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How to Study as a Mathematics Major

Lara Alcock
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Annie Selden
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This small paperback book is directed at U.S. undergraduate students who may be wondering how to get along in, or even survive, a mathematics major. It is written in a very conversational style by a sympathetic university mathematics teacher. The author, now a Senior Lecturer in the Mathematics Department at Loughborough University in the U.K., has a Ph.D in mathematics education from Warwick University and has also taught at Rutgers University in the U.S. She gives her own experiences as an undergraduate who struggled at first but ended up getting a “first class” honors bachelors’ degree in the U.K. — a distinction that is considered harder to get than the designation summa cum laude in the U.S.

Before I continue, in the interests of full disclosure, I should say that I know Lara (the author) from going to hear her mathematics education research presentations, from talking with her informally, and from being in an ICMI working group together.

For each chapter in How to Study, there is a short paragraph (in a different font) that tells you what will be discussed. And at the end of each chapter, there is a short summary (also in the other font), followed by suggestions for further reading. In essence, the author tells you what she is going to say, then says it (in an elaborated, but down-to-earth, no-nonsense way), and then briefly summarizes what she has just said.

The book has two parts. The first part concentrates on what a beginning, or even an upper level, undergraduate mathematics major would find useful to know about being a math major, but is usually never told. Lara gives examples of how to write mathematics, even of the calculational sort, so it can be read and understood by one’s professors. She gives reasons to learn to write mathematics well: (1) It gets you better grades. (2) Whatever career you go into, you will need to write well. (3) It will help you communicate as a mathematician would. She gives advice on how to use arrows and various kinds of brackets correctly. Sometimes she sounds a bit like a schoolmarm, “Don’t be lazy. Do it right. You can’t claim to like the fact that mathematics has right answers, then also claim you can’t be bothered with this sort of detail [i.e., precise writing].”

She tackles students’ fears of proof “head-on”, by saying in a matter-of-fact way, “some students get the idea that proof is a mysterious black art to which only the privileged have access”, but “all it really involves is writing in a more mathematically professional way — writing more like a textbook and less like a student, if you like.” There are sections on logic and notation and on proving that a definition is satisfied, on proving general statements, and on proving theorems using definitions, each illustrated in detail with a specific mathematical example. For the last of these, she provides a proof of the “sum rule” for differentiating the sum of two differentiable functions.

The second part of the book concentrates on study skills more generally, on what lectures are like, on time management, on what to do when you get behind in your studies (besides panic), and on what professors do. After discussing time management, she says reassuringly, “The last thing to remember about time management is that everyone fails at it, on a regular basis.”

This is not a book that anyone needs to read from beginning to end. Students can benefit from just “picking it up” for a short time — now and then — and reading just about any section. The sections are relatively short — sometimes just two or three pages, but are very informative. One could easily recommend it to undergraduates. Also, a math department might buy a few and put them in the “coffee room” or whatever space is provided for undergraduates to get together and talk mathematics. And for students who find the paperback too bulky to carry, there is a Kindle version which is even cheaper.

I do, however, have a few quibbles with the book. While Lara made a valiant attempt to translate this book (which was first written for U.K. undergraduates) into “American”, some “Britishisms” do creep in. Perhaps the most puzzling one for U.S. undergraduates is her use of the British term “revision” for studying material again for a test or exam — something which would be termed “review” in the U.S. There are a few other Britishisms, such as the word “crosses” for what we would term “x’s”, and the mention of a “postgraduate teaching assistant”, rather than a “graduate teaching assistant”, but most of these can be easily sorted out.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development. 

Part 1: Mathematics
1. Calculation Procedures
2. Abstract Objects
3. Definitions
4. Theorems
5. Proof
6. Proof Types and Tricks
7. Reading Mathematics
8. Writing Mathematics
Part 2: Study Skills
9. Lectures
10. Other People
11. Time Management
12. Panic
13. (Not) Being the Best
14. What Mathematics Professors Do