65 Uses for a Paperclip
Conjectures and Counterexamples in a Thinking Classroom
Dan Finkel
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I have been developing a structure to help teachers invite students into a genuine mathematical process, starting with their own understanding. We call it Making and Breaking Conjectures. Recently, we talked about this structure in our Math for Love/WXML Math Teachers’ Circle in Seattle. First, what are conjectures? What are counterexamples? A conjecture is a mathematical hypothesis, a guess of the underlying structure or pattern based on what we know so far. A counterexample is an example that proves a conjecture false.
Mathematics as a field progresses by way of conjectures and counterexamples. The good news is, we can use them even with very young kids.
We played the game “Counterexamples: to get a sense of how this works. The game is supersimple: The teacher makes a false conjecture, and the students prove it false with a counterexample. For instance:
Teacher: All pets have four legs.
Students: No! Because birds have two legs!
Teacher: Okay – refined conjecture: All pets have two or four legs.
Students: What about a snake?
Teacher: That’s a pet with no legs. So I’ll refine my conjecture again. All pets have at most four legs.
Students: What about a spider?
And so on. So compelling is the game that our MTC almost got off track when we considered examples of conjectures about area and perimeter. But there was work to be done on generating our own conjectures.
To warm up our observing, noticing, wondering, and conjecturing muscles, we started by asking, “What can you do with a paperclip?” This is a classic exercise, and kindergarteners tend to beat the pants off of adults. After spending ten minutes or so, groups had come up with everywhere from 20 to 65 uses for a paperclip. That was just the warmup, though, so we didn’t go too deeply into what the uses actually were; the main event was still on its way.
What Can You Do With This Grid?
Considering this question was trickier, and required a little more discussion to draw out observations and questions. By the time the conversation was done, however, we had a tidy collection of questions to consider:
 How many rectangles can be formed by connecting dots with straight lines?
 How many sides could a polygon have on the grid? Could it include all 16 dots?
 Can you make a square with any number of dots on its perimeter?
 How many different paths are there from the bottom left to top right?
 How many lines would there be if you connected every dot to every other dot?
 How many different lengths can you find by connecting two dots?
 How many symmetric shapes can you make with corners on the dots?
 How many fractions can you represent on the grid?
 How many different areas can you get if you form a triangle with three points on the grid?
 Can you find two triangles with the same area that are not similar?
Take in that list for a minute. There are weeks of beautiful, highlevel problems to explore here. For each, you can start by casting around. Soon, you’ll find you have conjectures. Once you have a conjecture, you can try to break it right away by looking for a counterexample. Refine and repeat until you end up with something that seems to be true, and then you can put together an argument, with luck and a little insight, into why it actually might be true. We’re really doing mathematics!
With all these questions to consider, we wanted to provide some guidelines. So we posed a choice: Teachers could choose the problem that inspired them most and work on that one. Or they could work on the question of how many different areas a triangle could have if its corners are on the grid. (Having a default question is something we’ve found can help prevent groups from becoming aimless. With an actual classroom, you might want to skip some of these steps, and start from a more tightly focused, teacherchosen task in the first place. Still, opening up the entire thinking process can definitely be worth it, if you’re ready to take the step.)
And with that, we sent the groups out to work on their own. However, we had one more idea ready at hand to ensure things went as well as they could go.
Transitioning to a “Thinking Classroom”
Peter Liljedahl has been developing a series of concrete steps to change the classroom to support genuine thinking in mathematics. The steps to encourage this transition are bizarrely simple to employ. Here’s how you begin, according to Liljedahl:
 Use “tasks” (that is, make real, meaningful mathematical experiences the heart of what you ask students to do).
 Use vertical, erasable surfaces (i.e., whiteboards).
 Assign groups in a visibly random fashion.
It seems almost too easy, but the room was electric with thinking energy. It’s hard to overstate how getting everyone standing and working together impacts the quality of thinking and engagement in the room. One of Liljedahl’s recommendations is to use just one marker per group. You can see the value of that suggestion in one of the videos in the blog post version of this article, when the carrier of the marker is drawn back up to the board to write in details of someone else’s computation.
With so many teachers considering such a range of problems, we decided to let groups pair up to share what they had discovered with each other.
All in all, everyone was totally energized by the workshop. The participants left with a whole bunch of great questions to ponder about grids, and the motivation to use tasks, whiteboards, and visibly random groups in their own classrooms.
And we organizers came away with a goal for this year’s meetings, especially for the middle school level: Create a series of professional development sessions that will help everyone turn their classroom into a Thinking Classroom, where students use Conjectures and Counterexamples to power genuine mathematical experiences.
Dan Finkel is Founder and Director of Operations of Math for Love, and coorganizer of the WXML/Math for Love Math Teachers’ Circle in Seattle. This article is adapted from a blog post at Math for Love.
 Peter Liljedahl’s website
 Read the original blog post on which this article is based.
This article originally appeared in the Spring 2019 MTCircular.
TABLE OF CONTENTS
SPRING 2019 MTCIRCULAR
The Roommate GameAn Exploration of Stable MatchingsEmily Dennett and Chris Bolognese 

65 Uses For a PaperclipConjectures and Counterexamples in a Thinking ClassroomDan Finkel 

Recruiting Change for a DollarPatterns Don’t Solve EverythingCraig Collins, Elizabeth Donovan, and Cynthia Cramer 

Contest: Turn Your World on Its SideWin a Free Book from MAA AMC 

Case Studies of MTC Participants Published in The Mathematics Educator 

Oklahoma State Department of Education Awarded Federal Grant to Support Native Youth and Implement Math Circles Statewide 

A Note from AIMRich Histories, Bright Futures 

Dispatches from the CirclesLocal updates from across the country 