To Borrow A Page

If you’re very like me you wonder sometimes about subtraction, like, where it comes from and how people have thought about it over time. I’m particularly interested in different ideas of what negative numbers have meant, but my amateur standings in the history of mathematics keep me from easily finding what I want to know. I’m not seeking pity; I know many interesting things as it is.

Pat Ballew’s On This Day In Math Twitter recently posted the above. It links to an article about how subtraction has been represented in history, with a particular focus on ways borrowing has been taught.

I have a particular horror for the mathematics books quoted in it that demand people work out borrowing problems without the use of any extra marks. That is, if working out “5276 – 3739”, no fair writing the first number as “52716” along the way. I can accept that, to someone experienced with arithmetic, the writing out of borrowing steps is unnecessary. And the steps do make for a pretty cluttered page. But it seems to me that especially in the learning stage this sort of false work is essential. Any new skill is hard, and it’s worth making some mess to be sure nothing essential is left out.

Ballew also mentions a fascinating point. The ordinary homeworks and assignments and preparation papers for teachers may have found their way into public libraries. These could be great guides to the ways people actually did calculations, or learned how to do calculations, in past eras. I don’t know how much material there is, or how useful it is. I confess that while I love mathematical history, it is a remote love.

Author: Joseph Nebus

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

6 thoughts on “To Borrow A Page”

  1. I agree with you – any method that helps a student grasp a new, and to them difficult, concept is fine!
    We learnt the “Borrow one pay one back” system for subtraction and I was amazed when I saw kids from other schools who crossed out and rewrote numbers. It looked very messy and confusing to me, but worked fine for them. :)


    1. Well, I suppose I should be cautious about going ahead with any method that helps someone learn. I can imagine cases where a scheme meant to learn one concept leads to problems later on. Maybe the hardest part of learning how to teach is figuring out how to say something that’s true but doesn’t mislead later on.

      But, for example, in middle school I learned this cute little magic-square-like gimmick for factoring quadratics. It was only good for integer-coefficient quadratics and I can’t really say that it offers any advantages on just learning the quadratic equation. But I can imagine someone taking that scheme to heart and then not understanding why it explodes when there need to be irrational or, worse, complex roots, and giving up the whole business of factoring polynomials as hopeless.

      It’s tough setting up falsework that will let people learn a technique and then not get in the way later on. Takes a lot of thinking through.

      Liked by 1 person

      1. As a scientist, I learnt that every thing we were taught was just the best Theory at the time, so it kept changing. Rather annoying.
        I think with teaching maths today there needs to be Fun! And also none of this ‘Girls can’t do Maths’ stuff.


        1. Stuff does change, yeah, and I admit I have a hard time giving up some of the earliest stuff I learned, especially about dinosaurs and moons. I’ve also been shocked to see how taxonomy has changed, mostly the result of genetic research being done. The evaporation of “rodent” as a family, for example, has been amazing.

          Absolutely, though, yes: the idea that “girls can’t do mathematics” must die. It should preferably die at the end of a long, grueling trial in which every aspect of the ides is held up, ridiculed, and shamed so that it will never reappear. I’d also like an end to the idea that most people can’t do mathematics, but girls and women seem to be the most unfairly treated right now so I’m content focusing on that. It’s a small thing but I am trying to use female pronouns for my default in speaking of what individual people doing mathematics might do, for example.

          Fun I’m honestly torn on. The field should be enjoyable, yes, and enjoyed. But sometimes one simply has to work at it, even at stuff that seems dull but proves useful. You can’t be a competent cartoonist without doing a lot of boring drawings of apples in bowls and knees at odd angles; why should people be mathematicians without doing at least some problems that are not all that fun but do teach or reinforce needed lessons?

          Liked by 1 person

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