View more books by Colin W. Clark: Math Overboard! - Part 1, Math Overboard! - Part 2

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ISBN: 9781457518089
326 pages
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ISBN: 9781457519482
326 pages
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Excerpt from the Book
How to use this book

Part 2 of Math Overboard! continues on from Part 1, discussing such topics as Trigonometry, Logarithms, Statistics, Vectors, Logic, and 3-dimensional Geometry. As before, the book emphasizes computational skills (algorithms) and comprehension (understanding). Trying to learn mathematics by simply memorizing techniques of calculation, with no effort spent on understanding what’s going on, is pretty much a waste of time. You soon forget what you memorized, and you gradually become confused and disoriented. Whenever someone tells me “Oh, I was never any good at math,” I figure that their teachers didn’t properly stress basic understanding. Understanding is more difficult than memorizing, but the payoff is in terms of self-confidence.

Methods for developing your understanding of basic math are discussed throughout Math Overboard! Look up the index entries under “mathematics – learning, solving problems, and understanding,” to review this advice. (The index for Part 2 actually covers both Parts of the book.)

Learning mathematics has always involved problem solving, both for practice and for testing your understanding of basic concepts. I suggest that you maintain a notebook, where you will write out Solutions (before peeking!) neatly. Use a pen, not a pencil. (You may first try to solve the problem using scratch paper, but unless it is completely routine, take the time to write down the solution in your notebook.)

For example, here is the solution to Problem 8.12:

8.12 Solution. To show that sin2 θ + cos2 θ = 1 for all θ.

Proof. By definition, Eq. 8.1,

Therefore

But by Pythagoras’s theorem [include a sketch] x2 + y2 = r2, so that

Going to this trouble may seem too fussy, but the process of writing this out carefully should make a permanent mark on your memory. The given equation sin2 θ + cos2 θ = 1 is fundamental in trigonometry, and it’s worth knowing that it follows from Pythagoras.

Notice that this problem (a) tested your knowledge of the definition of sin θ and cos θ, in terms of a basic diagram; (b) refreshed your grasp of Pythagoras’s theorem; and (c) used some basic algebra. Working this out for yourself (no peeking) may have taken some effort, of course, but that’s what learning math is all about. If you remembered this argument from school days, great – I found that most of my Calculus students had forgotten it – if they ever knew it at all!

There are not enough Problems in Math Overboard! for it to serve as a textbook. As mentioned in Part 1, you can often make up practice problems for yourself. Otherwise, search for math problems on line, or find a suitable textbook in your local library. I think you’ll find that Math Overboard! is more concise, but more useful and comprehensive than any standard textbook; it is also much more concerned with understanding than most on-line approaches.

 

For more excerpts, click on these links: 

Statistics:  http://www.MATHOVERBOARD.com/cms/uploads/Statistics.pdf

Vectors:  http://www.MATHOVERBOARD.com/cms/uploads/Vectors.pdf