Make-up Blog! Edward B. Burger, "History of Numbers"

I absolutely loved this Lecture! There is no other way to say. I enjoyed Dr. Burger and his enthusiasm for numbers and mathematics. He was very entertaining and gave a fabulous presentation about numbers.

I understand the disclaimer he gave at the beginning of the lecture stating that it is probable that at some of the information and stories are mostly true. But even then, I couldn't help but believe what he was presenting! I found it fascinating that it is very possible for literature, language, science and other subjects to stem from simply people learning to count. It made complete sense to me how the ancient people may have counted and how that lead from impressions in clay, and more detailed drawings, to writing and the development of our numbers today. I also found it fascinating that the Latin word for pebble is calculi. It makes so much sense!

As the lecture progressed, I found it more difficult to follow him. For some reason I did not quite make the connection between numbers and space. I think I may have missed some of what he said. But for most of the lecture I was glued to what he was saying about the possible future of numbers. I enjoyed it so much that I am convinced of the importance of the history of mathematics. As much as time and curriculum will allow, I am going to make sure that I add some history to my mathematics classes when I head off to teach.

Section 7.7, March 19, 2012

Difficult:

The most difficult part for me in this section was keeping up with the notation. Quotient rings were not really my favorite subject before this section and quotient groups are just as difficult for me to follow. It is difficult for me to conceptualize what exactly does G/N mean or what Na means. Overall, I was able to follow the reading fairly well, but I still do not understand exactly what I am dealing with.

Reflective:

I have noticed that this class basically just builds on itself over and over again like a snowball. Especially when you have already learned what a congruence class is, a quotient ring is, what a subring is, ect. Now we are learning about everything all over again with groups. It is wonderfull. Just wonderful. :)

Section 7.4, Due March 9, 2012

Difficult to understand:

At this point most of abstract algebra is what you would call "difficult" to me. In fact, the difficult thing about it is trying to really narrow it down to what is difficult. :) I comprehended most of the section, but I know that I really do not understand it yet because I know that I will be confused as soon as I start the homework problems. I think what is really difficult for me right now is to visualize how all of the groups and numbers relate to one another. I was confused about what an automorphism is because I do not think it was explained very well. Is it just an isomorphism from G to G?

Interesting:

Abstract is definitely getting to abstract for me at this point. Right now it seems like they are just throwing around letters and making up groups and connections and relations. The kind of math that I really appreciate is the math that relates more with the physical world rather than the imaginary, patterned world.

Questions in Prep for Exam

What do you think are the most important topics and theorems?
I think that the second theorem that we need to be able to prove on the exam is very important to know. This is because you can gather a lot of information about a problem and how to prove it by knowing the implications of this theorem.
I also think that the different group definitions are important to know. Knowing the definitions is a great tool on the test even if you did not memorize all the different types of examples of the groups, you can still find them if you know the definitions.

What kinds of questions do you expect to see on the exam? 
I expect to see questions about the theorems that we are being asked to memorize. I expect to see questions about definitions of important concepts. I expect to see questions asking to prove something that was on the homework. However, I expect the test to be reflective of our homework and excluding those problems that were the most difficult on the homework.

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


I need to review exactly what must be done on the test to prove that something is a (insert defintion). I need to go over the how to prove this, material for the test. Also, if possible, in class could you give us the exact numbers of all the theorems that we will have to prove, or should know. It helps me and other students better when we know the actual theorem number rather than what it is about.
Also, please do the sample problem, 5 in class. Thank you so much!!!

Section 7.3, Due March 4, 2012 (see below for a make-up blog post)

Difficult:

I understood about three quarters of this section, so I did understand most of it, but it was a little longer than usual and my brain somewhat stopped working at about the end of page 185. One thing that I found difficult was the proof of theorem 7.10 on page 182. It was a little confusing to me why they only needed to prove that e was in H. I think I may understand why, but it would have been nice to have a little more of an explanation after the proof or something. I also don't really grasp the understanding of cyclic groups. So the first example on pg. 185 was confusing to me. In general, I think I could easily understand all of the material in this chapter, I just need a little bit more time and experience with it in my homework.

Reflective:

I think I have discovered from reading this book that one of the key underlying features in math is patterns. Patterns make up everything in math. Just when you think that they have talked about every different kind of arrangement of numbers, the sprouts another one. That is what I feel about this section. Math really is amazing. I know that I have said that before a lot on these blogs, but it really is true. It is astonishing to me how many different patterns and kinds of "groups" of numbers there are in this world.

MAKE UP BLOG POST-Guest Speaker on Sep. 28, 2012 "Puzzles"

There were a couple of things that I enjoyed about the Guest Speaker on Tuesday, Sep. 28. (Sorry, I completely forgot his name). I really enjoyed the last puzzle he did with the triangles and the zeros and the ones. It made a lot of sense that that is how a computer would make different patterns.

Something that confused me was the one pattern that he put on the overhead but then told us that he was not going to tell us how to finish it. He said that he would just let us finish it. That was confusing to me because then he ended up moving the puzzle off the overhead. I did not even see it long enough to finish the puzzle so it was somewhat pointless. It was neat that this man has discovered a lot of ways to find patterns, but I found myself thinking at the end. So what? What does this mean for us? That was all that I thought the presentation lacked.

The rest of section 7.1, Due Feb. 28, 2012

Difficult:

I do not know enough about groups yet to really define was is most difficult about them that I have learned so far. I know that once I start doing problems, certain difficulties will arise and then I find myself saying, "Goodness, the most difficult thing about groups is (this)!" But until then everything is kind of smashed into a ball of knowledge in my head. I guess you can say the most difficult thing about this section was the whole section. :) I mean I understood everything that I read, but don't think I will honestly understand what I have read until I do some homework problems.

Reflective:

It is amazing to me how many different ways we can find patterns in mathematics. I really enjoyed looking at the shape example with the square and the different ways to move the square. It was a great way to introduce the idea of dihedral groups. I am fairly enjoying this section and I hope to grasp its ideas better than I did with the rings.