Every year, I see a few students who just can't seem to pull it together, and eventually sink faster and faster. But when I try to help them, I find that more often than not, the problems are deeper than I first thought. These students aren't just unprepared, or unwilling to work. Most often, the problem is a combination:
- On a purely mathematical level, their skills are weak, and their conceptual understanding is even weaker. They pull out a calculator to subtract 72 from 180, can't measure an obtuse angle with a protractor, and believe that when the sides of an angle are extended, the angle's measure increases.
- More generally, they have trouble understanding complex tasks. For example, one student, when asked to draw four quadrilaterals, measure their angles, and compute their angle sum, instead freehanded four rectangles. The student didn't realize that shapes without straight lines have angles that are at best poorly-defined; didn't realize that the experiment is essentially pointless with rectangles anyway; and, when he got an angle sum greater than 360 degrees, didn't try to resolve the discrepancy.
- On the level of practical reasoning, they have trouble connecting present behavior to future results, especially when positive results require sustained effort. Because of their poor preparation, on the occasions when they do put in effort, they don't get good grades on tests or quizzes. They quickly learn that these sporadic efforts don't get them the results they want, and then decide that those efforts were not worthwhile. A student gets a 15% on a quiz; after lots of studying, the test score is a 60%, and instead of seeing a 300% gain, the student says "I studied and I still got a D, so why bother?" Who would blame him?
- They don't have the support of families who can help them with the mathematics, or even help them connect their efforts (or lack of efforts) to results in any concrete way. These parents may not come to parent-teacher conferences, or if they do come, profess an inability to actually change what their children are doing. I tend to believe them: as a parent myself, I've come to realize how hard it is to actually make anyone do anything.
Personally, I'm able to help one or two of these kids per year, to the point where they're actually reasonably successful: passing classes, not eternally frustrated. But I have at least half a dozen, and that's at a super-selective public high school. I don't see any way to increase my own capacity, both because success requires so much time and energy per student, and because which students I "connect" with seems, at this point, a matter of luck more than anything else.
Fellow teachers: have I missed anything? Any thoughts on a way to do this better?