A Summer 2015 Mathematics A To Z: step
On occasion a friend or relative who’s got schoolkids asks me how horrified I am by some bit of Common Core mathematics. This is a good chance for me to disappoint the friend or relative. Usually I’m just sincerely not horrified. Much of what raises horror is students being asked to estimate and approximate answers. This is instead of calculating the answer directly. But I like estimation and approximation. If I want an exact answer I’ll do better to use a calculator. What I need is assurance the thing I’m calculating can sensibly be the thing I want to know. Nearly all my feats of mental arithmetic amount to making an estimate. If I must I improve it until someone’s impressed.
The other horror-raising examples I get amount to “look at how many steps it takes to do this simple problem!” The ones that cross my desk are usually subtraction problems. Someone’s offended the student is told to work out 107 minus 18 (say) by counting by ones from 18 up to 20, then by tens from 20 up to 100, and then by ones again up to 107. And this when they could just write one number above another and do some borrowing and get 89 right away, no steps needed. Assuring my acquaintance that the other method is really just the way you might count change, and that I do subtraction that way much of the time, doesn’t change minds. (More often I do that to double-check my answer. This raises the question of why I don’t do it that way the first time.) Though it does make the acquaintance conclude I’m some crazy person with no idea how to teach kids.
That’s probably fair. I’ve never taught elementary school students, and haven’t any training for it. I’ve only taught college students. For that my entire training consisted of a single one-credit course my first semester as a Teaching Assistant, plus whatever I happened to pick up while TAing for professors who wanted me to sit in on lecture. From the first I learned there is absolutely no point to saying anything while I face the chalkboard because it will be unheard except by the board, which has already been through this class forty times. From the second I learned to toss hard candies as reward to anyone who would say anything, anything, in class. Both are timeless pedagogical truths.
But the worry about the number of steps it takes to do some arithmetic calculation stays with me. After all, what is a step? How much work is it? How hard is a step?
I don’t think there is a concrete measure of hardness. I’m not sure there could be. If I needed to, I’d work out 107 minus 18 by noticing it’s just about 110 minus 20, so it’s got to be about 90, and a 7 minus 8 has to end in a 9 so the answer must be 89. How many steps was that? I guess there are maybe three thoughts involved there. But I don’t do that, at least not deliberately, when I look at the problem. 89 just appears, and if I stay interested in the question, the reasons why that’s right follow in short order. So how many steps did I take? Three? One?
On the other hand, I know that in elementary school I would have had to work it out by looking at 7 minus 8. And then I’d need to borrow from the tens column. And oh dear there’s a 0 to the left of the 7 so I have to borrow from the hundreds and … That’s the procedure as it was taught back then. Now, I liked that. I understood it. And I was taught with appeals to breaking dollars into dimes and pennies, which worked for my imagination. But it’s obviously a bunch of steps. How many? I’m not sure; probably around ten or so. And, if we’re being honest, borrowing from a zero in the tens column is a deeply weird thing to do. I can understand people freezing up rather than do that.
Similarly, I know that if I needed to differentiate the logarithm of the cosine of x, I would have the answer in a flash. It’d be at most one step. If I were still in high school, in my calculus class, I’d need longer. I’d struggle through the chain rule and some simplifications after that. Call it maybe four or five steps. If I were in elementary school I’d need infinitely many steps. I couldn’t even understand the problem except in the most vague, metaphoric way.
This leads me to my suggestion for what a “step” is, at least for problems you work out by hand. (Numerical computing has a more rigorous definition of a step; that’s when you do one of the numerical processing operations.) A step is “the most work you can do in your head without a significant chance of making a mistake”. I think that’s a definition that clarifies the problem of counting steps. It will be different for different people. It will be different for the same person, depending on how experienced she is. The steps a newcomer has to a subject are smaller than the ones an expert has. And it’s not just that newcomer takes more steps to get to the same conclusion than the expert does. The expert might imagine the problem breaks down into different steps from the ones a newcomer can do. Possibly the most important skill a teacher has is being able to work out what the steps the newcomer can take are. These will not always be what the expert thinks the smaller steps would be.
But what to do with problem-solving approaches that require lots of steps? And here I recommend one of the wisest pieces of advice I’ve ever run across. It’s from the 1954 Printer 1 & C United States Navy Training Course manual, NavPers 10458. I apologize if I’m citing it wrong, but I hope people can follow that to the exact document. I have it because I’m interested in Linotype operation is why. From page 308, the section “Don’t Overlook Instructions” in Chapter 7:
When starting on a new piece of copy, or “take” is it is called, be sure to read all instructions, such as the style and size of type, the measure to be set, whether it is to be leaded, indented, and so on.
Then go slowly. Try to develop even, rhythmic strokes, rather than quick, sporadic motions. Strive for accuracy rather than speed. Speed will come with practice.
As with Linotype operations, so it is with arithmetic. Be certain you are doing what you mean to do, and strive to do it accurately. I don’t know how many steps you need, but you probably won’t get a wrong answer if you take more than the minimum number of steps. If you take fewer steps than you need the results will be wretched. Speed will come with practice.