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From looking at undergraduate mathematics programmes it's quite apparent that mathematics degrees are demanding, one could even say the work load is gruelling. However I'm certain that there are things that one could do to prepare in advance for the rigours of such a degree.

What I'd like to know is what foundations must be in place so that the experience of learning mathematics at university is an enjoyable one. Enjoyable in the sense that if you're exposed to a new topic you aren't floundering and you can dive straight in and enjoy the exposition, without having to go backwards plugging in numerous gaps and addressing other deficiencies in your knowledge. I'm certain that a good grounding in pre-calculus mathematics and calculus are a prerequisite but aren't all that's needed.

What are the things must one know to have a solid grounding in mathematics with the aim of studying mathematics at a higher level? What are the things you wish you knew at the beginning of your degree?

Let's assume it's a quite a demanding degree programme: MIT, Harvard, Cambridge, etc.

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    $\begingroup$ I feel like one of the most valuable experiences I received from my mathematics undergraduate degree was a temporary sense of floundering. I think if you're enjoying "the exposition with relative ease" then you're a lucky soul. $\endgroup$ – Andrew Sanfratello Jan 13 '15 at 14:57
  • $\begingroup$ @AndrewSanfratello I see what you mean, perhaps I should have phrased myself better. Don't get me wrong, I enjoy wrestling with new concepts and I expect it to be a challenge no matter how well prepared I am, I don't think the experience would be worth it if it were any other way, but what I don't want is to feel like a fish out of water when I get there. $\endgroup$ – seeker Jan 13 '15 at 15:01
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    $\begingroup$ It may be worth emphasizing, given the diverse group of people here, that most U.S. undergraduate math programs (i.e. overlooking honors track programs at a top university such as Princeton or Harvard) tend not to be particularly demanding in comparison with those programs in many other countries. In fact, U.S. math tends to be quite a bit less damanding than physics, chemistry, and certain engineering programs (especially mechanical and electrical). On the other hand, in the U.S. there is typically quite a jump from undergraduate to graduate programs, and for that your question is also good. $\endgroup$ – Dave L Renfro Jan 13 '15 at 15:38
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    $\begingroup$ Be aware that there might be quite a big difference between US universities and Cambridge, just because it's a different system. For Cambridge the usual preparation is to study the syllabus of a Further Mathematics A-Level and the so-called STEP. This will put you in the same place most of their mathematics undergraduates are. Of course, they also have experience coping with furriners who've studied a different syllabus. I don't know whether or not Cambridge makes everyone take the STEP regardless. $\endgroup$ – Steve Jessop Jan 13 '15 at 18:22
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    $\begingroup$ It sounds like you're a student in such a program? Here's a word of warning: at least in my state university math B.A., I would never have called the workload "grueling". Everything seemed pretty straightforward, even while double-majoring (and graduating with high distinction). But the step into grad school was like a light-year beyond that; it was almost unsurvivably brutal IME. $\endgroup$ – Daniel R. Collins Oct 7 '15 at 2:32
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Don't denigrate "pre-calculus mathematics and calculus." Many of the problems of students is a lack of a solid foundation in the lower mathematics, especially algebra. Make sure you can do the algebra reliably and quickly.

I would say the next most important knowledge is that of basic logic--not necessarily mathematical logic, which will be taught in college, but the basic ideas. Can you quickly tell which logical expressions are equivalent? Do you know the major proof strategies? That used to be emphasized in high school geometry, but the logic part of geometry has been almost completely dropped these days.

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    $\begingroup$ +1 for Many of the problems of students is a lack of a solid foundation in the lower mathematics $\endgroup$ – Dave L Renfro Jan 13 '15 at 15:39
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    $\begingroup$ @DaveLRenfro - I'm skeptical about the causality. My old university routinely allowed able students to skip prerequisites without worrying about solid foundations. First year courses often fill in foundations. Assuming a university student doubles their knowledge of maths in their first year, how significant is a small hole in knowledge compared to the volume of new material that needs to be absorbed? Is it not likely that they are missing curiosity, personal work skills, or even just ability, that stopped them from learning it in high school and will continue to hamper them in university. . $\endgroup$ – Richard Jan 14 '15 at 11:43
  • $\begingroup$ @Richard could you elaborate on the "person work skills" you think are necessary? $\endgroup$ – seeker Jan 14 '15 at 18:06
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    $\begingroup$ @seeker Should have written "personal work skills", as discussed in dtldarek's answer. Patience and perseverance. Add to that time management, ability to work with temporary ignorance (before the Eureka moment hits), and humility to ask for help. These might be new skills that a smart kid didn't need at high school, but they get more important as you progress through university. Probably could also mention general healthy habits (eating, sleeping, exercise). If you are sick for the exam, no amount of intelligence will help you! $\endgroup$ – Richard Jan 14 '15 at 23:01
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    $\begingroup$ @seeker I was bad at this as a student. I wanted to do everything to my best ability. It would have been smarter to work according to the 80/20 "rule", make sure I get the easy 80% of the marks first, and then decide what I am going to be expert on without skimping on a healthy amount of sleep, exercise etc. There is little advantage to being expert at everything, in workplace or grad school. With this strategy you may loose some marks short term, but you are going to be studying for 4+ years, so a smart, sustainable approach is going to be more important for your degree. $\endgroup$ – Richard Jan 15 '15 at 0:17
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Disclaimer: You assume a "quite a demanding degree programme". In turn I will assume "quite good student", that is, a person who is able to handle such studies within reasonable time frame.


I would place the following three things as most important for enjoying higher mathematical education:

  • curiosity: it's pointless if you don't really want to know the things you are learning;
  • patience: math can be frustrating, yet the delayed gratification makes doing it worth it, just make sure you don't waver too much in-between;
  • be able do do hard (mathematical) work when it's required: if drawing 20 complicated diagrams is what it takes to understand some concept, just do it.

In my opinion if these foundations are in place, then the student does not have anything to worry about. However, if I would have to list some practical mathematical skills, then there are two:

  • quantifiers manipulations: in my opinion there is a big difference between high school and university-level definitions, where the latter are much more quantifier-heavy; being able to easily understand them seems important.
  • reliable substitution: substitution is used everywhere and doing it reliably greatly speeds up some derivations.

I hope that helps $\ddot\smile$

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  • $\begingroup$ Do you have any tips towards handling "such studies in a reasonable timeframe"? I have a habit of trying to find out everything about something I'm learning and that tends to be quite time-consuming.. $\endgroup$ – seeker Jan 16 '15 at 12:08
  • $\begingroup$ @seeker That would make a great question by itself. Unfortunately, no I do have any such tips, I can only tell you what often worked in my case: 1. If it is late, you go to sleep no matter what (you need to find out by yourself what late means). 2. Beyond basic facts focus on problems, that is, learn to solve things (if you like some area, just switch to harder problems). $\endgroup$ – dtldarek Jan 16 '15 at 13:43
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I think this is a difficult question to answer usefully because the notions of "fun/enjoy/..." vary so wildly among individuals. There are also extremely disparate impulses about delayed gratification, long-term versus short-term this-and-that...

Nevertheless, especially after the questioner's edit that indicates that the context should be "high-end" programs:

First, I think there is considerable utility in realizing that some difficulties are genuinely mathematical, and some are essentially completely artifactual (even if traditional!) created by curricula or instructors. Unless we go so far as to believe that mathematics itself is altogether nothing more than an artifact, which I think is too extreme, it can be important to learn to distinguish genuine complication from contrived. This is not to claim that contrived issues needn't be fun or entertaining or profitable. Indeed, doing well in contest-math, if one has the inclination and the capacity, is effective PR and can be an ego-boost. Doing well in homework assignments or exams can have similar benefits. However, I think one should not lose sight of the fact that such things are mostly quite deliberately contrived by other people as challenges to others, without regard to deeper meaning.

This distinction might seem pointless... but I have seen many cases of failure to appreciate the distinction between genuine and contrived lead to vast wastes of time and effort in misguided attempts to defend against sniping and prank questions. Sure, prepping for contests may warrant such, but at a certain point one should realize that it is not only essentially impossible to pre-emptively defend against deliberate prank-questions, but it is very expensive... and displaces much more worthwhile work.

I claim that this argues for moderation in line-by-line reading of texts, for moderation in doing exercises at the end of the chapter, and for moderation in thinking mathematics is only legitimate to the extent that it is formalized.

Rather, most mathematics is based in physical intuition (if of a refined sort), and one can come very close to "correct answers", at least if one is allowed to work iteratively (which also seems disallowed in most formalized settings), without necessarily playing the game of formal arguments. I'm not trying to argue against formal arguments per-se, but only that formality as an alleged minimal criterion for mathematical correctness ought not be a foregone conclusion.

I mention this sort of thing as a push-back against the (understandably convenient) formality of most texts and most curricula...

As an action item, I strongly recommend working to get an overview of the mathematics that is reflected in the standard curriculum, since that larger picture often explains the quirks of small details, thus avoiding any seeming need to "memorize" formal things whose motivations would only arise later. That is, read lightly through textbooks... just to see what happens later, how things re-appear or don't, etc. Especially beneficial is the bit of psychological familiarity this confers, in advance of appearance of new ideas in classes. Many people greatly underestimate the role that a sense of prior familiarity can play in "comfort". Discomfort is obviously an inhibitor, for most people...

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Caveat: From a UK university perspective.

I think the first thing to say, as perhaps other answers have said, is that almost all students at some time wrestle with one concept/theorem, or another. After all, mathematics is doing...and doing takes energy and effort.

Secondly, for many mathematicians and students alike, the challenge of mathematical understanding forms part of the enjoyment - or even is the enjoyment. An integral part of university mathematics is to make the transition from learning the tools of mathematics to applying those tools to solve new mathematical problems. So learning to enjoy mathematical challenge is important (if not a prerequisite for a university mathematics course!).

Even if you do not venture down the pure mathematics / analysis route, it usually forms the foundation of university mathematics - and having attended a demanding course from one of the institutions you mentioned, I can say that the largest knowledge gap (for many students) seemed to be around basic set theory, including the notion of mathematical proof, and algebra and geometry.

I have no affiliation to either author of these books. I did, however, work through similar books before the start of my university course. I own these two books now, and would recommend them as pre-university reading.

Sets and Proofs, and in algebra and geometry Algebra and Geometry.

Being familiar with the concepts presented in these two books would certainly ease the transition from high-school/secondary school to university mathematics.

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  • $\begingroup$ nice answer, I'll check those books out! $\endgroup$ – seeker Jan 14 '15 at 23:31
  • $\begingroup$ Also I guess you went Cambridge, do you know of any books that would be useful in preparing for STEP? $\endgroup$ – seeker Jan 14 '15 at 23:33
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    $\begingroup$ @seeker I put some in this answer here. With regards to books for STEP, the best preparation is to work through the past questions + STEP booklet, using the detailed answers to guide you. Over time you will get faster at answering the questions. An excellent (but old) book is Porter it is full of wonderfully difficult exercises at roughly the right level. And available online, it seems.... $\endgroup$ – Rusan Kax Jan 15 '15 at 2:29
  • $\begingroup$ Neat, thank you! $\endgroup$ – seeker Jan 15 '15 at 13:36
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    $\begingroup$ @seeker, seek and you will find Hammack's "Book of Proof" for free. $\endgroup$ – vonbrand Oct 5 '15 at 22:18
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"Enjoyable in the sense that if you're exposed to a new topic you aren't floundering and you can dive straight in and enjoy the exposition, without having to go backwards plugging in numerous gaps and addressing other deficiencies in your knowledge."

I'm not sure that ever happens (it certainly hasn't for me). Mathematics is an immensely rich and deeply interconnected topic, and learning a new topic always changes how you understand previous topics, which means going back and rethinking them and learning parts that you'd overlooked or hadn't appreciated before.

(But math is enjoyable anyway!)

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    $\begingroup$ You're right, but there's a distinction to be made here: going back because you've missed a prerequisite, i.e. Going back to learn calculus because you can't do Analysis without it (extreme example) – you don't want to be doing this kind of going back. And then there's the kind of going back you're talking about, which I feel is good, going back in light of new knowledge to enhance your understanding of things you already know. I just want to be on a sound footing before I start my degree is all.. $\endgroup$ – seeker Jan 13 '15 at 15:37
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    $\begingroup$ "Analysis" at the seemingly simple level of "single-variable calculus" is much subtler than many people realize... I.e., even "rigorous" calculus courses prevent themselves from looking at the issues in as sophisticated way as it merits (by which I mean L. Schwartz' theory of distributions, and perhaps a bit of Robinson's (and E. Nelson's) non-standard analysis...). $\endgroup$ – paul garrett Oct 5 '15 at 23:52
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With the benefit of hindsight, the one thing I wish I knew was the following: "Bits of this are hard. Ask for help with them. Share what you know (don't know)."

The last part (about sharing what you know) was easy for me, especially since I ran into fewer hard bits than many of my students. However in the latter part of my undergraduate and graduate studies, I ran into hard bits and did not ask for help (and did not share with others that I was lacking). Floundering, addressing, and overcoming challenges single-handedly can be quite satisfying and emotionally appealing (and even character building), but is not always a healthy mindset. Thinking that you have to do it by yourself is unnecessarily overwhelming. If you are trying to learn mathematics to achieve a reasonable goal, I think you will enjoy it more if you learn to ask for help along the way. Form study groups, talk to graduate students, participate in the online forums.

Gerhard "Sharing Isn't Just One Way" Paseman, 2015.10.05

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  • $\begingroup$ I note that Richard includes this as a comment (see "humility to ask for help") to Rory Daulton's answer. His comment is worth rereading. I think this particular aspect deserves emphasis, but I don't mind if you read this and give Richard's comment the upvote instead. Gerhard "Credit Where It Is Due" Paseman, 2015.10.05 $\endgroup$ – Gerhard Paseman Oct 5 '15 at 16:28
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There is much that can be learned about the kind of mathematics one encounters at University from taking part in high school level mathematical competitions. The key difference between HS and university maths is increased rigour; less focus on methods, and more on explanations and justifications; and as you say, a wider diversity of topics. These new topics are not encountered in isolation, and require the same persistence, logical thinking and creativity that a student will learn by completing challenging and unfamiliar problems in even elementary maths.

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The Greek Alphabet

A lot of mathematics professors prefer using greek symbols over the roman alphabet (a-z). Learn to recognize the names and both the small and capital symbols, and practice writing them.

One of my first lectures in Maths at Cambridge, UK used small xi - ξ and small zeta ζ for preference rather than x and y; challenging lectures are not a good place for a first encounter with these!

Matrix calculations

The other non-obvious thing is matrix calculations (how to multiply matrices etc). Since the algorithm for doing this is straightforward, a fast course will touch on it lightly and then move on to the complex things that you actually use it for. However, most students require a lot of practice before they can do this reliably.

Calculus and other Things You Still Won't Have Memorized

Experienced mathematicians will 'just know' all of the formulae they may want to know. Don't flounder around trying to keep up - that cheat sheet you made for revising? The one with all of the common calculus derivatives & integrals, trigonometric formule etc? Keep it and take it with you!

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