| I think the parent comment + yours (and others off parent) provides a perfect encapsulation of one of the dimensions of teaching / learning: what's often referred to as "style"*. One way to summarize, specifically, might be something like "inductive" vs. "deductive". As my experience has ... accumulated ... through the decades, I've come to feel that these sorts of differences / preferences likely don't have much impact on ultimate (potential) "level"**. And, I think you see this and related notions of "what mathematics 'actually is'" echoed (in a very fractal-like way, +1 to the universe in achieving a consistency we'll never rival) across the development of individual mathematicians as well as through the history of mathematics [1-6]. These distinctions are important in "pedagogy" - can be very helpful for teachers and students to be aware of and work at, especially at the more "basic" levels. This can make a massive difference in how an individual's arc unfolds - with extremes of "F this subject" vs. "I'm willing to accept low pay in exchange for torturing myself with this material for the rest of my life!" But, aside from trying to be mindful of the differences - and all involved, ideally, trying to USE awareness of knowledge and "EQ" and all of that in making the mutual learning enterprise work for everyone involved, many other aspects of the differences can just be outlets for time-wasting if focused on IMO (/ experience). * AFAIK, not really my field though and it has been ~15 years since I did any significant reading / study in the area - for the sake of 'full disclosure' ** The effects end up more in details of notes, problems and areas people are drawn to more or less, etc. [1] https://terrytao.wordpress.com/career-advice/theres-more-to-... [2] Polya's "How to Solve It", in particular, I think of (from the intro): "The title of the very short second part is 'How to Solve It.' It is written in dialogue; a somewhat idealized teacher answers short questions of a somewhat idealized student.") - many options for accessing / buying, but, for this text, it's in the (unfortunately images) here - https://math.hawaii.edu/home/pdf/putnam/PolyaHowToSolveIt.pd... [3] https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&d... [4] https://www.maa.org/sites/default/files/pdf/upload_library/2... [5] https://en.wikipedia.org/wiki/Galois_theory#A_non-solvable_q... [6] https://en.wikipedia.org/wiki/Hilbert%27s_program ... and, so many more, of course... |