I'm a Mom who is homeschooling one of my three children. My husband and I decided to home-school Scott because his daily experience in the local school was destroying his self esteem and, quite frankly, we were frightened to think how far it would go.
The other thing is he did not seem to be learning anything that "mattered" to him, and my husband and I thought we could do a better job than his teacher who had to deal with thirty kids at a time.
Scott is an unusual boy in a couple of ways, but the Internet is too public for me to share personal details. (And it's not necessary for my question.)
You (and others who are known for teaching teachers and parents HOW to teach math) seem to emphasize the importance of the category of skills known as "problem-solving". But for me this raises the question of how and when and why it was determined to be so valuable that it's worth diminishing the amount of time and energy remaining to go over the basics of real math?
After all, neither my husband nor I recall this "problem-solving" even being brought up in the math classes we had, unless it was a specific word problem that came from our textbook. Wouldn't you place a higher priority on first making sure the student has mastered each of the operations that the curriculum of his school year is supposed to include?
My second question, which is kind of the other side of what I just asked, is this: IS PROBLEM-SOLVING really as teachable as the more standard areas of math? The reason I ask is that I wonder if it's a talent a person is born with, like the talent to be a musician or artist, which would mean that it is Not really teachable.
And regarding the importance of the whole topic, my husband and I, along with most of our generation, managed to graduate perfectly fine without hearing a word about it in school.
Since I won't ask you your age (so I won't have to reveal mine!), I can't know exactly the period you're referring to by your "generation", but I can still respond to your confidence that your generation did "perfectly fine".
There are various ways of determining success, for individuals as well as for societies, but one simple, inexpensive and fast approach is to type up a short questionnaire, print out a hundred copies, get a clipboard and a pen, and make a trip to the nearest public place, like a shopping mall. Then, stop a hundred people at random; make sure they are the generation you are referring to, and say Hello. Then present the questionnaire.
Through the questionnaire, ask them how they feel about math in general, and how they feel about the kind of 'problem-solving' that their children or nieces and nephews or whoever they know are probably being taught at least once per week. My bet is this: after the pause it takes them to let down the façade and reveal the truth, you will find a surprisingly high percentage of them cringe at the subject. I've heard highly educated and very intelligent people tell me that they'd rather be required to strike a large town bell with their skull than do math problems, because at least the bell they know they might stand some chance of occasionally getting it right.
To me, such a pervasive negative attitude is not convincing evidence that a generation did "quite fine". Some did, and perhaps you did, and I did, but I've known intelligent adults who will do anything to escape meetings to hide in the restroom until the mathematics part of the meeting is over. This is not success for a society.
Next: Is this whole problem-solving thing really important?
It's VERY important. More so now, in fact, than in previous generations. People now get hired and earn a salary and receive health insurance to solve problems, not to take up space and do work that could be completed more accurately by a seven-dollar calculator. The world has changed. Computers are the most powerful tool most learners have, and whether you agree or not, computer people will readily tell you and show you that it is more important and more accurate to know how to find answers to questions than to memorize ones which are often outdated and inaccurate by the beginning of the next school year. Also, problem solving does not preclude having to master the basic skills. Take away the calculators, and the students will have to problem solve AND calculate.
One more point: In the real world, no one is asked a "pure math question". No, there are words and numbers vined around each other and knotted together, and it is the Problem-solver's role to come up with the correct answer mathematically. No one asks an adult if he/she can divide a certain number by twelve, but a banker will inquire about making those monthly payments on your property, and you'd be better off if you sort of know that the key number of months in a year is twelve. Then divide.
Finally, the most interesting part of your question to me is the last part, as it touches on something I have often found myself analyzing: Is problem solving truly "teachable", or is the whole idea built upon a fake eggshell? It is certainly true that there are some things which are more "teachable" than other things, and to show this I often give parents an example: You and your friends could hire the greatest opera teachers the world has to offer and get them to work full-time teaching me to sing opera. The experience will not be good for their self-esteem. They will never succeed, and they would never let the newly 'trained' me to go on stage in one of their productions and do ANYTHING that involves sound from my mouth. Genetics influence every aspect of human life, but to varying degrees. Most adults who live in suburbs, for example, learn to drive an automobile But some do not. Some discover that the range of skills the task involves, which most of us take for granted, is not within everyone's grasp. There are bicyclists everywhere, and the ozone layer appreciates them.
Still, everything is teachable to some extent, and even I could learn to improve my singing, though it might still subject me to an occasional interrogation from the authorities regarding my motivation.
I think that the golden rule on the topic of "teachability" is this: ANYTHING done consciously can be taught. Whatever opera singers do 'on purpose' that they would otherwise refrain from doing, can probably be taught, regardless of the quality of the result. But their vocal cords are a gift from nature and is not 'done' consciously. Therefore, one cannot be 'taught' to have such cords. 'done' consciously and therefore cannot be taught. Likewise, there are parts of problem solving that are conscious and which one develops from studying the way others have solved problems, and by years of practice developing as much flexibility in thinking as a person's genetic makeup will support; those aspects can be taught.
As it happens, problem-solving is probably my favorite area of most schools' mathematics curriculum, and I have found that the teaching of it can be among the most rewarding of all topics one can teach in school. Furthermore, it's doubly exciting because there are students who come alive for the first time all year when the subject of perplexing problems comes up and their brain gets juiced.
QUESTION: If you were stuck in an elevator of a cruise ship which was sinking into a cluster of hungry crocodiles, would you rather be sharp on your multiplication tables or confident in your ability to come up with solutions to real problems? For some that's a tough call.
SO, there is in fact so many important and interesting skills of problem-solving that can be taught to almost anyone, and enjoyed by them, that your question inspired me to assign myself the task of devoting an entire "answer" to sharing these techniques and giving examples of how they can be incorporated on a day-to-day basis to improve your life.
I will write that problem-solving letter sometime within the next few days and will post it sometime within the next couple of weeks. So, if you are interested in checking it out, keep your eyes and screen open and on.
Hope that helps,