> 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 wasn't aware of the extent to which I have subtle > mental imagery until I started tutoring and paying > more attention to what I'm doing when I solve an > equation, or a tough AMC problem. It can be below the > level of easy, direct awareness, but it is there, and > it is critical. > > 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). > > > 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. > > 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.) > > - -Mike
I teach students with learning disabilities, and my mantra is "use the strengths, remediate the weaknesses." If your student has difficulty creating, using, and manipulating mental imagery, then you must improve her ability to do so for her to be successful in math. Math, more than most subjects in school, requires the use and interconnection of several areas of the brain. You may be interested in the following articles:
Excerpt: **************** Recent brain research shows that the complex abilities apparent in individual kids are reflected on the inside, as well as the outside. Parts of the brain involved in reading, math, music, and personal relationships are different -- larger or smaller, more or less active -- in every child. These circuits are independent, so even if a child struggles in one domain, like reading, he may have a neurological advantage in others. And perhaps most surprising, scientists have established that learning and practicing certain skills can cause the corresponding brain areas to morph and grow. IN OTHER WORDS, BY HELPING A CHILD HONE HER ABILITIES, YOU CAN ACTUALLY CHANGE HER BRAIN. [my emphasis]
One disability that is associated with difficulties in math is a Nonverbal Learning Disorder (NLD or NVLD). "Nonverbal" in this case means they do not exhibit general weaknesses with language-based tasks (although they can have some specific language deficits); some students with NLD have highly developed language skills, scoring up to the 99th percentile on tests of verbal intelligence, such as verbal portions of the WISC or the SAT. They do relatively poorly on tests of visual-spatial skills and nonverbal forms of communication. There is a similar discrepancy between verbal memory and nonverbal memory tasks. Academically, they struggle the most with math, although sometimes their difficulties do not show up until middle or high school, because they are able to do well in elementary arithmetic by using their strong verbal memories to compensate.
Conversely, students with dyslexia, who typically have strong visual-spatial skills, have trouble with arithmetic, but often do well, even very well, in higher levels of math.
Correlative only, but the unique combination of strengths and deficits of these learning differences indicate that visual-spatial skills are vital to success in math, therefore I would absolutely recommend helping your student to develop these skills. I find that students who have difficulty with mental imagery fall into one of three categories:
1) Those who can and do create mental imagery but on a subconscious level. They are verbal thinkers with mental imagery running in the background (like watching a movie and listening only to dialogue, then realizing there is a beautiful scene or music in the background). This is easy to fix - you only need to help the student to become more aware of the mental imagery by discussing it. (Like you noticed, Mike, when you wrote: "I wasn't aware of the extent to which I have subtle mental imagery until I started tutoring and paying more attention to what I'm doing when I solve an equation, or a tough AMC problem. It can be below the level of easy, direct awareness, but it is there, and it is critical.")
2) Those who think primarily in pictures and have difficulty with language (Einsteinesque), especially "translating" between words and imagery. This is a little more difficult to remediate but again simply discussing imagery, actively moving back and forth between images and language ("How would you describe that picture in words?" or "Draw a picture based on what you are reading" or "Describe your mental imagery after reading that paragraph" along with "This is what that passage made me visualize", etc.) These students often have underlying language weaknesses, and an observant teacher might notice how wording of a math problem can trip them up, even though they understand the math. I have had the experience of simply rewording a math problem - syntactical changes - and the student says in an exasperated tone, "Why didn't they say it that way in the first place?!" then solves the problem without difficulty.
3) Those who have a great deal of difficulty creating mental imagery. This is the most difficult to remediate with perhaps more limited success possible. These are also the students who typically struggle the most in math. You start small and work in tiny steps. Have the student describe something that is very familiar - a favorite pet, their house, a character from a show, etc. Encourage them to add more and more detail. Move onto reading a simple sentence, such as "The book is on the chair," asking the student to visualize and describe the color, form, etc., of the book and chair. Then move onto paragraphs, initially pausing at the end of each sentence to form mental imagery. Ask questions to prompt recall of visual details.
I am purposely avoiding math problems in these exercises - I think you work your way up to math, then hone those visualization skills beginning with pre-algebra story problems. Algebra introduces a level of abstraction, so I wouldn't start there.
Hope this helps. I can give you more information if you are interested.