Math Practice Isn’t Like Practice in Sports or Music but It Should Be
Practice should look more like performance.
Okay so somebody made a stamp that rolls out practice problems in arithmetic.
It’s a beautiful bit of craftsmanship that I think is bad for math students. In my usual spirit of comity, I encouraged its inventors to invest themselves in a project with greater returns to society.
This prompted a bunch of people to rush to the defense of fluency practice in general. Fluency practice doesn’t need defenders, least of all from me.
I mean, check out the TIMSS study which surveyed teachers on their instructional activities in 2011. They found the US leading the world in students working problems.
Fluency is doing fine.
Thanks for reading Mathworlds! Subscribe for free to receive new posts and support my work.
People used different metaphors in their defenses of fluency practice, though, including basketball, carpentry, music, and others. Fluency is every bit as important in math as it is in each of those disciplines but those disciplines develop fluency in ways that are distinct from math drill sheets, and those distinctions are worth a quick look.
The math drill sheet offers worse feedback.
When you whack a nail into a board or take a jump shot or press a key, the world returns to you a bounty of information about your efforts. Is the nail crooked? How many swings did you need? Does the ball go in, go long, go short, go sideways? Did the note sound flat or sharp?
If carpentry or basketball or music were like the drill sheet, you’d shoot a shot and the ball would disappear as it left your hand. The nail would evaporate on contact with the hammer. You’d press the piano keys and hear nothing. You’d move to the next shot or swing or song having grown no wiser from the last.
The math drill sheet is lonely.
There are digital math drills that offer students certain meager amounts of feedback but even here they tend to isolate students in ways that are distinct from the disciplines people namecheck when they defend math drill sheets.
It’s possible to work on a carpentry project by yourself or shoot drills alone in a gym. But those are less common than working with someone on a project, participating in group drills, or, at least, observing other people doing their drills, gaining inspiration and knowledge from others.
The math drill sheet conceals the performance of mathematics.
When people practice sports or music, they generally do so with a strong understanding of the performance that practice is meant to support. They’ve watched other people play the games or listened to other people play music. They have very likely messed around clumsily on a court or at a piano. They understand its point and like it well enough that the practice feels welcome and necessary.
Define the performance of math however you want and ask yourself, first, how well do your students understand it? Have they watched other people play math? Have they messed around clumsily with math themselves? Do they understand the point of math and like it well enough that the practice feels welcome and necessary?
We need skill fluency that blends practice and performance.
When you look at the ways professionals (and even highly engaged non-professionals) develop fluency, they do so in a blend of practice and performance.
Read up on some of Steph Curry’s shooting drills. He isn’t standing stock still from different positions in the court, shooting over and over again, assuming (as many do in math) that fluency in those discrete areas will lead to expert performance in a game.
Rather, he’s moving between positions as he would in the performance of basketball. He’s trying to do it quickly because the performance of basketball (unlike the performance of math) is timed and requires speed.
What I appreciate about games like Garbage, which I’m playing with my own kids right now, task structures like Open Middle, ideas like variation theory, task fields, etc, is that they take practice seriously while also insisting that the practice should testify to the performance. They realize that even while students are learning lessons about fluency, they are learning other lessons as well, including lessons about the nature of performance.
Math drill sheets—whatever they teach students about operational fluency—also teach students a distorted view of the performance of mathematics.
You might be interested in David Perkin's extended riff on this metaphor: https://www.gse.harvard.edu/news/uk/09/01/education-bat-seven-principles-educators
<isthereHTML?><blockquote>Perkins identifies two unfortunate tendencies in education: One is what he calls “elementitis” — learning the components of a subject without ever putting them together. The other is the tendency to foster “learning about” something at the expense of actually learning it. “You don't learn to play baseball by a year of batting practice,” he says, but in learning math, for instance, students are all too often presented with prescribed problems with only one right solution and no clear indication how they connect with the real world.
The way to let young learners play the whole game is to find or construct a junior version of it. A junior version of baseball may involve fewer innings, a diamond that is smaller than standard, or teams consisting of whatever neighborhood kids show up in the park on a given day. Yet the junior version conveys the essence of baseball — swinging at and hitting a ball and then making your way around bases while the opposing team scrambles to put you out.
In teaching math, drilling children in multiplication or long division or even giving them “word problems” is likely to lapse into “elementitis.” But giving a child some money and asking her to calculate whether it's enough to buy the items in her shopping basket is a “junior version” of the way math skills are used in the real world.</blockquote></whatnoHTML?>
This is a really important distinction. When I set up a drill in basketball, the kids know why we are doing it and what aspect of the game we are trying to work on. Plus, it is dynamic and kids react to things that their teammates are doing. Kids don’t necessarily have that same intuition about a fluency topic and don’t often have the bigger picture and narrative about how that skill fits in to bigger questions they are tackling and why they might care about improving it. If someone wanted to take a 100 free throws after practice it was usually the direct result of them missing a bunch of free throws in a game or knowing they get fouled a lot. If I had multiple practices or even one where all I had everyone do was shoot free throws there *might* be some benefit but there would rightfully be a mutiny because it misses the mark of what it means to play basketball. Seems like a pretty big and exciting job to keep thinking about how to get every kid excited about the full and messy performance of math.