Lack of Mathematical Ability or Maturity?

[S.R]

Lack of Mathematical Ability

I'm currently in Grade Nine (by the Canadian Education System) and enrolled in Tenth Grade mathematics, scoring top-marks. I've self-studied Algebra I-II, Trigonometry and Calculus I (not including integration). However, I've noticed when I participate in competitions my results are rather low, relative to my course average. When I try to derive formulas, for instance: cosine or sine law, I become impatient and study the solution. This however, expresses a lack of comprehension. My problem lies here. I fear that when I enroll in University leveled courses, the level of interpretation required, will exceed my mathematical ability. This, of course, is frustrating since I might pursue a degree in engineering or mathematics. My question is, how can I improve my mathematical reasoning abilities and avoid impatience? Or is this an difficulty that will resolve once I'm older? Any resources or advice would be appreciated, thank-you.

-S.R

 

[Quantumjump]

As far as I know, engineering mathematics requires a good training in exercise solving, rather than being able to produce proofs. It is mainly a question of calculational training, and you can reach a relatively good level if you take the time to do as many exercises as you can, starting from the simplest, then gradually improving difficulty. True mathematical maturity is really a matter of mathematicians, while for physicists (like me) or engineers, what matters more is whether one is able to perform valid calculation, and to know how to interprate them with respect to a certain "real system". When saying "physicist", I include both theoretical and experimental physicists, I am myself a Msc student in theoretical physics.

 

[S.R]

As far as I know, engineering mathematics requires a good training in exercise solving, rather than being able to produce proofs. It is mainly a question of calculational training, and you can reach a relatively good level if you take the time to do as many exercises as you can, starting from the simplest, then gradually improving difficulty. True mathematical maturity is really a matter of mathematicians, while for physicists (like me) or engineers, what matters more is whether one is able to perform valid calculation, and to know how to interprate them with respect to a certain "real system". When saying "physicist", I include both theoretical and experimental physicists, I am myself a Msc student in theoretical physics.

Thank-you for replying, however, I forgot to mention my concern isn't based solely on my potential profession, but rather my own interest in mathematics.

 

[chiro]

I'm currently in Grade Nine (by the Canadian Education System) and enrolled in Tenth Grade mathematics, scoring top-marks. I've self-studied Algebra I-II, Trigonometry and Calculus I (not including integration). However, I've noticed when I participate in competitions my results are rather low, relative to my course average. When I try to derive formulas, for instance: cosine or sine law, I become impatient and study the solution. This however, expresses a lack of comprehension. My problem lies here. I fear that when I enroll in University leveled courses, the level of interpretation required, will exceed my mathematical ability. This, of course, is frustrating since I might pursue a degree in engineering or mathematics. My question is, how can I improve my mathematical reasoning abilities and avoid impatience? Or is this an difficulty that will resolve once I'm older? Any resources or advice would be appreciated, thank-you.

-S.R

Hey SR and welcome to the forums.

The thing with maths is that often when you starting out, things won't make complete sense in a way that you deeply understand: it usually takes a bit of time for you to connect all the dots in a way that gives the deeper understanding.

Just remember that nowadays you have a lot of different resources so don't be afraid to use them. If you are stuck, post a question on these forums, grab a textbook from the library, do a google search and read documents (especially on university websites by professors in a particular field), ask a friend or your teacher, lecturer, professor and do more or less that kind of thing.

If you take a look at the kinds of questions asked on this forum, you'll see that there are lots of different kinds of questions for all topics asked here. The only real requirement that people like to see is that you have an effort to a) craft the question so that it's coherent and specific and b) made effort to try and solve the problem yourself or think about something (if it's not a normal problem per se) and provide your own thinking so that the rest of us can find out specifically what you have misinterpreted if you have done so.

 

[S.R]

Hey SR and welcome to the forums.

The thing with maths is that often when you starting out, things won't make complete sense in a way that you deeply understand: it usually takes a bit of time for you to connect all the dots in a way that gives the deeper understanding.

Just remember that nowadays you have a lot of different resources so don't be afraid to use them. If you are stuck, post a question on these forums, grab a textbook from the library, do a google search and read documents (especially on university websites by professors in a particular field), ask a friend or your teacher, lecturer, professor and do more or less that kind of thing.

If you take a look at the kinds of questions asked on this forum, you'll see that there are lots of different kinds of questions for all topics asked here. The only real requirement that people like to see is that you have an effort to a) craft the question so that it's coherent and specific and b) made effort to try and solve the problem yourself or think about something (if it's not a normal problem per se) and provide your own thinking so that the rest of us can find out specifically what you have misinterpreted if you have done so.

Thank-you for the advice, Chiro! I use several resources for mathematics, both online and through literature. The quote "what I cannot create, I do not understand," explains my concern. However, in my situation, I find it difficult to solve contest-type questions, where interpreting the problem from a different perspective is essential.

 

[chiro]

Thank-you for the advice, Chiro! I use several resources for mathematics, both online and through literature. The quote "what I cannot create, I do not understand," explains my concern. However, in my situation, I find it difficult to solve contest-type questions, where interpreting the problem from a different perspective is essential.

I understand your concern and it is definitely good that you are thinking in terms of this.

Just be aware though that for some things, it is hard to re-create a proof of someone else in the same way with the exact same understanding that they had.

There are many reasons for this, but my own thoughts for why include that a lot of the actual context for the proof is lost because we only see a highly polished final result and not everything that preceded it.

To get context, you usually have to look at many different sources from all different angles and generate that context yourself if you can't find the context that has been generated from the author of a particular proof.

One thing that might really help understand this statement, is if you go on google and search for all known proofs of Pythagoras' Theorem. You will find that there are not just dozens, but more than a hundred ways to prove this. All of these different things add context in their own unique way. There might be redundancy in a lot of them, but the idea still stands.

I guess the best advice I can give you to get the context is to always seek the advice of an expert. They will have spent a large chunk of their life building the context and will be able to disseminate advice in the best way possible.

People naturally congregate with one another, so it's actually easier to find expert knowledge than you might otherwise think. There are journals, committees, and specific fields for study.

Once you find these you can find out the people involved and the resources involved that relate to these people like textbooks, blogs, journals and articles amongst other things.

It's not an exact science of course and generally you will have to wade through a lot of stuff to find the important things (happens to all of us as truth is the hardest commodity not only to find, but to realize when you actually see it).

Also one other thing: sometimes you things you wish to find aren't always where you expect them to be and they don't always come in the form you expect them to come in.

So yeah in short, find ways to find context for a specific thing and then take it from there.

 

[S.R]

I understand your concern and it is definitely good that you are thinking in terms of this.

Just be aware though that for some things, it is hard to re-create a proof of someone else in the same way with the exact same understanding that they had.

There are many reasons for this, but my own thoughts for why include that a lot of the actual context for the proof is lost because we only see a highly polished final result and not everything that preceded it.

To get context, you usually have to look at many different sources from all different angles and generate that context yourself if you can't find the context that has been generated from the author of a particular proof.

One thing that might really help understand this statement, is if you go on google and search for all known proofs of Pythagoras' Theorem. You will find that there are not just dozens, but more than a hundred ways to prove this. All of these different things add context in their own unique way. There might be redundancy in a lot of them, but the idea still stands.

I guess the best advice I can give you to get the context is to always seek the advice of an expert. They will have spent a large chunk of their life building the context and will be able to disseminate advice in the best way possible.

People naturally congregate with one another, so it's actually easier to find expert knowledge than you might otherwise think. There are journals, committees, and specific fields for study.

Once you find these you can find out the people involved and the resources involved that relate to these people like textbooks, blogs, journals and articles amongst other things.

It's not an exact science of course and generally you will have to wade through a lot of stuff to find the important things (happens to all of us as truth is the hardest commodity not only to find, but to realize when you actually see it).

Also one other thing: sometimes you things you wish to find aren't always where you expect them to be and they don't always come in the form you expect them to come in.

So yeah in short, find ways to find context for a specific thing and then take it from there.

I appreciate your advice. I will definitely research more than one proof of a formula or theorem, to gain context. I will just have to practice and practice!

 

[20Tauri]

I wouldn't worry about it too much. A lot of people don't learn how to prove or derive things (other than geometric two-column proofs) until college. You've still got plenty of time to build up the mathematical maturity you're hoping for.

 

[homeomorphic]

As far as I know, engineering mathematics requires a good training in exercise solving, rather than being able to produce proofs. It is mainly a question of calculational training, and you can reach a relatively good level if you take the time to do as many exercises as you can, starting from the simplest, then gradually improving difficulty. True mathematical maturity is really a matter of mathematicians, while for physicists (like me) or engineers, what matters more is whether one is able to perform valid calculation, and to know how to interprate them with respect to a certain "real system". When saying "physicist", I include both theoretical and experimental physicists, I am myself a Msc student in theoretical physics.

I would have to disagree. I am more of a mathematician by training, but I have studied a lot of physics and engineering. It's true that they don't have to learn to do proofs. But it's not just a matter of doing calculations. If you have some insight into what's going on in the calculations, that's where you actually learn things that can be taken away from the calculation. In electromagnetism, for example, Maxwell's equations have a lot of intuitive meaning that cannot be obtained solely from plugging and chugging away with them. It takes some imagination. Perhaps, you can get by in classes by just plugging and chugging, but you would be left with no deep understanding of nature.

To the original poster: don't worry about it. I'd give examples of how mathematically immature I was, but it's actually too embarrassing to even say it (now, I am finishing a PhD). From what I can tell, this sort of thing is common among mathematicians. You have plenty of time.

Attaining mathematical maturity is kind of a long process, so it can be difficult to describe without writing a book. Some good starting points are the books Lines and Curves: A Practical Geometry Handbook (meant for high school students) and Polya's book How to Solve It. The first book is good for developing some skill at seeing visually why mathematical facts are true, and the second discusses good problem-solving strategies. It might also help to look at books about creativity or on learning how to learn (that covers things such as how to transfer things into long term memory). So, you need to look into better ways of understanding how math works, and the kinds of thought processes needed to solve harder problems.

 

[S.R]

I would have to disagree. I am more of a mathematician by training, but I have studied a lot of physics and engineering. It's true that they don't have to learn to do proofs. But it's not just a matter of doing calculations. If you have some insight into what's going on in the calculations, that's where you actually learn things that can be taken away from the calculation. In electromagnetism, for example, Maxwell's equations have a lot of intuitive meaning that cannot be obtained solely from plugging and chugging away with them. It takes some imagination. Perhaps, you can get by in classes by just plugging and chugging, but you would be left with no deep understanding of nature.

To the original poster: don't worry about it. I'd give examples of how mathematically immature I was, but it's actually too embarrassing to even say it (now, I am finishing a PhD). From what I can tell, this sort of thing is common among mathematicians. You have plenty of time.

Attaining mathematical maturity is kind of a long process, so it can be difficult to describe without writing a book. Some good starting points are the books Lines and Curves: A Practical Geometry Handbook (meant for high school students) and Polya's book How to Solve It. The first book is good for developing some skill at seeing visually why mathematical facts are true, and the second discusses good problem-solving strategies. It might also help to look at books about creativity or on learning how to learn (that covers things such as how to transfer things into long term memory). So, you need to look into better ways of understanding how math works, and the kinds of thought processes needed to solve harder problems.

This is the reply I was hoping for. Thank-you for the reassurance! It seems Polya's book aims toward my concern; especially by improving my reasoning and problem-solving abilities.

 

[genericusrnme]

mathematical reasoning abilities and avoid impatience

Mathematical reasoning;
How to Prove it - A Structured Approach by Daniel Vellerman
avoiding impatience?
that part just takes will power

As for mathematical maturity I'd say The Theory of Sets by the Bourbaki group is one of the best books for endowing someone with mathematical maturity. I'm not sure if this book might be a little too advanced for you at the moment though (not that it requires any previous knowledge, it just requires a certain ability to work with axioms which for some people takes a while to get used to).

I wouldn't say getting older alone will solve the impatience problem but getting older will (as long as you continue to study maths) solve the lack of mathematical maturity problem.