How can I make myself like math more?

[Sean O]

Hey everyone,

I'm in my first year of electrical engineering, so as you can guess I'm not really taking any serious engineering courses yet - just general math and science. I'm pretty good at math... but I don't enjoy it that much. I've gotten by so far on my talent alone, but now that we're getting into more complex integration problems, I know that I'll need to do practice homework as well as the assignments we get, in order to do as well as I want to. The thing is, I find it hard to concentrate on math homework for a long time. Taking periodic breaks helps, but I still end up taking way longer than I should to finish my homework, just because I need to keep coaxing myself into concentrating. Once I do concentrate, I can usually solve the problems, though.

Anyway, like I said in the title, how can I make myself like math more? I don't expect to end up loving it with a passion, but I would like to be able to see it in a way that makes it more satisfying to me. Any input would be appreciated.

~Sean

P.S. Someone might bring this up, so I'll address it in advance: the upper year courses look very interesting to me, and I know that I can do well in them if I build a solid foundation in my first and second year, so there's no doubt in my mind that electrical engineering is the right degree for me. When the math is applied to something that I'm interested in, staying focused is easy, and I rarely need to take breaks. My concern at the moment is mainly just to find a way to be more interested in pure math, for as long as I need to take pure math courses.

 

[qntty]

What math course are you taking?

 

[Mike.]

The harder you try now sean the more you will get paid later on.
I've just started a mechanical degree, Its only 1 day a week (work the other days on plant) And maths is the hardest (compaired to mechanical science and Other subjects)
But companies like degrees with skills so keep in there.
Just think of the money.

 

[Dadface]

Draw up a timetable and stick to it.Practise your maths in short sessions with regular breaks between.

 

[FredericGos]

In my opinion, the way math is taugth in elementals & high schools is a complete disaster. The human mind is not wired up to start at arithmetics and elemental algebra. At least if you are a bit like me, which your post suggests. ;) The human brain is far better at abstract patterns than exactness.

Let me tell you a little story and try to explain why I mean that.

When I was young, I was pretty good at math. It came easily to me, but it wasn't interesting. Why? Because I allmost allways understood everything I was taught. boooring. And also probably because my teacher was crap. Attention span failure. The result was that, I got the impression that math was boring and had no application to real life apart from calculating the total of my shopping cart at the supermarket. But hey, yes, I also was a lazy teen that rather wanted to play with my commodore 64. So i became a selftaught programmer, and has so far done very well.

At some point, I got interested in 3d graphics. Read some books and found out that i needed the math after all. This started a long life selfstudy of mathematics and physics. Why? Because I didnt understand the math behind 3d at first and I became curious. Today I'm reading all kinds of stuff I don't understand... Doing that is what motivates me to learn the underlying boring stuff like Calculus, Linear algebra, differential geometry, abstract alegbra, etc etc. So, for me, it becomes a recursive process of reading something, no understanding, reading some more elemental stuff and working with it, then back at the interesting stuff I didnt get, and wupti, understanding it. Then on to something more advanced I don't understand.

So, basically what I'm saying is that, if you are like me, you should read some advanced/abstract stuff first (without getting frustrated of not getting it)

The only problem I have doing this is that sometimes, my 'elemental' knowledge is full of holes and I have to go back to something REALLY elementary before proceding. Just the other day, I was reading some group theory, and had to take my elemental agebra book form the shelves and read something embarassely elementary in order to proceed. ;)

/Frederic

 

[Tac-Tics]

Here is the one thing that always irritated me about math in school.

The teacher always starts out a class by writing some formula, set of rules, or algorithm on the board. Then the teacher procedes to prove what he or she wrote was true implicitly by the following argument: "If you don't believe it's true, you will fail the test."

To me, the fun part of math is that, at least in theory, it is all verifyable. Other than a few basic premises (set theorey, integer arithmetic, etc), you can prove to yourself what is true and what isn't. A teacher should simply help you understand the methods used to prove theorems in math. They are not an academic authority. They just tend to make fewer logical mistakes.

How much you can work out yourself varies from subject to subject. Engineering disciplines usually follow the form of high school classes. In high school, you learned x^n / x^m = x^(n-m) whenever x /= 0. College level calculus does the same damn boring thing. If y is a function of x and x is a function of t, then the chain rule says that dy/dt = dy/dx * dx/dt. Of course, these "dx's" aren't numbers. They aren't even formed in any technically sense, so rules like this are WORSE than the rules for algebra, in a sense. How can the student verify that it is OK to multiply them in this sense? Answer: they can't.

The more interesting subjects are the ones which focus less on calculations and methods and more about proofs and theorems. Basic analysis is a beautiful subject, because it takes all the nasty crap you find in calculus and shows you how to prove it's true. It starts off showing the "proper" way to formulate the real numbers (hint: a real number isn't just it's decimal expansion). Then it goes into explaining how to formulate limits, followed by the thousands of ways you can use limits: derivatives, infinite sums, Riemann integrals.

Additionally, set theory is absolutely essentially for any maturity in mathematics. It seems like almost anything at all can be formulated in terms of sets and functions between sets. Once you understand the ins-and-outs of sets, everything else becomes much simpler.

 

[future_phd]

I think when it comes down to it, you're not going to enjoy every single subject that you study even if it's related to the field that you want to study. You've said that you know that the field your in is right for you, so I think it just comes down to grinding through the areas you're not enjoying, using the fact that you will use this stuff in areas that you DO enjoy in upper years. It's not so much about "liking" the stuff you're doing now (if you don't like it, not much we can say here will change your mind), it's more about staying "motivated" enough to grind through the areas that you don't like by realizing that they wiill be used later on.

 

[turbo]

Hi, Sean. Are you putting yourself under time-constraints, and feeling pressure? That can certainly dampen your enthusiasm for a subject. I was bored to tears in HS, and when I entered college (engineering school with an Honors program to keep up with) I found out that some courses were not very fun, but it was mostly because they were throwing out advanced material very quickly. The math and chemistry courses were very compressed, and engineering students were expected to absorb those materials at roughly twice the rate as students in non-engineering curricula. I found out that I enjoyed the classes MUCH more when I could take the time to get ahead of the lectures, recitations, homework, etc. If you have a friend in a problematic class and he or she really seems to enjoy the material, perhaps you could form a little study-group. The extra motivation (and insights) that you might gain from studying with another person might help.

Good luck!

 

[Sean O]

Hey guys, thanks for your replies so far.

@ Dadface: That's what I already do. It works well enough, but having to take so many breaks makes it take longer to finish. I expect to want to take at least a couple of breaks, but the fewer I have to take the better. That's why I'm asking about finding a way to enjoy math more.

@ Frederic: Well, you're definitely right about the elemental stuff being boring. I think a part of the reason I'm having difficulty doing math homework without taking lots of small breaks is because it's all so repetitious and procedural. I definitely prefer to learn things as a conceptual framework, and then apply that framework to deal with specifics, later. Also, I haven't been reading up on abstract math specifically, but I have been reading up on some of the upper-year EE subjects I'll eventually be taking, and even though much of it doesn't totally make sense to me yet, I still find it interesting. Seeing an application helps me to focus, I guess.

@ Tac-Tics: Yeah, I'm definitely not a fan of the way math is being taught to us. Basically what my profs do is tell us what the concept we're about to learn is, write a proof that shows why it's true, go through a couple of really easy examples, and then give us challenging homework afterwards. The "homework" we are given is actually a series of assignments that all end up being marked, which annoys me because a) there's no transition between the very basic outline we're given in class and the more challenging homework problems, and b) we have no practice homework whatsoever, so we don't get much of a chance to become fully comfortable with the subject matter before we're marked on it.

This wouldn't bother me so much if they actually spent time trying to teach us how to think like mathematicians, or at least tried to give us a broader conceptual framework to approach problems from. But instead, all they do is tell us a procedure to use for specific kinds of problems (IBP, substitutions, etc.) and have us repeat the procedures over and over without properly explaining the intellectual/conceptual aspects of it. It honestly seems like we're being taught as though we're circus animals; we're just trained to do certain things over and over again, with no effort on the part of the trainer to help us understand what it is we're doing or why it's important.

Granted, after doing extra homework and analyzing the questions, I've started to build up a conceptual framework for the subjects we've covered, and it seems to work pretty well for me most of the time. Still, I find it hard to stay focused on the course itself since all we're doing is this mindless input-output routine, and this sucks since less focus = lower marks, even if I have much more potential than that.

I would love to be taught/teach myself a proper conceptual framework and/or systematic way of thinking about mathematics (this, I'm assuming, is partially what Frederic was getting at when he suggested reading up on the abstract stuff). Are you saying that learning Basic Analysis and Set Theory would give me a stronger mental foundation to learn math concepts from, Tac-Tics? If so, then I'd really appreciate it if you could direct me to any resources I could use to teach them to myself. I get the impression that the entire rest of my engineering education would be much, much easier if I could teach myself to have a solid foundation in mathematical thinking skills.

@ turbo-1: The pace of the course isn't so bad - it's more that I don't think we're being taught properly, as I described in my response to Tac-Tics. Too much repetition of mindless procedures just makes me zone out. I find that I'm very motivated when I can learn something based on a conceptual understanding, rather than regurgitation. The thing is, at the moment I find that I have to do extra work and analysis on my own in order to give myself a conceptual framework, which I guess does produce more pressure for me since I have this extra responsibility to teach myself, on top of all the assignments I get.

The thing is, once I have a solid conceptual understanding of a subject, I'm instantly a master. One of the reasons I always did very well in high school math, in spite of the fact that I never did homework and never even took notes, is because the concepts we were learning were pretty basic, and easy to ascertain simply from listening to the teachers' lectures. In university, however, the subject matter is deeper and more complex, as you know. This would normally excite me - however, we're not being taught the concepts, only the procedures, and finding a balance between the time I need to teach myself the concepts, and the time I need to do the assignments, has been tricky so far. It's frustrating, and it definitely has an effect on my motivation/enthusiasm.

 

[chiro]

Hey guys, thanks for your replies so far.

@ Dadface: That's what I already do. It works well enough, but having to take so many breaks makes it take longer to finish. I expect to want to take at least a couple of breaks, but the fewer I have to take the better. That's why I'm asking about finding a way to enjoy math more.

@ Frederic: Well, you're definitely right about the elemental stuff being boring. I think a part of the reason I'm having difficulty doing math homework without taking lots of small breaks is because it's all so repetitious and procedural. I definitely prefer to learn things as a conceptual framework, and then apply that framework to deal with specifics, later. Also, I haven't been reading up on abstract math specifically, but I have been reading up on some of the upper-year EE subjects I'll eventually be taking, and even though much of it doesn't totally make sense to me yet, I still find it interesting. Seeing an application helps me to focus, I guess.

@ Tac-Tics: Yeah, I'm definitely not a fan of the way math is being taught to us. Basically what my profs do is tell us what the concept we're about to learn is, write a proof that shows why it's true, go through a couple of really easy examples, and then give us challenging homework afterwards. The "homework" we are given is actually a series of assignments that all end up being marked, which annoys me because a) there's no transition between the very basic outline we're given in class and the more challenging homework problems, and b) we have no practice homework whatsoever, so we don't get much of a chance to become fully comfortable with the subject matter before we're marked on it.

This wouldn't bother me so much if they actually spent time trying to teach us how to think like mathematicians, or at least tried to give us a broader conceptual framework to approach problems from. But instead, all they do is tell us a procedure to use for specific kinds of problems (IBP, substitutions, etc.) and have us repeat the procedures over and over without properly explaining the intellectual/conceptual aspects of it. It honestly seems like we're being taught as though we're circus animals; we're just trained to do certain things over and over again, with no effort on the part of the trainer to help us understand what it is we're doing or why it's important.

Granted, after doing extra homework and analyzing the questions, I've started to build up a conceptual framework for the subjects we've covered, and it seems to work pretty well for me most of the time. Still, I find it hard to stay focused on the course itself since all we're doing is this mindless input-output routine, and this sucks since less focus = lower marks, even if I have much more potential than that.

I would love to be taught/teach myself a proper conceptual framework and/or systematic way of thinking about mathematics (this, I'm assuming, is partially what Frederic was getting at when he suggested reading up on the abstract stuff). Are you saying that learning Basic Analysis and Set Theory would give me a stronger mental foundation to learn math concepts from, Tac-Tics? If so, then I'd really appreciate it if you could direct me to any resources I could use to teach them to myself. I get the impression that the entire rest of my engineering education would be much, much easier if I could teach myself to have a solid foundation in mathematical thinking skills.

@ turbo-1: The pace of the course isn't so bad - it's more that I don't think we're being taught properly, as I described in my response to Tac-Tics. Too much repetition of mindless procedures just makes me zone out. I find that I'm very motivated when I can learn something based on a conceptual understanding, rather than regurgitation. The thing is, at the moment I find that I have to do extra work and analysis on my own in order to give myself a conceptual framework, which I guess does produce more pressure for me since I have this extra responsibility to teach myself, on top of all the assignments I get.

The thing is, once I have a solid conceptual understanding of a subject, I'm instantly a master. One of the reasons I always did very well in high school math, in spite of the fact that I never did homework and never even took notes, is because the concepts we were learning were pretty basic, and easy to ascertain simply from listening to the teachers' lectures. In university, however, the subject matter is deeper and more complex, as you know. This would normally excite me - however, we're not being taught the concepts, only the procedures, and finding a balance between the time I need to teach myself the concepts, and the time I need to do the assignments, has been tricky so far. It's frustrating, and it definitely has an effect on my motivation/enthusiasm.

When it comes to maths there's usually a lot of different perspectives to teach so trying to encapsulate a whole contextual understanding of a topic in the space of a lecture can be very hard to do, even for experts in their field (although an expert should have the capacity to perform this very task).

The thing is, there are tonnes of ways to understand something and because areas of math often overlap, sometimes understanding something in terms of one area of math say group theory, can help you understand something to do with say polynomials or calculus. Now this just takes practice. You usually have to do the boring work because often more than not, you might recall an instance where you worked out a problem and suddenly with some insight you understand the mechanics of what you were doing and relate everything back to a nice aesthetic way of understanding it in terms of the associated mathematical content.

If the professor just cheated and told you how to think then you would probably be a lot smarter but you wouldn't really be any better because you still don't have the contextual understanding that comes with years of experience. I'm not saying it couldn't be done because there are some brilliant minds out there that can turn absolute complexity into sheer simplicity, but I wouldn't bet too much on it with the exception of experts in their field.

Its like the poster said above. You can't be a programmer without writing code. You can't be an engineer without doing problem sets and you can't be a mathematician without doing math.

I'll give my 2c on some things to possibly think about when dealing with mathematical topics:

For a standard engineering degree its nothing more than using the definition of a limit and coming with a differential and then moving to the reverse limit of sums for an integral. It's as simple as that for beginning calculus. That covers the mechanics behind calculus. The rest is basically language manipulation. That comes with practice. The more practice you do with language manipulation, the better you will be able to predict the right integral or solution to a simple DE or ODE (PDE's are another story).

Just remember to take note of the mechanics of what you're learning. If you're learning about something involving a proof, take note of all the transformations of the mathematical objects (example equations, groups, sets, etc) involved in the proof. Basically everything in mathematics is transformed somehow by some set of operations on the object (the basic concept with groups in group theory). You can apply the same method for proofs because these things are just equations or set relationships or whatever. Try and interpret what the transformation is leading up to. Visualize each transformation.

It doesn't matter if your working with proofs or transforming a DE the same applies. If its a rotation group then try and visualize what happens with those matrices. Take notes of all the properties given and always interpret them. They mean something and although a lot of them are fairly obvious, its actually important to know them because a lot of properties depend on these things and some properties are not so obvious.

If you're stuck with a definition ask your professor to clarify it. Mathematicians strive to write clear unambiguous definitions, so asking them will signal to them that its not clear enough.

If you're doing statistics, you will get a few topics. Estimation and hypothesis testing is one, and probability is another. For probability learn to count (no sarcasm intended, I'm serious most probability questions are about counting situations). Draw tree diagrams of the scenarios and list them out if you're unsure (you'll be able to shortcut as you get better). Calculus based probability is simply summing probabilities with limiting sums. Discrete variables are nothing special.

If you have to cover programming, I strongly recommend that you have programmed enough to be able to model any program at the coarsest level (not including details with specific algorithms initially) and are able to do so in a structured manner. I say this because you're doing engineering and sometimes you need to program a computer. It's not a bad requirement anyway because eventually you'll have to do this possibly for exams.

If you don't understand what things really mean (ie you don't understand or you don't have perspective and depth to your understanding or both) then it will clearly show. You need to understand all basic calculus techniques because change is how we model things. If you need to calculate volumes for example, then you will need every bit of calculus to aid you possibly including parts, substitutions and anything required to solve that integral. Often things won't be in analytic form so you still need to understand what those limits mean and use numeric integration and be aware of the round off error that it generates.

The best thing I can tell you is if you think something is boring and not useful then lookup something more advanced and you will most likely find that you need that boring non useful thing to get through it.

 

[Tac-Tics]

Basically what my profs do is tell us what the concept we're about to learn is, write a proof that shows why it's true, go through a couple of really easy examples, and then give us challenging homework afterwards.

I remember this. Sometimes it's true that the homework is harder than the examples in class. But something I remember getting tricked a lot by this in college. The examples given in class are easier because you don't actually have to think about them. The professor gives you the problem, then the answer. The hardest you have to work at it is to VERIFY what they say makes sense.

This happened to me in Chemistry all the time. The professor would explain how to solve a problem. Take a log here, convert the units there, do a little playing with the formula to balance it, do some algebra, convert the units again, and you're done. It seemed so easy in class, I never did the homework problems. Then I did poorly on the tests.

In school, professors will generally only teach you the correct way to do things. Homework is vital because it teaches you the WRONG way to do things, which greatly outnumber the right ways.

(As a note, I never did my homework in school, because I was a lazy bum).

...we have no practice homework whatsoever, so we don't get much of a chance to become fully comfortable with the subject matter before we're marked on it.

I'd say you should go see the professor during his office hours or steal a T.A. or classmate to help you out. It's amazing how much having a homework parter can help. At the very least, you don't feel like you're the only dumb kid in the class =-P

This wouldn't bother me so much if they actually spent time trying to teach us how to think like mathematicians, or at least tried to give us a broader conceptual framework to approach problems from. But instead, all they do is tell us a procedure to use for specific kinds of problems (IBP, substitutions, etc.) and have us repeat the procedures over and over without properly explaining the intellectual/conceptual aspects of it. It honestly seems like we're being taught as though we're circus animals; we're just trained to do certain things over and over again, with no effort on the part of the trainer to help us understand what it is we're doing or why it's important.

This reminds me of practicing music. I get an hour of instruction every week from a wonderful teacher (just as a hobby). A student musician might take a class every other day. But only 1 to 3 hours a week of instruction will do nothing for you! Almost all your learning has to come from the dozens of hours you spend a week between those classes.

Class time, unfortunately, is very limited. Additionally, the professor has a number of constraints working against them. They need to move at a pace at the class's average. If a student is confused, it's very hard to answer questions satisfactorly without confusing everyone else.

But still, I think the most important role a teacher should play in your education is a motivational one. Dull teachers are the worst, because they would have you believe their subject is invariantly boring. If you can't communicate your enthusiasm for a subject to your class, you're not doing them much more than their textbook.

Are you saying that learning Basic Analysis and Set Theory would give me a stronger mental foundation to learn math concepts from, Tac-Tics? If so, then I'd really appreciate it if you could direct me to any resources I could use to teach them to myself. I get the impression that the entire rest of my engineering education would be much, much easier if I could teach myself to have a solid foundation in mathematical thinking skills.

Engineering and physics style mathematics is a messy ordeal, but always have clear applications. Pure mathematics on the other hand is very neat and orderly, following a definition-theorem-proof routine, but it's applications are often obscured by the generalization. My personal feeling is the best education you can have is to find a sweet spot in between: a rigorous, proof-based understanding of topics which have obvious practical applications. For this, nothing is better than elementary analysis, since physics is more or less just the study of applied calculus.

My favorite text on analysis is Introduction to Analysis (https://www.amazon.com/dp/0486650383/?tag=pfamazon01-20). It's a cute little book and only $3 used. I'm not sure how good the early chapters are, as it was my second book on the subject, and I mostly used it as a review. But it goes over the basic topics: a crash corse in set theory, the real numbers, metric spaces and continuous functions, limits, derivatives, Riemann integrals, and a few advanced topics.