What with Robert Pondiscio’s welcome and well-written article extolling the benefits of Direct Instruction (Zig Engelmann’s method for instruction) and thereby praising direct instruction in general, there are indications that others may be following suit. I just read a blog piece by a math teacher who has reached the eye-opening conclusion that conceptual understanding doesn’t always have to precede procedural fluency. In fact, procedures may not be the bogeyman that math reformers have been saying they are for the past hundred years or so.
And it isn’t as if math teachers have routinely refused to teach the conceptual understanding. It’s just that if you’ve spent any time at all in a classroom, you will have noticed that your students glom on to the procedures. And unless the conceptual understanding piece was part and parcel of the procedure (as is the case with adding and subtracting with regrouping) few if any remember the underlying concepts. This has led to math texts now “drilling understanding” by making students do the conceptual understanding piece as if it were the algorithm itself; i.e., 3 by 5 rectangles and shading the appropriate parts to represent 2/3 x 4/5 as a means to “understand” what fractional multiplication is.
The belief still persists that in order for students to understand, math must be made relevant. It just can’t be that if students know the procedures and can do them, and use them to solve problems, they really do not care if the problems are relevant or not. And so we have statements like this which appeared in a recent Education Week testimonial/polemic that passes as evidence-based, research-based, relevance-based, brain-based and any other kind of “base” you can think of:
Math lessons, on the other hand, have historically focused less on real-life connections. Like many students, I excelled in math by memorizing rules and tricks. In college, I trained to teach social studies, but became a math teacher by accident because I had earned enough math credits to qualify for a math teaching certification.
Never mind that the author of the article may have benefitted from what she calls “rules and tricks”.
In any event, it appears that there may be more teachers who had insisted we are at war with Eastasia now coming out from the woodwork to say that we’ve always been at war with Eurasia, though it comes out more like: “Hey, procedures aren’t that bad, and most kids don’t really get the understanding til later.”
I will leave you with the words of a math teacher I know from New Zealand who puts it this way:
A few years back I started explicitly telling my students “I don’t care if you understand it, provided you can do it” when they complained that they “didn’t understand”. I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding because they never have. To the fury of many reformers, most kids really don’t want to understand very much.