Structuring mathematics

• In school, math is something I’m better at compared to others.

• I’m thinking about the difference between me and others.

  • I recently thought that it might be because I can imagine “if…” situations. (aka)
    • i.e. Having an attitude of actively using inductive thinking based on generalized concepts? (blu3mo)
  • Is imagination equivalent to visual thinking? (visual thinking https://en.wikipedia.org/wiki/Visual_thinking))
    • In differential calculus, for example, I can visualize the derivative of a certain point (called the derivative function, I think) in my mind, and when it matches the desired answer, my understanding deepens. (aka)
      • Visualization can lead to mathematical discoveries, but it seems unable to prove them. (aka) https://plato.stanford.edu/entries/epistemology-visual-thinking/#Con
      • It’s like mathematical induction, right? (takker)
        • Predicting the correct answer for $n=0,1$ and then proving it with a proof.
        • Drawing a graph and estimating the range of the solution before solving it.
      • It’s exactly the same as the hypothesis-experiment-verification cycle in science.
      • It says the same thing.
        • 5fefe06479e11300001805b2
  • It may be related to chemistry, but in the case of electrolysis, if we take the electrodes and the cell out of the electrolytes, the electrons don’t complete a full cycle. Therefore, the electrode with a positive charge (anode) and the electrode with a negative charge (cathode) allow the ions to move in a time-dependent manner. (aka) (From Zara)

• I think it’s necessary to structure knowledge.

  • Just a hypothesis.

• Rather than having different solutions for various types of problems as separate weapons, finding the structure is the approach.

  • Connecting and generalizing the weapons.
  • (This is specifically about solving high school math problems.)
  • In fact, if you don’t understand the principles and structure, you can’t solve math problems… (takker)

• Example:

  • Instead of memorizing different types of problems involving trigonometric functions, generalize and find the key.
    • Key points:
      • Trigonometric ratios loop, so there can be multiple solutions.
      • Trigonometric functions are projections of a circle onto the x-axis and y-axis.
  • There are problems in trigonometric functions and logarithms where you can approach them by considering them as quadratic functions, treating them as a unit.
    • Where to abstract (=extract the structure) from.