How to Plan My Self-Teaching of Higher Maths

[Lewis M]

Earlier this year, I became bored and decided to begin learning math beyond my grade level. I soon discovered that I had greatly underestimated my abilities, and I could teach myself well into calculus (currently in grade 7). However, after losing my initial interest, I slowed down dramatically. The New Year has reminded me that, if I want to pursue my current interests, I first need to learn basic maths (calculus, ode, pde, etc.), and I have a limited amount of time to do so. This leads to my original question: How can I plan my self-teaching in a way such that I am proficient in partial differential equations by the end of high school?

 

[Thewindyfan]

Make it a habit to read/practice a concept everyday for at least 1 hour. Once it's become a habit for you, it'll be hard to stop unless you lose your interest in learning more.

 

[Mark44]

Earlier this year, I became bored and decided to begin learning math beyond my grade level. I soon discovered that I had greatly underestimated my abilities, and I could teach myself well into calculus (currently in grade 7). However, after losing my initial interest, I slowed down dramatically. The New Year has reminded me that, if I want to pursue my current interests, I first need to learn basic maths (calculus, ode, pde, etc.), and I have a limited amount of time to do so. This leads to my original question: How can I plan my self-teaching in a way such that I am proficient in partial differential equations by the end of high school?

It's hard to give meaningful advice without knowing which areas of mathematics you are compentent in right now. You didn't list several areas of mathematics that are important, such as geometry, right-triangle trig, circular trig, algebra, and precalculus functions. You can learn most or all of these on your own, but how do you measure that you have really learned these subjects?

 

[S.G. Janssens]

I first need to learn basic maths (calculus, ode, pde, etc.)

ODE and PDE are not "basic maths". Or maybe I never got beyond that.

How can I plan my self-teaching in a way such that I am proficient in partial differential equations by the end of high school?

What is your criterion for being proficient in PDE?

 

[Lewis M]

Make it a habit to read/practice a concept everyday for at least 1 hour. Once it's become a habit for you, it'll be hard to stop unless you lose your interest in learning more.

That seems as if it would be incredibly helpful to me.
To answer your other questions, I suppose I incorrectly assumed that there was a mathematical "trail" to follow, so I thought it was implied that I was proficient in Algebra, Geometry, Trig, and Pre-calculus. At least I know better now. I have tested my proficiency with MIT's online calculus work. Also, my justification for learning these maths is my interest in quantum physics and electrodynamics. I intend to pursue these after school, but I guess it's still a bit early for thinking like that. Anyways, thank you all for your help.

 

[Andreol263]

Hi Lewis M, 2 years ago i got the same dream, when i was in the first year of the high school i decided that i should pursue higher grounds in physics and mathematics, not only because the math of the first year is simple and boring, but because of curiosity and love with these 2 disciplines, and i had the same aim, understanding 'advanced' eletromagnetism and quantum mechanics, i know that it won't be a simple task, but i started with learning basic calculus(1), i taked so long to learn integrals that i almost give up, but my will is strong, and i don't give up in that moment, some time after i learned the integrals(finally!), after that I've pursued ODE, PDE, Linear Algebra, and other things.today I'm reading(trying!) Jackson's Classical Eletrodynamics, a famous book for graduate students, and I've read some chapters Griffiths Introduction to Qua Mechanics one year ago, and it's not that difficult, and next year is my final year in high-school, so, before i even started the university, i learned things that before appears simple impossible to me..,now i have new aims!, learning Solid-State Physics and QFT, it's hard, but with willpower and curiosity, YOU WILL GET THERE!:D

 

[Lewis M]

Hi Lewis M, 2 years ago i got the same dream, when i was in the first year of the high school i decided that i should pursue higher grounds in physics and mathematics, not only because the math of the first year is simple and boring, but because of curiosity and love with these 2 disciplines, and i had the same aim, understanding 'advanced' eletromagnetism and quantum mechanics, i know that it won't be a simple task, but i started with learning basic calculus(1), i taked so long to learn integrals that i almost give up, but my will is strong, and i don't give up in that moment, some time after i learned the integrals(finally!), after that I've pursued ODE, PDE, Linear Algebra, and other things.today I'm reading(trying!) Jackson's Classical Eletrodynamics, a famous book for graduate students, and I've read some chapters Griffiths Introduction to Qua Mechanics one year ago, and it's not that difficult, and next year is my final year in high-school, so, before i even started the university, i learned things that before appears simple impossible to me..,now i have new aims!, learning Solid-State Physics and QFT, it's hard, but with willpower and curiosity, YOU WILL GET THERE!:D

Your story is quite inspiring, as the beginning almost replicates my mindset throughout the last few months. Integrals have been a horrible challenge, but I believe I can manage them. Thank you for the support.

 

[Calaver]

Hi Lewis M,

Just to let you know, I have been doing things similar to you, except that I am in tenth grade and started in ninth grade (by the US system - even if it may not look like it because of the time when I am writing this) so you started two years ahead of me (perhaps wrongly assuming that you are in the US)! So my situation is similar to Andreol263, but it looks as if I am a year behind him. All this is to say, it can be done!

However, there are a few questions that you should ask yourself. One of these is what Mark44 asked: Have you gotten all the prerequisites down and how do you know for sure? When you say you are testing yourself via MIT OCW, what do you mean by this? Are you taking the tests and doing the homework, or just watching the videos?

There was a mistake I first made when I was teaching myself via OCW and other sources - that I needed problems to practice! Although in theory you should be able to learn the entirety of mathematics without doing a single problem, humans are fallible and thus problems are a great way to test the material as well as learn it much more quickly than simply hearing a lecture.

The next is more obscure, but still necessary: How is your thinking? Are you practicing critical thinking, problem solving, and logical deduction in your studies? It is important that you ask yourself "Why?" at each step of learning, because even if you don't understand the full explanation at the time (e.g. the phenomenon of magnetism is something you will just have to accept until you learn more about physics, Feynman mentioned this in an interview once), this question is what science is based upon and thus, since you are learning science, it is (at least it seems to me) of greater importance that you learn the way of thinking than the results.

An aspect that you might not have gotten to yet (if you have not taken High school geometry) is mathematical proof. This may not be an obvious thing to study by yourself, which is why I mention it. If you do not know how to write one or what one even is, then most certainly at least look them up, as I'm sure I would not do them any justice here.

The final question you should ask yourself is one you have already touched on: Why do you want to learn this material? You stated previously that you were going down this trail for quantum mechanics and electrodynamics. Why do you want to learn those, specifically? Also, what is preventing you from learning physics right after you learn the math? Certainly, math will make things easier, but I don't think that one could pick up a quantum mechanics text without a decent understanding of classical mechanics, even if they have learned all the subjects that you listed. So you may want to consider learning physics along the way.

Anyhow, these are a few questions that after nearly a year of going through similar self studies as you I thought might be valuable. I certainly don't claim to be the final authority on the topic, and someone with more experience may very likely scrap all that I said and offer you (and me) better advice. But I hope that these might spur your thoughts until then.

Oh, and one last piece of advice that I wish someone would have told me a year ago: People (even friends and family) may not give you any type of congratulation or encouragement for what you are doing, at least in the first year or so. However, you will gain something that I cannot explain sufficiently in words, but most closely it is a new appreciation for the structure of nature. Once again, someone with more experience can probably phrase this more precisely and elegantly, but I think this explains the gist of what I am trying to get across.

 

[Lewis M]

Calaver:
In order of your points made and questions asked:
I am in the US.
I am taking the tests and doing the work.
I may need to clean up my prerequisites, but I'm getting along fine now.
I am interested in the smaller characteristics of matter that make our world what it is.
I will try to incorporate physics in my learning, which may get rid of some of the boredom of studying a single topic for days on end.
I appreciate your help and encouragement, and I am glad to see that I am not alone.

 

[Calaver]

Calaver:
In order of your points made and questions asked:
I am in the US.
I am taking the tests and doing the work.
I may need to clean up my prerequisites, but I'm getting along fine now.
I am interested in the smaller characteristics of matter that make our world what it is.
I will try to incorporate physics in my learning, which may get rid of some of the boredom of studying a single topic for days on end.
I appreciate your help and encouragement, and I am glad to see that I am not alone.

Sorry that I haven't gotten back to this thread until now! It sounds like you're on a good track, assuming that you are doing well on the tests and problem sets. These questions weren't only meant for you to answer right now, but also as examples of what you can ask yourself as you are going down this path during each step of the way. Question 1 secures the fact that you can proceed into the material, Question 2 ensures that you don't become a quack while learning the material nor are you bamboozled by a false presentation, and Question 3 makes sure you don't waste time on areas that a) don't interest you or b) aren't necessary or aren't recommended.

 

[Lewis M]

Sorry that I haven't gotten back to this thread until now! It sounds like you're on a good track, assuming that you are doing well on the tests and problem sets. These questions weren't only meant for you to answer right now, but also as examples of what you can ask yourself as you are going down this path during each step of the way. Question 1 secures the fact that you can proceed into the material, Question 2 ensures that you don't become a quack while learning the material nor are you bamboozled by a false presentation, and Question 3 makes sure you don't waste time on areas that a) don't interest you or b) aren't necessary or aren't recommended.

Thanks, though I think it helped to write it out and organize my thoughts.

 

[Calaver]

Thanks, though I think it helped to write it out and organize my thoughts.

Glad that it could be of some help to you!