Literature on mathematical intutiion

Question

Is there any literature on mathematical intuition?

I would like to read the following, for example.

• How to train intuition?
• How to test intuition?

Answer 1

How to train intuition?

I'll answer your first question with a couple quotes from the literature.

From K. Anders Ericsson, one of the most influential researchers in the field of human expertise and performance (the quote is a comment in Anderson et al., 1998):

“...[M]odern educators have trained many generalizable abilities such as creativity, general problem-solving methods, and critical thinking. However, decades of laboratory studies and theoretical analyses of the structure of human cognition have raised doubts on the possibility of training general skills and processes directly, independent of specific knowledge and tasks.

For example, research on thinking and problem solving show that successful performance depends on special knowledge and acquired skills, and studies of learning and skill acquisition show that improvements in performance are primarily limited to activities in the specific domain.”

From Sweller, Clark, and Kirschner (2010):

“Recent ‘reform’ curricula both ignore the absence of supporting data and completely misunderstand the role of problem solving in cognition. If, the argument goes, we are not really teaching people mathematics but rather are teaching them some form of general problem solving, then mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general, and that will make them good mathematicians able to discover novel solutions irrespective of the content.

We believe this argument ignores all the empirical evidence about mathematics learning. Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach was never the most effective or efficient way to learn mathematics.
…
[L]ong-term memory, a critical component of human cognitive architecture, is not used to store random, isolated facts but rather to store huge complexes of closely integrated information that results in problem-solving skill. That skill is knowledge domain-specific, not domain-general. An experienced problem solver in any domain has constructed and stored huge numbers of schemas in long-term memory that allow problems in that domain to be categorized according to their solution moves.

In short, the research suggests that we can teach aspiring mathematicians to be effective problem solvers only by providing them with a large store of domain-specific schemas. Mathematical problem-solving skill is acquired through a large number of specific mathematical problem-solving strategies relevant to particular problems. There are no separate, general problem-solving strategies that can be learned.

References

Anderson, J. R., Reder, L. M., Simon, H. A., Ericsson, K. A., & Glaser, R. (1998). Radical constructivism and cognitive psychology. Brookings papers on education policy, (1), 227-278.

Sweller, J., Clark, R., & Kirschner, P. (2010). Teaching general problem-solving skills is not a substitute for, or a viable addition to, teaching mathematics. Notices of the American Mathematical Society, 57(10), 1303-1304.