Does order matter? To many of us, it does. The speed of learning something can be drastically affected by the order by which you learn them, and since we all have limited time, the time we spend learning is quite relevant to our concerns.

This post is merely a collection of thoughts (I hope some people will find them useful as a framework for developing their own hypotheses). I'd just like to hear your responses to my thoughts.

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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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learning how it’s done vs why it’s done: what’s the optimal order of learning these things

in grade school, you’re taught how it’s done. why it’s gone come later

in college, opposite happens. but sometimes it’s a lot more confusing that way. and requires absorption of more details and multistep processes

what is optimal? it obviously depends from person to person. it’s more “natural” to learn how it’s done first. but it’s only “natural” for phenomena that are discovered through hypothesis->observation or derivation. However, it’s not “natural” for phenomena that are discovered through serendipity, in which one learns the result/how to get it before one learns why the result is the way it is. and in some cases, like quantum mechanics, one may never learn why it is the way it is. of course, it feels more “natural” and “satisfying” to learn how it’s done, and promotes habits that are helpful to further discovery, but learning the process AFTER learning the result is ALSO curiosity-satisfying, and does not necessarily lead to the sense of “helplessness” that could allegedly come after learning the result first time after time. That “helplessness” could come, but if one has internalized both approaches, then it is far from inevitable, and then learning the result before the explanation can be faster and more efficient.

But in the end, it depends on person and context. Sometimes I feel more stifled when I learn result before explanation; sometimes I feel more stifled when I learn explanation before result. It is much easier to trick oneself into thinking that one has learned the material if one has only learned the result (without learning the explanation); it is also easier to forget the material if one has only learned the result (but learning the explanation along with the result shouldn’t take too much more time); and learning only the result is also less challenging (so familiarity with the process carries better “signalling” value and makes one more absorbed into the process so that one internalizes it better ). But again, once one has learned BOTH the process and the result, then the signalling value/internalization value is irrelevant. The only point of relevance is when one has learned one but not the other (which can happen, especially when people are lazy, slow, or time-constrained), or when one has partially learned one and learned another more (although this is very common). So perhaps in an environment where one has partially learned one and learned another more, then learning the process first may be more optimal, especially when people forget easily and quickly. But when one learns things completely, then the order should not matter much (or the order should depend on how much time more time one spends doing it one way vs how much time one spends doing it the opposite way; or on how rewarding the two orders are relative to each other)