# How to stay interested in less-tangible math

I've graduated high school and I am joining college soon. The problem with me is that I'm not finding less tangible math interesting at all.

Some people find abstract math to be very beautiful, and I'm exactly the polar opposite. I am turned off by seemingly pointless abstract mathematical structures with no use whatsoever. For example, some people find group theory to be very interesting after learning the definition of a nilpotent group; I couldn't care less as I never got why would someone make those weird definitions in the first place.

I have found that I will find a topic to be (very) interesting iff that has mathematical "real-life"/concrete/tangible/physics related applications (e.g., proving isoperimetric inequality using Fourier Analysis or Prime number theorem with complex analysis). But to be able to appreciate how that theory relates to the more concrete applications (e.g., understanding the proof of PNT with complex analysis), I need to navigate through the material I find "boring", which turns me off since if an exposition of the topic doesn't routinely provide examples of such concrete applications, I get bored very quickly.

For example, currently I'm now working my way through Stein and Shakarchi, Complex Analysis and Stein and Shakarchi, Fourier Analysis. What really intrigued me at first is that you can prove really cool number theoretic result with complex + fourier analysis and motivate the Olympiad coloring proofs (see this link for the details of what I mean by this) with discrete fourier analysis. So when I started reading I was very interested and breezed through the first few chapters (Chapters 1,2,3 in both books). But now I'm on the section "The action of Fourier transform on F" and I find the chapter to be too much boring so I am now feeling disinterested.

So, math educators: What are some tricks/strategies to keep students like me motivated with the material even when it feels less tangible to me?

PS: It's not that I have problem with understanding the material. In the chapters I read in SS, I didn't face any difficulty with any of the exercises/unstarred problems (though I did face lots of difficulty with starred problems.)

• I'm voting to close this question as off-topic because this is not a question about mathematics education. – Steven Gubkin Jul 15 '19 at 10:58
• This may simply be a case needing to gain a bit of mathematical maturity and mathematical breadth (different things), both of which very few high school students will have much of. Indeed, the fact that you're looking at nilpotent groups and complex analysis suggests that you might have leapt forward quite a bit in the last two or three years with your personal studies, and what you're feeling could be a by-product of this. To me, "motivate the Olympiad coloring proofs with discrete fourier analysis" is putting the cart before the horse, which seems to support my first sentence. – Dave L Renfro Jul 15 '19 at 13:01
• I love that you think of the PNT or the isoperimetric inequality as "tangible results". I hope that I have not misread your post. We need more students like you in this world. – James S. Cook Jul 15 '19 at 14:51
• It's not obvious to me at all that this is off-topic. The self-learning tag doesn't automatically make a question off-topic. – Chris Cunningham Jul 15 '19 at 21:49
• @Namaste I never intended to imply that I am the sole arbiter of whether something is on-topic. However, OP did make an effort to explain what they meant by "tangible" using multiple examples. Further, small edits to the question (I made them) easily make it more clearly on-topic. You are right that we can remove the attempt to poll users on whether they feel the same. Please feel free to make further edits. – Chris Cunningham Jul 15 '19 at 22:47

Your remarks about group theory ("I couldn't care less as I never got why would someone make those weird definitions in the first place") suggest that you need to see the ultimate goal to appreciate the path to that goal.

(1) For nilpotent groups, a natural goal may be to understand which polynomial roots can be expressed in terms of radicals. Once you see that solvable groups are a key to this question, and that nilpotent groups are solvable, the pursuit makes more sense.

(2) Continuing with group theory, a natural goal is understanding the symmetric wallpaper patterns. Then studying the crystallographic groups, ultimately showing there are exactly $$17$$, is motivated.

(3) And just for fun, Group Theory in the Bedroom gives an interesting application of groups to turning a mattress to balance usage. In fact, this whole book is one example after another of interesting questions/goals that lead to substantive mathematics. The math is motivated by the questions in a way not always evident in textbooks. (I see this is along one of the lines suggested by @DaveLRenfro.)

Hayes, Brian. Group Theory in the Bedroom and other Mathematical Diversions. Hill and Wang, 2008. AMS review (PDF).

(expansion of a comment made a week ago)

When you get to college I recommend actively discussing math topics with undergraduate peers and graduate students (assuming a university with a graduate math program). This will help insert you into the mathematics community --- undergraduates will tell you about the Putnam Exam and which professors to approach (and which professors to actively avoid) and the like, while graduate students are useful for giving you an idea of what graduate school is like and advice on applying to graduate school and the hurdles for beginning research in math.

Also, to maintain interest, it helps to read books that could serve as inspiration, such as "What is Mathematics?" by Courant/Robbins, "A Course of Pure Mathematics" by Hardy, the 3-volume "Feynman Lectures on Physics" by Feynman (yes, it's physics, but despite Feynman's sometimes disdain for aspects of pure mathematics, I find his writing to be very VERY strong with what Terrance Tao calls the post-rigorous stage), "Visual Complex Analysis" by Needham, "A Concrete Approach to Abstract Algebra" by Sawyer, "Adventures of a Mathematician" by Ulam, etc.

Spend some time browsing university library shelves to get an idea of what exists in mathematics (besides what random googling provides) and to help regain interest in mathematics when you're too study-tired and jaded to do any formal work (working problems, studying texts, etc.). Look through back issues of expository and undergraduate level journals you find on library shelves for things that might catch your interest and which you otherwise would not have encountered. Here's a possibly dated list of such journals that I posted back in 2005. Probably the 3 MAA journals and The Mathematical Gazette are what I'd most strongly recommend browsing. Also, keep in mind that unlike the case with many other academic fields, math articles from 70 or 80 years ago can be just as useful as those written in the last 5 or 10 years.