A Summer 2015 Mathematics A To Z: step


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.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

18 thoughts on “A Summer 2015 Mathematics A To Z: step”

  1. The Common Core idea is that kids will “discover” ways of doing subtraction that are simple to understand, and the one you show is an example. What has happened is that the people who write the curricula have managed to turn this into “this is how we do subtraction”, so it becomes just another method, which can equally well be applied without knowing why it works. Pity really.

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    1. I hope it doesn’t disappoint you too much that I can sympathize with both sides in this dispute. That is, I see the value of having people discover methods to do something. But I also appreciate that sometimes you just need a process you can follow.

      Generally, I’m fond of offering a couple techniques to doing anything. I tend to think several approaches makes it easier to understand what the others are doing. And sometimes people will struggle with one approach while really getting another, and they should be able to do problems the way they do best. Of course that has the drawback that someone who doesn’t know how to pick an approach can freeze up, unsure of what the “right” approach is and not confident in just taking any one they can do.

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    1. There is debate about whether such a “dyscalculia” exists. My understanding is that it’s controversial whether it’s a consistent describable common problem, though. I mean, it’s easy to imagine people who are not able to consistently read symbols in the correct order.

      But mathematics isn’t all about reading symbols. For example, a geometric proof might show that the area of a particular crescent moon-style-shape is the same as the area as this square, constructed from the crescent by straightedge and compass. Grant that someone can’t consistently tell the difference between 2038 and 2308 in an arithmetic problem; how would that even touch the mathematical reasoning involved in that?

      And would arithmetic or geometric reasoning have anything to do with (say) thinking of what the eigenfunctions for a particular differential equations operator might be, or how they might resemble or differ from one another?

      There might be. People are complicated things and it seems plausible that there are kinds of reasoning folks consistently can’t do. But isolating that as a consistent, describable, common problem that isn’t just related to unfamiliarity or inexperience seems hard to do. I suppose this is why they pay the experimental psychologists the big money.

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        1. Do you need to do it any faster? Getting the right answer — or an approximately right answer — is the important thing. Getting it fast is nice but not essential.

          Do you ever try estimating or breaking up problems? For example, 78 minus 33 looks lousy to me because that 33 is unpleasant. 30 would be much nicer. If you took three off the 78 and three off the 33, though, you’d get the same difference. 78 minus three is 75, and 33 minus three is 30. So 78 minus 33 has to be the same number as 75 minus 30. And that’s an easier problem, since you just have to do 7 minus 3 and get 45.

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          1. This is helpful, thanks.
            I suppose I don’t need to be able to answer sums quickly, it’s just that I don’t know anyone else who has such ease with language and words to have such a hard time with numbers, hence the possibility of the existence of some kind of condition………….like how dyslexics CAN spell and read, it just takes them a lot of work.
            :)

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            1. It’s possible that there is a condition that makes calculation harder than it needs to be. But it might also be that you just never happened to find calculating fun enough that you wanted to do it regularly, so you still do it as an inexperienced person.

              Are there things you like calculating, or logic puzzles that you enjoy doing?

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              1. Does Tetris count?
                Other than that, not really, I play Scrabble a lot, which has numbers involved, but is mostly word based.
                I think you’re right about me being inexperienced at it, as soon as we were allowed to use calculators in high school, I pretty much gave up on the pure number stuff.

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                1. There is neat stuff to be said about Tetris, mathematically. The obvious thing is the study of tilings and symmetries: how many different ways can you cover a flat surface using only these blocks, or using some subset of them, or using the blocks in only certain directions? (That L-shaped figure gets a lot more difficult to use if you can’t rotate it to fit.) And there’s neat other puzzles; for example, suppose you dropped a bunch of Tetris pieces at random into the tube. Surely you’d get at least a handful of completed lines by luck; how many? How much empty space would you expect to leave behind? What are the longest stretches of empty spaces you’d expect to leave behind?

                  Granted, this doesn’t help you calculating much of anything. But they’re neat questions to ask and to try answering. (I admit I don’t have much of an answer for any of them, although I can imagine working out estimates by computer simulation.)

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                  1. That’s exactly the sort of thing about mathematics I like thinking about in depth; the theoretical stuff that doesn’t involve numbers and sums and stuff like that to the layperson.
                    One thing I learned from a show I watched a few years ago;
                    if you bounce a chrome ball off of a big ball of quartz, the vibrations that occur, mimic almost exactly, in graph form, the same vibrations that the number Pi has in graph form……….that blew my mind, and to my knowledge, no one knows why that happens. Amazing!
                    Pi is an irrational number with no real symmetry or order, yet it’s replicated in a huge number of natural phenomena; even London taxis tend towards following patterns that mimic Pi (something like that)……it’s just unreal!

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  2. I read this post Wednesday via my phone (if WordPress doesn’t count mobile devices in your stats, I haven’t helped you much of late, I’m afraid!). I wanted to sit down at the computer to comment, and am finally here. The new math program rolled out as my children started school. I didn’t like it very much as it came in, and my husband (who works in geomatic surveys) complained that the new program wouldn’t give a solid foundation for a career in applied mathematics. I was working in an elementary school as the program rolled out. I guess I got to see school mathematics transformation through many perspectives, as parent, educator, and from my husband I saw a professional perspective.

    I did like how the process of thinking about mathematics forced a change in the schools. As a kid, I was a bit of a self-learner, and I often thought about math in a different way than most of my teachers. I noticed this wasn’t always a good thing for me, as it sometimes landed me in a bit of conflict with a weaker teacher if my questions seemed too challenging. The new math offers a greater flexibility for differing processes, I think, and I would probably have loved it as a child. I did notice that the new program also caused considerable stress for teachers without strong math skills. In three schools over eight years I saw two teachers give it up and request a math specialist to take over their math classes, which I think resulted in a better experience for everyone. As a parent, I’m glad that this change has forced a review within our education systems and has required teachers to have stronger skills in their subject areas; teaching is not just about teaching, after all! One has to know the material well enough before sharing that information.

    As a former employee in the education system, one of the greater struggles with the new math was how unwieldy it is for the large classrooms we have (usually 26 to 30 students). Distributing the manipulatives, for example, requires time and space that is simply not available. Storing the bins in a classroom that was at maximum capacity is another problem. We’d often move the kids throughout the school, using any available space we could find. The time we spent simply managing the lessons cut into learning time, and left us struggling to keep up. In those first few years as the process worked itself out, we always worried that we wouldn’t cover all the material in a given year. Most of the teachers adapted by borrowing time from other subjects whenever possible.

    As for the professional perspective, my husband finds that new college grads are not mathematically prepared for his field, but the new staff he receives now were students in the old system. I’m not sure if it could possibly be any worse, as very few of them seem to be mentally engaged in their work processes. Relying on the computer system to step through their work, it isn’t always a given that they’ll notice illogical results in their data. That, I’m afraid, is not necessarily a problem that teachers or curriculums are responsible for solving.

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    1. I want to thank you, so much, for such a long and detailed comment about this. As I say, it’s a level of mathematics teaching I’ve never done myself, but that friends often expect I should know something about, and I’m embarrassed that my perspective isn’t as informed as friends would like.

      I had suspected that a large part of what went wrong with the New Math, and that is frustrating Common Core mathematics, is that there’s a simultaneous drive to teach with as few people as possible. Teaching something new, or in a new fashion, seems to me something that has to be practiced. A performer tries out new material on small audiences several times and several different ways before putting it into the big show; how can we reasonably expect teaching to be different?

      I suspect that with enough time spent teaching smaller classes, New Math or other experimental methods might be developed that could be given out to big, 30-or-more student classes. But that does take time and money and if we could provide enough teachers to put students in (say) ten-person classes, why would we stop?

      I’m not well enough in touch with current college graduates, or teaching staff, to have a sense for how well prepared they are to calculate. Being able to work out exact results is nice but not that essential. Being able to estimate and tell whether an answer could plausibly be right is necessary, though.

      (And I get that wrong myself, sometimes, usually when I don’t stop to think about whether a particular number is credible. One that embarrassed me recently was a line in … I think it was Danny Danziger and John Gillingham’s 1215: Year of the Magna Carta, which gave an estimate of how many stately families in England went extinct each year. The rate was so high that it would imply nearly no families could make it through a decade, much less the century. I didn’t notice until someone called me on the implication.)

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      1. I suspect that the parents who suggest these learning tasks are irrelevant to adult work might not know for themselves what is causing their frustration. I recently read about the European reaction to Arabrian numerals and how some cities banned the new methods as the Fibonacci ‘text books’ began to circulate. Yet, here we are. ;)

        On another note, your humble confession gives me relief; I’ve been reviewing high school math to prepare for starting a new degree in the fall (in math) and sometimes I startle myself with the silly errors I can make in the calculations. I suppose mathematics is enough like writing in that a good proof reading is essential ( and sometimes best done by someone else).

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        1. I hadn’t thought of the similarity between resistance to new teaching methods and the suspicion that Arabic numerals faced when they were introduced. (They were suspected of being ways that merchants and bankers could cheat people, since whereas anyone could follow addition and subtraction with Roman numerals, Arabic numerals inflicted all these bizarre new symbols and rules that nobody could follow.)

          And, yeah, everybody makes silly errors in calculations. Back in grad school one of my fellow TAs was driven to madness by the number of students who on a test kept reducing 1002 to 2. I would like to laugh more at that, but as I recall, that was the same month I was stumped for a week on a differential equations assignment because I kept writing the derivative of “ex” as “x ex”, which is possibly even dumber than turning 100*100 into 10.

          (The derivative of “ex” is “ex” again. Indeed, “e” is pretty much defined so that the derivative of “ex” is “ex” again. All I can say is I eventually caught my mistake.)

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