There are four general perspectives one can analyze and explain mathematical thinking and learning.These views should be seen not as competing but complementing each other. The four are:

- Mathematics: where the focus is on the rules and norms of the mathematical research community.
- Mathematics education: where the focus is on the individual and social processes in a community of learners, in and out of the classroom.
- Cognitive psychology: where the focus is on the universal characteristics of the human mind and behavior, which are shared across individuals, cultures, and different content areas.
- Evolutionary psychology: where the focus is on the evolutionary origins of human cognition and behavior and their expression in “universal human nature.”

-from Uri Leron and Orit Hazzan in their paper *Intuitive and Analytical Thinking: Four Perspectives* published in Educational Study of Mathematics (2009) 71:263–278.

Let us differentiate these perspectives in terms of following questions:

*What is multiplication?*

Mathematically, it is not correct to define multiplication as repeated addition and some mathematicians think we should not teach it that way. You can read about the controversy around this in my post Math War over Multiplication. However, from the cognitive and evolutionary psychology point of view, it is but natural and perhaps to be expected that majority of the young students will make this deduction that multiplication is repeated addition. From the math education perspective of course, teachers are expected to eventually challenge this conception.

*Are mathematical errors good or bad? *(Errors here actually refers to misconceptions, that is, common errors). The following analysis is from Leron and Hazzan’s paper:

- The mathematical perspective typically views errors (misconceptions) as bugs, something that went wrong due to faulty knowledge, and needs to be corrected.
- The mathematics educational perspective typically views errors as partial knowledge, still undesirable, but a necessary intermediate stage on the way towards attaining professional norms, and a base on which new or refined knowledge can be constructed.
- Cognitive psychologists typically view errors as an undesirable but unavoidable feature of the human mind, analogical to optical illusions, which originate at the interface between intuitive and analytical thinking.
- Evolutionary psychologists, in contrast, view errors as stemming from useful and adaptive features of human cognition. According to this perspective, people make mistakes (at least of the universal recurring kind) not because of deficiencies in their intelligence or their knowledge but because the requirements of modern mathematics, logic, or statistics clash with their “natural” intelligence.”

For those thinking of doing a research on mathematics thinking and learning, you must be clear about which of these different perspectives you will be analyzing your data and explaining your findings.

Recommended readings for your research: