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> There are places where math is hard... This is the realisation I had (20 years late). Especially beyond high school, maths is hard. You have to put the effor

• bonoboTP 7y ago
  > I think its expressiveness and conciseness makes it more dispiriting to study: one page of a maths textbook can contain so much information, and it feels like you're just stupid when it takes an hour to understand that page fully. This is very important. Apparently some people get frustrated by this because they are used to studying lower-information-density textbooks. Maybe there would be value in teaching these meta-skills, like how to approach learning each subject. People may apply techniques, like rote learning or brute-force cramming, reading the same page for hours on end, expecting to magically grok it, where these are ineffective of even counterproductive. Not sure how well this is teachable though.
  • K0SM0S 7y ago
    My personal experience says two things: - illustration — whether drawings, gestures, animations, using objects, whatever works, but have a visual equivalent to 'translate' notation; the typical example would be fractions which are as simple as it gets with drawings and hard as hell using only numbers, for most people discovering them. The same is true for most common functions and transformations (which, when you reduce it to the maximum, are the entire body of applied math). - real-world application (typically the best candidates are simple physics or statistics in high school). The point here is to give a point of reference, a "feeling" of the mechanism or logic or paradigm. The deeper problem is that this application must feel 'interesting', like a puzzle, and not everyone is interested by the same things. That's where, imho, narrow AI could help craft customized courses with ad hoc examples that fit people's "world view" and "approach" to problems, and show animations for every variable, equation, let people play with those and see the results. Both probably fit in the "faster feedback" approach, i.e. have the teacher/teaching material give feedback as fast and as often as possible, to guide the learning mind on the right path. This is an extremely important discovery of UX in the last 50 years and I believe education has much to learn from these very applied insights into human cognition.
    • chongli 7y ago
      illustration — whether drawings, gestures, animations, using objects, whatever works, but have a visual equivalent to 'translate' notation This approach breaks down when you go into higher math. You end up working with these abstract objects that have no visual representation. You may be able to find an equivalent object from geometry or graph theory, but it may be so complicated you could never draw it. Heck, even basic shapes like the Platonic solids are difficult to visualize, let alone draw, correctly. I solved a problem recently requiring me to count the number of vertex colourings of an icosahedron, up to rotational symmetry. Trying to draw one of these things and visualize all of the possible rotations was basically impossible for me. I ended up loading Blender and playing around with one in 3D, using the coordinate planes to see all the axes of symmetry. If I had been given a higher dimensional object like a tesseract then all bets would have been off. I would have had to find another approach entirely, likely algebraic.
      • bonoboTP 7y ago
        > This approach breaks down when you go into higher math. It doesn't have to be literally visualization. Just some "feeling" for the objects. Some conception of them as actors, actees, things interacting, or having emotions or whatever. Again, whatever works. Maybe some people can work with concepts just based on strictly memorizing the definitions and drilling through proofs, but others need to conceptualize it all into a narrative. I don't mean dumbing it down, but making it easier to "grab" mentally. Hard to describe.