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How to Teach Mathematics

Steven Krantz
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
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‘… I’ve never had a teaching mentor. I’ve learned how to teach by making every mistake in the book and then trying to learn from those mistakes’ (Steven Krantz)

Initial teacher training doesn’t produce ready-made teachers — it merely prepares students for entry into the teaching profession. Having helped a few thousand students through this process during the past 40 years, I’ve seen some who failed the course, many who have turned out to be competent teachers, and others who became outstanding classroom practitioners. This is because, no matter what initial or in-service training is undertaken, success in the classroom is also down to the personality and academic integrity of the individual. This widely observed state of affairs raises the question as to whether it is actually possible to teach someone ‘how to teach’. Clearly there are very good mathematics teachers who have undertaken no such initial training (e.g. Steven Krantz).

So here is a book that seeks to provide a basis for those who teach at university level, and it may be unique in this respect. Therefore a minor observation on this book concerns its title, which perhaps should have been something like ‘Perspectives on the Teaching of Mathematics’. A subtitle might also indicate that it is intended for those who teach in universities (although it should also be of value to teachers of high school mathematics).

Consisting of 146 pages, the book is half the length of the previous edition. And yet, despite this severe pruning, its underlying aims are more or less the same. Specifically, its purpose is ‘to emphasize the nuts and bolts of good teaching’, which are said to emanate from careful preparation of lectures and ‘respect for students’. One additional feature is discussion of much learning material that is now available ‘OnLine’; chapter 4 provides a fulsome account of relevant developments in this field,

The central aims are mainly addressed in the first three of the book’s five chapters, and there is much discussion of the tension between two pedagogical approaches:

  1. Discovery, cooperative group learning and use of technology,
  2. Lectures, traditional exercises, drill.

To my mind, the most important message emerging from this book is that students should be made to feel that they are participants in the teaching/learning process, as opposed to being passive note-takers in formal lectures. Steven Krantz puts this message across very clearly, and he introduces many relevant procedures for achieving this goal (particularly the use of questioning and ways of generating classroom discussion). He also believes that mathematics should be taught ‘inductively’, which means starting with the particular and proceeding to the general. Results can then be formalized after they have gained intuitive acceptance. For example, the notion of ‘proof’ is relative to particular stages of learning, as expressed in the following sequence:

convince oneself

convince others

convince everyone

(examples and exercises)




(formal proof)

I very much agree with the approach to teaching mathematics that is outlined in this book. The book is full of useful ‘teaching tips’ relating to many topics such as class management, preparation, marking, examining and matters of discipline. Some of these issues should not be left solely to the discretion of individual teacher and should be dealt with according to departmental policy, but entrants to the profession of university teaching should know that they will be required to resolve them.

However, despite repeated reference to various teaching approaches, and despite the book’s extensive bibliography, there may be readers who require a more explicit outline of a cognitive basis for the learning of mathematics. One exemplar for this is provided in The Psychology of Learning Mathematics, by Richard R. Skemp. This is built on the distinction between relational understanding (method 1 above) and instrumental understanding (method 2 above). Relational understanding is defined in terms of concept maps, schematic learning, assimilation and accommodation. For instance, teaching based upon repetitive drill exercises, would result in instrumental understanding, while relational understanding invokes the use of graded, open-ended exercises.

This book is highly recommended as a source of self-improvement for teachers of university mathematics.

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Peter Ruane is retired from the field of mathematics education, which involved the training of primary and secondary school teachers. His postgraduate study included of algebraic topology and differential geometry, with applications to superconductivity.