On Jun 16, 2013, at 1:01 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> Your position is utterly bizarre. There are books and papers written about visualization in mathematics. I don't care how much you want to torture the meaning of the word with your idiosyncratic talk.
Visualization? No, we weren't talking about visualization, we were talking about "visual thinking". That is when the torture began. Since I make my living through visualization I think I know a thing or two about the difference between visualization and the mathematics behind it. While the animation mentioned in this thread might allow a student to accept that the result of multiplying can be smaller or larger, it will not and can not communicate the significance.
I used to take visualization for granted, although I never thought of it as the "thinking" in mathematics. I only started questioning the (proper) use of visualization in teaching mathematics when I saw really well done demonstrations of topological principles (a decade or so ago). While they certainly satisfy some innate craving for an intuitive explanation, they did nothing to convey the underlying theory or its significance. Can I at least say that? Or in those books and papers do you also have a picture book that teaches topology? Nonetheless, I was amazed enough by the ingenuity of the animations that I withheld judgment, and besides, it's topology and it is difficult/impossible to visualize anyways. Surely, they weren't trying teach topology that way (I hope).
But I didn't connect the problems of teaching topology visually to teaching algebra and geometry visually. Not until I started studying the way teachers are teaching algebra and geometry and not until I started teaching my son. It doesn't take long before you recognize that even though your son followed your perfectly illustrated explanation he certainly doesn't get it because he is unable to answer simple key questions. And when you retrace his steps you realize why. He recites back to you, rather nicely, your explanation and he refers to the illustrations correctly. This line from here to there and that line from there to there and so on.
And then you recognize the problem.
Those are not "lines".
The lines he is talking about are the lines I drew on the board.
The lines I am talking about, you will never see with your eyes.
Surprisingly, it was easy enough to correct. I am not saying that he has to immediately understand that a line is an infinite set of points following a certain relation that we "represent" visually by drawing a line on the board. In fact, I found that it was sufficient enough in the beginning that he just understand that the lines he sees are not the lines I mean. Over time, with the right prodding, he becomes abstract.
We take this very important distinction for granted because either we got it quickly or we don't remember well how we got it.
From my observations, the majority of the problem with mathematics attainment is the inability to abstract. In a way, that is the same as saying that the problem is an inability to think but my son was thinking when he recited back my explanation. He even corrected himself a couple of times. But he wasn't thinking abstractly. Visualization is no substitute for that nor is it even a pathway if you don't address the problem directly. I guess it takes a certain knack as a teacher to keep the importance of the abstract theme elevated above the visual theme on the board. For example, my son has a good grasp of the abstract qualities of a point and a line, but if I were to draw a parabola and tried to explain to him how the mapping of the x values is in such a way that the points on the parabola are infinitely many like the line and that the parabola is like the line, he might very well reply "Oh, you mean that the parabola is a solid line?" and I would reply "No, the parabola is a solid line because I drew it with a marker in one continuous motion." I would ask him to dig deeper and tell me more of what he means by "solid".