User:IssaRice/Mental representations in mathematics

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I think people don't talk enough about mental representations of objects in math, and I think it's horrible! This is closely related to how people don't talk enough about their stream of consciousness when thinking about math.

Examples:

  • rows of a matrix linearly independent vs rows of a matrix spans domain
  • mental arithmetic -- actually, this is one place where i think there has been a lot of discussion...
  • inf/sup stuff in analysis. I find this so much easier to think about when I draw a line segment and label points on the segment. But many books don't talk about this!
  • sup/inf/liminf/limsup: Tao's piston analogy!

Why don't people talk about this? Some things that could be happening:

  • It happens in-person rather than in writing, so I don't have access to it.
  • Mental representations are too personal. I don't buy this.
  • People have low metacognition.
  • People are elitist or otherwise don't want to share "cheat codes" to make the subject easier for others.
  • I'm wrong about the value of mental representations being important to share. One argument could be like "to really learn math you have to struggle to make your own mental representations".