User Case Study: Professor of Mathematics redefines math education with the help of Maple

A series of textbooks based on Maple being published to engage university students

“Mathematics is not old or static; it is alive and students need to experience it,”exclaims Professor Laureano Gonzalez-Vega, describing his philosophy on math education. In his years of teaching, Prof. Gonzalez-Vega has realized that students spend a lot of time trying to understand the mathematic and scientific formulas that are the basis for concepts in scientific or engineering programs, often without much success. About 15 years ago, he introduced mathematics software tools including Maple™ into his math classes to add an element of excitement and experimentation. As a result, he noticed an increased interest in math as well as a desire to explore the experimental side of the subject.

Prof. Gonzalez-Vega is a professor of Algebra in the Mathematics, Statistics, and Computing Department of the Universidad de Cantabria in the northern part of Spain. He is also the founding Director of the Centro Internacional de Encuentros Matematicos (International Center of Mathematical Meetings).

As he taught with Maple, Prof. Gonzalez-Vega noticed positive changes in his students’ approach and in their results. With the new software, the students found it much easier and faster to solve problems and understand the mathematics. This left them more time and opportunity to experiment further, to gain a deeper understanding of the practical aspects of mathematics, and to start learning new, complex topics.

Building on this success, Prof. Gonzalez-Vega has now undertaken a project to bring this benefit to students across Spain. With help from Addlink, a Maplesoft™ reseller partner in Spain, he is publishing a series of five books called “Mathematics Laboratory.” The books cover university topics from basic mathematics to more complex concepts.

The five books are:

- Numbers and Equations: starts with students’ basic knowledge and introduces new concepts such as polynomials, complex numbers, inequalities, linear systems of equations, matrices, and eigenvalues.
- Limits and Derivatives: covers basic introductory calculus topics including functions, limits, derivatives, Taylor series, and optimization problems.
- Vectors, Matrices, Points, Lines, and Planes(co-authored with Cecilia Valero Revenga): devoted to basic linear algebra and geometry concepts.
- Integrals and Differential Equations: provides a hands-on recipe list to computing areas, volumes, and lengths through the determination of symbolic primitives, plus a short introduction to solving (elementary) ordinary differential equations, demonstrated through examples such as trajectories computations.
- Visualizing and Manipulating Data: explores tools for advanced use, including statistics and engineering mathematics. Introduces new mathematical concepts not covered in standard mathematics courses but needed in engineering, physics, and chemistry.

Each book consists of a series of lessons, which are full of examples, followed by Maple exercises which students and teachers can use. These exercises usually reinforce lessons students learn in the classroom. They begin by introducing Maple and some simple examples that become more complex.

For instance, an example in the first book deals with finding a closed-form solution for the general term of the Fibonacci sequence. For this sequence, the first number is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers in the sequence. By setting up a matrix representation of the *n*^{th} and (*n+1*)^{th} terms, you can repeatedly multiply the matrix by itself and compute the values of the sequence. These calculations can become tedious, so a better method is introduced: the initial matrix A can be diagonalized, giving an easier way to write its *n*^{th }power. To do this, the eigenvalues and eigenvectors of A are computed to get an “equivalent” but diagonal representation of A whose *n*^{th} power is very easy to determine. From this, a simplified matrix can be generated, giving the general expression for the *n*^{th }term in the Fibonacci sequence.

One of the main examples in the second book deals with the computation of the maxima and minima of the function:

for different values of *n*. This elementary-to-state problem produces a series of nice pictures allowing students to ask themselves about the behavior of these extrema when *n* becomes very large.

The first and second books are now available for purchase and Prof. Gonzalez-Vega expects the others to be ready in a few months’ time. With this new approach to learning, he is currently teaching more than 100 students in first-year physics and second-year computer science classes. Other universities like the University of Burgos are considering the books with great interest and are evaluating the benefits of introducing them into their curriculum.

With assistance from Addlink and Maplesoft, plans are also in progress to provide web-based material to students. Tests and assignments for students will be provided online using Maple T.A.™, an easy-to-use web-based system from Maplesoft for creating exercises and assignments that automatically assesses student responses and performance. The book’s Maple content will be also published using MapleNet™, a tool for publishing live Maple documents on the Internet. This will provide convenient, interactive access to the content for students and will provide instructors a variety of ways to share the content.

“With Maple, I can do unexpected things in an easy way,” says Prof. Gonzalez-Vega, summing up fifteen years of experience with Maple. “It is very simple, yet very powerful. Apart from my teaching, it also plays a very important part in my research on algorithms for dealing with geometric entities. The utilities in Maple are exactly what I need. I have been a faithful user of Maple all these years because my experience with it gets better every year.”

Contact Maplesoft to learn how Maple can be used for your projects