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.

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Setting up Equations the Old Fashioned Way

Posted by mark schwartz on June 6, 2016


In Colaw and Elwood’s School Arithmetic Advanced Book (1900), they presented an interesting strategy for students to use in setting up equations, at least when first learning how to set up equations. They showed an interesting alternative.

The Story.

I noted something similar in several other texts of that time, but I don’t believe it was the universal approach, like today’s “let the unknown = x”. Colaw and Elwood don’t begin with “let x = ” but make an algebraic statement with words taken from the problem. It may not be as simple with problems that don’t translate as directly, but it is a very interesting approach.

Several example (pg. 172) are:

Problem: If 10 dollars is taken away from 5 times A’s money it would equal 20 dollars plus 3 times A’s money. The equation:

5 times A’s money ─ $10 =  $20 + 3 times A’s money

Problem: of my money is the same as subtracting  of my money from nine dollars. The equation:

The equation:   1/4 of my money = $9 ─  1/5 of my money.

Other examples are presented. They solve these equations without substituting “x’ for the words. Rather, they carry the words throughout the problem, including the dollar sign. In this case, they abbreviated “of my money’ to just “my money” but generally they don’t abbreviate the words, noting that having to write out the words every time helps students see the correctness or incorrectness of their solution. Another effect of this, I believe, is that students have to slow down to write everything out and perhaps gives them more time to reflect not only on their problem but also on their strategy for solving equations in general.

This strategy provides a step not typically available in today’s strategies, which is to set the unknown equal to “x”. Another aspect, I believe, of the Colaw/Elwood idea, is that it makes the process less abstract, which I believe is helpful. I’d also like to note that in other texts at that time, the use of the first letter of the item would be used rather than “x”. For the above problem  “5 times A’s money ─ $10 =  $20 + 3 times A’s money”, it could simply be “5a – 10 = 20 + 3a”.

You might want to give these ideas a try and see if it helps.


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