Having listened to Meyer’s Ted Talk, attended one of his workshops, and read several of his articles, I consider myself somewhat familiar with his work and while I don’t agree with all of his philosophies, I generally hold him in high regard. Meyer is renowned in the math ed world and although I’m sure he’s deserving of much praise, I find the arguments he puts forward in this blog to be flimsy, setting a trap for fools.
He begins his short essay by mentioning a stamp that rolls out arithmetic practice problems. This, he feels “is bad for math students.”
As a teacher, I never delivered randomized problem sets, nor do I recommend that other educators use them. However, I think elementary mathematics trainers are often quick to criticize our profession’s orthodoxy and in doing so jump to assumptions about traditional educational tools as well as the instructors delivering them.
It’s presumptuous to think that teachers who use “Math Drill Sheets” are not providing meaningful practice for their students. Too often, we as educators connect a page filled with equations to instructors mindlessly selecting topics, running off copies, and then putting them in front of students, who have no strategy to solve the problems in front of them.
Consider a teacher who provides their students with several dozen addition within ten facts after they’ve taught strategies to solve every problem on the page. That worksheet reflects weeks or maybe months of learning that the instructor wants students to enter into long term memory. There’s a big difference between a child understanding adding within 10 strategies, and being comfortable and fluent using them to arrive at accurate answers. How do they make the leap from understanding to fluency?...Deliberate Practice, a phrase coined by late Swedish Psychologist Anders Ericcson in his acclaimed book Peak: Secrets from the New Science of Expertise.
Meyer went on to write that mathematical fluency is every bit as important as it is in disciplines such as carpentry, basketball, and music. But - there are 4 distinct ways in which developing fluency in those domains is different than using drill sheets to do so in math. I’m sure that Meyer, like I, realizes that it’s a stretch to compare developing fluency in math with any of the aforementioned fields of study, but since he took a stab at it, so will I.
His first assertion is that math drill sheets offer worse feedback than hammering a nail, shooting a jump shot, or pressing a piano key. I would fully agree with Meyer if the students were practicing problems on a worksheet without using strategies or receiving feedback from their teacher. But - there’s no reason to assume that an instructor who provides math drill sheet practice is sitting at their desk feet propped up, hoping that their students drill and kill their way into becoming mindless computational bots.
Picture a teacher probing their classroom as students practice problems on a math worksheet. They pay extra special attention to a few children who are struggling with their confidence, reminding them of strategies that they’ve been working on in the days leading up to their practice. That same teacher might stop the students every 25-30 seconds to review answers or direct them to check their own work after every three to five problems they complete. When used this way, the feedback is close to immediate, returning (as Meyer puts it) “a bounty of information about their efforts.”
His second argument is “The Math Drill Sheet is Lonely”, stating that while it is “possible to work on a carpentry project by yourself or practice shooting a basketball alone in a gym…those are less common training methods than working with someone on a project, participating in group drills, or at least observing other people practice, and gaining inspiration and knowledge from them.”
Respectfully, I disagree with this entire assertion. I’ve never heard of any successful athlete, musician, or craftsperson arriving at their expertise without the guidance of a mentor and hours of solitary, sometimes tedious practice. That doesn’t mean that they never play or practice collaboratively, but one common thread that binds masters of all disciplines is that achieving their greatness was often a lonely endeavor. University of Memphis Research Mathematician, Ben McCarty - my guest for Episode 5 of Centering the Pendulum - said that extensive periods of working alone is a major part of his work.
Meyer’s third point is that drill sheets conceal the performance of mathematics, and students never see a connection between written practice and their lessons. Again, there is an underlying implication that teachers who use math drill sheets do so without first teaching conceptual understanding of the topic being practiced AND never communicate its purpose to students. To be sure, there are many teachers who fall under the layabout net that Meyer is casting, but we shouldn’t assume that paper and pencil practice always complements thoughtless teaching.
Meyer concludes by saying “We need skill fluency that blends practice and performance.” I completely agree with this statement, but disagree with the examples he cites.
He begins by writing “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. He goes on to cite the practice regimen of Golden State Warrior superstar Stephen Curry, who is widely regarded as the greatest shooter in NBA history.
There are two major flaws with this example. First, Curry would be the first person to admit that his practice drills are inappropriate for young players. If you don’t believe me, watch his master class on shooting and you’ll get an idea of how tedious - not to mention repetitious - learning to shoot a basketball really is.
But comparing basketball and math practice is really not the point. What is relevant to the discussion is that experts practice differently than novices and as a result of their expertise have divergent ways of taking in information and organizing knowledge. Renowned education author, Doug Lemov has written extensively on this topic.
Math learning activities can and should look differently depending on the age and knowledge of the students. Lower elementary math classes, for example, need to be more weighted towards arithmetic than junior high, high school or college lessons. The foundational scaffolds established in those grades serve as a springboard to learn more complex concepts when they’re older. That’s not to say that K-2 students shouldn’t be immersed in inquiry, discussion, and discovery. Quite the opposite. But - different phases of mathematical learning development call for different instructional emphases.
I understand and support Math drill sheets not being Dan Meyer’s preferred method for developing fluency. They aren’t mine either. Still, even though I never used tools such as the one Meyer spent 850 words criticizing, I would argue that a teacher can do far worse than carefully selecting a math drill sheet that matches in class learning, pointing out to students how it connects to past and future lessons, and then having them spend a few minutes intensively practicing the problems on the page.