#### in the abstract

There's an interesting article in Scientific American about cognition and symbol-processing in children. Numerous tests have shown that very young children can't really tell the difference between a thing and a representation of that thing (like a picture or a scale model)... so what does this tell us about what is going on in their heads? Judy DeLoache, on her experiments:

This inability to map back and forth between an object and its referent is not strictly an age-based or intelligence-based phenomenon. When I was tutoring college trig I had a student who was having a lot of difficulty with concepts like sine and cosine, and I was running out of new ways to present the info to him. I decided that instead of trying to get him to understand the math, it might be a good idea to step back and have both of us try to understand what his mental process was.

I asked him to just talk out loud while he was solving some problems, and while he was doing so I noticed that when the problem was presented in a certain way, he always got the right answer. This always occurred when the drawing showed a triangle whose angle of interest was on the left, with the 'right' (i.e. '90°') angle on the lower right. When I gave him problems with that same triangle mirror-imaged, he could still solve the problems but it took him longer because he was mentally mapping the triangle to a 'right' (i.e. 'correct') triangle.

When I showed him a triangle whose base was its hypotenuse (i.e. the 90° angle up in the air) he stared at it for awhile before telling me it couldn't be done... there wasn't a 'right' angle, or even a 'wrong' angle that could be mapped to a 'right' angle. The idea of 'right'='correct' was so strong that doing something squirrelly like rotating the image was unthinkable.

This guy wasn't unintelligent. He'd made a reasonable connection in his mind and unfortunately had that connection strengthened over the course of hundreds of problems presented in the exact same way by the teacher... every example triangle was drawn the same way, the 'right' way, and nobody had ever showed him how to solve 'wrong' triangles. When I finally figured out what the hell was going on I gave him a problem that he couldn't solve, then set that one aside and gave him the exact same problem, but oriented the 'right' way. He solved it without a hitch. When I picked up that piece of paper, rotated it and set it down next to the (identical) one he couldn't solve, I could just see it all tumbling into place in his head, a real satori experience, momentary mathematical enlightenment.

It makes me wonder what other abstractions have faulty representations in our heads. Actually, scratch that... make that *my* head. I don't even fucking want to even know what's going on in other people's heads. Your process is *all* fucked up.

About 20 years ago I had one of those wonderful moments when research takes an unexpected but fruitful turn. I had been studying toddler memory and was beginning a new experiment with two-and-a-half- and three-year-olds. For the project, I had built a model of a room that was part of my lab. The real space was furnished like a standard living room, albeit a rather shabby one, with an upholstered couch, an armchair, a cabinet and so on. The miniature items were as similar as possible to their larger counterparts: they were the same shape and material, covered with the same fabric and arranged in the same positions. For the study, a child watched as we hid a miniature toy--a plastic dog we dubbed "Little Snoopy"--in the model, which we referred to as "Little Snoopy's room." We then encouraged the child to find "Big Snoopy," a large version of the toy "hiding in the same place in his big room." We wondered whether children could use their memory of the small room to figure out where to find the toy in the large one.

The three-year-olds were, as we had expected, very successful. After they observed the small toy being placed behind the miniature couch, they ran into the room and found the large toy behind the real couch. But the two-and-a-half-year-olds, much to my and their parents' surprise, failed abysmally. They cheerfully ran into the room to retrieve the large toy, but most of them had no idea where to look, even though they remembered where the tiny toy was hidden in the miniature room and could readily find it there.

Their failure to use what they knew about the model to draw an inference about the room indicated that they did not appreciate the relation between the model and room. I soon realized that my memory study was instead a study of symbolic understanding and that the younger children's failure might be telling us something interesting about how and when youngsters acquire the ability to understand that one object can stand for another.

This inability to map back and forth between an object and its referent is not strictly an age-based or intelligence-based phenomenon. When I was tutoring college trig I had a student who was having a lot of difficulty with concepts like sine and cosine, and I was running out of new ways to present the info to him. I decided that instead of trying to get him to understand the math, it might be a good idea to step back and have both of us try to understand what his mental process was.

I asked him to just talk out loud while he was solving some problems, and while he was doing so I noticed that when the problem was presented in a certain way, he always got the right answer. This always occurred when the drawing showed a triangle whose angle of interest was on the left, with the 'right' (i.e. '90°') angle on the lower right. When I gave him problems with that same triangle mirror-imaged, he could still solve the problems but it took him longer because he was mentally mapping the triangle to a 'right' (i.e. 'correct') triangle.

When I showed him a triangle whose base was its hypotenuse (i.e. the 90° angle up in the air) he stared at it for awhile before telling me it couldn't be done... there wasn't a 'right' angle, or even a 'wrong' angle that could be mapped to a 'right' angle. The idea of 'right'='correct' was so strong that doing something squirrelly like rotating the image was unthinkable.

This guy wasn't unintelligent. He'd made a reasonable connection in his mind and unfortunately had that connection strengthened over the course of hundreds of problems presented in the exact same way by the teacher... every example triangle was drawn the same way, the 'right' way, and nobody had ever showed him how to solve 'wrong' triangles. When I finally figured out what the hell was going on I gave him a problem that he couldn't solve, then set that one aside and gave him the exact same problem, but oriented the 'right' way. He solved it without a hitch. When I picked up that piece of paper, rotated it and set it down next to the (identical) one he couldn't solve, I could just see it all tumbling into place in his head, a real satori experience, momentary mathematical enlightenment.

It makes me wonder what other abstractions have faulty representations in our heads. Actually, scratch that... make that *my* head. I don't even fucking want to even know what's going on in other people's heads. Your process is *all* fucked up.

## 1 Comments:

## Tom said...

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