## Archive for the ‘Learning Math’ Category

### How to Solve IT, by George Polya

Saturday, October 6th, 2012

This post is the first of a series; in each one, I will describe a book on science or mathematics that I have found to be particularly interesting and informative, and which the reader might want to have a look at. Since I will choose only “good” books, each of my reviews will be generally positive. In a conventional book review, a book that does not seem to be a good one gets criticized. Here, if a book that I have read seems flawed, I will simply ignore it.
How to Solve It, by George Polya
In How to Solve It, Polya sets out to teach the reader to improve his/her ability to come up with solutions to problems that require some insight or creativity. His techniques and examples are mostly taken from mathematics (especially geometry), but he believes that these habits of thought apply also to physics, or any other area where the problems are quantitative in nature.
Before he gets specific, Polya makes some general observations:
1) People often enjoy expending mental effort, as the devotion to crossword puzzles, Scrabble, and other activities shows.
2) This willingness to think does not often include mathematical things. Indeed, says Polya, mathematics is the most disliked subject in the schools. Then, many of those people who come to dislike math become elementary school teachers, where they induce a new generation to dislike it also.
3) One reason for the unpopularity of mathematics may be the schools’ emphasis on rote learning and techniques. Students, Polya believes, should be given more practice work with problems requiring at least some creativity. Then, with some guidance from a teacher or tutor, the student can feel the reward of thinking through a problem to its solution.
4) The mental discipline of thinking about a problem, trying different things, and finally solving it, is an important life skill for any field of work,. Polya thinks that solving mathematics problems successfully may help to develop that discipline.
After this introduction, Polya immediately list his four phases in solving a problem:
1) Understand the problem. Draw diagrams, pictures, or graphs. Define the variables, including those given and the one we are required to find.
2) Devise a plan. Have you seen something similar? Is there a formula that connects the problem’s variables? Is there a simpler problem whose solution might help with this one? etc., etc.
3) Carry out your plan.
4) Look at the solution. Does the answer seem reasonable? Can I check the answer for correctness?
The remainder of the book is mostly elaboration on these points, using many example problems. These examples are at the high school level, except for two or three that involve a little calculus.
To give a sense of the difficulty level, here are two sample problems:
1) Bob has 10 pockets, and he has 44 coins, all alike. Can he place the coins in his pockets in such a way that each pocket has a different number of coins?
2) The perimeter of a right triangle is 60 centimeters. The altitude perpendicular to the hypotenuse is 12 centimeters. What are the lengths of the triangles sides?
How to Solve It has been continuously in print since 1945, so that it is a sort of minor classic. From my experience as a tutor, the book and its techniques would be lost on a recalcitrant math hater. Those with more interest and motivation will find some excellent hints here.

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### Great Formulas: the Binomial Series, part 2

Sunday, July 22nd, 2012

In the last post, I described Newton’s Binomial Series formula, one of the most important results in mathematics. In this post, I will go through three practical examples of how the binomial series can be used.

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