A Mathematician Comes of Age by Steven Krantz

I’m a big fan of Steven Krantz. His book on teaching mathematics is the one I recommend most to beginning maths lecturers. The second edition is a must have since it contains short replies to his book from other lecturers, some of them highly critical.

Cover of A Mathematician Comes of Age

Cover of A Mathematician Comes of Age

Currently, I’m reading his recent book A Mathematician Comes of Age. This is concerned with how a mathematician becomes mathematically mature. From the back cover: “It describes and analyzes how a student develops from a neophyte who can manipulate simple arithmetic problems to a sophisticated thinker who can understand abstract concepts, can think rigorously, and can analyze and manipulate proofs.”

I must admit I’ve been diving in and out of the book at random and although I don’t agree with everything he says (or agree with the inclusion of certain topics – why are North Americans so concerned about “Math anxiety”?) there are thought-provoking passages every page or so.

The parody of the Evolution of Teaching Math on page 49 is very funny and includes

1980s: A framer sells a bag of potatoes for $10. His production costs are $8 and his profit is $2. Underline the word “potatoes” and discuss with your classmates.

Leaving aside the humour, I particularly like the section on Reading and Thinking (p95):

It has been observed [No reference given – KH] that the key things that a good teacher does are

  • engage the students in the learning process
  • pace the students
  • teach the students to read

Now, reading mathematics is something I’ve thought about and try to get my students to do, see Chapter 2 of How to Think Like a Mathematician (follow the link for free samples of chapters 3 and 4). Like in HTTLAM Krantz mentions the importance of reading with pen and paper:

“You must concentrate, you must have a pencil in your hand, and you must frequently be stopping to cogitate and calculate and try things.”

He does however, mention one point that is not stressed in HTTLAM: “Reading mathematics develops in stages”. Essentially, the type of reading changes over time and in fact, perhaps counter-intuitively, gets slower. For calculus books he says that “if you can read four pages an hour, then you are really tearing up the joint”. For books where the proofs are much more dense, then “If you can read one page per hour you are doing well”.

Even though I haven’t finished it, the book is highly recommended. I may write again, perhaps about his comments on writing mathematics or Uri Treisman’s teaching techniques.

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