Friday, March 25, 2016

Cyphering books

In my casual research on mathematics and its teaching I came across something interesting -- I'm just sharing it here in order to keep track of it myself, and to offer it up to anyone else who may be interested.

I found the latest fragment in a book called Rewriting the History of School Mathematics in North America 1607-1861: The Central Role of Cyphering Books, which I will probably never buy because it costs over $100. 


A cyphering book is something like a copy book, only for rules of computation and examples of how each rule works, plus exercises which the student solved himself. Each student wrote out his own cyphers in his own notebook, copied from work the teacher gave him. The cypher book was intended to serve him the rest of his life as a reference manual.

A page from Abraham Lincoln's cyphering book

Back to Rewriting the History . . . . It turns out that something I had been suspecting is true -- which is not surprising, because there's nothing at all revolutionary about it, but it's always fun to find actual proof -- and that is this: The way we teach arithmetic today has more to do with book-keeping than with mathematics.

Remember last summer when I mentioned that the ancient Greeks made a distinction between arithmetic and logistics? Logistics is skill in computation for practical purposes. There is nothing at all wrong with teaching logistics. After all, we want our kids to be able to function in our society, so of course they need to know how to keep a budget, how to double or halve a recipe, how to buy enough paint or carpet or lumber for a project, how to figure out what kind of insurance they need, or whether they can afford a mortgage, and all those things. Many of our kids will need more complicated math for programming computers or analyzing data. So I'm not saying we classical/CM educators shouldn't teach our kids that kind of math.

But I do think it's lopsided for that kind of math to make up the bulk of our curriculum.

The bit of Rewriting History that's available for viewing on Google gives a rough of idea of the development of the modern situation.

Beginning in the 1200s, trade between city-states and republics proliferated to the extent that successful merchants needed to hire skilled "reckoners" to calculate profits, predict risks and control losses, figure weights and measures, deal with simple and compounding interest, keep track of partnerships, and all kinds of complicated things. 

As demand for this skill increased, reckoning schools sprang up around Europe. But get this. Boys of ten or eleven years of age would be sent there for a two-year course which prepared them for work in the actual business. And they didn't have calculators.

Of course, the universities were still concerned with the mathematics as liberal arts, and the book goes on to describe the changing attitudes there, but that's the extent of what I can read online for free.

Maybe I should as for this book for Christmas.



  1. Hello there Kelly! I randomly clicked on a link from Gretchen Joanna's and came over here! So glad I did- this is utterly fascinating!!! I love the thought of having my own cipher!
    SO interesting that our maths method is based on business! Ha, typical!xx

    1. Welcome, Kezzie! Thanks so much for all your comments.

      Yes, it's interesting and annoying that it's that way because it makes people think that that's what math really IS, and not just one way of presenting it.

  2. I hope you write more about this because I definitely want to hear more! Fascinating book!

    1. I'll definitely be writing more on it -- it's kind of my hobby right now. :-D

  3. Write you more!, yes, please. I still don't know very well what you are saying, but I enjoy reading about it.
    Of course, I will ask you in person. So, the conversation will bounce from the Aeneid to Math.


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