Saturday, January 11, 2014

"An engineer,,, can hire a mathematician to solve his mathematical problems for him" -- Pólya on word problems

George Pólya writing in the book Mathematical Discovery:
I hope that I shall shock a few people in asserting that the most important single task of mathematical instruction in the secondary schools is to teach the setting up of equations to solve word problems. Yet there is a strong argument in favor of this opinion.

In solving a word problem by setting up equations, the student translates a real situation into mathematical terms: he has an opportunity to experience that mathematical concepts may be related to realities, but such relations must be carefully worked out. Here is the first opportunity afforded by the curriculum for this basic experience. This first opportunity may be also the last for a student who will not use mathematics in his profession. Yet engineers and scientists who will use mathematics professionally, will use it mainly to translate real situations into mathematical concepts. In fact, an engineer makes more money than a mathematician and so he can hire a mathematician to solve his mathematical problems for him; therefore, the future engineer need not study mathematics to solve problems. Yet, there is one task for which the engineer cannot fully rely on the mathematician: the engineer must know enough mathematics to set up his problems in mathematical form. And so the future engineer, when he learns in the secondary school to set up equations to solve “word problems,” has a first taste of, and has an opportunity to acquire the attitude essential to, his principal professional use of mathematics.

Friday, January 3, 2014


[I'm putting together an e-book collection of puzzles from Golden Age comics. This post is taken from the introduction.]

From Wikipedia's Famous Funnies article:
That same year, Eastern Color salesperson Maxwell Gaines and sales manager Harry I. Wildenberg collaborated with Dell to publish the 36-page one-shot Famous Funnies: A Carnival of Comics,[4] considered by historians the first true American comic book; Goulart, for example, calls it "the cornerstone for one of the most lucrative branches of magazine publishing".[5] It was distributed through the Woolworth's department store chain, though it is unclear whether it was sold or given away; the cover (see left) displays no price, but Goulart refers, either metaphorically or literally, to Gaines "sticking a ten-cent pricetag [sic] on the comic books".[6] 
When Delacorte declined to continue with Famous Funnies: A Carnival of Comics, Eastern Color on its own published Famous Funnies #1 (cover-dated July 1934), a 68-page periodical selling for 10¢. Distributed to newsstands by the mammoth American News Company, it proved a hit with readers during the cash-strapped Great Depression, selling 90 percent of its 200,000 print run; however, its costs left Eastern Color more than $4,000 in the red.[6] That quickly changed, with the book turning a $30,000 profit each issue starting with #12.[6] Famous Funnies would eventually run 218 issues, inspire imitators, and largely launch a new mass medium.

If you look at The cover of that first issue of famous funnies, you will See the subtitle "100 COMICS AND GAMES – PUZZLES – MAGIC".

The idea behind those early comics was simply to make cheap reprints of some of the most popular features of the daily newspapers. There was at least one major difference, while newspapers were seen primarily as an adult medium with some content thrown in for children, comic books were seen as primarily a kids medium.

This difference in target audiences was particularly notable when you look at the puzzle sections. In a newspaper or magazine of the time, these sections could be extremely demanding (such as the New York Times crossword puzzle) and even mathematically sophisticated. Starting in the late 19th century in magazines such as the Strand, creators like Sam Loyd, Henry Dudney, and, of course, Lewis Carroll were composing puzzles that often opened up areas of serious mathematical research.

You wouldn't find much in the way of cutting edge mathematics in the pages of famous funnies and other early comic books. These were puzzles intended for children. What's more they were cranked out at considerable speed and were usually fairly standard and often redundant.

Within those limitations, however, some creators really did manage to shine. Two in particular stand out, both as puzzlers and artists. The first was Art Nugent, a gifted cartoonist who produced a steady stream of charming and often quite clever puzzles for about a half a century. The second, though less prolific as A puzzlemaker, is sometimes considered one of the premier cartoonist and graphic artist of the 20th century, George Carlson.

Carlson was an extraordinarily successful commercial artist. Among other notable accomplishments, he painted the dust cover for the first addition of gone with the wind. As a comic book artist and writer, he created the highly influential jingle jangle comics which writer Harlan Ellison claims puts him in the company of artist such as Windsor McKay and George Herriman.

In terms of puzzles, Carlson was equally gifted. Martin Gardner said of Carlson's Peter Puzzlemaker series "no better collection of puzzles for young people was ever published."

Monday, December 30, 2013

A Golden Age Comic Book introduction to Recursion -- The Pyramid of Eternity

From 1947. this Captain Marvel Jr. story has a surprisingly mathematical bent.


And a little later...

and a bit later still...

"The Pyramid of Eternity" is better known as the Towers of Hanoi and it's often used as example of recursion, particularly in programming courses. Here's the Wikipedia explanation:
A key to solving this puzzle is to recognize that it can be solved by breaking the problem down into a collection of smaller problems and further breaking those problems down into even smaller problems until a solution is reached. For example:
  • label the pegs A, B, C — these labels may move at different steps
  • let n be the total number of discs
  • number the discs from 1 (smallest, topmost) to n (largest, bottommost)
To move n discs from peg A to peg C:
  1. move n−1 discs from A to B. This leaves disc n alone on peg A
  1. move disc n from A to C
  1. move n−1 discs from B to C so they sit on disc n
The above is a recursive algorithm: to carry out steps 1 and 3, apply the same algorithm again for n−1. The entire procedure is a finite number of steps, since at some point the algorithm will be required for n = 1. This step, moving a single disc from peg A to peg B, is trivial.

Sunday, December 29, 2013

Puzzle ideas for K through 2 -- odd and even

From Suzie Comics, 1954

These basic formats would be fairly easy to apply to other properties of numbers such as being prime or divisibility by, say, three.

Saturday, December 28, 2013

Talking about the How to Solve It list -- part 1

How to Solve It starts with what you might call a summary. It covers two pages (printed sideways to form one large sheet) and it is, by far, the best known part of the book. This is both a good and a bad thing. On the good side, the sheet contains an extraordinary amount of useful ideas; on the bad side, the teachers who explain it often know little about Pólya's philosophy and treat the list as a kind of an algorithm for general problem solving.

Pólya disliked cookbook approaches and he believed that the search for a generalized approach to problem solving, while interesting and likely to yield useful insights, was a doomed effort. He certainly didn't mean for this list to be treated as an algorithm to be checked through. Here's how he puts it in the introduction:

Put another way, this is a list of things you can try when you're having difficulty solving a problem. They have an excellent track record over a wide range of situations and difficulty levels but they have to be applied with common sense in context and, even when used appropriately, they are meant to help you think through a problem, not to do your thinking for you.

Monday, December 16, 2013

Polya -- found in translation

Whenever my students complained about word problems, my standard response was "life is word problems. No boss is ever going to walk up and ask you to solve this quadratic equation." I've logged a lot of cubicle hours since then and the observation still holds.

As for the actual mechanics of actually translating prose into algebraic expressions, you will find few explanations better than this one from George Polya's How To Solve It. Pay particular attention to the distinction between word-for-word translation and idiomatic translation.