How does advancing through the math major work?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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

Answer 7

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 $(a^m)^n = a^{mn}$).