Keegan Finlayson, struck me. He wrote, in part,
" As a math teacher, I have seen the memorization of steps first hand. Let's take a low-hanging fruit as an obvious example. For years, distribution of binomials has been taught using the acronym "FOIL" (First, Outer, Inner, Last). This is a set of memorized steps to finding the product. The problem is, this does nothing for the student's understanding of why distribution works in this context. It most certainly creates a student that on the outside is capable of multiplying binomials, and later allows for the students to "reverse FOIL" to factor, but once this method has been taught and tested, the memorization fades until years later, there is nothing left. Conceptual understanding is far superior to memorization of steps because it lasts longer, is more generalizable, and creates stronger connections among mathematical ideas." (my underline)
The last sentence is the philosophy that I started teaching with and tried very hard to stick to: conceptual understanding, by itself, was enough. Memorization somehow diminished conceptual understanding, would actually harm it, and if you needed to have the students memorize something, then you hadn't taught it correctly. I actually found myself discouraging students from memorizing formulas or procedures.
But after 20+ years of teaching math, I find that conceptual understanding doesn't always "last longer." As I mentioned in my previous post, there are lots of topics that my students explore during the year that they discuss and understand but have forgotten 6 months later: formulas for geometry, 5^0, etc. I have also had a significant number of students who do not internalize or generalize ideas for themselves as easily as others. For those students, development of concepts and memorization can proceed in parallel, and need not take away from each other. A combination of good thought-provoking questions (designed to uncover some of the misconceptions pure memorization might produce) and some well-designed practice can be more effective to build understanding than either alone. We need to be more discerning about when memorization can hurt and when it can help: I agree that memorizing FOIL or "do the same thing to both sides of an equation" before students understand the concepts behind them will likely short-circuit some important thinking. But memorizing formulas or facts after we have been immersed in them is an important by-product of the thinking.
So what does well-designed practice look like? As both Keegan and Jo Boaler pointed out, memory fades. Can we do something about that? I recently was introduced to a Quizlet-like website called Cerego which is designed to help you remember better and longer. You or your students can create sets of facts for drill, but Cerego takes it to another level by tracking your success and scheduling review sessions for you to optimize the rate of retention. They have a great video that explains their theory. I can't swear that the science is right, but it fits my experiences as a teacher and as a late-bloomer music student. I have made a couple of sets for my students, but mostly I have been inspired by a couple of take-away ideas to redesign the practice we do in class:
1) Practice doesn't have to be a lot or long. I can give students practice in quick little shots, literally 2 or 3 minutes.
2) Spaced rehearsal - very short intervals at first, and longer intervals later on, for a long time. That means the first time I review that formula for the area of a trapezoid or ask them about 6 to the power of -2 will be a day after they learned it, then a day after that, not two or three weeks after the unit test. But I will review it after the test, and then another week or so after that, and so on.
I think this has the power to transform my teaching. Clearly there are lots of ways to review material - pure drill, math talks, patterns, quizzes, word problems, find the one that's different in a multiple choice - you just use it in some form or other. By putting some thought into how I pace the sessions, I can invest just a few minutes a day to help my students keep those tools at hand.