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I am an undergrad math major that just completed Calculus 3 last semester. This semester I signed up for Discrete Mathematics, and will be taking Intro to Advanced/Abstract Math next.

Of course-- I expected the numbers and computation to be larger and much more complex but instead am finding that there is hardly any number-crunching at all. Just a lot of proofs and logic skills. What gives? I thought the more advanced math got, the more complex the number-crunching would get.

Is it always going to be like this from now on? For both pure and applied math majors?

And secondly-- after completing four years of this stuff, how in the world do you guys remember every rote memorization technique taught step-by-step during freshmen year for all of your past courses in trigonometry, college algebra, calculus 1-3 classes? There are usually like 4-5 steps per technique, with 3-4 techniques per section w/ 10 sections per chapter of a book!

Take for instance my College Algebra book is 500 pages long-- I can't even remember every single section's memorization of how to answer the problem even after a year of doing Calculus, let alone the three years it will take me to graduate as a math major!

Maybe I just don't understand but it seems like all of the earlier useful rote memorization technique is going to get lost. IS this supposed to be the case?

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user152810
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    One of my math professors joked that the only numbers in his latest paper were the page numbers at the bottom of the page. – GEdgar Feb 11 '16 at 21:15
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    When I make an arithmetic error nowadays, as an excuse I sometimes say that as a math major I was never required to count higher than 3. – David K Feb 11 '16 at 21:50
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    As for those rote procedures, IMO they're a cancerous growth on math education. I'm trying to avoid starting a rant on that, because I might never stop. – David K Feb 11 '16 at 22:08
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    @DavidK I couldn't agree more. I have a lot of students who claim to hate math, and I am always tempted to grab them and yell that they don't hate math, they hate base computation. Instead, I usually try to tell them that "how you were taught math is comparable to if you had an art class where all you learned was how to use a paint-roller to paint a wall white. They have sucked all the beauty and creativity out of it and instead force-feed you only the things that have been deemed 'practical' and 'useful' by people who don't understand math." – Sean English Feb 11 '16 at 22:23
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    @DavidK So I guess my question becomes: without the rote stuff how are we supposed to solve math problems? I know it sounds dumb. But that's the only way I've been taught how to solve. Nobody ever explained to me in highschool why the Gaussian method of elimination gives the answer, but they just told us to use it to find the intersection of two graphs. – user152810 Feb 11 '16 at 23:43
  • @SeanEnglish I guess I'm just confused as to how rote isn't the way to solve math problems. In my limited experience, I just cant fathom how logics and proofs would help out in trigonometry for example. – user152810 Feb 11 '16 at 23:43
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    The idea is that if you truly understand the concepts, you can figure out a procedure that you think should work, then prove that it will work. Rather than memorizing rote procedures, you learn how to build them yourself. EG with Gauss-Jordan Elimination, if you understand what operations preserve the solution set and you understand that to solve a system of linear equations you can use their relationships to reduce the problem down to (hopefully) the single variable case, you can come up with a method that works similarly to GJE, then use your proof skills to show it works how you want it to. – Sean English Feb 11 '16 at 23:53
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    @SeanEnglish that is amazing. I really wish I had known this stuff before. My mind is blown tonight! Can't wait to tell my classmates-- no more computation! – user152810 Feb 12 '16 at 00:00
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    @user152810 Be careful now, there will be some computation still, but it will become less emphasized. Instead, the classes you will soon encounter will tear apart everything you thought you "knew" in math, and rebuild it from the ground(which are our axioms, or assumed truths) up. Soon you will be able to answer "why does -1(-1)=1" or "why is 2+2=4" without saying "because that is what I was told." It is a long, and usually difficult journey though, but in my opinion, a truly beautiful and enriching one. – Sean English Feb 12 '16 at 00:05
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    @user152810 Also, if you do enjoy, sticking around on this site and simply reading answers can really get your mind thinking about how to do the math you encounter. As a bonus, you can attempt to answer other peoples' questions, building upon all the knowledge you gained from others, using comments to guide your answers to better forms. – Simply Beautiful Art Feb 12 '16 at 00:09
  • @DavidK "Let $S = {1, 2, \ldots, 4}$..." – pjs36 Feb 12 '16 at 01:12
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    Perhaps the question is in fact serious, and I hope the asker manages to disabuse himself of these misconceptions, but is this really a good question for the site? Right off the bat it it makes several blatantly wrong assertions (in most other forums on the net I'd suspect it's a troll) and the answer (if one can be given) is unlikely to do anyone much good. The asker needs to first ask "Is all math just rote memorization and number-crunching" and go from there. – Superbest Feb 12 '16 at 03:08
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    @pjs36 That's "let $S = {1,2,\ldots, n}$" where I come from. – David K Feb 12 '16 at 03:20
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    @user152810, once you get the hang of it you practically won't need to remember anything by rote memorization because you'll be able to derive it from scratch whenever you need it – sig_seg_v Feb 12 '16 at 05:30
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    @user152810: Question: What are your career goals? What led you to enter a math program in the first place? – Daniel R. Collins Feb 12 '16 at 05:31
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    One of my favorite apocryphal stories goes something like: A professor is lecturing on number theory, when a student raises their hand. "I'm sorry, what is that squiggle on the bottom of that board?" After some confusion about which squiggle is being referred to, the professor looks at it, looks up the board, down the board, to the prior board. At this point he is quite bewildered and steps back a few paces, continuing looking around at his proof. Minutes pass. Finally, his face lights up and he gives a sigh of relief, walks back to the board, points to the squiggle and says "That is a 5." – Eric Stucky Feb 12 '16 at 07:22
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    One of my favorite math professors used to say, "We don't memorize, we derive." – littleO Feb 12 '16 at 08:04
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    This joke made the rounds when I was taking math classes: "Professor, all these concepts are too abstract for me. Can you give us an example using concrete numbers?" - "Certainly! Take for example the rational numbers a and b ..." – waldrumpus Feb 12 '16 at 09:29
  • @Superbest No, I'm not a troll. Yesterday a TA had spotted me carrying around a College Algebra book in my Discrete Mathematics course and inquired about it.

    I told him since I had trouble remembering every technique I had memorized in my freshmen College Algebra I thought that I just needed to start all over again and carefully devote everything to memory again, from highschool Alg.1- Calc. 3. He laughed and then proceeded to tell me it was a waste to do so as the REAL math starts in Abstract Alg.. As he was the only one to ever tell me this-- naturally I didn't believe him. So I came here.

     – user152810 Feb 12 '16 at 14:14
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    I was about to write the same thing as your TA said. Do you have a particular career plan (reason for entering math)? – Daniel R. Collins Feb 12 '16 at 17:17
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    A model "Mathematical Task Analysis Guide" in 4 steps from Teachers College, Columbia U. Consider the difference between "memorization tasks" at the lowest level, and "doing [real] mathematics tasks" at the highest level -- https://tn.gov/assets/entities/education/training/attachments/TPD_Summer2013_Math_TaskAnalysisGuide.pdf – Daniel R. Collins Feb 12 '16 at 22:04

7 Answers7

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Questions like the OP's aren't rare, especially among maths majors (or simply bright young students interested in mathematics). The truth is that you've been lied to for most of your life as to what mathematics are. Mathematics aren't a bunch of formulae or dry steps that you must memorise, or tedious calculations that you have to perform for hours on end.

That "lot of proofs and logic" is what mathematics are. Those formulae and calculations weren't just given to us by Prometheus; they were discovered, over hundreds and thousands of years, by great minds labouring within the frameworks of logic and pattern-seeking that have come to be known as mathematics. Each time you perform a calculation, you aren't really doing mathematics...you're applying the mathematical work of someone who's gone before. Every math problem you've ever solved up to this point was really a specific example of a more general theorem, which you can only rely on to be true because someone verified it by a proof.

Real mathematics is just that: going from a small set of rules (often called axioms), constructing objects that fit those axioms, and seeing what patterns emerge when those objects are manipulated. When solid patterns can be verified by logical steps, we call these patterns theorems, and use them to refine the framework we're working with, build new objects, and survey new patterns.

If that's not your cup of tea, it's perfectly alright; if you really are interested in calculation, there are lots of engineering disciplines that will give you as many hours of algebraic recitation as you could ever want. And it isn't really your fault for having this impression...the entire public school system of North America (if not the whole of the world) is geared toward instilling this gross misapprehension of mathematics into all students' minds. If you like patterns and solving puzzles, stick with mathematics...if not (and, again, that's fine!), you may want to find a discipline more suited to your tastes.

EDIT: Here is a link to an excellent paper by William P. Thurston, which gives a much greater insight into the workings of real mathematics (and the thinking of real mathematicians) than I have here exposed. I heartily recommend reading it if you want to get a better feel for what you're setting yourself up for with a mathematical education.

S. G.
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    Might also consider linking Lockhart's lament but note that it is somewhat polemical. – Kevin Feb 12 '16 at 02:34
  • Great post! Thanks. They briefly touched on axioms, proofs and theorems in highschool geometry but when it came to the Algebra, Trig and Calc besides how to derive a couple of formulas, I had never seen any of those three until I hit a College Calculus course and the professor required us to know certain proofs, such as Rolle's Theorem.

    Even though I thought it was important to derive, I thought it was silly for the professor to ask us to devote any real activity to understanding the proofs instead of just doing the regular homework problems. Boy was I wrong!

     – user152810 Feb 12 '16 at 14:03
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Did you sign up to be a math major hoping that you would be doing tedious computations all day? If so, why?

Math is all about logic and proving things in a rigorous way. What you're seeing in discrete math is a taste of what true math is really about. Maybe if you go down the applied track there are more computational things (I honestly don't really know, I just do pure). But you should get used to the idea of proving things, it's much more fulfilling than just performing computation once you get into the swing of things.

Alex Mathers
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    This is what it's about. Even in applied math there should be an emphasis on "proofs and logic skills" (quoting the OP) so that you can correctly instruct a computer to do the big number-crunching. – David K Feb 11 '16 at 22:02
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    @mathers101 Maybe so. It just feels really strange. Me and a bunch of other guys who just got out of Calculus-- found ourselves saying at every lecture '' this logic stuff is okay but when is the REAL math stuff going to start''? – user152810 Feb 11 '16 at 23:45
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    @user152810 whats the difference in difficulty between $1+1$ and $12361246704529+21371426558938$? More important knowing how to add the second one, right? – Simply Beautiful Art Feb 12 '16 at 00:21
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    @user152810: You got yourself a misleading view of higher mathematics, then. That's unfortunately not unusual but you're starting to discover reality now. – Nate Eldredge Feb 12 '16 at 00:22
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    Not just doing tedious computation, but doing it from memory. – Superbest Feb 12 '16 at 03:03
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    While of course mistaken, my impression is that this is fairly common for a certain cohort of math majors. – Daniel R. Collins Feb 12 '16 at 05:21
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    I shared very similar thoughts in an answer elsewhere on this site http://math.stackexchange.com/a/67772/15328 – Bill Michell Feb 12 '16 at 10:09
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Mathematics beyond introductory courses (like Calculus) involves less and less hand computation. There are sort of two tracks you may follow. An emphasis in "applied" mathematics will likely involve learning to do "the hard computations" with a computer. For example, simulations that use extensive linear algebra and or solve differential equations. These are far too complicated to solve by hand and typically don't have nice solutions that can be easily written down. An emphasis in "pure" mathematics will be focused on proofs, logic, and deduction.

Personally, I never memorized the "wrote techniques" that you mention. I instead focus on understanding the motivation, and if I need to recall a method or result, I can look back. For example, in Calculus there is a process (taught) for finding min/max values of a function. It goes something like: take a derivative, set it equal to zero, solve, then use 2nd derivative test to determine if it is a min/max or inflection point. I don't worry about "the process" instead, I just remember that derivatives tell you the slope. At a min or max, the slope is flat (i.e. 0). So if I want to find a min or max, then I want to know where the derivative is 0. To find that, first I need the derivative, etc, etc.

Trying to memorize everything, and just regurgitate it later, will get you almost nowhere in mathematics.

TravisJ
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    this is awesome information. So-- as a TA if I were asked something about finding a z-statistic in Stats 2 or continuous compounded interest formula taught in highschool algebra, I wold just pretty much just look it back up and proceed instead of trying to memorize and regurgitate? I just hope forgetting all that useful stuff won't hurt me in Advanced math.

    This whole time I was worrying because I couldn't recall for example the half-angle formula for sine, cosine, and tanget and thought I had to remember everything from 3 years ago!

     – user152810 Feb 11 '16 at 23:57
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    You shouldn't forget completely how to do these computations. This is difficult to elucidate exactly, but you need the remember the 'sense' of these techniques: you need to know what they're for, when you can use them, and something of the character of the computation process. Personally, I have found it useful to have memorized the half-angle formulas, because they are frequently quite useful. – Sinister Cutlass Feb 12 '16 at 00:01
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    @user152810 If you want to remember the trigonometric identities, just learn the complex extension of the trig. functions and you will be able to derive not only all of your identities, you will have an extension for complex input and other identities you may never have known. – Simply Beautiful Art Feb 12 '16 at 00:32
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    @user152810, if it makes you feel better, I have a PhD in pure mathematics and I only began remembering trig half angle formulas after I taught calc I a few semesters in a row. The things you use a lot you'll remember. For everything else, it's just helpful to know what's out there and where to find it if/when you need it. – TravisJ Feb 12 '16 at 01:29
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Yes, this is exactly how it works. There is no 'Calculus 10' class that us super advanced math majors ace with a minimum of effort. It is all arguments and proof (and some computation when necessary) from here on out.

Computation and calculations will still be used, but they won't be the heart and soul of mathematics. They are only part of the package: you use them to produce illustratory examples for yourself and others, and also to clarify a more complex theorem or result.

If you go to grad school in mathematics and become a teaching assistant - or even if you don't, and occasionally do tutor work instead - you will have ample opportunity to remind yourself of, and remaster, the techniques and processes you learned in high school and calculus long ago.

Sinister Cutlass
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  • so you actually DO lose ALL of the math that you had studied such as Calculus and Trig?! Doesn't forgetting that information hurt your ability to do the high-level math later on though?

    Maybe it's because my major was originally Biology (where everything is rote-memorized) that I'm confused, because in Bio if you didn't remember everything from the first year and highschool there was no way you would do well in the advanced stuff.

     – user152810 Feb 11 '16 at 23:51
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    @user152810 Just because there aren't later classes that depend slavishly on the more basic classes from before doesn't mean you have to lose that knowledge. Part of becoming a scholar is becoming more auto-didactic - more adept at teaching and reteaching yourself complex things when necessary, with little to no institutional assistance. This is no skill to be sniffed at; sometimes, one only truly begins to master it when working for a master's degree or PhD. – Sinister Cutlass Feb 11 '16 at 23:57
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    @user152810 If you need help remembering Calculus and Trig, try answering questions on this site. This site has become my 'extra homework' so that I don't forget side math courses like trig. – Simply Beautiful Art Feb 12 '16 at 00:34
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    @user152810 - You don't lose it all. You forget little details here and there and look them up real quick -- or derive them on the spot. Anyway all of the earlier stuff becomes second nature over the years. I know this from experience and I'm just an engineer. – MackTuesday Feb 12 '16 at 02:25
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    I have a colleague (tenured math professor) who says he went into math because it was obviously the discipline that required the least amount of memorization. Which is of course true -- if not a motivation that I would completely condone. – Daniel R. Collins Feb 12 '16 at 05:27
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    @user152810: my experience (UK university, so different set-up) was that yes, by the time I took my final examinations I had forgotten a lot of the clever integration-by-substitution tricks that I learned in first year calculus. No, that doesn't hurt the ability to do high-level maths because anywhere except exams you can look things up, and exam questions are carefully designed to be completed using only the rote-learned facts that the examiners consider worth retaining. Besides which, the things from early on that I repeatedly needed to use later, I didn't forget. – Steve Jessop Feb 12 '16 at 12:49
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Why would you think that you would have bigger numbers and harder things to compute? That's why you have calculators!

And just as a thought, when do you think you truly encountered math? The point of the classes you're taking is to understand math at its deepest nature, not just the outside number stuff.

For example, what's more important, the quadratic formula or how you derive the quadratic formula? The summation formula for $\sum n^a$ or how you derive the formula?

At a deeper level, is it true math to be able to do $1+1=2$ or to explain why?

What do you think mathematicians do? They don't take given formulas and plug in really big numbers, rather, our job is to make the formulas and explain how they work, why the exist.

As you have noticed, rote memorization of every single darn page of your math book isn't going to be of much help. Imagine redoing Calculus, but instead of being given the formulas, you were taught how to make the formulas.

If you were taught that, Calculus would be majorly reduced as far as memorization. I taught myself the first half of Calculus, and, fearing I would forget (forget before I actually take Calculus), I learned a brutal lesson about math.

You have to understand the formulas, not just what they are.

Makes math a lot easier in the long run in my opinion.

Simply Beautiful Art
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The short answer to your question is yes. Things are proceeding normally in your undergraduate mathematics education.

My math studies were typical and looked like this:

  • Calc I - IV
  • Foundations (Logic, Set Theory, Proof Methods)
  • Modern Geometry
  • Probability and Statistics
  • Linear Algebra
  • Abstract (Modern) Algebra
  • Real Analysis
  • Discrete Math
  • Topology
  • Electives, Independent Studies, etc

My favorite courses were Abstract Algebra and Topology. Also there was a bunch of computer science in there. Your mileage will vary. TL;DR whatever.

Here is an excellent video on the history of mathematical thought. In just 50 minutes Dersch touches on about 5,000 years of material. Your education will parallel his exposition, sort-of.

Anyway, different programs at different schools might assign prerequisites among these as they please to suit their curriculum, but generally you will find that the "arrows" point downward (e.g. Calc is a prereq for the Algebras).

What I call Foundations above is the rite of passage of mathematics - it transitions you out of childhood and into adulthood. I know you're confused by these sudden and unexpected changes. It's OK, grasshopper, let it happen, and when it is done you will be a man.

You see, pure math is about what is true, how we know it is true, and how to make sense of it all. In high school you learned to find the roots of a quadratic polynomial, but in college you will learn why it is impossible to do the same thing for a polynomial in the 5th or higher degree.

As for lots and lots of calculation, well, you will be doing lots and lots of calculation, but it won't be numbers as such. Check out the famous problem of the Seven Bridges of Königsberg. There are no numbers involved (except seven, which is important), but to find the answer you must do computation in some sense. It is that sense in which things get interesting in higher mathematics. A once girlfriend of mine once pulled a math book off my shelf and said "there are no numbers". That means that it is a true math book, that you are a man.

If you really want to just learn to calculate stuff and solve real problems, applied mathematics is better for you. At Harvard, Bill Gates declared as an Applied Math major because it allowed him to take courses from all over the college that were otherwise separated from each other: economics, business, science, nothing was off-limits.

Math is the study of what we know, Applied Math is the study of how to apply it to the real world. For example, in Linear Algebra you will learn the theory of vector spaces, but in Applied Mathematics you will use vector spaces to model the human face.

The choice is yours. If you want to understand the universe and be smarter than everybody in the room and have them hate you because you're right about everything all the time, or if you want to teach for the rest of your life, then study pure math. If you want to make eigenface generators or do hyper-fast quantitative Forex trading or build shoe-computers to beat the roulette wheel in Vegas, then you should study Applied Math, or you could just completely douche out and study Physics.

It is a vast world, but it is modern; you conquer it with mathematics. Good luck.

QED

QED
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  • Excellent post! Thank you so much for the video! – user152810 Feb 12 '16 at 13:51
  • You really answered quite a bunch of other questions I had also! Thanks so much!

    I had been hearing the whole time from more senior math majors that the ''REAL math'' starts once I get to Intro to Advanced Mathematics and that I'll never look at math the same.

    Had no clue what they meant and didnt give too much thinking to it as they would never elaborate! I wish I would have gotten a more proper intro into this stuff, but thanks to this forum-- at least now I do!

    Thanks again.

     – user152810 Feb 12 '16 at 14:25
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I thought the more advanced math got, the more complex the number-crunching would get.

Yes, it's so advanced that you crunch arbitrary numbers¹! (Like $n$ or $x$ or $z$.) Sometimes you also crunch $n$-tuples of arbitrary numbers (called vectors), or more advanced number-like things called cohomology classes, or ideals, or...

¹a/k/a variables

And there are so many number crunching techniques that you really have to have a firm understanding of basic logic. You get a glimpse of the real fun in your final year of your undergraduate, but it really starts during your Master's or PhD and just gets better from there.

Enjoying hands-on computation with real numbers is important for understanding how to crunch arbitrary numbers, so I'd say it's great that you enjoy number crunching.

Maybe I just don't understand but it seems like all of the earlier useful rote memorization technique is going to get lost. IS this supposed to be the case?

Oh, rote memorization is for kindergarten. You need to remember very little, because you'll be able to quickly derive the formulae you need. (To derive the sine/cosine equalities – like double angle formulae, etc. – all you need is Pythagoras' Theorem and the law for exponentials $(a^m)^n = a^{mn}$).

Earthliŋ
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    And also knowing the restrictions on the law for exponentials: http://math.stackexchange.com/questions/1628759/what-are-the-laws-of-rational-exponents – Daniel R. Collins Feb 12 '16 at 17:24