One trouble with learning math is that children who are accustomed to thinking of learning as retention of information (no they wouldn’t put it that way) have a hard time adjusting to an art.
In the art of mathematics, you’re goal is not to remember facts. Your goal is to be able to do things with those facts – to develop skill and understanding.
It seems important to me, therefore, that in the early grades the teacher deliberately combine the three different aspects of learning:
- Drill the facts into their heads (five minutes two or three times a day for 12 years is probably enough)
- Give students problems that demand imagination. These problems need gaps that the mind has to leap over, even when there is no given process. The older and more confident students get with math, the bigger these gaps can and need to become.
- Make sure they know the facts well enough to remember them before you start drilling them.
1 and 3 create a dynamic process in which the teacher needs to be aware of her students state. If he does not know a math fact, he should not be drilled on it. If he does know it, he should be drilled.
Drilling is necessary for speed of recall and, even more, for adaptability and use.
So you need all three steps.
1. Tell them the facts as isolated facts. Over and over again. 2+3=5, 2+3=5, 2+3=5.
2. As they gain recall (and this can take a long time – these are the hardest lessons a person ever has to learn in math), begin to drill them by mixing up the facts they have to recall.
3. Present them with challenges that demand mathematical imagination and both challenge and expand their understanding/perception of how numbers behave.