Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

How to Study Mathematics (2017)

math.uh.edu

Post reply on HN
Print view View original

61–70 of 76 posts

Re: How to Study Mathematics (2017)

#61

Post by Karrot_Kream »

dwohnitmok wrote: post #59
stevegalla wrote: post #46

Earlier quoted context omitted.

In all seriousness, I stopped going to lectures* and studied at my own pace. My learning isn’t linear, so there will be some topics that are easy that you can move quickly on. Other topics take more time. Sometimes you can take a detour and dive deeper kn interesting topics. Going to lecture, sitting in lecture, waiting around, etc. provided very little information per unit time for me. I can’t learn math or anything…

I found lectures to be valuable only if I had reviewed the material beforehand. This was mainly because otherwise I just wouldn't be able to keep up in lecture. If I had reviewed the material beforehand lectures were often extremely valuable for gaining new intuitions about a subject at hand that could be gleaned by an instructor's choice of explanation. And being able to ask questions in real-time was also quite val…

I wish I had realized this while I was still a student. By the time I was in grad school, it was too late. Now I make sure that I at least skim the material a bit before the lecture so I can see the other perspective, but also ask the professor/teacher any questions I had from my own pre-reading.

Re: How to Study Mathematics (2017)

#62

Post by Karrot_Kream »

ACosmicDust wrote: post #17
ZephyrBlu wrote: post #14

Although the title is "How to Study Mathematics", I think a more accurate title would be "How to Study Mathematics as a Mathematician ". I am studying some maths right now with the goal of understanding some statistical methods. Having a rock solid understanding of all the underlying maths is counter-productive to my end goal though (Applying the statistical methods), because it would be extremely time consuming. If…

I am sort of facing the same problem. On one hand I like math, abstract algebra, calculus etc but on the other hand I know I am not great with it. I am slow, I am not creative. So, when trying to learn something for programming like differential geometry, when I go deeper than I need to, I feel like I am wasting my time. And sometimes it is particularly hard to read a book written for mathematicians because my knowle…

I highly suggest you push through it. Mathematical knowledge ties in together very well. It's like snowboarding. Once you get over the initial hump, it'll come fairly easily to you (what's often called "mathematical maturity"). Then you'll be able to easily learn about statistics and ML if you want to experiment, or scene rendering, or differential geometry, or audio processing, and on, and on. Computers were originally created to be calculating machines, and they remain great ways to blend math with instructions.

Re: How to Study Mathematics (2017)

#63

Post by bollu »

Karrot_Kream wrote: post #60
bollu wrote: post #5

Also, use computers! I use SAGE extensively to get a feeling for lots of mathematical objects! 1. Grobner bases: http://bollu.github.io/computing-equivalent-gate-sets-using-... 2. Localization: https://github.com/bollu/bollu.github.io/blob/8cd335687ff3ef... 3. More broadly, an answer on math.stackexchange on how to debug math: https://math.stackexchange.com/questions/1769475/how-to-debu... 4. (WIP) continued fraction…

Does SAGE work well for analysis as well, or is it mostly for algebras/objects?

Good question. I mostly use it for algebra. It has varying levels of support for analysis. There is a SageManifolds project that allows for computations over differentiable and Riemannian manifolds, for example.

Re: How to Study Mathematics (2017)

#64

Post by MaximumYComb »

hansvm wrote: post #49
baby wrote: post #26

Earlier quoted context omitted.

When you spend most of your time in class, when do you have the time to do the assigned work? That’s what I struggled with.

If you have other significant demands on your time then school is going to be hard no matter what strategy you choose and might just take a lot of years to finish (please vote for UBI to help change that status quo). Assuming you can commit to school full-time though, it's less of an issue: - You can finish a typical bachelor's degree in 5y by devoting 48h/week to school (12h for classes, 36h for assigned work). That…

I'm nearly completion of a Bach Comp Sci that I've spent ~20 hours per week over 6 years (by the time I finish). I've done this while working full time as a single father who studies part time.

The time aspect is brutal. I'm ready to have regular hobbies. I'm ready to have a serious girlfriend. I'm ready to have regular social events. 6 years is a long time to just stop having a fulfilling life.

EDIT: I'm 1 grade in 1 unit off a perfect GPA. I'm at the point where I'm willing to have my GPA drop in order to free up some time to actually not be consumed by uni for the remainder of the time.

Re: How to Study Mathematics (2017)

#65

Post by hnglt »

hansvm wrote: post #49
baby wrote: post #26

Earlier quoted context omitted.

When you spend most of your time in class, when do you have the time to do the assigned work? That’s what I struggled with.

If you have other significant demands on your time then school is going to be hard no matter what strategy you choose and might just take a lot of years to finish (please vote for UBI to help change that status quo). Assuming you can commit to school full-time though, it's less of an issue: - You can finish a typical bachelor's degree in 5y by devoting 48h/week to school (12h for classes, 36h for assigned work). That…

> It's quite a bit more efficient to take more courses at once, especially if they're closely related (e.g., topology, abstract algebra, real analysis, ...).

I agree that the interplay between related topics helps me form a more robust of understanding of the material.

On the other hand, it might be worth considering proactive and retroactive interference, (the difficulty of storing similar, long-term memories). The layman's takeaway is that it's generally better to learn a variety of non-related topics concurrently instead of similar ones in order to facilitate better long-term recall.

Re: How to Study Mathematics (2017)

#66

Post by TheTrotters »

qsort wrote: post #9
erichahn wrote: post #4

Bad rules. My suggestions ask yourself the following all the time 1) Do I ~really~ understand what the definition / theorem is supposed to tell me. 2) A proof is just a reason why something is true in maths. Do you understand ~why~ a statement is true? 3) Exercise all the time. You won't learn math by memorizing definitions, theorems and proofs.

How is this disagreeing with the OP? It specifically says that memorizing e.g. theorems is bad.

Rote memorization is of course not enough but you need to remember relevant theorems. It's necessary but not sufficient for deeper understanding.

What would it even mean to "understand" a theorem but to be unable to state it (and prove it)?

Re: How to Study Mathematics (2017)

#67

Post by Synaesthesia »

vagrantJin wrote: post #57
cambalache wrote: post #51

I am going to be honest, these kind of articles are useless. 1. Make sure you understand the concepts 2. Mark what you dont know clearly 3. Write down what you know 4. Use 3 to find out 2. Oh gee, thanks! This is a more realistic approach. 1. Have a reasonable amount of mathematical talent 2. Study materials appropriate to your current level and mathematical maturity. 3. Work hard at it. 4. Ask for help when you are…

What does mathematical talent mean? I was bang average at high school algebra(computation) but untouchable at geometry/trig. At uni linear algebra was a breeze but Calculus II was a nightmare. What does mathematical talent mean?

I believe q certain amount of mathematical talent may exist but also that maths is accessible to every person of normal intelligence.

Re: How to Study Mathematics (2017)

#68

Post by mtzet »

I think this misses a pretty important part of mathematics, which I find is nessecary to grokking fields like linear algebra, algebraic geometry, algebraic topology, and many more.

What's missing is /why/ the definition is the way it is. What is it trying to encapsulate? A lot of mathematics is built up from simpler ideas being generalized until finally a purely algebraic definition is reached. The algebraic definition is completely void of context and any intuition, but it is very powerful to work with. The definition is in a sense the /result/, not the starting point!

For example, a beginning student of algebraic topology might start reading Hatcher's Algebraic Topology. Here simplicial homology is defined in terms of abstract functions d with certain properties. How did they arrive at this definition?

One answer is to start with complex analysis. Here you can notice that the complex plane and the plane without zero can be told apart by the function 1/z. This function also doesn't have an anti-derivative. Thus you begin to see how to define topology of the complex plane. Now extend this line of reasoning to the calculus of differential forms and you end up with the de Rahm Cohomology. Finally, you can realize that you can get the same results without having any interpretation of your functions. Thus begins the purely algebraic theory. The proofs may be different, but you can now /guess/ the theorems.

Of course in the above I omitted that differentials forms are defined in terms of a wedge product, which is also an abstract algebraic definition. This also has an explanation...

Modern mathematics is rife with explanations like the above, but they are often hidden away. It's very remniscient of the simple vs easy discussion of programming. The algebraic definition may be "easy" but it carries a lot of baggage.

Re: How to Study Mathematics (2017)

#69

Post by johbjo »

Similar to immersion when learning a language, you need to to do the exercises from the ground up until the proofs seem self-evident. If there are uncertainties about the exact meaning of notation and operations, you will probably fool yourself into believing that you understand.

Re: How to Study Mathematics (2017)

#70

Post by Syzygies »

bollu wrote: post #5

Also, use computers! I use SAGE extensively to get a feeling for lots of mathematical objects! 1. Grobner bases: http://bollu.github.io/computing-equivalent-gate-sets-using-... 2. Localization: https://github.com/bollu/bollu.github.io/blob/8cd335687ff3ef... 3. More broadly, an answer on math.stackexchange on how to debug math: https://math.stackexchange.com/questions/1769475/how-to-debu... 4. (WIP) continued fraction…

Sage is a remarkable project. For Grobner bases it relies on the Singular computer algebra system. I cringe every time I'm at a talk where someone credits Sage when they should have cited Singular.

It's hard work writing a system like Singular, and not an obvious path to glory, so this stings.

Post reply on HN