Puzzles in Math Curriculum

A puzzle is a relationship between the puzzle constructor and the puzzle solver. There is an unwritten contract between the two. Here are some of the contract’s clauses:

The puzzle must be solvable and fair.

The puzzle must be challenging.

The solution must be satisfying.

Of course, these requirements depend on context. The same puzzle may be too easy to be satisfying for one solver, while another solver might deem it unsolvable, and yet another may consider it "just right". Still, these guidelines may be helpful to puzzle constructors, as they provide some direction on how to think about this. For a puzzle to be solvable, it must be possible to imagine some path to the solution. Fairness is harder to determine, as it depends on matching the puzzle difficulty to the solver’s probable skill and experience. What complicates matters is that insufficiently challenging puzzles are not as satisfying to solve. The purpose of a puzzle is for the solver to “win”, but not to win easily.

Alas, these guidelines do not provide a blueprint. Here are some ideas that may (or may not!) help the actual process of creation.

• The puzzle should be interesting to you, the constructor, even if you consider it easy to solve. If you’re bored, the solvers will be bored.
• You should mentally inhabit the mind of the solver, and imagine how they might get to the solution. If there are multiple paths to the answer, all the better. If there are partial solutions along the way, those help to keep the solver engaged. (Alas, not all puzzle solvers appreciate partial solutions. Some completists would rather not ever have tried a puzzle they cannot fully solve...)
• You should also try to imagine what a frustrated solver would feel if they break down and look up the answer. Would they think “Darn, I should’ve gotten that”, or “How did they expect me to figure this out?” (The first reaction is the one you want.)

Admittedly, those ideas are abstract and general. I will try to make them more concrete with an example from cryptic crossword construction, which is one of the things I do when I'm not doing math education. (I am a long-time member of the National Puzzlers' League, and co-construct the Out of Left Field puzzles.)

A cryptic crossword, of course, is a puzzle, but each clue therein is its own mini-puzzle. That structure already allows for multiple paths, as solvers can decide the order in which they solve the clues. This means that there are different entry points for solvers with different skills and backgrounds. Moreover, each individual clue contains three paths to its solution. For example, consider this clue:

Tech pioneer: "I know A-Z, but in a different order" (7)

(The 7 indicates that you're looking for a seven-letter word.) Let's say that you already have

W _ _ _ _ _ K

in the diagram. The unusual letters at the start and end of the entry may suggest the answer. Or, you may get it from the definition of the answer ("Tech pioneer".) Or, you may get it from the wordplay part of the clue "I know A-Z, but in a different order", which to a solver of cryptic crosswords suggests anagramming (rearranging) the letters IKNOWAZ. One of those three paths to the answer, or more likely a combination of two of them, or all three, will lead you to the solution: WOZNIAK.

As a constructor of cryptic crosswords, I have some choices. For example, I could make the solution easily researchable: replace "tech pioneer" with "Apple founder". But that would not be satisfying to the solver: whether they already know it or look it up, the answer is obvious, and they would not need either of the other two paths to the answer. Or, instead of "but in a different order", I could write "anagrammed", but that too is just too blatant, and moreover it would take away from the humor of the clue, which is part of what makes the solution satisfying. So I would say that this clue hits the sweet spot, and satisfies all the guidelines I suggested above.

But we should get back to math education. Constructing puzzles for the classroom brings with it additional complications and challenges. I will now discuss specific examples of classroom math puzzles, and explore those to help flesh all this out for the readers of this article, who are probably not particularly interested in cryptic crosswords. (If you want to find out more about cryptic crosswords, go to Out of Left Field Cryptic Crosswords, and scroll down to Cryptics: How to.)

I will zero in on the specifics of creating puzzles for the mathematics classroom by analyzing some examples.

A characteristic of all classrooms is that they are constituted of students whose backgrounds and talents vary widely. Offering multiple puzzles simultaneously can help, as it allows students to find their own way through the set, by selecting puzzles at the appropriate level of difficulty, and/or by pursuing partial discoveries. This addresses classroom heterogeneity, while having all students work on closely related problems. Here are some examples along these lines:

• Staircases: find sets of consecutive whole numbers whose sum is 3, 4, 5, etc. For example, `2+3+4=9`
• Egyptian Fractions: find three fractions with numerator 1, whose sum is 4/3, 4/4, 4/5, etc. For example, 4/5 = 1/2 + 1/5 + 1/10
• Make These Designs: find linear functions whose graphs create interesting designs.

All three activities allow the students to find their own path through them. They avoid a common pitfall of curriculum development, which is the hubristic belief that one is capable of writing a single sequence of puzzles that will work just as well for all students. This is a common failing of both traditional and contemporary curricula. For example, the consistently brilliant Desmos environment offers teachers and curriculum developers the ability to craft one-path-fits-all sequential lessons in the Activity Builder. The best Activity Builder lessons, such as Marble Slides, incorporate many excellent puzzles. This is vastly better than most supposedly "intelligent" educational software, which tries to eliminate the need for teachers and is based on reductionist and insulting memorize-the-algorithm-and-practice sequences. Still, one can hope that a future version of the Activity Builder will allow the creation of choose-your-own-path activities.

In addition to the availability of multiple paths, the above three examples also share other properties.

1. They are reversals of standard classroom activities. Instead of the mind-numbing request to "add these numbers", "add these fractions", "graph these equations", the questions are reversed: "find numbers whose sum...", "find fractions whose sums", "find equations whose graphs...". Reversal, in fact, provides a powerful mechanism for the construction of classroom puzzles: start with what you're trying to teach or apply, and reverse the question. Voilà! You've created a puzzle.
2. They offer non-random practice of important skills. Drill is not necessarily a bad thing, but random drill is boring and thus can be counter-productive. In these examples, drill is in the context of an interesting overall quest, and thus much more motivating. Also, unlike random drills, it lends itself to reflection, discussion, and generalizing.
3. They are each a set of related puzzles, rather than one-of-a-kind puzzles that rely exclusively on "aha" insights. Therefore, solving some of the puzzles helps the student develop skills and intuitions that can then be applied to other puzzles in the set, and more importantly, contributes to their mathematical maturity. This also means that they provide an excellent environment for teachers to provide hints, and scaffold student learning. For example: "solving this easier puzzle will help you make progress on the one you that is currently frustrating you."
4. They are interesting to both kids and adults. I have used these in the classroom with students at various levels, and in professional development sessions for teachers, and found that they are just as engaging for all. This is in part due to their "low threshold, high ceiling" quality: all include simpler and more difficult puzzles. Moreover, they suggest additional questions, such as the creation of similar puzzles, or the generalization of results, or the need for a proof.
5. They involve significant mathematics and carry a substantial "curricular" load. They are about the math teachers and students already know they should teach and learn. Using non-math puzzles as a "change of pace" is a waste of precious class time, and gives students the wrong impression that math is not fun.

One cannot expect all these criteria to apply to every classroom-bound puzzle or puzzle set, but hopefully they are helpful guidelines for teachers and curriculum developers.

Geometric Puzzles

As a young elementary school teacher, in the 1980's, I encountered geometric puzzles in Martin Gardner's books and columns. At the time, there were nice tangram-based materials for elementary school, such as a fantastic set of puzzles by EDC, but there was not much using pentominoes. I decided to create my own sets of pentomino puzzles, suitable for students. The key insight was that puzzles that did not require the use of the full set were much more accessible than the 12-piece puzzles discovered by Solomon Golomb and popularized by Martin Gardner. More accessible, but still interesting, and in many cases extremely curricular! I started with well-known puzzles from recreational mathematics, explored them on my own, and translated the fruits of that interest into classroom materials. This was an ongoing creative obsession over many years. You can read more about this work on my Geometric Puzzles page, though in fact this has infiltrated many other parts of my work as a curriculum developer.

Algebra Manipulatives

One of the features of the lessons I developed for algebra manipulatives in the 1990's involves a crucial re-envisioning of their role in the classroom. The standard algebra tiles lesson is based on the idea that the tiles illustrate what is going on with the symbols. In my Lab Gear materials, I turn this around. Start with a geometric puzzle: arrange these blocks into a rectangle. Then interpret what you accomplished with the help of the rectangle model of area. This is more fun, more accessible, and in the end more effective. I also introduced a whole genre of perimeter puzzles (e.g. use an xy-block and a 5-block to create a figure with perimeter 2x+2y+2), and visual patterns based on these blocks (what is the 10th figure in the sequence? the nth?)

I will not comb through my (freely downloadable) Geometry Labs and Algebra: Themes, Tools, Concepts to find all the puzzles they include sprinkled throughout, but I should mention my puzzle-based approach to geometric construction.

As you know if you've read this far, I'm a big fan of puzzles in math education. However, there is no one way: while puzzles are an essential ingredient in effective teaching, they are not everything. There are very interesting and fruitful explorations that cannot be described as puzzles. Still, even topics as dry as factoring a sum of cubes, or function behavior, or rate of change, can come alive in puzzles! Teachers, curriculum developers: stay alert to those possibilities!