Difficult:
This whole section was a little difficult for me. I read through it seeming to somewhat understand what they were talking about but I really didn't. I do not really know what they mean by the product of two groups. I do not know what it means to multiply two groups together. The proof of theorem 8.1 was also difficult to follow.
Reflective:
I liked the example starting at the end of page 245. It helped to be able to look at the tables and visually see why the two groups were isomorphic to each other. I am not anticipating how this new section will complicate our homework. I think at this point I am ready to only focus on what we have learned to prepare for the final. Though I am glad that we did not have to learn about a brand new entity. I like working in groups. If we stay in groups then that will be a good thing.
Thursday, March 29, 2012
Tuesday, March 27, 2012
Section 7.10, March 27, 2012
Difficult:
I do not really understand the concept of alternating groups. It is difficult for me to understand what they mean without doing a couple of examples. Reading through the proofs I think I understand what the theorems mean and why they are true. But I still do not think I have grasped the concept of alternating groups.
Reflective:
I think mathematics is this never ending way to group numbers together and name them something. Anything that has a pattern it seems like you can have a different name for it. I would even venture to say the there is an infinite number of patterns that you can find using numbers and therefore mathematics is a kind of infinite subject. :)
I do not really understand the concept of alternating groups. It is difficult for me to understand what they mean without doing a couple of examples. Reading through the proofs I think I understand what the theorems mean and why they are true. But I still do not think I have grasped the concept of alternating groups.
Reflective:
I think mathematics is this never ending way to group numbers together and name them something. Anything that has a pattern it seems like you can have a different name for it. I would even venture to say the there is an infinite number of patterns that you can find using numbers and therefore mathematics is a kind of infinite subject. :)
Sunday, March 25, 2012
Section 7.9, Due March 25, 2012
Difficult:
I understood most about cycle notation and the product of permutations, etc. But it is a little difficult for me sometimes to see what the product between two permutations will look like. Especially when there are more than two permutations to combine into one. It was also difficult for me to follow the proof of Lemma 7.49. I am sure there is an easy way of explaining it to someone, but the proof was difficult to follow.
Reflective:
I like the idea of cycle notation. I know that it will make it easier for me to write permutations. But I also know that I can potentially get very confused and start mixing up all the numbers. This will call for some serious careful calculations when I am doing my homework. It is sections like these that make me see mathematics as more of a history of ideas; one day someone was thinking about simpler ways to write permutations and then BOOM! he thought of cycle notation.
I understood most about cycle notation and the product of permutations, etc. But it is a little difficult for me sometimes to see what the product between two permutations will look like. Especially when there are more than two permutations to combine into one. It was also difficult for me to follow the proof of Lemma 7.49. I am sure there is an easy way of explaining it to someone, but the proof was difficult to follow.
Reflective:
I like the idea of cycle notation. I know that it will make it easier for me to write permutations. But I also know that I can potentially get very confused and start mixing up all the numbers. This will call for some serious careful calculations when I am doing my homework. It is sections like these that make me see mathematics as more of a history of ideas; one day someone was thinking about simpler ways to write permutations and then BOOM! he thought of cycle notation.
Thursday, March 22, 2012
Section 7.8, March 22 2012
Difficult:
This section was a little more confusing than past sections. A couple of concepts were more difficult for me to grasp. I do not know what is meant by the fact that the kernel measures how far the homomorphism f is from being injective. That does not make any sense to me. I do not understand the concept of "how far" when dealing with numbers in abstract algebra.
Reflective:
I know that a lot of what we are learning about groups is supposed to connect with what we have already learned about rings. But I am having a difficult time making that connection. It is a wonderful thing that they can connect in such logical ways, but that connection is hard to make. I think part of it has to do with all the new names and notation. But other than that this new material is great!
This section was a little more confusing than past sections. A couple of concepts were more difficult for me to grasp. I do not know what is meant by the fact that the kernel measures how far the homomorphism f is from being injective. That does not make any sense to me. I do not understand the concept of "how far" when dealing with numbers in abstract algebra.
Reflective:
I know that a lot of what we are learning about groups is supposed to connect with what we have already learned about rings. But I am having a difficult time making that connection. It is a wonderful thing that they can connect in such logical ways, but that connection is hard to make. I think part of it has to do with all the new names and notation. But other than that this new material is great!
Wednesday, March 21, 2012
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.
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.
Monday, March 19, 2012
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. :)
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. :)
Thursday, March 8, 2012
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.
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.
Tuesday, March 6, 2012
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!!!
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!!!
Sunday, March 4, 2012
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.
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.
Saturday, March 3, 2012
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.
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.
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