Response: Several Ways to Become Better at Teaching Math

For most of my career, I’ve encountered a fair amount of variance about mathematical pedagogy, especially as it concerns the idea of mistakes. There are those who will pick a student apart for being off by one digit, and there are others who will dilute their curriculum because the “poor kids” don’t have enough computational or procedural understanding to even get intermediate concepts. At times, I’ve been guilty of siding on either extreme, but after a few years of this conversation, I’ve come to a concrete solution that may satisfy both the math gods and the sympathizers alike.

I follow a three C’s rule for mathematical answers: complete, correct, and consistent. So long as these three components are present in a student’s answer, I can’t see why the answer isn’t legitimate as a response. Let me expound.

“Complete” means that the student has described his or her response to the point where anyone else who follows the response can pick it up and clearly see the logic from problem to solution. For example, most mathematicians are OK if a student answers 6 x 8 with 48 without showing tally marks or numerical grouping. Most of us wouldn’t be OK if the student was asked to find 20% of 68 and simply put 13.6 on their paper. We would immediately ask “how” or “why” and would hope to get an appropriate response.

“Correct” means that the student actually answered the question correctly, whatever correct means for that particular problem. For instance, we would still be OK with students solving 6 x 8 as 48, but if we added units to those numbers, like 6 in. x 8 in., we would see the correct answer as 48 in.2. 48 on its own would not be correct because it’s not correct. Obviously, the concept of completion and correctness are intertwined here. I envision completeness as a matter of explication, and correctness as a matter of a finished product.

“Consistent” means that the student can use a similar methodology for arriving at the answer for one problem to any similar problem. For instance, the student may have taken ten steps to solve what should be a three-step solution. However, if the student can consistently use the ten-step method for solving such a problem with any given problem, I find it acceptable. That obviously merits a discussion of efficiency because, at some point, we want the student to see some more efficient methodologies, but we ought not to disregard a student’s answer because it doesn’t follow the proper procedure per se.

We want to encourage students to see their thinking in math. No longer can some of us take on the idea of math as this immovable object created by ancient mathematicians. If anything, we should promote them to challenge our thoughts so they too can arrive at creative solutions. We can find the balance between rigor and acceptance. The three C’s may be one of the keys.

The absolute best way to improve as a math teacher is to overhaul the way you assess your students. Math class is ripe for assessment reform, and reform must be done because math is the proving ground for most students’ abstract thinking skills.

Sadly, math class is often the time when students learn to play the game of school the best. Students learn that cramming and purging are valid academic strategies, and math instructors do little to stop this process. Often we berate our students when they can’t remember material from two chapters ago, but can we really blame them? The assessment strategies used in most math courses lend themselves to creating a poisonous culture that demotes learning to an accidental byproduct.

The solution to this problem is simple: Math teachers must give up the love of scheduled quizzes and perfectly spaced exams in favor of an assessment scheme that allows students to show their learning when they actually achieve it. This scheme should include periodic reassessments of all of the learning targets in the course, such that students are sent the message that retention truly matters.

In addition to a sensible reassessment policy, teachers must also make learning targets explicit. As a teacher, I claim to demand rigor and remediation from my students, but without clearly stating the goals and philosophical hurdles students need to grapple with, they are left to guess what I think is important, and end up developing the venomous psychology of I-hope-he-doesn’t-put-that-on-the-test.

There are a few simple fixes that any teacher can implement tomorrow. First and foremost, fight the urge to record assessment titles in your grade book. Students, parents, and often teachers have no idea two months down the road what “Quiz 5: Chapter 2" assessed. This is debilitating to a student who needs helps with the material that may have been assessed on that quiz.

Instead, record items across your grade book that read, “Slope of a line: computation,” and “Slope of a line: identification in real life.” I can guarantee that a failing grade in either of these targets can be remediated effectively by parent and student. To maximize the impact of this simple change, allow the scores for these targets to be overwritten by new assessments. Don’t damn students to the average of their past and present performances; this sends the tacit message that failure is to be avoided instead of learned from.

Finally, go read A Mathematician’s Lament by Paul Lockhart, take a nap, and come out swinging.

Dear new teacher:

The “dear new [anything]” genre may be doomed. Instinct tells me I should lay out a list of maxims and explain each one briefly. Something like, “Good curriculum will make classroom management mostly unnecessary.” I would elaborate on that, explaining how early in my teaching I invested myself into routines and norms that’d occupy my students well enough that I could escape the hour alive. I’d go on to say that routines and norms are still important but their value to a student peaks several hundred miles beneath the place where great, perplexing curricula is just beginning to reveal its value. “A perplexed, engaged student is a well-behaved student,” I’d say.

That list would be fun to write. It’d be useful reflection for me and perhaps interesting also to the teachers I run around with online -- most of them between three and twenty years of experience. But that’s the thing, really. They’ve all had between three and twenty years of experience. They’ve all experienced the limitations of classroom management. They’ve all had that flash of insight that “this is really as good as classroom management gets, isn’t it” followed by the question, “What else is there?”

It’s at that moment of intellectual need that my list of maxims, my answers to the question, “What else is there?”, would be most valuable to you. If you’re just looking ahead to your first year of intern teaching, that list of maxims will be much less valuable to you.

But there is one maxim that might strike you at the right moment of your intellectual need:

As much as you’re currently learning about learning in your preservice program, that’s just a raindrop in the ocean of what you’re going to learn over the rest of your career. Find ways to manage and profit from that learning.

If you receive some kind of certificate at the end of this process, that certificate doesn’t qualify you for success, as it does if you were studying to be an MRI technician in a hospital. It qualifies you for failure. The people who gave you that certificate know you’re nothing now like the teacher you’ll be in five years. They know you’re going to fail. They hope you’ll fail productively.

I failed productively by writing down my failures on pieces of paper. Seeing all my failures in one place helped me see their common features. (“Too overbearing. Poorly organized. Careless selection of examples.”) At another point I started writing those failures down online, publicly. I didn’t write with the expectation that other teachers would read my words but teachers stopped by anyway -- better teachers than I was -- and they helped me analyze my failures in ways that I couldn’t.

Eight years now into math education, I can call up any kind of issue like classroom management or assessment and watch my status progress from “beginner” to “not bad at all.” That’s how I managed and profited from my learning about teaching. You need to do the same.

Because the amazing and frightening fact about the task you’re taking on is that there is no way to stop improving. Teaching will happily accept any investment of a minute, an hour, or a year you want to give it. Your best insights about the job will come at times when you wish you were thinking about anything other than your job. Manage and harness that learning. And find some way to enjoy it, above all else.

Dan