[tps5]

People always say that math students don't learn about applications of math.

This was never the case for me. When I learned trig ratios, I always understood some basic things that trig ratios could be used for. The teacher always introduced some applications, we always had a lot of word problems, and I could fill in the gaps myself.

Same for calculus. When I learned calculus, I always understood some things that calculus could be used for.

So I understood how those things could be applied to general, everyday sorts of problems. What was missing, though, was that I had nothing to which I could apply those techniques, besides homework.

Learning math (and reading STEM papers) has become easier for me since I now have actual problems to solve. Don't get me wrong: I'm not solving particularly challenging problems or using particularly advanced math. Nothing that tens of thousands of people haven't done before me. But I do need to understand the problems, solutions, and some of the context in order to successfully implement them. This provides a motivation that was always missing before.

I suspect this general narrative is true for a lot of people: that having an actual problem to solve is almost necessary to get a student to really learn the material, instead of just coasting along for a grade.

[jacobolus]

High school trigonometry and introductory differential & integral calculus are not the kind of books being described in this discussion.

The example in the original post is books about group theory (or the group theory sections of abstract algebra books more generally). I can attest that this subject is very rarely described in textbooks with clear examples shown before definitions and theorems; usually the presentation is entirely abstract, following a pure definition–theorem–proof kind of structure. But many other areas of pure mathematics at the undergraduate level and above are presented in a similar fashion.

(I recommend Nathan Carter’s book Visual Group Theory for a lovely counter-example to the prevailing trend, which starts with the concrete, and is very accessible. http://web.bentley.edu/empl/c/ncarter/vgt/)

[ianai]

We used to "run through" books like that. Their reasoning was to prove a theorem you only need the definitions/axioms. They really wanted us to be able to grasp the truths of a logical system from just its theorems and definitions. It was horrifically difficult. (Not all professors taught that way there.)

I feel that a lot of blame lies at modern academias curriculums. They feel every student needs to graduate in X number of years with a pretty long list of courses. It leaves little time for students who need or want more time with topics.

[douche]

A lot of trig and geometry never really clicked for me until I had to use them in shop class. For instance, planning out the dimensions and cuts that you need to make in a rafter to get the desired roof pitch for a shed of certain dimensions. Or laying out the stringers for a set of stairs.