Am I the only one that find ODE hard?

[Treadstone 71]

I love math, but it seems love does not conquer all: last semester, my average turns out to be B+, the bare minimum required to stay in the honours program. I told my professor that I study already a lot, and he told me that perhaps I was studying the wrong way. So, how does one study for math?

p.s. I got B+ in ODE, when everyone told me I should have gotten much higher because it's an extremely easy class. But it's not! All of analysis, algebra and adcal combined is easier than ODE! Am I the only one that find ODE hard?

 

[ranger]

So, how does one study for math?

I study (or practice) math by doing as many problems as possible. Knowing the theory is one thing, but applying it to a situation is another. This is where practice come in. Sometimes I search the internet for different tutorials on the same topic. This allows me to get a "better" point of view. I even search for math lectures online and found some damn good ones too, packed with examples and everything.

 

[Lisa!]

I love math, but it seems love does not conquer all: last semester, my average turns out to be B+, the bare minimum required to stay in the honours program. I told my professor that I study already a lot, and he told me that perhaps I was studying the wrong way. So, how does one study for math?

p.s. I got B+ in ODE, when everyone told me I should have gotten much higher because it's an extremely easy class. But it's not! All of analysis, algebra and adcal combined is easier than ODE! Am I the only one that find ODE hard?

1-sided love always sucks! I agree with ranger. Doing some problems(not easy ones ) let you know whether you've learned something or just memorized it!
Anyway if you tell us what's your way of studyng, perhaps we'll be able to tell you what's wrong with it.

 

[Treadstone 71]

The first thing that confuses me is the fact that my professor puts a strong emphasis that "math is not med", and that we should not memorize things. On the other hand, he is EXTREMELY tough on grading exams, especially about definitions. That is, if we don't reproduce a definition almost WORD PER WORD, we lose all marks. Seems to me the only way to do this is to actually MEMORIZE the definitions, isn't it?

I usually study by doing the old exams/assignments. However, the problem with these higher math classes is that you can be certain that the questions on previous exams will NOT be on the one you're taking. And since they deal mostly with proofs, either you can do it, or you can't. Doing old exams doesn't seem to be helping.

 

[Benny]

Just out of interest, why would an ODE exam have questions mainly on proofs? I would've thought that the emphasis would have been on computation with a bit of theory thrown into the mix.

Edit: What sorts of proofs are you referring to?

 

[Treadstone 71]

The higher math classes I was referring to were algebra and analysis. The difficulty of ODE, to me, comes from the fact that it's mostly an unecessary exercise in algebraic manipulation, rather than the actual material of "ODE". That is, you move the equation around until you have something that will fit the equation-solver, then you turn the crank. Too boring for me. There were no proofs in ODE.

 

[Lisa!]

The first thing that confuses me is the fact that my professor puts a strong emphasis that "math is not med", and that we should not memorize things. On the other hand, he is EXTREMELY tough on grading exams, especially about definitions. That is, if we don't reproduce a definition almost WORD PER WORD, we lose all marks. Seems to me the only way to do this is to actually MEMORIZE the definitions, isn't it?

I know what you mean! You know some people think that if you understand something, you're able to reproduce it word per word.

I usually study by doing the old exams/assignments. However, the problem with these higher math classes is that you can be certain that the questions on previous exams will NOT be on the one you're taking. And since they deal mostly with proofs, either you can do it, or you can't. Doing old exams doesn't seem to be helping.

Sounds like professors become more and more strict every year!
You knowthat's what I did too but only on the night exam. And for sure I didn't expect to see them on my exams. In fact, I just wanted to test my learning. And once I was able to solve these problems, I did well at my exam . Although the problems on my exam were usually more difficult!
Anyway I hope the problems of old exams wouldn't be the only problems you do for your studies?
and just 1 question: Have you asked some of your classmates who do well at the exams, the way of their studies? or perhaps the professor who said there might be something wrong with your way of studies?

wish you success in your studies

PS I hope 1 of the math professors reply to this thread. Certainly they can be a good help!

 

[mathwonk]

well i am teaching ode starting tomorrow. i expect people to have perfect ability to reproduce definitions in all math classes, because imperfect definitions are useless. one word omitted from the definition of a continuous function, or anything in amth really, makes it wrong usually. that is because in mathn the definitions are very precise and very economical. no word is there that does not matter.


as to ode, i always disliekd it as a student because it was so compoutationally oriented, turn the crank, get an answer. no thjeory to illuminate it, so I will include some theory myself.


the best book on ode i have found for understanding it is the one by arnol'd.


an ode is simply a field of vectors, one at every point of your surface, or of space, or of the line, or whatever is your ambient space. a solution is a particle flowing along your space and having excatly the right velocity vector at every point, namely the one specified by the given field.


its like a road with speed signs everywhere, a solution is a car driving along at exactly the right speed at all times.

then the compoutational, solution is a formula for the position of the car at all times. this is hard to get, since we know from calculus that the formula for an integral is often mroe complicated than the formula for the integrand. in fact in the whole world of functions, the integral of a known functiuon is usually an unknown function.

thus it is likely that in ode, even if the field of vectors is specified by known functions, i.e. those to which we have given names, the function giuving the solution, i.e. the position of the moving particle, will be given by some new function.

so in ode, unless we want to makew the unnatural restriction that we will only study problems with answers given by functions we already know, we must fgace up to the most difficult issue for begionning students: namely what is a function and how much dow e need to nkow aboput it to know it?


a student almost always thinks a function is something given by a familiar formula, but in ode this is not so for most equations. so he/she must accept that a function is given by some limiting proicess that yields a result that ahs no nkown name. this is tough.'

hence in ode one should be as flexible as possible about functions. the first step toward solving an ode should by just to be able to sketch a rough idea of the graph of the solution.


e.g. let all the vectors be of unit length and horizontal. what do you think the motion looks like having those vectors as velocity vectors?


now let all the vectors at each point be perpendicular to the radius from that point to the origin, and counterclockwise in direction. what would the motion be?

then ask yourself if you can really tell the difference between these vectors and ones that tilt ever so slightly in towards the origin, i.e. making angle 89 degrees with the radius. if this is the case, what would the motion be?

then in the final throes of the subject, put forward a few specific formulas for the vectors and ask whether the motion also has similar formulas.

almost the only cases where the answer is known and simple is where the formula for the vectors is just given by linear functions (with constant coefficients), such as one asked elsewhere on this forum recently:

y' = x+y.


try sketching the vectors for y' = -x and y = x'.

then try y = x' and y' = -sin(x).

 

[shmoe]

The definitions must be precise or you can end up with total nonsense. The more you use a specific one, the more you should understand what properties it's trying to capture, and the better the chance that you could make up an equivalent one from scratch from a colloquial description.

It's instructive to see what happens when you change a definition slightly and see how it affects things. For example, in the epsilon delta definition of the limit from first year calc, there's lots of < signs. Some we can change to <= without affecting the definition, but not all, find the ones that are flexible this way. This can help your understanding of why a given definition is the way it is, rather than simple memorization.

For studying 'proof based' calsses, really problems, problems and more problems is the way to go. The more you do, the more you should begin to understand how the theory fits together and how to apply it. Working with others can be a great help too, especially when it comes to explaining your own solutions. Studying the proofs of theorems is also important to give insight on what makes them work as well as the techniques used.

Doing poorly in a computational ODE class due to lack of interest isn't such a big deal in an honours program if you're managing in the more abstract courses better. ODE's are typically the 'hardest' plug'n'chug math course required in an honours program (yours may vary of course), and honestly the least interesting (when taught the plug'n'chug way). Aim towards the abstract proof based stuff with any electives you have if that's where your interest is.