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Is there a cognitive psychologist in the house?

The comments to my post on math prerequisites that aren’t being met are the reason that, despite all of the comment spam, I leave these pages world-writable. Some excellent stuff there, and I hope that some curriculum developers in my province are reading what my readers have to say. I have more to say in the prerequisites thread, but first I wanted to address some comments from high school math teachers who assure me that their students aren’t leaving their classrooms without knowing what an equation was, how to add fractions, and so on.

I believe them. To look at the BC high school curriculum, it’s hard to find anything specific that is explicitly wrong with the course. (Aside, that is, from the “This book is brought to you by the letters T and I” thing, which I suppose is a big one.) By the time British Columbian teenagers leave grade twelve, they’ve been exposed to fractions, exponents, quadratic equations, graphs, and logarithms – the prerequisites for college math. They’ve even been exposed to questions that are a lot more open-ended than I expected – questions that demand a modicum of creativity.

The problem is that many students – perhaps most – never properly learned this stuff. Others knew it to some degree at some point, and promptly forgot it.

I’ve been tutoring a student in that second category – he can’t remember how to solve a linear equation (when do you divide both sides by the same number? When do you add stuff to both sides?), he can’t remember how to evaluate powers, he can’t remember…much of anything he learned. And nothing makes me feel like a failure as a teacher than hearing that the mathematics I’ve been trying to teach has come across as something merely to remember.

Can anyone recommend any reading material, accessible to a layperson, on how knowledge is stored? Why, and how, some information gets stored in short-term memory, some finds its way into long-term, and other types of data – a first language, for instance – is not something that people think about as knowing and not as remembering? Because I don’t think it’s something intrinsic about mathematics that it seems to get stored into the most unreliable section of my students’ brains.

As I was thinking about this post, reader Susan made an insightful connection between math (something students memorize) and language (something students learn):

It’s possible for a non-expert to determine whether a child is literate by asking him or her to read something unfamiliar (ideally both outloud and silently) and to explain what they’ve read in their own words.

We need to define and expect something similar as far as being numerate.

To which I add: not expecting anything similar for math gives students no reason to processand learn math instead of just remember it.

I’m only now remembering that I actually did do something similar last term, although I didn’t draw the connection to reading that Susan did. Last term, after one particularly miserably-done test, I gave my students an opportunity to submit (for credit) corrections to the word problems. I gave them a template to follow: they had to describe, in words, what information they were given in the problem, and what they were missing.

They had to describe, in words, how those things were related, and then, based on their previous work they had to provide the equations they needed. A few students actually came up to me and said that that exercise really helped them understand what they were doing with word problems. I was shocked – this was the horrible precalculus class – and encouraged. A month later, however, I found that those same students drew a blank when faced with the word problems on the test.

We’ve got a long way to go.

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