I want to try math tutoring, not expecting a whole lot of work, but I've always loved math since doing high school math competitions (up to the AIME) and as I've gotten older I've developed a patient and nurturing side, so math tutoring seems like a fit.
I have one student so far and we did about ten sessions in pre-algebra and algebra. She liked me and was motivated to show up every week (twice a week, actually), so that was nice.
So, this is my very early attempt to be a math tutor. I already have a basic question. I know that a lot of tutors say that they adapt to the style of the student (different students have different learning styles). While that may be very important, I wonder if it's also important to recognize that mathematicians think and learn in ways that make math easier, and maybe we should be helping students to practice those ways.
In the art world, I'm thinking of that book "Drawing on the Right Side of the Brain" which pointed out that people have both a left-brained way of seeing the world and a right-brained way. The conclusion of the book was not "embrace your half of the brain"---no, it was "Artists use the right brain--so we'll teach you to do it, too."
So what thinking/learning style is math? I've encountered some evidence that math involves a lot of visual thinking. I know in myself that I have a mental picture to go along with most math concepts, sometimes a mental animation. When I start to work a problem, I develop a sense of where equations are laid out on the page.
So one obvious thought is, I could, perhaps for certain problems, teach my student to think like me. And I did a little of that. She wondered why multiplying by a number less than one would make something smaller (because MULTIPLYING makes things BIGGER, right?). I created an animation.. three bars going up and down. The left two bars are the multiplicands, and the right bar is the result. Usually the middle bar is fixed, and the leftmost one varies between 0 and 2, passing through 1 on the way up and again on the way down. My student could see that as the left bar approached 1, the result approached the fixed center bar -- and she already knew that "anything multiplied by 1 is itself," so this confirmed it. When the bar dipped below 1, it made complete sense that the result bar would go down and get smaller than the fixed multiplicand. And when the left bar got to 0, then you could see WHY "anything multiplied by 0 is 0."
It took her about five seconds to grasp this and she said "Oh, now I know what multiplying by less than 1 makes something smaller."
Right now I'm working on making a series of videos which could turn into computer software, teaching how to recognize the "form" of an algebraic expression or equation, and teaching why it's useful and powerful to be able to transform an equation.
And it's going to be highly visual. I'll use pictures, diagrams, animations, pulsing letters and numbers, etc. to create an organized use of visual space, and to connect concepts.
So, I'm a bit excited about my idea, but I still realize that not everyone is a visual thinker. My videos will be teaching people to think the way I think is most organized and powerful.
But what of thinkers who don't feel drawn to this style? Let's say a student comes to me who has a hard time in math. (Like my current student.) And they reveal some of the their learning style and it is not an adaptive style for math (like my current student). Is it my job, then, to help them do math from their current perspective, or is it my job to introduce them to a much more powerful perspective, even if they only get a little bit of it, long enough to pass math class?