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How to Study Mathematics (2017)
- jdlyga 6y agoI would love to get back and really learn math. I was really interested back in High School with Geometry and Algebra. But my interested totally burned out with poorly taught and punishing Calculus classes in College.
- SkyMarshal 6y ago“Calculus Made Easy” might help you get back into it. Lots of discussion over the years on it: https://hn.algolia.com/?q=calculus+made+easy https://hn.algolia.com/?q=calculus+made+easy
- ipnon 6y agoIt's a long journey with many paths. The best you can do is set aside time everyday. I do an hour and it adds up quickly.
- dvfjsdhgfv 6y agoThat's a shame, Calculus is a fascinating subject and a basis for many interesting applications. Basically, there are two ways to approach it: to pass your exams (and there are just a couple of rules to learn, it's not that complicated), and to really understand what it's about. If you choose the second approach, you don't even need to memorize any formulas, because you will be able to reconstruct all of them as you need them. Moreover, you won't be conceptually limited to the geometric interpretation (which is invaluavle in giving some intuition in the early phases but might get in your way later).
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- no_wizard 6y agoWhat would anyone recommend to someone who really only had high school math[0] to get up to speed on enough math to handle more advanced computer science concepts? 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
- flir 6y agoI'm working my way through this: https://pimbook.org/ https://pimbook.org/ 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.
- dvfjsdhgfv 6y agoIn theory, any introduction to discrete mathematics would do. In practice, there are many differences in depth and breath of the material. You probably will do well if you choose Rosen, but there are several great alternatives, such as Levin[0] that you can start right away with. http://discrete.openmathbooks.org/dmoi3.html http://discrete.openmathbooks.org/dmoi3.html
- mindcrime 6y agoI highly recommend the Susanna Epp book[1][2] for Discrete Math. I found her book to be much better written, and more understandable, than Rosen. [1]: https://www.alibris.com/Discrete-Mathematics-with-Applications-Susanne-Epp/book/1748023?matches=4 https://www.alibris.com/Discrete-Mathematics-with-Applicatio... [2]: https://www.youtube.com/watch?v=FPr5-X9nZc4 https://www.youtube.com/watch?v=FPr5-X9nZc4
- ivan_ah 6y agoI have two books that might be a good fit for you since they are specially written for adult learners in mind. (disclaimer: I wrote these books and I have a financial interest in promoting them) 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_preview.pdf https://minireference.com/static/excerpts/noBSmathphys_v5_pr... [2] https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf https://minireference.com/static/excerpts/noBSLA_v2_preview.... See the reviews on amazon for what people say.
- benrbray 6y agoFor abstract algebra self-study, I highly recommend "A Book of Abstract Algebra" by Charles Pinter. Each chapter starts with a few definitions and examples, and the rest of the chapter is a series of exercises meant to help you discover algebraic concepts for yourself.
- dorchadas 6y agoThis is what I started with, then I moved on to Dummit and Foote. It really did help as a build up.
- erichahn 6y agoBad 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.
- qsort 6y agoHow is this disagreeing with the OP? It specifically says that memorizing e.g. theorems is bad.
- raegis 6y agoIf you're in grad school taking qualifying exams, memorizing theorems is required. There's just too much stuff to digest. Undergrads and younger students have ample time to learn mathematics with serious depth. This is not true for the average math grad student. The comfort you had in learning mathematics as a youngling just instantly disappears when you get to grad school. This is painful, but you have to learn to accept that you won't be competent at all the mathematics you are exposed to. Some very fine mathematicians experience this.
- TheTrotters 6y agoRote 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)?
- bollu 6y agoAlso, 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-grobner-bases.html http://bollu.github.io/computing-equivalent-gate-sets-using-... 2. Localization: https://github.com/bollu/bollu.github.io/blob/8cd335687ff3ef6ab9c62c7ebf17bdfa1a092f71/localization.md 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-debug-math/1769492#1769492 https://math.stackexchange.com/questions/1769475/how-to-debu... 4. (WIP) continued fractions to compute pi: https://bollu.github.io/fractions/index.html 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.
- spekcular 6y agoYou might enjoy Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by David Cox.
- bollu 6y agoThanks! I'm studying this right now, after I got a taste of algebraic geometry (schemes) from Ravi Vakil's fantastic "algebraic geometry during the time of COVID" last year!
- Karrot_Kream 6y agoDoes SAGE work well for analysis as well, or is it mostly for algebras/objects?
- bollu 6y agoGood 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.
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- yoaviram 6y agoSlight tangent - any suggestions on none conventional ways to teach children math?
- harry8 6y agoDepends on the kid(s), age, progress so far etc i would think. 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.
- DavidSJ 6y agoMath from Three to Seven — The Story of a Mathematical Circle for Preschoolers: https://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf https://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf
- kinghtown 6y agoWhen my daughter is a bit older I’m going to buy her the Singapore school books for young math and pay her a salary to complete them. I don’t doubt their effectiveness.
- ZephyrBlu 6y agoAlthough 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 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.
- ACosmicDust 6y agoI 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 knowledge of it is not as "continuous" as theirs.
- kinghtown 6y agoWell, a mathematics degree is one of the highest income earners for a reason so its usefulness is beyond dispute. I personally have life and artistic goals which take priority of my time over math but I’m willing to allow mathematics (together with high-level scientific understanding) a place in my life as a tertiary goal or drive. We are all a little too hard on ourselves to get everything done before we turn 30 or 40 as though it were some kind of deadline and then why bother, right? 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.
- the_only_law 6y agoI want to go back to school, and if I do, it will probably end up doing CS because it’s about the only “useful” degree accessible to working professionals, but if I had a choice, I’d probably pick a mathematics degree given how widely applicable the learned concepts can be.
- hyperpallium2 6y ago> How to make sense of a proof > 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.
- danck 6y agoI dropped out of my maths degree after struggling for six years (!) at university - I had fallen too far behind. I can't just blame my part-time jobs for that. 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...
- alisonkisk 6y agoDoing the assigned work is good way to learn the material. Most teachers know better than most students what is need to do to succeed in the class.
- baby 6y agoWhen 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.
- ectopod 6y agoHow much class did you have? My first year at university had three hours of tutorials and ten hours of optional lectures per week. Subsequent years had less. There was plenty of time.
- rfrey 6y agoMy engineering degree had six classes per term at 3 hours of class time per course. Two of those would be labs with a 3h lab each week. So 24 hours of in person time per week, with an expectation of 2-3 hours or outside work for every in class hour.
- galaxyLogic 6y agoI think that lot of math curriculum does not really make clear to students the rationale for why they cover the topics and proofs in the order they do. The students lose the big picture of the path they are on, why is it important for me to be working on this right now, so that I will be able to tackle the next subject when that comes along. The structure of theorems and proofs and how they all build upon one another is not presented clearly. The point of a proof is not to prove something, but what further proofs this proof makes possible. NO?
- TrackerFF 6y agoMy only real critique of academic books, was that the exercises varied A LOT in difficulty. Some books had exercises that seemed to assume that the reader had been a honors student, and thus able to pre-process/setting up the problem with intricate identities and what not, before coming to the part where you apply some theorem. 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.
- dorchadas 6y agoI've also found it way too easy to just convince myself I know the material and move on without actually doing any problems when I was self-studying. Thankfully I came across a Discord server that is run by a math PhD where he assigns problems and corrects proof for self-learners. Best thing that has ever happened to me in learning maths, and I have since finished several undergraduate textbooks and am working on upper undergraduate/graduate ones currently, with problems assigned from him and guidance/proof critiques. 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.
- gtpedrosa 6y agoHey, this Discord server seems to be a valuable resource. Do you mind sharing where to find it? I would love to try to get a deeper dive into some math subjects sometime in near future.
- tunesmith 6y agoAre there any resources for taking a self-assessment to know what to learn next as a self-studier?
- dorchadas 6y agoIn math? You could follow a standard undergraduate curriculum, such as the one used at MIT (with their OCW stuff). Or you can just pursue what topics interest you, and trace back their prereqs to make sure you've covered everything you need to know.
- codeproject 6y agoThere is one trick to master mathematics: Proactive. you can't be lazy. You have to ask question relentlessly. Always ask question, always find an example.
- fizixer 6y agoAs an experienced programmer, I have spent enough time trying to understand and make sense of some aspects of pure/higher math, specifically logic, that I feel qualified to say this: - 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.'
- cambalache 6y agoI 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 stuck There.
- vagrantJin 6y agoWhat 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?
- Synaesthesia 6y agoI believe q certain amount of mathematical talent may exist but also that maths is accessible to every person of normal intelligence.
- cambalache 6y ago" Grothendieck's mathematical education changed radically by his arrival, in 1949, to the Cartan seminar hosted at the *Ecole Normale Superieure". As Grothendieck himself confessed(reference), he had not heard by then about topological spaces, groups, rings,modules, homology!!. The astounding capacity of Grothendieck is revealed in the giant mathematical leap achieved between his naive ignorance in 1949(in Paris) and his spectacular technical nous in 1953(in Nancy). At that year he finished his doctoral thesis about "nuclear spaces" and had become according to Schwartz in "the number one worldwide specialist" in the theory topological vector spaces.According to Dieudonne, Grothendieck was author by then of a life work "only comparable to Banach's" A rough translation from https://www.pesmm.org.mx/Serie%20Cursos_archivos/smm2018Cursos5Zal.pdf 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.
- mtzet 6y agoI 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.
- johbjo 6y agoSimilar 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.
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