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Comment by jamesy0ung | Hacker News Reader

As someone who just finished school, I’m trying to figure out how to get genuinely interested in mathematics. I’ve never been particularly strong at it, yet I’m planning to enter a university program that demands a high level of math. The problem is, it’s hard to motivate myself to study math for its own sake. For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.

How do you develop a lasting interest in math when it doesn’t feel immediately useful?

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Make it practical! Graphics programming involves linear algebra. Databases involve relational algebra. Machine learning involves requires calculus. You’ll naturally encounter hands-on tasks with tangible goals that involve learning new math.

One of my undergrad degrees is in math. As you study it, you learn to identify your assumptions (axioms), find or build interesting abstractions, prove properties about them (theorems), and then map all sorts of other things into those abstractions by figuring out that they're really the same thing. It's even more interesting when you start to find things that are different or question things you always took for granted.

Math gives you the ability to leverage the very structure and relationships of pure abstraction. It's quite the super power.

None of the specific things you learn studying math will be nearly as useful as the ability to think mathematically.

N=1 datapoint here. I studied physics in university and before I started I was not aware that physics is basically just math where the results sometimes relate to reality. The pure math courses I took were the most difficult and in the beginning I loathed them, because it felt so unattainable to get any intuition, let alone real proper comprehension for all the concepts they threw at us. For a long time I felt like I was just hanging on by threads and especially if I compared myself to those who had some innate interest in math or generally some really good intuition on the abstract concepts (or even prior knowledge) it was really demotivating. But I also felt like I had no choice but to continue and as time went on the I grew fond of it. And the feeling of being overwhelmed changed - that is to say I still was completely lost every time a new topic was breached and I could not understand even half of the proofs in class - but I did not feel so defeated about it. And I grew to like the feeling of actually completing the work sheets they gave us every week. The process of solving them was often excruciating but if you did the sense of accomplishment is real. I think for most people higher math is really difficult and that is part of why it is interesting. Another aspect I had to accept over time is that even though you can state a mathematical fact or conjecture in just a hand full of symbols or a plain sentence it does not mean that truly understand it, its implications or how you got there can be understood the same way that other prose can be. Sometimes you have to stare at, contemplate and scribble around one equation for days until you understand whats up.

If there was any advice I would give, then it's probably similar advice on how to stop procrastinating on anything that is difficult. Establish a routine first - find a spot that you will only use for studying this (like a spot in a library), start small, divide and conquer, accept that you will not understand most things easily, reward yourself for the small wins along the way, find an accountability partner or someone to study with if that's your thing, make a regular schedule with regular times where this is what you do - consistency is key, even if its just for 5 minutes, stack it onto other habits, see yourself as a scholar of math - it is what you do, lean into the discomfort, as enduring that is a valuable skill in itself.

Don't study it for usefulness, study it for beauty. Look for amazing insights.

Yes, you need some practical math as well. I did engineering, there's a lot of inelegant stuff there.

But that stuff actually tends to be right next to some very interesting things.

Here are three things you can find out.

First, there's more than one kind of infinity. You can't make a map from natural numbers like 1, 2, 3 etc to real numbers like e, 0.632268, sqrt(2) etc. Look for Cantor diagonalization.

Second, a random walk like a heads vs tails comes back to zero almost certainly. It also does so in two dimensions, like walking randomly in Manhattan. In three dimensions, it does not, and so for higher dimensions. Look for Polya.

Third. There is a way for you and me to communicate secretly, despite everyone in HN being able to see our entire exchange. Look for Diffie Helmann.

These days, there's a whole industry of people doing math videos with interesting stuff.

Find math that interests you!

I didn't particularly find (at the time) calculus, multivariable calculus, physics, etc. interesting as I didn't find the applications interesting at the time. I find these subjects representative of what you traditionally learn at school.

When I entered uni I discovered my passion for discrete math, algebra (groups, rings, fields, etc.), number theory, cryptography, theory of computation, etc. as they have a lot of application in CS.

That's really what did it for me - and also I had great uni lecturers. I wish they would have taught the subjects I like in highschool - the difficulty level is about the same.

It's easier to appreciate math when you are disinterested in the results or applications, because the nature of academic topics near the core grouping of math/philosophy/empiricism is that they are discovered with a lot of meandering at first, and then sometime down the line they become repurposed into a direct application that can be learned by rote. School tends to instruct in some of the most directly applicable stuff first - the "three R"s" plus some civics and training aligned with national goals. And that means that school predominantly teaches associations between math and rote methods, to the disgruntlement of many mathematicians. The "meandering" part is left to self-selected professionals, so it doesn't get explored to much depth.

So I think a good motive for math study is really in games and puzzles, where the questions posed aren't about win/lose or right/wrong, but about exploring the scenario further and clarifying the constraints or finding an interesting new framing. Martin Gardner wrote a long-running column and a few books in this vein which are still highly regarded decades later.

> For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.

Consider doing something that actually needs it. You like computer programming - consider making a game engine. It might be easier to learn when you can actually see that it is useful.

Keep in mind though that math is a lot of things. People obsess over calculus but that is just one type. Math is just as much the different types of symmetry in wall paper patterns as it is finding the derrivative. Don't be afraid to try different areas. If you dont know where to start, consider picking up "A Concise Introduction to Pure Mathematics" by liebeck which introduces a bunch of different math concepts and see if any feel more interesting to you.

I'm going to share my anecdote, because it may help, but everyone is different and your mileage may vary.

I'm a MechE by classical training (professionally I actually work doing software/network stuff, don't ask, DNS (screams internally)), so here's where it stood out for me:

https://en.wikipedia.org/wiki/Hydraulic_analogy

Internalize what this simple example represents, think about why that's mathematically interesting, and start looking for where it applies elsewhere. You too could be roped into doing systems engineering at scales you didn't think people haven't already figured out.

University is not a good place for learning mathematics as most of your math instructors there will be very good at math and very bad at teaching people that are not already very good at math.

Sorry, no. Universities are great places to learn math. You’re misrepresenting the genuine passion for teaching that many university instructors have.

Except maybe not calculus. I remember my calc class kind of being terrible because it was a weed out for other majors. Every math class that wasnt required though was great.

Sorry, no.

For whatever reason, many University programs use high level math classes as a filter to weed out 1st year students from that program. If university instructors had a genuine passion, and ability, for teaching high level math then they wouldn't accept that as an outcome.

That is unfortunately true; not only in the US, but all around the world. The particulars do depend on the instructor, and many if not most instructors try to be motivational, but the syllabus is perfectly clear: "this is a weed out class". And when it comes to test time, the syllabus wins.

The only thing I disagree with in your comment is about the instructors: they want to be employed, and they have to accept the syllabus and testing standards. It is not about passion and ability to teach (most, especially younger ones, are full of those); it is about meeting the departmental requirements.

Sorry no.

You are misrepresenting what's happening. Other departments use beginning math classes as a way of weeding out students they feel won't succeed in their fields because they can't pass basic mathematics classes. Most math departments would absolutely love to have more students in them.

The problem is that these students aren't prepared properly by K-12 mathematics courses and math builds upon itself. If you don't have a good grasp of algebra, you just won't succeed at calculus. We're sticking people in the equivalent of Spanish 4 without having learned Spanish 1 properly.

If you love programming, there's quite a lot of programming where math is vital. Graphics, optimisation problems, cryptography, neural networks, figuring out if a hash works, projecting if an algorithm will scale...

The tricky bit is often that you need to learn some of the math before you can see how it's useful, but if you need stronger motivation, you might try diving into a slightly math heavy programming problem and learn the math as you go

Probability/Statistics is a good excuse to learn mathematics, because paying a little attention one finds lots of day-to-day situations where is possible to apply it. For example, see the secretary problem[1].

[1] https://en.m.wikipedia.org/wiki/Secretary_problem

I wish I couldve excluded everything past basic algebra and hopped right to statistics at a young age - I *loved* everything about the practicality of it, how it explained tangible relationships and illuminated the world. Algebra and calculus were so un-engaging I had those teachers calling me everything but a stupid child.

Except you can't really understand statistics without calculus.

*statistical theory notice I had a heavy emphasis on my love of practicality…which theory, to me, is not.

Perhaps think of them as solving logical puzzles. It's fun. Even though not always related to everyday tasks.

For me, it began many years ago when reading about Hilbert's hotel paradox. Turns out our laymen's understanding about infinity isn't as really refined.

I write mobile apps for living and indeed these stuffs are irrelevant for my work.

I feel like a lot of platitudes are being said here.

If anyone had a guaranteed way to make people enjoy math, we'd already be applying that method.

Just read ahead to figure out what you'll need to learn, and do some advance reading. Anything thag make the courses easier will tend to make them more fun.

Haigh's Mathematics in Everyday Life [0] provides modern examples.

[0] https://plus.maths.org/content/john-haigh

as someone who loved maths first but then do programming for a living, it's because solving puzzles is fun. I get the same dopamine hit whether it's a math problem, a coding task, or a video game level. but I think forcing yourself to like something is not the correct approach; you either like it naturally or you tolerate it for some other goals

https://betterexplained.com might prove helpul?

Get a good teacher. They make it fun, or interesting.

Have you ever watched a video of a highly skilled tetris player? Where they fill the screen most of the way to the top and then suddenly they just combo the whole thing down and everything wraps up cleanly, and then they start fresh.

The feeling of "oh yeah, that was nice watching that mess turn into something clean and squared away" is where I get a lot of my joy from math.

But also, there are uses to math that you might be able to play with through every day, but you've never thought of those scenarios in a mathematical way.

I was walking today, and on the street there is a right angle turn. The inner portion of the turn is just a square right angle, but the outside of the turn is a radius. I started wondering to myself, if I want to be on the outside of the turn going into and exiting the turn, what would be different ways I could walk this, and what would the distance differences be.

Crossing directly across, to the inner corner and crossing directly across to the outer side again, would be 2w (for the width of the road w). Following the edge of the radius would (assuming perfectly circular), be 1/4 of a circle, so 1/42piw = 1/2 pi * w. The shortest route is a straight line, which would make a right triangle, so w^2 + w^2 = c^2, 2w^2 = c^2, sqrt(2) w = c

So crossing twice is 2w, following the edge is 1/2piw, and shortest path is sqrt(2)*w. Not super applicable, and I didn't need to do math to figure it out, but I was walking and bored, so I found joy in it. The fact that they all boil down to having w as a factor means I could figure out a nice ratio between all of them. And then I needed to mentally figure out what 1/2 pi was. 3.14/2 = 1.57... And I know that sqrt(2) is roughly 1.41 ish.

So now I know that crossing twice has a cost of 2, following the edge is 1.57, and direct line is 1.41. Following the edge is vaguely close enough to the ideal path to warrant not walking into the street to optimize the route, 1.57 / 1.41 is about ~110%. Whereas by defintion, a cost of 2 is going to be sqrt(2) times sqrt(2), so ~141% more than shortest path.

A few things to note here. First off, I'm aware that not everyone finds the same joy in doing simple mental math and thinking about problems mathematically even when there is no need to do it, but trying to think of things more minor trivial things mathematically may cause you to at least appreciate it more, which can grow into joy. And second, I wasn't doing any complicated math in my head. I just thought to myself "is it faster to cut to the inside corner and then cut back out... of course not, right?" and I was able to answer that definitively to myself. Did it matter? Was the answer probably obvious anyway? Probably, but I was able to _prove_ that. And I value facts. Finding joy in the simple things lets you build up more of a familiarity and view it more as a problem solving tool than a tedious thing to rote memorize.

A great way to build up math familiarity and see how other people find joy in mathematics would be to watch Numberphile videos on YouTube[0]. It's a bunch of mathematicians sharing things they find interesting about math. Some times are REAL hard to grasp, but some are just very interesting puzzles[1]. The puzzles don't always have clear immediate usefulness, but can often be described as "a mathematician wanted to know an answer, so they did some math to find out and prove something to themself."

Sorry, end of spiel.

tl;dr - find the joy in the simple things and use math as a tool to answer (even simple) questions to help highlight the usefulness.

0: https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A 1: https://youtu.be/ONdgXYEBihA

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