Final Exam, Due April 13

• Which topics and theorems do you think are important out of those we have studied?

Honestly. I have no idea. I'm going to spend the next few days trying to learn (not even re-learn) this entire course, because hardly anything had made sense to me.

• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday.

As I said previously, I'm SO far behind in this class, that I don't think I could make a list of things I need to better understand. I could probably make a list of things I DO understand, but it would be a very short list. However, one thing that comes to mind is I'd really like to know what the heck it means when you have G/H or A/B or K/I or any of those things involving a /. It always seems to mean something different.

• How do you think the things you learned in this course might be useful to you in the future?

 I've had this discussion with quite a few people. I don't know why Math Ed. majors are required to take a course like 371. It's SO much harder than anything we'll have to use in a high school. Perhaps it's there purely for the benefit of those who are considering going on to get their masters in Math Ed? If this course was helpful, it was because it showed me that there are MAJOR flaws in my study habits, and abstract/analytical math is NOT for me.

8.3, due April 11

• Difficult:  The sylow theorems actually seemed to make sense to me. Of course, I won't really know until I try to put them into practice. However, I'll admit that I still don't really understand the x^(-1)Kx. I feel like we've seen that a lot, but I have no idea what it means.
• Reflective: I feel like Cauchy's Theorem in this section is very similar to LeGrange's theorem. I guess the difference is that p must be prime, and then G DEFINITELY contains an element of order p, rather than it could contain an element of order p. Yes?

 

8.2, due April 8th

• Difficult: Woah. This whole chapter made so little sense that I don't even know where to begin! First of all, I didn't know why any of this was important (let alone what it means). But then again, I don't really know why any of this is important (at least in my discipline). But I guess what confused me the most is it seemed like they were pulling letters out of nowhere. They were sure to clarify that a was in g, but then they were saying ak=0 and sa=k_1, etc. But what is k? What is s? Where are all these things coming from?
• Reflective: I think I've reached that point of no return. I wasn't able to grasp the earlier concepts quick enough, and now all of these new concepts don't make ANY sense to me. I'm still trying to figure out what exactly an abelian group IS!

8.1, Due April 6th

• Difficult: I was confused by Theorem 8.1. When it says N_1, N_2 ..., N_k, does it mean that those are ALL of the normal subgroups of G, or can it be any number of normal subgroups of G? Or does that second part of the proof say it doesn't matter how many normal subgroups there are so long as ever element is covered? It wasn't clear. Also, because I didn't understand Theorem 8.1, I don't understand why Lemma 8.2 is important.
• Reflective: A lot of this was pretty straight forward. In fact, i wasn't really sure what we were supposed to find in this section that was NEW. It seemed like it was just showing you how to take cartesian products of groups, and isn't that what we've already been doing?

7.9, due April 1

• Difficult: I was a little confused by the example in the middle of page 231. It says (2 4 3)(1 2 4 3) = (1 4 2 3). No... to me this says 1 goes to 2, and 2 goes to 4 so one goes to 4. Then 2 goes to 4 and 4 goes to 3 so 2 goes to 3. Then 3 goes to 1, and 1 goes to 1, so 3 goes to 1. Then 4 goes to 3, and 3 goes to 2 so 4 goes to 2. Thus I get (4  3 1 2) which is NOT the same answer they got. What am I doing wrong?
• Reflexive: I think I really like this new notation. It totally makes sense to have (2465) mean 2 -> 4 -> 6 ->5. However, I haven't been able to make it work. So... maybe it isn't that great.

Make Up Post-- ME Social

I attended MEA's activity where we got together and met with other majors who's initials were ME (Math Education, Music Education, and Mechanical Engineering). The reflective part of this post is easy: I enjoyed discussing with the mechanical engineers the similar classes we have taken, and the different classes. Their major is MUCH more computational than mine (as made evident by this class). I'm not sure if I can find a "difficult" part, though. It was just... fun.

Make Up Post-- MEA Core Curriculum

• I attended a lecture by Dr. Peterson on the current math core curriculum in schools and how it is changing.
• Difficult: I had a really hard time understanding if Dr. Peterson thought that the core curriculum was good or if it needed improvement, or both. He seemed... certainly not abivalent, but abiguous and disjointed. He frequently gave his opinion on the core, but sometimes his opinions would contradict themselves. Very confusing.
• Reflective: I've noticed that the BYU math education program seems to be full of a lot of ideals, but I have a feeling most of them are vain. Yes, there are so many wonderful ways to incorporate math into the classroom so that it becomes more than just memorizing formulas. But with the way the core is currently set up, I don't know if it's possible. And I don't think the system is ready for a complete overhaul of the core. I wish he had talked about that. How will we incorporate these ideals, not just the ideals themselves.

7.6, due March 25th (Part 2)

These blog posts are SO hard for me to remember! I don't know why. Anyway. Here's today's:

• Difficult: I'm TOTALLY lost. I thought I understood what normal meant, but then it says that Na=aN does NOT imply that na=an. Then what DOES it imply? How do you test for normalcy if not like that? That's how I did most of my homework that's due Friday.
• Reflective: It seems like congruence classes always seem to multiply the same way. Which is nice, so long as it never changes.

7.3, due March 11

• Difficult: I didn't understand if a subgroup had to contain elements of the group G, or if it just had to be a group under the operation in G. Because if the subgroup H doesn't have to contain elements from G, then it doesn't seem like a subgroup to me, it seems more like a... co-group. What exactly makes group H "sub"? I think that 7.11 was saying it has to be a subset of the group G, but that's not what the definition said. So I'm confused.
• Reflective: If a center is every element in a such that ag=ga, then does that mean that every element is a center in an abelian group? That'd be nifty.  

6.2 part 2, due February 28

• Difficult: I understand how to find a homomorphism and all that, but what I don't understand is why? Why would it be important to find an ideal homomorphism? I guess I probably shouldn't ask the why question, because I feel that way about almost everything in this class, but that's the main thing I was wondering while reading this chapter. Ideals are important because they generalize congruence, but what benefit comes from being a homomorphism?
• Reflective: This section seemed to be the one that varied the most. In the past few sections I've easily been able to recognize parallels to the previous chapters, but not so much in this one. Did I miss it? Or are we staring to change topics?

6.2 part 1, due February 25

• Difficult: This chapter actually seemed pretty straightforward. Once I was able to wrap my head around the whole a+I thing, the rest was easy. However, I'm still a little confused on what R/I means. Is that just another notation for a+I? Why do we need the two notations?
• Reflective: This is pretty much exactly the same as what we learned before, which is nice. So long as I don't allow myself to get hung up on the new notation (which is actually really hard to do) then this doesn't seem so bad.

6.1 part 2, due February 23

• Difficult: The most difficult thing for me in this section was understanding exactly what I is. Because I is a subring. And so wehn they have things like a is congruent to c (mod I) iff a+I=c+I it doesn't make any sense to me. How can you have a+I? What does that look like? I just can't visualize it. Also, I wasn't sure what a coset was.
• Reflective: I like that the relation of congruenec modulo I is an equivalence relation. Things always seem to work out nicely when that happens. I also liked the example at the top of pg. 139, that was very easy to understand (unlike most of the examples in this book).

6.1 part 1, due February 22

• Difficult: This chapter actually made a surprising amount of sense. I guess the most difficult thing for me is actually visualizing an ideal. Because even though I could tell you what one should be like, I don't know if I could actually recognize or create one. I look forward to examples in class.
• Reflective: Theorem 6.1 states that if a,b in I, then a-b in I. But if both a,b are in I, then that means that a,b, have a common factor, right? Just like in the first example where it was mod 3. If a,b in I, then a-b=3(c-d)=3(e) so of course c-d 3e would be in I, so a-b is in I. I'm I thinking of this correctly?

5.3, due February 18

• Difficult: In Theorem 5.10 it says that p(x) is irreducible in F[x]. But why is that necessary? A field didn't JUST have prime numbers, so why does this one have just irreducible polynomials? It doesn't make sense to me. Also, I have no idea what an extension field is. I've read that part a bajilion times, and it's still jibberish. Looking forward to the explanation in class.
• Reflective: Hm... For once, it's actually the reflective part that's hard. "What was the most interesting part of the material?" Um... well... I didn't understand what an extension field was, and that's pretty much what it was all about. So... "How does this material connect to something else you have learned in mathematics?" well, so far as I can tell it doesn't! "How is this material useful/relevant to your intellectual or career interests?" I'm not sure. I wish someone would tell me how AA is going to help me teach high school math.

5.2, due February 16th

• Difficult: In theorem 5.7 it ends by saying "Furthermore, F[x]/(p(x)) contains a subring F* that is isomorphic to F." This doesn't make sense to me. It's just so vague. Why does it contain a subring? What's in this subring? Why does that subring have to be isomorphic? This theorem just doesn't make sense.
• Reflective: I feel like we're learning the same thing over and over again just with little twists. You'd think that that would make things easier. And I guess in some degree it does. But I do kind of feel that there's a lot of "duh" stuff that the book explains, but then it doesn't explain some of the harder stuff. I know this seems like it should be more under the "difficult" bullet, but this is what I was "reflecting" on while reading this section.

9.4, due February 11

• Difficult: Woah. There was a whole lot of weird punctuation going on in this section. But I guess it's about time we addressed division in rings/fields. I think the most confusing part for me was the idea of dividing equivalence classes. Because you can't have an equivalence class of 3/5, can you? Equivalence classes have to be whole numbers. So that doesn't make sense.
• Reflective: The book writes it as (a,b)~(c,d), and at first this really confused me because it then said ac=bd which looks like it's doing some kind of weird F.O.I.L. thing. But then I realized that it's just the first number in the parenthesis, divded by the second number. So a/b=c/d. This was easier for me to understand than ac=bd, and once I got that, the chapter read much easier.

Exam 1, due February 9

Which topics and theorems do you think are the most important out of those we have studied?

• I think it's most important to know the definitions of all the different terms for the test. If you know exactly what all the terms mean, then you can more fully use that knowledge on the exam. Also, theorems with names are usually pretty important.

What kinds of questions do you expect to see on the exam?

• I honestly have no idea. This class has been an odd mix of both computational problems and proofs. The sample questions have a lot of "prove or disprove." So... probably a lot of that. But I HOPE they are computational.

What do you need to work on understanding better before the exam?

• I really have no idea how to do proofs. I just lack that intuition to know where to start or what to do. I really hope that by learning the definitions better I'll know what I need to do on a proof. Because right now I am scared to death.

4.3, due February 4th

1. Difficult: I understood that if something is irreductable that means that it is basically a "prime" polynomial, but some things in the definition confused me. What does it mean that its divisors are its "associates"? I read the definition of associate, but I didn't get it. Also, even though something is irreductable, can it still be divided by a constant term? Because the definition said "its only divisors are its associates and the nonzero constant polynomials." But that doesn't sound very prime-ish to me.
2. Reflective: I love how closely related these polynomials are to the "normal" rings we've been studying. I've found it very interesting to extend the idea of a ring to a new form. And I'm glad things haven't changed too much because I really feel like I'm building up my knowledge rather than having a bunch of new concept trying to take up space in my brain.

Questions, due January 28

How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?

Each homework assignment takes me about 2 hours. The lecture and reading lets me know what I'm supposed to do, but I never am really sure how to do it. For example, when proving subrings I know that I'm supposed to prove that it's closed under subtraction and multiplication, but I get stuck on how. The computational problems are usually easy, but the proofs I find very difficult. Very rarely can I do a proof without any assistance. This is usually because I'm not sure what is relevant to that proof and what isn't. I lack that "intuition."

What has contributed most to your learning in this class thus far?

Even though they don't really help me when it comes to proofs, I do think that the lectures have been very helpful. You're very thorough-- always making sure people understand.

What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)

Well, I REALLY wish there was a math-lab section for 371, because I would like to have a casual place I could to go and be surrounded by other 371 students. I learn best when I can discuss with others. I know you have office hours, but I usually feel really uncomfortable working through a problem (especially a proof) with a professor. It'd be really nice if we could do more proof-ish examples in class. And I don't mean theorem proofs (those are nice, but aren't most in the book?), I mean proofs similar to those in our homework. Then I might know how to do them. I know we do this a little bit, but more couldn't hurt, eh?

Thanks!

What has contributed most to your learning in this class thus far?

3.1 part 2, due January 21

1. Difficult: It was actually difficult to find something difficult this time. Our lecture yesterday (1/19) really cleared up a lot of my questions. I guess the biggest thing I didn't understand this time is the difference between a subring and a subfield. The book didn't really clarify that. Also, when we say r & r', and s and s' Is r' and s' related to r and s? Or are they any other number? That's really confusing me too.



2. Reflective: I'm glad that the rules of cartesian products are the same. I don't know if I could handle another thing that was just slightly different.

3.1 part 1, due January 19

1. Difficult: I really didn't understand this chapter. Probably because it's all new to me. I'm really confused on what a ring is. I read it a couple times and I just don't get it. The first definition said that a ring is a set equipped with two operations, but then it went on to list 8 axioms. What are the properties? Are they just closure under multiplication and addition? I don't know.
2. Reflective: I'm having vague memories of Linear Algebra. it seems like this was very similar to... what were they called? Linear systems? I'm not sure if this was just because they had some matrices, or if the long list of axioms were familiar. I hope they're similar, 'cause then it'll be easier to understand. 

Packing Primes, make-up post

1. Difficult: I'm afraid I have to admit that the majority of Dr. Friedberg's lecture made very little sense to me. If it had gone a little slower, I probably could have gotten it. But by the time I wrapped my mind around one concept he had jumped past two or three more concepts. The thing that was most difficult for me to understand was the example he gave in the beginning with the ATM. I know he was trying to tell us how primes can be used in a real-world application, but I just got lost. Something about taking a prime, and multiplying it by 541 and then finding the remainder of the division of the prime? What the what? I was so lost.
2. Reflective: However, I really enjoyed his sand-metaphor. The idea of-- if you blow grains of sand up so that they're huge, then they don't fit together very well at all. But if you have small grains of sand, you can pack MILLIONS of them into a bucket with no problem at all! This metaphor was probably the thing that really made it so I knew what he was talking about. I loved it!

2.3, due January 14

1. Difficult: There was a part of the proof of corollary 2.9 that I didn't understand. When it's proving existence it says that a(w-ub)=aw-aub=b-b=0 but this seems a little presumptuous. Because it stats thta x-ub, so this really says that a(w-x)=aw-ax and we already supposed that w=x, so why isn't it that it's a(0)=b? I'm so confused.
2. Reflective: I thought it was very interesting that for non-prime equivalence classes, if [a][b]=0, then it doesn't mean that [a]=0 or [b]=0 BUT for a prime equivalence class if [a][b]=0 then [a] or [b] DOES =0. It took me a minute to wrap my head around that, but it makes perfect sense. Because the only time you have [a][b]=0 is when [a] and [b] are the two parts of the factorization of n (in mod n). But since primes have no factorization, the only way to get 0 is to multiply by zero! It makes perfect sense but I had never thought of it before. 

2.2, due January 12

1. Difficult: I'm a little confused. It seems like the book is contradicting itself. It asks at the beginning of the section if [a] + [c] = [a + c] and it gave some examples that worked. But then it later gave an example where 1+7=8 (or B+C=D) and then -3+25=12 (or B+C=A) thus this obviously doesn't work. But then it gives the definition which states "Addition and multiplication in Zn are defined by [a]+[c]=[a+c] and [a]*[c]=[a*c]. What did I miss? I feel like they just barely said this did NOT work, but now they are putting it in a definition so it obviously DOES. I've analyzed the theorem that comes between the two statements, but it doesn't seem to help me segue between the two ideas. And now I'm just really confused.
2. Reflective: Even though I don't really understand the property as mentioned above, I do like that it seems to follow all of the basic arithmetic rules. It's commutative, associative, distributive. This will make it a lot easier for me in the future.

2.2, due January 10

1. Difficult: I have a hard time understanding when a problem says _n. I often mix up what exactly that "n" means. I know that it literally means "integers mod n" but unless an equation is actually WRITTEN in the form of I easily get confused. I also have a hard time understanding when equivalence classes are written as equal to each other. I.E. I'm not exactly sure what it means when I see something like: [4]=[8].
2. Relative: I already knew theorem 2.3, which shates that a is congruent to c (mod n) if and only if [a] is congruent to [c]. However, I don't think I EVER realized how easy it is to find all of the different elements in an equivalence class! The examples in this section state that if you want to find all of the values in equivalence class [2] most 3, then you just take all values 2+3k for k in the integers. Looking at that I thought, well DUH, because you'll have a 2 remainder, but I never thought it was that easy! I remember being in 290 and testing value after value after value to see which equivalence class it was in. If I had known this I would have had my homework done in half the time!

1.1-1.3, due January 7

1. Difficult: The hardest thing for me in this entire class is going to be proofs. I already know that. I often have a hard time knowing where to start, what theorems to use, etc. The hardest part for me in this specific section was a) the uniqueness part of the division algorithm proof, and b) The proof of theorem 1.8. The concepts themselves make sense to me, but the way the proof is worded is hard. I have a hard time following the logic. The division algorithm proof is a beast, and it doesn't help that I've seen it proved many different ways. After a minute my eyes just glaze over. And I had to read theorem 1.8 a few times before I had some idea what it was saying. It seems like a simple enough concept, but the way the book has worked through the proof is making it really hard for me to grasp it. I don't know how crucial either of these theorem proofs are to future exercises, but I hope they're not too heavily used.
2. Reflective: I LOVE THE EUCLIDIAN ALGORITHM! I remember that when we were introduced to this in math 290 I got so excited! Finally! Something that was computational! I also love the way it's kind of like a puzzle, with all the numbers shifting around, and how you can solve it backwards to find the linear combination. That's just plain cool! However, the EA can get a little messy when you get up to huge numbers without a calculator. WITH a calculator it makes finding the GCD of big numbers infinitely times easier than the grade-school way of listing out all the factors of both numbers.

Introduction, due on January 7

• Year In School: Sophomore (57 Earned Credits)
• Major: Double in Mathematics & Mathematics Education (Minoring in Theatre Education)
• Courses Taken: Math 290, Math 313, Math 334 (Current Enrollment)
• Why I'm taking this class: It's required to graduate, and I love math! I figured this would be a good class to take concurrently with 334 because I believe this class is very... well... abstract! Whereas 334 is more computational.
• Dr. Erin Chamberlain-Martin: Dr. Martin was a fabulous professor. She was very thorough in her explanations, and always looked out to the class to make sure we weren't lost. She wasn't afraid to have fun in class, but we always got done what we needed to get done. Her enthusiasm for the subject was contagious.
• Something Unique: I am a professional young adult book reviewer. I work with a number of big publishers (Simon & Schuster, HarperTeen, Penguin, Bloomsbury, etc) and review their YA books, primarily fantasy. I publish my reviews online. I've been doing this for nearly four years. I also design custom websites/graphics for myself and other book reviewers.
• Office Hours: Your scheduled office hours are PERFECT for me.