4.2, Due February 2

Difficult:

I think I am still struggling with what a 'field' is in context of the book, so I get a bit confused when it talked about fields in regards to some of the theorems in this section. I'm also having trouble understanding the GCD in regards to a function. The theorem made sense to me right up until they gave the example of the GCD of 3x^2+x+6 and O. I don't really understand why they multiplied it by 1/3.

Reflective:

As the book mentioned, most of the results from divisibility and greatest common divisor are almost the same as they were for regular arithmetic. The necessary modifications to apply the theorems to F[x] were intuitive, for the most part.

4.1, Due January 31

Difficult:

When I read Theorem 4.2, I was pretty confused at first. I'm not sure what I thought it was referring to, but I already know that x^4*x^2=x^6 because you add exponents when you're multiplying them together. I don't think I've ever heard them referred to as "degrees" before, so when I read deg[f(x)g(x)] = deg[f(x)]+deg[g(x)] I was thinking more of regular arithmetic than exponents.

Reflective:

The Division Algorithm In F(x) seems almost exactly the same as the regular division algorithm, but they don't require r(x) to be greater than zero. I didn't totally follow the proof, and perhaps it was addressed in there, but I'm curious as to why r(x) doesn't have to be greater than zero...

Questions, Due January 28

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

I probably have to spend as much as 2 hours per assignment and even then it's not necessarily done. That's just how proofs are for me, they don't come easily. I feel like the lecture and reading are basically the same. I generally don't get much more from the lecture besides what I already got from the reading.

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

The interaction with others (outside of class) - working with peers on homework or one-on-one attention from Dr. Jenkins.

What do you think would help you learn more effectively or make the class better for you?

I think I need to continually review past chapters and their theorems to help me remember which ones to use. I also think it would be valuable for Dr. Jenkins to go over a few homework problems at the beginning of class. If time is a concern, we can skip some proofs, as we already read them in the book, and I don't think all proofs need to be rehashed in class.

3.3, Due January 26

Difficult:

I'm not sure if I understand why we even care about rings being isomorphic. It was strange to see the + and x tables of Z10 written "out of order" from what we would originally do, but I could understand why they chose to do it that way. However, what do we get from 2 things being isomorphic? Why does it matter that Z5 and Z10 are isomorphic?

Reflective:

This particular section brought up a few mathematical ideas that I haven't seen for a while, namely surjection, bijection, and image. I only vaguely remembered those concepts, so I ended up having to go look those definitions up again!

3.2, Due January 24

Difficult:

I am still thinking of operations in Z when I'm looking at some operations on rings. I was watching the example on pg 59 where they have (a+b)^2 and all seemed well and good until they mentioned that we can't necessarily say "ab+ba=2ab" because ab might not equal ba.... I have to keep reminding myself that in Z6 it's ok to have 1-2=5.

Reflective:

I found it interesting that this book uses the term "cancellation" but only in light of what has been talked about in my practicum class (as I'm a math ed major). We had talked about how "cancelling" gives the wrong message, like we can just arbitrarily take out or add in what we need to, when really we are "simplifying" not "cancelling."

3.1, Due January 21

Difficult:

It was hard to follow the example of the Cartesian product Z6xZ. I was confused when they said that a,a' are both in Z6 and z,z' are both in Z. It seemed like it would make more sense to have a,z in Z6 and a',z' in Z. It wasn't until reading in Appendix B and reading theorem 3.1 that it actually made sense to me.

Reflective:

In class I think I was a little confused about subsets of set that are rings, but I think now I understand. While Z is a ring, and E is a subset which also happens to be a ring, O is not a ring. It's ok for subsets to not have all the properties of a set. I think I had always thought of it the other way around.

2.3, Due January 14

Difficult:

I was able to follow the theorems and their proofs, but when they applied it in the example problem I realized I didn't really have a grasp of the usefulness of the concept. After reading the example of 24x=5 in Z95, I had to go back and re-read the theorems again to understand what they had done to arrive at their answer of x=20.

Reflective:

I definitely appreciated the footnote directing me to Appendix A so I could remind myself what it means when they say "the following conditions are equivalent" as I had forgotten what exactly that meant. I didn't realize how much reading the various Appendixes would help me review my basic logic and proof skills!

2.2, Due January 12

Difficult:

The most difficult part to wrap my mind around is the way ordinary arithmetic might look when using modulos. It may be true that in Z(mod 5), 4+1=0 or 3x4=2 or 4+4=3 - usually when we see an equation set up like that, we intuitively assume that they are just regular integers and the equations just feel wrong!

Reflective:

It was definitely easy to recognize all the basic operations that could be performed on Z(mod n). It was also easy to follow the proofs the book presented for valid operations in Z(mod n). I found this chapter much easier to connect to previous chapters and theorems than any other section.

2.1, Due January 10

Difficult:

I found it difficult to remember what "a=b(mod n)" means. As they were going through various theorems and proofs, I had to keep going back to the definition of congruent modulos. Even more difficult was the concept of the congruence classes of (mod n) and it's accompanying theorems and proofs. I will likely need to re-read section 2.1.

Reflective:

I did find it curious that when we talk about 2 sets of congruence classes, it's only a matter of disjoint vs. identical. Usually 2 sets can have some elements in common but not be the same sets, but with congruence class sets, that can't happen.

1.1-1.3, Due January 7

Difficult:

I had a hard time following the back substitution portion of the Euclidean Algorithm. I think I understand the reasoning and process when they apply the division algorithm to find (324, 148), but I don't understand why or how they applied back substitution. If I understand the algorithm, 4 is the GCD (324,148)? Then why the back substitution?

Reflective:

In my Math history class that I'm taking through independent study, I need to prove that the square root of a prime number is irrational. I've been struggling to understand or create a proof for such. Now that I've learned more about the division algorithm and it's application, as well as learning more about primes, I think I will have a much easier time creating the proof!

Introduction, due on January 7

What is your year in school and major?

I'm a MathEd major in my senior year here at BYU.

Which post-calculus math courses have you taken?

I've taken Math 190, 214, 315, 334, 343, and 362.

Why are you taking this class?

Well, it's required by my major. Honestly, I would never have picked it myself. I know it's going to be a lot of work for me - I struggle with courses that rely heavily on proofs. Proofs have just never come easily to me.

Tell me about the math professor or teacher you have had who was the most and/or least effective. What did s/he do that worked so well/poorly?

I think my favorite teacher was my Pre-Cal teacher in high school. It was obvious that she was enthusiastic about the subject and that she cared whether or not the students were learning. She tried very hard to relate math to the students - often through movie clips! It was an enjoyable class :)

Write something interesting or unique about yourself.

I've been married for 5 years and I have 2 kids. Benjamin is 3.5 and probably the smartest kid you'll meet. Tristan is 17 months and a little terror. By the end of the month I'll be finishing my basement - by the end of the year, I'll be graduating - and sometime in the next 5 years I will become the owner of a tax practice that supports about 600-700 clients currently. It's going to be a crazy next few years!

If you are unable to come to my scheduled office hours, what times would work for you?

The perfect time for me would be MWF 10-11 (just before 371 starts).