Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Hannah Solves a Problem

Posted by mark schwartz on April 1, 2016


This posting shows just how imaginative students can be. We had been studying algebra in a Contemporary Math class and the text we used had the simplest of algebra problems. This course is a survey of a range of topics; not an in-depth study of any topic. I decided that it would be interesting to see if the algebra learned in this class (as well as some students in class having had algebra previously) would work as well with older problems, meaning a problem taken from one of my 1800s collection of math texts. Could the students visualize how to address this problem because it’s a series of actions and yikes!! … it has fractions in it. I said to use the strategy of their choice and if they wanted to work alone rather than in their usual group that was okay.

The Story:

A problem presented in class from Introduction to Algebra (W.C. Colburn, 1839, p. 40) was:

A person in play lost 1/4 of his money, and then won 3 shillings; after which he lost 1/3 of what he then had; and this done, found that he had but 12 shillings remaining; what had he at first?

The expectation was that students would apply a problem-solving strategy we had studied, namely, identify the unknown, identify the relationships and write and solve an equation. Algebraically, the problem sets up as a fussy equation:

x – x/4 + 3 – (x – x/4 + 3)/3 = 12


Hannah, however, presented the following:

6   6   6                 18 − 3             5     5   5   5                 20

When asked how she came to this answer and the meaning of the string of numbers, her explanation was (somewhat paraphrased):

I worked backwards. I started with the “remaining” 12, and 12 was 2/3 of something since 1/3 was taken away. If it was 2/3 then 2 equal parts of the 3 equal parts of 12 is 6 each, so the three 6’s gives 18. He gained 3 shillings, so I had to take three shillings away, so I got 18 ̶ 3 or 15. And then I did the same thing with 15 that I did with 12. Since 15 was 3/4 of something since 1/4 was taken away, 15 is 3 equal parts of the 4 things, or 3 equal parts of 5 each. Since there are 4 equal parts to start, I added one more 5, and the answer is 20.

Hannah, by the way, was very proficient in solving equations and very good at applying traditional problem solving strategies. She just chose to play with this problem this way. If I were a rigid traditionalist and unaware of her math skills, I would have to have concluded that Hannah can’t do Algebra. Rather, she reinforced the idea that there are alternative paths to solving problems and allowing students some freedom provides some novel and interesting outcomes.


One Response to “Hannah Solves a Problem”

  1. sarablack said

    Reblogged this on Sarablack's Blog and commented:
    Think you don’t love math? You do…

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