Am I Missing the Right Approach to Studying Math?

[Heisenberg7]

Hello,

This month, I've solved around 700 problems in trigonometry. And, I probably went over 200 problems again to internalize a few concepts, but I still have that feeling that I'm missing something. Last week, I was practicing math in cafeteria (early in the morning) and my math professor came to my bench. He wanted to know what I was doing and I told him that I was going over some problems again. What he said was kinda strange to me. He told me that I shouldn't be doing that. He said, "Why are you doing that, you are not a fraud?". I didn't think of it that much at that time. I thought that I was doing the right thing. Later in the day, when we had math, he gave us a tough problem(relative to the frame of reference of my class, the problem is below, you can try it if you want). That problem was the second on the board, and as he was writing the fourth problem, I finished the proof for the second problem. I felt proud of myself, but what he said next kind of shattered everything. He told me, "We will see how good you will do on the test because on the test I will give you the problems you've probably never seen". Now, I can't for sure say if he said this to provoke me or not, but that's beyond the point. Through my whole math journey, I felt like just practicing problems would help me score great on everything. And it worked, till now. Now that I've changed schools, I have a chance to work with the best professors in my country (including the professor above). They work with our olympiad team. Now that I have a chance to work with possibly one of the greatest math minds in my country, I'm beginning to feel like I am nothing more but a trained product. My ideas are poor and I just feel like I'm not smart enough for math. I feel like all the past success of mine was just because of my training. I feel like my classmates would do 10 times better than me if they studied as much as I do(they are quite lazy). Is there anything I can possibly do to get my mind going somehow? I mean, there is no way this is about hard work. My classmates are extremely lazy yet they are a lot smarter than me. Is this just the matter of intelligence?

 

[Heisenberg7]

By the way, here is the problem:
Prove that for a sharp angle $\alpha$ this inequality holds:
$$\sqrt{\sin{\alpha}}\sqrt[4]{\tan{\alpha}} + \sqrt{\cos{\alpha}}\sqrt[4]{\cot{\alpha}} \geq \sqrt[4]{8}$$

 

[ohwilleke]

Studying enough has value and most people don't study enough.

But, the point of doing practice problems is to make sure that you have internalized the mathematical rules that they illustrate, including corollaries of those rules that aren't explicitly spelled out. It also confirms that your operationalization of the mathematical rules is correct and that you don't have a conceptual error that may not have been clearly stated in the presentation of the rules.

Once you have done that, however, you've done enough problems. There are diminishing returns to doing more problems when it is fresh in your mind, although you might want to do more problems if you are rusty and haven't done a certain kind of problem for a while.

Once you've done enough problems, perhaps work ahead, because often the hardest problems in your current lesson are elementary when you master the next lesson, and are intended to motivate your next lesson.

 

[Heisenberg7]

Studying enough has value and most people don't study enough.

But, the point of doing practice problems is to make sure that you have internalized the mathematical rules that they illustrate, including corollaries of those rules that aren't explicitly spelled out. It also confirms that your operationalization of the mathematical rules is correct and that you don't have a conceptual error that may not have been clearly stated in the presentation of the rules.

Once you have done that, however, you've done enough problems. There are diminishing returns to doing more problems when it is fresh in your mind, although you might want to do more problems if you are rusty and haven't done a certain kind of problem for a while.

Once you've done enough problems, perhaps work ahead, because often the hardest problems are elementary when you master the next lesson.

Thank you for your response. Yes, I agree with you. I followed that path 99% of the time. And, to be honest, it worked 90% of the time. But, this 10% is making me a huge problem. Our professor is planning to give us problems that we've possibly never seen. And, whenever I have to do a new tough problem, I rarely find an elegant solution. Often times, I spend 3+ hours just solving one problem. That's when I get this gut feeling that I am not smart enough. And, to be fair, I am not a very smart man. I've always been called out for my intelligence(sadly in the negative way) which kind of did get me to think this way. Surely I can internalize the most important problem solving methods, but when I have to invent a new way to solve a problem, it gets quite messy and often times I don't succeed. I am working really hard on improving myself and I simply don't want all this time to go into nothing.

 

[Heisenberg7]

Also, I can clearly see that something is wrong in my way of thinking, when my classmates who work 15 times less than me often see things before me.

 

[gleem]

Forget about comparing yourself to others. Are you doing well in your courses? If so do not worry. So they do not work as hard as you. Maybe you are overdoing it. As far as being trained that is why you go to school. If you didn't need training why go? How do you know how much time your classmates are studying?
.
your professor said: "Why are you doing that, you are not a fraud?"

I would take this as a compliment. To me, this suggests that you are doing well. It looks like he thinks your extraordinary effort is unnecessary and what are you trying to prove? Give yourself some credit. If you need more effort to be successful so be it. Most of us had to work harder than some of our classmates.

 

[ohwilleke]

Thank you for your response. Yes, I agree with you. I followed that path 99% of the time. And, to be honest, it worked 90% of the time. But, this 10% is making me a huge problem. Our professor is planning to give us problems that we've possibly never seen. And, whenever I have to do a new tough problem, I rarely find an elegant solution. Often times, I spend 3+ hours just solving one problem. That's when I get this gut feeling that I am not smart enough. And, to be fair, I am not a very smart man. I've always been called out for my intelligence(sadly in the negative way) which kind of did get me to think this way. Surely I can internalize the most important problem solving methods, but when I have to invent a new way to solve a problem, it gets quite messy and often times I don't succeed. I am working really hard on improving myself and I simply don't want all this time to go into nothing.

For better or for worse, you can't change your IQ. So there is no point in worrying about that.

There are always going to be people out there who are smarter than you, no matter who you are.

I taught myself high school geometry and precalculus, the entire three semester calculus curriculum, college level linear algebra, and college level discrete math, through self-study before going to college, and I was a tutor in advanced math subjects. I was a low key math prodigy. But, in my math major, there were other students who were as bright as the Sun when I was only as bright as the Moon. The smartest of them got her PhD and is now a maths professor, whose research specializes in non-commutative geometry and non-commutative algebra at the University of Edinburgh, in Scotland, right now. (This specialty is critical, for example, for a lot of the theoretical work being done now in quantum gravity work.)

So what?

I can do what I can do, and she can do what she can do, and you can do what you can do. School is about maximizing your own personal potential. It isn't ultimately about where you rank compared to everyone else.

My college professor father's goal when he was teaching was for his students to get everything right (rarely perfectly successfully even in his graduate program), not to sort the wheat from the chaff.

Go as far as you can. Look for opportunities to study with strong students in your subject and pay attention to how they handle novel new problems - ask how they did it if it isn't clear to you.

All other things being equal, students who view academics as about hard work and not primarily about ability do better than students who focus on having or not having the brainpower needed.

Take some time away from studying to watch the anime "We Never Learn: BOKUBEN" whose multiple seasons are primarily about this issue, if you need inspiration and to get a feeling for how to deal with the times when you are feeling inadequate, and to learn how to ask for help. (It's available at Crunchyroll, which isn't too expensive if you get the cheapest available subscription.)

Your time won't be wasted. I ended up not going into STEM and becoming a lawyer. But it is amazing how many people can't handle compound interest calculation, or working out how much tax is due on a tax form when you have the numbers to fill in and just have to add them up, or working out the math to make a divorce settlement or to compute the ownership interests in a corporation with several owners. And none of that math is even as sophisticated as trig. Every little bit of quantitative competence you develop will pay off in the long run, no matter what you end up doing, even if you don't end up actually doing trig calculations or integrals themselves.

Take one step at a time. You don't need a five year plan. Keep working and if you keep getting decent enough grades and understand enough to follow each new lesson, you're fine. Keep going until you can't do that any more.

 

[Heisenberg7]

For better or for worse, you can't change your IQ. So there is no point in worrying about that.

There are always going to be people out there who are smarter than you, no matter who you are.

I taught myself high school geometry and precalculus, the entire three semester calculus curriculum, college level linear algebra, and college level discrete math, through self-study before going to college, and I was a tutor in advanced math subjects. I was a low key math prodigy. But, in my math major, there were other students who were as bright as the Sun when I was only as bright as the Moon. The smartest of them got her PhD and is now a maths professor, whose research specializes in non-commutative geometry and non-commutative algebra at the University of Edinburgh, in Scotland, right now. (This specialty is critical, for example, for a lot of the theoretical work being done now in quantum gravity work.)

So what?

I can do what I can do, and she can do what she can do, and you can do what you can do. School is about maximizing your own personal potential. It isn't ultimately about where you rank compared to everyone else.

My college professor father's goal when he was teaching was for his students to get everything right (rarely perfectly successfully even in his graduate program), not to sort the wheat from the chaff.

Go as far as you can. Look for opportunities to study with strong students in your subject and pay attention to how they handle novel new problems - ask how they did it if it isn't clear to you.

All other things being equal, students who view academics as about hard work and not primarily about ability do better than students who focus on having or not having the brainpower needed.

Take some time away from studying to watch the anime "We Never Learn: BOKUBEN" whose multiple seasons are primarily about this issue, if you need inspiration and to get a feeling for how to deal with the times when you are feeling inadequate, and to learn how to ask for help. (It's available at Crunchyroll, which isn't too expensive if you get the cheapest available subscription.)

Your time won't be wasted. I ended up not going into STEM and becoming a lawyer. But it is amazing how many people can't handle compound interest calculation, or working out how much tax is due on a tax form when you have the numbers to fill in and just have to add them up, or working out the math to make a divorce settlement or to compute the ownership interests in a corporation with several owners. And none of that math is even as sophisticated as trig. Every little bit of quantitative competence you develop will pay off in the long run, no matter what you end up doing, even if you don't end up actually doing trig calculations or integrals themselves.

Take one step at a time. You don't need a five year plan. Keep working and if you keep getting decent enough grades and understand enough to follow each new lesson, you're fine. Keep going until you can't do that any more.

Thank you for your response. Your story is simply fantastic and your writing is so fine. I really had to compliment you on that. I will definitely try to dedicate some time into watching that anime of yours. Anyway, I have watched a few lessons on psychology and thinking by Jordan B. Peterson and I have come to a few conclusions. I simply do not know how to think properly. He mentions that you should first learn how to write and by doing so, your thinking skills will follow. And as you mentioned, there really is nothing one can do about their IQ but adapt to the situation. I've been putting a ton of pressure on myself for the past few years and I do believe that I need to let some steam off. I believe I've learned my lesson and if not, well, there will be more opportunities to learn in the future. Thank you for dedicating your valuable time to help me!

 

[Heisenberg7]

Forget about comparing yourself to others. Are you doing well in your courses? If so do not worry. So they do not work as hard as you. Maybe you are overdoing it. As far as being trained that is why you go to school. If you didn't need training why go? How do you know how much time your classmates are studying?
.
your professor said: "Why are you doing that, you are not a fraud?"

I would take this as a compliment. To me, this suggests that you are doing well. It looks like he thinks your extraordinary effort is unnecessary and what are you trying to prove? Give yourself some credit. If you need more effort to be successful so be it. Most of us had to work harder than some of our classmates.

Thank you for your response. I suppose this was more of a self esteem problem, but there certainly are some ways I can improve the final result (read my reply above).

 

[WWGD]

I suggest, for the sake of flexibility and your peace of mind/mental health, that you just accept that people at times can just be weird and random , and not overthink things.