# If I can spare one or two hours daily (how to improve my higher math skills)

Question: If I can spare one or two hours daily for my own, what would you suggest to improve my math maturity? I am referring to higher math, not computational math.

Here is my little background: I am a HS calculus teacher by profession. Since I picked up graduate math study one year ago, I have been discipline in setting aside at least one hour each day to read and do proofs, especially in abstract algebra since I was told that it is the backbone of all other branches. I do see small improvement in maturity, but I would like to see more.

This thread is prompted by @mathwonk 's response to my posting in abstract algebra forum. Thank you very much for your time and suggestion.

Simon Bridge
Homework Helper
Question: If I can spare one or two hours daily for my own, what would you suggest to improve my math maturity? I am referring to higher math, not computational math.
Help others.

psparky
Gold Member
Help others.

I agree. You don't really know a subject unless you can teach it to others.

However, you are already a teacher!!!

Here's a thought, if you are only doing math, your maturation process will be slow. However, if you start applying your math to things of importance, I believe that's when your maturation will increase.

What I mean by that is to use math to solve real world problems. As an engineer myself, math seems fairly easy to me. However, setting up the math for a specific problem is always the challenge.

If you are already doing all that stuff, perhaps take more advanced college courses if it makes you happy.

Here's another question for you personally, if you do take major leaps in math....what are you going to do with it? Do you plan on advancing beyond the high school level or are you just doing this for the "fun of the math"?

Stephen Tashi
to improve my math maturity? I am referring to higher math, not computational math.

We need your definition of "math maturity". If you were a complete novice then it would help to study logic and set theory. Since you have experience in graduate level math, I think "maturity" in different branches of mathematics could mean different things.

My idea of a person with "basic" mathematical maturity is that they understand how to read and use mathematical definitions (as opposed to substituting their own concepts for those definitions), that they understand how to read and formulate basic logic (for example, how to deal with the quantifiers "for each", "there exists" and how to negate statements involving quantifiers). and that they are reasonably precise when they write-up mathematical questions, proofs, claims etc. (Of course, this includes using precise terminology.)

You can have basic mathematical maturity and still have a rough time with a new mathematical topic. A typical problem is that you encounter definitions and theorems and are left wondering "What is the point of all that?". Perhaps people with superior maturity effortlessly surpress this reaction. (I don't.)

esuna
Gold Member
Here's a thought, if you are only doing math, your maturation process will be slow. However, if you start applying your math to things of importance, I believe that's when your maturation will increase.

What I mean by that is to use math to solve real world problems. As an engineer myself, math seems fairly easy to me. However, setting up the math for a specific problem is always the challenge.
He is talking about abstract algebra and other proof based math topics. He specifically said "not computational math." The only way he's going gain mathematical maturity is to do proof after proof after proof. So spending an hour or two every day to read and do proofs is a great thing to do. Having someone who can act as a motivator and can help you develop refined proof solving techniques is also a good thing to have (maybe a professor/mentor?).

He is talking about abstract algebra and other proof based math topics. He specifically said "not computational math." The only way he's going gain mathematical maturity is to do proof after proof after proof. So spending an hour or two every day to read and do proofs is a great thing to do. Having someone who can act as a motivator and can help you develop refined proof solving techniques is also a good thing to have (maybe a professor/mentor?).

@esuna : I am sorry I came back to you late, for unknown reasons I did not get the email alert. Thanks for your response, I will keep on doing proofs. As for having somebody acting as a motivator or mentor, that is exactly the reason I sign up with this forum. Hope you have had a great Thanksgiving yesterday. Thanks again.

Help others.

@Simon Bridge : Good idea. But I think helping my fellow classmates is not a good idea, they have totally different attitude. They just wanted to have a "sneak preview" of homework and assignment, they do not want to learn, they need only the degree for job. The only possible arena would be online forum like this one. Thank you again.

We need your definition of "math maturity". If you were a complete novice then it would help to study logic and set theory. Since you have experience in graduate level math, I think "maturity" in different branches of mathematics could mean different things.

My idea of a person with "basic" mathematical maturity is that they understand how to read and use mathematical definitions (as opposed to substituting their own concepts for those definitions), that they understand how to read and formulate basic logic (for example, how to deal with the quantifiers "for each", "there exists" and how to negate statements involving quantifiers). and that they are reasonably precise when they write-up mathematical questions, proofs, claims etc. (Of course, this includes using precise terminology.)

You can have basic mathematical maturity and still have a rough time with a new mathematical topic. A typical problem is that you encounter definitions and theorems and are left wondering "What is the point of all that?". Perhaps people with superior maturity effortlessly surpress this reaction. (I don't.)

@Stephen Tashi : I am sorry for getting back late to you, for unknown reasons I did not get email alert. I think I am quite comfortable using ##\forall##, ##\exists##, ##\lnot##, etc., and doing set theory, therefore your second paragraph is probably not reflecting what I am looking for, but the third paragraph is exactly what I am going after.

This is especially true with graduate level algebra, most of its theorems are so convoluted (not as beautiful as those in the computational math I used to have) that left me wondering: "What is it all about?" and "Did anybody ever use this theorem before?"

Thanks again for your input, hope you have had a great Thanksgiving yesterday.

Simon Bridge
Homework Helper
Help people who are not classmates... like here.
Look for threads where you don't know the answer and try to figure it out.

This is especially true with graduate level algebra, most of its theorems are so convoluted (not as beautiful as those in the computational math I used to have) that left me wondering: "What is it all about?" and "Did anybody ever use this theorem before?"

You might find this interesting:

http://pauli.uni-muenster.de/~munsteg/arnold.html

Maybe slightly over-stated, but hey, that's part of the fun. I always sort of "knew in my heart" that the way math is taught is often wrong or at least missing something. Many very high level mathematicians at least partially agree with me. Advanced math textbooks are often a good reference for the final form of a theory and as a source of exercises to practice on, but they tend to be pretty weak at conveying an idea of what the subject is all about and why anyone would care, and it can also be difficult to extract the intuition from them.

Unfortunately, most of the books on abstract algebra that I suspect are good for that purpose are books that I have not actually read. Most of my own ideas of what abstract algebra is about come from using it in other subjects and thinking about the significance of the definitions and theorems for myself.

I'll give you the list of algebra books that I should have, but have not read, here:

Symmetry by Hermann Weyl
Visual Group Theory by Nathan Carter
Elements of Algebra: Geometry, Numbers, and Equations by John Stillwell
A Book of Abstract Algebra by Charles Pinter
Abel's Theorem in Problems and Solutions: Based on the lectures of Professor V. I. Arnold by V.B. Alekseev

I'm least confident about the Pinter book because I'm not familiar with that author, but the rest seem very promising, based on who the authors are, and things I've heard or read and so on.

Here are a few more that I have read parts of:

Galois Theory by Ian Stewart
Abstract Algebra: The Basic Graduate Year
http://www.math.uiuc.edu/~r-ash/Algebra.html
There's also a number theory book by Stillwell that's a companion to his algebra one that I mentioned, Elements of Number Theory.

Another book that's more of a special topic, but involves amazing geometrical intuition that would help with group theory in general is this:
Mirrors and Reflections: The Geometry of Finite Reflection Groups. It assumes you already know the basics of group theory, but it would be a great place to apply it in a more geometric context. Personally, my interest in the subject comes from the fact that it clears a lot of the fog surrounding the classification of Lie Groups/Algebras that I studied in grad school, and found lacking any intuitive glue to hold it together and make it memorable or explain where the results come from. I gave up on trying to do math research, but it's my hobby to try to put right a lot of the things that were thrown at me in grad school that were lacking in motivation and intuition.

I originally learned graduate algebra from Dummit and Foote, which has a lot of nice aspects to it, but in some ways falls into the trap that I mentioned at the beginning of not having the best motivation.

You can also consider looking into the history of math. Morris Kline's book is one example. He talks about the origins of some of the concepts in algebra.

If you are interested in applications, you should consider looking into coding theory, cryptography, and possibly physics or crystallography.

Another thing to keep in mind is, as I mentioned, sometimes learning other branches of math makes things about abstract algebra click more. If you study differential forms from a good source, then stuff like exterior algebras might make more sense, although if someone explained it well enough, you wouldn't really need to know it. Lots of stuff can be that way. Makes more sense in a different context. From topology, there's the notion of a fundamental group, which is another way to think about groups.

So, you can see that there's a lot of stuff to read if you want to get to the bottom of "what the subject is all about", which might be one reason why some mathematicians are tempted to skip most of the fun stuff and stick to the more boring, formal aspects. The good news is if you read everything I listed, there would probably be a lot of redundant material.

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Oh, and I meant to link to this

just as a concrete example of how many results have been robbed of the intuition that can explain them. If you find it difficult to follow, I do, too, but if you keep thinking about it and re-watching, eventually, Fermat's little theorem should become an obvious result.

@homeomorphic : Thank you so very much for your two responses, I totally agree with your view and I will surely keep your book list as future reference.

(1) Most professors (if not all) loathe to go informal, they prefer to stick to rigid formalism. Perhaps they just behaving like lawyers: Either out of habit or fear of getting caught up in their own words, they always talk in legalese-speak. But in fact what I look for is intuitive meanings, at this learning stage.

(2) You mentioned John Stillwell twice, I agree he is a great author. In fact, I used to read his history book from time to time in order to get a big pictures of all of the terms. Very friendly style.

(3) Also, you mentioned Morris Klein's history book. I know one Isreal Klein is the author of The History of Abstract Algebra. Is there any typo here?

Thank you again and again, hope you have had wonderful Thanksgiving Day couple of days back.

PS. I typed in your screen name at the beginning of this reply: "@homeomorphic". Does the system will alert you of this response?

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It's Morris Kline. History of Mathematical Thought from Ancient to Modern Times. You'd want to figure out which volume has the abstract algebra stuff.

I had a great Thanksgiving. Thank you.