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 markdotmath@gmail.com. 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.

Staring

Posted by mark schwartz on May 31, 2016

Introduction.

This is a short piece but one which I believe is worth reading. One thing I heard repeatedly from students in remedial/developmental courses was that doing math fast was important. They recall speeded exercises and tests and the contests to be first with the answer. This feature of the math classroom became part of the horror for students who weren’t fast. For some reason, the doing-math-fast-all-the-time myth becomes important and over-rides the essence of math. I call it a myth because although the answer to the problem posed will be right or wrong, how one gets to right or wrong is, in my mind, more important than speed … and there are as many ways to get to “right” as there are to get to “wrong”; well maybe not as many ways to get to “right” but allowing students to free themselves from the time boogie man can and does result in some very interesting outcomes.

The Story.

So, what does staring have to do with this? Regardless of the topic we were about to do, when it came time for students to do the work, I would tell them lay down their pencils and pens and then just stare at the problem for a while. Some students considered this silly, but for those who had the “hurry up” in their heads, this seemed to reframe the work for them. I prompted them to do this for every problem, not just for every exercise. I roamed around the room prompting them to do this … it wasn’t an easy thing for them to learn.

Here’s the point. Slowing people down to stare at the circumstance, be it a math problem or something else, provides time for reflection. “What’s really happening here? What are the relationships? What’s important? I wonder if I can …?” and a host of other “what” and “why” questions arise. This may cause discomfort at first, particularly if it’s never been experienced before, but it becomes a pattern of response enabling students to see that math is more than getting the answer quickly. Remarkably, many students started seeing alternative paths to the solution. They got invested in playing with possible solutions rather than searching for the correct, traditional path to solution. When one student did the following, I stopped and recorded it; it was a great example of imagination. The problem was  5 ( ̶ 2x + 9) ÷ 6 + 3 = 1/2

Before seeing his solution take a moment, if you like, and consider how to solve this the traditional way, starting by multiplying every term in the equation by the LCD 6. The order of operation rules then would direct you to do the indicated multiplication, “getting rid” of the parentheses. The rest is adding, subtracting and finally dividing to get the answer, x = 6. Now look at his step by step imaginative approach. The annotation in parentheses is his telling me what he did and why.

5 ( ̶ 2x + 9) ÷ 6 + 3 = 1/2

5 ( ̶ 2x + 9) ÷ 6  =  ̶  5/2         (subtracted 3 from both sides … wow, seriously violated order of operations)

10( ̶  2x + 9) =  ̶  30               (cross-multiplied, just as in solving proportions)

̶  2x + 9  =   ̶  3                   (divided both sides by 10 … saw it was possible and did it)

̶  2x =  ̶  12                               (subtracted 9 from both sides … standard stuff)

x = 6                                     (divided both sides by    ̶ 2 … yeh)

This student had mastered a lot of Algebra basics and used them cleverly. I asked him how he came to his solution and he said: “because you give us time to play with stuff”. One example does not prove the case for my proposition of staring, since this is only one example, but it is an interesting example of setting classroom conditions for the blossoming of imagination. Even though the traditional method may be more “efficient”, staring causes students to slow down, to reflect on and consider what’s being presented, and to think about possible alternatives to the traditional path to solution. Further, this student was very motivated to get on to the next problem, which seemed to be an unexpected outcome for many in the class. And by the way, there were those who chose to stick with the traditional. Yes, teach the traditional but allow for alternatives. Staring lets them do this and it only takes about a minute.

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