Common math misconceptions
Recently I had the opportunity to tutor a college student in geometry. She was taking the class for a third time, to achieve a "C" grade; a requirement for her teaching credential. Her past attempts had only resulted in "D"s. Her case was particularly unique because she suffered from a genetically induced "non-verbal learning disorder" relating to a visual/spatial perceptual impairment. To help her, I attended the class sessions with her, just like any other student would. I have to tell you, it was a real learning experience for me too, in more ways than one might expect. As it turned out, I had a few misconceptions about math that needed some reconciliation.
Geometry is of course math, and it and its' trigonometric counter part, may well represent the most abstract form of mathematic discipline, because these two math forms require you to think in three dimensional terms. It is a very visual form of math, and for someone who doesn't see things, or at lease perceive what they see the same way that most of us "normally" would, given the ambiguity of the word normal, it has to be down right frustrating. I would liken it to somebody who was colorblind taking a course in color photography, a blind person taking art appreciation or someone who is deaf, music appreciation. And yet, almost all of us harbor some anomalous facet of perception which makes us see things just a little differently. Add to that the paradoxical attribute that most people who teach math are gifted at it, love it and in the most extreme cases live with it in almost every aspect of their lives. In other words, they have a collection of neural synapses that has been fine tuned for dealing with mathematical complexities. In contrast, people who don't like math, don't have an aptitude for it, don't have an absolute love of it, don't teach it.
From the perspective of the student striving to deal with concepts foreign to his or her own perspective or sense of reality, and an instructor who deals with mathematical constructs as though they were as plain as the nose on their face, it is understandable that common misconceptions about math arise. Add to that, any level of mental dextral impairment on the part of the student, and you have an untenable situation. I was faced with trying to resolve just such an enigmatic perplexity, how to find some common vernacular to bring about a symbiotic union of a brilliant mathematician and well experience math instructor, with a student who otherwise exhibited above average intelligence. To exacerbate the problem, I had to do it in such a way as to not alienate the teacher, as it was almost instantly obvious that my mere presence was to some degree intimidating to her, and congruently, to avoid further demoralizing the student, who was by this time extremely frustrated with her own ability to grasp the complexities of geometric perception.
I have long held the perception myself, that learning is a matter of knowing the right questions to ask, and then asking them, regardless of how stupid they may seem. Ironically, many students have a hard time learning math, simply because they, or any one else in their class, neglects to ask the right questions. There are of course, some instructors who don't like to be asked questions, and it is my opinion that such should be expelled from the profession. In this case, the instructor welcomed questions, and spent extra time explaining what to her was obvious, but to her students had yet to achieve any level of perspicuous clarity. Ironically, as it turned out, 35 years having elapsed since I myself had first conquered geometric discipline, an opacity of misconceptions had been allowed to accrue within my own sagacious enterprise. Many of the questions I found myself asking were therefore necessary to bring about my own enlightenment, or a rejuvenation of it, as it were.
The misconception we evoke in our minds with respect to the challenge of learning mathematical disciplines, that we are somehow stupid if we don't get it, that others seem to be more gifted then we, or that math is simply to hard, are simply barriers we erect for ourselves to give us an excuse not to try. The greatest enlightenments that humanity has realized, have been connected by one thread or another to mathematical disciplines. Albert Einstein, who himself had great difficulty with the rigor of math, none the less went on to master it, at least to the degree that it enabled him to perceive entirely new dimensions of the universe.
Finally, I should note, that all those questions I asked, or perhaps some less conspicuous attribute of my presence in the class at least, produced the needed symbiotic relationship between instructor and student, and the young lady I was tutoring got her "C". I too received a passing grade, and had to overcome a few personal misconceptions, about old minds being able to deal with the complex constructs of math.