Studying Mathematics Thread

[Aliza]

Galois theory is actually why group theory got going. There is some wonderful structure here relating roots of polynomials, isomorphisms of fields into themselves, and the structure of groups.

In any case, the progression you give looks good.

if you have any questions, feel free to ask! BTW, what book(s) are you using?

I didn't realize that Galois theory got group theory going.  I really enjoyed group theory,  so now I have  more motivation to get underneath Galois theory.

I'm primarily using Abstract Algebra by Saracino.  For whatever reason, his method of instruction and his style of proof-writing work well for me; I'm grateful that he has written his book, as it has made abstract algebra pretty accessible to me.  On occasion, I'll open up A Book of Abstract Algebra by Pinter or A First Course in Abstract Algebra by Fraleigh if I find that a particular concept is a little vague in Saracino. Also, if I have any questions and I'm able to express them in a precise and coherent manner (IMO, this is an art-form in itself), then I'll post them here.  Thanks.

What attributes do you like about one particular book over another? Do you favor a more narrative description where the author writes like they're speaking to you, or do you like step-by-step solutions with notes on the side telling you what's going on? Or is it another attribute entirely, like examples being more like just general equations to solve vs. examples having a practical, real-world application that you can relate to?

[Fireball]

That's a helluva good question! Every one of those methods is used because different people find the different ones useful. A lot of people just can't deal with word problems to save their souls. So to speak. I know that when I had to "regurgitate" a theorem or proof in a math or physics class, having some little comment on a line explaining what I was doing went quite well with the professors. To simply memorize and regurgitate on demand isn't going to help one understand the material, but just to maybe get a good grade. I crammed for a test maybe 3 times in the course of getting my B Sc. I knew that anything I did later depended on that foundation.

Bottom line, it's just what people like. For my first two semesters of Calculus, I had Leithold's text. He's really wordy. First semester, we just did the work. Second semester, the prof complained that Leithold "Talks too much!"  

Applications always seemed to give people trouble, but I had kids in class who asked, "What will we ever use this for!?" (in a disparaging way). I was more than happy to provide an example, because that's the kind of thing that cemented it into their little concrete heads...not that I think that you are in that category- I'm certain of the opposite.

[Aliza]

That's a helluva good question! Every one of those methods is used because different people find the different ones useful. A lot of people just can't deal with word problems to save their souls. So to speak. I know that when I had to "regurgitate" a theorem or proof in a math or physics class, having some little comment on a line explaining what I was doing went quite well with the professors. To simply memorize and regurgitate on demand isn't going to help one understand the material, but just to maybe get a good grade. I crammed for a test maybe 3 times in the course of getting my B Sc. I knew that anything I did later depended on that foundation.

Bottom line, it's just what people like. For my first two semesters of Calculus, I had Leithold's text. He's really wordy. First semester, we just did the work. Second semester, the prof complained that Leithold "Talks too much!"  

Applications always seemed to give people trouble, but I had kids in class who asked, "What will we ever use this for!?" (in a disparaging way). I was more than happy to provide an example, because that's the kind of thing that cemented it into their little concrete heads...not that I think that you are in that category- I'm certain of the opposite.

You didn't reply to anyone, but was that a reply to my post? It seems like it follows.

[Fireball]

That's a helluva good question! Every one of those methods is used because different people find the different ones useful. A lot of people just can't deal with word problems to save their souls. So to speak. I know that when I had to "regurgitate" a theorem or proof in a math or physics class, having some little comment on a line explaining what I was doing went quite well with the professors. To simply memorize and regurgitate on demand isn't going to help one understand the material, but just to maybe get a good grade. I crammed for a test maybe 3 times in the course of getting my B Sc. I knew that anything I did later depended on that foundation.

Bottom line, it's just what people like. For my first two semesters of Calculus, I had Leithold's text. He's really wordy. First semester, we just did the work. Second semester, the prof complained that Leithold "Talks too much!"  

Applications always seemed to give people trouble, but I had kids in class who asked, "What will we ever use this for!?" (in a disparaging way). I was more than happy to provide an example, because that's the kind of thing that cemented it into their little concrete heads...not that I think that you are in that category- I'm certain of the opposite.

You didn't reply to anyone, but was that a reply to my post? It seems like it follows.

Guilty as charged! didn't want to get the quote chain too long.

[Kernel Sohcahtoa]

I didn't realize that Galois theory got group theory going.  I really enjoyed group theory,  so now I have  more motivation to get underneath Galois theory.

I'm primarily using Abstract Algebra by Saracino.  For whatever reason, his method of instruction and his style of proof-writing work well for me; I'm grateful that he has written his book, as it has made abstract algebra pretty accessible to me.  On occasion, I'll open up A Book of Abstract Algebra by Pinter or A First Course in Abstract Algebra by Fraleigh if I find that a particular concept is a little vague in Saracino. Also, if I have any questions and I'm able to express them in a precise and coherent manner (IMO, this is an art-form in itself), then I'll post them here.  Thanks.

Looking at the Pinter book, I had forgotten to mention the questions about ruler and compass constructions, which are related in some interesting ways to Field and Galois theory.

So, the question of which regular polygons can be constructed by ruler and compass has been completely answered (well, except for some number theory issues). The fact that trisecting a general angle and doubling a cube cannot be done via ruler and compass also follows from this directly.

Just scanning the table of contents suggests Pinter's book will cover this.

Definitely feel free to ask questions either publicly or via PM. I'm always willing to help with math.

Thank you, polymath.  Saracino's book also has a section on ruler and compass constructions, which is right in between unique factorization domains and normal and separable extensions.  I'll be sure to add that to my study list, as it sounds cool based on your description.

(1) What attributes do you like about one particular book over another? (2) Do you favor a more narrative description where the author writes like they're speaking to you, or do you like step-by-step solutions with notes on the side telling you what's going on? Or is it another attribute entirely, like examples being more like just general equations to solve vs. examples having a practical, real-world application that you can relate to?

(1) In regards to higher level mathematics (proof-based), I like a book where I am challenged to fill in the missing details of the author's proof: IMO, filling in the details and really getting underneath the material, allows me to gain an appreciation for the subject matter, which enhances the learning process for me.  For example, in Saracino's Abstract Algebra, he writes plenty of proofs that are condensed, and as a result, he may not mention that he negated a particular definition in his proof or that his chosen proof technique is actually the contra-positive of the theorem or of a definition.  Thus, in these cases, the reader would have to carefully analyze each sentence in order to identify key mathematical concepts and proof techniques that would  clarify the author's conclusions, which can seem to come out of nowhere, especially  if one has not taken the time to really get underneath the proof. 

(2) I do like books that sound like the author is talking to you; I'm not talented at math, so I need a book that is accessible and explains the material while still allowing plenty of opportunities for the reader to challenge himself or herself  .  I like Saracino's instructive approach, because once he proves a key theorem in a chapter, he will usually connect the generality of the theorem to specific examples. For example, if Saracino proves a particular theorem about groups, then he may talk about the set of integers, the set of real numbers, the integers modulo n, the set of all 2X2 invertible matrices, etc., and how they have all the properties of the theorem.  Thus, his approach works for me.  


That said, how about you, Aliza? What type of math books do you like?  You mentioned earlier that math is more of a means to an end for you. Would you be willing to elaborate? Thanks for your response.

[Aliza]

You didn't reply to anyone, but was that a reply to my post? It seems like it follows.

Guilty as charged! didn't want to get the quote chain too long.

Applications always seemed to give people trouble, but I had kids in class who asked, "What will we ever use this for!?" (in a disparaging way). I was more than happy to provide an example, because that's the kind of thing that cemented it into their little concrete heads...not that I think that you are in that category- I'm certain of the opposite.

Then to answer your question, I'm as dense as a bag of bricks. Trust me.  

[Aliza]

Looking at the Pinter book, I had forgotten to mention the questions about ruler and compass constructions, which are related in some interesting ways to Field and Galois theory.

So, the question of which regular polygons can be constructed by ruler and compass has been completely answered (well, except for some number theory issues). The fact that trisecting a general angle and doubling a cube cannot be done via ruler and compass also follows from this directly.

Just scanning the table of contents suggests Pinter's book will cover this.

Definitely feel free to ask questions either publicly or via PM. I'm always willing to help with math.

Thank you, polymath.  Saracino's book also has a section on ruler and compass constructions, which is right in between unique factorization domains and normal and separable extensions.  I'll be sure to add that to my study list, as it sounds cool based on your description.

(1) What attributes do you like about one particular book over another? (2) Do you favor a more narrative description where the author writes like they're speaking to you, or do you like step-by-step solutions with notes on the side telling you what's going on? Or is it another attribute entirely, like examples being more like just general equations to solve vs. examples having a practical, real-world application that you can relate to?

(1) In regards to higher level mathematics (proof-based), I like a book where I am challenged to fill in the missing details of the author's proof: IMO, filling in the details and really getting underneath the material, allows me to gain an appreciation for the subject matter, which enhances the learning process for me.  For example, in Saracino's Abstract Algebra, he writes plenty of proofs that are condensed, and as a result, he may not mention that he negated a particular definition in his proof or that his chosen proof technique is actually the contra-positive of the theorem or of a definition.  Thus, in these cases, the reader would have to carefully analyze each sentence in order to identify key mathematical concepts and proof techniques that would  clarify the author's conclusions, which can seem to come out of nowhere, especially  if one has not taken the time to really get underneath the proof. 

(2) I do like books that sound like the author is talking to you; I'm not talented at math, so I need a book that is accessible and explains the material while still allowing plenty of opportunities for the reader to challenge himself or herself  .  I like Saracino's instructive approach, because once he proves a key theorem in a chapter, he will usually connect the generality of the theorem to specific examples. For example, if Saracino proves a particular theorem about groups, then he may talk about the set of integers, the set of real numbers, the integers modulo n, the set of all 2X2 invertible matrices, etc., and how they have all the properties of the theorem.  Thus, his approach works for me.  


That said, how about you, Aliza? What type of math books do you like?  You mentioned earlier that math is more of a means to an end for you. Would you be willing to elaborate? Thanks for your response.

I like publishers and curriculum that use a lot of mixed media and incorporate a variety of different approaches to reach multiple learning styles. I especially like publishers that publish sets of material using longitudinal learning which cover similar topics in a consistent order, but with progressing difficulty with each advancing book. It gives the student the opportunity to master the basics, and then build on the skills they learned and apply those skills to the next level as they step up through an overarching curriculum. It’s kind of like learning the material and mastering the basics, then applying that mastered material to learn the next level, and so forth.

I also like when my textbook directs me to some YouTube content the publisher has made available where I can get a better idea of what the writer is trying to convey. I also like books that provide worked out examples online in addition to the answers in the back of the book. I don’t like reading, even if it’s in a narrative style. I want to see the equation worked out with all the steps clearly demonstrated (even the steps for stupid people). But as Fireball said, a good book uses various teaching techniques to appeal to a variety of learners.

[Fireball]

Guilty as charged! didn't want to get the quote chain too long.

Applications always seemed to give people trouble, but I had kids in class who asked, "What will we ever use this for!?" (in a disparaging way). I was more than happy to provide an example, because that's the kind of thing that cemented it into their little concrete heads...not that I think that you are in that category- I'm certain of the opposite.

Then to answer your question, I'm as dense as a bag of bricks. Trust me.  

I can assure you that getting my B Sc wasn't easy. I beat my head on some books for hours, trying to get the stuff to soak in.

[Kernel Sohcahtoa]

IMO, regardless of one's skill level, a student of mathematics will encounter concepts and ideas that are challenging: it seems that, for the time being, there will always be problems to solve and statements to prove. Hence, for any students who are interested in studying mathematics, please don't get discouraged when you hit a bump/boulder in the road: you are not alone.  Like so many mathematicians and students of mathematics before you, embrace the fact that you are being challenged, ask questions, and allow yourself to grow from this experience, regardless of whether you are successful or not.

[Aliza]

IMO, regardless of one's skill level, a student of mathematics will encounter concepts and ideas that are challenging: it seems that, for the time being, there will always be problems to solve and statements to prove. Hence, for any students who are interested in studying mathematics, please don't get discouraged when you hit a bump/boulder in the road: you are not alone.  Like so many mathematicians and students of mathematics before you, embrace the fact that you are being challenged, ask questions, and allow yourself to grow from this experience, regardless of whether you are successful or not.

Sound advice.