its

applications, 302-3, p.601-613)

Mediation and Internalization Process

- DG can be used to favour an internalization

process moving from external operations in the

environment to internal ones at the theoretical

level - Semiotic mediation (Vygotsky) a process

transforming an external tool (externally

oriented) into a psychological tool (internally

oriented)

How to favor an Internalization process?

- Adequate tasks creating a problem to the students

to be solved in the environment - The solution of the problem requires the use of

actions in the environment being the counterpart

of mathematical knowledge to be appropriated - Interventions of the teacher and interactions

with students to make explicit the correspondence

between what is done in the environment and the

theoretical knowledge

Flexibility between representations

- Every mathematical activity is mediated by

external representations - Using various representations of a same concept

and moving between them are part of the

construction process of the concept (Duval) - Doing mathematics requires processing and acting

with and on various representations - This is one of the declared aims of our national

program of study in France for some concepts such

as function. - But students encounter difficulties in linking

different representations and this flexibility

must be developed through adequate tasks

New kind of tasks made possible in DG

- Tasks centered around
- the representations and the links between two

different kinds of representations - and resorting to mathematical knowledge
- Interpreting the behaviors of variable

representations - Producing variable representations
- Examples
- Moving a point in space until reaching a hidden

spatial object.

Function and Graph

- Several kinds of representations of a function,

in particular - Symbolic expressions
- Graphical representations (graph) (HK Learning

Objective 2.2) - Each kind of representation brings to the fore

different features of functions and is relevant

for different uses and problems - The variations and extremal values are better

seen on a graph - Numerical problems are better solved with the

algebraic expression

Students difficulties with graphs

- However the genesis of the notion of graph is

lost for students (Vinner Dreyfus, Eisenberg,

Trigueros, Markovits, Schwarz) - The graph is just seen as an entity attached to

the function - Students have difficulties in conceiving the dual

aspect of the graph - Set of points (x, f(x))
- A curve with geometrical properties
- A source of the problem is in paper and pencil
- It is impossible to draw all points (x, f(x))
- Often only some points are drawn and linked with

a smooth curve, but students have no idea of what

represents this curve.

Functional dependency

- The relationship of dependency between the two

variables x and f(x) is not grasped by students

in the paper and pencil graph - It is not visible in the graph
- The graph is static
- The two variables play a symmetrical role
- As a result, lack of operational relationship

between function and graph for high school

students

Correspondence between algebraic and graphical

representations

- The correspondence between the two

representations is based on the idea of

representing a variable number by a variable

point on an axis

Euler proposed this geometrical way of

representing a function

Introductio in analysis Infinitorum Tomus

secundus Theoriam Linearum curvarum Lausannae

MDCCXLIII

Eulers thought experiment

M

M

M

M

P

P

P

P

For each value of x, a point P For each point P a

perpendicular segment PM representing y When x is

increasing from 0 to infinity, one considers all

points M which give rise to a curve.

DG materializing the Eulers thought experiment

- A variable number x is represented by a point on

an axis - In dynamic geometry the variation can be

represented by motion - In DG a variable point can leave its trace
- The graph of a function is the trajectory of

point M - The dual meaning of the graph is restored in the

notion of trajectory

Mediation of dependency through DG

- It is possible to distinguish between the

independent and dependent variables through

dragging - The independent variable is represented by a

point which is directly movable - The dependent variable is represented by a point

which can be moved indirectly by dragging the

point representing the independent variable - In a first step, dragging is an external tool

enabling students to distinguish between

dependent and independent variables - Then with the guidance of the teacher, this tool

can be internalized by students who construct the

concept of independent and dependent variables

(semiotic mediation after Vygotsky)

From a numerical function to a geometric function

- The graph can be seen as the image of a geometric

function which maps P to M - Instead of starting from numerical functions, the

idea is - to first introduce students with geometric

functions in DG - then to move to numerical functions and use

geometric function as a way of representing a

numerical function

Teaching experiment

- Design of three sequences of activities for 15-16

year old students in France and Italy (2000-01,

2001-02, 2002-03) (Falcade 2005) - Lycée in a suburb of Grenoble
- Liceo Scientifico in Forte dei Marmi (Lucca)
- We wanted to investigate more deeply the process

of mediation and internalization that we assumed.

Structure of the sequence

- 1) Introduction to the notion of geometric

function in Cabri-geometry - A problem find an unknown function
- Imagine a new geometric function
- 2) A problem how to represent geometrically the

co-variation of two numerical variables? - Reading and discussing the historical solution

proposed by Euler about the notion of graph of a

function - Implementing the solution of Euler in Cabri
- 3) Working on graphs with Cabri
- Solving extrema problems

What does change DG?

- Increased visualisation
- Makes mathematical objects more tangible
- Links different representations
- Particularly important for weak students who lack

mental imagery and flexibility - Students have more questions because they can

experiment more - HK curriculum it is important for our students

to think critically, analyse and solve problems - Increased role of the teacher