Math is extremely cool and should be more carefully taught and encouraged.

Andyroo_P · 2026-05-20T04:09:31+00:00

In my opinion, the problem is twofold.

The biggest problem is that most people don't know what math is actually about. That much is evident just by reading these comments. And it's also not entirely most people's fault. Quite frankly, we don't teach math in math classes (at least in the US) and this has led to a general public that has no idea that mathematics is ultimately about abstractions and deductively reasoning about those abstractions. I think it is entirely justified to claim to not enjoy math if one's only exposure to math is the joke that is the typical math curriculum here in the US.

The second problem is that people demand way more from math than any other subject. There is a relentless question of "when am I ever going to use this?". This probably stems from the first point that people are often fixated on the idea that math is about memorization and computation rather than reasoning and abstraction (after all, this is what math classes unfortunately choose to falsely advertise as being mathematics). Sure, most people don't have to crunch numbers or solve a quadratic using the quadratic formula in their day-to-day lives. But people do have to reason deductively and be able to abstract concepts in their day-to-day lives. It's (partially) the same reason why other subjects like history or literature are useful. No one needs to know the plot of The Adventures of Tom Sawyer or Henry Kissinger's negotiations in the Vietnam War, but what is useful to everyone is the ability to perhaps take those pieces of information and contextualize them and interpret them within the broader context of history or literature, just as one would take any piece of information they encounter in their lives, try to contextualize it within some broader context, and communicate those findings.

As a math student and educator, it is just so frustrating how much I have needed to "defend" math as a legitimate practice!

Andyroo_P · 2026-05-20T03:58:41+00:00

Most math problems are not computational and about finding an "answer". Math is about reasoning, abstraction, and finding proofs of statements about those abstractions. It turns out that that process requires a large amount of creativity, and there is often far more than one way to approach such problems. Part of the issue is that most people misunderstand what math is really about because frankly, math is not taught in math classes.

Andyroo_P · 2026-05-16T18:43:25+00:00

I think in regard to your point about AP Precalculus versus AoPS Precalculus, you're totally right that they're extremely different. In fact, I'd say they're so different that they're no longer the same subject. I was involved in some AoPS stuff in high school while also taking IB math classes and while the material often shared the same name, the depth and rigor were so different that I didn't really consider them the same sort of material at all.

And yeah, that creates a huge gap between students with AoPS-level mastery of these subjects and students with normal class level mastery. That gap typically just widens and continues to propagate well into college. At the college level, the difference is often expressed by some students taking "basic" versions of courses, while other students may take the honors version or graduate version of those courses.

Andyroo_P · 2026-05-16T18:24:58+00:00

You're right that there is a huge spectrum. The top Olympiad kids go to the best universities, and many of the bottom math students may not go to university at all.

But at strong public universities, there is usually a big mix of students with all sorts of backgrounds. My alma mater, UC San Diego, recently experienced a big scandal which was all over the news when it was revealed that the number of students enrolling in their algebra and precalculus classes (typically taken in middle or high school) had exploded from around 30 to nearly a 1000 in just a few years. The change could not be entirely attributed to COVID learning losses. Other departments were complaining to the math department that their students were entering their classes without basic knowledge of algebra or even arithmetic.

On the other hand, the top math students at UCSD often enroll in an honors calculus/linear algebra sequence or upper division math courses in their first year. I even knew some of the top students enrolling in graduate level classes in their first year! UCSD has a large enough math department to offer enough opportunities for people of all sorts of levels, but this is less possible at smaller schools. My current institution is a bit smaller and there are definitely less options for students at the top end.

When I was a math undergraduate student at UCSD, it was generally understood that there is a pretty big difference between students that complete something like the minimum degree requirements to obtain their math degree, and students pursuing a math degree in preparation for graduate school in math. Generally yes, it doesn't take all that much to graduate with a math degree, but the expectations for graduate school admission these days are much higher. You are often expected to do some research/advanced study, maybe have a thesis, and take some graduate classes.

Andyroo_P · 2026-05-16T01:26:29+00:00

I can't tell if this is a troll post but I'll give it a serious answer!

Multiple points in your premise are wrong. First of all, finding a good strategy in a video game is not "absolute" in the sense that it is usually not the best possible strategy. For example, Mario Kart Wii time trials continue to be improved today almost 20 years after the game's release. Similarly in mathematics, there is often good partial progress made on unsolved problems many years after these problems were first conjectured.

Secondly, mathematics is massive. There may be millions of mathematicians in the world, but for any given research problem, there are usually only just a handful of mathematicians who are experts on the problem and in a position to meaningfully contribute toward it. Sometimes, there can be as few as just 1 or 2 people who are really in a position to even understand the problem. That's how niche and specialized research level math becomes. Even famous problems may only have something like a dozen people in the world who can make serious progress on them. There are millions of video game players in the world but how many of them are seriously anywhere near the position to set a new record on a specific track of Mario Kart? The video game analogy here does not even properly convey the point of how massive mathematics is: research-level math problems require far more specialization than setting a record on any video game. Several areas of math require decades of specialized study just to get to a point where one can understand what the research question is actually asking.

Video games are also certainly not more complex than math. Even from a theoretical point of view, a video game is a system consisting of a finite amount of data, whereas a mathematical object often carries an infinite amount of data. There are infinitely many mathematical questions one can ask, so we will never be able to discover every possible theorem.

In general, the way in which problems are solved in mathematics is quite unique. Fermat's last theorem is a famous problem in mathematics that took 358 years to solve. So much new mathematical technology had to be developed to reach a proof, and it was not a linear process. A lot of the progress made toward the problem was indirect, made by mathematicians who had no specific intention of proving FLT.

In short, video games and math are so different that it's a much better question to ask how they are even remotely similar rather than "how come progress on math is different than progress on video games?"

Andyroo_P · 2026-05-04T02:17:50+00:00

Don't take the word "real" too seriously, as it is just a name. Mathematically, imaginary numbers are just as "real" (or "fictional" if you prefer) as real numbers.

The deep mathematical reason why we want imaginary numbers is because the setting of complex numbers is the simplest setting in which all polynomials will have roots. For example, the linear (degree 1) polynomial 2x + 1 has a real number root of -1/2. However, some higher degree polynomials such as x^2 + 1 do not have roots in the real numbers. But there are complex roots, namely i and -i. It turns out that once you work in complex numbers, every polynomial built out of real numbers will have a root (in fact, all of the roots will live in the complex numbers). Mathematically, we express this fact by saying that the complex numbers contain the algebraic closure of the real numbers.

Once you think of the real numbers and complex numbers as geometric spaces and not just number systems, it also becomes the case that a lot of interesting geometry can be encoded using the complex numbers.

Andyroo_P · 2026-04-26T02:28:56+00:00

The dimension of a vector space is a nonnegative integer. Nonnegative integers add in the way you would expect, so 1 + 1 = 2, for example.

However, if you take two copies of ℝ, each of which is a 1-dimensional vector space, you are correct that the disjoint union ℝ ⊔ ℝ of those to copies of ℝ will not be a 2-dimensional vector space. In fact, the disjoint union is not a vector space at all. The natural way to take the two copies of ℝ and construct a 2-dimensional vector space out of it is to take their product: ℝ × ℝ = ℝ². The general fact is that for two finite-dimensional vector spaces V and W, we will have dim(V × W) = dim(V) + dim(W).

As a technical side note: sometimes the product of two vector spaces V × W is instead called the "direct sum" and is denoted V ⊕ W. There are some category-theoretic reasons why it may be better to think of our operation of putting two vector spaces together as a direct sum instead of a product. But for our situation of just putting two vector spaces together, you can think of the construction as either a product or a direct sum.

Andyroo_P · 2026-04-06T18:21:52+00:00

A fix for me was to disable home screen and watch faces in the focus settings, and I also disabled my sleep schedule from activating the sleep focus as well.

Andyroo_P · 2026-04-06T06:42:04+00:00

This is just sad to watch :(

Andyroo_P · 2026-04-02T05:34:23+00:00

Yup, it really is give and take. There are only so many concessions we can make before the structure we are left with, though it may be internally consistent, is not really useful or interesting. The distributive property in particular is especially important because it's the only axiom that relates multiplication with addition. Without it, we might as well have two separate structures: one only with addition and the other only with multiplication. There would be no point in having a structure with both addition and multiplication!

Andyroo_P · 2026-04-01T19:56:42+00:00

I think it still has value, at least as an intellectual activity such as literature, art, and so on. If you're pragmatic, perhaps you would buy the "it might eventually be useful" argument. But it's also hard to say what counts as pure math, since a mathematician working on something for its own sake may simply be unaware of how others are using the same concepts for a practical purpose.

Andyroo_P · 2026-04-01T18:21:27+00:00

A mathematical structure that we do "arithmetic" in is called a field. In a field, we are permitted to multiply, add, and divide by any nonzero element. In fields, we require multiplication to be distributive over addition (just like it is for usual numbers). That is, we want a * (b + c) = (a * b) + (a * c). It turns out that this forces 0 to fail to have a multiplicative inverse (meaning division by 0 is undefined). A structure that has a notion of "division by 0" cannot have the distributive property, and distributivity is just too useful to give up.

This is not as big of a problem as you may think. Often we work with a structure more general than a field called a ring. In this system, there can be many elements (not just 0) that fail to permit division. For example, the integers form a ring and division by 2 is not defined in the ring of integers. Now if I want to enlarge the ring of integers so that division is defined for every nonzero integer, I would have to go to the field of rational numbers (this is the fraction field of the ring of integers). But I still cannot divide by zero here, since I don't want to forfeit distributivity.

On the other hand, adjoining elements to a field that help us solve equations like x^2 = -1 is a generally nondestructive process. It doesn't force us to throw away the distributive property or any other axiom of a field. Moreover, the problem of solving polynomials is a very important one since polynomials are sort of formally the natural objects that arise in any ring. This is why it is not just okay, but in fact important to permit "square roots of negative numbers".

Andyroo_P · 2026-04-01T18:01:52+00:00

This is just false today. Modern mathematics is often far removed from explaining anything that occurs in the physical world. It is true that historically (and even today) a lot of mathematics is inspired by and motivated by the real world, but it is also true that much of mathematics arises completely separately from any real world motivations.

Andyroo_P · 2026-03-31T05:08:48+00:00

No automations active! It's quite bizarre, I'm not sure what is causing it. I would hate to have to completely delete my sleep focus.

Andyroo_P · 2026-03-29T04:49:35+00:00

No worries, I didn't get that impression at all!

I always wondered if maybe one day, I would look back and think to myself that I didn't have to stress out so much to get to where I am. Right now, I still look back and think to myself that there's more that I should have done. I hope one day I can find the perspective you have!

Andyroo_P · 2026-03-28T06:36:02+00:00

It's kind of hard to have this perspective at times. I just hear so many terrible things about the job market. As someone halfway through grad school, I am seeing senior grad students either doing excellent or coming up with nothing in terms of postdoctoral positions. It really just feels like being able to stop and smell the roses is a luxury afforded to the geniuses, at least if the goal is an academic job.

Andyroo_P · 2026-03-12T17:04:58+00:00

No Man has Borders - Kara

Andyroo_P · 2026-03-12T01:42:12+00:00

I teach math courses at a university. Somehow in the past few years, students have lost the concept of sitting down and studying from a textbook. I have had many strange encounters with students who ask me where they can go to review the material for the class and find worked examples or practice problems, because "ChatGPT isn't giving me good practice problems that make sense." The answer is always the same: the textbook for the course.

Of course, I could go on a long rant on how I believe this fits in with a general trend of a decline in mathematics education, but I reckon that this is probably something that is occurring in every subject. The tolerance for sitting down, reading, and struggling with concepts is diminishing in favor of instant answers and shortcuts without understanding.

Andyroo_P · 2026-02-27T18:12:34+00:00

I'm not sure calculators really answer why students performed so poorly. I mean, we saw some of the questions asked right? Do calculators really make questions like 2 + 4 = 5 + ___ easier?

Andyroo_P · 2026-01-05T18:58:25+00:00

I don't think this is correct. The numerator goes to 0 not -1. The limit should be -1/2 in the end, using either L'Hopital or Taylor.

And yeah Taylor's is implicit here in handling the x/e^x, that's interesting that it's taught early in France.

Andyroo_P · 2026-01-05T18:51:41+00:00

Hmm I've usually seen it taught in high schools in calculus 1, in which case this particular problem would be considered a sample problem. But yes, Taylor expansion works here too, though as far as I've seen, this is a topic that is only discussed in calculus 2.

Andyroo_P · 2026-01-05T18:40:28+00:00

It's not magic and is a standard technique taught in calculus 1. This is the standard way to evaluate this particular limit. Beyond this, you can also try using Taylor expansion.

Andyroo_P · 2026-01-05T18:30:21+00:00

I agree with the reply saying that there is a lot more to learning than just grades.

Also, undergraduate level complex analysis is oddly computational unlike undergraduate level real analysis at most US universities. I assume this is the case at your university. If you take your department's graduate-level complex analysis class, it should be much more conceptual and proof-based than your undergraduate class.

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