How to Teach Mathematics [Paperback]

Steven G. Krantz (Author)

Book Description

ISBN-10: 0821813986 | ISBN-13: 978-0821813980 | Publication Date: January 1, 1999 | Edition: 2

This expanded edition of the original bestseller, How to Teach Mathematics, offers hands-on guidance for teaching mathematics in the modern classroom setting. Twelve appendices have been added that are written by experts who have a wide range of opinions and viewpoints on the major teaching issues. <P>Eschewing generalities, the award-winning author and teacher, Steven Krantz, addresses issues such as preparation, presentation, discipline, and grading. He also emphasizes specifics--from how to deal with students who beg for extra points on an exam to mastering blackboard technique to how to use applications effectively. No other contemporary book addresses the principles of good teaching in such a comprehensive and cogent manner. <P>The broad appeal of this text makes it accessible to areas other than mathematics. The principles presented can apply to a variety of disciplines--from music to English to business. Lively and humorous, yet serious and sensible, this volume offers readers incisive information and practical applications.

Editorial Reviews

Review

"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

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: 10.8 x 7.2 x 0.9 inches
• Shipping Weight: 1.3 pounds (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #290,984 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

17 of 17 people found the following review helpful:

5.0 out of 5 stars How to teach math to community college students, September 6, 2002

By 

Tim Porter (Hebron, Illinois USA) - See all my reviews

This review is from: How to Teach Mathematics (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.

15 of 16 people found the following review helpful:

5.0 out of 5 stars A fantastic book for the beginning mathematics teacher, September 23, 2001

By 

"c_r_garner" (Conyers, GA USA) - See all my reviews

This review is from: How to Teach Mathematics (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.

17 of 21 people found the following review helpful:

3.0 out of 5 stars Fairly useful, March 16, 2009

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: How to Teach Mathematics (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." He grasped the counterexample just as well as we do.

And how is Krantz's "that formula is wrong, this is right" any better than the original "here's the right one, and here's the proof"? In fact, it is much worse, since if the student did not already know that his formula was wrong and the other one right he could not even have asked the question in the first place.

My answer to the question would be as follows. The calculus is not a game of formulas so one should not expect to find the right answer by such considerations. The misunderstanding (fg)'=f'g' stems from thinking of (fg)' as a string of symbols rather than the idea that it represents. What is this idea? We can think of it like this. fg is the area of a rectangle with sides f and g. (fg)' means d(fg)/dx, the rate of change of this area with respect to x. So let's say that x changes by dx. What happens to the rectangle fg? The sides increase by df and dg respectively, so the change in area is (df)g+(dg)f+(df)(dg). The last term can be neglected since it is infinitely small compared to the others. Dividing by dx then gives the answer.