Strand
1:
Integration
of IC technologies into learning processes
ICTMT
5, Klagenfurt, 6-9 August 2001
(Schedule, tentative as of 8.6.2001)
Chair:
Jean-Baptiste Lagrange
Monday 11:30 - 12:15 Chair: Jean-Baptiste Lagrange
Plenary:
The
construction of meaning for abstract algebraic concepts
Tommy
Dreyfus (Israel, tommy.dreyfus@weizmann.ac.il)
Monday 15:15 - 16:00 Chair: Jean-Baptiste Lagrange
The impact of training for
students on learning mathematics
Roger
Fentem (UK, rogerfentem@netscapeonline.co.uk)
Monday 16:15 - 17:00 Chair: Jean-Baptiste Lagrange
Experimental Mathematics. Someone invented the knife - everybody uses it
Christian Thune
Jacobsen
(Denmark, ct@ghg.dk)
Monday 17:00 - 17:45 Chair: Jean-Baptiste Lagrange
The ODE Curriculum:
Traditional vs. Non-Traditional - The Case of One Student
Samer
Habre (Lebanon, shabre@lau.edu.lb)
Tuesday 8:30 - 9:15 Chair: Jean-Baptiste Lagrange
Remedial Education of
Quadratic Functions Using a WWW-based On-line Exercise System
Hitoshi
Nishizawa (Japan, nisizawa@toyota-ct.ac.jp)
Tuesday 9:30 - 10:15 Chair: Jean-Baptiste Lagrange
Observing student working
styles using Graphic Calculators
John
Berry*, Andy Smith (UK, jberry@ctm1.freeserve.co.uk)
Tuesday 10:30 - 11:15 Chair: Jean-Baptiste Lagrange
Data Collection and
Manipulation using Graphic Calculators with 10-14 year olds
Ruth
Forrester (UK, ruth.forrester@education.ed.ac.uk)
Tuesday 15:15 - 16:00 Chair: Jean-Baptiste Lagrange
Evaluating the Effectiveness
of Computer-Based Learning in Mathematics
Neil
Pitcher (UK, neil.pitcher@paisley.ac.uk)
Tuesday 16:15 - 17:00 Chair: Jean-Baptiste Lagrange
Functional Algebra with the
Use of the Graphing Calculator
Henk
van der Kooij (Netherlands, H.vanderKooij@fi.uu.nl)
Tuesday 17:00 - 17:45 Chair: Jean-Baptiste Lagrange
The Role of the Graphic
Calculator in Early Algebra Lessons
Jenny
Gage (UK, jag43@tutor.open.ac.uk)
Wednesday 8:30 - 9:15 Chair: Jean-Baptiste Lagrange
Thinking the Unthinkable -
Understanding 4 Dimensions
Claus
Meyer-Bothling (Germany, cmb@zum.de)
Wednesday 9:30 - 10:15 Chair: Jean-Baptiste Lagrange
The Role of Technology in
Mathematical Diagnosis
Neil
Challis (UK, n.challis@shu.ac.uk)
Wednesday 10:30 - 11:15 Chair: Jean-Baptiste Lagrange
To learn from and make history
of maths with the help of ICT
Marie-Thérèse Loeman (Belgium, mth.loeman@pandora.be)
Thursday 8:30 - 9:15 Chair: Jean-Baptiste Lagrange
Integrating Mathematics,
Physics and Interactive Digital Video
John
Pappas (Greece, me00410@cc.uoi.gr)
Thursday 9:30 - 10:15 Chair: Jean-Baptiste Lagrange
Cognitive and didactic ideas
in ICT environments for the learning and teaching of mathematics
Gisèle
Lemoyne (Canada, lemoyne@SCEDU.UMontreal.CA
Thursday 10:30 - 11:15 Chair: Jean-Baptiste Lagrange
The Visualisation of a
parameter
Carel
van de Giessen (Netherlands, carelvdg@tref.nl)
Thursday 15:15 - 16:00 Chair: Jean-Baptiste Lagrange
Mapping for Learning:
Differentiating Mathematics Instruction for Personalised Learning
Mara
Alagic (USA, mara@math.twsu.edu)
Thursday 16:15 - 17:00 Chair: Jean-Baptiste Lagrange
Mathematical software in the
educational process of the French and Hungarian teachers
Maria
Bako (France, aszalos@irit.fr)
Abstracts:
Plenary:
Tommy Dreyfus, Israel:
The construction of meaning
for abstract algebraic concepts
The
teaching and learning of algebra, whether elementary, linear or
modern algebra, seems to virtually cry out for computer support,
for several reasons: A large variety of multi-representational tools
are available, the heavier calculations can easily be taken over by
the computer, and most importantly, appropriate software can be used
to bridge the existing gap between the concrete and the abstract
(see, e.g., Schwarz & Dreyfus, 1995). Indeed, there are examples
of success in using technology for students' construction of meaning
for abstract algebraic concepts but there are also examples of
failure. In the lecture, I will examine a number of possible reasons
for failure, including inadequate task design (Sierpinska, Dreyfus &
Hillel, 1999) and the ambiguity of representatives for mathematical
objects (Dreyfus & Hillel, 1998; Schwarz & Hershkowitz, in
press). I will conclude that there is no simple explanation. I will
then make the point that in order to deepen our understanding of the
relevant learning processes, a re-conceptualisation of abstraction
is in order, as well as a research program that allows describing
processes of abstraction. Such a re-conceptualisation will be
proposed and a research program will be outlined (Hershkowitz,
Schwarz & Dreyfus, in press).
Mara
Alagic, USA:
Mapping
for Learning: Differentiating Mathematics Instruction for
Personalised Learning
In
the context of WHAT? - HOW? - WHO?, if the WHAT is a mathematics
and/or technology standards-based curriculum and the WHO? are
learners, could the HOW explain our way of thinking, our
teaching/learning/reflecting philosophy, and/or our sense-making
processes? Where is the place of technology in these processes? This
paper attempts to give some answers/examples and pose more questions
about the power of technology in the learning of mathematics: How
technology can make a difference in the way we differentiate
instruction for personalised learning in mathematics classroom?
Maria
Bako; France:
Mathematical software in the educational process of the French and Hungarian
teachers
The
French and Hungarian education systems spend a lot of energy to keep
up with the new developments in the field of technology. Informatics
is taught through out high school all over but the computers had no
enough role yet in the teaching process of various subjects. The
poll's aim, presented in the article, is to show how much and how
well the college professors and their students knows and uses
mathematical programs. The subjects of this poll are the professors
and the students at the Faculty of Mathematics of the University Paul
Sabatier of Toulouse and the University of Debrecen of Hungary. The
parallel study of this two, culturally and economically different
countries brought our attention to some very interesting particular
and general problems, which are presented in details in this paper.
This and the ideas on the questionnaires can help to set new goals in
the application of the computers in the teaching process of
mathematics.
John
Berry, Andy Smith, UK:
Observing
student working styles using Graphic Calculators
When
students are working with hand-held technology, such as a graphic
calculator, we usually only see the outcomes of their activities in
the form of a contribution to a written solution of a mathematical
problem. It is more difficult to capture their process of thinking or
actions as they use the technology to solve the problem. In this
paper we describe an empirical investigation of student working
styles with a graphic calculator using software that captures the
keystrokes that are used. In this way the students were able to work
naturally without the feeling of 'being observed'. After the student
problem solving session we were able to playback the sequence of
keystrokes to explore how the students actually used the technology,
whether they used 'trial and error' mode and how their working
related to the training they had received.
Neil
Challis, UK:
The
Role of Technology in Mathematical Diagnosis
In
the UK and elsewhere, access to higher education is widening.
Students arriving on the same course can have widely different
mathematical backgrounds. The issue arises of identifying students'
individual mathematical needs, and following up appropriately, as
well as making courses appropriate to those students. We report on a
project at Sheffield Hallam University addressing this issue,
particularly examining the role that technology, for both learning
and doing mathematics, can and cannot play.
Roger
Fentem, UK:
The
impact of training for students on learning mathematics
Training
for teachers in the use of graphing calculator technology is widely
accepted. To what extent are the training needs of the users of the
technology addressed i.e. the students? This paper introduces a
research project designed to investigate the issues of technology
training for both teacher and student in studying mathematics post
16. Attitude, relative achievement and practice are studied, recorded
and analysed.
Ruth
Forrester, UK:
Data Collection and Manipulation using Graphic Calculators with 10-14 year
olds
A
teacher researcher group at the Edinburgh Centre for Mathematical
Education is currently investigating the use of graphic calculators
in Mathematics classes for pupils aged 10 -14 years. One focus has
been on the development of data handling skills. Activities have been
devised where pupils use graphic calculators in the collection of
data and its subsequent analysis. Classroom implementation has
produced positive results. Evidence has been found of gains in
understanding of statistical concepts attributable to the use of this
technology. Positive motivational effects were also seen. The graphic
calculators enabled the use of pupils' own data and allowed the
teacher to pace and vary the learning experience appropriately.
Jenny
Gage, UK:
The Role of the Graphic Calculator in Early Algebra Lessons
This
is a study of first algebra lessons at secondary school using the
lettered stores of a graphic calculator to form a model of a
variable. The calculator provides a tool for thinking and for
building up concepts. In this paper is a discussion of what happened
in the classroom, and how the calculator helped in the remediation of
a specific misconception without any need for teacher intervention.
There is also discussion of what ideas the students bring to the
work, and how these ideas change during the lessons.
Samer
Habre, Lebanon:
The
ODE Curriculum: Traditional vs. Non-Traditional - The Case of One
Student
A
Traditional course in ordinary differential equations consists of
tricks to find formulas for solutions with very little emphasis on
the geometry of the solutions or on an analysis of the outcomes.
Since differential equations are important in many fields, educators
have come to believe that this approach is obsolete. With the
advancement of computer graphics, it is now possible to offer a
course on differential equations using a qualitative approach. This
paper examines the two approaches as offered by the same instructor
at the Lebanese American University in Lebanon. In particular, the
point of view of one student who took the course twice using a
different approach each time is presented. Results show that the
qualitative approach is more appreciated, and that technology plays
an essential role in the understanding of the material.
Gisèle
Lemoyne, Canada:
Cognitive
and didactic ideas in ICT environments for the learning and teaching
of mathematics
Over
the past few years, we have designed computer environments for the
teaching of arithmetic, pre-algebra and algebra. We describe some of
these to demonstrate how cognitive and didactic ideas are put into
practice and how these environments engage both learners and teachers
in non trivial problem-solving activities. The first environment is
devoted to additive and multiplicative problems. Three different
tasks were planned:
construct
an iconic representation of a problem, using the tools in the
environment
write
a mathematical sentence that corresponds with an iconic
representation of a problem
write
a problem that corresponds with a mathematical sentence.
In
the second environment, teachers have access to a calculator and can
create problems by specifying numbers and operations and then
choosing on the key pad of the calculator which keys will be non
functional. Each subgroup of students receives specific calculations.
The third environment consists of a task of abstraction of properties
and characteristics of numbers and operations.
Marie-Thérèse
Loeman; Belgium:
To learn from and make history of maths with the help of ICT
Results
from the EEP Comenius Action 1 : "The history of some aspects of
mathematics like: history of mathematical persons, symbols,
algorithms..." Looking through different aspects of history of
maths, in co-operation with people from other nationalities and
cultures, convinced our students that maths, having its special
common language and symbolic notations, has no boundaries. Digging in
history of maths and working cross-subject ( English, religion,
philosophy, chemistry, geography, physics...) revealed to them that
as it comes to solve a problem, not only the solution is to be
appreciated but certainly getting to know a nice, perhaps different
and original way of reasoning can be a source of inspiration for the
scientist being superior to the machine ! In addition they were
encouraged to learn from the stronger elements in each partner
country.
Claus
Meyer-Bothling, Germany:
Thinking
the Unthinkable - Understanding 4 Dimensions
The
existence of a fourth spatial dimension is confirmed by the Theory of
General Relativity. Furthermore some simple properties of
4-dimensional objects, say of a 4-D-cube, can be deduced by analogy.
The 3-D-projections of such objects can even be illustrated. Although
we can state the properties of a 4-D-cube, we cannot picture the
object itself. Our brain is not equipped to do that - following
today's accepted wisdom anyway. My claim is that with the aid of
modern resources we will probably be able to overcome this obstacle:
With today's technology of illustration it should be possible to
train our perception in such a way that we will be able to imagine
4-D-bodies.
Hitoshi
Nishizawa, Japan:
Remedial
Education of Quadratic Functions Using a WWW-based On-line Exercise
System
The
method and the effectiveness of remedial education using a WWW-based
on-line exercise system are reported. The system displays a graph of
a quadratic function and requests the student to express it in a
symbolical expression. Six students were selected to attend the
remedial course using the system. Although they used only one formula
to express the graphs before the exercises, they have extended the
variety of their expressions through the exercises.
John
Pappas; Greece:
Integrating
Mathematics, Physics and Interactive Digital Video
Previous
research on Digital Interactive Video Technologies (DIVT) is limited
to the domain of kinematics and graph interpretation in particular.
This pilot study is part of a full-scale research that aims to extend
the field of investigation using Digital Video Technologies as a
connecting link for the Integration of Mathematics and Science. Five
students participated in this study, which consisted of two parts,
one without and one with DIVT support. The analysis of data gathered
indicate that being able to manipulate the reference frame in the
environment of the DIVT software and notice how it affects
co-ordinates, graphs and equations improves the students' conceptual
knowledge on this subject, in two levels:
By
bringing the reference frame to particular positions of 'special'
interest, such as positioning one of the axes to be parallel to an
inclined level, they can deal with their misconceptions and
gain a better understanding and insight to the role of a co-ordinate
system.
Neil
Pitcher, UK:
Evaluating
the Effectiveness of Computer-Based Learning in Mathematics
This
session will discuss effective ways of integrating computer-based
learning environments into university Mathematics courses. The system
'Mathwise' will be used as an exemplar. Mathwise contains materials
both for learning and for assessment. Such a system needs to be used
carefully if it is to promote good study skills. Different teaching
methods will be examined and some evaluation results presented.
Christian Thune Jacobsen, Denmark:
Experimental Mathematics. Someone invented the knife - everybody uses it
Computer algebra systems (CAS), such as Derive and Maple, will naturally be an
integrated part of teaching mathematics in the future - just as the use of
calculators has been for the last two decades. The question is only how to implement CAS.
Carel
van de Giessen, Netherlands:
The
Visualisation of a parameter
Based
on the ideas of David Tall we, Piet van Blokland and I, have
developed a program to investigate graphs and formulas. Two aspects
may be of special interest: variables and parameters. For the young
students (12-14 years) it is easier to understand the concepts
involved with graphs and formulas when using word-variables. The
concept of 'parameter' in formulas is difficult to grasp, because the
mathematical level needed to understand a parameter is high. We
therefore introduced a so called 'sliding parameter'. In the
programme this concept arises interactively using a scrollbar: the
parameter value changes and so does the graph. This is a dynamic way
to investigate a graph and the role of a parameter. One graph, one
value of the parameter.
Henk van der Kooij, Netherlands:
Functional
Algebra with the Use of the Graphing Calculator
Algebra
is a very important topic in mathematical programs for upper
secondary education, but a vast majority of students is weak in
understanding and using formal algebraic tools. This paper discusses
some ideas about using the graphing calculator to support the
learning of algebra in the context of functions and to help students
overcome algebra-anxiety. Accepting the graphing calculator as a
supportive toolkit in the learning of algebra has far-going
consequences for the way in which what kind of algebra should be
learned and taught.
| 
The Fifth International
Conference on Technology
in Mathematics Teaching
