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How to Teach Mathematics 2nd Edition

4.4 out of 5 stars 8 customer reviews
ISBN-13: 978-0821813980
ISBN-10: 0821813986
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Editorial Reviews


"Since the first edition of How to Teach Mathematics the increasing maturity of both traditionalist and reform movements has given Krantz more insights into the teaching of mathematics. The book is intended primarily for the graduate student or novice instructor; however, the book is also valuable for others. Post-secondary instructors ... Mathematics department heads ... Teaching Development Centers ... university administrators. In the appendices twelve other mathematics teachers comment in some way on Krantz's text and give some insight into other approaches to teaching. This book is a must read for instructors preparing their courses for next semester." ---- MAA Online

"An original contribution to the educational literature on teaching mathematics at the post-secondary level. The book itself is an explicit proof of the author's claim `teaching can be rewarding, useful, and fun'." ---- Zentralblatt MATH

"Unlike secondary school teachers, college and university teachers usually have no preliminary theoretical background in the teaching of mathematics. [This book] is written in a lively and humorous style, even though the points discussed are entirely serious and sensible. The author succeeds in elucidating the fine points of excellent teaching and offers a lot of important practical advice. The book is strongly recommended to everybody who teaches mathematics." ---- European Mathematical Society Newsletter

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Product Details

  • Paperback: 307 pages
  • Publisher: American Mathematical Society; 2 edition (January 1, 1999)
  • Language: English
  • ISBN-10: 0821813986
  • ISBN-13: 978-0821813980
  • Product Dimensions: 0.8 x 7.2 x 10.2 inches
  • Shipping Weight: 1.3 pounds
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #268,042 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

Format: Paperback
This book has been a tremendous help to me to identify some of the problem areas on my teaching. Mistakes I have made in the past with ideas about why I made them and how to avoid them. I am new to teaching math but not to teaching in general and the thoughts laid out in this book can be applied to most any field of teaching.
I would HIGHLY recommend this book to anyone teaching in a community college. You have the widest and most difficult range of students in the education business, but they are all there to learn. Do whatever you can to make your efforts effective!
I would love to see much of this material presented in a workshop for adjunct faculty.
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Format: Paperback
The author's focus is on college teaching, but is also readily applicalbe to high school or other secondary teaching. Chapters include fundamentals such as how to lecture and other pedagogical ideas, to extremely practical items such as writing and grading tests and tutoring. Many ideas are presented for immediate adapting/absorbing into your own teaching framework. Being a high school calculus teacher, I was entertained by the glimpse this book provides of a college or university teaching situation. This book is readily available from the American Mathematical Society at ams.org.
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Format: Paperback
This book is fairly useful, but I want to comment on some things that annoyed me.

Krantz is critical of teaching substantial applications such as Kepler's laws and predator-pray systems in calculus classes, for this reason:

"How will you test them on this material? Can you ask the students to do homework problems if their understanding is based on such a presentation?" (pp. 29-30)

The direction of implication is deeply alarming: by *assuming* the mode of examination, Krantz *infers* what material is to be taught. Apparently he does not see a problem with this.

Here is an example where Krantz is trying to teach us how to "answer awkward questions in a constructive manner":

"Q: Why isn't the product rule (fg)'=f'g'? The answer is not 'Here is the correct statement of the product rule and here is the proof.' Consider instead how much more receptive students will be to this answer: Leibniz, one of the fathers of calculus, thought that this is what the product rule should be. ... Because we have the language of functions, we can see quickly that Leibniz's first idea for the product rule could not be correct. If we set f(x)=x^2 and g(x)=x then we can see rather quickly that (fg)' and f'g' are unequal. So the simple answer to you question is that the product rule that you suggest gives the wrong answer. Instead, the rule (fg)'=f'g+g'f gives the right answer and can be verified mathematically." (p. 17)

Krantz's answer is worse than the dummy answer he is trying to improve upon. It perpetuates the highly misleading myth that "the language of functions" has some mysterious power to make insights appear. This is nonsense. Leibniz's error was due to haste and negligence, not a lack of the talismanic power of "the language of functions.
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Format: Paperback
This book is well worth reading from start to finish for a starting college instructor who hasn't taught in North America before. An instructor who has been educated in North America has experience with standard ways instructors present material and is familiar with the expectations that students have for university courses, and will not benefit as much from reading this book. Much of the material in the book is obvious to someone who is reflective about their teaching and aware while teaching (you should have some feeling for a room, at a minimum to sense from pained faces if the students are not absorbing anything). The essays appended to Krantz's book are mixed but several are well worth reading even by instructors who have taught for years.

I assert that the biggest psychic obstacle to learning is thinking that your level of talent in a subject is fixed like fingerprints. Students should be explicitly told not to feel pride for mastering something with little practice, and should also be explicitly told not to feel shame for doing terribly at something when they first try it. I think it is one reason why students don't work as much as they could on mathematics.

The book seems to me to pose the following dilemma: either an instructor speaks and writes notes on a board that cover the material in the course, or students work in groups and the instructor probes them and gives them targeted help. I have two alternatives. One is tell students what to read in a textbook, so that the instructor might never mention some topics for which the students will nevertheless be responsible. The instructor could then focus just on the harder and more unifying ideas in the subject and repeat them several times.
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