On Jun 22, 2014, at 11:50 PM, Ray <email@example.com> wrote:
> I realize that mathematics evolved and develop a life of its own independent of nature; however, I was only pointing out that nature provided the inspiration for mathematics. > > There are simple examples of geometric shapes existing in nature. The triangle is a natural derivative of nature. The pentagon and how it relates to the golden section. Then you have the studies of Fibonacci. The notion of a right angle is a natural phenomenon. > > As man started to write down these shapes, he noticed certain patterns. Along with the triangle and the notion of a right angle, the Pythagorean theorem was discovered. It was discovered because it is there in nature. Without the Pythagorean theorem, there would be no calculus. I believe that the concepts of Ratio, Proportion and equality are natural concepts that are observed.
The notions of measure and angle certainly have roots in our physical senses. Without the experience of shape and form in nature, we would never dream up such notions. So we agree there. But the act of refining and making these notions into mathematical concepts is a mental exercise and a rather unique one at that. It is that exercise that leads eventually to the significance of right angles and the Pythagorean theorem. You could put a million kids into nature and not one of them is going to come up with the concept of measure or angle, let alone something built on these concepts like the Pythagorean theorem. The Pythagorean theorem was the result of thousands of years (at least) of mathematics, and the notions of measure and angle were also the result of thousands of years of mathematics. I think we agree that without nature none of this would be, but I don?t think even the smallest mathematical thoughts are direct consequences of nature.
> I understand that the mental processes are not there for children but they can go through the physical processes through exercises and exploration. Then as their minds develop, they will have a more natural grounding of those concepts.
We are on the same page. But isn?t that a phase in the beginning and doesn?t it become less of the focus by 3rd grade? The majority of curriculums I have studied have ample physical examples and activities in the early grades. When I said that I haven?t seen a curriculum succeed that based itself on physical activities I meant in later stages or when they tried to substitute pursy physical activities for mathematics.
> I am only indulging in the more philosophical aspects of basic Mathematics; and in particular, why so many people have no clue as to what a proportion is.
All I can say, and this is just my opinion, is that curriculums have become very remedial in nature. They have broken the subject into a hundred little check boxes and while in the very beginning, that approach may have merit, like phonics in the beginning of learning to read, it falls apart as soon as the subject starts getting whole, which occurs before fractions and is in full force during fractions. Put another way, they do not teach the subject as if the students are getting it and even the students that might of gotten it, don?t.