Integrating Computer Algebra Systems in Post-Secondary Mathematics Education: Preliminary Results of a Literature Review

Chantal Buteau , Neil Marshall , Daniel Jarvis , Zsolt Lavicza

Brock University, CANADA; Nipissing University, CANADA; University of Cambridge, UK
cbuteau@brocku.ca, nm00ay@brocku.ca, danj@nipissingu.ca, zl221@cam.ac.uk

We present results of a literature review pilot study (326 papers) regarding the use of Computer Algebra Systems (CAS) in tertiary mathematics education. Several themes that have emerged from the review are discussed: diverse uses of CAS, benefits to student learning, issues of integration and mathematics learning, common and innovative usage of CAS, and integration scope in university curricula. Our analysis suggests that, perhaps contrary to popular belief, CAS integration in tertiary mathematics teaching occurs most frequently in courses for mathematics majors as opposed to service courses designed for non-math majors. The types of paper contributions indicate that the theoretical framework proposed by Lagrange et al. (2003) for literature reviews on technology use in mathematics education needs to be adapted to better address tertiary education, in particular for use in our upcoming comprehensive literature review that will build upon the pilot study review reported herein.

Using Symbolic TI Calculators in Engineering Mathematics: Sample Tasks and Reflections from a Decade of Practice

Michel Beaudin and Gilles Picard

École de technologie supérieure (ETS), 1100 Notre-Dame Street West, Montréal, Québec, Canada, H3C 1K3
michel.beaudin@etsmtl.ca , gilles.picard@etsmtl.ca

Starting in September 1999, new students at ETS were required to own the TI-92 Plus or TI-89 symbolic calculator and since September 2002, the Voyage 200. Looking back at these ten years of working with a computer algebra system on every student’s desk, one could ask whether the introduction of this hand-held technology has really forced teachers to reassess their goals in teaching mathematics. For some teachers - in fact, fewer than expected - the answer is “yes”. But what has really changed? Although some exam questions are different because students almost always have access to their calculator during tests, the curriculum is essentially the same. The power of computer algebra generally remains underused; it is considered merely a substitute for pencil and paper techniques or a way to illustrate concepts. This paper will give our personal perspective on this ten-year experiment based in part on informal discussions with colleagues as well as positive feedback from students, who have often thanked us for showing them how to use their CAS calculators efficiently. We will give examples of what daily use of computer algebra in the classroom should produce: a better appropriation of (many) mathematical concepts. Computer algebra systems are here to stay. Not using them won’t make them disappear.

Scilab and Maxima Environment: Towards Free Software in Numerical Analysis

Ángel Mora, José Luis Galán, Gabriel Aguilera, Álvaro Fernández, Enrique Mérida, Pedro Rodríguez,

Málaga University, Spain
amora@ctima.uma.es

In this work we will present the ScilabUMA environment we have developed as an alternative to Matlab. This environment connects Scilab (for numerical analysis) and Maxima (for symbolic computations). Furthermore, the developed interface is, in our opinion at least, as powerful as the interface of Matlab.

Towards the Development of an Automated Learning Assistant for Vector Calculus: Integration over Planar Regions

Yuzita Yaacob , Michael Wester and Stanly Steinberg

Industrial Computing Department, Faculty of Technology and Information Science, Universiti Kebangsaan Malaysia, Malaysia, 43600 UKM Bangi, Selangor, Malaysia. yy@ftsm.ukm.my
Department of Mathematics and Statistics, Center for High Performance Computing,
University of New Mexico, Albuquerque NM 87131-1141 USA.
Department of Mathematics and Statistics, Cancer Research and Treatment Center, University of New Mexico,

This paper presents a prototype of a computer learning assistant ILMEV (Interactive Learning - Mathematica Enhanced Vector calculus) package with the purpose of helping students to understand the theory and applications of integration in vector calculus. The main problem for students using Mathematica is to convert a textbook description of a problem into a form that Mathematica can understand and then choose the correct solution technique. ILMEV is designed to help students with this process. The typical presentation of this material in textbooks is not easily adapted to an interactive interface, so we developed a model of vector integration that allows ILMEV to present a structured overview of this material that helps students choose a correct solution method. Mathematica can solve the translated problem using its cylindrical algebraic decomposition (CAD) algorithm, but does not provide any explanation of what is being done. To overcome this, we implemented a simplified CAD algorithm which is used to reduce integrals appearing in vector calculus to sums of iterated integrals which Mathematica can then compute. This allows students to interactively compute closed form solutions to many two dimensional textbook examples using the ILMEV interface. ILMEV is built on important pedagogical guiding principles - interactivity, visualisation, experimentation, White and Black Box Principle, multiple representations and step-by-step technique with explanations. The user interface was critical for the implementation of the guiding principles.

Getting From x to y Without Crashing: Computer Syntax in Mathematics Education

David J. Jeffrey

Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7
djeffrey@uwo.ca

When we use technology to teach mathematics, we hope to focus on the mathematics, restricting the computer software systems to providing support for our pedagogy. It is a matter of common experience, however, that students can become distracted or frustrated by the quirks of the particular software system being used. Here, experience using the systems Maple and Matlab in the classroom is described with a view to highlighting the places where students are most likely to experience frustration. Strategies to help them avoid the common pitfalls are given.

A Comparison of Equality in Computer Algebra and Correctness in Mathematical Pedagogy (II)

Russell Bradford, James H. Davenport and Chris Sangwin

Department of Computer Science, University of Bath, Bath BA2 7AY, United Kingdom
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
R.J.Bradford@bath.ac.uk, J.H.Davenport@bath.ac.uk, C.J.Sangwin@bham.ac.uk

A perennial problem in computer-aided assessment is that “a right answer”, pedagogically speaking, is not the same thing as “a mathematically correct expression”, as verified by a computer algebra system, or indeed other techniques such as random evaluation. Paper I in this series considered the difference in cases where there was “the right answer”, typically calculus questions. Here we look at some other cases, notably in linear algebra, where there can be many “right answers”, but still there can be answers that are mathematically right but pedagogically wrong. We reformulate the problem in terms of articulating the sought-after properties, which may include both mathematical equivalence and algebraic form.

Using Forensic Investigations and CAS to Motivate Student Interest in Mathematics

Patricia Leinbach and Carl Leinbach

Adams County Coroner’s Office, Gettysburg, Pennsylvania, USA
accoroner@embarqmail.com
Department of Computer Science , Gettysburg College, 300 N Washington St., Gettysburg, PA, USA, 17325
leinbach@gettysburg.edu

In this paper, we are proposing the use of forensic case studies as a means to provide students with interesting problem solving opportunities that capitalise on the popularity of several TV series and shows. It also satisfies their natural curiosity about how answers are found to seemingly complex real life problems. We begin with a very brief explanation of our choice of forensic science as a vehicle for teaching mathematical topics. Next we define the term forensic science or forensics, give a brief overview of the duties of the coroner, and where mathematics can be used in coroner’s and police investigations. We conclude with a description of some of the activities we have in our collection of case studies. All case studies have a basis, but are not exact presentations, in cases with which the first author was involved.

Mathematics Education with a Handheld CAS – The Students’ Perspective

Karsten Schmidt

Schmalkalden University of Applied Sciences, Blechhammer, 98574 Schmalkalden, Germany
kschmidt@fh-sm.de

After carrying out a project over several years in eight upper secondary schools in Thuringia on the effects the use of a handheld CAS (Computer Algebra System) has on the students’ skills in mathematics, the former ban on the use of such technology in mathematics education was revoked. Late in the project (2002) a survey of all students in the project schools was carried out to find out students’ attitudes regarding the handheld CAS device they were using. Since 2003, each school in Thuringia can decide for itself if a handheld CAS is used in mathematics education or not. One year later, more than a quarter of all Thuringian upper secondary schools used handheld CAS in mathematics classes. The 2002 survey was repeated in 2005. The results of the two surveys are analysed, with special focus on the question as to whether certain characteristics of the students (e.g., their sex or how good they are in mathematics) influenced their answers.

 

 

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