Abstracts:
Experiences with CBL and the TI-92 in Austrian High School classes integrating Math, Physics and Chemistry
Experiments carried out by students are very motivating and lead them to a better understanding of principles and processes in sciences. Most of the experiments have only qualitative aspects. The students can only see that something has happend in a certain way.
However, for visualization and a mathematical analysis of experimental data, quantitative investigations are required. The main objective is to find functions fitting best to the experimental data and to interpret the special values of the parameters of these functions. The students have to combine knowledge about the different types of functions with the knowledge about chemical and physical facts.
As an additional aspect, the students also have to take care of the accuracy in working for obtaining good results. The collection of large lists of data is supported by the TI-CBL system. Mathematical experimenting, complicated mathematical computations and visualization are supported by the TI-92.
In this presentation we will report about the experiences made with students at the age of 17 to 18 carrying out experiments in chemistry and physics in science courses. We concentrate on how the students were able to use their knowledge from mathematics and sciences in these practical situations.
Exams in the perspective of an intensive use of CAS in a regular course
In this paper I present my own experience with exams in the context of a very intensive use of DERIVE with 18-20 years old students in a regular courses with groups from 20 to 50 students, namely not with experimental groups or with gifted students. This experience began in 1994 in a mexican University and will continue in the future. All courses are of one semester and follow an almost traditional treatment of Calculus, Vector Calculus, Linear Algebra and elementary Ordinary Differential Equations, from the beginning of the experience the use of computer was allowed in all exams. I present several examples of:
· my own process of gradually adaptation of traditional exam questions;
· practical and institutional obstacles and how we overcome them;
· some changes in my teaching style and in the learning style of my students;
· assessment in a regular course;
· global opinions from the students about their computer-based course.
The experience set up a very important basis for a second step in my University: the beginning in 1996 of a series of teachers' training workshops about the integration of technology information in Mathematics Education. After several workshops we find that when computer is present is more easy to teach students than teachers. I present too examples about
· some obstacles in teachers' training and;
· materials designed to support teachers' training.
Alenka
Cvetkovic & Dezider Ivanec & Toncka Spegel-Razbornik & Karolina Starin
Gimnazija
J. Plecnika Ljubljana
Slovenia
gjp@guest.arnes.si
In our paper Quadratic function is chosen to compare methods
of teaching mathematics with or without CAS. We focus on exam questions nevertheless
we make some general observations.
Some of the existing (traditional) exam questions will be adapted to the
use of CAS or substituted by the new ones.
Both groups of questions will be analysed according to the skills and abilities
required from the student.
We will show the advantages and disadvantages of using CAS in mathematics teaching
according to the learning goals.
We will refer to the possible or necessary changes in the mathematics curriculum
as the consequence of using CAS.
We will point out that using CAS requires a process embracing various educational
levels.
"FUNCTIONS" with hand-held calculators and dynamic geometry in our teaching, our tests and our exams (TI83, TI92, Cabri) – new problems with new aims – new exams with new shapes
We will show:
1/ How one can perceive "functions" if one uses calculators in a primitive way.
2/ The hidden tools of calculators or softwares that we must discover behind the specific tools labeled in the utilisation mode. Our mathematical and information knwoledges permit us to create them and consequently, new ways of teaching and new exercices with numerous changes of surroundings.
3/ Exemples of exercises and problems needing new technologies, allowing teachers to develop new abilities for their students: to experiment, to conjecture to validate and also to proove.
4/ Exemples of new mathematical works to evaluate students replacing exams and :
exemples of problems that can be asked for students as exam to evaluate "level" and new abilities.
We detail a study in which a class of year 7 students were given a concentrated block of lessons with spreadsheets, including word problems. Students were tested before and after the spreadsheet work for their ability to solve problems algebraically, and the results were compared, for memory priming effects, with a control group of the same size, who had no classroom exposure in that period to either spreadsheets or algebra. The results demonstrate that, to a high degree of probability, the spreadsheet group made significant gains in their algebraic skills and understanding of variables. Post-treatment interviews with students added weight to the statistical results.
Most students have difficulies to understand the function concept, its graphical representation and interpretation, eg. word problems, graphical and symbolic representations of various situations and events, etc. However, the finding of current theories of mathematics education suggests that hand-held technologies, e.g. graphics calculators, computer algebra systems, have the potential to assist students in develop structural conceptions of functions and use the language of graphs in communication. In this paper, the authors will reflect a group of 9th grade high school students’ perceptions on the use of TI-92 and their performance in understanding of the linear function concept and the language of graphs.
More specifically, we will report a sample of designed activities on TI-92/Derive supported algebra and pre-calculus teaching materials on linear and quadratic functions as well as the responses of students from Uskudar American Academy in Istanbul. Findings in the present study supports the views that the graphics calculator has some effect in bridging the gap between operational and structural conceptions in understanding functions, namely tabular, symbolic and graphical representation. Moreover, the applications of graphics calculators into schools mathematics face with some difficulties and obstacles in teaching methods, assessment etc. Besides the access to the technology and new roles of teachers, the faced difficulties are related to the contents of what we teach and what we consider basic skills that students have to obtain. Thus, in the technology supported mathematics teaching 15-16 year old students, there should be clear shifts in the goals, contents, methods and assessment as can be seen in this short presentation.
Thomas Himmelbauer
Gymnasium Neulandschule, Vienna
Austria
j.himmelbauer@chello.at
The application of Cabri-Geometry of TI-92 in maths lessons is not wide spread in Austria, although many classes in Autria are using the TI-92 in maths lessons. There are four important reasons for that:
The handling of Cabri-Geometry of TI-92 is complicated.
The time of one single lesson often is too short to explain even only one problem.
If pupils miss a lesson using Cabri-Geometry, it is hard for them to learn the subject matter by themselves.
Maybe it is difficult to create exam questions using Cabri-Geometry.
So I will demonstrate a way to solve these problems. I will show some examples for lessons preparing an exam using Cabri, some examples for exam questions using Cabri and some examples of tests of pupils.
Calculator modulated arithmetic in elementary school
A study of currently available elementary school mathematics curriculum suggests that the curriculum is computationally oriented. Computational algorithms are nothing more than a set of rules for efficient paper and pencil answer findings. Although it is important for children to be able to perform arithmetic algorithms, time spent teaching long algorithms does little to help children understand mathematics and its application.
The definition of computation must be broadened and we must become more effective at teaching computation by shifting from a computationally based curriculum to a conceptually oriented curriculum using the calculator as an instructional tool. For example, dropping long division could create time to be devoted to application and problem solving where the focus would be on when to divide, not how to divide. Arithmetic itself can be learned as a powerful mathematics topic, not just a tool to assist in learning other mathematics topic.
The paper will investigate the problem of f knowledge of arithmetic and suggest how calculator can be used in an exploratory and investigatory way, and how it can help children in constructing their own understanding of arithmetic. All three methods of computation : written computation, mental computation (also estimation), and calculator should relate to conceptualisation and problem solving. We see calculator modulated arithmetic as a possible solution for redefining school arithmetic in terms of enhancing understanding and problem solving.
Vedic (also pronounced as Vaydik) Mathematics is an ancient system of mathematics originating in India in Vedic times. It was rediscovered between 1911 and 1918 by Sri Bharati Krsna Tirthaji Maharaja(1884-1960) who studied the ancient Indian texts called "Vedas". The Vedic system is extremely refined. The methods are simple and complementary, so that for example, ‘long’ division is a simple reversal of the one-line multiplication process; similarly with squaring and square-roots.
Anyone familiar with the Vedic system will be aware of the remarkable Vedic techniques: ‘ difficult’ problems or huge sums, which can be solved immediately by the Vedic method. These striking and beautiful methods are just a part of a complete system of Mathematics, which is far more systematic than the "modern" system. Vedic Mathematics manifests the coherent and unified structure of Mathematics and the methods are complementary, direct and easy. The simplicity of Vedic Mathematics means that calculations can be carried out mentally and this is very much encouraged in the Vedic system. There are many advantages of using a mental system. One of the very important advantages is that "It trains the mind with these elegant methods".
The Vedic system works on the base of 16 Sutras (aphorisms) and their sub- Sutras(corollaries). These Sutras and sub-Sutras can be individually applied or applied in various combinations to a problem to arrive at it’s solution.
In short, Vedic Mathematics is the fastest and the most elegant route to solution.
( I plan to discuss a few applications of Vedic Mathematics).
Symbolic
Problem Generation in a Computer Algebra System
Since it is generally considered that the role of a Computer Algebra System (CAS) in education is to enable students to solve hard, real-life problems without getting bogged down in minutiae of computation, the issue of generating plain drill exercises within CAS environment is usually not viewed as important. The premise of this paper is that certain fundamental skills of basic algebra have to be mastered the old fashioned way - through repeated practice of drill exercises. Today, typical high-school algebra teachers are faced with either manually creating their own problem sets or using workbook / software combinations that often do not meet their specific didactical needs. This is a surprising state of affairs, considering the power and flexibility of existing symbolic manipulators. In this work we will describe how an educational CAS (Algebrator) can be used to generate problems that are based on teacher defined templates. Methods of template problem design, multiple test creation and generation of correct and incorrect solutions will also be discussed.
Can procedural and conceptual mathematical knowledge be linked through computer assisted learning?
It has been recognized that computers can be used to introduce a new balance of instructional time by decreasing the time for procedural skills and increasing the time for conceptual understanding. If we agree that a main goal of mathematics education is developing both procedural and conceptual knowledge and making links between the two, a very important research question regarding computer-based mathematics education proposed by Kaput is "how different technologies affect the relation between procedural and conceptual knowledge". However, just few CAL studies have examined (usually secondarily) whether their computer-based treatments linked procedural and conceptual mathematical knowledge. This study summarizes their findings, explains how these knowledge types may be linked according to some papers examining this question in fuller detail, and proposes two constructivist CAL environments enabling these knowledge types to be connected. The study reports some outcomes of research that is currently being done with Lenni Haapasalo from University of Joensuu, Joensuu, Finland.
An beruflichen Schulen in Baden-Württemberg (Deutschland) finden CA-Systeme langsam auch Eingang in die Lehrpläne verschiedener Schularten.
Zunächst wurden für das Fach Datenverarbeitung
vier Grundkurse "Computer-Algebra-Systeme" entwickelt (Grundlagen,
Programme und
Datenstrukturen, mathematische , wirtschaftliche Anwendungen). Typisch für diese
Lehrpläne ist,.dass die Schulen selbst wählen können, ob sie mit einem portablen
Rechner oder mit einem PC-Programm arbeiten wollen.
Es folgten Lehrpläne für Mathematik an den Technischen Gymnasien (Grund- und Leistungskurse); hier wird an verschiedenen Stellen auf den Einsatz von CA-Systemen hingewiesen; an ausgewählten Schulen wird auch der Einsatz in der Abiturprüfung erprobt.An den sog. Berufsoberschulen (vergleichbar den Fachoberschulen anderer Bundesländer) ist der Mathematikunterricht ohne CAS jetzt praktisch nicht mehr durchführbar. An dieser Schulart wird angestrebt, dass "jede Schülerin oder jeder Schüler Zugriff auf einen CAS-Taschenrechner oder auf einen PC mit geeigneter Software hat."
In mehreren Bereichen müssen jetzt Erfahrungen gesammelt wrden:
 Die Methodik des CA-Einsatzes im Unterricht.
 Die Erstellung von zentralen Prüfungsaufgaben
 Die Organisation der zentralen Abschlussprüfung
 Die Finanzierung der Rechner oder der Software
In verschiedenen Veranstaltungen zur Lehrerfortbildung wurde zunächst ein Überblick über die gängigen CA-Systeme gegeben; dann wurden gezielte Kurse zum Einsatz DERIVE und zum TI-92/TI-89 angeboten. Wir sind jetzt dabei, Prüfungsaufgaben für Mathmatik mit CA-Einsatz zu entwickeln und zu erproben.
Since Fall 1999, in Quebec, the Science Program of Cegeps “Collèges d’Enseignement Général et Professionels”, where students come after 5 years of high school, was reformed in order to ensure that science students will become computer-literate during their two years at college. As an example, for the two courses of Mathematics: Integral Calculus and Linear Algebra, it was advised to introduce the students to the Canadian CAS (Computer Algebraic System) Maple. Inevitably these new guidelines have not been popular and well accepted by older teachers, close to their retirement and unwilling to change their pedagogical methods, and also have been disdained by “purist” mathematicians who hate to see any hardware infiltrating Mathematics. With a team of other mathematicians, I give personal tutorials to our colleagues, organize workshops and edit pedagogical materials to help them to get familiarized with Maple environment.
I would like to illustrate the benefits brought by Maple in the teaching of Linear Algebra. For this discipline, students are asked to work in successive frameworks Vector spaces, Systems of linear equations, Algebra of matrices, determinants and linear transformations. Very often they fail to observe the equivalencies between these various modes. Using the power of animation of Maple and as well its capacity of symbolic calculation, I will show a sequence of tasks dealing with the parameter time, that should succeed to exhibit unity in Linear Algebra representations.
We present the Alice Maple Web Server ( http://calculus.rug.ac.be:8080/ ) whose purpose is to automatically administer and grade math homework.
The key feature of this system is its use of Maple to create and evaluate questions. This means that one can easily ask questions requiring symbolic solutions, not just multiple-choice or numeric answers. The use of Maple also makes it easy to design, display and automatically grade mathematical questions. In addition, any web features (such as graphics, java applets) can also be incorporated into quizzes.
The system also incorporates features to encourage student feedback such as a feedback button and student surveys.
This system will be used in Spring 2000 as an aid in teaching vector calculus at University of Gent, as well as a course on ODE's at Vrije Universteit Brussel. We expect to be able to report on this experience.
Historical Curves and New Technology
In order to solve problems (e.g. the 3 classical problems)
the ancient Greeks invented special curves (conchoid, cissoid, quadratrix, Archimedian
spiral). These curves, which are rather complicated to draw by hand, can easily
be demonstrated with the help of new technologies. Hence the opportunity comes
up to reanimate these interesting objects in mathematics education and to discuss
their historical background. With technology we can also show the advantages
of other coordinate systems beside the Cartesian system (and the - overemphasized
- graphs of functions).
When implementing these curves into classroom teaching an important question
arises: How to pose related exam questions that fit into the tradition of assessment
in mathematics education? Proposals will be made for such exam questions as
well as proposals for changing the tradition.
Two-Tier Exams as a Way to Let Technology In
We believe that two-tier exams would be a well-balanced compromise meeting both the desires of technology supporters and the reservations of those who are concerned about the use of technology in the classroom. Some fundamental thoughts about two-tier exams are presented.
Bernhard
Kutzler
ACDCA, Leonding,
Austria
b.kutzler@eunet.at
Introduction to the Symbolic Calculator TI-89/92
The TI-89/92 is a handheld calculator that combines numeric
computation and graph plotting with the techniques of computer algebra (à
la Derive). The TI-92 also includes computer geometry (à la Cabri Geometre
II). After the numerical and the graphical calculators, the TI-89/92, as a symbolic
calculator, introduces a new dimension in mobile mathematical problem solving.
You learn working with the TI-89/92 through selected examples, most of which
are related to school mathematics teaching
Solving Systems of Linear Equations with the TI-89/92
We describe experiments for using the algebraic calculator TI-89/92 for teaching and learning how to solve systems of linear equations by means of the numeric method of table refinement, the graphical method of intersection point determination, and the algebraic methods of Gaussian elimination and substitution. Our approach follows the educational concepts of experimentation, visualization, and the scaffolding method. In addition the scaffolding method is described in further detail.In order….
Introducing Differential Calculus with Mathematica
We intend to present a short series of lessons that we have developed to introduce informally the concept of a limit, particularly the concept of a gradient of a Ôwell behavedÕ function. We have developed a Mathematica notebook with several user defined functions that will display graphically mathematical functions, approximations to the gradient of a function at a given point and the graph of the Ôgradient functionÕ. We will explain how we use this system to introduce the concept of a differential for the first time and how we go on to encourage the students to find empirically the rules of differentiation. We intend to discuss how we assess the students progress through this activity, both with assessed computer/calculator work and with more conventional classroom test questions.
In the talk, which is mostly intended as discussion between presenter and participants, some thoughts and questions concerning the usage of technology in teaching mathematics are going to be presented. Unfortunately the talk will not give answers to those questions. It will just try to provoke the discussion about those issues.
Introduction of technology often leads that people lose certain capabilities - how to avoid this to happen when introducing computer algebra systems? Mostly all computer algebra systems have much too powerful capabilities, especially at the certain moment of teaching. It is easy to forbid using Solve for example - but does not this introduces to much negative attitude from students towards subject taught at that moment? What is basic knowledge in mathematics - how to identify it? Does the change in teaching a certain subject gives rise to mathematical knowledge on one side but drop the certain other skills on another side.
Do we really want to teach how mathematics is used: how to reach all different aspects of it (mathematics in chemistry lab, mathematics in shops, mathematics at the sports training) – are we not better off with developing the logical reasoning, pattern recognition, ... Are "artificial" examples perhaps not better?
What tools to use in teaching: general purpose tools or specialized tools? How to identify the benefits of one or another? How to measure success in introducing technology into learning process and to eliminate the effect of "more interested because is something new"
Slovene final external examination in the view of Computer Algebra Systems
In Slovenia as a prerequisite for studying at University students have to pass the External Examination at the end of high school, consisting of five subjects (three mandatory and two chosen). Mathematics is the mandatory one. At the present moment usage of graphic calculators as well as those with symbolic algebra capabilities is forbidden during the examination.
In the talk we will take a look on the last few exams. We will classify the questions according to the scheme proposed by V. Kokol – Voljè in Exam Questions when using CAS for school mathematics teaching, ACDCA 5th Summer Academy, Gösing (Lower Austria). We would try to draw some conclusions about what these External examinations really measure and test.
In the second part participants will on the basis of one chosen examination paper try to produce a paper, which will cover the same mathematical topics as previous one, but in the way students could use CAS tools.
Many adults remember their mathematics lessons at school as a bad time in their life. "Why should I learn this?" is a question posed frequently. Now you can get an answer on CD ROM made by Andreas Stoeckl and Juergen Maasz.
They use multimedia to show how and where mathematics is used to understand and solve real world problems. Five examples are presented with video, computer animations, text, formulas, grafics and interactive parts.
The CD ROM has two other chapters. One is a glossar with a short summary of the mathematics that is presented on this CD ROM. It has some interactive parts. For example the variation of graphs of typical functions are shown. If you change a parameter the computer changes the graph simultaneously. The third chapter contains a lot of training tasks about functions, linear and quadratic equations, elementary and analytical geometry, calculus and linear optimization. The feedback for each task is not only "right" or "wrong" but a solution and sometimes a hint what else should be learned.
Assessment in the CAS age: an Irish perspective
Computer Algebra Systems (CAS) are becoming more widely available at reasonable prices. Hand-held devices, graphic/symbolic calculators, are in widespread use. The effect, in particular on mathematics curricula and examinations, is only beginning to be seen.
In the Republic of Ireland the Department of Education administers statewide examinations at the end of second-level schooling; these examinations are called the Leaving Certificate. The results of these examinations are used by employers and also for entry into third-level education. The most prestigious subject, for various reasons, is honours mathematics.
The purpose of this study is to analyse how the availability of CAS would affect the Leaving Certificate mathematics papers. The most recent honours mathematics papers are analysed in the following manner: the questions are answered with the help of DERIVE; a classification of the types of question is then made; fundamental questions about the curriculum arise. The idea of a CAS-index for examinations is introduced; it consists of two components:
(i) How CAS-proof are the questions?
(ii) How well do the questions exploit the possibilities of CAS?
Following on from the use of networked computer assisted assessment for 9 years, on-line assessment has been used in the School of Computer Science and Mathematics for over two years. Initially It was introduced for self-assessment only, but is now in its second year of use for formal tests and examinations.
The key to its successful introduction has been the exploitation and extension of a sophisticated new suite of programs, Question Mark Perception (http://www.qmark.com/perception) which supports rapid authoring, flexible delivery, secure management and results analysis of tests.
Basic mathematics test questions may be MCQ, MRQ, numeric, text, match and other types. An example of these are the Mathletics tests, developed at Brunel University in the UK, which have been converted for on-line delivery at Portsmouth (http://l62.csm.port.ac.uk/mathletics.html) and are openly available.
In some tests random parameters have been added as an extra feature to enable greater question variety.
Other test questions use additional software in support of on-line tests. For example, Maple/Matlab skills and maths problem solving are formally assessed on the Web at Portsmouth. (see http://meat.csm.port.ac.uk/cmp108/assess.htm)
Recent work is focusing on extending question banks, developing new random question types, exploiting MathML and advancing the use of Maple via Java servlets.
The purpose of our talk is to show how students can learn the basic properties of arithmetic operations on polynomials using the TI-92 calculator with CAS.
The related questions can be devided into two subgroups: similarities with operations on numbers and main differences. To make the connection between arithmetics of integers and polynomials more transparent, we recommand the use of vector notation of a polynomial (without explicite indication of the variable). The connection between both notations is quite simple and unique in both directions. The vector notation of the polynomials allows students to apply their knowledge of linear algebra when working with such nonlinear objects as polynomials are. Viceversa, some abstract definitions and theorems from linear algebra can be better understood by students through their experience of work with more familiar polynomials.
In any discipline, teaching and learning methods are conditioned by the available tools which is also the case for Mathematics. In this way, two thousand years ago, Geometrical aspects of Mathematics were studied by means of Euclidean Geometry (Ruler and Compass) and calculus were made by using the Calculus Ruler which has been used until the sixties. In particular, it has been necessary to make complex calculus in order to carry out any mathematic study. This feature has distracted the student from the mathematical theoretical basis of his study, and so, from his main goal.
The appearance of the personal computers (Pc´s) allowed us to access to a powerful numerical calculus tool in an easy way. Later, the Computer Algebra Systems (C.A.S.) were developed for the Pc´s (ten years ago). This technology frees us to deal with the heavy calculus and it allows us to operate with algebraic expressions like a student does, i.e. by applying directly the theoretical results.
In this sense, this technology is going to increase both the number of questions per exam, as well as, to permit more complete and closer to the reality ones (due to the lack of slowly calculus).
Our teaching work is developed on the following subjects: Mathematics, Physics and Computer Science, over 16 - 20 years old students. Due to this fact we have chosen an interdisciplinary C.A.S. like Mathematica.
We have developed some packages and notebooks over these subjects:
- Error calculus.
- Function graphic representations.
- Torsorial Statics.
- Network analysis.
- Dynamical System analysis.
Also, we have described the basic mathematic knowledge for each of them, without which the software becomes useless.
They make up an useful basis for developing exam questions, which can be used by the students in order to make self-evaluation.
Graphic Calculator Problems in the Israeli Math Matriculation Examination: Analysis of Questions and Students Answers
Beginning in 1995 an experimental group of 20 high schools were allowed to use graphic calculators in the final mathematics examination at the end of grade 12.
The math group in the Science teaching center at the Hebrew University was involved in planning the curriculum of the experimental program and in the preparation of some of the problems of the final examination.
In the final examination the ministry was interested to test the students abilities to use the technology in new types of problems and ensure their capabilities to solve standard questions. The use of graphic calculators in the exams raised two major issues for committee, which prepared the matriculation:
One. Some of the standard problems gave advantage to students using graphic calculators and had to be changed.
Two. New problems had to be written such that students had to show actual use of the technology.
In this presentation we will show some of our special exam problems focusing on the following issues:
1. The evolvement of the special graphic calculator problems.
2. The influence of the students answers on the phrasing and content of the problems.
3. The time needed to solve meaningful problems using technology.
4. Recommendations towards new standards for technology – supported exam questions.
An Introduction to new methods of assessment using the TI-92
The presentation will focus on the way traditional exams are enriched or replaced by different, new formats of examination in a project currently in use at an Austrian high school (9th grade level). In addition to explaining the new exams, the presenters will also introduce the idea of yearly "focus papers" and oral presentations thereof done by each student. The independent research, experiments and reading for the focus papers help students integrate calculating and problem-solving skills across the curriculum as they rely on material from the areas of physics, finance and economics, to name a few.
The project thus interrelates different competencies accumulated by the students in their various subjects and fosters independent work as well as the taking on of responsibilities on the part of the students. The new formats of examination and presentation constitute a link between assessment and the broad spectrum of mathematical proficiency that is actually taught, thus foregrounding the significance of reasoning, interpretation and problem-solving skills.
‚Representing‘ in CAS-supported Mathematics Teaching
‚Representing‘ is a basic mathematical activity besides ‚manipulating‘ and ‚interpreting‘. The aspect of represention is paid particular attention in papers concerning the use of CAS in mathematics teaching. (Special terms such as ‚multiple linked representation‘ or ‚window-shuttle-principle‘ are used, numerous teaching examples are developed.) Arguments for emphasizing this aspect are mostly very general and superficial, profound educational analyses can hardly be found.
In this presentation the meanings of representations in CAS-supported mathematics teaching are analyzed according to the following questions:
What is the meaning of representations for mathematics teaching from a social and from an individual point of view?
Which specific contribution does CAS make to represent mathematical subjects? What are the differences between ‚CAS-representations‘ and ‚representations by hand‘? What is the innovation?
Does the meaning and the relevance of some representations change by CAS?
Which basic abilities and basic skills are necessary for using CAS-representations in an adequate way?
How can the issues mentioned above be reflected in exams?
The role of technology in improvement of testing
In Mathematics, knowledge is tested by written exercises and oral questions.
In its own way, each testing is used to measure the level of student's understanding of a particular topic.
Nowadays, the two ways of testing are clearly distingueshed by teachers as well as by students. Thus students prepare in two different ways: for oral testing they study mathematical facts whereas for written test they practice doing exercises.
Both ways, however, often include little Mathematics, little learning by comprehension. The use of modern technologies can connect both ways of testing. What is more, it can provide a new quality. Exercises can be made that comprise theoretical knowledge and doing exercises that are realistic and connected with real life.
We would like to show on examples how this is possible.
Widespread computer use is introducing new possibilities in teaching science. There are programs that perform powerful numerical and symbolic operations which reduce user's (student's) work. Such tools need to be considered by science educators. How can we develop mathematical reasoning and problem solving techniques if many of them can be solved by computer programs?
I use Scientific Notebook's Exam Builder to tremendous benefit. The program picks question variations and random variable values and produces solutions according to rules I define. This way I can concentrate more on the concepts of exam questions instead of a particular exam. It is easy to create multiple choice questions because Scientific Notebook can calculate both correct and wrong choices. So I prepare such exams more often, whenever I believe that basics or some fragments of knowledge should be tested. After students gain some confidence with basics, they can take tests with complex problems. Students like variety, even in testing. And by placing tests on a WWW server, students get a huge test bank available wherever they might be.
The introduction of new technologies has brought on many changes in contents, strategies, and attitudes in education: the computer can be used as a complementary tool for educational training as well as a learning tool on its own.
A key aspect of the use of computers as aids to learning mathematics is the possibility of presenting the subject in an interactive form more accessible to novice exploration than the traditional presentation by marks on paper .
Modern learning theories emphasise the importance of constructivism when integrating technologies in learning. Constructivism based learning is seen as a building process in which learners have an active role and obtain new knowledge by constructing it on the basis of previously acquired knowledge.
Computer based education (C.B.E.) represents a link between traditional teaching techniques and computer technology and also offers a unique opportunity to enlarge their learning possibilities. Efficient computers based tutors can be realized inside a Mathematicaä 4.0 Notebook which is a complex system of numerical and symbolic analysis, and thus a great aid to all those people who make mathematics at every level.
The work's aim is to describe a prototype of package for teaching of the mathematics focused on a trigonometry lesson: the triangles resolution. The package is implemented using the Mathematica programming language and library functions provided by Mathematica. It is structured
in a main menu which contains: theory, exercises and examples. Buttons and hyperlinks allow the student to see again during lesson, definitions and concepts that he cannot remember at the moment.
This package has been used as a subsidiary instrument in the learning process of secondary school students. At first the theoretical concepts on triangles resolution have been introduced. Then the class has been allowed to use the package during some practice sessions.
Students have received very gladly the didactic innovation and step by step development of the exercises turned out to be very profitable.
The Vienna International School has been using the TI-92 with select classes in the secondary school, particularly in grades 6 – 10. Our class sizes range from 15 – 27 in these grades. A model for learning and reinforcing manipulative skills through the use of group work will be presented, including the assessment of these skills. Potential difficulties in group work situations, such as varying language and technical skills, and diversity of mathematical abilities and backgrounds, will be discussed with particular attention to using these differences to maximize skill acquisition.
In addition, limitations of the calculator as a pedagogical tool will be discussed, as well as ways to circumvent the problems encountered using CAS to introduce some algebraic topics. For example, teaching rearrangement of formulas (particularly important in the sciences) is often counter-productive on an introductory level using the TI-89/92 since it renders expressions that are not immediately recognizable as equivalent to those obtained from pencil and paper methods (e.g., it will alphabetize variables, factor out negatives, etc.). When consolidating this topic, however, it affords an opportunity to examine algebraically equivalent expressions through self-assessment. The students rearrange using traditional paper and pencil methods, then obtain CAS solutions. They must evaluate their own solutions by determining if their answers are equivalent to those the calculator renders.
In the last few years more & more teachers in Austria have allowed students to use personal and pocket computers in mathematics when writing tests. Consequently, examples were gradually adapted to the new demands of computer algebra systems. Assessment, however, was still based on the traditional rules of test writing.
In 1997/98 a lot of teachers in Austria used the new TI-92 with CAS in teaching and for tests. At their final meeting in August seventy of them summarised their most important and sometimes unexpected findings. In spring 1999 a team of teachers developed some variants of a new model of assessment under the leadership of H. Heugl (ACDCA). In accordance with the Ministry of Education experimental studies to test the new model are carried out in the present school year (1999/2000). I chose to use the following variant in form 11:
The pre-set time for written tests in a school year - 350 minutes in form 11 - can be used in different ways:
· Short tests - up to a maximum of 25 minutes - to check reproductive skills or reproductive knowledge (possibly without CAS too).
· One longer test per term, e. g. 100 minutes, to check problem solving skills. There should be sufficient time to experiment, and to use materials which have been worked out in school or at home.
· Each student should work on a short chapter of mathematics - which has not been dealt with in school - and present it to the other students.
In my presentation I am going to show both, an example of a short as well as of a longer test. I am going to report about the different reactions of the students and the new experiences I have made so far.