Michael Pershan recently wrote this glowing reflection on the work of Henri Picciotto. If you don’t now Henri, he’s something of a math educator’s math educator – not as well known as some, but consistently thoughtful and producing high-quality materials and ideas.

Michael was thinking about what we can learn about lesson design from Henri, and highlighted three points:

1. A single set of instructions
2. The materials are organized to create opportunities for greater generality
3. The materials are easy to use (i.e., you can download and print them without messing with them, there’s enough white space, etc.)

These are all nice points, but there was a detail that struck me about them. Of the three, points 1 and 3 are about form, or format, or user experience. Part 2 is about something deeper, I think – the mathematics involved, or the experience of mathematics that Henri is trying to provide to students.

All of which makes me think of a dynamic in math education relating to what’s most vital to focus on when it comes to creating mathematical experiences/curricula/lessons/etc. We could consider a dial between two points of view: does math class just have a PR problem, and need to be marketed better, have better user experience, better graphics, better production values, etc., but not require changing what we teach? Or does math class have a fundamental issue in the content itself, and need a deeper overhaul?

Dan Meyer gave a well-known TED Talk that articulated the former perspective: Math class needs a makeover. And Paul Lockhart’s Mathematician’s Lament argued from the other end, that the stuff we teach in math class is fundamentally off, and needs a total overhaul; we might call this the “Math class needs a heart transplant” perspective.

You can probably place your favorite math educator somewhere on this spectrum. (Where’s Pam Harris? Maybe the middle? What about Fawn Nguyen? A bit right of center?) And you can place yourself on there too! (I’m definitely on the right side of center. I’m sensitive to the need for ease of use and clarity of instruction,  but my instinct is more bent toward content than form.)

We don’t have to pigeonhole the people involved, to be clear. I’d argue that Dan Meyer is thinking about content too. But what we emphasize matters. His three-act structure was a framework that emphasized format and user experience. People went out and made a bunch of them, and some were good, and some were weak, because the math content in them was weak. The system glommed onto the format, and everything but the essential points were stripped away, and what remained was the defining elements, and they didn’t include strong mathematical content, even though Dan Meyer’s original examples tended to have strong mathematical content.

I sometimes think about this when I talk about math games in the classroom. I have a few points about what makes them useful:

1. Games should involve opportunities for players to make meaningful choices (content)
2. Math should be the engine of the game (content)
3. They should be simple to learn and quick to play (user experience)

What I leave off this list is what feels obvious: games should be fun. I leave this off because it feels obvious, but also subjective, and also hard to define. And ideally, if a game involves meaningful choices, it will be fun. But game design is it’s own subtle art. Without going too deep into it, I want teachers choosing a game to feel like it’s at least pretty fun to play, and that’s part of what draws them to using it.

All this is makes me think about how Michael is approaching Henri’s work. Based only on his reflection, it seems to me that Michael is coming from the left of center. He may be thinking about content and the types of experiences students have, but that’s one third of what he’s writing about.

For example, Michael make this point about an activity of Henri’s that has this instruction: “The instructions are simple: draw hands on the clock to represent times.” Below are a bunch of blank clocks. Michael has this to say about it:

It’s not hard to imagine a more finicky version of this activity. “Draw 5:00.” “Draw 5:30.” “Where is the hour hand at 5:30?” and so on. This might help us feel confident that the student will encounter important ideas, but with a cost—it’s taxing to read new instructions, and it interrupts some of the fun and flow of the core puzzle.

From my perspective (from the right side of the dial?), there’s something deeper going on here than ease of use: students are being invited to make their own choices. Experiencing mathematics as a place where you have a degree of authorial control is, to my mind, fundamentally different from answering a bunch of teacher questions. Michael and I both like the same thing, but our reasons feel entirely different. (Or at least, they’re coming from the makeover side in Michael’s reading and the heart transplant side in mine.)

This feels important! Because yes, ease of use and simple instructions are clearly good, but experiencing mathematics as a place where you make choices and set the frameworks and are involved with the questions as well as the answers is about something much deeper. If we can do that, we’re offering students an entirely different experience of what mathematics is! And once you notice that Henri is doing this, you notice that he’s doing it everywhere.

Henri (in my reading) is right of center on that spectrum, and he’s thinking more about what mathematics he wants his students to be doing, and how he wants them to experience the doing of mathematics. What makes him worth looking at is that he sees mathematics as fundamentally rich and interesting, and is trying to craft experiences that give students meaningful choices and greater ownership, that invite them to make conjectures and explore deeper generalities (Michael did mention this), and, maybe most vitally, he’s choosing interesting content to explore! I see slanted squares and polyomino area and perimeter and my first reaction is: “Wow – it’s great to see that Henri is offering such excellent content to students. Those are such interesting environments to explore, and so rich.” There are a lot of user experience moves that are good too (asking for a formula for maximum perimeter of a polyomino, but a pattern for minimum perimeter shows that someone has spent some time with that particular structure. Explore it and see why!), but just like I’m forgetting to say that the games should be fun, I feel like Michael is forgetting to point out that the math here is deep and interesting and real.

Maybe Michael takes that point for granted. But I’ve seen enough math class activities in need of a heart transplant, and too many situations where systems pick up the trappings of a great activity but miss the actual point (the interesting math). It’s worth underlining just how important the content is.