Math's pedagogical curse – Grant Sanderson [video] (2023)

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• Hacker News
• (2023)
  by traes
• unpopular opinion: imho there is no substitute for deep learning and doing problems. Math is unlike other subjects, where you have to actually do it head on; you cannot just passively consume it like listening to a history podcast or watching a video.
  by paulpauper
• I recently came to know of the following which others might also find useful;

  1) Math Made Visual: Creating Images for Understanding Mathematics by Claudi Alsina and Roger Nelsen - https://bookstore.ams.org/CLRM/28

  2) Roger Nelsen has also written other similar works like the Proofs without Words(3-vols), Nuggets of Number Theory: A Visual Approach, Cameos For Calculus: Visualization In The First-year Course, etc. - https://bookstore.ams.org/browse?Author=%22Roger%20B.%20Nels...

  by rramadass
• Writing "3Blue1Brown" would be more recognizable than "Grant Sanderson" :)
  by Razengan
• ”The roots of education are bitter, but the fruit is sweet.” — Aristotle
  by estetlinus
• > After scratching the itch of knowing that something's been written completely rigorously, it's easy to call a work done when maybe we shouldn't call it done yet. And there's contexts where this is maybe OK, research comes to mind...

  It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.

  It's a great video though. He gives a little checklist of ways to check if your work is clear:

  - Do definitions have motivating examples?

  - Do proofs [or other results] feel rediscoverable?

  - Is there personality?

  - Are core ideas illustrated with diagrams?

  - Is relative importance highlighted?

  ...

  He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him

  by qnleigh
• Mathematicians lie.

  The three lies (by omission) are:

  1. This solution is how I solved it. If you look at problem solving as A* search then the solution is not what was done to solve it. Learning mathematicians are supposed to learn to search trough the space of mathematics by examples of solutions. Each step presented on the blackboard is a step in the right direction, omitting steps in the wrong direction, and backtracking in the initial efforts to solve (or even define) a problem.

  2. This idea is abstract and has nothing to do with reality. Mathematics is a distillation process of abstraction. In this purification process all preconceptions of reality are filtered out, so the result is rigorous and universal. This often leads to a denial of any connection to reality. (A well known counterexamples is Conway's Analysis of go games, leading to the surreal numbers.)

  3. This is the only and right way to do it. You are presented with a single solutions, definitions or axioms, omitting alternative definitions. This get better in higher education (think of how many ways there are to define numbers).

  This is not what it looks like doing math. It leads to frustration in students who learn (wrongly) that math is about doing everything right the first time, intuitive thinking is wrong, and basically only about what you are not allowed to do (as the feedback they get are highlights of mistakes in exams).

  Our task should be cleaning that up and reteach exploration, reteach that math - like any universal language - can describe everything however you want, finding solutions is trial and error, intuition is good and fast but fuzzy and can mislead you, to not stop with a solution but explore around it more for better understanding, and while we have good cause to define things as we do, it's not necessarily the only way to do things (looking at you, axiom of choice).

  To use an analogy: Otherwise we stick to teaching orientation to scouts by fastest satnav routes.

  by pxagntuvzt