On Jun 21, 2014, at 9:20 PM, Ray <email@example.com> wrote:
> Man has learned mathematics by observing nature.
I would disagree. I think the majority of mathematicians would disagree. I would also point out that I have seen no successful pedagogy with that notion as its core principle. I have seen pedagogies that play heavily on that notion, but they aren?t successful, by the very measures you mentioned in your post. The kids never get the picture, big or small. Mathematics has visceral roots and relies entirely on inquisitiveness, logic and formal thought from that point forward. It is a theory of real numbers and measure constructed entirely in the mind. Physics derives from nature, and while mathematics plays a big role in physics, they are separate endeavors in the mind. While physics may need math, math doesn?t need physics.
People seem to confuse visceral with physical but visceral refers to an internal sense, not a physical sense. For example, knowing that if A > B and B > C then A > C is a visceral sense. While a student may agree with this statement after you show them several examples that illustrate it, this (usually) doesn?t mean that they have developed a visceral sense of its truth. Mathematics requires that you sense the truth not just agree with the truth. I have had a lot of difficulty getting this point across here to some but it has changed my approach to teaching mathematics to my son with noticeable effect.
I don?t think you can teach visceral sense, or any sense for that matter, but you can do things to aid its development. I think your only control is to teach in such a way that is best conducive to the development of the sense. First and foremost, the student has to become accustomed to the abstract world in which the sense operates. The student has to engage with numbers and measure and not just by chance, but with a plan, and you must ensure that this engagement occurs regardless of whether they would rather play video games instead. In many cases, you will have to drag them through this part. I don?t care whether the student likes math or not, without that experience, there can be no sense. Secondly, as the sense develops, seize on those opportunities to have dialogs with the student based on that sense. This involves examples and counter examples of truth, not just stating the truth. As much as is possible, make them choose, but when needed, mentor. And finally, remember! that math is just one art, there are many others.
The ultimate goal is enjoyment of this life and education is necessary for that goal from a practical standpoint but developing any specific art, including mathematics, is not. This means that arithmetic and fractions are certainly things that need to be gotten right, but algebra and calculus are not. At the very least, when teaching arithmetic, focus on arithmetic, not algebra and calculus.