Mathematics and learning

Like many other students, I can’t deny that I spend sizeable amounts of my time on the Internet. I may not actively participate on any major forums, but I enjoy reading just about any such posts: it’s beneficial, because you can often pick up advice from people who are older than you and have more experience. I also feel there is much that can be learned from people younger than I am. Everyone has their own interesting perspective on things, and unique way of thinking.

A topic I’ve noticed a lot of interest in is how a person’s IQ relates to things like academic performance, the chances that person has at being successful in academia, and so on. Surprisingly, I’ve seen a lot of posts babbling things to the effect of “not everyone was created equal, only certain people are intelligent enough to do X”, and so on. It is of my personal opinion that this is, for the most part, nonsense. I’m not denying that there is some subset of living people who truly lack the capacity to, say, become mathematics professors. My point is that the last thing people should be telling these high-schoolers is “choose another field to work in, you probably wouldn’t make it in that.” Is this really the kind of thing we want to be pouring into ambitious children’s ears? Sure, in today’s digital society, it seems like everyone craves numbers and statistics. Scoring a high number on an IQ test doesn’t make you better than anyone else, nor is it a reliable estimator of how much you’ll achieve in life, or how fat your salary will be. There are plenty of people who are freak geniuses on paper and still contribute little to nothing notable to society. Let’s face it, the people who prance around displaying their membership to some “High IQ Society” are insecure, arrogant, and in general, you probably don’t want anything to do with them. Hard work will get you far, and that’s more or less the recurring theme of this post. To anyone reading this who is thinking of entering mathematics, I offer all my encouragement. Don’t think that you’re unfit just because you didn’t spend your childhood training for math contests.

In late high school, after correspondence with fellow students and a few books, I realized what mathematics was. It wasn’t about calculating dot products of vectors, or finding solutions to quadratic equations, or any of the other computational nonsense that pollutes the present-day Canadian high school math curriculum. It was about logic, deduction, structure, and beauty. From this point on, after reading from a couple of books that didn’t have too many pictures in them, I became cynical of everything. I didn’t particularly care about anything my Calculus teacher said; I learned the methods, the equations, whatever else I needed to get the marks in the course. And on the side, through these books, I was giving myself a small supplement of the superior mathematical background endowed upon many who came before me. Canada’s school system has folded. There is no longer anyone concerned about teaching critical thinking, abstract reasoning, or other important skills. It’s as though the country is begging to be left in the dust of intellectual progress. As students here are encouraged to be academically lazy, the theorists of tomorrow trickle out of schools at an ever-slowing rate. All the while, elsewhere in the world, the exact reciprocal is taking place.

This brings me to my next topic: success in university. It is not natural that someone can attend class all term, occupy themselves only marginally with their course material, and score near-perfect on exams. I’m not one of the “effortlessly near-perfect” people I mentioned. I’m one of many students who is burdened by making careless mistakes on exams or being presented with the odd exam problem that seems to be “just beyond” my reach. A lot of people are in search of some academic success secret. After searching for it myself, I feel ready to announce it. The good news is that the secret is very straightforward. The bad news is that for most people, this lifestyle change is not one that takes place easily. After enough analysis of my own learning patterns, I’ve concluded that the solution is solitary exploration. Studying with others is great in certain situations, but what I’m talking about is something you have to do alone, and most likely in front of a blackboard after class hours are over or something (unless you really don’t mind using up stacks upon stacks of paper). In the average day, a university student learns plenty to keep him or her occupied for a few hours just exploring the concepts, testing the methods learned, and so on. It’s almost spiritual. What class of problems can be solved by a given method? Can you show beyond a reasonable doubt that any problem solvable by a given method must be of that class? If not, perhaps you are missing something, and the method is more broad than the examples suggest. If this is the case, construct new problems; problems that have fundamental structural differences from the ones given to you. Try to solve them. If you have time, develop your own way of solving them, just for fun. If you’re in math like I am, come up with theorems about stuff – anything that seems intuitively plausible to you – and try hard to prove it. If you can’t, maybe you should try looking for a counterexample. You literally have to get your hands dirty, feel every ripple, every bump, every scratch in the concepts you’re studying. Become uncomfortably intimate with them. My point is that practicing things like this will train you a lot better to think critically about the material, the type of thought that often eludes you until the exam rolls around. Poking around in the dark forces you to develop some pretty hardcore concept-synthesizing skills. Assignments help with this, but at least in my case, my own exploration pays off a lot more. If you’re not being challenged by the assignments, this isn’t good either, and you should find some alternative source of problems that will truly push you beyond your comfort level.

Remember: if you’re not spending a significant amount of time each week struggling through problems, then you’re doing the wrong problems. The pain is necessary, and there is no alternative. So eat dinner, go to campus, find an empty lecture hall, and cover some blackboards. Godspeed.


About mlbaker

just another guy trying to make the diagrams commute.
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10 Responses to Mathematics and learning

  1. Future Math Student says:

    This is a great post. I plan on entering the University of Waterloo next fall to study Math and CS too, and I’ve actually been going through a similar experience to what you’ve described – my interest in calculus is minimal compared to my interest in math books and the like. I’m about halfway through “How to Prove it” and am loving every second of it. I was wondering, do you have any other recommendations for books that would be good preparation for next year?

  2. dx7hymcxnpq says:

    If you’re interested in pure math, you’ll love the advanced courses here, because they’re taught at a really theoretical level that doesn’t even exist at most universities. In this sense, you actually get to do “real” mathematics, for example even in MATH 147 (Advanced Calculus 1) you’ll be seeing concepts that are presented in third year courses elsewhere. The assignments for these courses really make you think a lot, and exams are almost exclusively proofs. There’s also an advanced section for first year CS courses (145/146).

    The textbook normally used for the calculus courses is “An Introduction to Analysis” by Wade, and for linear algebra, it’s “Linear Algebra” by Friedberg, Insel, and Spence. Another book I like is “A First Course In Real Analysis”, by Protter and Morrey (it’s a Springer book), which you could easily use as a supplement to Wade (Wade’s book is considered too terse by a lot of students here).

  3. dx7hymcxnpq says:

    They’re expensive books, but if you can somehow manage to get a copy just to look through it, and make sure you’re comfortable with the way it’s presented, that would probably be a good idea.

  4. Future Math Student says:

    Thanks for the advice. I’m definitely planning on taking the advanced math and CS courses, and I’m actually really looking forward to the switch from math classes based on computation to math classes based on rigor and proof.

    One Calculus/Analysis book I’ve heard a lot about is Calculus 3rd edition by Spivak, I was actually under the impression that the advanced calculus stream at UW used this book as its text… I think I’ll be able to dig up pdf versions of all the books you’ve mentioned, and Spivak, and have a look at all of them before I buy one for the summer :).

    Another question, do you know what text the Math 145 course uses? Its content seems the most interesting to me at the moment (not to say Math 147 doesn’t sound interesting!), but I can’t seem to find a copy of “Integers, Polynomials, and finite fields” ( says that’s the textbook) on amazon or any similar sites… I did notice that the author happens to be a professor at Waterloo, so maybe it’s just in use there? Would it be possible to get my hands/eyes on a copy of it, if that’s the case? If not, could you recommend a similar book?

    Also, I’m kind of curious as to what CS 146 entails. I’ve seen it mentioned several places, but when I try to look it up here it’s nowhere to be found.

    Anyways, thanks for the advice! Looking forward to next year 🙂

  5. dx7hymcxnpq says:

    Spivak’s “Calculus” (which I have much heard of but have not read) is a standard advanced calculus text; perhaps more standard than Wade, although Spivak’s most celebrated work is certainly “Calculus on Manifolds”. Another text I’ve seen used in analysis is Walter Rudin’s “Principles of Mathematical Analysis”, which has earned the nickname Baby Rudin (I own this book, and haven’t yet made it through the second chapter: “basic topology”). The last two books I’ve mentioned are considerably dense and as such are (probably) quite useless if you’re learning the material for the first time.

    The MATH 145 title you’re referring to is not a textbook, but a set of course notes. If you’d like, I can give you a list of topics covered, but I don’t have an electronic version of them. It is a unique course in that it whizzes over elementary number theory in half of the term, and spends the other half covering concepts like polynomial rings (irreducibility, etc.) transcendental numbers and field extensions. I took this course last term with Patrick Ingram, and found it extremely difficult compared to any other first-year advanced math course. I’d recommend “Abstract Algebra” by W.E. Deskins (which actually contains some elementary number theory), or “Modern Algebra” by Seth Warner (which does not). They are both Dover books (cheap).

    CS 146 will only exist starting this Fall 2011. It used to just be a single course, CS 145, which allowed you to finish first-year CS in one term (otherwise, you had to take CS 135 followed by 136). I took it last term and the class had about 65 students. It’s two courses now, so the acceleration aspect of it is gone. The man in charge of this course is Prabhakar Ragde so if you have any questions about it, he’s the one you want to talk to.

    Dr. Ragde is quite selective when it comes to admitting people into CS 145, so you may have to send an email and convince him. On the other hand, anyone can get themselves overridden into the advanced math courses, even with poor contest scores.

  6. Future Math Student says:

    Thanks for the advice, again! I’ve heard that Spivak is an excellent introduction to advanced calculus if you’re familiar with writing proofs, but looking at the reviews/description of “A First Course in Real Analysis” makes it also seem like it would be helpful. I think I’ll go through the first chapter or two of each book before I make a decision.

    If it’s not too much trouble, I’d be grateful for an outline of the topics covered! Thanks for the book recommendations too, they both look promising.

    Interesting that CS 145 is being split up into two courses. What kind of things would I want to say in order to convince Dr. Ragde to let me take the course? I’ve achieved high scores in all computer science courses taken but my junior contest score wasn’t great and I wasn’t signed up for the senior one.

  7. dx7hymcxnpq says:

    I sent an email to the address you entered with a table of contents for the MATH 145 notes. Gordon Cormack was the instructor for CS 145 in Fall 2010 when I took it. My CCC score was mediocre as well (I was obsessed with Python back then, and was forced to write it the contest in Visual Basic 6, because I wasn’t comfortable enough with C++) so I was a little scared I wouldn’t be allowed to take the course, but I sent him an honest email saying I was looking for a more challenging and mathematical introduction to computer science, talked about my interest in abstract reasoning and so on, and he was nice enough to give me the benefit of the doubt. Anyways, you could always just go talk to Dr. Ragde about the course, during your first week here. I can tell from your posts you’re pretty keen, and profs can sometimes see that easier if you go speak to them in person.

    One funny thing you’ll find about those courses is that during the first few weeks, your lectures will be absolutely packed. The walls will be lined with students who are there just to see if the flavor of the courses suit them. After those few weeks, the courses shrink by at least a factor of 2. I actually started out in mathematical physics (through Science), taking the non-advanced courses. After I made the switch, one thing I noted is that there is a tightly knit atmosphere that develops because everyone there is truly interested in math, and the professors may even get to know you, whereas in the normal courses, a lot of people are just like “blah, I hate this class” and are really only there to get the mark and get out.

  8. Future Math Student says:

    I got the e-mail and sent you a reply. Thanks for doing that!

    When do you normally sign up for courses? If I were to talk with Dr. Ragda during my first week would I have to transfer from one course to another? Would it make sense to talk with/e-mail him before I sign up for courses, and simply choose CS 145 then?

    I think I came across this sentiment in one of your previous posts, and am very keen on it. I’ve always been around the top of all my math classes, so it will be very interesting (and a blow to my ego I’m sure ;P) to take a math class with some of the brightest people in the country. Can’t wait!

  9. dx7hymcxnpq says:

    I think it’s in July for new students, but really there are a lot of people in your same situation who won’t actually be enrolled in the right courses until after lectures have started. It’s no big deal. You probably won’t be able to just autonomously enroll in CS 145 if you didn’t make the cutoffs for contest scores or whatever, but you can get him to add you to the course manually. Whether you talk to him about it ASAP or at the beginning of the term doesn’t really matter, but I’m sure an email wouldn’t hurt.

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