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Why Do We Teach Algebra Anyway?
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Kevin Drum is talking about algebra.

He's framing the current struggle over gradeschool curricula this way:

The basic outline is pretty simple: reformers, led by the National Council of Teachers of Mathematics (NCTM), wanted to put more emphasis on "discovering" math and real-world problem solving , while traditionalists wanted to keep the emphasis squarely on computation skills and "basics."

And a lot of the debate is swirling around a book by Addison-Wesley informally dubbed "Rainforest Algebra" because it has a great deal of multicultural readings and sociological and environmental tie-ins along with the algebra problems. It was up for approval in Arizona and apparently is up for review in Texas now.

Now I taught high school algebra for 2 years in Texas. I don't remember the names or publishers of the textbooks we used, but they were pretty standard fare. Early chapters dealt with the concepts of variables and coefficients. Then they learned how to perform arithmetic functions they already knew (addition, multiplication) to terms with variables. This scaled up in a logical way to more complicated problems involving similar concepts.

Each chapter generally had a 1-3 page explanation of the concepts with sample problems stepped through. Then there would generally be 30-40 homework problems, with 5-7 word problems at the end. Answers to the odd-numbered problems were in the back of the book.

The idea was for the student to learn the basic concepts and practice them before applying them in a more complicated situational problem. This seems pretty sound to me. I haven't seen the book they're talking about, and I have no problem with an interdisciplinary approach to any subject, but there does need to be an initial approach of learning the basic concepts before they can be applied to a situational problem. There's just no way around it.

Drum presents the following problem:

Peter bought 70 items, and Sue bought 90 items. Each item cost the same and the items cost $800 altogether. How much did Sue pay?

Which he says can be rephrased as:

What is 70 + 90? (A: 160)
What is $800 รท 160? (A: $5)
What is $5 * 90? (A: $450)

Okay, but how is a high school freshman supposed to know how to rephrase the problem? Drum's rephrasing doesn't use the concept of a variable. But that's kind of the key to understanding and reframing the problem, isn't it? If a student understands that a variable can represent an unknown, they're one step closer to reframing this problem in a meaningful way. What is the unknown in the problem above? The price of an item, right? So let that be x. Then the total amount Peter paid would be 70x and the total amount Sue paid would be 90x. Then the problem would be framed as:

70x + 90x = 800

Which would then be easy to solve for x, and then since Sue bought 90 x's, all that's left is to multiply the price times the number she bought.

If you don't teach the students how to use the abstraction of a variable to represent an unknown quantity or rate, then how the hell are they supposed to get from the original problem to Drum's rephrasing? How do they know how to even start?

And in the early part of the course they need to simply work problems like: 10x + 5x = 60 so they know what to do once they've framed the problem in the right way. There simply is no shortcut around learning some of the basic concepts and skills. There are different ways of presenting them, and I don't know if the text in question does a good job or not, but there's no way around teaching the basics of any discipline.

Drum then quotes Richard Neill, a member of the Texas State Board of Education:

My point is this: Addison-Wesley's watered down algebra destroys the true beauty of mathematics. You see, math helps mold children. It teaches them perseverance, attention to detail, critical thinking skills and discipline.

To which Drum says:

Here we get to the core issue: not teaching math but fighting moral decay. Neill doesn't seem to care whether this textbook does or doesn't teach kids how to use algebra. What he cares about is molding children via perseverance and discipline. Apparently, if math is tedious and hard to learn, that's a good thing.

Count me out. If math can't be made fun and interesting, that's too bad. Kids have to learn it anyway. But if it can be made fun and interesting, surely we should welcome that. After all, if Texans want to teach their children perseverance and discipline, there's always football.

Ah, very cute.

Drum conveniently avoids Neill's reference to critical thinking.

Could we all at least agree that one of the reasons we teach algebra in the first place is for critical thinking skills? Most of us don't use the quadratic formula on a daily basis, but the point of teaching higher levels of math is not for direct practical application, but for critical thinking and general problem solving skills.

Drum almost seems to be saying that the focus should be on the end result, that however the kids reach the answer doesn't really matter...maybe I'm putting words in his mouth, but that's what he seems to be saying. But in the example he gave, the process of reframing a word problem is predicated on understanding fundamental concepts. And he just doesn't seem to acknowledge that.

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