### Multiple Paths

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.
- 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 these designs.

*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.

### Features of Effective Classroom Puzzles

In addition to the availability of multiple paths, the above three examples also share other properties.- 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. - They are
*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. - 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." - 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. - 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 "normal" math is no fun.

### More Examples

**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. [Note to Northern Californians: I'll be talking about Geometric Puzzles in the Classroom at the Asilomar meeting, on Dec 2. See you there!]

**Algebra Manipulatives**

One of the features of the lessons I developed for algebra manipulatives in the 1990's involves a crucial re-envisioning 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

*n*th?)

**And yet more examples**

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, which I presented in multiple blog posts and on my Web site.

### Puzzles Throughout?

**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 be turned into puzzles! Teachers, curriculum developers: stay alert to those possibilities!

--Henri

Henri,

ReplyDeleteFunny, I taught 42 years before retiring! I think math could use another Dale Seymour Publication type blitz where lots of fun problem solving is sent out to math educators. Keep it Up

Richard "Dick" Seitz

Yes, there were many great small publishers back in the day. They've been swallowed up by giant corporations with little interest in anything besides huge textbook sales.

DeleteThanks so much for these two posts. Interesting in the details and the bigger implications for curriculum writing. What strikes me is that these puzzles do such a good job of emulating what mathematicians do. There's a time for open play, but so often we're chasing a conjecture, following the rules of the particular situation. We don't know how to get there, but have faith that there is a path. My students were working on halving puzzles this week (dividing a shape into two congruent halves) and some needed to know that there was a solution. They weren't comfortable really trying until they knew there was an answer. Which we don't have while we're being mathematicians, but maybe necessary as a support for learners?

ReplyDeleteIn short, thanks for all the amazing work!

I think you're quite right: beginners need to know there is a solution. It takes much greater mathematical maturity to jump in without knowing that. For one thing, with more maturity comes the understanding that one can change the question in order to get an interesting answer.

DeleteI enjoyed your "make all the squares by combining these odd-area polyominoes". It was easy for me, as I knew enough to quickly choose my pieces, but it was probably well calibrated for your students. How did they do?

My students got the puzzles relatively quickly, but were slower to the underlying idea you saw. I made the pieces close to square to make the puzzles less obvious, and the students who made the 9 with 1, 3, 5 got it faster than the people who just said they have the 9 already. Interesting. Want to try with elementary, but haven't had the chance yet.

Delete