The following guest post comes from my friend and colleague, Robert Talbert. It is thanks to the success of the University of Michigan (MS '92) in the March Madness tournament that I earned this honor. I do not know if Robert is aware that I also attended Northern Michigan University (Teaching Certification '87) and that my grandfather was the registrar there, but these connections make this post all the more special. Thanks to Robert and GO BLUE!
“It makes sense when you do it, but when I try it, I have no idea where to start!” In other words, it never made sense. The sense-making was only an illusion and did not actually happen.
Anyone who has taught mathematics to a real human being for more than 30 minutes has encountered the above statement. I think the sooner we realize that sense-making is not something that you feel but something you do, and the sooner we mathematics instructors shift our focus away from providing feelings (can I say “sensations”?) of sense-making and onto the actual making of sense in the classroom, the better off we will all be.
I’m currently at the annual meeting of the Michigan section of the Mathematical Association of America, where a number of the talks so far have centered on sense-making, none moreso than a plenary talk by Peggy House of Northern Michigan University titled “Reasoning and Sense-Making”. I appreciate Peggy’s perspective on reasoning and sense-making, since she is a veteran classroom educator and math education specialist. Regular readers of my blog over at the Chronicle of Higher Education know that I spend a lot of time thinking about various instances of the flipped classroom, including the adaptation of Eric Mazur’s model of peer instruction to university mathematics. The more I work with these pedagogical models, the more I realize that sense-making is really what they are all about.
A good peer instruction question, for example, is one that is aimed squarely at the biggest misconception on a fundamental concept — which is to say, it unearths the very thing that students most urgently need to make sense of. And then we ask them to make sense of it. And these days, when I discuss the flipped classroom with someone and they ask, “What do you do with the class time once the lectures are moved out?”, I just say: sense-making activities. I want students not only to make sense of the things they are learning but also learn how to make sense of things.
In Peggy’s talk, she posed the following problem: Find the area of the octagon in the picture below, which is formed by the intersection of the various segments connecting vertices and midpoints of a square:
One of the first points Peggy made in her talk was that sense-making is in the eye of the beholder. When instructors approach mathematics as pure procedure and then focus on showing students how to perform a procedure, this not only is not sense-making, the thing we are talking about will only make sense to the students insofar as their ways of seeing the problem coincide with our ways of seeing it. Which is to say, most of the time it won’t make sense at all, even though after a clear lecture it will seem to make sense. This is a dangerous position for students. It would be better to have concepts not make sense and for students to know that they don’t make sense, than it would be for the concepts not to make sense although students think that they do.
In the octagon area problem, for instance, I could certainly lead students through a clear discussion of a solution through direct instruction. This would not be all bad. There is some value to seeing an expert learner (that’s me) explicitly model his or her decision-making and thought processes as he or she works through a problem. But the benefit ends there, because this problem is not about “the right answer”. Very few nontrivial mathematics questions are! Peggy went on to give examples of some of the ways her geometry students solved this problem:
Some of them were wildly creative:
And some of them were creative in all the wrong ways — for example, one group got an answer by assuming that the octagon is regular, which as it turns out, it isn’t. And that’s where reasoning comes in to sense-making. When a student is solving a problem, it’s not enough to be creative. There also has to be good reasoning involved as well. If one’s reasoning behind a solution or a conjecture is flawed, then the concept being thought about has not fully made sense yet.
One of the great lessons learned from this talk, something I knew already but can always use encouragement on, is that my job as a mathematics teacher is not to download by brain into the students’ brains, or even to wire their neural pathways to look like my neural pathways. Indeed this is not really even possible, because — thank goodness! — every student is a unique human being, with a unique backstory and ways of looking at the world. Sense-making is in the eye of the beholder. My job instead is to provide them with an environment that is rich in sense-making activities and in which students are held to high standards in the ways in which they argue for their solutions.
Someone in the audience asked, not cynically, how instructors in K–12 schools can be expected to teach in this sort of way while still preparing kids for the content on standardized tests. Peggy’s answer was, you just have to do it, and perhaps there are ways to design classes where you can teach content in this way and not lose much. I have nothing to add to that answer, except that I hope that people everywhere begin to realize that it doesn’t do much good for students (K–12 or university level) to know a lot of content that makes no sense to them.
Epilogue: I’m writing this guest post because I lost a bet that I never made. My wife’s alma mater Indiana University was making a good run of it in the NCAA mens’ basketball tournament back in March. Dave, John Golden, and I thought it would be fun to see whose Big Ten alma mater (or proxy thereof; I went to Vanderbilt, which posted a relatively miserable 16–17 record and didn’t even make the NIT) went the farthest. Needless to say the Hoosiers didn’t get it done. But at least some good can come out it.