By Steven Klee, Contributing Editor, Seattle University
During my freshman year of high school, my geometry teacher came into class one day and challenged us to trisect an angle with a compass and a straight edge. Anyone who was successful would receive an A in the class for the rest of the year. We wouldn’t have to do any more homework or take any more tests. Nothing. Of course, this should have seemed too good to be true. But I was in ninth grade and didn’t know any better, so I set off to solve this seemingly innocent problem.
I came up with a dozen or so false proofs, all of which included reasoning like “well, now you just move the compass a bit over here and then you draw this line, and it works!” Of course it didn’t work, but this is the kind of non-proof you would attempt to make if you had only just learned what a proof is.
But rather than simply tell me I was wrong and insist that I was doomed to failure, my teacher let me share the ideas behind every failed proof so that I could see the shortcomings in my arguments. He sat with me and we talked more broadly about what does and does not constitute a proof. He knew I was going to be wrong. He knew this was an impossible assignment. But he still listened.
My teacher’s openness to hearing my ideas inspired me to keep working and to keep trying new approaches. As I learned more math, I kept coming back to this problem. I tried using trigonometry. I tried using calculus. I tried making up a unit distance that I would call “1.” After watching Good Will Hunting, I decided that it would probably help if I drew all of my diagrams on mirrors. None of these things helped. Along the way, I learned about quantifiers. I learned about proofs. I learned to identify the errors in my attempted proofs on my own. Ultimately, I think I shed a tear of joy when I finally saw the proof of impossibility in my graduate algebra class.
This story can lead to a lot of different discussions. Ben Braun wrote a beautiful article for this blog about the value of having students work on difficult and unsolved problems, which I highly recommend. Instead, I’d like to explore the value of talking about mathematical ideas informally, especially when they are ill-formed and possibly incorrect; the value of encouraging our students to share such ideas with one another; and the value of participating in these discussions with our students.
Why should we take time to talk about math?
The practices of active learning, inquiry based learning, project-oriented group learning, and others have become quite popular as means of addressing the fact that simply lecturing about math is not effective for students (Deslauriers 2011), (Freeman 2014), (Lew 2016). Encouraging mathematical communication is a byproduct of many of these methodologies, and I would like to start by arguing in favor of talking about math because there are added educational and cultural benefits to encouraging open discussion of mathematics both inside and outside the classroom.
- By discussing mathematics together, our students develop their own language and intuition for mathematical ideas. As I circle around the classroom listening to my students as they work, I sometimes hear conversations that can best be described as the opposite of the game of “telephone.” One student tries to describe a solution in a way that, to me, is completely incoherent. But his groupmates kind of get the idea and someone else chimes in with a more coherent explanation. Then someone else adds more clarity. And by the end they have a solid basis upon which we can introduce more formal language and definitions. If I had simply interjected with a litany of corrections and edits after the first incoherent attempt, I would be robbing my students of the opportunity to learn by developing their own ideas. I think we would be fooling ourselves if we claimed that our private research moments and meetings with collaborators did not follow this similar “telephone-in-reverse” phenomenon.
- Mathematical conversations encourage multiple ideas, multiple perspectives, and different solutions. It seems fair to say that most of us, as educators, want our students to appreciate that a single problem may have many possible solutions. Through traditional lecture, we may only present one such solution, leaving students who had different approaches to wonder if their solutions are correct — or worse yet, to believe that their different ideas are wrong.By giving our students time to work together and talk together, we give them the time to learn from one another by discussing different solutions and approaches to the same problem. Students are more likely to ask the question “I did the problem in such-and-such different way. Does that still work?” in a small group or private conversation than they are in front of an entire lecture section.
- By discussing our students’ ideas, we can provide more personal attention to their learning. When we talk with our students we can very quickly assess the difference between someone who can complete all but the hardest problem on a homework set and someone who is struggling with the first problem. Talking about the first problem with the former student or talking about the hardest problem with the latter student is a waste of everyone’s time. By talking individually with our students, we can concentrate on what they need to learn based on what they already understand.
Most of all, we empower our students’ learning by giving value to the questions and mathematical ideas that are at the front of their minds, rather than by taking a scatter-shot approach and hoping that we address something that is meaningful to everyone at some point during each lecture.
Mathematical (mis)communication extends beyond the classroom
William Thurston wrote an article “On Proof and Progress in Mathematics” for the Bulletin of the AMS (Thurston 1994) where he says:
Mathematicians have developed habits of communication that are dysfunctional…we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models.
He goes on to explore this idea further in an example. If Alice and Bob are researchers within a given subfield, Alice may be able to communicate the overall ideas behind a recent research development to Bob over coffee. But in contrast, Bob may struggle to glean similar insights from an hour-long colloquium talk or over the course of several hours of reading Alice’s paper. Thurston continues:
Why is there such a big expansion from the informal discussion to the talk to the paper? One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use gestures, they draw pictures and diagrams, they make sound effects and use body language. Communication is more likely to be two-way, so that people can concentrate on what needs the most attention.
In contrast, research talks and written papers require far more formalism, and they prevent the audience from interacting with the material in such a personal and intuitive way.
As professional mathematicians we have all experienced this. We have all sat through talks without understanding anything after the first five minutes. We have all read the same sentence in a paper 20 times without understanding its meaning. And we have all asked a question over coffee to find illumination in a well-phrased answer from a colleague, collaborator, or friend. So if this is the case when we, the so-called experts, are trying to learn new material, how then can it not be the case for our students as they are trying to learn mathematics?
How can we facilitate mathematical discussions?
In theory this argument may resonate with a lot of people, but implementing these ideas may seem difficult for any number of reasons. Here are a few concrete tips that can be implemented anywhere:
- Set aside 5 minutes of each class for your students to work on an example problem. This example can be as simple as “What is the derivative of \((3x+1)^2\)?” Have them compare their answer with their neighbors and give each other a high five if they agree. If you have more time, set aside more time and give the students more problems. An example the students work out together is far more valuable than another example you do on the board.
- Encourage students to attend your office hours, your TA’s office hours, and a campus math help center. Remind them about these resources every day. Be open and approachable. Your students are human beings, and many of them are interested in doing cool things. If you engage with them on a personal level, they will feel more comfortable in asking you math questions.
- Share your mathematical struggles with your students. One reason that many of us have been successful as mathematicians is that we are willing to keep working on a problem that seems impossible at first. But in our students’ eyes, we can seem to be omniscient solutions manuals who know how to solve every math problem. We need to strive to bridge this divide.
- Solicit student input in helping you present solutions to problems. Ask them to articulate why they did certain things, and develop diplomatic reactions to incorrect ideas. Rachel Levy posts some great suggestions for accomplishing this here.
And Mr. Pelzer, if you’re out there reading this — thanks for letting me share my ideas with you. Failing to trisect an angle sparked a lifetime of mathematical curiosity.
Deslauriers, L. et. al. “Improved Learning in Large-Enrollment Physics Class.” Science 332 (2011): 862-864.
Freeman, S., et. al. “Active Learning Increases Student Performance in Science, Engineering and Mathematics.” Proceedings of the National Academies of Sciences 111, no. 23 (2014): 8410-8415.
Lew, K., et. al. “Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey.” Journal for Research in Mathematics Education 47, no. 2 (2016): 162-198.
Thurston, W. “On proof and progress in mathematics.” American Mathematical Society. Bulletin. New Series 30, no. 2 (1994): 161-177.