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GeoGebra-konference
The use of GeoGebra as a didactical tool to
support pupils in the development of
conversions between Duval’s mathematical
representation registers
24 Oktober 2015
Introduction
Adrian Rau Bull associated professor of mathematics at VIA
university college – teachers college Aarhus. Writer: Multi 7
This shorttalk describes how a program like GeoGebra can be
used as a didactical tool to support pupils in the development of
mathematical proof schemes in mathematical education. This
development of pupils’ mathematical skills is specifically
developed to help pupils convert between Duval’s mathematical
representation registers. It is also an approach connecting the
pupils’ own idiosyncratic representations and society’s
conventional representations. Through empirical inquiries, the
pupils can get experience with the conventional representations
and acquire them as empirical or deductive proof schemes. The
objective is for pupils to expand their mathematical skills.
Education in mathematics
The idiosyncratic representation that children create
vs.
Cultural and conventional representation
(Smith, 2003)
Guershon Harel & Larry Sowder: Towards comprehensive perspectives
on learning and teaching of proof
Proof scheme: A person’s (or a community’s) proof schemes
consists of what constitutes ascertaining and persuading for that
person (or community). (Harel and Sowel, 2007).
Proof schemes (both community’s and person’s) can be divided
into tree classes.
Empirical proof schemes, are marked by their
reliance on either (a) evidence from examples or (b) perceptions
External Conviction proof schemes, depends on
an authority such as a teacher or a book it is marked by the
person dont do the reasoning themself
Deductive proof schemes, are marked by their
reliance on generality, operational thought and logical inference.
Raymond Duval: A Cognitive Analysis of problems of comprehension in a
learning of mathematics
A danish example:
Geometry
Reasoning
Problem
tackling
Modelling
Communic
ating
…
…
Algebra
Statistik
…
A danish example Reasoning
inspiret by John Mason
Reasoning and arguing intuitively on
concrete mathematical activities and follow
others' oral arguments
( reasoning skills)
Devise and implement informal and simple
formal mathematical reasoning and follow
oral and simple written arguments
( reasoning skills)
Devise , implement, understand and
evaluate oral and written mathematical
reasoning and work with simple proof
( reasoning skills)
Proof
Direct Proofs
Proof by Contradiction
Proof by exhaustion
Visual proof
Reasoning 2 inspiret by John Mason
Reasoning 3 inspireret af John Mason
Raymond Duval: A Cognitive Analysis of problems of
comprehension in a learning of mathematics
Treatment: Mathematic operations inside a registre
Conversion between registres
Implementet in GeoGebra
Examples
Areal of a cirkel :
Dependent variable
Variable
GeoGebra / IKT
Det visuelle aspekt ved IKT
Der giver mulighed for at illustrere matematikken i flere forskellige registre samtidigt. Og dermed
også støtter mulighed for at man kan undersøge og opdage ændringer i flere registre samtidigt og
dermed blive opmærksom på sammenhængen mellem registre.
Kodning eller programmerings aspektet
Der giver mulighed for at sammenkoble forskellige registre i Duvals model, og derved simulere
denne sammenkobling, som den eksistere i matematikken. Derved kan elever bruges deres
erfaringer fra et registre af matematikken, til at blive opmærksom på hvordan disse ser ud og
opfører sig i andre registre.
Det dynamiske aspekt ved IKT
Der giver mulighed dynamisk at ændre på de matematiske objekter og giver mulighed for at
simulere ændringerne. Således at elever gennem empiriske undersøgelse med IKT, kan opdage
matematikken.
Litt.
Raymond Duval: The cognitive Analysis of problems of comprehension
in then learning of mathematics (2002)
Raymond Duval: A crucial issue in mathematics education the ability to
change representation register (2004)
Raymond Duval: A Cognitive Analysis of problems of comprehension in
a learning of mathematics (2006)
Guershon Harel & Larry Sowder: Towards comprehensive perspectives
on learning and teaching of proof
Smith, S.P. (2003). Representation in school mathematics: Children’s
representations of problems.
Højgaard, Tomas (2012). Competencies and the fighting of syllabusitis