Practice makes perfect, right? If we read more, we become more fluent and can seamlessly comprehend higher level texts. If our runs each week increase in mileage, eventually we can meet our goal of running a half-marathon. If we practice piano each day, we can read music a bit faster and make the music a bit more enjoyable. If we practice twenty math problems in class and thirty for homework, we will master the content.
Wrong. Usually. Raise your hand if you felt you were a good math student growing up? I know I did. I loved math homework and felt such joy and pride when I got an answer correct the first time or an A on a test. I thrived off grades and the attention which followed because of them. When I was in 11th grade, I had my first year of Physics and therefore the math which accompanied the subject. It was hard. I felt stupid and as though everyone around me who was succeeding was clearly smarter than me. I slept through most physics’ classes. On assessments, I wrote pages of work in hopes the teacher would be so confused I would at least get some credit. I just assumed that I had hit my wall in math; and that was okay with me. I was planning to go to school for Psychology and didn’t need high-level math courses.
What I now know as a teacher is that I wasn’t “good” or “bad” at math. In fact, in the past few years I have relearned content from middle school math and finally understand the reasoning behind all the steps necessary to solve. You see, what I was good at was practicing and memorizing. After performing the same procedure sixty-eight times, I was able to rattle off the process for solving. Unfortunately, I did not truly understand the procedures I was following, and it eventually caught up with me.
As math educators, it is critical to understand practice versus understanding. At first, this was a tough concept for me as a teacher. I assumed students would only understand how to divide multi-digit numbers if they memorized the steps (Does McDonalds Sell Cheeseburgers ring any bells?) and then practiced over and over until they could do a division problem in their sleep. What I found though was students would do well on the summative assessment and then three months later forget how to divide. Sounds like a them problem, right? Maybe they need more practice? And then the cycle repeats itself.
But it wasn’t the students, it was an issue with my instruction. I wasn’t teaching the why behind the steps to dividing, only the procedure. And the countless practice problems I assigned led to memorization of procedures, which led me to developing a false sense of believing students understood the concept. Instead, what students need is less repetitive meaningless practice. Students need more conversations, more real-world connections, and more conceptual knowledge. Imagine if we required students to understand each step and describe it in detail. Not just what to do, but why. And imagine if they could then explain it to their peers and listen to their peers’ explanations and make sense of their thinking. This type of instruction takes time but is extremely valuable to student success in mathematics.
I cringe when I see students with math homework consisting of 20+ problems. I cringe even more when I see the problems are all procedural and require no real understanding in order to solve. We need students to make sense of mathematics in order to advance to higher level mathematics. We need students to take in-depth looks at procedures and understand the basis for them. Or otherwise, they will hit their wall too.
