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Portal Through Mathematics: Journey to Advanced Thinking

O. A. Ivanov
MAA Press
Publication Date: 
Number of Pages: 
Anneli Lax New Mathematical Library
Problem Book
[Reviewed by
Jack Chen
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Many high school students have a mild interest in mathematics, but something is lacking to grow their interest and expand their confined understanding. In Portal Through Mathematics, Oleg A. Ivanov provides a book that does just that. It is a resource for secondary teachers to inspire their students, a gateway for secondary students to advance their knowledge, and a collection that would be of interest to any reader. Through his translation, Robert G. Burns attempts to bring the book to American audiences for the same benefits. Though the text is excellently translated, the difficulty of the ideas remains; American audiences will surely find Ivanov’s book to be a significantly challenging — although profitable — read.

Portal Through Mathematics is appropriately subtitled Journey to Advanced Thinking. The book contains problems and discussions based on the topics of Russian secondary schools and provides students with insights into advanced mathematical ideas. The author indicates in his preface that the target audience consists of “high school teachers and students in schools offering more advanced mathematical instruction,” but continues and states that the audience also includes “every fan of mathematics.”

With 29 themes and over 250 problems, Ivanov touches upon all aspects of mathematics, all the while placing an immense emphasis on surprise and pleasure. Many problems are not what they seem: difficult-looking problems are actually deceptively simple after being recast, while seemingly simple problems are in fact difficult and require much ingenuity. The problems on each theme are presented methodically, with subsequent problems stemming naturally from what was previously presented. Through the problems and the discussions, curricular knowledge is applied in intriguing ways that truly do act as portals to further knowledge. Always constant is the theme of taking current knowledge, expanding it, and finding surprises wherever they may be.

Ivanov’s experience in teaching is evident and brings an extra dimension to his book. Various footnotes are directed at secondary teachers to aid in classroom instruction, many classroom anecdotes are included, and a whole chapter focuses on a topic that the author finds lacking in mathematical curriculums.

Despite these benefits, American secondary teachers may find the book to be useful for very advanced mathematical students. Each theme certainly “begins at a level approachable with minimal background requirements,” but for many themes (especially in the latter half of the book), a student will find that full enjoyment of the material requires much more than just a minimal background. Part 3 of the book, with its discussion on topics such as Taylor’s formula and Euler’s formula, is extremely challenging; in this part of the book, even the wording of the problems can take quite some time to digest. That is not to say that an average student can’t find joy in all sections the book; rather, they should expect to be challenged and be able to use supplementary resources to assist them in their reading.

Throughout his book, Ivanov assumes much knowledge on the reader’s part. Rolle’s theorem, mathematical induction, and the Halmos symbol are just the very beginning of what the reader is expected to know, and while this expectation may be fair for a Russian audience, it is much less fair for their American counterparts. The notation of Ivanov’s book is formal, efficient, and neat; by all means it is a proper and effective mathematical presentation, but for the target audience the notation can prove quite difficult at times.

A further point worth noting is that while the author always presents the solution, he does not always present how he arrived at the solution. Many proofs are ingenuous and surprising, but sometimes the proofs seem to have been conjured out of thin air. Thankfully, in the later and more difficult sections, the book’s pace is slower and the process of arriving at a proof for many problems is explained. If would have nice — at least from the perspective of this reviewer — if the discussion throughout Ivanov’s book was more in-depth, but certain readers may have an easier time following the author’s logic and thus enjoy the pace of the book.

Portal Through Mathematics is an excellent resource for students, teachers, and casual mathematicians. Ivanov gives secondary teachers and students a taste of beautiful mathematics — beauty that is rarely observed at the level of secondary education. That beauty often came from the focus on “surprise,” which also gave Ivanov’s book a flavor that is sometimes frustrating, often interesting, and always instructive. Overall, this reviewer found it to be an enjoyable read, albeit a difficult one that was at some moments immensely challenging. It is certainly worth perusing over and can prove useful to readers of all levels.

Jack Chen has just begun his first semester as a student of Engineering Science at the University of Toronto.