Saturday, October 6, 2007

Learning linearity from scratch

I'm teaching Algebra I for the first time this year, and having occasion to take a closer look at possible sources of great confusion and misunderstanding in later courses. Teaching Algebra II, I've often resorted to the number line to work with negative numbers and with fractions, always presupposing that the students would have a basic grasp of the very idea of a number line. Now I'm seeing what it looks like when students don't quite understand how a number line works.

In graphing scatter plots, it turned out a number of students - a significant fraction in a talented class - were 1) varying the distance between the numbers, and 2) placing the numbers according to their order in the table rather than sorting them in increasing order. I returned the homework assignments with a new deadline for redoing the plots, directing students to keep the spacing between the numbers on the number line constant. Several conscientious students redid the graphs, dutifully keeping a constant spacing of, say, 4 graph paper squares between the numbers - but the numbers thus placed would still be, say, 41, 37, 58 - the order would not be strictly increasing, and the differences between the numbers were not constant.

So the "simple" task of creating a scatter plot in the beginning of Algebra I actually presupposes some grasp of the Ruler Postulate, which they won't see for a year - the postulate that it is possible to assign numbers to the points on a line in such a way that number differences measure distance. After reiterating again and again that a certain distance on the axes must correspond to a fixed number difference, and copying student graphs onto transparencies and critiquing them in class, and conferring with students one-on-one - the graphs are finally looking pretty good.

I was initially surprised by what I saw as a puzzling deficiency in PreAlgebra skills. But my husband, a theoretical Physics grad student, pointed out that I was expecting students to understand linearity as a prerequisite for this unit on linear relations. The idea of a correspondence between distance and number difference contains what I am supposed to teach them.

After this lesson learned I am wondering about the Algebra II students I taught last year, whether many of them also never really knew how number lines worked either, and I just didn't get it - or whether a bit of Geometry taken in the meanwhile at least takes care of that issue.


jonathan said...

The more of these little details you are aware of, the more you can 'cover' them with throw-away comments or quick exercises. They serve as a reminder for kids who knew them and forgot, and even a 30-second riff on the number line can be informative for those who never formally encountered it.

All of this, of course, presupposes that you know where the gaps are likely to be. That, unfortunately, is gained mostly through experience. (but listening to other teachers and reading this blog-stuff can really help).

H. said...

Reading math teacher blogs really has been one of the most helpful things. I'm wondering, though, whether there isn't some nice handy compilation of standard misconceptions in Algebra that can help hurry up the process? Physics Education has generated, like, libraries of expected areas of confusion for each Physics topic, with quite detailed descriptions of typical misunderstandings. It would seem likely there'd be something of the kind for Math too, but my coursework was in Physics Education rather than in Math Ed, and I just don't know the relevant literature.

Sarah said...

Despite having read this last year, I was still surprised when my Algebra students had difficulty with number lines yesterday.

The frustration with Algebra 2 was that because this was the first time we'd drawn our own axes and the first time we'd scaled the axes for the coordinate plane I'd freehanded my version on the board. It still took 10 minutes for some students to draw and label theirs. I think I need to work on classroom expectations, but it was a good reminder that there's so much I expect students to already know.