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.