> 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.
I agree but that is not really what I mean. I keep trying to type something meaningful but it is not easy. Perhaps I should just explain by way of my personal experience. Maybe that will help to get my point through, instead of philosophizing about things I am not an expert in. I also realize there are a lot of variables that makes education difficult.
I am one of those students, after completing High School, who was practically illiterate. I could recite the multiplication tables but that?s about as far as it goes. I did learn to type which proved to be useful over the years but it ends there. So I understand the lack of knowledge that one can have after 12 grueling years of schooling.
I realized at the end of taking Calculus II in College, I didn?t understand what I was doing. Since I had a poor math background, I took a very brief and useless geometry class, Algebra I & II, Trigonometry, pre-calculus and Calculus I & II.
I became very proficient in manipulating formulas but I didn?t understand anything, I later realized. It is amazing what you can do when you put your mind to it. It is also amazing that I could even get to Calculus II and ended with a B in it without really understanding anything. I really must emphasize "anything". So I decided Math was just not something I should pursue.
I also had an interest in Art. I had previously had some success in Art school. So after College I thought that perhaps I had a poor knowledge of Geometry. So I picked up this excellent Geometry book by Harold Jacobs and became engrossed in it. I did all the exercise. I definitely had a poor understanding of Geometry.
I became particularly fascinated with the Polyhedrons which were in that book. So I started to create all the polyhedrons with cardboard. I would paint them very colorful. I then decided to create a 5 foot mobile by inscribing all the polyhedrons in the 5 foot metal rings I purchased.
I decided to use wood which made it more difficult because I needed to also solve the angles in how the shapes are joined together. I purchased a table saw, so that I could get the precise rip cut, so the polygons would fit together properly. I also then started to expand the polyhedrons by cutting the tips or adding pyramid type shapes on the polygons. As well as stretching the polyhedron.
I then proceed to figure out what the sizes would be needed to fit those polyhedrons in that 5 foot ring. I must amazed at how the cube fit nicely inside the Dodecahedron. From this I was able to figure out the diameter of Dodecahedron, as well as the cube and how the Icosahedron had a reciprocal relationship with the Dodecahedron. How neatly the Tetrahedron fit in the cube and lastly the Octahedron.
After doing all this, along with following the book, Trigonometry started to make sense to me. I was so clueless that I didn?t even understand what Pi really was. Or even a radius. I now understood how important the Pythagorean Theorem is. What a square root is. I understood what those numbers in the trig tables mean. What a Sine, Cosine, tangent is. It just all came together. I began to understand how the Pythagorean Theorem related to Calculus in solving for the slope of a tangent line.
I also realized that I really didn?t totally understand what a ratio and a proportion is. I also realized how the concept of equality wasn?t really clear to me. I think that my confusion with equality was that I saw equality as sameness and didn?t focus on the bigger picture of proportion. So I never understood how a ratio, a proportion and equality all related to each other. I then realized they are different but they are also similar. They weave in and out of each other.
A also had a Zen experience with perhaps the most simple formula of all: If A = B, and B=C, then A=C. It just came to me how important that formula is. Perhaps this is the heart of Mathematics. I could give many more examples of concepts I lacked and how I came to understand them by myself just by wanting to understand.
I didn?t continue any academic Mathematical classes after that. I went on to other things. I got married and ended up with a career in mental health. Now If I had all that information as I was taking all those classes, I would have gotten farther. Of course, if is a big word; Life just takes us with what have at the moment.
So what can you say about this? The problem with college is that it is not a place for learning. Once you are on this merry go round, you can?t get off till it stops. There is no one to help you fill in the gaps. You are expected to know all this.
I once had this thought that it would nice to be a Math doctor. To be able to diagnose what someone needs in order to understand. You have to understand at some basic level, not necessarily in any deep sense but just basic things. You just can?t continue to manipulate formulas and numbers without understanding.
I say this, of course, from my personal experience and have not been a part of any studies. So I hesitate to generalize but I doubt that the importance of what I mention does not play a part in the overall understanding of Mathematics in a general sense. The contact with something real that makes sense to you. It's like we get thrown into this very abstract world much to soon in our development. So many students just get left behind and become totally confused.