Or you might have tried something far less time-consuming such as, "Multiplication makes things larger? What about one-half of something? One-half of everything? One-tenth of everything? Followed by, "That's multiplication by a positive number between zero and one." Better NOT to say simply "less than one" because (perhaps already?) the student is supposed to be algebra-ready enough to be proceeding to deeper stuff where multiplication by negatives (all less than one) will need to be taken for granted. Try to identify - and as quickly as possible correct - other inhibiting deficits (competence in arithmetic through ordinary fractions and ratio and proportion are probably enough). And then get on to some real math.
At 03:51 AM 6/12/2013, Michael Mossey wrote: >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? > >Mike