When I was a kid learning math seemed weird and officious. Half of the time I thought the teachers were nuts when they explained how numbers could be used, as though they were trying to convince us to buy something. When I was excited to "discover" something about math, something which we would be later taught in some methodical fashion, the teachers would often wonder what I was talking about. Most seemed unable to relate to math outside the context of a lesson. This sounds very computationally elite and snobby of me but sometimes I think I understood the nature of numbers better than they did because I didn't have the good sense to be terrified of math.
I thought I would probably stop taking math after high school. It seemed to be more of memorization than anything. It was boring and nobody ever bothered to explain the connections. When I was older, the larger concepts began to unfold more sensibly. I finally saw the grander picture--or pieces of it. Why don't more high school teachers encourage kids to think about math conceptually? Probably the better question is, why does math have to be such a terrifying experience in grade school?
Many elementary-level teachers aren't enthusiastic about math. I took a lot of math in college with the goal of teaching it. While on this path, I also took various courses on elementary math instruction because I thought it could be helpful. My adviser told me he could waive the elementary math methods classes because of the amount of upper-division math I had already taken. However, understanding how educators embraced the development of math concepts in younger kids was of interest to me.
What I discovered shocked me. Future educators were being taught some truly dynamic math development concepts. Why didn't they teach it like this to kids? A big problem was that the university professors were excited about their subject and the students, future elementary-level educators, were not. A good percentage of my fellow students were on their 2nd time in the classes (after receiving failing grades initially.) This was not the case universally but it was true far too much of the time.
Many of these future teachers admitted being terrified of taking these classes because they "always hated math." It only takes one teacher's clear dislike of the subject to alter a child's perception irrevocably. Clearly, "hating math" is a case of the chicken and the egg.
We've somehow raised generations of math phobics. However, the previous generations are just as queasy about math and so on and so on. As a result, subsequent generations of teachers are often very single-minded in their approach to mathematics education, relying on memorization of formulas and even confusing the use of manipulatives. Too many kids are learning rote methods of computation without any clue that there are conceptual equivalents. If a kid has a question that is not directly dealt with in the teacher's manual, it may not be addressed and a curious child will become a hardened one. They don't get the play with the numbers and see what they can do. No monkey bars with 4, no tag with 7, nothing.
Kids need to learn that it's ok to take their numbers out to play, that there is more than one way to divide or multiply. Dividing by a fraction isn't about "flipping over" the divisor and multiplying. What if you can't remember the rote method for dividing by fractions? What do you do then? If you are comfortable rooting through the mathematics toolbox, there are numerous tools to help you reinvent the wheel so you don't have to memorize strict methods of computation.