How many exercises should I do after each math lecture? (Calc 1/2)

[TGV320]

Hello,

I am currently self studying MIT 18.01 in my technical college, having just finished watching the 3rd lecture. I do my best doing the assignments and pre-reading of every lecture. I find the lectures by professor Jerison to be really rich and intuitive, and I write down loads of notes after each of them.

The courses have assignments and some simple exercises courses coming with them but I do not know how much exercises I should really do to master the subject well. The book that comes with the course(I was able to get it ) has really quite a lot of exercises. I read on the web that an average of 3 hours for every lecture hour of math is required. I am willing to use quite a lot of time on math (spending weekends in the library from 9 am to 9 pm is ok for me).

Should I over-do each lecture in order to master the subject? Are the assignments enough or should I add up more of them?

Thanks

 

[symbolipoint]

I cannot say about "MIT" nor of "MIT 18.01". But some comments...
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The courses have assignments and some simple exercises courses coming with them but I do not know how much exercises I should really do to master the subject well.

Do not limit yourself. Do as many as you want, choosing as much variety as you can. Remind yourself, that your goal is "master the subject well."

The book that comes with the course(I was able to get it ) has really quite a lot of exercises. I read on the web that an average of 3 hours for every lecture hour of math is required. I am willing to use quite a lot of time on math (spending weekends in the library from 9 am to 9 pm is ok for me).

Good! You then have PLENTY of exercises to choose and you can choose to work on as many as you can, of as much variety as possible. That time ratio is a bad guide. Do you believe that 3 hours of study + homework for every 1 hour of "class"/lecture is enough? Even worse, one must not restrict study time to just "weekends" and one must not try to stuff all of a week's study time into some restriced time period once or twice per week. You need to STUDY EVERY DAY!
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[PeroK]

Should I over-do each lecture in order to master the subject? Are the assignments enough or should I add up more of them?

It really depends. I tend to know when I've mastered something. Some additional exercises may help, but it can also be a waste of time - you should be moving on to the next topic.

Personally, I prefer frequent revision. Go back a week or two later and test whether you still understand the previous topics. That's why it's good, IMO, to get more complicated problems that test a range of what you've learned.

Two tests I use are: could I explain the topic to someone? And, can I see how to solve the problems entirely without writing anything down? You don't need to be able to do all the calculations in your head, but see the whole solution in principle. If I can do those things, then it's definitely time to move on.

 

[Vanadium 50]

How smart are you? If the MIT 18.01 courseware assigns 20 problems, if you're twice as smart as an MIT student you can get by with 10. Four times as smart? You'll only need 5. Ten times? Only two.

 

[TGV320]

Hello,

Thanks a lot for the advice. I am actually starting to spread my study time on each day. Not very easy, but I can manage to get at least 3-4 hours per day during the week. I did overlook the importance of revisions, they seem to be much more important than I ever thought. The topic explaining method does seem quite effective. Ain't that called the Feynman method?

I (very) obviously am not as smart as an MIT student, therefore, I shall therefore double the efforts, and increase the focus on doing the exercises.

Thanks very much for the advice

 

[malawi_glenn]

Depends on what kind of excersices

 

[TGV320]

Hello,

Well, everyone here was just so right. The first two lectures were really just an introduction, and now after doing the homework of the third one, I really start to understand the need for much more practice and understanding of the course material. Just like in sports, understanding and being able to do the task yourself are two very different things.

Thanks

 

[PeroK]

Just like in sports, understanding and being able to do the task yourself are two very different things.

I don't find this to be the case. If you understand a problem and choose the right solution, then there is nothing physically stopping you writing that down. With sports on the other hand, once you've made the right decision, you still have to execute something physically challenging.

Sports is essentially rote learning for your muscles. I don't learn mathematics by rote.

 

[CrysPhys]

I am currently self studying MIT 18.01 in my technical college, ...

Perhaps we should back up a bit. If you are enrolled in a technical college, why are you self studying MIT 18.01? Doesn't your college offer calculus?

What else are you studying? "Do a zillion exercises until you master the material" is too simplistic a response. You need to balance your time commitment to 18.01 with your time commitment to other courses, and other aspects of your life.

 

[vela]

I don't find this to be the case. If you understand a problem and choose the right solution, then there is nothing physically stopping you writing that down. With sports on the other hand, once you've made the right decision, you still have to execute something physically challenging.

Sports is essentially rote learning for your muscles. I don't learn mathematics by rote.

The point is you don't get good at a sport by watching or reading about it. You actually have to do it yourself and practice. In the same way, you don't learn math by watching others do calculations or writing proofs. You have to do solve problems yourself.

 

[PeroK]

The point is you don't get good at a sport by watching or reading about it. You actually have to do it yourself and practice. In the same way, you don't learn math by watching others do calculations or writing proofs. You have to do solve problems yourself.

Perhaps, but an idea is an idea. I certainly can learn mathematics just by reading about it. I don't have to practise it necessarily. Understanding is often enough.

 

[Dr Transport]

All the exercizes in the textbook

 

[hutchphd]

It is also a good idea (IMHO) to spend a bit of time previewing upcoming exercises before the lecture. I call this process "quantifying your ignorance" .

 

[PhDeezNutz]

This is a tough question when dealing with freshman level classes because Each section has 100 problems or so. In higher level classes it is much more feasible to “do as many problems as possible” because an entire chapter has maybe 40 problems.

https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/pages/exams/

Since you’re self studying you don’t have the pressure of needing to make a grade. Do the assigned HW problems, take the tests, grade yourself and go from there. It may turn out you don’t even need to do more problems if your test grades are good enough.

 

[PeroK]

For example, suppose you are learning completing the square for quadratic equations. How many problems do you do?

I would do it once with $a = 1$, then once dividing through by $a$. Then I would have the idea.

The alternative is to do 10, 100, 1000 (?) problems all with different values for the coefficients.

The point is that a pro tennis player does practise hitting tens of thousands of routine forehands every year. A pro mathematician does not have to practise completing the square over and over every year.

That's the fundamental difference and why learning mathematics shouldn't be about rote memorisation but about understanding the algebraic techniques.

I reckon I would only do more than a handful of exercises ( literally no more than 5) unless I was struggling with the material and getting things wrong.

 

[PeroK]

What I do find is that for more advanced material I need to go through it three times to really nail the concepts. E.g. SR, wave mechanics, relativistic EM etc. I find it better to leave some of the harder exercises for the second and third readings

 

[hutchphd]

Perhaps, but an idea is an idea. I certainly can learn mathematics just by reading about it. I don't have to practise it necessarily. Understanding is often enough.

I take issue with that because unless you understand something, you are not capable of truly assessing your level of understanding. This presents an obvious conundrum. I know of three good ways out:

1. The toughest is to try to teach the material to some bright human.
2. Have an extensive probing dialogue with a good teacher
3. Do illustrative problems

One of the great surprises to me upon reaching my Biblical three score and ten years is that I remain perplexed about most things.

.

 

[TGV320]

Hello,

Thanks a lot for all the advice. I'll do my best . Actually my school does offer courses of math, but they are extremely basic and shallow, last only a year, and focus entirely on the computation, whereas the courses of western universities such as MIT focus much more on the reasoning behind.

The most probant example I have found is when the MIT calculus course teaches me how to prove that lim sin(x)/x =1 through two different very thorough geometric approaches, the thin textbook in my school tells me to just memorize that as a fact by directly by enumerating that as x approaches 0, sin(x)/x becomes 0.9, 0.998 then 0.99999 and so on.

Such is the state of things, I can't change them, but I can still have some control over what I study.

I haven't started grading myself yet, perhaps the time has come.

Thanks a lot

 

[mathwonk]

The fact that sin(x)/x approaches 1 as a limit as x-->0, is a consequence of the meaning of arc length. I.e. by definition the length of an arc of a circle is the limit of the sum of the lengths of secants joining points along the arc as those points become more numerous and closer together. Since the arc length is also the sum of the lengths of the subarcs determined by those points, and the dividing points may be taken equidistant along the arc, it follows that the length of each individual secant also approaches the length of each individual subarc, and that their ratio approaches 1. This is the statement that sin(x)/x approaches 1, since in radian measure, x is an arclength and sin(x) is the length of (half) the secant defined by the arc obtained by doubling x. I.e. this method shows that 2sin(x)/2x approaches 1 as x-->0.

oops, just noticed this thread is a bit old, but had fun making up this argument. In reference to the original question, when I first taught calculus, I was so afraid of being made to look bad by a question, that I worked all the problems at the end of every section in the book (Thomas) before the class on that section, at least until they began to seem repetitive.