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August 7, 2008: Corinne: Last night I asked Colan to give me the slope-intercept form of a line, y = mx + b, and the equation for the slope of a line, ytwo - yone over xtwo - xone = m, he knew both of these equations. I rewrote for him the equation for the slope to be the point slope form of the equation for a line, y - yone = m(x - xone), and I asked him to make this last equation look like the slope intercept form. What he saw was a pair of linear equations in two unknowns and he immediately jumped to the template for solving for the point in common and so he substituted the slope intercept form into the point slope form and then rapidly got lost. It wasn't until he'd pushed that template as far as it could go that he was willing to listen to me repeat the question and to see that I was actually asking him to do something quite different. It's interesting to me that when I tried to explain this to Emily, she immediately knew what it was that I was going to ask him to do and insisted on pursuing that template before she was willing to listen what it was I had to say about what Colan had done. I think in Colan's case this was an immature version of something which will eventually evolve into a very successful professional sense making strategy, which is that when presented with an unfamiliar set of equations the professional will explore them until they've unpacked the meaning enough to have called up a cluster, until they've unchuncked a rich cluster of understandings about those equations, until they've seen how these equations are related to things they already know. In Emily's case, she tapped in to the correct relationship and so by taking the time out to make sure she had unchuncked her knowledge about that situation in her head all of that chunked knowledge was in working memory and she was then better prepared to understand what it was I was going to say about the situation that might be new to her. In Colan's case, he happened to tap in to the wrong relationship and so it was a useless side venture. So, the connection to the class is that I knew clearly that the game that was being played, the template that was being acted out, was to look at what the transformation did to ALL the vectors all at the same time and furthermore, I knew that since it was a linear transformation that there were only four choices: rotation, reflection, projection, and combinations of the above. The students didn't know that that's what the template was and so they were busy describing in a natural geometric sense what happened to a single vector. So they correctly described something that wasn't relevant to the question at hand. I think this happens all the time at this middle division level where students are getting their first exposure to a bizillian new templates. Most homework problems involve the application of a new template but students don't know what they're being asked to do. Faculty who know clearly which template they intended to evoke are startled by how many wrong directions the students can go. (EVZ: see Alan Schoefeld's paper on mathematics decision making 198? In this paper, he reports on interviews with small groups of students, pairs?, who were asked to solve problems. The students had comparable subject matter knowledge of the relevant mathematics. The difference between those who succeeded in solving the problem in twenty minutes and those who did not, seemed to be in their ability to step back and think about whether what they were doing was relevant. Those who checked frequently (metacognitivity) would stop going down the wrong path and try another one; others kept going without checking and spent the twenty minutes without making much progress. I think this is relevant because it shows the tendency to jump to a template and try to use it rather than scanning available templates and selecting the most appropropriate one)