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Excerpt from the Book
Part 2 of Math Overboard! continues on from Part 1, discussing such topics
as Trigonometry, Logarithms, Statistics, Vectors, Logic, and 3dimensional
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 selfconfidence.
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 sin^{2} θ + cos^{2} θ = 1 for all θ.
Proof. By definition, Eq. 8.1,
Therefore
But by Pythagoras’s theorem [include a sketch] x^{2} + y^{2} = r^{2}, 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 sin^{2} θ + cos^{2} 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 online
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
