The Social Jump-Start
IntroductionWhen you give your student a math problem to solve, hopefully the student thinks about the problem. In this process, the student hopefully develops the mental models needed to solve the problem. When the student solves the problem (or makes progress on the problem), the student experiences enjoyment.
Obviously, there are failures, and you should try to minimize the failures. However, you can't avoid them, and they probably do no great harm -- humans are built to enjoy "partial reinforcement".
This assumes that when you give the student a math problem, the student will have an interest in solving the problem and will actually think about the problem and try to solve it. In fact, it probably requires the student to become absorbed in the problem.
Students who like math easily become interested in problems and absorbed in them. But some students have been exposed to math only as memorization. They have little reason to be interested in a math problem, and perhaps little ability to become absorbed in a math problem.
Social FactorsWhen I introduce students to math, I use social factors to make it fun. One-on-one, it is most likely to be a contest (that I end up losing). For example, I might say to a student, "I bet you can't think of two numbers that add up to 10." When the student does, the student is a winner. Then I might say, "Here is one that for sure will be too hard for you... and do a slightly harder problem."
This of course is difficult to describe, and you can't hope to imitate it just from my description. I think the important thing is having a social interaction, and the exact style doesn't matter. You could be someone looking for help, and probably the basic "praise-as-reward" will work. High fives also fit in there somewhere.
Usually I teach the students who are good at math, but one time I tried my beginning lesson on a third-grader who was not good at math. He was essentially sitting around with nothing to do, because he couldn't attend gym, yet he had to be cajoled by me and his teacher to try it. By the end, he was having a great time and really getting into the problems.
A Whole Class
When I faced a class, I couldn't do the one-on-one social interaction, and I feared just giving math problems on a piece of paper. So my first lesson was a class discussion on a nonmathematical topic -- how to define things. I have a little skill with creating interesting discussions, and the class was naturally participative. So we had a good time. Meanwhile, defining things is a very difficult skill, so they had to think hard.
The next class was again a class discussion, this time on defining geometrical concepts (mostly "rectangle"). This again was fun but also took a lot of tough thinking.
The third class started with the Wharf-Sapir idea of what things do we have names for. We did a few nonmathematical concepts, then listed the numbers with names. Then we tried to list the numbers and define fractions. Class four we set out as discoverer or inventers and tried to discover/invent as many numbers as we could.
The fifth class was the feared "working-by-themselves-on-math-problems." It went fine. Maybe it would have worked fine from the start, I don't know. I would like to think that my problems-first method is so great that from the start, an entire class of 13 fifth-graders would enjoy math problems.
But I think I will use my social factors to get things started. The fact is, I had conditioned the class to think that any problems I gave them would be difficult, interesting, and worth trying to solve. There some basic problem-solving skills -- like that you don't give up just because you don't quickly know the answer. We had practiced those skills in a nonmathematical setting, and in a math setting with lots of social factors. They needed those skills to become absorbed in the math problems.