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 - PowerPoint PPT Presentation


PPT – 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 PowerPoint presentation | free to download - id: 3db9f0-OThmZ


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation

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


Research results about the contribution of dynamic geometry to the teaching of mathematics Their possible use in implementing the new senior secondary curriculum in ... – PowerPoint PPT presentation

Number of Views:31
Avg rating:3.0/5.0
Slides: 39
Provided by: webEduHk


Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: 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

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

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

  • Essential feature of dynamic geometry
  • Offers visualization of phenomena ruled by
    mathematical laws
  • Representations of mathematical objects behaving
  • 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,
  • 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
  • of quantifiers on variables
  • Ex HK curriculum framework Learning objective
    10.2  The angle in a semi circle is a right

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,
  • Starting in 1990
  • Research in mathematics education at the
    international level on the use of dynamic
  • 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
  • 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

  • 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
  • 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
  • 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
  • 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
  • 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
  • Semiotic mediation (Vygotsky) a process
    transforming an external tool (externally
    oriented) into a psychological tool (internally

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
  • 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

Correspondence between algebraic and graphical
  • 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
Eulers thought experiment
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
  • 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
  • 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