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.

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.

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.

Pólya on Pedantry and Mastery

From How To Solve It:

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.

One very limited argument against learning your multiplication tables

When I was in elementary school, I refused to learn my multiplication tables. I detested rote memorization and memorizing answers struck me as cheating (after all, you were still looking up answers;  you were just doing it in advance). Fortunately, (though it may not have seen that way if the time) I was at and unstructured school that allowed students a great deal of freedom.

The immediate result was that I became very slow at doing math assignments. I had always been moderately slow doing worksheets. Teachers had commented earlier to my parents that though all the questions attempted were usually correct, I seldom finished my assignments. When multiplication entered the picture, my pace became glacial.

After a while, though, I started to pick up some tricks that helped greatly. Since doubling a number in my head was easy and tripling wasn't too bad,

I soon started factoring multipliers where possible and then multiplying by the factors. Thus four became double double and six became triple double. Since five and 10 were also easy to multiply by, that covered everything but seven. For seven and for larger numbers, I often relied on the distributive property though I had no idea that's what I was doing. 11 Times a number is the same as 10 times a number plus the original number. 19 times a number is equal to 20 times the number minus the original number.

This created an entirely new "problem" or at least extra step when I did my math homework. Whenever I had to multiply two numbers together I had to decide which forms of the numbers to use. Eight could be 2×2×2 or 4×2 or 2×4. Nine could be 3×3 or 10-1. Some forms made the problems easier than others but, amazingly, no matter which form I used the answer was always the same.

In the short term, my self imposed pedagogical experiment could not be called the success. My work was slow and my test performance dropped. Fortunately I had supportive parents and exceptionally understanding and flexible teachers and administrators.

I say "fortunately" because in the end things did work out well. The turnaround started with fractions. Ideas like simplifying and finding the lowest common denominators came naturally because of the groundwork that had been laid earlier. With algebra, the effect was even more pronounced.

This was partially because of the specific concepts I had been using, but the more important factor was the general approach to problem solving which I have been forced to take. Questions that would have been trivial had I done what I was told to do, instead required a great deal of thought. I had to think about all the different ways I could find to state the problem, then I had to decide which approach would be the easiest and quickest. In the long run, it turned out that this method was a pretty good way of approaching most mathematics.

Just to be clear, I am not saying that we should stop teaching multiplication tables. I'm not even saying that I would not have been better off had I simply done what I was supposed to do.

Polya's approach to anagrams

[You might want to check out the first two posts in the thread before going on]

George Polya uses a wide range of example in How To Solve It, ranging from some relatively advanced math problems to basic puzzles.

This example of the latter might make an interesting problem to do as a class. Here's how I'd set it up.

1. For the opening, you can pretty much just stick with what Polya has here.

2. Then have the class come up with words using at least some of the letters in the anagram.

3. You'll probably end up with relatively few words using X (and maybe Y). If so, suggest that since words with X (or X and Y) are fairly rare, it might be a good idea to list all of these the class can think of. Note that Polya gave the similar advice.

4. If the class comes up with the word EXTRA then loses steam, you might ask if there are any words that start with EXTRA.

5. If you run out of time and hints and if you can call up the internet, "when I Googled 'anagram solver' this is what I got" and go here. Remind them that finding an answer online or in a book is yet another problem solving strategy.

Discussing Polya's approach to helping students

[basically picking up from where we left off in the previous post]

Simple prose to George Polya was like short prose to Pascal, a sign of hard work and careful thought. Polya kept his popular writing clear and direct. That readability has a potential downside, though; it can cause readers to go too fast, to miss subtleties and mistake insightful points for pleasant truisms. For that reason, I'm going to slow down and spell out some points implicit in the original text:

1. We want mathematics education to produce problem-solvers, but we need those problem-solvers to be observant, resourceful and self-reliant. It is possible for a student to be quite successful without these traits, at least for a while, but outside of the classroom they are essential.

2. Attitude is a major factor in all of these traits, so in addition to cultivating them, you also have to cultivate the belief in them. Teachers work toward their own obsolescence. Insert whatever metaphor you want (crutch, training wheels), but when the process is over, we want students to think independently. That requires ability but it also requires faith. We develop that ability by having students work as independently as is feasible; we develop that faith by reinforcing the perception of independence, even if it is, very occasionally, illusory.

3. As with physical training, there's a sweet spot between too much and too little, both in terms of difficulty of the work and level of assistance. We want to challenge the students, but we don't want to make things so difficult that they give up or become so timid that they ask for help for problems they would have figured out on their own in another couple of minutes.

4. Doing all of this requires teachers to think both objectively and subjectively. Part of this is for empathy and for getting a read on an appropriate level of difficulty. The main reason, though, is given away by the last sentence. You want to ask questions that "could have occurred to the student himself" because at some point you want them to become questions that the students do ask themselves.

Polya's approach has been called Socratic, but that's not a good fit. The Socratic method is usually concerned with testing hypotheses. Polya is more interested in how we find hypotheses. From the preface:

The author remembers the time when he was a student himself, a somewhat ambitious student, eager to understand a little mathematics and physics. He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: "Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such facts? And how could I invent or discover such things by myself?"

For Polya, the questions students asked themselves (like "what's the unknown?" or "would it help to draw a diagram?") were an internal resource, a way of coming up with approaches to a troublesome problem by turning a problem over, examining it and restating it, looking for patterns and inspiration and lucky guesses.

And finally

5. If done correctly, Polya's approach to teaching is not only remarkably, perhaps even optimally effective; it is also surprisingly enjoyable on both sides of the table. The teacher gets to engage in a stimulating conversation instead of just mindlessly running through an algorithm while the student comes away with the sense of accomplishment that comes from having solved a problem instead of simply having been told how to solve it. Perhaps as important, the solutions arrived at through this method tend to make much more sense to the student. That helps with retention and conveys the essential message that math always makes sense if you look at it right.

Now, after too much delay, here's the opening to How to Solve It.

George Pólya's preface to the first printing of How to Solve It

A preface to Polya's preface

I'm going to admit to a personal bias here. How to Solve It was a real revelation to me when I first encountered it. The book managed, in wonderfully simple and direct language, to fully form ideas that I had been struggling to bring into focus for years. I discovered the book in my early twenties just as I was getting serious about mathematics. It significantly improved my analytic thinking. More importantly, it made me more self-aware of the process. Polya taught me (or at least, helped me start teaching myself) how to study the clockwork of the process of problem solving, to see how a good solution worked or why a bad solution didn't and how it could be fixed.

The following preface can more than stand on its own, but I do want to single out a few points, more for future reference than anything else.  

1. Mathematics education should be stimulating. It should be fun and  should cultivate a lasting taste for mathematics in particular and problem solving in general;

2. Mathematics education should also cultivate intellectual self-sufficiency, teaching kids how to find their own answers;

3. When mathematics education is so boring and repetitive that it kills the curiosity, it may do more damage than good.  

4. Though mathematics is usually presented as a straightforward deductive process -- A therefore B, B therefore C, C therefore D -- the actual processes of discovery are often indirect and inductive. Studying this messy half is an important part of studying mathematics. 

More on all of these points soon.

Preface to the first printing 

A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.

Thus, a teacher of mathematics has a great opportunity. lf he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.

Also a student whose college curriculum includes some mathematics has a singular opportunity. This opportunity is lost, of course, if he regards mathematics as a subject in which he has to earn so and so much credit and which he should forget after the final examination as quickly as possible. The opportunity may be lost even if the student has some natural talent for mathematics because he, as everybody else, must discover his talents and tastes; he cannot know that he likes raspberry pie if he has never tasted raspberry pie. He may manage to find out, however, that a mathematics problem may be as much fun as a crossword puzzle, or that vigorous mental work may be an exercise as desirable as a fast game of tennis. Having tasted the pleasure in mathematics he will not forget it easily and then there is a good chance that mathematics will become something for him: a hobby, or a tool of his profession, or his profession, or a great ambition.

The author remembers the time when he was a student himself, a somewhat ambitious student, eager to understand a little mathematics and physics. He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: "Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such facts? And how could I invent or discover such things by myself?" Today the author is teaching mathematics in a university; he thinks or hopes that some of his more eager students ask similar questions and he tries to satisfy their curiosity. Trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others, he was finally led to write the present book. He hopes that it will be useful to teachers who wish to develop their students' ability to solve problems, and to students who are keen on developing their own abilities.

Although the present book pays special attention to the requirements of students and teachers of mathematics, it should interest anybody concerned with the ways and means of invention and discovery. Such interest may be more widespread than one would assume without reflection. The space devoted by popular newspapers and magazines to crossword puzzles and other riddles seems to show that people spend some time in solving unpractical problems. Behind the desire to solve this or that problem that confers no material advantage, there may be a deeper curiosity, a desire to understand the ways and means, the motives and procedures, of solution.

The following pages are written somewhat concisely, but as simply as possible, and are based on a long and serious study of methods of solution. This sort of study, called heuristic by some writers, is not in fashion nowadays but has a long past and, perhaps, some future.

Studying the methods of solving problems, we perceive another face of mathematics. Yes, mathematics has two faces; it is the rigorous science of Euclid but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. But the second aspect is new in one respect; mathematics "in statu nascendi," in the process of being invented, has never before been presented in quite this manner to the student, or to the teacher himself, or to the general public.

The subject of heuristic has manifold connections; mathematicians, logicians, psychologists, educationalists, even philosophers may claim various parts of it as belonging to their special domains. The author, well aware of the possibility of criticism from opposite quarters and keenly conscious of his limitations, has one claim to make: he has some experience in solving problems and in teaching mathematics on various levels.

The subject is more fully dealt with in a more extensive book by the author which is on the way to completion.

Stanford University, August 1, 1944