My students and their PFCs

Well, I'm sitting here getting ready for another week out on the island of misfit boys. Our population recently went up to the max and the last week out had me pretty exhausted at the end of it. It is amazing what a difference even one kid can make to the state of things on the island, having three new kids is pretty remarkable. Curiously compared to other times we've had this much change I haven't really seen much in the way of chimpanzee fighting for the top of the heirarchy. This is rather unusual and perhaps says something about the type of kids we have these days.

I'm not really looking forward to going out to the island. I often feel this reluctance, most especially when there are certain risks regarding my proposed learning activities for the week. The last week I went out I thought that I had the perfect stuff planned for one of my most advanced students and he didn't take to it at all. He's one of the few students I've had in recent months who is studying algebra. During previous weeks I'd noticed that he did a lot better when I released the various algebra techniques one relatively small bit at a time. In this particular instance I had found a text that broke up certain work in solving fractions into pieces that were small enough that it would have annoyed the piss out of me when I was in high school, but it seemed to be exactly the kind of steps that he would be able to do and figure the whole thing out. The problem was that it was just enough for him to start having trouble remembering to do all the stuff he'd already been working on. It's that magical point where things get hard and the student starts calling the math work "gay" or "stupid." A kind of irony when it is usually the student who feels stupid at that point. It also seems to be one of two points that my students seem to fairly consistently run into where some kind of real cognitive deficit seems to rear its ugly head. This is the point where we've maxed out the kid's working memory capacity.

I've been reading up on this whole working memory thing lately and it seems that it correlates very strongly with all kinds of intellectual tasks including the sort of reasoning tasks that mathematics requires. The issue is being able to keep simultaneously in mind the ultimate goal of the problem solving venture, a mental roadmap of how to get to that goal, and the performance of the operation immediately at hand. As the road map gets more complex or has more steps added to it, or the immediate operation gets more complex the sort of mental blackboard gets cluttered beyond the individual's ability to read it and they start making mistakes that they actually know better than to make. One way of dealing with this problem is to develop what's called automaticity. Automaticity is the ability to perform certain mental tasks automatically and without thinking. This is the kind of thing marital artists train for in forms practice. It is also the thing your elementary school teachers were trying to give you with all those timed times tables tests. Really the only way to develop automaticity is to do the same things over and over and over a million times. This is something my students really resent me telling them they need to do.

Interestingly working memory is supposedly trainable. There have been a couple studies that have shown that people who practice at it and work on memory tasks of increasing difficulty can improve their working memory and show improvements in all kinds of reasoning tasks. It is one of the few things that can be taught that demonstrates a good deal of transfer into other tasks. The classis way of training this is the old electronic Simon Says game where you have to remember the sequence of flashing colored lights. Another is the Memory card game where you have to remember the location of matching pairs of cards that have been revealed one at a time.

With this in mind I've started up a regimen of memory training tasks for our evening study hours on the island. Sadly I don't think that it is going to be enough to make the kind of difference that we'd really notice. (This whole business underscores another reason I believe we need to rethink our entire system of education, but I digress.) Anyway, I'm giving it a shot and it seems that it has been effective to the point that the kids were willing to do it for a while. They're getting bored of the old task though and so I'm going to shift over to another task.

I mentioned earlier that there are two points of cognitive deficit that my students seem to be regularly afflicted by. If the first is diminished working memory the second is abstraction of proportional reasoning, or maybe just proportional reasoning. This is something that I don't know that much about. How this manifests itself though is that you can demonstrate to a kid with physical objects the concept that if you cut it into halves and then fourths that 1/2 is the exact same amount as 2/4 and they still don't seem to get it, and they certainly can't generalize it to proportions in general. This is a mystery that has been a nuissance for a long time and I really haven't gotten any closer to figuring it out. What I've found myself doing is teaching kids how to work out these and other sorts of fraction problems over and over. Somehow, I always come back around after enough time to giving students the exact same work that I've already given them, they've worked out all the problems but still can't remember how to do it. It may be that part of this is derived from them not having obtained automaticity in division. It may also represent something related to their behavioral problems. If a person cannot intuit about proportions in math can they understand proportionality in behavioral choices and outcomes?