I hope you remembered my simplistic explanation of the three philosophies of education some blogs back, the one involving circles. Let me explain the math problem from that point of view. There are two basic methods in teaching mathematic. One is from top down--the idealistic method. You start by explaining what numbers are and how to manipulate those numbers. You work from the simple to the complex, from adding of small numbers to the big bang theory and quantum physics. It is primarily a rote method of instruction.
The other method (note that I did not say the second method as that might imply the 1st being better then the 2nd--see how numbers can be viewed?) is the realistic method of teaching where the students have a problem to solve and use numbers in a way to "discover" or "own" the solution to the problem. They may actually use something else besides the numbering system we now use but in essence they then understanding "numbers" as well as the solution to their problem.
So what we have then is the "rote" method vs. the "ownership" method. Which is best? In spite of studying the mathematics curriculum problem for most of forty years, I still haven't the foggiest idea which method is right. And if you think "politics" or "religion" are super sensitive subjects, try asking the Mathematics Department at your university as to which is best and watch the fur fly. A hot question to ask any math prof is...."How much mathematics does a K-12 teacher need to know to be able to teach?" And....."how should they teach mathematics?" This is one of the more perplexing problems of the educational system.
I have a sense after teaching all these years that the answer to this question of learning mathematics is in our genes. Some children seem to understand the abstraction of numbers and enjoy manipulating them. Other children see no fascination in arithmetic and are bored with the subject. It seems to last throughout one's lifetime. But I have no proof (pun intended).
I wonder what the Seattle School Board decided? [blogger's update] Seattle decided last night to forego the discovery approach to mathematics and by a 4 to 3 vote picked a more top down approach. Many reasons for the change, many people unhappy. You can read Seattle P.I. article about it at: http://www.seattlepi.com/local/405961_math07.html [May 7, 2009]
Do you know how to count and make change? Remember to thank a teacher for that skill.