Conjectures and Counter-Examples: Centering Student Thinking with Genuine Mathematical Conversations

Without question, introducing the terms “Conjectures” and “Counter-examples” into my classrooms has been one of the most impactful changes to student engagement in my classes over the past few years. And just as exciting–it’s super easy for you to put it in your classroom too!

Although I’ve used these terms day-in-and-day-out while I was studying mathematics, using them in K-12 classrooms was inspired by Math for Love–they’ve got a great write-up about it here. I’ve presented about this framework at the National Council of Teachers of Mathematics conferences in Boston and Salt Lake City this fall (slides here), and now I’m putting it online to share with anyone and everyone.

Rather than defining the words, let’s just dive right in:

Conjecture: Everything that has legs can walk.

I’m sure you’ve already thought of a handful of counter-examples (tables, chairs, babies, etc.) Of course, you only need to tell me one in order to convince me that my conjecture was false. Students are super eager to share counter-examples when they find them–and it’s precisely this eager engagement that is largely absent in most math classrooms across the world.

When you put rich math tasks and rich math problems in front of students, they naturally notice patterns, and they are eager to talk about those patterns. And when they’re primed with the language of Conjectures and Counter-examples, that eagerness to talk becomes eagerness to pose conjectures and to find counter-examples. The talk becomes rigorous mathematical discourse using the exact same vocabulary that professional mathematicians use.

Let’s look at one of my favorite problems to make this a bit more concrete. It dates back at least as far as Pythagoras and seems to engage learners at all levels from 4th grade and up:

Choose a positive integer.
List its factors.
Add up all the factors, including 1 but not including itself.
Repeat the process with the number you found.

Example:
Start with 16. The factors are 1, 2, 4, 8, and 16.
We add 1 + 2 + 4 + 8 and get 15.
Now, we continue from 15: the factors are 1, 3, 5, and 15.
We add them up and get 9.
Continuing, we see the factors of 9 add to 4.
And the factors of 4 add to 3.
And the factors of 3 add to 1.
And then we stop, because adding an empty list of numbers seems problematic.

To summarize that example, we went from 16 to 15 to 9 to 4 to 3 to 1. If you’re a curious creature, all sorts of problems are likely popping into your head.

Do the numbers always get smaller?
16 and 4 both went down by 1–is there something to that?
The pattern went even, odd, odd, even, odd, odd–is there something to that?
We ended up hitting every square number along the way from 16 to 1… does that always happen?

Although I’ve paraphrased them a bit, these questions are all directly out of the minds and mouths of my 4th and 5th graders. And 4th and 5th graders are quite capable of turning these questions into conjectures!

Numbers always get smaller.
Powers of 2 always go down by 1.
The pattern will always go even, odd, odd, even, odd, odd.
If you start with a square number, you’ll hit every square number smaller than the one you started with.

Students find counter-examples to most of these conjectures within a few minutes of playing around. Wait–I found a counter-example! 30 gets bigger!

The classroom bubbles with genuine engagement with student-posed ideas about one of the most oft-discussed math problems of human history.

The way this operates in my classroom is really quite simple: I have a stack of Conjecture Pages and Counter Example Pages available at all times. Whenever a student wants to, they grab a Conjecture page, write down their idea, and use a magnet to post it on the whiteboard. At this point, we say the student has entered a conjecture into the classroom conversation. And it really is part of the classroom conversation–when a student can’t make sense of someone’s conjecture or thinks they have a counter-example, peer-initiated, peer-moderated, and peer-resolved conversations about rigorous mathematical ideas arise naturally in the classroom. And if you play your cards right, they’ll continue outside the classroom too!

A bit later, I also introduce Definition Pages, but asking students to write definitions is substantially more ambitious. To get started, I’d suggest just printing out some Conjectures and Counter-examples and diving in.

And amidst all this intense curiosity about deep mathematics, students are working on grade-level content skills: they’re working with peers to make sure they find all the factors and adding long lists of numbers. They’re relying on peers rather than an instructor or an answer key to check their work, and as an added twist, I’ve tasked my students with drawing scale rectangles to represent the factor pairs they find. And because I happen to know just how surprising it ends up, I’ve asked them to draw rectangles for each step of the process starting with 30. They’ve found that 30 goes to 42 to 54 and then to 66! (But Mr. Gafni, how can we draw a scale 1 by 66 rectangle on graph paper? There’s not enough boxes!) I’ll let you play around with it yourself to see what’s going on with this pattern. Does it keep counting by 12? Does it go on forever? The rectangles might shed some light on what’s going on: 30, 42, 54, and 66 all yield a very similar collection of rectangles…

Here are some Student Posed Conjectures from a high school Number Theory course I taught in the Summer of 2018. I’d like to put up some 4th and 5th grade conjectures too, but I’m having scanner issues at the moment… Feel free to ping me about it if they’re not up by the end of October!