# Are puzzles a good way to interest pre-college students?

Nothing more quickly disspitates the myth that most people aren't interested in math than hitting them with a good puzzle and watching the instinctive human urge to solve it get to work. To be fair, this doesn't apply to literally all people, but there's certainly a lot more people that enjoy riddles and puzzle games than there are mathematicians.

I've always thought that if I were to teach students at the pre-college level (who we can therefore assume are there because they have to be, not because they already like math), I would use puzzles to get them interested in mathematics, emphasizing that the fascination they (hopefully) feel towards these puzzles is exactly how mathematicians feel about their work.

But does this actually work? Or, if I gave a class of pre-algebra 13 year olds a puzzle like "Given the sum and difference of two numbers, can you work out the numbers themselves?", would I just get a roomful of groans and no attempts at a solution?

My definition of a "puzzle" is roughly as follows:

1. A puzzle is not a routine application of the class material, it requires insight and creative thinking.
2. A puzzle can be solved using techniques learned in class or about to be learned in class, although the exact connection might not be obvious.
3. It may be too hard for the entire class to solve it, the goal is more to get them thinking about the problem (to get them interested in the class, and motivate the techniques you're teaching them).
4. The statement of a puzzle is elegant and designed to be inherently interesting (rather than "here is a very specific geometrical scenario, calculate this very specific paramater relating to it").
• I'm afraid this is too vague as it stands. Do you want to know if there is evidence puzzles work? (I'd say they do, what else could explain the "problem of the month" section in assorted journals, and the many websites catering to math puzzles). Do you want good puzzles? (If this, you'd have to narrow the question to one (or a few) areas, and give background information on the ones to be puzzled). – vonbrand Apr 11 '14 at 16:12
• @vonbrand I'm interested in evidence, anecdotal or otherwise, that this kind of method works (or doesn't). For instance, experiences from someone who taught with this type of policy for several years. I'm definitely not just looking for examples of puzzles. – Jack M Apr 11 '14 at 18:31
• I'd suspect this works with some (perhaps even the majority?) of people, others won't be interested. You have to be prepared to offer a full menu... – vonbrand Apr 14 '14 at 18:33

(This is more an extended two-part comment than explicit answer.)

Part one. Finding "puzzles" (often with a mathematical flavor) is not too difficult. Martin Gardner is probably the main go-to, but "The Art and Craft of Problem Solving" (Zeitz) or the classic "Polyominoes" (Golomb) might fit the bill, too. Some other books I like are "The Tokyo Puzzles" (Fujimura) and "The Simple Book of Not-So-Simple Puzzles" (Grabarchuks). There is yet another book, which has a title meant somewhat as a "jape," called "More Games for the Super-intelligent" (Fixx). Here is an excerpt:

The mere fact that you are reading this book [on puzzles] by no means grants you license to urge its contents on those who are not similarly committed. The only safe procedure is to acknowledge the fact that in your liking for puzzles you are different from other people. Revel privately, if you want to, in that difference, but do not - under any circumstances - undertake even the gentlest of proselytizing. That way lies nothing but disaster. Remember: you are an intellectual at play, not a missionary... I think, we must choose our puzzles companions with care. We must not assume, until we have seen irrefutable evidence of it, the presence of an enthusiasm that may not exist at all. Rather, those of us who share an interest in puzzles should stick quietly together, welcoming outsiders whenever they chance to turn up, but never taking it for granted that they will (vi-vii).

Whether or not I agree entirely with Fixx, I can at least say that there are plenty of students for whom puzzles are not enjoyable, so one must be cautious in expecting any sort of interest-transfer.

Part two. Looking at your definition of puzzle, it essentially accords with the idea of a problem as contrasted with an exercise (in the Mathematics Education literature). To this effect, I think books on problem solving or old examinations (such as the Putnam - but maybe stick to A1, A2, B1, B2 for pre-college...) could be useful sources for what you are looking for.

In distinguishing between a problem and exercise, one way to define the former is:

a question for which the method of solution is unknown at the outset

This ought to satisfy your first three conditions, and asking that the problem be good (elegant, interesting, etc.) should be enough for the fourth and final condition.

• I might also suggest any of Raymond Smullyan's puzzle books for Part one. – user37 Apr 12 '14 at 15:38
• @Mike Yes; good call. There is some worthwhile stuff by John Conway, too... – Benjamin Dickman Apr 12 '14 at 18:21
• haha i cant believe you referenced games for the super intelligent! I found that years ago at a used book store and was turned off by exactly the excerpt you used – celeriko Oct 31 '14 at 2:28

As a grade 8, 9, and 10 maths teacher I can testify that good puzzles often do motivate my students. Here are a few observations I have about the puzzles I've given in class:

1) A good puzzle must be framed very well. A puzzle about the sum and product of two integers is (as stated) too boring for all but my most highly motivated students. However, an engaging puzzle has a real world context, a game, an argument, a manipulative, open ended questions (but still structured), etc. I've found that there is relatively little correlation between how interesting I find a puzzle and how interesting my students find it.

2) A puzzle must attract and keep my weaker students' attention. It's relatively easy to gain the attention of my top students (if I just give them a Martin Gardner book they can often occupy themselves with puzzles) but much harder with my weaker students. For them, point (1) is critical, but it's also important for them that I give the puzzle good structure so they don't give up. By that I mean that I try to present puzzles with lots of encouragement, defining terms clearly, providing a simple example and counterexample if possible, and sometimes -- but never before at least a little frustration has set in -- giving one or two good hints.

3) The serious math discussion is best when it comes with a surprise ending. For example, we discussed the U.S. men's college basketball tournament and at the end I demonstrated that the probability of picking 63 games in a row correctly at random is 1 in $2^{63}$. If I begin the exercise by talking about scientific notation, I will most likely get groans, but by discussing basketball, gambling, competition, etc. first some of my students were at least a little impressed when they found out that it would take everyone on the earth counting one number per second for over over 8 months to reach $2^{63}$.

• Do you have examples of times when a puzzle seemed satisfactory for your better students, but didn't seem to work with weaker students, to give an idea of the kinds of problems that can arise? – Jack M Apr 14 '14 at 22:02

When I was in high school we had a brilliant teacher who gave us an assignment every second week and a test on the alternant weeks. In the assignment he has 10 or so boring questions about whatever we were learning, then he had his alpha, beta and gamma questions. The Greeks were worth bonus marks, in fact if you answered all the alphas you usually made enough bonus marks to not need to do the boring questions, and a solid attempt at a gamma or 2 would also be sufficient to pass. I am not sure how he sold it to us, but I always started with the Greeks, at least reading them and thinking about how I would start solving the problem, I never remember doing any of the boring questions and always got at least a pass mark.

Reflecting on the problems years later I realised the Greeks were also structured so they generally either required a deeper understanding of whatever we were learning, or provided a historical or practical usage of the underlying concepts, or he introduced whatever we were going to learn next, or merged something we had just learnt with something we had previously learnt.

Interestingly in the class of 25, 20 came in the top 100 in the state in our final exam. I am not sure if this was because of the fun we had, or the fact our teacher loved maths, and taught us how much fun it was and how easy it was. When I got to university I discovered that I had learnt most of what they were trying to teach 2 years before and never realised this was meant to be hard.

I also don't know how my teacher dealt with the slower students, except perhaps they didn't start with the Gammas like I insisted on doing.

Well, your sum/difference question is awful, but then the 'puzzle' definition is spot on.

There's a first layer where real life math makes sense. Ex - as an interim step, a student had to calculate half of 1500. I told him he could do this in his head, no calculator. He protested, and I said "what's half of \$15?" He replied, '7.50' instantly. (I can't make this up)

The trick to get to the next level is to have the puzzle be really interesting, that they'll want solve for the sake of the challenge, and will push them a bit to learn something new, or apply what they know, but in a new way.

• I think the sum/difference question is great because I think the simple statement automatically invites you to try and solve it, although a middle-school student might not agree with me. It also has a pretty visual solution. – Jack M Apr 11 '14 at 18:32
• @JackM - if you can add context to it, making it apply to something, I'd agree. Else, it's just an equation to solve, not sure the kids will jump on it or find it clever. – JTP - Apologise to Monica Apr 11 '14 at 19:07
• I think the issue there is that if kids have already been taught how to solve equations, they'll jump to seeing it as routine application too quickly. I was imagining posing it before the class had seen any algebra. – Jack M Apr 11 '14 at 19:29