Dan’s comment on this exchange got me thinking about procedures (the mathematical kind, like FOIL, not the general classroom kind, like how to turn in homework). The student in that exchange was a senior who had given up on ever learning to multiply polynomials. I said “Oh, no problem” and began to explain, and she literally covered her ears. She said the words don’t help. So I said “Ok. I’ll write on the board. You ask questions.” I did a silent, large-format, multi-color demo of FOIL, one step at a time. At each step she looked intently at the board for a minute and then I saw her make up a hand gesture for each step. I told her to go practice. The next day she came in for assessment and I put one on the board and left the room. When I returned she had done that one and three others that she’d made up on the spot, including one with huge coefficients, just for fun.

There’s a magic moment between hard and boring. FOIL is a *rote procedure* — anathema — that for this student had always been hard. When she constructed her own way of organizing it, it became fun. The success was fun. Knowing what to do was fun.

This year I taught few if any memorized formulas or procedures, and I downplayed the ones students presented. I am firmly in the “if you understand it, you don’t have to memorize it” camp. I think Dan’s comment describes precisely why that student finds math “boring or hard.” But, I experienced this student’s mastery of FOIL as a successful educational moment. Put these together and I am confused.

Maybe I need to examine my aversion to teaching rote procedures. Are they like sugar in the diet? Insubstantial and unhealthy as a main course, but in small amounts, an enjoyable and useful way to increase confidence and willingness for more serious inquiry? The hard-line anti-proceduralist in me says “That was a pyrrhic victory. If she hasn’t built up a solid mental model of what it means to multiply sums, it’s just another arbitrary algorithm.” But another part of me saw in her exuberance–finally learning to FOIL–a move from I Can’t to I Can, a step in the direction of smart and free.

So I am left with questions about rote procedures. Ever? When? Which ones? Why? How often? For whom?

Keep up the great work Dan! I love these questions, they are so rich and important.

I have a lot of thoughts, not answers, just thoughts, that I do not have time to write down. Let me try to bullet them a little:

* I feel you. I have a powerful knee-jerk anti-memorization thing that has always made me want to bulldoze over the subtleties of this question, but the question is vital in both senses (central / alive).

* The important thing is for students to understand what true / flexible / deep / conceptual / integrated understanding looks and feels like, to know what it offers them, and to know when they do and don’t have it. It doesn’t mean they have to have such understanding before they’re allowed to learn a procedure every single time.

* The question is made hard by the fact that for so many kids (and teachers), they don’t even know the difference between rote and real understanding. In this context our task is to teach so that this difference becomes a central part of their experience. This is what militates toward insisting on more real understanding and probing and not being satisfied with procedural competence much of the time. But also, this distinction should be an explicit topic of discussion. I virtually always do a lot of work with my tutoring clients around recognizing that they’re in “how do I do it?” mode and they skipped straight past “what is it?” mode, and how do you expect to be able to do it when you don’t even know what it is? So, we err on the side

* My own thinking about this is being influenced by my last 4 years of experience learning math intensely, and how it’s been different under different circumstances (on my own under no time pressure; in class; studying for the math GRE; working with friends; preparing for the written exams of my approaching doctoral program; etc.; also, different subjects). When do I give myself the luxury of investigating all my questions about something and getting complete satisfaction before moving on? When do I accept a claim without proof? When do I work through a proof only to find that I only remember the result and not any reason for it? Do I then discipline myself to work through a proof again or in a different way? If I do, what controls whether it sticks? What is the difference between understanding something and being able to solve a problem with it? If I can solve the problem but know I don’t understand everything I did, when is this good enough, and when is it important for me to dig through and try to understand everything?

* Procedural proficiency and other kinds of skill (not necessarily based on conceptual understanding)

can supportthe growth of conceptual understanding. (Esp. in the presence of students understanding the difference between the two.) How this happens is thatprocedural competence allows students to inhabit the landscape of mathematical reality more concretely.I recently discovered that there is a body of research literature on this process, going under the headingreification. Reification is the learner’s process of getting intimate enough with a process that they can start to (a) visualize / predict how it will end without carrying it out all the way, and (b) regard it as anobjectrather than just a process. The idea is that if at a later date you and your student got to have a conceptual conversation about the nature of polynomial multiplication, her procedural mastery would be concrete substance for her to hang that conversation on, whereas if you tried to have it before she had obtained the procedural mastery she wouldn’t have a reference point for the conversation. But all this is valueless if the student does not recognize or seek real understanding.* Another perspective I’m bringing to this is the contrast between the way the question is thought about in K-12 and in math grad school. In grad school, you already know I understand what real understanding is, and that I’m seeking it. You don’t have to worry I’ll get the misapprehension that I’m really here to memorize stuff. So perhaps the question of me practicing something till I’ve got it down cold is less fraught. The need for me to practice a procedure till I’ve got it, even if I already understand the rationale, or even

on the way to understanding the rationale, doesn’t feel like it’s in conflict with the need for me to properly understand the nature and value of math. So on a given occasion we can approach the question of how to best proceed in building my understanding without feeling like my misunderstanding the nature of math hangs in the balance.Oops, like always, I went on too long! Thanks for starting this conversation! I guess the takehome lesson is this: it would be good to separate the choice about how to proceed with a given piece of content and given students on a given occasion, from the question of how they perceive math as a whole. (The rich, inventive, coherent, sensible, awesome evolving city we know it is, or an activity akin to trying to memorize the yellowpages.) How will we control their perception of the nature and value of math and understanding as a whole? Once that question is answered satisfactorily (as though that were easy), the pressure is off of individual pieces of content as far as procedure first vs. concept first vs. etc.

As with Ben, I have thoughts but not really answers. Here they are:

I think it’s really interesting that between boring and hard is fun, but it seems like you can only get to fun by going through hard. Math became fun because the student struggled, but now she gets it. If she just understood it from the beginning then it would have been boring. Maybe what that tells us is that it’s worth trying to emphasize the importance of persistence and getting your students to understand that struggling is not only okay, but good.

As for the question you actually asked, I think Ben has a really good point about the students’ need to know the difference between memorization and real/deep understanding. And then they need to actually want to get to a deep understanding. For a student who wants a deep understanding, it can be okay to start with a rote procedure, because after she masters it she’ll still be trying to reach the next level of understanding. The danger is with the students who think that memorization is all there is to learning math, because they’ll stop as soon as they can successfully implement the procedure.

I totally know where you’re coming from for being anti-proceduralist (I feel the same way), but I’ve been thinking about this all day, and I think that maybe for some people the conceptual stuff just doesn’t make sense until they’ve mastered the procedure. For myself as a learner, I know that I like to start conceptual/abstract, and then once I understand it I can work on doing the procedure. But I know other people for whom the conceptual stuff just won’t stick until they’ve seen the concrete procedural stuff. So maybe it can be a good thing to let the students memorize the procedure first, and then help them generalize it to the larger mathematical concept you want them to understand? But that definitely ties back to the earlier point about whether they recognize that there’s a deeper understanding to aim for and are willing to do the work to get there.

Unrelated: I can’t believe you didn’t tell me you have a blog when we were talking about teacher blogs (in January). I found your blog about a month ago because of Riley’s Virtual Conference, and I’ve been meaning to comment since then. I read through all your old posts and really enjoyed them – they reminded me how much I enjoyed being in your classroom and how much I learned from every conversation with you. I’m glad I found this! I’m looking forward to continuing to hear what you’re doing in your classroom.

Thanks Ben and Becca (Becca! Great to hear from you!). Like you, Becca, I also know people who describe themselves as “how before what” people (using Ben’s terms), while others (like me and you) are “what before how” people. I’ve noticed we’re all stringent about it: I catch myself refusing to listen to someone who is trying to show me how to do something if I don’t understand what and why first. Maybe a way to teach the difference between rote and real understanding (again using Ben’s terms) is to let students self-identify as what-then-how or how-then-what. Announce that everyone is responsible for knowing both how and what, but then say “Ok: here are some steps. how-then-what people, copy these down and ask questions as we go. What-then-how people, don’t copy them down, just follow us and see if you can guess the ‘what’ as we go. We’ll talk about your guesses in 5 minutes, clarify the ‘what’, then go back over the ‘how’.”

Dan,

I read this Friday morning and did not have time to respond before taking off for the weekend. Much to say and I might take a few rounds of comments to get organized. First thought is with regard to mnemonics. I have long thought that mnemonic devices are helpful for facts where procedures or ideas are not involved. Spent the weekend at the St lawrence river in NY and my friend who hosted us told me that the river originates in Lake Ontario. I immediately remembered the mnemonic HOMES to remember the five great lakes. The thing is, if I don’t remember them, I cannot create that knowledge in any way other than looking it up. With FOIL (which I always refer to as ‘that four letter F word”, there IS a way to conceptually organize this idea – the distributive property. Given the fact that this mnemonic – this acronym – only applies to binomial on binomial multiplication anyway, I roundly avoid using it. I just fear that it becomes another filter between understanding and applying.

The larger question on rote procedure is one that I have to be less tired to attack.

Right to the heart of several tough questions, as usual! I have no answers about “rote procedures — which ones and when.” But here’s another lens. This wasn’t “a step toward serious inquiry.” It

wasserious inquiry — just not about distributivity. For this student, it was serious inquiry about how she learns. Words don’t work — she tested that hypothesis, discerned a pattern, articulated it. So you worked together on some more trials. What made this work? Was it because it was silent, colour-coded, or large-format? I’m curious to know what your student would say if you asked her. The most interesting part of the story for me was about your student making up hand gestures. Did you suggest this to her? Or did she independently invent a learning technique, then test it? Sounds like inquiry to me — inquiry into pedagogy. Has she done this before? Does it only work in math? Would it help others? If so, who? What other kinds of learning situations could she try it out on? Why does it help? Are there other learning techniques that would work for the same reason? Maybe simple procedures can be a way for students to do research on their own learning (maybe the procedures don’t even have to be about math).@Jim: I hear what you’re saying about mnemonics helping with facts that would otherwise just have to be looked up. I wonder, though, about banishing them from ideas. I started off where you are – FOIL is a shortcut around understanding that students rely on to their detriment. But if I combine your point—the idea here is the distributive property—and what Ben is pointing to around reification, I get something like this: the idea of the distributive property is something like “the product of sums is the sum of all the products”. For a student who doesn’t yet see that, something like FOIL might be a stepping stone to help remember that there aren’t two products in (3x+2)(4x+3), there are actually four of them to deal with. Now, I would rather rely on more general-purpose tools than FOIL. But I’m not quite willing to completely banish learning-by-rote from the process of absorbing a new concept. As you say, though, it often turns into a filter between understanding and applying, and that’s what concerns me.

@Mylene: Yes – an important dynamic there. I didn’t discuss it with her, but my hypothesis is that she already knew what works for her (hand gestures to help memorize things) and what doesn’t (hearing verbal explanations). This one-on-one was the first time she’d had a chance to educate me about what she already understood about how she learns. I like the idea of having students trade strategies about how they learn or remember things.