When Did They Reinvent Math?
One of the things I'm often required to do as a parent is to help the kids with homework. Most of the time that takes the form of help with math and/or science.
I've discovered recently that sometime between when I was in the 8th grade and now, they apparently reinvented middle-school math. I'm not sure who "they" are - my suspicion falls on the textbook publishers - but part of the exercise seems to be to make parents useless.
Last evening, my daughter was attempting to factor some simple quadratic polynomials. I learned to do this task many years ago using the quadratic formula, which is burned into my memory the same way that the opening theme from Sesame Street is.
When I asked my daughter if they'd talked about the quadratic formula in class, she looked at me as if I was growing a third eye in the middle of my forehead. Not a good sign.
Further investigation indicated that while they had briefly mentioned the quadratic formula, their "official" method of factoring these equations was through the use of some peculiar graphical method where they filled in squares on a grid with coefficients, sprinkled some fairy dust on the page, and (apparently) divined the answer by magic.1
To me, it seems that the "new" way of grinding through exercises like this is much more complicated than it needs to be.
To make it worse, grades tend to be based more on whether a student cranks through the annointed process correctly, rather than whether the student actually understands what she is doing. Consequently, turning in an assignment with all the correct answers, but using a technique that I learned almost 30 years ago (!) won't get her many points, even if she can fully explain the approach.
I can hardly wait until she hits high-school math next year...
-Jay
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1I think what's going on in the chart is that they're actually calculating the individual terms used within the venerable quadratic forumla - b2, 4ac, and 2a - and then operating on them as prescribed by the forumla, but apart from taking up more space on paper, I'm not really sure what that accomplishes.
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January 14th, 2010 - 11:00
I run into that kind of problem helping with math homework a lot too. I am sure the educators who create the methods and texts believe they are providing a better way of learning, but your post makes me wonder when using an obsolete and inferior (for the sake of argument) method is better if it can be better taught. Like the way a lot of older technologies remain standards even when newer and better technologies are available. There has to be a break even point somewhere there.
January 14th, 2010 - 11:14
I’m curious about the actual process being used. I’ve been tutoring my 11th-grade nephew, and they did some strange things on factoring polynomials – there was one that was visual, with a big square – they called it “completing the square”. It didn’t make sense at first, but when I trudged through it, I saw the utility.
Also, along the same lines, my 3rd grader was learning some strange ways to multiply, but when I looked it over, it, too, made more sense than what I had learned. I think it’s easy to knee-jerk that it’s just more complicated, but sometimes the more complicated process leads to greater understanding of the concept — so many kids go through school just following the steps and not understanding a bit of what it means.
And speaking of interesting ways to factor polynomials, check out Bluma’s Method for polynomials where the first coefficient (on the square) is not 1: http://www.dupontmanual.com/manualteacher/rash/misc/Bluma-Method.pdf
January 14th, 2010 - 11:39
I think “completing the square” is one of the strategies in her book. I’ve asked her to bring the book home this evening so I can sort through what’s going on.
Personally, I’d rather the students actually understand what’s going on, and if that means a different technique than what I learned, fine. However, I’m not too crazy about the mindset that “the way it’s done in the book is the ONLY way to do it.”
January 14th, 2010 - 11:43
I’ll agree with that. I got frustrated with part of my nephew’s homework when he HAD to use Method X (he was learning several), and I KNEW that Method Y (which he had also learned) made solving it a breeze.
In fairness, later in the homework was “solve using any method”, and I get the point of practice them all so you know them all, but at least make them problems where it makes SENSE to use Method X…!!
January 14th, 2010 - 16:09
It sounds like they are doing some form of completing the square as others have noted. The idea of completing the square is that it is easy to solve equations of the form m(x+k)^2 + n =0. The point of completing the square is to get them in that form. The method that your daughter is using is likely the geometric way of completing the square (which was actually known to the ancient Greeks). In fact, to actually derive and prove the quadratic formula it really is easiest to complete the square in the general case.
January 14th, 2010 - 17:42
Subsequent interrogation of the offspring confirms that it was indeed the completing the square approach.
That raises the disturbing question of why I don’t recall specifically learning that approach in school.
Perhaps I’m getting old…