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I am a fourth semester mathematics student. I have obtained great grades in Calculus/Analysis II & III. The first two semesters, I would say top 5%.

Have I cheated myself, as I did not use the notes from the professor, I used the books Calculus II & III from Gilbert Strang, that had many interesting examples. Thanks to this book I was able to ace my calculus exams. Examples give me the courage to solve the exercise sheets and proposed exercises. On the other hand, I have obtained mediocre results on my other modules where solutions where not "available" to me.

Now that more advanced modules are being taught to my cohort, the solutions are harder to come by, and this really demoralizes me, making me procrastinate more often than not, obviously leading to poorer grades.

Are amateur mathematicians that use this study method ie. that learn through examples and solved exercises, doomed to fail their mathematics career? Are we cheating ourselves? Mathematicians are supposed to be problem solvers, not plug and chug machines am I right?

What can I do to improve?

Thank you for your answers.

4 Answers 4

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In a lot of ways you seem to be doing the right thing, and if the exams are sufficiently challenging then more so. I'll assume that your goal is learning, and not just grades in the following.

In math, doing a lot of exercises is a good way to learn, and eventually to gain insight into the inner workings of something like analysis. Doing exercises give you reinforcement, but there is always the possibility that you will reinforce the wrong things. That is why feedback is also important to obtain. Normally, instructors should give you that feedback on assignments so that you don't go off the track.

You ask about solution manuals or solved problems online. If you use them for feedback then they are fine, though not perfect. But reading solutions to a difficult problem is a lot less efficient for learning than doing it yourself, no matter how hard. You can delude yourself, actually if you take shortcuts to learning. Ask yourself, can you extrapolate your learning to solve other problems that aren't so similar to the one's whose solution you've seen.

You don't say why you don't use the instructor's notes, so it is hard to comment on that. They might be useful or not, and they might be more useful to supplement your thinking before you go to a solution manual.

If you ask the instructor for hints when you get stuck then you might move faster to learning. But a wise instructor will try to give you minimal hints to help you avoid wrong turns, but letting you find the path yourself. Don't ask for answers. Ask for hints. Try to explain your reasoning.

But, do lots of exercises, more than required. And get feedback on your solutions even if you have to ask for it during office hours. Especially when you think you have an insight, talk to the instructor about that.

However, if the exams closely follow the assigned problems and you need help to solve those, then, yes, you might be deluding yourself. It would be a shame to finish a course only knowing how to solve some relatively small set of problems that had a structure that you had prompting for. Eventually, problems will arise that don't have simple online analogues. So, while it is useful to have "training wheels" as a kid learning to ride a bike, you need to take them off eventually, even if it means skinned knees for a bit.

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  • Well I try to do all the assignments, but when facing an exercise my first instinct is to try to find one similar on the internet, rather than trying to solve it on my own. Which I find shameful. I guess I don't have much confidence in my problem solving abilities.
    – user140047
    Feb 16, 2022 at 20:02
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You're comparing two different skills

  • On the one hand, you are expected to understand the nature of mathematics.

  • On the other hand, you are expected to understand how to research a solution.

I'm not a mathematician, I'm an engineer - and I do a lot of programming in my work, which I believe is a salient example of what you're talking about.

  • On the one hand, I'm very good and knowing how to express my needs in research to rapidly discover a solution to my programming problem.

  • On the other hand, I occasionally (and if I'm really being honest with myself... frequently) must re-research the same solution multiple times before I remember the solution because I'm not actually learning to better understand programming — I'm just finding a solution to my problem quickly.

But from a practical point of view, what's the difference?

These are two different skills: knowing the essential nature of your field of study vs. knowing enough about your field of study combined with great research skills to solve a problem.

If your intended career is in academia, you're probably causing yourself more harm than good because you need to know those gruesome details to effectively teach the subject to someone else. Why? Because the first time a student asks you "why?" you won't be able to derive an answer without depending on research.

But if your intended career is in not in academia, you're probably teaching yourself a more valuable skill — how to get to the correct answer as fast as humanly possible. Because, for example, in the corporate world, speed + correct = paycheck. How you achieve speed is often (perhaps too often...) irrelevant in the corporate world.

So it's important to understand that while your instructor may have a preference as to which skill he/she thinks you should be learning, both skills are valuable, both have their place, both have their pros and cons, and neither one alone will make you a good mathematician.

And to underscore that last statement: if your test scores are flagging due to a lack of good examples, that suggests you've been depending a bit too much on your research skills and not quite enough on your essential understanding. That's a valuable piece of information! and I'm glad you're learning it now rather than mid-way through your career when you can't do much about it. To be great at what you do, you need to be good at both skills.

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I think nearly everyone does better if there are example-solutions, example-proofs available. Prototypes! So, I'd say that the reason you do better in courses where worked-out examples are available is that you learn the material better and more easily. That's desirable! :)

Yes, there is some traditional viewpoint that math students are supposed to solve lots of problems "in a vacuum", but that is very inefficient. Having examples that one can more-or-less imitate is much more efficient, and is a reasonable way of learning. The teaching component of that is choosing and giving the worked-out examples.

There is no reason for students to have to figure everything out on their own... Sure, it is very important to engage with the material (rather than thinking that it's adequate to "memorize" worked-out examples), but engagement does not entail working in a vacuum, trying to re-invent the wheel.

Still, yes, it can be interesting to "try to do it yourself", if you have the time and inclination. Even so, success is not guaranteed, and even if you succeed to some degree, your solution may be inferior to the "best" version... which could be learned from someone else.

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You are not cheating.

It is nothing wrong in learning from examples or solved exercises. In fact in some courses like Probability, statistics , vector calculus( applied mathematics course) a student has good chances of learning subject and getting good grades if he has gone through examples and /or exercises. You are just putting your effort and time and there is nothing bad about it.

Also, let me tell you solving exercises and reading examples will always help you even if it is a pure mathematics course. Answers/ Proofs of some courses in Pure mathematics will take more time and effort to come by but that is due to abstractness, I am sure everyone goes through this. Mathematics is learnt by doing. Even if by solving exercises and reading examples you are getting good grades then what is the harm in that?

Probably, in you class there might be students who are able to get same marks as you but by solving less exercises. Don't get discouraged, everyone has different way of catching knowledge. I also learn by solving exercises and examples. But, the problem will certainly arise, if you compare yourself with your peers by the amount of effort which you see they put. What you are doing is not teaching.

Also, your mathematics career can't be judged by this only.

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