Two quotes stood out for me. The first is about the nature of "doing" mathematics:
I think many students give up on mathematics because they don’t see how they could possibly come up with the solutions to problems they see in their textbook, or which the teacher gives on the board, or which some of their classmates produce. What they don’t understand is that the clever argument they have just been presented with was not arrived at by deliberate, rational thought. It was constructed after-the-fact. And so the student misses the crucial lesson that the secret to doing mathematics is not an unusual brain but sheer persistence, trying one thing after another and failing each time until eventually the light comes on.
and the second about the "multiplication as repeated addition" furor:
Using a brittle metaphor (multiplication is repeated addition, for example) inevitably leads to problems later, when the metaphor no longer holds but gets in the way of a better understanding of the concept. It’s hard enough grasping the abstractions of mathematics without compounding the issue with brittle metaphors. One problem is that metaphors inevitably lead to natural inferences. For instance, thinking of multiplication as repeated addition leads to the belief that multiplication makes things bigger. This false belief often persists throughout people’s lives. It’s particularly hard to eradicate since it is often something the child observes him or herself, and as we all know, knowledge we generate ourselves tends to stick like glue. When it comes to mathematics, I think it is probably always unwise to use metaphors as “interim definitions”, which is what often seems to be done, since they all break sooner or later. Rather we should present the student the same instances, but as motivational and illustrative examples of, not metaphors for. Mathematics is abstract. It does students no good in the long term to present it as something concrete. Moreover, there is no need to do so. There is plenty of evidence that children can handle abstraction, particularly when the learning is scaffolded by a range of concrete examples.
Do you think that we make math too tidy for students? Do you have a favourite example of a "brittle metaphor"?