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Arithmetic versus Problem Solving
June 12, 2004Last time, we wrote about the false dichotomies which Progressive education reformers use in their fight against "old fashioned" methods of teaching basic skills.
We'll fisk a bit of exposition from Tim Stahmer's Assorted Stuff:
There’s a trend among some critics of public instruction to try and push creativity out of the teaching of reading and math.He's misreading us critics. We don't want there to be no creativity, we'd rather not have creativity displace actual facts and skills.
Especially in math, they want students to drill on the mechanics, learning the step by step algorithms . . .Whoa. He doesn't want students to know the mechanics of math? No algorithms?
. . . so that students can recall them for the standardized problems on the next "big test".Ah, he said it. Those evil standardized tests.
What he doesn't consider is the simple fact that kids might have to know these algorithms in order to go deeper with the study of math in future years.
As a result, most students are taught the "one right answer and one way to get there" approach to math . . .He says it like it's a bad thing. In a "hard science" field like Math (unlike soft subjects like Social Studies or English) there are, in fact, absolutes.
. . . and miss out on the creative aspects of problem solving inherent in the subject.He's all too willing to sacrifice children on the altar of creativity. One of the joyful aspects of studying Mathematics is that there may be any of a number of approaches to a given problem, but that there may be only one right answer!
There is a great deal of creativity inherent in the mental approaches to really good math problems, even if it turns out that there's only one right answer. But what makes his criticism of "the one right answer and one way to get there approach" so pernicious is that it contains a grain of truth.
True, in the vast majority of primary-level math problems, there is one right answer, and true, students are taught "one way to get there." (For example, using the standard long division algorithm we examined two months ago, even though there are several good algorithms for doing it.)
But it is absolutely false that students are taught that there is only one way to get there.
In any field of specialty on the planet, if an experienced elder wants to teach a novice a new skill, something the newbie's never done before, the best way to do it is to teach one tried-and-true method, a step-by-step routine that's guaranteed to get the person to the desired end.
It could be the proper way to hoe a row of soil for planting seeds, the right way to hold a football when running through blockers, applying mortar to a brick when building a chimney, or converting an improper fraction to a mixed number.
But the worst way to do it is to try to fill the novice's head with the multiple ways of doing it--methods in which the elder is well versed. If the elder does her job correctly, the "newbie" will gain knowledge and skills, and in time will have the foundation for exploring other avenues.
The foundation comes before the skyscraper.
I don't know if this is corroboration of your idea or not, but when I was a kid, my mom taught me the "tried and true, old-fashioned" method of doing long division - essentially the algorithm you had in the earlier post.
Well, when I got to fourth grade math, they had instituted one of the forms of New Math (this would have been 'round about 1979), and the direction was to do long division by somehow breaking out each number into its hundreds, tens, and ones "places" first. There was also something that involved writing something off to one side and circling it (I don't remember the method at all, just my hatred of it).
Two problems came up almost immediately. First, the New Math method confused me, and I could not remember all the steps, as the traditional method of long division had settled in my mind. And secondly (more importantly, I think) I rebelled against the New Math method, because I saw it as requiring much more work than the method my mother had taught me (which I understood, and I could explain why and how it worked to the teacher). But the teacher, poor hidebound thing, felt required to get me up to snuff on the New Math method, so I spent more than one recess in the classroom, fighting with that danged long division.
The funny (or not) part of the story is the next year (5th grade) the math teacher was a much older woman (close to retirement) and she told us she would "be darned" if she was teaching the New Math method...and proceedeed with the old tried-and-true method. (I loved that teacher, and not just for that reason. She was also the history teacher and she was the only person I've ever met who remembered- and told us - that she heard the radio announcement of Pearl Harbor happening. She had a way of making history come alive, and not just the history she had lived through).
Ricki, thanks for your comment.
There are a bunch of ways to do long division, including that one which involves writing things off to the side and circling. Maybe it's just as good. But as E.D. Hirsch argues in Cultural Literacy, wouldn't it be easier if we all had common ground in the body of knowledge and skills upon which we build each year of school?
I say, stick with the standard long division algorithm (and bravo to you for being able to explain how it worked when you were in fourth grade--I never really considered the mechanics until I was an adult).
chett June 15, 2004 02:06 AM