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Exploring Mathematics

Daniel Grieser
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
Problem Book
[Reviewed by
Jack Chen
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When presenting mathematical problems, authors often present the solution, but rarely do they explain how the solution can be arrived at. In Exploring Mathematics, Daniel Grieser provides a systematic approach to mathematical problem-solving. Problems presented throughout the book have thorough investigations and invitations to “think about it!”. The solutions come naturally out of the investigations, and occasionally, proof-writing and mathematical language are discussed.

This book is a valuable resource for undergraduate students, keen secondary students, and the recreational mathematician. It provides an excellent balance between rigour and approachability. Every chapters begins with basic introductions and provides problems with rapidly increasing difficulty. The problems provided in Grieser’s book are excellent for stimulating the reader.

Though the necessary knowledge and tools are technically all available in the book, some problems will certainly require the reader to have previous experience and skills. Helpful hints are provided, but only for selected exercises. No solutions are included in the book. There are explorations of many topics, but one subject is notably missing — geometry. Save for a small problem involving geometric symmetry, the topic is not broached at all. A small chapter on basic geometry would have been nice for a more complete exploration of mathematics.

Exploring Mathematics can be used as an introduction to mathematical problem-solving, but may be better used as a supplement. I personally found that this systematic review of mathematical topics was refreshing, helping to fill gaps in my problem-solving toolkit. The reader may find it beneficial to use the book to review the fundamentals and enrich their problem-solving knowledge.

Jack Chen is an Engineering Science student at the University of Toronto planning to major in Robotics Engineering. His current mathematical interests are in algebraic combinatorics.

See the table of contents in the publisher's webpage.