Special Group 1:

Derive, TI-92, and other CAS

ICTMT 5, Klagenfurt, 6-9 August 2001

(Schedule, tentative as of 28/6/2001)

Chairs: Josef Böhm, Vlasta Kokol-Voljc, Bernhard Kutzler

Tuesday 10:30 - 11:15 Chair: B Kutzler, M Torres-Skoumal

Using a CAS to Teach Algebra - Going Beyond the Manipulations

Carl Leinbach (USA, leinbach@cs.gettysburg.edu)

Standardizing the Normal Probability Distribution - an Anachronism?!

Josef Lechner (Austria, lejos@aon.at)

Tuesday 15:15 - 16:00 Chair: B Kutzler, M Torres-Skoumal

Movies From the TI-92+

Detlef Berntzen (Germany, berntz@uni-muenster.de)

The Evolvente Curve of a Circle, used for Gear-Wheels. You need it everyday.

Hans Dirnböck (Austria, hans.dirnboeck@uni-klu.ac.at)

Tuesday 16:15 - 17:00 Chair: B Kutzler, M Torres-Skoumal

Ortskurven - Loci

Wilhelm Weißkirch (Germany, w.weiskirch@t-online.de)

Elimination of Parameters and Substitution With Computer Algebra

Guido Herweyers (Belgium, guido.herweyers@yucom.be)

Tuesday 17:00 - 17:45 Chair: B Kutzler, M Torres-Skoumal

Fermat's "Little" Theorem

John Cosgrave (Ireland, john.cosgrave@spd.ie)

How to Make Tests for Students Using CAS Tools (TI-89)

Bengt Ahlander (Sweden, ba@ostrabo.uddevalla.se)

Wednesday 9:30 - 10:15 Chair: M Torres-Skoumal

The Use of CAS in the Thuringian School System: Present and Future

Karsten Schmidt (Germany, kschmidt@wi.fh-schmalkalden.de)

Mathematics With Graphic and Symbolic Calculators - Teacher Training in Lower Saxony

Heiko Knechtel (Germany, HKnechtel@aol.com)

Wednesday 10:30 - 11:15 Chair: M Torres-Skoumal

Theorema-Based TI-92 Simulator for Exploratory Learning

Youngcook Jun (Austria, yjun@risc.uni-linz.ac.at)

Advantages and Dangers in the Teaching of Stochastics by Using CAS

Otto Wurnig (Austria, otto.wurnig@kfunigraz.ac.at)

Thursday 9:30 - 10:15 Chair: J Böhm

Introducing Fourier Series with Derive (Part I)

Alex J Lobregt (The Netherlands, a.lobregt@icim.fnt.hvu.nl)

Introducing Fourier Series with Derive (Part II)

Alex J Lobregt (The Netherlands, a.lobregt@icim.fnt.hvu.nl)

Thursday 10:30 - 11:15 Chair: J Böhm

From Pole to Pole

Joseph Böhm (Austria, nojo.boehm@pgv.at)

Curvature of Functions as an Aid of Treating Sequences, Limits, and Derivatives in Mathematics Lessons

Karl-Heinz Keunecke (Germany, kh@keukiel.netzservice.de)

Thursday 15:15 - 16:00 Chair: J Böhm

Computers in Engineering Education

Rolf Wasen (Sweden, r_wasen@algonet.se)

The TI-89/92 as a Tool for Analytic Geometry

Wolfgang Pröpper (Germany, wproepper@wpro.franken.de)

Thursday 16:15 - 17:00 Chair: J Böhm

Issues on Integrating CAS in Teaching Mathematics: A Functional and Programming Approach to some Questions

Halil Ardahan*, Yasar Ersoy (Turkey, ardahan@bote.fedu.metu.edu.tr)

Teaching Elementary Number Theory with a Software System

Mykola M Kolodnytsky etal (Ukraine, kolod@ziet.zhitomir.ua)

Abstracts:

Bengt Ahlander, Sweden

How to Make Tests for Students Using CAS Tools (TI-89)

In my school, Ostrabogymnnasiet an upper secondary school in Sweden, I work with a math class where every student uses the TI-89. The age of the students is 17 year. My thoughts about how to examine students using this powerful tool and still testing the understanding of mathematics will be explained. Questions such as “What are the roots of the equation x^2-6x + 5 = 0?” are not testing the understanding if you use the TI-89. But if you give the students the answer (the roots of a quadratic equation are x = 5 and x = 1), you can ask the students to give examples of equations that will give this answers. This is a kind of jeopardy in maths and really tests if they have the understanding behind the solutions of quadratic equations. We can also give questions with some solutions and ask the students to control and explain the steps in the solution. That will also test if the students can explain in proper way mathematical thinking. I will give some more examples in my presentation from my classroom experience.

Halil Ardahan, Turkey:

Issues on Integrating CAS in Teaching Mathematics: A Functional and Programming Approach to some Questions

In recent years we have attempted to study main issues and various research questions about integrating and implementing cognitive tools such as computer algebra systems (CAS) environments, in particular TI-92 calculator in both teaching and learning mathematics in Turkey. In this presentation, after overviewing the main issues and obstacles on the subject matter very briefly, we construct a new function, named digit spare function (dsf), a functional approach to two digit prime numbers and a programming approach to find the greatest common divisor (GCD) of integers. Finally, we present a few instructional materials, which were designed and developed in the viewpoint of new learning theories and models, namely constructive and discovery learning.

Detlef Berntzen, Germany

Movies from the TI-PLUS

Screenshots from the TI-92PLUS can be arranged to little movies (storage capacity of less than 30 KB) by using a GIF Construction tool. The technical details are easy to use and therefore of interest for pupils activities in math lessons. The lecture will be used to show the technic as well as to discuss the usage in math education.

Josef Böhm, Austria

From Pole to Pole, A numerical journey with an analytical destination

The TI-89/92 Data - Editor is an excellent tool to have a numerical approach to basics of calculus. We show how to combine numerical and graphical means to introduce discontinuities, differentiability and curvature. We find not only numerical, but also analytical solutions without using any calculus. Our starting point is a pole of a rational function and our destination is a pole of an evolute. This teaching unit can easily be presented with any other CAS.

John Cosgrave, Ireland

Fermat's 'little' theorem

To mark the 400th anniversary (on 17th August 2001) of the birth of Pierre de Fermat I will present a survey paper - using Maple - on his renowned 'little' theorem. I will treat the theorem itself, and present ideas relating to its applications to periods of decimal expansions, solutions to congruences, primality testing, Pollard's p-1 factoring method, and public-key cryptography. I will also consider some open questions relating to Fermat's 'little' theorem. I will pitch my talk at a general, non-specialist audience.

Hans Dirnboeck, Austria

The Evolvente-Curve of a Circle, Used for Gear-Wheels. You Need It Everyday

Gear-wheels are an important chapter of Kinematic Geometry. The terms to construct or to plot the evolvent curve of a circle are given. The fundamental law of gearing is explained. On two wheels we fix two evolvent curves; we proof that they can work as profiles of two gear-wheels. Special case: An evolvent curve fixed on a wheel and a straight line fixed on a rack are working as profiles. This gear-wheel mechanism You are using everyday in Your car, in the railway, the aircraft; in Your coffee-mill etc. You need it and You need Geometry. DERIVE, drawings, models to visualize it.

Guido Herweyers, Belgium

Elimination of Parameters and Substitution with Computer Algebra

Elimination of parameters and substitution with computeralgebra. Starting with the geometrical concept of parametric equations of lines and planes, we illustrate the method of elimination to obtain a cartesian equation. This elimination can be done in a direct and simple way by using the procedures "solve" and "substitute" (the basic algebraic manipulations of formulas) of a CAS. Without a CAS this method is difficult to realize by hand (e.g. solution of a system of two equations in a context with different "letters"). Therefore it was necessary to introduce in advance more elegant (but also more sophisticated) algebraic techniques like determinants. The result was that, for a lot of pupils, the meaning of the elimination process disappeared behind these algebraic manipulations. Later on in the educational process, we have the opportunity to show the equivalence and strength of the new algebraic techniques. These ideas will be illustrated in a few (geometric) examples.

Youngcook Jun, Austria

Theorema-based TI-92 Simulator for exploratory learning

One of the Theorema system¡¯s capabilities provides computing environment which can simulate the existing graphing calculator such as TI-92. Moreover, the deductive reasoning facility of Theorema allows the simulator to deal with propositional and predicate logic for pedagogical purposes. We present how to apply the use of such a simulator to help students explore mathematical ideas in terms of black box/white box principle. This experimental approach is demonstrated with our prototype by explicitly generating the sequences of calculator keystokes. Exploratory learning as a part of cretivity cycle is realized with algorithmic and logical empowerments built in the Theorema system.

Karl-Heinz Keunecke, Germany

Curvature of Functions as a Limit

A road sign "Dangerous Curve" can introduce to the problem. A car driving through a curve must not "cut" but osculate the road. For a short while, when the steering wheel is in a certain position the car moves on a arc of a circle. From this discussion all the expressions are available to define the curvature of a function by means of the radius r of the osculating circle as k = 1/r. We will realize the teaching unit using DERIVE 5´s new features to enable the students producing their own "notebooks" combining text, graphs and calculations.

Heiko Knechtel, Germany

Mathematic with Graphic and Symbolic Calculators - Teacher Training in Lower Saxony, Germany

History - organisation - contents of teachertraining in Lower Saxony: In Lower Saxony a new concept of teacher-training was developed from the mathematics advisers: Every math-teacher at highschool have to take part in 4 math workshops within 3 years. They should learn, how to integrate the new technology of the handheld calculators and dynamic geometry in their own math lessons. Interested teachers were trained within 2 years for math-multipliers. The math-multiplier-groups were divided in teams of two persons. Each team is responsible for six schools in their region. Each team focussing on special interests for each school and go ahead for four times with the groups. They will visit the colleagues in their own school and give several workshops there. Items of the workshos are handling with graphic and symbolic calculators and dynamic geometry; developing units with the new technology basing on their traditional math lessons by their own. After testing their own units during half a year the last 2 workshops give them a view on new possibilities in math lessons, specially in advanced or real-world mathematics. Supplementary every year in each region there are Regional T³-Conferences with a main lecture and up to 15 workshops all over the day.

Mykola M. Kolodnytsky a.o., Ukraine

Teaching Elementary Number Theory with a Software System

In this paper we show how to teach and to solve some computational problems of elementary number theory including modular arithmetic using the software tool "DSR Open Lab 1.0" designed and developed by the authors. We consider such computational problems as follows: to run the prime number test, to determine all prime numbers in some range ("the sieve of Eratosthenes"), to factorise a number into primes, to compute the GCD for a pair (or more) of numbers, to solve the systems of linear or polynomial congruences, i.e. polynomials modula m, to compute residue classes, i.e. modulo m, as well as the Euler phi-function, quadratic and power residues, reciprocal number modulo m, primitive roots modulo m, modular exponential, indexes, discrete logarithm, etc. We also give the comparison of the user interface implemetation of our software with the following: Maple V release 5, Mathematica 4 and DERIVE. The shown examples convince that the process of elementary number theory problem solving and teaching became easier now due to the visual interface of the presented software.

Josef Lechner, Austria

Standardizing the Normal Probabilitiy Distribution - an Anachronism?!

Numerical calculators have replaced all function tables (like tables for sin, cos, tan, ln, lg and so on) from textbooks and classrooms. Nowadays there is only one exception remaining: tables for the standard normal probability distribution (i.e. normal distribution with mean 0 and variance 1) can be found in every student textbook used in statistic courses. What are the reasons for this anachronism? Are there traditional or technical reasons or is it something else? What does it mean if the more or less time consuming process of scale transformation can be skipped?

Carl Leinbach, USA

Using a CAS to Teach Algebra - Going Beyond the Manipulations

In this paper I will examine two of the basic theorems from a first year algebra class, the Division Algorithm and its corollary, the Remainder Theorem for polynomials. These two theorems are the basis of much of the teaching and learning in a first course in algebra. Unfortunately, most of the students efforts are devoted to factoring polynomials and finding their roots with little gained in terms of insight as to why they are performing these tasks. In this paper we will show how we can use these theorems to write expansions of polynomials about x = a for a ¹ 0. Once this is done, students can learn about the idea of local linearity and tangent lines to the graphs of polynomials. I intend to develop two applications of these ideas. One is an application to pure mathematics, the other is to more real world settings.

Alex Lobregt, Netherlands

Introducing Fourier Series with DERIVE

In Electrical Engineering Courses functions such as the square wave Sq(t) and the sawtooth Saw(t) are frequently used. These periodic functions may well be approximated by a so-called Forier Series. In a workshop we will present some examples leading to an application, which can be shown by means of DERIVE as a first step in the filtering theory.

Wolfgang Pröpper, Germany

The TI-89/92 as a Tool for Analytic Geometry

The CAS calculators by Texas Instruments seem to be primarily suited for algebra and calculus at a first glance. The home screen menus give special emphasis to operations like "factor" and "comDenom" or "limit" and "taylor" respectively. For problems that typically appear in Analytic Geometry assistance is scarcely found. Solving vec­torial equations can only be achieved after a large-scale (and by that faulty) rewriting into systems of equations or into matrices. Functions of vector algebra are not available in the home screen but must awkwardly be looked for in a catalog. Texas Instruments however took care for a way out of that dilemma when designing the operating system. The user can easily create customized menus and complete not available functions by pro­grams of his own. In the contribution a menu together with some desirable functions is presented and shown how it can be put into action for solving problems that usually occur in classical Analytic Geometry.

Karsten Schmidt, Germany

The Use of CAS in the Thuringian School System: Present and Future

Based on a recent survey carried out in all 450 secondary schools in the state of Thuringia, Germany, the following questions will be investi­gated: Which level of computer equipment is available for classroom use? Which kinds (simple / scientific / graphical / symbolic) of pocket calculators are used in which grades? Does the school possess a license for a CAS? In a second part of the survey, the person filling in the questionnaire is asked to give some of his/her personal attitudes, which will also be analysed in the presentation: Which kinds (simple / scientific / graphical / symbolic) of pocket calculators should be used in which grades? Which knowledge does he/she have of symbolic calculators and CAS? What are the advantages and disadvantages associated with the use of symbolic calculators and CAS in the classroom?

Wilhelm Weiskirch, Germany

Ortskurven - Loci

Kurven sind mehr als Graphen von Funktionen. Dass die verbreitete unterrichtliche Reduktion des Kurvenbegriffs auf das Bild einer Funktion dessen mathematische Bedeutung und das didaktische Potential nicht annähernd ausschöpft, ist unbestreitbar. Insbesondere geometrische Zugänge zu nichttrivialen Kurven und deren analytische Betrachtung werden durch DGS und CAS ermöglicht und können dazu beitragen, die gegenwärtige Starrheit der Oberstufenmathematik zu durchbrechen. Am Beispiel nichttrivialer Kurven als Ortslinien abhängiger Punkte, bzw. Massenpunktbahnen sollen unter Ausnutzung der genetischen Methode deren Bedeutung und Potential für den Mathematikunterricht erörtert werden.

Rolf Wasen, Sweden

Computers in Engineering Education

I will present experiences from 1 ½ years at a mathematical Study Center and the use of computers and computer algebra in project works in the basic analysis courses. A model of how to use computer algebra in mathematical education was developed and will also be presented. It turned out that the computer was an indispensable tool for illustrating and testing mathematical ideas ­ this not at least for the teacher ­ and that the objections can be met with. There is an attractive possibility to continue these project works into research at different levels of ambition.

Otto Wurnig, Austria

Advantages and Dangers in the Teaching of Stochastics by using CAS

The use of CAS in the teaching of stochastics can be dangerous because the students like to use standard functions and functions which the teacher programmed as a tool without thinking. In student oriented thinking, however, CAS can well be used to gradually develop definitions and to help with the understanding of formulas and ways of solutions. The simulation of experiments by direct input of CAS commands makes it possible to put a stronger accent on the building of models.