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How to Study Mathematics (2017)

661 points by kenny87 9y ago

kenny87 9y ago
One of the best explanations of the subject that I've seen.
deleted 9y ago
[deleted]
HackinGibsons 9y ago
Love it
miobrien 9y ago
Awesome. A related thread from two days ago for those who missed it: https://news.ycombinator.com/item?id=16372454 https://news.ycombinator.com/item?id=16372454
applecrazy 9y ago
This advice also applies to more advance mathematics at the high school level, such as AP calculus BC or even linear algebra. Source: have followed this technique indirectly for the past few months
gtani 9y ago
Some books on proof, theorems/axioms, set theory, epsilon/delta/continuity/limits/differentiability, natural/rationals/reals/countability etc before heading into your first proof based LA or analysis sequence: - Kevin Houston "How to Think Like a Mathematician" - Keith Devlin "Intro Mathematical Thinking" - "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and Math Degree titles)
- dvfjsdhgfv 9y ago
  My favorite is Burn Math Class: And Reinvent Mathematics for Yourself
  - goldenkey 9y ago
    Seems well liked but whats is up with the cover? Most people who are into mathematics aren't zealous rebels bent on burning the world. I feel like the marketing director for the book screwed up big time. I won't judge the book by its cover though. http://amzn.to/2CrgFDI http://amzn.to/2CrgFDI
    - gowld 9y ago
      Huh? where did you get "zealous rebels bent on burning the world" ?
    - dvfjsdhgfv 9y ago
      I actually don't remember the cover. The title is provocative but adequate. You could rephrase it as "Could you reinvent modern mathematics if it disappeared?" - but then probably less people would buy it.
- jacobolus 9y ago
  Even more useful as a first step in my opinion is to think about problem solving per se. E.g. see: Mason, Burton, & Stacey, Thinking Mathematically Pólya, How to Solve It; Mathematics and Plausible Reasoning (2 vols); Mathematical Discovery (2 vols) Schoenfeld, Mathematical Problem Solving Larson, Problem-Solving Through Problems
dboreham 9y ago
Something I would add : Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today? In my mathematical education I noticed that I had a more pleasant time and felt more motivated to learn the material when the teacher gave us this kind of background story. Since most teachers don't do so today, the student typically needs to get on the Internet to do their own research.
- jnbiche 9y ago
  > Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today? For some of us, this is incredibly useful advice re: math study habits. I also figured out that once I had the context and history, it was much easier for me to learn and apply.
  - jdironman 9y ago
    The 'Why' lends to the 'How'. I agree. I usually like to know why even in things that are not math before attempting to apply or understand them.
- mbag 9y ago
  This happened almost through all my formal education. The knowledge was just presented, as something that was obvious. There was very little discussion of motivation, or what problems was person/people trying to solve. I understand that there is so much knowledge to cover during high school and college, but IMHO, this causes students to develop mindset, that you either see solution instantly, or you are just not smart enough to crack the problem.
  - soVeryTired 9y ago
    I think Bourbaki are partly to blame for this state of affairs. Vladimir Arnold had a lot to say on the subject: he was a big fan of keeping the intuition and motivation for development in the subject clear. At times he went a bit overboard, but he makes some valid points in this lecture: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
Philipp__ 9y ago
I always strugle with memorizing part. I am never able to reproduce word for word what I’ve learned or read, but I can put it into context and show my understanding pretty well. Here, with math theory, there is no alternative. I failed exams so many times only because of that vocal reproduction of learned theory that required word for word knowledge. I am having math exam on integral theory in 2 days and here I am sitting and lookin at screen. :)
- jolmg 9y ago
  > with math theory, there is no alternative. You can do the exact same thing you mentioned with math. Theorems aren't arbitrary; there's a way they're derived. If you learn how the theorem comes to be, even if you can't remember exactly, you can often remember enough to come up with the theorem again. I think the way you learn is the ideal. Knowing the context behind the facts you learn gives them worth, and the redundancy of being able to derive the fact from the context also makes it better integrated with the rest of your knowledge and easier to remember in the long term.
  - Philipp__ 9y ago
    I totally agree with you, but I feel like with math theory there is always good chunk of knowledge that needs to be somehow inserted in your memory as is. That’s the most irritating part for me personally. If i find a way to overcome it or make it more accessible, I think I will enjoy math theory much much more. :)
    - Smaug123 9y ago
      I'm afraid it's just practice. Lots and lots of solving problems until the definitions, methods, and techniques are burned into your mind. Once you have the background knowledge, everything is a lot simpler. Learn whether each theorem has an easy proof or a hard proof. If it has an easy proof, you can forget it (deriving it when you need to; an example is the contraction mapping theorem [1]). If it has a hard proof, you can forget it and look it up when you need to. [1]: https://www.patrickstevens.co.uk/how-to-discover-the-contraction-mapping-theorem/ https://www.patrickstevens.co.uk/how-to-discover-the-contrac...
Myrmornis 9y ago
It should emphasize more that you have to read very slowly, and often re-read the same section.
cle 9y ago
Memorization is so underrated. Memorizing the fundamentals and having them available for instant recall is hugely valuable, especially when trying to grok a new concept. I generally buck the standard advice and memorize first, before trying to understand. Understanding is much easier for me if I can easily hold everything in my working memory.
- Philipp__ 9y ago
  I agree, but I still haven’t found a way to make that process more enjoyable or less fatiguing.
- newman8r 9y ago
  I think people tend to conflate "memorization" with "cramming". You're right that memorization is underrated. Are you familiar with the 'method of loci' (i.e. memory palace). It's an amazing technique.
- fooker 9y ago
  Proper mental indexing is even more so. You can not memorize everything, but you can certainly index almost everything. That is why we can have databases that do not fit in 'memory'. Do you see the analogy here?
  - goldenkey 9y ago
    Luckily we have brains that expose this interface quite readily with intuition. Learn the logic, and the partial indexes, the foreign keys, etc..they all follow. Reason before RAM. It is often much faster to derive data than to store it, and look it up. This applies to CPUs and to our brains. Our brains have instruction cache, and often instructions are better use of memory than their final executed output.
- joe_the_user 9y ago
  I wouldn't personally find advice like this useful for me. I suspect it varies with "personal style". I grasped mathematical concept easily from an early age and I never made an effort to memorize things. If anything, I saw those coming from less-advanced math founder on advanced math (arithmetic to algebra, algebra to calculus, calculus to advanced subjects) because they attempted to deal with the subject based on memorization rather than grasping the basic point. Generally, I did wind-up committing a lot of content to memory but it was and is much easier, even "effortless", when I knew the reason for each part of the content. Of course, language is slippery and we might be talking about exactly the same thing. So I wouldn't give blanket advise - each person has to find their own learning style (or styles, if it varies by discipline).
e0m 9y ago
By far the best explainer of mathematics I've seen anywhere is 3Blue1Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.
- legionof7 9y ago
  Also the book Burn Math Class.
  - haskellandchill 9y ago
    That book got me back into math. Using functional equations to teach concepts is very powerful. You get to say I want these properties and derive things like the exponential function.
- sophacles 9y ago
  In a similar vein, I've been enjoying PBS Infinite series. The original host recently stepped aside to finish her dissertation and the new hosts are doing great but still finding their footing a bit - I suggest looking at some of the older videos to get a good sense of the channel. https://www.youtube.com/channel/UCs4aHmggTfFrpkPcWSaBN9g https://www.youtube.com/channel/UCs4aHmggTfFrpkPcWSaBN9g
- nothis 9y ago
  It always saddens me how mathematicians seem to look down on “intuition”. Maybe higher math as a full-time job is just hard work and stubborn precision but for me, an intuitive, visual look is probably getting me closer to understanding than any cold-hard-facts book does. Not to mention how much easier it is to appreciate the beauty of it.
  - pflats 9y ago
    There's nothing wrong with intuition, but intuition does not write proofs. It can help you write proofs, but you need more than that. To pick an example: Intuition is great for understanding the intermediate value theorem. If the temperature was 40 degrees at 8 am, and is 60 degrees at noon, it must have been 50 degrees somewhere in between. Why, though? How can you guarantee that there will always have to be a time where it was 50 degrees[1]? [1] provided that temperature is real-valued and temperature change is continuous, the debate of which I'll leave to the science people
rublev 9y ago
I've always wanted to learn mathematics but have no idea where to start, I have no formal education in anything. I tried the Ivan Savov books and they seem great but I have no way to judge the quality of any resources. Where do I start as a complete beginner? I want books not online resources like Khan etc. I would eventually like to get into Geometry but I have no foundation whatsoever, just basic algebra.
- rmidthun 9y ago
  The web site https://betterexplained.com/ https://betterexplained.com/ has physical books that you can purchase "Math, Better Explained: Learn to Unlock Your Math Intuition". The explanations on that website are excellent, I imagine the book is the same.
- dboreham 9y ago
  You could try : What Is Mathematics? An Elementary Approach to Ideas and Methods https://www.amazon.com/dp/0195105192 https://www.amazon.com/dp/0195105192
- hollander 9y ago
  I find the online courses the most entertaining and stimulating. Many are available for free, like on Coursera, MIT.edu. Some offer real classes, where you follow live classes I believe. You cannot start when you want, you have to follow the program. On Udemy you can find nice courses as well for a good price. They have discounts all the time, each course for $10-20. Never pay the full price. See the reviews, and you can always take a risk for that price, plus they have a 30 day money back guarantee.
- nextos 9y ago
  Start with: * Algebra, Gelfand * Geometry, Kiselev
- mkl 9y ago
  Measurement, by Paul Lockhart, may be helpful to start with. It carefully and clearly derives mathematical principles from basic ideas, in a very approachable way. Here's how it describes its content: Part One: Size and Shape In which we begin our investigation of abstract geometrical figures. Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves. Part Two: Time and Space Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader.
dsacco 9y ago
This is good. I think two principal components should be emphasized when studying mathematics. 1. Proper preparation. There are textbooks at even the graduate level which have no formal prerequisites and which are largely self-contained. Technically speaking, someone with no prior background but a strong mathematical maturity could tackle these, but it might take them an inordinate amount of time to really grasp the material. For example, if you understand things like mathematical induction and proof by contradiction, you can learn analysis before you've been exposed to calculus, or category theory without abstract algebra. But it's far from ideal because you'll probably need to go over the same material several times and struggle with it. Furthermore, even the same subject within "advanced" mathematics can have wildly different depths of coverage depending on the author. Pinter's A Book of Abstract Algebra is probably approachable for anyone currently reading this comment, or even high school students. Dummit-Foote is a step beyond that, and appropriate for undergraduates who are already immersed in a math degree. But Lang or MacLane-Birkhoff would be significantly more challenging without first building up to them. Sometimes this is not just a question of depth, but also of pedagogical style. You can get a lot of satisfaction by learning analysis from Rudin for the first time, but it's really a rough go of it if you're not prepared for the terse definition->theorem->proof->remark->definition->theorem->proof->remark style of writing. On the other hand, Tao's Analysis I and Analysis II are much more approachable (similarly, some writers, like Halmos or Munkres, are praised for their exposition in introducing otherwise complex material). Ideally someone looking to study a subject should introspect about whether or not they are prepared for that subject overall. Once they've confirmed they are, they should read the first 10 pages of five or so well-recommended textbooks on the subject at their level, then choose to stick with the one that has the most approachable exposition style for them. 2. Proper study. When studying any given textbook (or videos, lectures, etc) it's really important to understand that mathematics is an active discipline. You cannot learn it by reading it. The process that has worked for me is the following: first, read through a chapter without taking any notes. Do so quickly, but not quite so quickly as skimming. When you come across things you don't know, compartmentalize them a bit and keep moving forward to the end of the chapter. The idea is to let the chapter's new material percolate a little before you begin actively tackling it. Next, start over at the beginning of the chapter and write down every single definition and theorem as you read. Before reading the author's proof of any given theorem, try to prove it yourself for at least 10 minutes. Then compare your work to the author's, and copy their proof meticulously in order to learn the method. Continue on to the end of the chapter. Finally, there will probably be anywhere between 5 - 20 exercises at the end of the chapter. Solve a meaningful fraction of these exercises, and don't look up the solutions to any of them until you've struggled with them for a good half hour or so (each). When you do look up the solutions, make sure you check multiple proofs for the same exercise so you can understand how the chapter's material can be applied in different, flexible ways. Mathematics has always exemplified a central belief of mine, which is that humans learn under conditions of optimal struggle. Even though it feels like being mired in hopeless complexity while you're struggling to complete a particularly difficult problem, you're actively learning the subject by doing it. But it's a question of efficiency. You want to aim for a subject and a presentation of that subject which is difficult enough to be just out of your current capabilities, but not so difficult that you can't follow its exposition.
commandlinefan 9y ago
One thing I figured out on my own that I wished I had realized sooner (which I'm trying - so far unsuccessfully - to impress on my 14-year-old son) is that, when reading math books, they follow a similar pattern. They describe a concept, show an example problem fully worked, and then discuss the ramifications of that concept, followed by another concept, followed by a fully worked problem, etc. I started making it a habit to try to work the example fully-worked problem by myself, based on the description that preceded it, before reading through the author's work. I was amazed how much better I was able to understand what he presented, and how much better I did working the exercises in the chapter afterwards.
skybrian 9y ago
One thing they don't talk about is how you decide what to study. Before spending a lot of time on a particular topic in math, you have to decide whether it's worthwhile studying it at all. It's apparently just assumed that you're taking a course so the decision is made for you.
- sincerely 9y ago
  I'm willing to bet that a course in post-secondary school is the context for the vast vast majority of people studying mathematics.
adamnemecek 9y ago
The Franklin method from yesterday still applies. https://news.ycombinator.com/item?id=16390046 https://news.ycombinator.com/item?id=16390046 Also a lot of books these days have wolfram mathematica code and it works surprisingly well even for some more abstract parts of math, To get a good intuition.
- aje403 9y ago
  The greatest book review of all time, sort of similar approach: https://www.amazon.com/review/R23MC2PCAJYHCB https://www.amazon.com/review/R23MC2PCAJYHCB Put the book down when you get to the example or proof of a theorem, keep all the theorems and examples before it in your head, and see if you can reason it out yourself. Can't promise that your head won't hurt afterwards
  - BadMathBook3 9y ago
    Those prices seem comical. Rudin passed nearly 8 years ago. Who is this benefiting? America is off its rocker. Google the title + PDF
iron0013 9y ago
Thanks for this, everyone should take the time to review this material
bluetwo 9y ago
I wonder if anyone here has a background in instructional design and what they think of this tutorial.
- hollander 9y ago
  Do you need a background in instructional design to have an opinion on this? ;-)
  - bluetwo 9y ago
    No. Not what I'm saying. I'm asking for an option from someone with particular experience. That doesn't invalidate the options of others, snowflake.
    - AnimalMuppet 9y ago
      Drop the "snowflake". Gratuitous personal insults don't belong on HN. You have a valid point without it. Leave it at the valid point, without the personal dig.
      - bluetwo 9y ago
        Anyone who claims to be offended by an honest question on behalf of some unnamed party is a snowflake, in my opinion.
        AnimalMuppet 9y ago
        Feel free to hold that opinion. But saying it is counterproductive. It leads at least some of your readers to dismiss you as a jerk, rather than hearing your message. [Edit: Oh, yeah. It's also against HN site guidelines. Saying such things repeatedly can get you banned.]
        bluetwo 9y ago
        When someone misinterprets a rather clear communication and claims to be offended by the result, I'm going to point that out. If that offends you, that's your problem. Or, look at it as a life-hack: Don't waste your time protecting the feelings of a non-existant group of people.
BadMathBook3 9y ago
I bailed on high school math, thinking I'm math dumb. In my late 20s I decided to try again, but jumped straight into calculus. And at first regretted that decision. However, I got lucky by stumbling upon this book: https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480 https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/... It "reads" like a book, with the ideas given context. I had an "ok" connection with Algebra, and the book explained the rest well enough for me. In school, the textbooks were loaded with symbols, but not enough description -- I guess they relied on bored teachers making minimum wage to do that part. I went to a school with poor academic showings (but connections to state superintendent of ed got them a grant for football facilities). Coincidentally, this book goes well with the technique described here: http://www.pathsensitive.com/2018/01/the-benjamin-franklin-method-of-reading.html http://www.pathsensitive.com/2018/01/the-benjamin-franklin-m...
sampo 9y ago
This is more "How to study pure mathematics", when the aim is to understand how the theory is build, so that you learn to contruibute to the theory by discovering and rigorously proving your own theorems. I don't think I've ever seen a "How to study applied mathematics". How do applied mathematicians, physicicst and engineers (who apply mathematics to real world problems) study mathematics, when they use it as a tool? How much or little emphasis do they give to proofs and theorems?
- dboreham 9y ago
  All mathematics gets applied sooner or later, no?
  - aaachilless 9y ago
    Maybe, but “applied mathematics” is conventionally more similar to “how to apply mathematics” than it is to “mathematics that is applied”. Studying how to apply mathematics is qualitatively different in many ways than studying pure mathematics.
    - dboreham 9y ago
      Ah. I genuinely did not know that. In Scotland we did not have Applied Mathematics (as a school exam subject) so I guess I just assumed it was about physics problems (there was an A-level with that name in England). It never came up again in my Mathematics career..
- yiransheng 9y ago
  Some personal thoughts: the key for me at least, is a bit like the hammer-nail mentality. An software engineer for example has a (small) mathematical toolkit (a hammer), sometimes forcing your problem at hand into the shape of nail is sufficient to solve it. A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d). Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount. An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :) [1] http://longnow.org/essays/richard-feynman-connection-machine/ http://longnow.org/essays/richard-feynman-connection-machine...
Hasz 9y ago
I cannot stress the importance of working problems yourself enough. There is literately no substitute.
lowtec 9y ago
Notation and syntax are areas that I have struggled with in higher level mathematics. Can anyone recommend a guide or resources useful to understanding things like converting set theory, sequence and series style problems into equations and solving them?
- tkzzbneig 9y ago
  Number theory's as good a place to start as any. You'll learn proof techniques and see a lot of notation in contexts that require varying amounts of background knowledge. It's feasible that an undergrad intro class would be up your alley if you're a software dev. https://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-spring-2012/lecture-notes/ https://ocw.mit.edu/courses/mathematics/18-781-theory-of-num...
- goldenkey 9y ago
  Bill Shillito's lecture series for Project Polymath is by a mile, the best introduction to all the notation necessary for higher level math. It requires absolutely no prerequisite knowledge. Introduction to Higher Mathematics: https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz... If you find he talks a bit slow, change your playback speed to 1.5x. Enjoy! :-)
nailer 9y ago
If you can't read the page: window.document.body.style['max-width'] = '650px'
alexbecker 9y ago
I left math after college for software engineering, but reading A Mathematicians Lament recently re-kindled my love for it. It is tragic how intuition, technique and mental models are left out of modern mathematics education and writing. It occurred to me that while I learned in college how to show, using Galois theory, that quintic equations are not generally solvable by radicals, I had no idea how Galois theory really relates to the process of finding roots. So I went back and derived the quadratic formula, the cubic formula, and sketched the quartic formula to see how the process used the ideas formalized by Galois theory and where it breaks down. I've tried to write the result up in a motivated and understandable way, instead of like a math textbook: https://alexcbecker.net/mathematics.html#the-quadratic-equation-and-beyond https://alexcbecker.net/mathematics.html#the-quadratic-equat...
bjourne 9y ago
What was the name of that 30-something woman who had never studied mathematics before, but then finished a degree in quantum physics in two years? Her name has been posted on HN at least two times before. I guess what she did is exactly how you should study mathematics if you want to become got at it.
- sebg 9y ago
  Think you're talking about Susan Flower and two of her articles: If Susan Can Learn Physics, So Can You - https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgling-physicist-archives-if-susan-can-learn-physics-so-can-you https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli... and So You Want To Learn Physics... - https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-physics https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-l...
  - badpun 9y ago
    Sadly (given her immense passion for physics), she ended up in software (doing SRE).
    - systemBuilder 9y ago
      Hey I am in SRE. At one point last year there were 7 ex professors sitting within 15 feet of my desk including me (and I did a PhD in theoretical computer science at the same school as this math professor whose article we are reading). I don't think it is an underrated career.
  - trendia 9y ago
    On the one hand, her account is very inspirational to people who might think that learning physics is out of reach or they're too old to do it. But on the other hand, her case is very out of the ordinary -- I think a lot of us took longer than 1.5 years to get through undergraduate physics. According to Susan's account, it took her 1.5 years to go from no college-level math to a full college-level physics curriculum. At the school I studied at, the Calc series (I, II, and III) takes 1.5 years. The intro physics (I and II) takes a full year, and the upper physics take at least a year (Mechanics I & II, E&M I&II, Relativity I&II, Quantum I&II, Lab I&II). (And you had to take intro physics before higher-level physics, and most courses had dependencies.) That's approximately 45 credit hours. If you took 18 credits a semester, that would take at least 2.5 years. And those are all tough courses -- no 3 credit gen ed "filler" courses. Even at 15 credits a semester, that's still 3 years. So, to get through all of those courses in 1.5 years would be like learning about 30 credit hours / semester. And add in the physics electives, like thermo, waves, statistical mechanics, quantum field theory, additional math courses, etc... that's a lot on top. I guess my point is that if Susan's account is discouraging, note that a lot of people take far more than 1.5 years to get through undergraduate physics.
perseusprime11 9y ago
Mathematics for the million is a great book if you are looking to reignite your passion for math.
backprojection 9y ago
Ha, Dr. Blecher's webpage on HN! That's great. I did my PhD in math, in large part because I enjoyed his class on advanced linear algebra so much, and later on real and functional analysis.
graycat 9y ago
I studied a lot of math, pure and applied, taught it, applied it, published research in it, etc. so developed some ideas relevant to the OP. For > To the mathematician this material, together with examples showing why the definitions chosen are the correct ones and how the theorems can be put to practical use, is the essence of mathematics. Is good, but more is needed. (1) Plan to go over the material more than once. The early passes are just to get a general idea what is going on. In such passes, for the proofs, they are usually the near the end of what to study and not the first. (2) When get to the proofs, for each proof and each of the hypotheses (givens, assumptions), try to see where the proof uses the hypothesis. Next, try to see what are the more important earlier theorems used in the proof. So, sure, in this way might begin to see some of how one result leads to or depends on another and have something of a web, acyclic directed graph, of results. And try to see what are the core, clever ideas used in the proof. (3) For still more if you have time, and likely you will not, can use the P. Halmos advice, roughly, "Consider changes in the hypotheses and conclusions that make the theorem false or still true." (4) But, note that to solve exercises or apply or extend the theory, need some ideas. So, where do such ideas come from? In my experience, heavily the ideas come from intuitive views of the subject. So, my best suggestion is to try to develop some intuitive ideas about the material. Definitely be willing to draw pictures, maybe on paper, maybe only in your head. In the end, a solution or proof does not depend on intuitive ideas, but finding a solutions or proof can make use of a lot in intuitive ideas. For research, most of the above applies, but IMHO there are more techniques needed.
HaoZeke 9y ago
So basically.. Mug up everything and hope you have exactly the same thought process as the course instructor. This is a disgrace. In any case. Definitions vary from book to book so mugging them up ad verbatim would hardly help.
Ldorigo 9y ago
This text should be required reading for ANY university-level math course. It made me go from "ugh, not math again" to "yay, I'm going to learn something new!" in a matter of days.

661 points by kenny87 9y ago

kenny87 9y ago

One of the best explanations of the subject that I've seen.

deleted 9y ago

[deleted]

HackinGibsons 9y ago

Love it

miobrien 9y ago

Awesome. A related thread from two days ago for those who missed it: https://news.ycombinator.com/item?id=16372454 https://news.ycombinator.com/item?id=16372454

applecrazy 9y ago

This advice also applies to more advance mathematics at the high school level, such as AP calculus BC or even linear algebra. Source: have followed this technique indirectly for the past few months

gtani 9y ago

Some books on proof, theorems/axioms, set theory, epsilon/delta/continuity/limits/differentiability, natural/rationals/reals/countability etc before heading into your first proof based LA or analysis sequence: - Kevin Houston "How to Think Like a Mathematician" - Keith Devlin "Intro Mathematical Thinking" - "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and Math Degree titles)

dvfjsdhgfv 9y ago
My favorite is Burn Math Class: And Reinvent Mathematics for Yourself
- goldenkey 9y ago
  Seems well liked but whats is up with the cover? Most people who are into mathematics aren't zealous rebels bent on burning the world. I feel like the marketing director for the book screwed up big time. I won't judge the book by its cover though. http://amzn.to/2CrgFDI http://amzn.to/2CrgFDI
  - gowld 9y ago
    Huh? where did you get "zealous rebels bent on burning the world" ?
  - dvfjsdhgfv 9y ago
    I actually don't remember the cover. The title is provocative but adequate. You could rephrase it as "Could you reinvent modern mathematics if it disappeared?" - but then probably less people would buy it.
jacobolus 9y ago
Even more useful as a first step in my opinion is to think about problem solving per se. E.g. see: Mason, Burton, & Stacey, Thinking Mathematically Pólya, How to Solve It; Mathematics and Plausible Reasoning (2 vols); Mathematical Discovery (2 vols) Schoenfeld, Mathematical Problem Solving Larson, Problem-Solving Through Problems

dboreham 9y ago

Something I would add : Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today? In my mathematical education I noticed that I had a more pleasant time and felt more motivated to learn the material when the teacher gave us this kind of background story. Since most teachers don't do so today, the student typically needs to get on the Internet to do their own research.

jnbiche 9y ago
> Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today? For some of us, this is incredibly useful advice re: math study habits. I also figured out that once I had the context and history, it was much easier for me to learn and apply.
- jdironman 9y ago
  The 'Why' lends to the 'How'. I agree. I usually like to know why even in things that are not math before attempting to apply or understand them.
mbag 9y ago
This happened almost through all my formal education. The knowledge was just presented, as something that was obvious. There was very little discussion of motivation, or what problems was person/people trying to solve. I understand that there is so much knowledge to cover during high school and college, but IMHO, this causes students to develop mindset, that you either see solution instantly, or you are just not smart enough to crack the problem.
- soVeryTired 9y ago
  I think Bourbaki are partly to blame for this state of affairs. Vladimir Arnold had a lot to say on the subject: he was a big fan of keeping the intuition and motivation for development in the subject clear. At times he went a bit overboard, but he makes some valid points in this lecture: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

Philipp__ 9y ago

I always strugle with memorizing part. I am never able to reproduce word for word what I’ve learned or read, but I can put it into context and show my understanding pretty well. Here, with math theory, there is no alternative. I failed exams so many times only because of that vocal reproduction of learned theory that required word for word knowledge. I am having math exam on integral theory in 2 days and here I am sitting and lookin at screen. :)

jolmg 9y ago
> with math theory, there is no alternative. You can do the exact same thing you mentioned with math. Theorems aren't arbitrary; there's a way they're derived. If you learn how the theorem comes to be, even if you can't remember exactly, you can often remember enough to come up with the theorem again. I think the way you learn is the ideal. Knowing the context behind the facts you learn gives them worth, and the redundancy of being able to derive the fact from the context also makes it better integrated with the rest of your knowledge and easier to remember in the long term.
- Philipp__ 9y ago
  I totally agree with you, but I feel like with math theory there is always good chunk of knowledge that needs to be somehow inserted in your memory as is. That’s the most irritating part for me personally. If i find a way to overcome it or make it more accessible, I think I will enjoy math theory much much more. :)
  - Smaug123 9y ago
    I'm afraid it's just practice. Lots and lots of solving problems until the definitions, methods, and techniques are burned into your mind. Once you have the background knowledge, everything is a lot simpler. Learn whether each theorem has an easy proof or a hard proof. If it has an easy proof, you can forget it (deriving it when you need to; an example is the contraction mapping theorem [1]). If it has a hard proof, you can forget it and look it up when you need to. [1]: https://www.patrickstevens.co.uk/how-to-discover-the-contraction-mapping-theorem/ https://www.patrickstevens.co.uk/how-to-discover-the-contrac...

Myrmornis 9y ago

It should emphasize more that you have to read very slowly, and often re-read the same section.

cle 9y ago

Memorization is so underrated. Memorizing the fundamentals and having them available for instant recall is hugely valuable, especially when trying to grok a new concept. I generally buck the standard advice and memorize first, before trying to understand. Understanding is much easier for me if I can easily hold everything in my working memory.

Philipp__ 9y ago
I agree, but I still haven’t found a way to make that process more enjoyable or less fatiguing.
newman8r 9y ago
I think people tend to conflate "memorization" with "cramming". You're right that memorization is underrated. Are you familiar with the 'method of loci' (i.e. memory palace). It's an amazing technique.
fooker 9y ago
Proper mental indexing is even more so. You can not memorize everything, but you can certainly index almost everything. That is why we can have databases that do not fit in 'memory'. Do you see the analogy here?
- goldenkey 9y ago
  Luckily we have brains that expose this interface quite readily with intuition. Learn the logic, and the partial indexes, the foreign keys, etc..they all follow. Reason before RAM. It is often much faster to derive data than to store it, and look it up. This applies to CPUs and to our brains. Our brains have instruction cache, and often instructions are better use of memory than their final executed output.
joe_the_user 9y ago
I wouldn't personally find advice like this useful for me. I suspect it varies with "personal style". I grasped mathematical concept easily from an early age and I never made an effort to memorize things. If anything, I saw those coming from less-advanced math founder on advanced math (arithmetic to algebra, algebra to calculus, calculus to advanced subjects) because they attempted to deal with the subject based on memorization rather than grasping the basic point. Generally, I did wind-up committing a lot of content to memory but it was and is much easier, even "effortless", when I knew the reason for each part of the content. Of course, language is slippery and we might be talking about exactly the same thing. So I wouldn't give blanket advise - each person has to find their own learning style (or styles, if it varies by discipline).

e0m 9y ago

By far the best explainer of mathematics I've seen anywhere is 3Blue1Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.

legionof7 9y ago
Also the book Burn Math Class.
- haskellandchill 9y ago
  That book got me back into math. Using functional equations to teach concepts is very powerful. You get to say I want these properties and derive things like the exponential function.
sophacles 9y ago
In a similar vein, I've been enjoying PBS Infinite series. The original host recently stepped aside to finish her dissertation and the new hosts are doing great but still finding their footing a bit - I suggest looking at some of the older videos to get a good sense of the channel. https://www.youtube.com/channel/UCs4aHmggTfFrpkPcWSaBN9g https://www.youtube.com/channel/UCs4aHmggTfFrpkPcWSaBN9g
nothis 9y ago
It always saddens me how mathematicians seem to look down on “intuition”. Maybe higher math as a full-time job is just hard work and stubborn precision but for me, an intuitive, visual look is probably getting me closer to understanding than any cold-hard-facts book does. Not to mention how much easier it is to appreciate the beauty of it.
- pflats 9y ago
  There's nothing wrong with intuition, but intuition does not write proofs. It can help you write proofs, but you need more than that. To pick an example: Intuition is great for understanding the intermediate value theorem. If the temperature was 40 degrees at 8 am, and is 60 degrees at noon, it must have been 50 degrees somewhere in between. Why, though? How can you guarantee that there will always have to be a time where it was 50 degrees[1]? [1] provided that temperature is real-valued and temperature change is continuous, the debate of which I'll leave to the science people

rublev 9y ago

I've always wanted to learn mathematics but have no idea where to start, I have no formal education in anything. I tried the Ivan Savov books and they seem great but I have no way to judge the quality of any resources. Where do I start as a complete beginner? I want books not online resources like Khan etc. I would eventually like to get into Geometry but I have no foundation whatsoever, just basic algebra.

rmidthun 9y ago
The web site https://betterexplained.com/ https://betterexplained.com/ has physical books that you can purchase "Math, Better Explained: Learn to Unlock Your Math Intuition". The explanations on that website are excellent, I imagine the book is the same.
dboreham 9y ago
You could try : What Is Mathematics? An Elementary Approach to Ideas and Methods https://www.amazon.com/dp/0195105192 https://www.amazon.com/dp/0195105192
hollander 9y ago
I find the online courses the most entertaining and stimulating. Many are available for free, like on Coursera, MIT.edu. Some offer real classes, where you follow live classes I believe. You cannot start when you want, you have to follow the program. On Udemy you can find nice courses as well for a good price. They have discounts all the time, each course for $10-20. Never pay the full price. See the reviews, and you can always take a risk for that price, plus they have a 30 day money back guarantee.
nextos 9y ago
Start with: * Algebra, Gelfand * Geometry, Kiselev
mkl 9y ago
Measurement, by Paul Lockhart, may be helpful to start with. It carefully and clearly derives mathematical principles from basic ideas, in a very approachable way. Here's how it describes its content: Part One: Size and Shape In which we begin our investigation of abstract geometrical figures. Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves. Part Two: Time and Space Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader.

dsacco 9y ago

This is good. I think two principal components should be emphasized when studying mathematics. 1. Proper preparation. There are textbooks at even the graduate level which have no formal prerequisites and which are largely self-contained. Technically speaking, someone with no prior background but a strong mathematical maturity could tackle these, but it might take them an inordinate amount of time to really grasp the material. For example, if you understand things like mathematical induction and proof by contradiction, you can learn analysis before you've been exposed to calculus, or category theory without abstract algebra. But it's far from ideal because you'll probably need to go over the same material several times and struggle with it. Furthermore, even the same subject within "advanced" mathematics can have wildly different depths of coverage depending on the author. Pinter's A Book of Abstract Algebra is probably approachable for anyone currently reading this comment, or even high school students. Dummit-Foote is a step beyond that, and appropriate for undergraduates who are already immersed in a math degree. But Lang or MacLane-Birkhoff would be significantly more challenging without first building up to them. Sometimes this is not just a question of depth, but also of pedagogical style. You can get a lot of satisfaction by learning analysis from Rudin for the first time, but it's really a rough go of it if you're not prepared for the terse definition->theorem->proof->remark->definition->theorem->proof->remark style of writing. On the other hand, Tao's Analysis I and Analysis II are much more approachable (similarly, some writers, like Halmos or Munkres, are praised for their exposition in introducing otherwise complex material). Ideally someone looking to study a subject should introspect about whether or not they are prepared for that subject overall. Once they've confirmed they are, they should read the first 10 pages of five or so well-recommended textbooks on the subject at their level, then choose to stick with the one that has the most approachable exposition style for them. 2. Proper study. When studying any given textbook (or videos, lectures, etc) it's really important to understand that mathematics is an active discipline. You cannot learn it by reading it. The process that has worked for me is the following: first, read through a chapter without taking any notes. Do so quickly, but not quite so quickly as skimming. When you come across things you don't know, compartmentalize them a bit and keep moving forward to the end of the chapter. The idea is to let the chapter's new material percolate a little before you begin actively tackling it. Next, start over at the beginning of the chapter and write down every single definition and theorem as you read. Before reading the author's proof of any given theorem, try to prove it yourself for at least 10 minutes. Then compare your work to the author's, and copy their proof meticulously in order to learn the method. Continue on to the end of the chapter. Finally, there will probably be anywhere between 5 - 20 exercises at the end of the chapter. Solve a meaningful fraction of these exercises, and don't look up the solutions to any of them until you've struggled with them for a good half hour or so (each). When you do look up the solutions, make sure you check multiple proofs for the same exercise so you can understand how the chapter's material can be applied in different, flexible ways. Mathematics has always exemplified a central belief of mine, which is that humans learn under conditions of optimal struggle. Even though it feels like being mired in hopeless complexity while you're struggling to complete a particularly difficult problem, you're actively learning the subject by doing it. But it's a question of efficiency. You want to aim for a subject and a presentation of that subject which is difficult enough to be just out of your current capabilities, but not so difficult that you can't follow its exposition.

commandlinefan 9y ago

One thing I figured out on my own that I wished I had realized sooner (which I'm trying - so far unsuccessfully - to impress on my 14-year-old son) is that, when reading math books, they follow a similar pattern. They describe a concept, show an example problem fully worked, and then discuss the ramifications of that concept, followed by another concept, followed by a fully worked problem, etc. I started making it a habit to try to work the example fully-worked problem by myself, based on the description that preceded it, before reading through the author's work. I was amazed how much better I was able to understand what he presented, and how much better I did working the exercises in the chapter afterwards.

skybrian 9y ago

One thing they don't talk about is how you decide what to study. Before spending a lot of time on a particular topic in math, you have to decide whether it's worthwhile studying it at all. It's apparently just assumed that you're taking a course so the decision is made for you.

sincerely 9y ago
I'm willing to bet that a course in post-secondary school is the context for the vast vast majority of people studying mathematics.

adamnemecek 9y ago

The Franklin method from yesterday still applies. https://news.ycombinator.com/item?id=16390046 https://news.ycombinator.com/item?id=16390046 Also a lot of books these days have wolfram mathematica code and it works surprisingly well even for some more abstract parts of math, To get a good intuition.

aje403 9y ago
The greatest book review of all time, sort of similar approach: https://www.amazon.com/review/R23MC2PCAJYHCB https://www.amazon.com/review/R23MC2PCAJYHCB Put the book down when you get to the example or proof of a theorem, keep all the theorems and examples before it in your head, and see if you can reason it out yourself. Can't promise that your head won't hurt afterwards
- BadMathBook3 9y ago
  Those prices seem comical. Rudin passed nearly 8 years ago. Who is this benefiting? America is off its rocker. Google the title + PDF

iron0013 9y ago

Thanks for this, everyone should take the time to review this material

bluetwo 9y ago

I wonder if anyone here has a background in instructional design and what they think of this tutorial.

hollander 9y ago
Do you need a background in instructional design to have an opinion on this? ;-)
- bluetwo 9y ago
  No. Not what I'm saying. I'm asking for an option from someone with particular experience. That doesn't invalidate the options of others, snowflake.
  - AnimalMuppet 9y ago
    Drop the "snowflake". Gratuitous personal insults don't belong on HN. You have a valid point without it. Leave it at the valid point, without the personal dig.
    - bluetwo 9y ago
      Anyone who claims to be offended by an honest question on behalf of some unnamed party is a snowflake, in my opinion.
      - AnimalMuppet 9y ago
        Feel free to hold that opinion. But saying it is counterproductive. It leads at least some of your readers to dismiss you as a jerk, rather than hearing your message. [Edit: Oh, yeah. It's also against HN site guidelines. Saying such things repeatedly can get you banned.]
        bluetwo 9y ago
        When someone misinterprets a rather clear communication and claims to be offended by the result, I'm going to point that out. If that offends you, that's your problem. Or, look at it as a life-hack: Don't waste your time protecting the feelings of a non-existant group of people.

BadMathBook3 9y ago

I bailed on high school math, thinking I'm math dumb. In my late 20s I decided to try again, but jumped straight into calculus. And at first regretted that decision. However, I got lucky by stumbling upon this book: https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480 https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/... It "reads" like a book, with the ideas given context. I had an "ok" connection with Algebra, and the book explained the rest well enough for me. In school, the textbooks were loaded with symbols, but not enough description -- I guess they relied on bored teachers making minimum wage to do that part. I went to a school with poor academic showings (but connections to state superintendent of ed got them a grant for football facilities). Coincidentally, this book goes well with the technique described here: http://www.pathsensitive.com/2018/01/the-benjamin-franklin-method-of-reading.html http://www.pathsensitive.com/2018/01/the-benjamin-franklin-m...

sampo 9y ago

This is more "How to study pure mathematics", when the aim is to understand how the theory is build, so that you learn to contruibute to the theory by discovering and rigorously proving your own theorems. I don't think I've ever seen a "How to study applied mathematics". How do applied mathematicians, physicicst and engineers (who apply mathematics to real world problems) study mathematics, when they use it as a tool? How much or little emphasis do they give to proofs and theorems?

dboreham 9y ago
All mathematics gets applied sooner or later, no?
- aaachilless 9y ago
  Maybe, but “applied mathematics” is conventionally more similar to “how to apply mathematics” than it is to “mathematics that is applied”. Studying how to apply mathematics is qualitatively different in many ways than studying pure mathematics.
  - dboreham 9y ago
    Ah. I genuinely did not know that. In Scotland we did not have Applied Mathematics (as a school exam subject) so I guess I just assumed it was about physics problems (there was an A-level with that name in England). It never came up again in my Mathematics career..
yiransheng 9y ago
Some personal thoughts: the key for me at least, is a bit like the hammer-nail mentality. An software engineer for example has a (small) mathematical toolkit (a hammer), sometimes forcing your problem at hand into the shape of nail is sufficient to solve it. A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d). Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount. An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :) [1] http://longnow.org/essays/richard-feynman-connection-machine/ http://longnow.org/essays/richard-feynman-connection-machine...

Hasz 9y ago

I cannot stress the importance of working problems yourself enough. There is literately no substitute.

lowtec 9y ago

Notation and syntax are areas that I have struggled with in higher level mathematics. Can anyone recommend a guide or resources useful to understanding things like converting set theory, sequence and series style problems into equations and solving them?

tkzzbneig 9y ago
Number theory's as good a place to start as any. You'll learn proof techniques and see a lot of notation in contexts that require varying amounts of background knowledge. It's feasible that an undergrad intro class would be up your alley if you're a software dev. https://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-spring-2012/lecture-notes/ https://ocw.mit.edu/courses/mathematics/18-781-theory-of-num...
goldenkey 9y ago
Bill Shillito's lecture series for Project Polymath is by a mile, the best introduction to all the notation necessary for higher level math. It requires absolutely no prerequisite knowledge. Introduction to Higher Mathematics: https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz... If you find he talks a bit slow, change your playback speed to 1.5x. Enjoy! :-)

nailer 9y ago

If you can't read the page: window.document.body.style['max-width'] = '650px'

alexbecker 9y ago

I left math after college for software engineering, but reading A Mathematicians Lament recently re-kindled my love for it. It is tragic how intuition, technique and mental models are left out of modern mathematics education and writing. It occurred to me that while I learned in college how to show, using Galois theory, that quintic equations are not generally solvable by radicals, I had no idea how Galois theory really relates to the process of finding roots. So I went back and derived the quadratic formula, the cubic formula, and sketched the quartic formula to see how the process used the ideas formalized by Galois theory and where it breaks down. I've tried to write the result up in a motivated and understandable way, instead of like a math textbook: https://alexcbecker.net/mathematics.html#the-quadratic-equation-and-beyond https://alexcbecker.net/mathematics.html#the-quadratic-equat...

bjourne 9y ago

What was the name of that 30-something woman who had never studied mathematics before, but then finished a degree in quantum physics in two years? Her name has been posted on HN at least two times before. I guess what she did is exactly how you should study mathematics if you want to become got at it.

sebg 9y ago
Think you're talking about Susan Flower and two of her articles: If Susan Can Learn Physics, So Can You - https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgling-physicist-archives-if-susan-can-learn-physics-so-can-you https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli... and So You Want To Learn Physics... - https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-physics https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-l...
- badpun 9y ago
  Sadly (given her immense passion for physics), she ended up in software (doing SRE).
  - systemBuilder 9y ago
    Hey I am in SRE. At one point last year there were 7 ex professors sitting within 15 feet of my desk including me (and I did a PhD in theoretical computer science at the same school as this math professor whose article we are reading). I don't think it is an underrated career.
- trendia 9y ago
  On the one hand, her account is very inspirational to people who might think that learning physics is out of reach or they're too old to do it. But on the other hand, her case is very out of the ordinary -- I think a lot of us took longer than 1.5 years to get through undergraduate physics. According to Susan's account, it took her 1.5 years to go from no college-level math to a full college-level physics curriculum. At the school I studied at, the Calc series (I, II, and III) takes 1.5 years. The intro physics (I and II) takes a full year, and the upper physics take at least a year (Mechanics I & II, E&M I&II, Relativity I&II, Quantum I&II, Lab I&II). (And you had to take intro physics before higher-level physics, and most courses had dependencies.) That's approximately 45 credit hours. If you took 18 credits a semester, that would take at least 2.5 years. And those are all tough courses -- no 3 credit gen ed "filler" courses. Even at 15 credits a semester, that's still 3 years. So, to get through all of those courses in 1.5 years would be like learning about 30 credit hours / semester. And add in the physics electives, like thermo, waves, statistical mechanics, quantum field theory, additional math courses, etc... that's a lot on top. I guess my point is that if Susan's account is discouraging, note that a lot of people take far more than 1.5 years to get through undergraduate physics.

perseusprime11 9y ago

Mathematics for the million is a great book if you are looking to reignite your passion for math.

backprojection 9y ago

Ha, Dr. Blecher's webpage on HN! That's great. I did my PhD in math, in large part because I enjoyed his class on advanced linear algebra so much, and later on real and functional analysis.

graycat 9y ago

I studied a lot of math, pure and applied, taught it, applied it, published research in it, etc. so developed some ideas relevant to the OP. For > To the mathematician this material, together with examples showing why the definitions chosen are the correct ones and how the theorems can be put to practical use, is the essence of mathematics. Is good, but more is needed. (1) Plan to go over the material more than once. The early passes are just to get a general idea what is going on. In such passes, for the proofs, they are usually the near the end of what to study and not the first. (2) When get to the proofs, for each proof and each of the hypotheses (givens, assumptions), try to see where the proof uses the hypothesis. Next, try to see what are the more important earlier theorems used in the proof. So, sure, in this way might begin to see some of how one result leads to or depends on another and have something of a web, acyclic directed graph, of results. And try to see what are the core, clever ideas used in the proof. (3) For still more if you have time, and likely you will not, can use the P. Halmos advice, roughly, "Consider changes in the hypotheses and conclusions that make the theorem false or still true." (4) But, note that to solve exercises or apply or extend the theory, need some ideas. So, where do such ideas come from? In my experience, heavily the ideas come from intuitive views of the subject. So, my best suggestion is to try to develop some intuitive ideas about the material. Definitely be willing to draw pictures, maybe on paper, maybe only in your head. In the end, a solution or proof does not depend on intuitive ideas, but finding a solutions or proof can make use of a lot in intuitive ideas. For research, most of the above applies, but IMHO there are more techniques needed.

HaoZeke 9y ago

So basically.. Mug up everything and hope you have exactly the same thought process as the course instructor. This is a disgrace. In any case. Definitions vary from book to book so mugging them up ad verbatim would hardly help.

Ldorigo 9y ago

This text should be required reading for ANY university-level math course. It made me go from "ugh, not math again" to "yay, I'm going to learn something new!" in a matter of days.