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.
Tuesday, February 28, 2012
Monday, February 27, 2012
Section 7.1, First Part, Due Feb. 27, 2012
Difficult:
The main thing that I found to be difficult with this section was understanding the difference between a group and a ring. At first, when I was reading the definition of a group it seemed to me that it was similar to a ring. But then I realized the bigger specification that a group holds for most of the ring axioms only with ONE operation. Whereas, a ring has different axioms involving a few different operations.
Reflective:
I enjoy compositions of permutations. At first they are a little tricky and I was a little confused trying to follow them in the reading. But once you see the pattern it is interesting to find inverses of certain arrays. I am sure that is what I will be doing in the homework. :)
The main thing that I found to be difficult with this section was understanding the difference between a group and a ring. At first, when I was reading the definition of a group it seemed to me that it was similar to a ring. But then I realized the bigger specification that a group holds for most of the ring axioms only with ONE operation. Whereas, a ring has different axioms involving a few different operations.
Reflective:
I enjoy compositions of permutations. At first they are a little tricky and I was a little confused trying to follow them in the reading. But once you see the pattern it is interesting to find inverses of certain arrays. I am sure that is what I will be doing in the homework. :)
Monday, February 20, 2012
Section 6.1-6.2, Due Feb. 21, 2012
Difficult:
Overall I think I understood most of the ideas behind this section. The difficulty for me is trying to become comfortable enough with this material to be able to complete problems on my own. For instance, the example on the top of page 141 was somewhat confusing to me. With most of the theorems and examples that I read I start out somewhat intimidated and unsure of how I would go about proving it. I am still wrestling with the definitions of rings, fields, ideals, etc. in my head trying to get myself more comfortable in thinking about them in abstract ways. But, almost always, as soon as I start reading the solution to an example or a proof I always think, "oh Duh! Of course that is how you would do it!" Which is very frustrating because then that means that I am so close to being able to think about these things on my own. I just hope that I grasp everything before the next test.
Reflective:
Math never ceases to astonish me! I just love how simple everything is explained in this book! Even though this material is still very difficult, it is quite comforting knowing that there is always a logical and simple way of approaching each problem. It helps give me a model for how I should go about proving my homework problems and problems on the test. :)
Overall I think I understood most of the ideas behind this section. The difficulty for me is trying to become comfortable enough with this material to be able to complete problems on my own. For instance, the example on the top of page 141 was somewhat confusing to me. With most of the theorems and examples that I read I start out somewhat intimidated and unsure of how I would go about proving it. I am still wrestling with the definitions of rings, fields, ideals, etc. in my head trying to get myself more comfortable in thinking about them in abstract ways. But, almost always, as soon as I start reading the solution to an example or a proof I always think, "oh Duh! Of course that is how you would do it!" Which is very frustrating because then that means that I am so close to being able to think about these things on my own. I just hope that I grasp everything before the next test.
Reflective:
Math never ceases to astonish me! I just love how simple everything is explained in this book! Even though this material is still very difficult, it is quite comforting knowing that there is always a logical and simple way of approaching each problem. It helps give me a model for how I should go about proving my homework problems and problems on the test. :)
Tuesday, February 14, 2012
Section 5.3, Due Feb. 14, 2012
Difficult:
Almost everytime I read a new section in this book it seems like the book presents things in such simple ways that I should be able to understand it. And while reading it, I feel like I comprehend what I am reading, but I don't understand it. I know what the book is saying, but I don't understand what it entails. The difficulty in these readings is trying to really understand the new concepts. But it seems that understanding only comes after expereicne with the new concept and trying problems on my own. If I can try to come to understand the concepts as I am reading than that would minimize the amount of times that I need someone elses help on the homework and I will be able to do the homework problems on my own.
Reflective:
It is interesting to note about past mathmeticians and what their views were of math. Especially understand now what they did not understand then. I find it fascinateing that abstract algebra can be used to define the complex numbers and proof of their existence. Though I do not fully understand how or why that works, it makes me smile thinking that it does make sense and that mathmeticians would not doubt the logic if they had known it then.
Almost everytime I read a new section in this book it seems like the book presents things in such simple ways that I should be able to understand it. And while reading it, I feel like I comprehend what I am reading, but I don't understand it. I know what the book is saying, but I don't understand what it entails. The difficulty in these readings is trying to really understand the new concepts. But it seems that understanding only comes after expereicne with the new concept and trying problems on my own. If I can try to come to understand the concepts as I am reading than that would minimize the amount of times that I need someone elses help on the homework and I will be able to do the homework problems on my own.
Reflective:
It is interesting to note about past mathmeticians and what their views were of math. Especially understand now what they did not understand then. I find it fascinateing that abstract algebra can be used to define the complex numbers and proof of their existence. Though I do not fully understand how or why that works, it makes me smile thinking that it does make sense and that mathmeticians would not doubt the logic if they had known it then.
Sunday, February 12, 2012
Section 5.2, Due February 12, 2012
Difficult:
It is difficult for me to fully grasp and understand how the the field F[x] modulo p(x) can act just like the Integers modulo n. It seems to make sense when it is explained in terms of proofs and generic letters representing numbers, but when I start to think of actual polynomials, the connection and the understanding starts to fall apart. I guess the reason for this confusion is that I always saw polynomials as these random sequences of numbers and variables that I needed to solve. I never really saw any kind of significance to their patterns.
Reflective:
Like, I have said before it is amazing how you can expand one simple idea into another realm of combinations. The whole congruence modulo has been expanded from the integers now into the field F[x]. Like, I have said before, it seems to make sense in the book. But it will take a little time and experience before I can become completely comfortable with understanding how everything works.
It is difficult for me to fully grasp and understand how the the field F[x] modulo p(x) can act just like the Integers modulo n. It seems to make sense when it is explained in terms of proofs and generic letters representing numbers, but when I start to think of actual polynomials, the connection and the understanding starts to fall apart. I guess the reason for this confusion is that I always saw polynomials as these random sequences of numbers and variables that I needed to solve. I never really saw any kind of significance to their patterns.
Reflective:
Like, I have said before it is amazing how you can expand one simple idea into another realm of combinations. The whole congruence modulo has been expanded from the integers now into the field F[x]. Like, I have said before, it seems to make sense in the book. But it will take a little time and experience before I can become completely comfortable with understanding how everything works.
Sunday, February 5, 2012
Questions, Due January 27--but cleared with Dr. Jenkins to turn in late :)
-How long have you spent on the homework assignments? Did Lecture and the reading prepare you for them?
I usually spend about an hour and a half on the homework. Unfortunately, this is not because I am really smart and consistent enough that that is all the time I need, it is simply that that is the all the time I have to spend on my homework every MWF. I should be starting the homework before the day that it is due, and I plan to do that from now on, but that is what I have been doing up to now. And that is why many of my assignments have been turned in incomplete. :)
Lectures and readings do help prepare for the homework, but not near as much as they could. I find that the lectures and the readings are very similar. As a student I go through both of them because I have to; i.e. I read the book because I need to blog, and I go to class because I want to do well in the class. But I often find myself and my classmates just focusing on copying down all of the lecture notes rather than trying to understand them better. In fact, there are times when it feels as though it would be against the flow of lecture to ask questions and really try to understand the material better. Does that make sense? This doesn't happen all of the time; questions are not unwelcome in class by any means. It just seems like the lecture should explore more discussion and discussion from the students rather than a repeat of the theorems that we just read, only now we are listening to them and frantically copying them down. :) I hope this helps.
-What has contributed most to your learning in this class thus far?
Honestly, working out the problems with my peers and having them decipher the book and give it to me in a lay person terms. That is how it is with almost all math classes at BYU it seems. The professor requires reading, regurgitates everything from the book onto the blackboard, answers a couple of questions from those daring enough to interrupt the flow of lecture, and then gives us the homework assignment to do where we have to completely re-invent how to solve problems so that we can make it through the homework and the midterms. :) Like I said before, honestly, if it weren't for the math lab and my friends, I would not have made it through my assignments in this class. I think it is great that we have the math lab, and I think study groups are awesome but I think that the professors should help the students prepare more to have the confidence needed to do the problems by themselves.
-What do you think would help you learn more effectively or make the class better for you?
I have obviously given you feedback above, so I will give myself some goals seeing as I obviously play a big part in my education as well. My goals are as follows:
-1. At least try all of the problems before the day that they are due.
-2. Try to understand the readings and the lectures more by paying better attention and taking notes as I read.
-3. Write down questions that I come across so that I can ask them in class, seeing as other students might have the same questions.
I usually spend about an hour and a half on the homework. Unfortunately, this is not because I am really smart and consistent enough that that is all the time I need, it is simply that that is the all the time I have to spend on my homework every MWF. I should be starting the homework before the day that it is due, and I plan to do that from now on, but that is what I have been doing up to now. And that is why many of my assignments have been turned in incomplete. :)
Lectures and readings do help prepare for the homework, but not near as much as they could. I find that the lectures and the readings are very similar. As a student I go through both of them because I have to; i.e. I read the book because I need to blog, and I go to class because I want to do well in the class. But I often find myself and my classmates just focusing on copying down all of the lecture notes rather than trying to understand them better. In fact, there are times when it feels as though it would be against the flow of lecture to ask questions and really try to understand the material better. Does that make sense? This doesn't happen all of the time; questions are not unwelcome in class by any means. It just seems like the lecture should explore more discussion and discussion from the students rather than a repeat of the theorems that we just read, only now we are listening to them and frantically copying them down. :) I hope this helps.
-What has contributed most to your learning in this class thus far?
Honestly, working out the problems with my peers and having them decipher the book and give it to me in a lay person terms. That is how it is with almost all math classes at BYU it seems. The professor requires reading, regurgitates everything from the book onto the blackboard, answers a couple of questions from those daring enough to interrupt the flow of lecture, and then gives us the homework assignment to do where we have to completely re-invent how to solve problems so that we can make it through the homework and the midterms. :) Like I said before, honestly, if it weren't for the math lab and my friends, I would not have made it through my assignments in this class. I think it is great that we have the math lab, and I think study groups are awesome but I think that the professors should help the students prepare more to have the confidence needed to do the problems by themselves.
-What do you think would help you learn more effectively or make the class better for you?
I have obviously given you feedback above, so I will give myself some goals seeing as I obviously play a big part in my education as well. My goals are as follows:
-1. At least try all of the problems before the day that they are due.
-2. Try to understand the readings and the lectures more by paying better attention and taking notes as I read.
-3. Write down questions that I come across so that I can ask them in class, seeing as other students might have the same questions.
Section 4.4, Due February 5, 2012
Difficult:
The difficulty of this section started at the very beginning. For some reason the very first paragraph was confusing to me. Ironically, understood mostly everything else from that point on. It took me a second, but I soon quickly understood what they meant by a polynomial function when I started to think of it as the same polynomial functions that I have been dealing with since I was in junior high. From there I came to understand the distinction between the intermediate x and the variable x. I also understood the concepts of roots, seeing as it is not a novel concept in math. As soon as they began to discuss these polynomial functions in terms of fields, it became more difficult. Fortunately, I was able to understand fairly well at first but it gradually got more difficult towards the end of the section. In particular I did not fully understand the proof of Corollary 4.16. Consequently, Corollaries 4.17 and 4.18 were not completely clear as well. But other than that, I pretty much understood the overall picture that this section was trying to depict.
Reflective:
I find it very interesting and almost humorous that they found the need to really make sure that we understood the difference between the intermediate x and the variable x, seeing as the intermediate x deals with polynomials in the polynomial ring R[x] and the variable x deals with polynomials in the ring R. Oh my goodness! Heave forbid we get them confused! I guess I understand the need for a distinction, because they are different, it just seemed like something obvious to a math major. But, nevertheless, as is expected in a Math major, I appreciate the attention to detail and the extreme thoroughness. :)
The difficulty of this section started at the very beginning. For some reason the very first paragraph was confusing to me. Ironically, understood mostly everything else from that point on. It took me a second, but I soon quickly understood what they meant by a polynomial function when I started to think of it as the same polynomial functions that I have been dealing with since I was in junior high. From there I came to understand the distinction between the intermediate x and the variable x. I also understood the concepts of roots, seeing as it is not a novel concept in math. As soon as they began to discuss these polynomial functions in terms of fields, it became more difficult. Fortunately, I was able to understand fairly well at first but it gradually got more difficult towards the end of the section. In particular I did not fully understand the proof of Corollary 4.16. Consequently, Corollaries 4.17 and 4.18 were not completely clear as well. But other than that, I pretty much understood the overall picture that this section was trying to depict.
Reflective:
I find it very interesting and almost humorous that they found the need to really make sure that we understood the difference between the intermediate x and the variable x, seeing as the intermediate x deals with polynomials in the polynomial ring R[x] and the variable x deals with polynomials in the ring R. Oh my goodness! Heave forbid we get them confused! I guess I understand the need for a distinction, because they are different, it just seemed like something obvious to a math major. But, nevertheless, as is expected in a Math major, I appreciate the attention to detail and the extreme thoroughness. :)
Friday, February 3, 2012
Section 4.3, Due February 3, 2012
Difficult:
This section required that I have a good understanding of fields and integral domains. But unfortunately I am having a difficult time understanding exactly what those are and what can happen in them and what cannot. So that made it difficult for me to understand most of this section. I do not completely understand all of the definitions that were given. I guess this is where this class starts to get more ABSTRACT wich is difficult for me to connect with because there is no physical visual aspect of it.
Reflective:
Once I started to try to understand the material better it began to make better sense to me. I find it interesting how everything is connected to each other through different proofs and theorems. I also find it helpful how everything is very clear cut in this book. The book explains things very well and and clearly. You would think I would understand it better, but it is still difficult for me to connect to.
This section required that I have a good understanding of fields and integral domains. But unfortunately I am having a difficult time understanding exactly what those are and what can happen in them and what cannot. So that made it difficult for me to understand most of this section. I do not completely understand all of the definitions that were given. I guess this is where this class starts to get more ABSTRACT wich is difficult for me to connect with because there is no physical visual aspect of it.
Reflective:
Once I started to try to understand the material better it began to make better sense to me. I find it interesting how everything is connected to each other through different proofs and theorems. I also find it helpful how everything is very clear cut in this book. The book explains things very well and and clearly. You would think I would understand it better, but it is still difficult for me to connect to.
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