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January 8, 2008

Understanding Middle School Math

Nick Diaz

The job of the typical middle and high school mathematics teacher is a challenging one, as evidenced by the kinds of questions and statements made by students, parents, and school administrators; these are the people who comprise our “constituency”, if I may.

For most of my 30 years with Frederick County Public Schools (FCPS), I was a mathematics teacher at the middle-school level. Most of those years were invested in the teaching of sixth-grade students, fresh from six years at the elementary school level. Sixth grade is a frightening time for so many students.

– Where will my locker be?

– Can I memorize my locker combination?

. – Will I get to class on time?

– Will I make friends? Will everyone hate me?

– What if the 8th graders pick on me?

Adjustment to a large public middle school for most kids is a chore, a career goal in and of itself.

We mean-old-sixth-grade math teachers make it doubly difficult in that adjustment, however, due to the nature of mathematics as a discipline; also, due to the background of middle and high school teachers, in contrast to that of elementary school teachers of mathematics.

Math, as a middle-school course, regardless of “level,” is much more difficult and demanding than elementary-school math courses. There are many reasons for this increased difficulty:

1.) Class time allowance. Typically, a math period at the middle school ranges from 50 to 90 minutes, depending on many scheduling factors. A 50-minute math period one day could be followed by a 90-minute math class the next.

2.) Amount of material covered in course. Middle school math teachers reasonably expect students to have mastered (not just having been exposed to) most of the more routine computational skills.

3.) Sixth-grade math is typically the beginning of the study of mathematics as a discipline, as opposed to the learning of arithmetic.

What is arithmetic? It’s the mastery of algorithms – a definite list of well-defined instructions for completing a task. Given an initial state, this task will proceed through a defined series of successive states, eventually terminating in an end state. Mastery of arithmetic and its common algorithms is what middle school math teachers expect students to demonstrate, right from the beginning. On the other hand, what is mathematics? Nineteenth-century math guru Benjamin Peirce referred to math (or “maths”) as “the science that draws necessary conclusions.” To me, math is the tool mankind uses to improve its understanding of the world and to make sense out of its perceived randomness. It is in sixth grade that students are introduced to mathematics, and taken beyond arithmetic.

So, why is it that middle-school math teachers consistently find themselves fending off questions and protests from students and parents, particularly the latter, such as:

– Why does my child have to learn this vocabulary, (or these axioms, or these laws, or these rules, or these abstractions)? He’ll never be a mathematician!

– Can’t you get through life without fractions? (It’s a commonly known fact that 5 out of 3 people do not understand fractions…)

– I’ve always been bad at math. How do you expect him/her to understand this stuff? (A common complaint by mothers, particularly about their daughters).

– Why does he have to learn this stuff when he can just use a calculator?

And the best of all, the most common ailment:

– My child always got A’s and B’s through elementary school, so why is he/she getting C’s and D’s (or worse…) now?

The implication of this last statement is that, of course, the child would be getting A’s and B’s if only you, the teacher, weren’t such a mean and/or bad teacher. If Johnny gets a “C” in 6th-grade math, he won’t get into Harvard or MIT… Laughable, but I’m not exaggerating that much, people!

So, why is it that so many students, who previously had performed reasonably well in elementary school math, now find math difficult to fathom, understand, and apply at a satisfactory level?

It could be because math, as a discipline, requires a student to perform two activities that cause great pain and suffering: reading and thinking!

I’ve been guilty for almost four decades of imparting such pain and suffering on poor, downtrodden little children by expecting them to read and think, as well as to listen, apply, conclude, deduce and induce, evaluate, elucidate (orally and in writing), interpolate, and extrapolate.

In other words, being able to multiply a four-digit number by a two-digit number is not enough to secure a good grade in middle-school math, although it certainly does not hurt. I don’t care if a student can turn a fraction like 3/40 into a decimal by blindly dividing, as much as I care that he understands that:

– 3/40 is half of 3/20, which means that 0.075 is half of 0.15 (or 0.150).

– 3/40 is one-tenth of ¾, so that 0.75 (75 cents) is turned into 0.075.

The student who thinks he’s so smart that he can divide 3 by 40 quickly and easily and get the answer will and should be met by the “lazy” kid who doesn’t want to work harder than necessary. Thinking mathematically is not computing, is not performing arithmetic, and is not performing empty, meaningless algorithms.

I want thinkers, not mechanics, in my math class. Mathematicians are basically lazy people; my best, most proficient and efficient math students have always been a bunch of lazy bums, like me. I’m the head bum. I take delight in being surpassed in laziness, the good laziness, by many of them, more often than I would care to admit.

By “laziness” I mean “efficiency,” which I define as “the ability and willingness to perform the greatest amount of work, in the least possible amount of time, and of the highest quality, and the lowest degree of effort.”

The next installment in this series should pose more questions and even fewer answers about secondary math education. In the meantime, do keep in mind that “pi” are not square – they’re round…

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