I actually gave some thought to a career in math, but I’m still uncertain it’s the right path for me. As you say, it takes a while to develop a good intuition for every mathematical concept or theorem; though what really bothers me is that professors (and students!) don’t generally care about making math intuitive because, as you say, it’s ten times more difficult to do that than to explain proofs “superficially” and focus on calculations.
Moreover, I daresay that you never truly understand mathematical ideas in all of their possible dimensions (say, rigorously, algebraically, geometrically, etc.) at a first glance. Take the power rule in calculus, for example. It was explained as a trivial matter in class, but to me, it wasn’t necessarily that intuitive. Of course, you could prove it with simple algebraic manipulation, but I found it sort of magical that there’s this function nx^n-1 that gives you the rate of change of the function x^n. Later, on my own and a long time after I finished my introductory calculus course, I found there are specific geometric proofs you can do to show that, for example, the derivative of x^2 is 2x using the area of a square (and I think you can do this for x^3 too, using a cube). And that made me realize that, everytime I wrote down that the derivative of a function x^2 is 2x, I didn’t have a strong picture of what was really going on.
And the fact that these kinds of things were so overlooked in class made me feel a bit inadequate for math. Sometimes I wonder if I’m using the wrong approach and wasting time trying to deeply understand stuff which can’t be quite taken on by a beginner in the field.
That’s true; however, seeing the foundations for yourself doesn’t mean you will find out how everything fits together as a consequence. All proofs are logical, which is not to say they’re intuitive. Countless of times I have stared at a proof, clueless as to why a particular route to prove the theorem was constructed that way. Sometimes I may even grasp very well what’s going on, but I’m sure I wouldn’t have been able to come up with the proof myself if I hadn’t been presented to it before.
Is it really the case that you don’t accept anything without understanding? You probably have a really sharp mind then! Or maybe mine is way below average, I don’t know. Most of the math I’ve had in college has been about regurgitating procedures and, sometimes, regurgitating proofs too. I tried my best to think about them, but I couldn’t do much on my own.
Applied math wasn’t that different either. I could see the applications, yet the emphasis was more on those applications rather than on proofs/intuition. Take the case of the measures of skewness. There are many coefficients you can use, but I was never taught why one was preferred over the other or if they’re just “arbitrary” formulas that were developed as needs go.
It could be I have a biased view of college math because I never took a class specifically for math majors. Do you think that may be the problem?
All in all, thank you a lot for your help! I was leaning more to the humanities, but now you’ve made me reconsider the sciences again
First of all, I have to thank you again Juillet for your advice! It’s been very illuminating for me, and I’m endlessly grateful to you for not giving up on me yet (I’m a tough nut to crack, I know).
I’m not sure I know why I’m this much of a skeptic. I think it’s because I don’t find anything to enjoy when reading something without being able to engage deeply with the material.
Maybe this feeling stems from my high-school years. Although I was a good student, I didn’t quite enjoy my classes, and even less studying for tests. Yet eventually, during my last year, something clicked in place for me. I realized that the reason I didn’t quite like studying (in spite of my curiosity) was that I attempted to memorize everything that was handed to me. I crammed successfully for most of my tests (I’m not proud of it) because I was a responsible kid, but I didn’t take the slightest pleasure in the process. That’s probably around the time when I found out about Khan Academy (trying to cram for a math test, probably), with a more pedagogical teaching approach (I think Sal emphasized intuition in many videos), and thus I delved into a (still incomplete) process of reshaping my math knowledge, which was mostly based on memorizing procedures (honestly, I had no idea what I was doing when I subtracted, say 496 from 1000).
I also read in philosophy class about the different kinds of discourse (myths, dogmas, philosophies, science), and figured out that, to succeed academically in the future, I had to change my approach to studying for all subjects. Because I couldn’t be dogmatic like believers (no offense) if I wanted to be a smart academic (I was thinking of my future career at the time). I had to think creatively, use my curiosity and whatever intelligence I still had.
So yeah, I think that’s the reason why I’m so concerned with memorization: because it takes me away from studying.
Sure, but it’s evidence based on authority. And while I respect those authorities, I don’t want to “learn” in that sense. There would be no real intrinsic difference between, say, listening to my professor’s lecture or the priest’s sermon at my local church. None at all. They’re just different ways of becoming devotees to the way other people perceive the world to be like.
I’m not sure I agree with this. Indeed, it’s easier to first listen and study and later do research in grad school or beyond. But it need not be the only way. Science could be taught in a different way, less like a dogma and more like the creative (and historical!) process it really is. There’s a reason why so few students are into the hard sciences (and engineering). Something must be going wrong in our education system.
I appreciate the role of textbooks in presenting new topics that students have never even heard about. But making them accessible doesn’t mean they should be dumbed down. In fact, I would say trying to make textbooks accessible has led to an overemphasis of exercises and problems, just to make the reader feel he/she is doing well up to the point (and so if they didn’t quite grasp the concept, a correct application of it will make them feel they can continue).
Oh, most probably! It must be the same thing that happens in economics: some economists (usually the ones that do policy) have a tendency to state facts more categorically (i.e.: when X happens, Y will definitely happen), as they need to look like they know what they are doing. On the contrary, researchers would be a bit more wary of what they say and predict (especially in their papers), as they know the economy is a complex machinery to affirm anything very precisely.
Ok, I’ll search for a good book when I’m in the mood to learn psychology!
(Though after what we said, doing psychological research is not crossed out of my list, unlike clinical psychology)
Will check all of these too! Mastermind sounds good, does it cover deep thinking in educational environments?
I’m going to talk about my experiences doing math since I didn’t explain everything thoroughly in my first post. Maybe math will turn out to be the right major for you, maybe not. This might be long…
Which math classes have you taken? If you’ve taken the big intro courses, like Calc I - Calc III, it’s pretty easy for this kind of stuff to fall by the wayside. The professors know most people are only there because they want to get on with their non-math majors. They probably prove the results, but realize that most people don’t care. They probably don’t put the “theory”, so to speak, on exams. Then this mismatched emphasis is transmitted to you, and you don’t get a good picture of what the subject is like.
OK, maybe I lied. It’s probably impossible for you to have 100% intuition about everything that is going on in math, no matter how hard you work. You’re never going to be able to geometrically visualize every result. I agree that these visualizations are cool to see, but they’re not always possible. When you derive the power rule, it’s an algebraic manipulation using the definition of the derivative and the binomial expansion. This doesn’t give you a good visualization. Maybe you can figure it out for the derivative of x^2, or x^3. But what about a complicated polynomial like 5x^17 + 12x^3 + 4? What if you work in n dimensions? How are you going to see that? What about if you work in algebra or a field of math that is very difficult to produce these nice “pictures” of what is going on?
I think what is characteristic of math is that you don’t HAVE to seek out perfect geometric proofs of each specific derivative. In math, you abstract. The point is not to figure out derivatives of functions. The point is to understand that the derivative can be thought of as a rate of change, or, especially in higher dimensions, a linear approximation of a function at a point. Once you understand this, it doesn’t really matter what the derivative of a certain polynomial is, unless you are trying to solve a specific problem where it’s important. The important point is that you have the concept of derivative, and you know you can tell how functions change. As an elementary example, when you multiply 54 by 7, you don’t picture 378 objects in your head. But it makes sense to you because you know what multiplication means.
However, it is a problem if you have no idea what the derivative is, like if you just don’t know what it means for the function f(x) = x^2 to have the derivative 2x at every point. Then you just need to work to understand the meaning. You should work on it until you do feel comfortable with the abstraction of the derivative.
Personally, I fell in love with math the day my calc professor showed us a geometric proof for the sine double angle formula. Geometry is cool and a very important tool. You will get to see lots of geometric intuition. But you’re not going to always have geometric pictures for everything, and lots of times it’s going to be too complicated to hold in your head while you work. If that bothers you, math may not be for you (but then the rest of the sciences are probably out too).
No, you’re supposed to try to deeply understand it. But I can assure you that you shouldn’t be worried if you still don’t feel like you have a complete grasp on it, even if you’ve gotten great grades. I can pretty much guarantee that you won’t get it the first time. Calculus (and the rest of math) is kind of subtle. Even if you feel like you’ve understood it, you’ll probably end up realizing that you haven’t, and you’ve missed something. This is totally normal.
My math professors assure me that since math is just a series of logical steps, anybody can follow it. I’ve doubted them from time to time - sure, I might be able to follow the steps, but what if I can’t fit the big picture together in my mind? I still struggle with that sometimes, but every time I’ve been certain that I’ll never be able to understand something, I’ve eventually figured it out. That’s anecdotal, of course.
About the logic of proofs: first, if you don’t understand the intuition behind a proof, it’s possible you’ll understand it in the future. Second, some proofs are just glorified algebra. There is not that much behind them to grasp, if you’ve thoroughly understood all the definitions. Third, sometimes, it seems like proofs pull some crazy trick out of nowhere, and you’re left wondering “where on earth did they come up with that? I could never do it!”. That’s fine. It’s probable that the way they came up with that proof is by seeing other people use similar techniques on other proofs. And it’s probable that you’ll see that technique over and over again, and eventually it won’t feel so crazy to you, and you’ll be able to prove things with it, and people will wonder how you ever thought of it. And so the circle will go on. It’s mostly a matter of practice and exposure, like a cultural thing. This is not because math is full of “tricks”, exactly, but it is a really old subject where you learn how many other people have solved problems. So you have to get used to that. Generally, I admire proofs for their cleverness, even if I know I couldn’t reproduce something similar.
Haha, it’s definitely not that case that I never accept anything without understanding. In my experience, I can accept some things without understanding and pass the exam. But typically, if I want to go further in math, I really do have to understand things. As an example, I thought I was fine producing Taylor series without a total understanding… well, I wasn’t! It was fine for Calc II, but it’s not fine anymore that I’m studying complex analysis. You go back and forth and learn things. Every time you revisit something, you understand it a little bit more.
On a deeper level, you have to accept some concepts like “number” without understanding. You can try to axiomatize these things… I don’t really know that much about it. But you could look into logic/philosophy of math. Still, I think it’s much better than accepting something like orbitals in chemistry without understanding.
I also recommend you find a professor to talk to. It’s hard to do things on your own. Finding a good professor can also help you see what math is like, and whether you like it or not. If you think you might be interested in math, I would definitely try a class oriented towards math majors because it’s a different flavor, which you should try before you decide if you like or dislike it. Try to take it with a good professor.
And I haven’t done any applied math, so I can’t comment much. It’s different though.
That was a very in-depth reply, Vinnatan! Thank you! I probably can’t say anything too relevant at this point of the discussion, so feel free to ignore some parts of my answer. You’ve already helped me tremendously. (By the way, I’m writing this in two different messages because it’s getting too long.)
I have taken Calculus I and II, along with Linear Algebra. None of these courses were for math majors, so you’re probably right in saying that theory was unimportant from the get-go.
However, I’m still unsure that things would change much in a class for math majors. While I imagine they are more willing to make the effort of learning these subjects deeply than other students, they probably also feel the pressure to keep a good grade and thus have to memorize a lot of stuff they don’t understand just to hang in there (unless they’re geniuses).
I might be biased though, because I’m speaking from my experience. But since all the swans I’ve seen are white, I’m inclined to believe the rest are white too (which is fallacious reasoning indeed ;))
That’s true, I can’t replicate my reasoning for polynomials (I would have to find a way to relate the area of a two dimensional body to that formula, which seems impossible) nor for higher dimensions. I felt happy having some visual interpretation of the derivative of x^2 and x^3, but I didn’t want to go further because I probably knew the same process wouldn’t work for any function (now I see it’s downright impossible).
Thus far, Algebra has been an impossible field for me to understand intuitively, but I know that there are certain branches of math, like Topology, that make some of the results (like N-dimensional equations) be a bit easier to grasp. I’m just speaking on hearsay anyway, feel free to correct me.
I don’t mean to have a geometric proof for everything. Sometimes they might not even help to understand the concept. I mean, just as an example, the cross product of two vectors gives you the area of a parallelogram constructed by those two. Why would that make the concept of a cross product useful? I have no idea.
Indeed, we don’t always need a visual picture of the math we do. However, multiplication needs no proof (unlike derivatives), as far as I’m concerned. Sure, we memorized the multiplication tables, but we still know perfectly well why they work. Because if we sum 7 five times, or 5 seven times, we get 35. Or if you want to see it geometrically, the area of a rectangle of sides 7 and 5 is always 35. Hence, we can use that intuitive result for an algorithm when multiplying bigger numbers (just like we do in sums and subtractions): we decompose the number 54 into 50 and 4, and do 4x7 + 50x7 (and if you want to, decompose the 50 into 10x7x5).
With stuff like the power rule, that’s not so obvious. The algebra works out, just like it does for the multiplication algorithm, but it isn’t self-evident (and you can’t resort to summing the number 7 fifity-four times if you feel like checking the result you got).
I know the concept pretty well, but when it comes to applying it, not so much. For example, the division or chain rule can be easily proved, yet they seem so magical! And this is especially the case with integrals, to the point where I know perfectly well what it means (area under a curve, infinite summation of rectangular areas - Riemann sums-) but I haven’t the slightest idea what I’m doing when I use it for calculations (integration by parts, partial fractions, etc).
It bothers me not to have any intuition at all (be it geometrical or whatever). Do you really think that STEM subjects are not for me, then?
Thanks, that motivates me to carry on with my learning approach. You pretty much feel this in all subjects, it’s very difficult to say “I know everything I could possibly know about this particular topic”. There are always different interpretations to be explored of every topic (be it from the sciences or humanities).
Well, as I’ve been discussing with juillet, I’m not very happy accepting something in the belief that I will understand it better in the future. It’s not very satisfying. I don’t mean to perfectly understand it, but I want a partial intuition or justification at the very least.
Regarding the algebra, it really bothers me to take it for granted. Sure, it might make sense, but if there wasn’t anything beneath it, then 50% of the subject is probably impossible to intuit. So many proofs involve algebra manipulation…
I agree with you about the third point. It must be a matter of practice too, though I’m sure it would slightly annoy me if I decided to take math seriously.
Ah, I also have lots of issues with the Taylor series. I could see why it generated a polynomial, but just as we were saying regarding proofs, I feel like I could’ve never come up with that on my own. When you’re presented to it, it’s logical and intuitive enough. Yet you aren’t truly getting it, in my opinion, if you wouldn’t be able to replicate the result yourself (unlike, say, derivatives, where it’s easy to see what motivated the concept).
Heh, it’s not like I want to get that deeply into the roots of mathematics to the point where I would question the concept of a number. It wouldn’t probably be of much use. I know math is about humans making sense of symbols we believe represent aspects of reality (counting is so natural to us), and I’m not bothered by that.
Other kinds of axioms, well, they deserve more than a fleeting thought. Imagine studying euclidean geometry before there were non-euclidean alternatives. If you had spent time pondering about and questioning the parallel postulate, your professors would’ve said you were beating your head against the wall. Now, however, we know that there are non-euclidean geometries, so those kinds of questions wouldn’t look so stupid in class nowadays.
I might end up doing that, I lose nothing for trying. Thanks again for your help!
