How to Study Mathematics (2017)
math.uh.edu
math.uh.edu
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 fractions to compute pi: https://bollu.github.io/fractions/index.html
And so forth. I find the computational aspects of most theories to be very rich, and it's really gratifying to code something up and "read off" the results.
It's hard work writing a system like Singular, and not an obvious path to glory, so this stings.
Especially early on I often dismissed problems and assignments that I felt I wouldn‘t be able to tackle in terms of time or intellect.
This wasn‘t what the successful students did. They took on every problem with determination and focus. No matter how hopeless. Along the way of failing they quickly developed their skills whereas I was falling more and more behind.
I then applied this lesson on my second attempt at university. A CS degree. Again I struggled at the beginning, but I became competent much quicker and in the end the degree was a breeze.
Still wish I‘d known this from the beginning...
Most teachers know better than most students what is need to do to succeed in the class.
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 quantitative from watching someone do it. By using lecture time as study time I was able to double the time spent learning the material.
When you get stuck, go to office hours. Or look for notes from similar courses for a different perspective.
At the start of the course look at the book and see what the prerequisite material is. Review that material during the first week when there is time.
* exceptions being courses that have a participation grade.
I tried my best to have profs that followed a textbook.
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 valuable.
- You can finish a typical bachelor's degree in 5y by devoting 48h/week to school (12h for classes, 36h for assigned work). That's more time than most teachers expect, and except for a few unlucky points in time where multiple large projects coexist you'll rarely have to actually spend that long studying.
- It's hard to express just how much of a difference it makes to start with a little easier content to ensure you have the requisite background; don't let your pride hold you back. E.g., if the university places you in math class X, start at class X-1. You'll spend less total time learning throughout your education, be far less stressed, get better grades, and understand the material better.
- If you just want the degree, a lot of the first 1-2yrs of college courses can be replaced with CLEP/AP/... tests. Anyone can take them (not just in high school), and it'll take a lot fewer hours and dollars to get a passing score on one of those than it will to get a good grade in an equivalent college course (at most universities such tests won't affect your GPA, so even a score of 40% on some of them still gets you a passing grade).
- 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, ...). If you have closer to 70-100h/w to spend in class and studying then consider finishing your degree in 2-3y instead of 4-5y. You'll still have plenty of time to do fun things over the summer and winter holidays, and I know I personally found it more motivating to have an end date in the near future.
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.
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.
[0] I stress academic here because at least in my case I was working a lot during undergrad. 40+hrs/week while in JC and 30+ when I transferred to a uni. But succeeding in those areas aren't the same as academic work ethic and I do want to acknowledge people that just ran out of energy.
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.
What would it even mean to "understand" a theorem but to be unable to state it (and prove it)?
I’m really interested but all the material I can find is either for kids (which just isn’t sufficiently stimulating for an adult) or aimed at college kids with a decent background in math that is fresh
[0]: not even calculus just what they called technical math which is like all practical example based curriculum. One of my life regrets here to be honest
I'm not very far in so I can't exactly recommend it, but I am enjoying it. In places where I'm rusty, I'm falling back on Khan Academy.
[1]: https://www.alibris.com/Discrete-Mathematics-with-Applicatio...
The No Bullshit Guide to Math & Physics [1] is a condensed review of high school math, followed by mechanics (PHYS 101) and calculus (CALC I and II). It's not as rigorous as other more proof-oriented textbooks, but it still covers all the material.
The No Bullshit Guide to Linear Algebra [2] is all about linear algebra and also includes three chapters on applications, so you'll learn the fundamental ideas but also what they are used for IRL.
Both books come with exercises and problem sets with answers, which is essential for learning. In fact one could say all math learning happens when you try to solve problems on your own, not just reading.
[1] https://minireference.com/static/excerpts/noBSmathphys_v5_pr... [2] https://minireference.com/static/excerpts/noBSLA_v2_preview....
See the reviews on amazon for what people say.
I’ll buy your book now I’m a man of means and ready to learn properly.
I recommend getting the print version because it's easy to read (and flip back and forth with page references). We have a free-eBook-copy-when-you-buy-print policy—just get in touch with me by email and I'll send you the PDF with matching page numbers.
You may want to pick a book on writing proofs to familiarize yourself with the concepts first, such as "How to prove it" (either the one by Velleman or Polya). Another good one for getting to some intuitions might be "How to solve it" by Polya.
Then, you might pick up any elementary book on Real Analysis, Linear Algebra, as well as Graph theory/some Algorithms. Most of these should be self-contained. These 3 areas should lay a very firm mathematical foundation, and other parts of mathematics will become a lot more accessible with them.
Just be mindful that you'll probably need a long time going through these books, and that's normal. If you gloss over things, you'll quickly miss important bits. It's not like other books where you can kinda grok things out of context if you just continue reading, at least for me. I wouldn't do more than max 2h per day, but be consistent if you want to see progress.
> When you finish you should know why each step follows from what came before. You may not see how anyone could have thought to do the proof that way, but you should be able to see that it is correct.
Knowing that something is true is not the same as knowing why it is true.
I don't know what the problem is with understanding, but here's three thoughts:
1. You need a deeper level understanding - you can't understand a problem at the same level you encountered it. Perhaps, understanding different but related areas, so you see the same problem from a different perspective. Perhaps understanding the formal system that is used to define terms used in the problem description.
2. You need familiarity, which creates the feeling of "intuition". If you know how it behaves in all situations, you will feel you understand it, even if you don't. So, just lots of practice/exercises.
3. You need to fully understand the components from which the problem is formed. For example, the natural numbers and addition, and build up from there.
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 you want to learn maths for the sake of understanding maths, then this could be the right approach. But it's definitely not a pragmatic approach.
If you find pure math interesting then give yourself the slack to learn it long term. Think of it like a rock garden which you tend to over the years.
Up to high school algebra level I can talk a little about but ymmv. Everyone is different.
The number Devil is an enjoyable story. Khan academy has a good problem bank with gamification of progress etc. Mathantics is a good substitute for school teaching of conventional stuff on video and has good worksheets. Mathific app is another source of practise and gamified progress.
Interested to hear other ideas and for other ages.
Now - no big problem if you worked with very keen math students, and they'd show you that, or if your TA could give some pointers. But you could easily get completely stuck, if you were using these texts for, say, self-study.
All of this to say, is it's really necessary to have guidance with it, and to make sure you actually do problems, not just convince yourself that you know the material.
like leetcode and other problem assesors, you learn a lot by trying, failing, and getting an explanation of where it went wrong.
Would be pretty useful to have an online platform for worked textbook problems from arbitrary textbooks.
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 stuck
There.
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?
A rough translation from https://www.pesmm.org.mx/Serie%20Cursos_archivos/smm2018Curs...
Either you believe there is no such thing as mathematical talent and you and me are in the same caliber of Grothendieck or you have to accept there are people with inborn qualities to excel way above the average human in that area. Same for Kasparov in chess, Mozart in Music, Jordan in BB. Blank-slatism will be the death of western societies.
- Experienced programmers are better trained at formality and precision than mathematicians, in some respects, and are able to ask questions that make experienced mathematicians go "why are you asking this question?" or "just get used to the idea (because that's what everyone does)"
- Much of higher math study advice (such as the one posted here) is aimed at laymen. And experienced programmers are no laymen.
- Mathematicians are laymen in many aspects compared to experienced programmers. An experienced programmer will have an easier time learning and using a proof assistant. Mathematicians (most of them) run away as fast as they can the moment they hear the phrase 'proof assistant.'