In An Introduction to Modern Mathematical Computing with Maple, Borwein and Skerritt show that computers are an excellent companion for learning mathematics. They do not do this with an essay on the advantages of computers, such as, fewer sign errors or quicker algebraic manipulations (both of which are true). Rather, they show readers that a particular computer algebra program, in their case Maple, is so flexible and powerful that it can work alongside students to show them insights that may be otherwise difficult to see.
To that end, the authors go through many aspects of Maple such as: algebraic manipulations, graphics, matrix manipulations, integration, differentiation, sums, limits, and number theory. Their treatment of the program is thorough, well explained, and instructive. This book is a good companion to the user manual and, maybe even better than the manual on some topics because the authors' examples are succinct and clearly illustrate many facets of the program. (It is not a replacement for the user's guide, however.)
The theme of the book is that Maple can supplement mathematics learning and, what is more, can do much of the mathematics for the students. That is certainly true. What is missing in this book is just how will Maple actually help students understand mathematics when the students are still learning the math.
Any computer algebra system can solve equations and plot functions. These operations are simple and students can type a formula (with the correct syntax) to tell the program to plot it, and presto!, the graph appears. Of course, students have to trust the program that the graph is correct. Does the graph correctly show the function around a singularity? Well, to know if it does, students have to understand singularities. The program cannot tell them that.
Maple can differentiate, integrate, and simplify expressions. Is the form that Maple provides useful? Students must know what they need and what the various forms are that meet their needs. In fairness, Borwein and Skerritt make some of these points. They say that Maple can do these tasks and note that it is up to students to know how to ask Maple to provide what they want.
Still, with Maple (and other similar programs) so readily available, one wonders if there is any incentive for students to actually work to gain an intuitive insight into mathematics. On that score, the authors are silent. I wish the authors had spent time telling us not only what Maple can do, but why students who are just learning mathematics should use it.
The temptation is tremendous for students to skip the real work to have a true understanding of mathematics. There should be more emphasis on students knowing mathematics, not just how to type an expression to see a result. This issue is not addressed in the book but it needs to be explored fully so that students know more than just how to use a program. Students and non-students alike should know when and why the results of the program work. Maple, for all its power, is powerless to tell them that.
David S. Mazel welcomes your feedback and can be contacted at mazeld at gmail dot com.