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There are two fundamental ways of learning

(a) general to specific

(b) specific to general (e.g. case studies)

Scientific hypotheses (general) are motivated by experiments (specific). With data, one can hypothesize a trend and see the general hypotheses

One can also try this process mathematically, as specific results can motivate a hypothesis of the general structure, which can then be proved.

Which way is faster? It depends on person. It might be plausible that learning styles are “bunk” and that smarter students are more efficient through learning of type (a), but it’s also quite plausible that this is not true (for one thing, learning is dependent on both intelligence and motivation/interest, and the motivation/interest component can make type b learning more efficient even for geniuses). I, for one, learn best through the “specific to general” method. As such, I believe that I learn math best when it’s motivated by physical phenomena (in other words, learning math “along the way of doing science”) than when pursuing math first and then learning science (which is what I did, which didn’t work as well as I hoped, especially since it killed my motivation). As I’m quite familiar with the climate trends of specific localities, I also learn the generalities of climate best through case studies.

And then after learning the applications of this math field, one is more motivated to learn the specifics of the math behind the math, and one even has more physical intuition through this learning route. It actually means something when one learns through the second route.

It is also true, however, that route (b) can be taken too far, as is evident in the “discovery-based” math curricula, which generally produce poor results. When one is self-motivated, route (b) can be especially rewarding, but the selection of case studies is important, as an improper selection of case studies can result in a very minute exploration of the general structure (it is also true that very few textbooks are written in a way as to make route (b) most exciting to learn about). Generally textbooks present their material as ends, not as means to an end (except in the crappy discovery-based math textbooks). However, one can most certainly learn calculus through physics (especially div/grad/curl), and linear algebra through its applications, and a very smart (or lucky) person can design such a curriculum that would work for many people (it is much easier to design such curriculums for oneself than it is for a wide variety of personalities).

Nonetheless, route (b) is often stultifying. In fact, I sometimes feel impatient and feel like I’d rather learn the math first. A person’s temperament may vary from time to time, and find type a rewarding at some types, and type b rewarding at other times.

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