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?
Monday, March 24, 2008
Saturday, March 01, 2008
More than a feeling?
When I was a kid we used to swim in a series of geothermal pools at the base of a volcanic hill a short drive east of town. The place was called Warm Springs, which seemed kind of inappropriate in that the water almost always seemed really cold in summer time. In winter time, however, it never froze (at least not to my knowledge) and it would frequently fog up half the valley.
Anyway, there were a few summers when it seemed like we went swimming there almost every day. I was a rather mediocre swimmer and I kind of enjoyed the chance to invest some mediocre efforts at some mediocre improvement. I also liked taking snorkeling gear which I inexpertly used to check out the guppies people had released from their fish tanks as well as the rocky bottom with the sorts of things that at the time only Huck Finn and I could have considered treasures. It was also exciting for the mystique of an abandoned mill for processing minerals or gravel or something mined from mountains out on the western end of the valley. From a distance it looked to me like a ruined castle with a giant green hand painted on a little rock face overlooking it. Eventually I learned that the hand was in fact a representation of a marijuana leaf, but I digress.
On one certain bleak summer day I had been out there swimming with my siblings and a couple of cousins. I call the day ‘bleak’ because it was one of those times when the sunlight has a way of coming down that makes everything look washed out and more barren than usual. At the same time every breeze on our wet skins made the water feel that much colder. It must have been one of those summers we went every day because I remember us getting bored with it sooner than usual. The weather may have been contributing to the fact that I just couldn’t get interested.
So we packed up our gear and our towels and what not and climbed onto the back of my grandmother’s little steel-blue Dodge pick-up so she could drive us back into town. My ears were filled with water producing an eerie sense of balance and an otherworldly half-deafness. As we bounced off the dirt road onto the highway I noticed that my shoes were missing. In those days I ran around barefoot most of the time and had developed some pretty terrific calluses so it was really easy for me to not notice the absence. A strange and uncomfortable sensation started to form in my gut because of it and I asked to go back and look for them.
We went back and looked around. I couldn’t find the shoes and the worry in my abdomen wouldn’t subside. There wasn’t much area to check for them so the search didn’t last long and we again left for home. The strange feeling didn’t go away and I started to identify it as a weird combination of guilt and fear. It’s not the usual kind of fear like that of heights or social anxiety. The fear seemed to me to be more of a supernatural kind of thing. Lovecraft wrote that he tried to create fear in his writing that was not corporeal but confronted the reader with a dread for the chaotic possibilities of an infinite cosmos. I think the fear I experienced was something like that, as if by losing my shoes I had somehow angered unknown gods and should shortly fall victim to the torment of their earthly instruments. I imagined some kind of witches with my shoes in hand using them in a ritual to slowly pick apart my mind like a knitter picking apart some mistake in the scarf she’s making.
The reason I tell this story is because it is the first time I experienced this feeling. It comes back to unsettle me on rare occasions and has done so recently. I purchased the Planet Earth series with my favorite wildlife documentary guy David Attenborough. I took it down to Woods Hole. I’d been showing some of them to the kids and lent them to one of my coworkers to see. One of the disks has come up missing. It was a bit expensive and the fact that I lost part of the set has me feeling a bit guilty. But as with the shoes, the disappearance is so far inexplicable. These facts make the connection pretty obvious and provide some context for explaining the dread I feel. Even so, it seems like too much for the magnitude of a problem like a DVD or a pair of shoes. Maybe the witches really are at work.
Anyway, there were a few summers when it seemed like we went swimming there almost every day. I was a rather mediocre swimmer and I kind of enjoyed the chance to invest some mediocre efforts at some mediocre improvement. I also liked taking snorkeling gear which I inexpertly used to check out the guppies people had released from their fish tanks as well as the rocky bottom with the sorts of things that at the time only Huck Finn and I could have considered treasures. It was also exciting for the mystique of an abandoned mill for processing minerals or gravel or something mined from mountains out on the western end of the valley. From a distance it looked to me like a ruined castle with a giant green hand painted on a little rock face overlooking it. Eventually I learned that the hand was in fact a representation of a marijuana leaf, but I digress.
On one certain bleak summer day I had been out there swimming with my siblings and a couple of cousins. I call the day ‘bleak’ because it was one of those times when the sunlight has a way of coming down that makes everything look washed out and more barren than usual. At the same time every breeze on our wet skins made the water feel that much colder. It must have been one of those summers we went every day because I remember us getting bored with it sooner than usual. The weather may have been contributing to the fact that I just couldn’t get interested.
So we packed up our gear and our towels and what not and climbed onto the back of my grandmother’s little steel-blue Dodge pick-up so she could drive us back into town. My ears were filled with water producing an eerie sense of balance and an otherworldly half-deafness. As we bounced off the dirt road onto the highway I noticed that my shoes were missing. In those days I ran around barefoot most of the time and had developed some pretty terrific calluses so it was really easy for me to not notice the absence. A strange and uncomfortable sensation started to form in my gut because of it and I asked to go back and look for them.
We went back and looked around. I couldn’t find the shoes and the worry in my abdomen wouldn’t subside. There wasn’t much area to check for them so the search didn’t last long and we again left for home. The strange feeling didn’t go away and I started to identify it as a weird combination of guilt and fear. It’s not the usual kind of fear like that of heights or social anxiety. The fear seemed to me to be more of a supernatural kind of thing. Lovecraft wrote that he tried to create fear in his writing that was not corporeal but confronted the reader with a dread for the chaotic possibilities of an infinite cosmos. I think the fear I experienced was something like that, as if by losing my shoes I had somehow angered unknown gods and should shortly fall victim to the torment of their earthly instruments. I imagined some kind of witches with my shoes in hand using them in a ritual to slowly pick apart my mind like a knitter picking apart some mistake in the scarf she’s making.
The reason I tell this story is because it is the first time I experienced this feeling. It comes back to unsettle me on rare occasions and has done so recently. I purchased the Planet Earth series with my favorite wildlife documentary guy David Attenborough. I took it down to Woods Hole. I’d been showing some of them to the kids and lent them to one of my coworkers to see. One of the disks has come up missing. It was a bit expensive and the fact that I lost part of the set has me feeling a bit guilty. But as with the shoes, the disappearance is so far inexplicable. These facts make the connection pretty obvious and provide some context for explaining the dread I feel. Even so, it seems like too much for the magnitude of a problem like a DVD or a pair of shoes. Maybe the witches really are at work.
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