Open Set

For years, math reformers have been talking about the Big Math Teaching Metaphor.  It takes a few different forms, but they are all basically the same:

 “The way we teach math in this country is like if we taught students to play scales on the piano, but never let them listen to a song.” 

“The repetitive practice students get in math class is like learning to play soccer by running drills, but never having a scrimmage or a game.” 

I really like this metaphor! We know how people learn, we see it when they successfully play a sonata or score a soccer goal. There was a skill they were not born with, but they spent time building their knowledge and skills, and eventually they were able to master it in a way that brings joy to themselves and others. But in the math classroom, we rarely aim for a sonata or a scrimmage. Instead, we just asked students to imitate the teacher’s steps, practice over and over until factoring or calculating a slope was as good as muscle memory. 

Recently, I’ve heard this same metaphor used in other ways. In particular the current discussion has people fuming about the lack of practice in math classrooms. You might hear an argument like, “Have you ever tried to teach someone how to do a jump shot or hit a tennis ball? You can’t do it without practice.” The debate is raging about how the new math textbooks don’t have enough practice problems in them, they only focus on conceptual understanding. And so the metaphor persists, and gets used in new ways. And I still think it is a valuable metaphor that holds up!! Absolutely, when learning how to do a jump shot, you need to practice. And when you are learning how to solve an equation, you need practice too, for analogous reasons. On the court, you need speed. In a complex math problem, you need working memory. It’s powerful how well aligned these ideas can be.

I find that this metaphor actually holds up better the more I think about it. Notice that the people advocated for more practice pick out a small skill. They are not talking about learning to play the game, instead they are talking about learning how to do a jump shot or hit a ball. Small, specific skills require more practice, absolutely true. On the other hand, if all you know is how to make a jump shot, you are not going to get very far in the game. You need to know when to guard closely and when to stand back, you need to be able to find the opportunity to push past the defenders, etc. Some things need to be learned through playing the game, not just through practice drills. And yet without the practice drills, players would be missing the skills they need for the crucial moments. The whole point is that you need both the drills and the play time. No one would dream of coaching a sports team without both drills and scrimmage.  

What worries me about the practice supporters is that they might forget that we are here to play the whole game, not just to learn how to do jump shots. I became a teacher so I can help students develop their ability to reason abstractly, to make sense of problems, and to become confident users of mathematics, in addition to developing specific skills. Because that’s the game. 

I agree that students will never be confident users of math if they never practice the skills they learned. And most people will never learn any useful skills if the teacher never pushes them to use a somewhat efficient method. But what really worries me is the general mindset that, to rely on our metaphor, there is no “game” in math, it’s just a collection of techniques that need to be memorized and practiced. And it worries me because I’ve come across it so often, from parents and admin and other teachers. I wonder how much of this discussion about practice is an outlet for a mindset that I fundamentally disagree with, through the guise of a teaching technique that I do agree with. 

And yet, we have not stretched the metaphor as far as it can go! I once heard the metaphor completely misused in a conversation about how to work with students who are behind grade level. I was in a discussion about this, and someone told me that we need to teach students exactly the same grade-level material as we teach everyone else, and give them questions that are complex and challenging within that grade level material, even when we know that they are missing huge swaths of relevant background knowledge. (They conceded that some quick background lessons would be okay, but not taking up much of our class time.) The justification was that we want all the students to be able to “play the game” instead of just “doing the drills.” 

Alas, this was a misuse or misunderstanding of the metaphor. If you think through the analogy, I think you reach a different conclusion. A student who is behind grade level in background knowledge is like a basketball player who does not know how to pass or dribble or shoot at all. What does the coach do with those players? They teach those skills through practice drills, AND they let them play games. They don’t just give them exactly the same drills as everyone else and then put them on the varsity team in week two. They take a group of players with somewhat similar levels of skill and let them play games against each other. This is part of why we have a JV team. Playing on the JV team lets you build up your skills and your knowledge of the game in a way that being on the varsity team just would not do for a player who is substantially behind. 

I think that we can accurately stretch the metaphor to shed light on our work as teachers. A student who is behind in math should learn the most important relevant material, and they should practice what they have learned. They should also have the opportunity to “play that game,” but at the JV level. That is, they should have thoughtful problems that require them to make sense out of a situation, figure out how to solve something, and use their reasoning skills, while utilizing and/or exploring some of that background information. We should not, for example, expect them to learn how to factor integers, factor out a GCF from a binomial, and then use the grouping method all in one day, just because the rest of the class walked into the room ready to learn the grouping method with all the necessary background. Nor should we treat factoring integers just as something to memorize and practice with no thinking or reasoning involved just because you were supposed to learn it four years ago. The earlier material can have its own “game” too, you don’t need to just throw the students into the hardest problems in order to let them “play the game.”

I think that there is an awful lot to be said for this metaphor. Why is it that we can get so much insight into math education from looking at how we teach sports or music? That is truly a question worth pondering.