Symbolic Computation and Education

The following sections are included:

• PREFACE

• CONTENTS

Math education in the United States is in crisis. Student performance and desire to learn is declining. Some groups feel that the cure is to drive math education back and increase rote memorization and teaching to the test. The fact is that the math taught 50 years ago will not prepare today's students for today's world.

In the last decade, the integration of technology in the teaching and learning of mathematics has increased substantially. As a result, many graphical ideas, some numerical ones, and some good modeling problems have found their way into the precalculus literature. At the same time, relevant concepts, methods, and approaches, now available via technology, are being ignored. In this article, we review examples of accessible concepts, approaches and applications, not yet present in most precalculus books, that illustrate how the numerical and graphical capabilities of graphing calculators together with the variety of data types that they provide, can further be used to enhance the teaching and learning of key calculus concepts.

While extraordinary gains have been made in the capabilities of and access to symbolic computation in many educational settings, there are still many secondary school mathematics classrooms in many countries in which student access to such facilities is either very limited or non-existent, either at home or at school. This paper focuses on secondary mathematics education for students and teachers who are without reliable and regular access to computers or to the Internet. While the arguments and analysis are of general relevance, the particular focus of this paper is on Asian contexts. The place of hand-held technologies, including scientific calculators, graphics calculators and integrated devices such as the ClassPad 300 will be considered, using examples from Casio, one of three major manufacturers. The computational support offered to students by such devices will be described and evaluated. Opportunities presented by hand-held technologies for students to learn mathematics in different ways than in the past and for teachers to teach differently will be described. The significance of hand-held technologies for the mathematics curriculum, its evolution and its assessment will be outlined and some issues associated with effective integration of symbolic computation into the secondary school curriculum will be explored.

Reports nationwide have shown that success rates in college mathematics classes have been dismal, particularly those prior to Calculus (Lutzer, Maxwell & Rodi, 2002). The current paper evaluates the utility of providing Supplemental Practice (SP) as a means to facilitate learning in an Applied College Algebra course. A pilot study revealed that students who earned more attendance points for SP sessions also received higher course grades. Additionally, students who earned the same number of SP points performed equally well regardless of their incoming skill level. A more controlled study revealed similar results, indicating that students who attended more SP sessions performed better on exams and earned higher final course grades than those who attended fewer SP sessions. Pragmatically, however, the only students for whom SP effectively facilitated success were those who passed a Pre-assessment exam of basic skills.

Some interesting examples are given in this article to show how one can learn and investigate mathematics through experiments with the help of computers. Each example begins with a problem. Trying to solve the problem one experiences exploration, invention and application of mathematics in the process.

CREACOMP provides an electronic environment for learning and teaching mathematics that aims at inspiring the creative potential of students. During their learning process, students are encouraged to engage themselves in various kinds of interactive experiments, both of visual and purely formal mathematical nature. The computer-algebra system Mathematica powers the visualization of mathematical concepts and the tools provided by the theorem proving system Theorema are used for the formal counterparts. We present a case study on the concept of equivalence relations and set partitions, in which we demonstrate the entire bandwidth of computer-support that we envision for modern learning environments for mathematics.

SSP (Super Sketchpad) is educational software in Chinese. Teachers and students can do almost everything for teaching and learning mathematics by using SSP. Even the free version of SSP still has a lot of surprising functions. This paper provides an introduction to the design, implementation, and capabilities of the free version of SSP.

This article describes through some selected case studies how symbolic computing can be used to improve dynamic geometry software. Focusing on some of the key problems around the dynamic geometry paradigm, such as general discovery and loci derivation, we review some of our algorithms and tools relating computer algebra based methods and dynamic geometry environments.

This article discusses some ways to combine software for interactive geometry and computer algebra, with an emphasis on using this software in teaching. Some arising mathematical problems and necessary software specific paradigms that have to be adjusted are considered, and solutions to these problems are presented. The article also highlights some of the aspects that drove the development of the mathematics software Cinderella.2, which integrates an interactive geometry software package, a physics simulation engine, and an all-purpose functional programming language.

T-algebra is a project for creating interactive problem solving environment for basic school expression manipulation exercises: calculation of the values of numerical expressions; operations with fractions; solving of linear equations, inequalities and linear equation systems; operations with monomials and polynomials. This article describes and motivates solution step interface and error diagnostics developed in T-algebra.

In this paper, two symbolic algorithms are given to generate readable proofs automatically for a class of limits of sequences and functions by definition. The algorithms have been implemented in the computer algebra system Maple and work effectively for a number of examples.

An improved Fibonacci-Sylvester algorithm for computing unit fraction expansions is presented. In general, the maximal denominators computed by this new algorithm are comparatively smaller than those obtained by the original algorithm. Discussions of its implementation in computer algebra systems, comparison of tested results, ideas for further research and its applications in undergraduate teaching are included.

In this paper, we obtain finite series expansions for positive integer powers of sine and cosine functions. The terms of the expansions consist of trigonometric functions of multiple angles. Such formulas do exist in the literature. However, more than the final results themselves, we are interested in the actual method of obtaining them. We use the method of successive differences of sequences and the computer algebra system (CAS) Mathematica to obtain our results.

In this paper, we present in detail a little known application of the fast Discrete Fourier Transform (DFT), also known as FFT. Namely, we first examine the use of FFT in approximating polynomials with sines and cosines (also known as Fast Fourier Fit or FFF) and then derive and solve the heat and wave equations.

The following sections are included:

• AUTHOR INDEX