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.