On Jun 14, 2013, at 6:11 PM, Michael Mossey <email@example.com> wrote:
> Visual thinking in mathematics, to my way of thinking, is not that my mind generates slick animations or beautiful typeset equations, but that simply there is subtle visual thinking, subtle images that fly through my mind as an aid to what I'm doing on the paper, or as an aid to understanding a concept. The subtle images *support* the effort to count, manipulate algebraic equations, or whatever. They aren't the whole thing, but I suspect they *are* an important ingredient.
I am not saying that you can separate visualization from mathematics. I am saying that visualization is by no means the active ingredient. Can you separate visualization from reading? I should think not. But does anyone here actually think that visualization is the active ingredient to reading? The active ingredient to reading is, brace yourself, reading!
> I found that providing animations (which I did for several topics) for my student was a tremendous *aid* to understanding (not the whole story, but critical).
I thought I pointed out that "understanding" isn't the point. Changing the way they think is the point. If your student isn't on their way to tackling unique problems, like those in the AMC exams, then they gained nothing. I am not saying that it is your fault, and I am not saying that the student will be a failure in life. But they did fail algebra. Which isn't the end of the world. You have to accept that this is a fact. If you don't believe that "understanding" algebra is worthless compared to actually being able to do algebra, then go talk to 100 adults who took algebra in school and passed it, but can't use it.
> What I am wondering about now, is whether people who don't do much visual thinking are going to have a hard time with algebra. If so, then one goal of videos or math educational software would be to stimulate their visual thinking.
Well, someone that doesn't do much thinking period isn't going to pass an algebra class. There are millions of animations and visual aids out there now, a thousand fold increase over 40 years ago before personal computers and the internet. And many of them are very good and yours are probably very good. Yet, almost no one disagrees that there has been nothing but a serious decline in the ability of our students, even our top students, to cope with algebra. And it shows in the technological global economy of the world today.
> The other possibility is that people who aren't good at visual thinking (in a mathematical way) should use the mode of thinking they *are* good at. What I'm tossing over in my mind right now is whether that is a good compromise, or whether visual thinking is so integral to algebra that it is simply more effective to train their visual thinking. If they aren't natural visual thinkers, this can only go so far. But maybe it is the best solution in practical terms. (An example of a practical goal would be passing the GE requirements in college.)
Getting someone to pass a class is not the same as teaching someone mathematics. Do we agree on this? Classes are notoriously easy to pass. Especially now. Especially after they ripped the art of the subject out and replaced it with "concepts". Like, understanding that keeping your eye on the ball is important, in place of actually proving that you can keep your eye on the ball. When I said I would teach a student just to pass the class if I had to, I only do that because the stupid school has a statutory requirement that every student take and pass algebra. I think, if anything, it is that statutory requirement that killed algebra.
Are you saying that these animations are the key to students tackling AMC problems or that they are the key to helping students pass algebra class? What I am hearing is that this student isn't getting algebra and you have found other ways relying on visualizations to help her pass the class. There is nothing wrong with helping her pass the class but sometimes you seem to be saying that you are succeeding in teaching her mathematics. I am not seeing that.
I am not criticizing how you are tutoring and I think you are probably an excellent tutor. But tutoring to pass a class and tutoring to teach mathematics are two different things. I will freely admit that when I change to tutoring to pass the class that I have stopped teaching mathematics. What I mean by that is very simple and very honest and should be very understandable to you. When I stop teaching mathematics, the goal is no longer that the student be able to tackle AMC problems at the end.