Of course, puzzling is not the only ingredient in my approach to curriculum development. As I pointed out in response to a request for a “pedagogical framework” some time ago, I believe that actual classroom teaching (and thus high-quality curriculum) cannot be imprisoned into a single framework. Teachers are eclectic, and curriculum developers need to learn that flexibility. That said, a math consultant I know, who often uses my materials when working with teachers, insists that contrary to my claims to the contrary, I do operate within a pedagogical framework. For example, I have zero interest in creating pages of random drill exercises. Fair enough, but I don’t think generalities like “guided discovery” and “student-centered” are particularly helpful.

I'll say that if I do have a pedagogical framework, there are different overlapping but distinct ingredients to it. One ingredient, for example, is what I call a

*tool-rich pedagogy.*Another ingredient, largely in line with the emphasis NCTM has been championing for decades, is putting problem-solving at the core. In fact, problem solving is where the connection with puzzles is most obvious. As I see it, not all instruction should be problems, and not all problems are puzzles. Still, even non-puzzle activities and problems gain from being created with a puzzle constructor's approach. That is what I hope to address in this post and the next.

### Puzzles as relationships

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.

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.)

*The Nation*magazine

*.*)

### An example

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 _ _ _ _ _ Kin 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. In my next post, I will discuss specific examples of classroom math puzzles, and explore those to help flesh all this out for the readers of this blog, who are probably not particularly interested in cryptic crosswords.(If you want to find out more about cryptic crosswords, go to my Puzzle Page, and scroll down to

**Cryptics: How to.)**

To be continued...

--Henri

Thanks for writing this. You're not uniquely positioned to see connections, but rare at least!

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