A big impediment to effective learning happens when we misunderstand the nature of what we're trying to learn. Here is an example of (amateur) mathematical thinking in action and some observations about what we're really doing when we're thinking mathematically.

00:00 Intro
00:22 The square-jumping story begins
2:40 A side-note about parity
3:40 A different way of thinking about the same thing
5:04 Another extension
6:24 What did we learn?

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Big shout-out to Presage Media on this one. This was an animation-heavy video and I love what they did.

References:

I don't have any direct citations, but here are two interesting pieces about mathematical thinking that I have read in the past year.

Stylianou, D. A., & Silver, E. A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences. Mathematical thinking and learning, 6(4), 353-387.

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational studies in mathematics, 52(3), 215-241.

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