Strand 4:

Changes in geometry and algebra via DGS and CAS

ICTMT 5, Klagenfurt, 6-9 August 2001

(Schedule, tentative as of 8.6.2001)

Chair: Hans-Georg Weigand

Monday 15:15 - 16:00 Chair: Hans-Georg Weigand

Cinderella: Software Tool for Euclidean and Non-Euclidean Geometry

Robert Mayes (USA, rmayes@unco.edu)

Monday 16:15 - 17:00 Chair: Hans-Georg Weigand

Expression Equivalence Checking in Computer Algebra Systems

Eno Tonisson (Estonia,eno@math.ut.ee)

Monday 17:00 - 17:45 Chair: Hans-Georg Weigand

Cubics and quartics on computer

Pavel Pech (Czech Republic, pech@pf.jcu.cz)

Tuesday 8:30 - 9:15 Chair: Hans-Georg Weigand

Self-correction in algebraic algorithms with the use of educational software: an experimental work

Michael Kourkoulos*, Marianne A. Keyling

Tuesday 9:30 - 10:15 Chair: Hans-Georg Weigand

Integrating MuPAD into the Teaching of Mathematics

Miroslav Majewski, Fred Szabo (Macau/Canada majewski@julia.iium.edu.mo)

Tuesday 11:30 - 12:15 Chair: Hans-Georg Weigand

Plenary:

Interactive web-based resources and a new perspective on algebra and geometry

Jean Flower (UK, j.a.flower@brighton.ac.uk)

Tuesday 15:15 - 16:00 Chair: Hans-Georg Weigand

A CAS project carried out in Scotland with 16-17 year olds using TI-92s

Tom G Macintyre (Scotland, Tom.Macintyre@ed.ac.uk)

Tuesday 16:15 - 17:00 Chair: Hans-Georg Weigand

Magic Polyhedrons

Bronislaw Pabich (Poland, pabich@edukacjazti.pl)

Wednesday 8:30 - 9:15 Chair: Hans-Georg Weigand

Maple and a unified approach

Li Ma (Schweden, mali@itn.liu.se)

Wednesday 9:30 - 10:15 Chair: Hans-Georg Weigand

Teaching and Learning Geometry: dynamic and visual

Hans-Jürgen Elschenbroich (Germany, Elschenbroich@t-online.de)

Thursday 8:30 - 9:15 Chair: Hans-Georg Weigand

A Microworld For Helping Students To Learn Algebra

Denis Bouhineau*, Jean-François Nicaud, Xavier Pavard, Emmanuel Sander (France, Denis.Bouhineau@irin.univ-nantes.fr)

Thursday 9:30 - 10:15 Chair: Hans-Georg Weigand

Tabulae and Mangaba: Dynamical Geometry with a Distance Twist

Rafael Barbastefano (Brazil, rafael@fgv.br)

Thursday 15:15 - 16:00 Chair: Hans-Georg Weigand

A CAS-index applied to engineering mathematics Ps

Eoghan MacAogain (Ireland, eoghan.macaogain@ul.ie)

Abstracts:

Plenary: Jean Flower, UK:

Interactive web-based resources and a new perspective on algebra and geometry

This paper will reflect upon the use of DaC (dynamic geometry and computer algebra software) in two contexts - two undergraduate Linear Algebra courses taught at different UK universities. The main questions of this strand will be considered in the light of this experience. It is hard to compare the two linear algebra modules and claim that one was "more successful" than the other. One covered more pure algebra topics, whereas the other included more applications of Linear Algebra. Both used DaC. One used Maple and JavaSketchpad, and the other used TI92's algebra and Sketchpad on the PC. The students on one module were mainly training to become teachers, whereas the students on the other were studying for a mix of maths degrees, heading for business.

Is it necessary to achieve widespread use of DaC throughout a course for best benefits? The students who had a wider exposure to Sketchpad in a range of modules over many semesters made better use of the Linear Algebra images than the students who were unfamiliar with DGS. How do the costs (time as well and money) of introducing DaC in a single module compare with the benefits?

Is it necessary to integrate DaC into assessment at the same time as its introduction to the teaching? The students whose assessment included a Maple test learned to use Maple mainly for the purposes of completing the test, whereas the students with TI92s used them more widely to shortcut rote algebra. Use of the handheld technology was not required for successful completion of the course, but the TI-92s were used more widely.

How can we tie in a DaC approach to a subject whose key texts take a more traditional approach? There is a mismatch between the students' experience of Linear Algebra in the classroom (and in the website) and the students' experience of Linear Algebra from books. Does this contribute to confusion? Can we make use of this contrast to deepen understanding of the different facets of a subject?

The use of DaC allows for revitalisation of some "tough" topics which were getting taught later on in a degree. Tasks which required intensive numerical calculation can now be completed quickly, allowing more space for understanding the results of the calculation. The use of technology itself can provide relevant applications for study (eg. computer graphics). Different approaches to proof and argument contrasts axiomatics (a traditional way in to Linear Algebra) with investigation (assisted by DaC).

What is the relationship between working on the computer and working with paper and pencil? This question is critical when introducing DaC into courses which maintain traditional assessment strategies like exams, where students may not have access to DaC.

Looking at the changing nature of algebra and geometry is like trying to gaze into a crystal ball. But we can have some fun looking there.

Rafael Barbastefano, Brazil:

Tabulae and Mangaba: Dynamical Geometry with a Distance Twist

We report on the ongoing development of two complementary DGS, for plane and space geometry. The design briefs of both softwares were tailored bearing in mind the needs of distance teaching and Web communication. The current implementation is described in some detail, and we also discuss some of the issues that brought about the decision to engage in the project, as well as the implications for the technology driven teacher training program that provided the initial motivation for it.

Denis Bouhineau*, Jean-François Nicaud, Xavier Pavard, Emmanuel Sander, France:

A Microworld For Helping Students To Learn Algebra

This paper describes the design principles of a microworld devoted to the manipulation of algebraic expressions. This microworld contains an advanced editor with classical actions and direct manipulation. Most of the actions are available in two or three modes; the three action modes are: a text mode that manipulates characters, a structure mode that takes care of the algebraic structure of the expressions, and an equivalence mode that takes into account the equivalence between the expressions. The microworld also allows to represent reasoning trees. The equivalence of the expressions built by the student is evaluated and the student is informed of the result. The paper also describes the current state of the implementation of the microworld. A first prototype has been realised at the beginning of February 2001.

Hans-Jürgen Elschenbroich, Germany:

Teaching and Learning Geometry: dynamic and visual

"A generation has grown up that may be far more visual than verbal ... . The state of mind of young mathematicians is not what it was fifty or hundred years ago ..." (Davis)

Dynamic Geometry Software like Cabri II, Cinderella or Euklid-Dynageo offers new chances by using dragmode and loci to learn and to teach geometry in a visual and dynamic way. Classical ideas can be brought to life.

DGS is not seen as a substitute, but as a complement to and an extension of the classic tools compass and ruler. Electronic worksheets will give a safe basis, which avoids lengthy phases full of mistakes and will support experimental and heuristic activities of the students.

After some basic reflections about visual learning and teaching, well-tried examples of electronic worksheets and pre-formal, visual-dynamic proofs will be presented.

Michael Kourkoulos*, Marianne A. Keyling:

Self-correction in algebraic algorithms with the use of educational software: an experimental work

Our work points out that self-correction is a complex but fruitful activity concerning the learning of elementary algebraic algorithms. Pupils who have worked with an adequate software («Arithm»), both in Greece and in France, present a significant improvement of their strategies of localisation of errors, which are an essential element of the self-correction procedures. Furthermore, the work done led these pupils to a significant amelioration concerning the treatment of the examined algorithms.

The software allowed teachers to be alone in their class (or in a half-class in the case of weak pupils) but nevertheless to offer adequate individual support to the pupils in their self-correction work, which is very difficult to realise in usual teaching conditions.

Miroslaw Majewski, Fred Szabo, Canada:

Integrating MuPAD into the Teaching of Mathematics

Computer Algebra Systems are becoming more and more popular in mathematics education. However, many teaching issues are still unresolved, and no one is able to give a simple recipe how to integrate computer algebra systems into the teaching process. In this paper, we discuss some proven strategies for using MuPAD in the teaching of mathematics.

Li Ma, Sweden:

Maple and a unified approach

This paper will discuss the use of Maple in teaching Linear Algebra and Calculus as a unified approach.

Eoghan MacAogain, Ireland:

A CAS-index applied to engineering mathematics Ps

A CAS-index is applied to a set of first year university engineering mathematics examination papers; the results are analysed. The CAS-index is an index of suitability; its purpose is to try to answer the following question: given a mathematics examination paper which was written for a CAS-free environment how suitable is that examination paper for use in a CAS-supported environment?

Tom G Macintyre, UK:

A CAS project carried out in Scotland with 16-17 year olds using TI-92s

This study explored the impact of using hand-held technology throughout a course of study in a year 12 mathematics course - leading towards the Scottish Higher Grade. Students in the study sample had dedicated access to Texas Instruments TI-92 calculators, utilising the built in Computer Algebra System (CAS) as they developed their knowledge of the various components of mathematics studied. Both quantitative and qualitative data was gathered from the study sample students and teachers, who were based in three secondary comprehensive schools. Additionally, data was gathered from the three paired-control groups, providing evidence of algebraic ability at the start and end of the period of intervention. Performance in algebraic skills was of particular interest in this study, ascertaining whether extended use of technology had a positive or negative impact on students' abilities. The quantitative findings, taken from the two assessments administered at the start and end of the one-year course, demonstrate a significantly better performance in the study sample compared with the control group. This affected performance in items that were common to both assessments, resulting in a 7% increase in the study sample compared to the control (p=0.004). A similar trend was noted in new items that assessed mathematics studied during the course of the year; taking the base level of performance into consideration there was a 5% increase in the study sample compared with the control (p=0.046). Some underlying reasons for these differences in algebraic ability are explored. The discussion includes consideration of: the teaching approaches promoted by the staff; the impact of mathematical rigour and syntax demanded by the technologies; the emphasis on equivalence when interpreting screen displays; and the general motivational effect that dedicated access to the technology has had on the students in the study. A number of questions remain, for current debate and future research into the use of a CAS in mathematics education.

Robert Mayes, USA:

Cinderella: Software Tool for Euclidean and Non-Euclidean Geometry

Although axiomatics account for a small part of the current boom in geometric research, the study of the axiomatic approach dominates the geometry taught in high school and college. The result is a curriculum where the geometry of plane figures is developed from a very narrow point of view. Students view geometry as an intellectual game of proof that has little or no relation to the "real world". In addition, many students do not see a connection between geometry and other areas of mathematics. If teachers present solely an axiomatic approach, they will propagate this approach among their students. The outcome is an isolated and outdated geometry course that serves to turn students off, rather than demonstrating the beauty and utility of geometry in our world. Breaking away form the current narrow curriculum provides for a variety of societal and mathematically desirable goals. Modern Geometry should aspire to attain some of the goals recommended by the NCTM in the Curriculum and Evaluation Standards for School Mathematics and the NCTM 1987 Yearbook: Learning and Teaching Geometry, and by COMAP in Geometry's Future.

Bronislav Pabich, Poland:

Magic Polyhedrons

Close your eyes and imagine that you are connecting the midpoint of a cube with its vertices by line segments, creating in this way six congruent square pyramids, which will completely fill this cube. Now duplicate each of these pyramids by reflecting each of them on the plane given by its base. You get now 6 square pyramids positioned onto the faces of the cube outside. The cube together with these six pyramids perform a new polyhedron. Draw this polyhedron in that way you can imagine it. Then answer the following questions:

How many vertices, faces, edges does have this new polyhedron?

Which kind of polygonal shapes are its faces of?

Are its faces congruent?

Is this polyhedron a regular one?

What's its volume? (Compare the volume of this polyhedron with the volume of the cube in regard with the method you did create it.).......

Pavel Pech, Czech Rep.:

Cubics and quartics on computer

In basic courses of geometry at universities are mainly linear and quadratic objects studied. Using computers enables us to include into this courses also objects, which are described by an algebraic equation of the order higher than two. With the co-operation with the students of the Pedagogical Faculty at the University of South Bohemia the software has been developed by means of which cubics and quartics (and conics as well) can be mapped in a high quality.

Eno Tonisson, Estonia:

Expression Equivalence Checking in Computer Algebra Systems

This paper investigates the possible educational application of equivalence checking and the capability of expression equivalence checking in some common computer algebra systems. The applications of equivalence checking can be analysed from the viewpoint of three types of users: that of the teacher, that of the student, and that of an Intelligent Tutoring System.

This paper deals with the way a computer algebra system copes with the checking of the basic equivalencies of algebra and trigonometry. It appears that the tools are far from perfect and require improvements.