January 30, 2002

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

On 30 January 2002, after my CT reviewed graphing linear equations in a warm-up exercise, I attempted to introduce linear inequalities. Students had previously been working on graphing lines by either using a t-table of x and y values, or extracting the y-intercept and slope in order to find two points. I introduced the term and then reviewed inequalities, e.g. 5 apples > 3 apples, and 3 apples < 5 apples, with the "alligator eats the bigger number" mnemonic and a botched attempt to put one student's left-right mnemonic on the board (as usual under stress, I confused left and right).

A question came up about "less (greater) than or equals," and I tried to explain that using the boolean OR. I blame my computer science days; it took me a few confused looks from students to realize I was confusing the issue. My supervisor suggested afterward that trying test cases for n <= 9 would have been more helpful in clearing up what was going on. It's a little frightening that I can't remember anything about how I learned inequalities other than the alligator mnemonic and shading in linear inequality graphs (how did I figure out what to shade?), which would be useful as a starting-point for what helps and doesn't help students.

I showed the class an example by starting with a line and trying to guide the class toward the idea of being "less than" or "greater than" a line. This concept wasn't at all intuitive for the students--frankly, it was mysterious to me when I learned linear inequalities, and I have only faint memories of coloring in graphs but none at all of how the topic was taught to me. In any case, I started out by testing points to see if they made the statement "true" or "false." Though it seemed like a good idea at the time, in retrospect it seems clear that students weren't sure where I was trying to take them (though many politely played along). I asked them to make a hypothesis after only two points--another nice idea that didn't work in practice because many students are not comfortable taking risks in this class, something that is fostered by the results-oriented curriculum.

This leads me to my question: how do you know in advance what will confuse students? I think when teachers assume something is obvious to the class, it's because that something is obvious to them at the time they teach. The paradox, then, is in being able to unravel those things that are obvious to you so you can see where they are not obvious to others. The trouble is that most assumptions evaporate when they are challenged or made explicit; and it's precisely the subterranean structures that shore up our paradigms without our awareness that are the most dangerous.

I've heard mention of research into common student misconceptions. Where is this research and why isn't it in those bloody teacher manuals? If there's not much of it, I begin to worry about reliability and research biases. If there's a lot of it, I won't know where to begin looking. Articles detailing research are also sometimes pretty heavy going for someone who isn't trained in statistics, let alone sociometry and psychological measures (like me). While it's possible that a "list" of common misconceptions for given topics might tend to pigeonhole a teacher's thinking and responsiveness to a class, this isn't a fault of the data but of the teaching, and such a list would be extremely helpful in planning activities or anticipating questions.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 21 May 2002