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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Exploring an 1864 Demonstration that ( – )( – ) = +

Posted by mark schwartz on May 1, 2016


When introducing (or reviewing) signed numbers to a remedial/developmental community college class, I try to identify how many in the class already have a mantra for some of the operations with signed numbers. My favorite way to check is to ask them to complete the sentence, out loud, that I’m going to say.  I make sure they’re ready and then say “a minus and a minus is a _____”. For those who voice the answer of “plus”, I pause and then ask them when. Here, I get muddled responses although that was the right answer. I also get a clue about the class because usually there are about only 5 to 10 who answer correctly. Students then take to the board to teach each other how this mantra works and this provides some more insight into how strongly this and other memorized shards of math interfere with learning and applying some core basic math resulting in blurry spots in their math experience.


In the instance of addressing multiplying two negatives to obtain a positive, many math instructors have created a variety of approaches – physical, tactile, and melodic. Of the non-physical, non-tactile, and non-melodic approaches I’ve used, Mr. Greenleaf in his 1864 New Higher Algebra (pg.25) provides a simple and direct demonstration that seems to make sense to students.

Here are his statements:

“When the quantity to be subtracted is partly or wholly, negative. Let it be required to take b  ̶  c from a.”

“Operation.  a  ̶  (b  ̶   c) = a   ̶   b + c.”

“If we take b from a, we have a   ̶   b. But, in doing this, we take away c too much, consequently the true difference will be a   ̶   b increased by c, or a   ̶   b + c.”

“Hence, the ALGEBRAIC DIFFERENCE between two quantities may be numerically greater than either quantity.”

I took license with Mr. Greenleaf’s algebraic demonstration and put it in a numerical demonstration. I didn’t go through his algebraic presentation with the class.

First, I presented to the class the statement 8  ̶   (5  ̶  3) and asked them to carefully and in slow-motion do the order of operations to get the answer. Almost everyone knew to do the work in the parenthesis first, getting 8  ̶   2, and subtracting gives 6. After everyone was satisfied that this was correct, I left this on the board and I wrote the original problem on the board again.

Now, as I started to step through this differently, the class pointed out to me that I was doing it incorrectly; that I was cheating, that I had to do the work inside the parenthesis first, etc.  Technically, they were correct but I proposed that I would do it a piece at a time in slow motion. After much interesting discussion, they granted me permission to continue. Based on Mr. Greenleaf’s method, I first did 8  ̶  5,  noting to the class that this would be the first operation to do. They accepted this, so the result at this point is 8  ̶  5 = 3.  Are we done, I asked? Several sharp-eyed students pointed out that if the problem were done the “correct” way, the answer would be 6, not 3. Another student then pointed out that we hadn’t finished the problem! We hadn’t yet done the 8  ̶   (  ̶  3) part.

So, it was determined that when we subtracted 5 from 8, we were subtracting too much (because the real problem was to subtract less because 5  ̶  3 is less than 5), we had to account for the 3. So, how do we do this?

Again, it caught the attention of several students that the difference between the real answer of 6 and the answer here was 3, so can’t we just add the 3? A lot of back and forth about this until there seemed to a consensus that it worked.

So, after doing 8  ̶  5 and getting 3 – which was too low since we took too much away – we had to add 3 back and this +3 came from  8  ̶   (  ̶ 3).  So, can we conclude that 8  ̶   (   ̶ 3) = 8 + 3?

Again, more fun discussion and finally acceptance! A lot of the students expressed thanks for showing them why a minus and a minus is a plus and I was careful to remind them that they need to modify their mantra to say a minus times a minus is a plus. The class was quite engaged in seeing the “rule” emerge rather than simply being told it’s true. They owned it!


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