We accept statements like “If something is understood … it can’t be forgotten” at face value, but I think it is a much more nuanced relationship. For starters, the “something to be understood” is typically actually different than the thing we want to remember.

If we are studying the area of triangles, I want my students to understand that area means counting squares, and that seeing triangles as half of parallelograms and rectangles can help find the area of triangles more precisely. What I want them to remember is b·h÷2. If we are exploring exponents less than 1, the thing I want them to understand is what the notation means and how we can extend, generalize and apply it. What I want them to remember is that anything to the zero power is 1 and negative powers give reciprocals of positive powers.

My experience in the classroom tells me that conceptual learning is necessary but not sufficient for procedural fluency. Helping students understand a concept doesn’t automatically produce that procedure or algorithm that is useful to remember. And my experience with math and music tells me there should be a blend of understanding and procedural fluency. Algorithms are not the unfortunate byproduct of math education. They are a valuable part of it. But to bring them about from and connect them to learning and understanding (and not unintentionally overvalue either learning or fluency), I have to want it to happen and I have to do some intentional stuff to make that happen. What? How do the two interact with and support each other? Or how might they work at odds (an algorithm introduced too early… or is it just so clearly established as the “real” objective, since that gets students the right answer)?

More acutely, for a significant number of students, there seems to be a divide between mathematical insights and algorithms: no real connection ever arises naturally between them in their mind. Unintentionally clumsy attempts to force the issue and create connections for them just results in the confusion that has hindered the attempt to make understanding central to math classrooms – the model made so much sense, we had such a great discussion, how come they still keep asking me to tell them whether or not they should add the denominators? Or we just knowingly or unknowingly proceed along parallel but separate tracks in math class: problem solving and memorizing. It seems to me there is a lot to clarify here to be more effective teachers. What is the nature of that divide? Is it bridgeable? How?

AND when it comes to remembering, again I think we have dangerously conflated two things. Students may remember what they understood for a lot longer than they will remember the procedure. Or at least the memory is constructed and reinforced in different ways. Back to the area of triangles, after we have discussed and justified the formula, the next day fully half the class will go back to drawing rectangles around triangles and cutting them in half. If you ask them the formula for the area of triangle, they have no idea. No judgment here about which is more valuable, just an important fact: remembering one does not lead to remembering the other. We want them to know/remember/be fluent/demonstrate expertise with both. So we need to understand how to promote the memory of concepts we learn (intuitively this seems to be automatic, which may be the source of some of the confusion) and the memory of generalized algorithms and rules that come from those concepts.

I would imagine there is some research out there, I just don’t know where it is. For instance, part of the organizing principles for Illustrative Math discusses using practice to develop procedural fluency. I wonder if they used recent brain and learning research to help design their practice structures, or based it more on just common sense and typical practices. If so, I think we can be more deliberate than that about what makes practice effective with lasting effects, ie. how much? how often? in what ways is it useful to tie it to the conceptual learning and what are the challenges there?