Conjectures and Counterexamples 1

February 11, 2021

I wrote recently about a growth ladder for mathematics teaching. It involved three steps, starting with the easiest, and moving from there, that could build momentum for changes in pedagogy in the classroom.

Openers are a great first step—easy, popular, fun, and obviously beneficial to kids. (My recently launched mini workshop on the topic can help you get started.) And games are relatively straightforward to continue (I have a webinar for pK – 2nd grade up on that topic as well.)

But what about rich tasks?

Notice the Australian use of “Maths”

I took a trip to Australia a few years ago to put together a rich task video series with Maths Pathway. It was a great

collaboration, and I’m very happy with what came out of it: a free video series breaking down rich tasks, and three additional videos on specific rich tasks to try.

Still, I’ve been thinking about the need to routinize more of the process of delivering rich tasks. When I began teaching, I loved to take an improvisational, free-form approach. (I still love to teach this way.) But I’ve come to realize that my natural approach is too idiosyncratic and specific to me. Over the years, I’ve been trying to think of structures that teachers would be comfortable moving into any classroom and making their own. Reading books like Twyla Tharp’s The Creative Habit turned my earlier sense of structures and routines around. Twarp argues that the more routinized your life outside of the creative part, the more freedom and energy you have to create. It’s a compelling argument, and I see why it makes sense in math class too. It’s hard to innovate in more than one area at once. If you can get used to a routine that enables you to introduce rich tasks to your class, you have more mental energy to focus on teaching the content. (Shades of cognitive load theory! But for teachers!)

The routine around openers is simple to learn. Ditto games. But what is it for rich tasks? And why does it feel so simple to do it, but seem so hard for teachers to jump into using them in their classes?

Observations and Approaches

Paul Lockhart talked about the absence of rich learning from math class (though he wouldn’t have used that phrasing) as the absence of mathematics from math class. The kind of thinking mathematics involved demands encountering certain kinds of challenges.

Peter Liljedahl observed that if teachers regularly use rich tasks (and use them well), their students end up learning the material MUCH more deeply, rigorously, and quickly. But he also observed that the center of gravity in classrooms would pull teachers away from using tasks, despite their best intentions. He began looking for concrete, repeatable steps—often related to changing the physical classroom space—that would shift the center of gravity and make rich tasks the natural go-to. For teachers ready to jump in to using tasks, his suggestions are excellent. I especially like using vertical, erasable surfaces. Here’s a video tour of a Thinking Classroom, to use Liljedahl’s phrase. (These are adults, but the ones with students look similar.)


Here’s a bit more of an in depth video with kids of many ages.

The obvious elements at play here, like the fact that people are on their feet, working at white boards, jump out. And they’re important! But they’re also not the end in themselves. Those structural changes to the classroom are meant to support engagement in rich tasks; that is, they’re about support mathematical thinking.

What’s been on my mind is what are the non-physical elements that make rich tasks work?

Conjectures and Counterexamples

When you begin using openers in the classroom, you can feel a shift as students feel the need to express their thinking, and understand what other students are seeing. (Openers are really rich tasks in miniature.) As a teacher, you get involved to demonstrate and scaffold productive ways to engage, or step aside to give students the space to do it on their own!

Rich tasks, before students are accustomed to them, can be challenging, particularly for older students, who have grown accustomed to teachers doing their thinking for them. Student sometimes need more help understanding how to engage in real mathematical thinking. So what do we make explicit to help them? How can we explain what success with mathematical engagement looks like? And how can we routinize the process of introducing rich tasks so everyone can spend their energy on the thinking, and not figuring out what’s happening?

I’ve been arriving, more and more, at a routine I call Conjectures and Counterexamples. (That’s what mathematicians call it too.) This routine can be introduced in the game Counterexamples, by making false claims (i.e., conjectures) and letting students prove you wrong by producing counterexamples. Once this game is comfortable for students, they’ve developed a habit that’s at the center of rich tasks.

The process of making conjectures and breaking them with counterexamples is the fundamental way we play with and think about rich tasks, and mathematics as a whole

Rich tasks, at their center, are about presenting questions or environments that are interesting (and hopefully irresistible), that students have the capacity to explore on their own, that resist easy solutions, and that slowly reveal more information about how to understand them as you play with them more. The process of making conjectures and breaking them with counterexamples is the fundamental way we play with and think about rich tasks, and mathematics as a whole. (Later, when we cannot find counterexamples, we turn to constructing proofs, but that’s not where we begin.)

An Example: 1-2 Nim

For many years, 1-2 Nim (lesson plan here) was a favorite way to demonstrate rich tasks. The “environment” to explore was the game itself, and the natural question to attack was simply: “how do you win this game?”

If you’re not familiar with the game, it’s delightfully simple. Start with two players, and a collection of 10 counters. Take turns taking 1 or 2 counters from the pile. Whoever takes the last counter wins.

Inevitably, we had instincts for what might work as a winning strategy. With a little encouragement, students can articulate those instincts as conjectures.

Some natural conjectures about 1-2 Nim

  • The person who goes first wins.
  • The person who goes second wins.
  • Who wins depends on how many counters there are.
  • It doesn’t matter what you do until there are 5 counters left.
  • You win if there are an even number of counters on your turn.
  • You win if there are an odd number of counters on your turn.

And so on. As we articulate the conjectures explicitly, we can start breaking them. If you think that you can win whenever you start with an even number of counters, then let’s play, and see if you can beat me with 2 counters; with 4 counters; with 6 counters. If you can’t, then something’s probably off with your conjecture. But that’s useful feedback, because now you can adjust your conjecture and try again!

The Universal Learning Structure

When students learn to articulate conjectures and break them with counterexamples, they’ve learned a kind of habit that will help them learn math deeply. This isn’t an accident. This structure mirrors what happens in many, many fields of study. In science we test hypotheses. We think something might be true, and we run experiments to get more data and see if our hypothesis holds up. We make similar kinds of arguments in the humanities. Those fields may have more room for interpretation or muddiness in the data, but the fundamental process is very similar.

I’d like to collect more examples of the types of questions and environments that make for excellent rich tasks, as well as the kinds of conjectures and counterexamples we might expect to see when we approach them. This post is much longer than I expected to write when I sat down, so I’ll save the list making for a future post. But if you have favorite examples of question, conjectures, and counterexamples, please leave them in the comments.

One last thing to share! I wrote a TED-Ed riddle on a Nim variation. If you enjoy 1-2 Nim, try this one!

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