There has been a lot said so far about Andrew Hacker’s argument for de-emphasizing algebra, and I’ll have more to say about that later this week. But one thing I’ve noticed anecdotally is that when students say they have difficultly with algebra, that’s usually not the entire story. Typically, that means they have also trouble with *arithmetic*. There’s a reason why the ability to do long division is correlated with long-term mathematics performance: you have to master the basics.

If a student can’t figure out what 2^{3} is, how is he going to understand *x*^{3}? Basic arithmetic skills matter. This problem continues into *college*:

But to step back a bit, this trail of links began with the release of new teaching guidelines by the National Council of Mathematics Teachers:

The report urges teachers to focus on three broad concepts in each grade and on a few key subjects — including the base-10 number system, fractions, decimals, geometry and algebra — that form the core of math education in higher-achieving nations.

I think this is exactly the right approach. It’s more important for students to develop specific competencies, such as fractions, decimals, geometry and algebra, than to develop the fuzzy skills often described in state educational standards–‘critical thinking’ being the worst of these. A story by Abbas describes exactly what I mean:

I sometimes tutor students for graduate admissions tests like the GRE or GMAT, and the first time I meet with them they often show me algebraic word problems they got wrong in a practice test. I ask how their junior high math is, and no one ever admits that they can’t do 7th or 8th grade math. Then I ask them to subtract one number from another for me, using a pen and a piece of paper I hand them: say -2 and 7/8ths minus 1 and 3/17ths. You’d be surprised how many of them are tripped up and make a mistake in a simple subtraction that any 8th grader should be able to do. The problem is they really cannot do ANY algebra until they are consistently and confidently competent in such simple tasks as adding, subtracting, multiplying and dividing numbers, and yes, this includes fractions, decimals, and negative numbers, but even these college graduates generally are not.

The problem isn’t algebra *per se*. It’s more basic than that.

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We had a similar experience tutoring geometry and advanced algebra. All three of the children had trouble with understanding grouping and the order of operations. This would show up as trouble dealing with fractions or radicals or dividing polynomials, but the underlying problem was with grouping. The brightest of the three did the best, but it seemed no one had explained that the bar of a fraction or a radical works like parentheses and groups the enclosed expressions. Maybe it seemed too obvious to mention.

All three students did go through the same elementary and middle schools, but with different teachers, so we aren’t sure how this critical knowledge was missed. Our guess is that, as one student noted, there was so much material to cover each year that by the time she started to feel she understood it, they had moved on to something else. We did find the math courses to be awfully crowded, and spent an awful lot of time on minor topics, such as kite shaped quadrilaterals, quadratic inequalities and asymptotes, while glossing over the proof structure of geometry and things like exponents and logarithms.

I loved teaching geometry and deductive proofs. I thought that our geometry course was the only place where proofs would be taught. When I taught pre-cal and calculus there were many instances where geometric proof thinking habits could be used with algebraic proofs.

I used to show off my multiplying skills and ask “You think this means I’m a smart guy, don’t you…but it’s just the DISTRIBUTIVE LAW!!!!!!” (heavy sigh)

I hated calculators because they hid numbers behind decimals. (I loved programmable graphing calcs however.)

I wanted a student to think of pi/2 when he/she saw 1.57 or square root of two instead of 1.414

I yelled “Hey pizza guy (or gal)…if you divide by two and then divide by three, you really just divided by (two times three).

I sang, I danced (neither of which was very entertaining). I promised to improve their social life by showing them how to prove 1=2

Now I’m old and don’t do it anymore…unless someone asks.

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Fractions are such a typical stumbling block that they should perhaps be first introduced in an algebraic context. One of the problems we face in the United States is that common measurements, like distances, are expressed as proper fractions like ¼” or 3½” instead of integers like 6mm, so we think of fractions as a basic skill, and it doesn’t help when the most manipulable expressions are called “improper”. Too many kids are lost before they even get to algebra.

We had a saying among the TAs when I was in grad. school TAing calc I and II: “Very few students fail calculus; they fail algebra or trigonometry while taking calculus”. Maybe we should add “arithmetic” to that list.

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I learned the japaneese abacus when i was in high school. This device is optimized for decimal arithmetic. I already knew arithmetic quite well. But this technique led to mental arithmetic. So, while it took over a month to learn ten digits of pi, i was able to divide a twenty digit number by another twenty digit number, and get twenty significant digits of result as mental arithmetic. And, my reliability went to 100%. Two features of the technique lead this way. First, carries and borrows in addition and subtraction are performed when they are needed, and don’t have to be remembered. Second, the penalty for guessing wrong for the next digit in division is minimal. We could do quite a bit better at teaching arithmetic.

The japaneese abacus has four beads below, and one bead above the bar to represent a number from zero to nine. But you can use your four fingers and one thumb. So you can count from zero to ninety nine on your hands. So you can get started. I started a write-up on my blog.

http://predelusional.blogspot.com/2006/05/counting-on-your-fingers-arithmetic.html