As computers have grown to be far more than elaborate calculators, computer algebra systems (CAS) such as Mathematica, Maple, and Sage have become essential tools in mathematics. Students must now learn how technology fits into the problem-solving process, but it is not always clear exactly how to incorporate this into a mathematics curriculum. 

Exploring Mathematics with CAS Assistance by Lydia S. Novozhilova and Robert D. Dolan is a collection of activities that could help. The approach is to offer a variety of problems and focus on how to view them from an algorithmic perspective. 

The book presents all algorithms in pseudocode, thus not tying itself to any particular CAS. The fundamentals and notation are covered in the first chapter with the rest of the book dedicated to mostly self-contained topics that could be used as assignments, projects, or computer lab activities. The chapters are written to be widely accessible but can easily be adapted to a variety of experience levels, both mathematical and computational. Each section includes expository narrative interspersed with exercises, with several designated mini-projects and labs that invite deeper dives. 

There are fifteen chapters divided into three parts: Algebra and Geometry (e.g., the three centers of a triangle, Pythagorean triples, solving polynomial equations), Calculus and Numerics (Newton’s method, splines, Fourier transforms), and Probability and Statistics (the central limit theorem, Monte Carlo methods, random walks). The activities could slot into the classes one would expect, but specific topics could find homes across the curriculum. For example, the short “Rigid Transformations in the Plane” section offers ideas that could fit in courses in geometry, linear algebra, modern algebra, or computer graphics. The book is well-organized, clearly written, and includes several thoughtful pictures (indeed, visualization is emphasized throughout as an important use of CAS). 

However, the breadth of content comes at the cost of depth. The sections are short and typically do not go far beyond establishing the algorithmic lens (although they point out in various ways how one might go further). The book is worth a look for any instructor looking for ideas on how to implement more CAS activities into their course, or it would be an interesting choice for a problem-solving or seminar course. 

Bill Wood is a professor of mathematics at the University of Northern Iowa. A couple of decades ago he worked at Wolfram Research as a Mathematica kernel tester for just a few months, but that was enough to establish a career-long interest in computer algebra systems.