1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material was understanding the proof of Theorem 4.4. There are just too many new definitions and properties and such in this section that make the proof really hard to understand because the new material with definitions and such have not sunk in yet. It is a proof that I am going to have to go over multiple times before I can really get it.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
I don't know why, but I have always found polynomials fascinating. So I am excited that we are applying ring stuff to polynomials. I just think they are so fun!
Sunday, January 30, 2011
Thursday, January 27, 2011
Random Questions, due on Friday, January 28
1. How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
I have spent between 3-5 hours on each homework assignment. Sometimes it is frustrating because I feel like a lot of the problems are repeats, or like all the parts aren't necessary to get the points across. Sometimes it could be shortened. Lecture and the reading do prepare me, but the problem is in lecture we go over exactly what is in the reading. We go over the exact proofs, examples, etc. that are in the book. I know how to read a book, so if lecture focused on more application of the theorems instead of repeating what the book said, then I would be able to be more prepared for the homework.
2. What has contributed most to your learning in this class thus far?
I have a clear understanding of the material because lecture is so clear. That really helps. You are one of the best professors I have had because you speak and write clearly, and you lecture at an appropriate pace and at an appropriate level of knowledge. I feel like I can really learn because it isn't over my head or unclear. I really appreciate that, and for me, how the professor teaches has a huge impact on my learning. So thank you.
3. 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.)
I think including in lecture more application of the theorems and definitions and stuff would be really effective because homework is a giant mathematical leap from the reading and the lecture. To bridge the gap, help us in lecture to understand application principles involved in the material to prepare us for the hard homework.
I have spent between 3-5 hours on each homework assignment. Sometimes it is frustrating because I feel like a lot of the problems are repeats, or like all the parts aren't necessary to get the points across. Sometimes it could be shortened. Lecture and the reading do prepare me, but the problem is in lecture we go over exactly what is in the reading. We go over the exact proofs, examples, etc. that are in the book. I know how to read a book, so if lecture focused on more application of the theorems instead of repeating what the book said, then I would be able to be more prepared for the homework.
2. What has contributed most to your learning in this class thus far?
I have a clear understanding of the material because lecture is so clear. That really helps. You are one of the best professors I have had because you speak and write clearly, and you lecture at an appropriate pace and at an appropriate level of knowledge. I feel like I can really learn because it isn't over my head or unclear. I really appreciate that, and for me, how the professor teaches has a huge impact on my learning. So thank you.
3. 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.)
I think including in lecture more application of the theorems and definitions and stuff would be really effective because homework is a giant mathematical leap from the reading and the lecture. To bridge the gap, help us in lecture to understand application principles involved in the material to prepare us for the hard homework.
Tuesday, January 25, 2011
3.3, due on Wednesday, January 26
1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material was following how the proofs work for some of the corollaries and theorem sthat deal with homomorphism and isomorphism. For example, I have a hard time connecting Corollary 3.13 with the work we did with subrings from the previous sections. There are too many definitions, rules, and theorems to keep up with to understand all these weird properties for homomorphism and isomorphism.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
I think it is interesting how functions have popped back into this mathematics. I would not have expected functions to be included in working with rings or with our new concepts of homomorphism and isomorphism. It is interesting how most mathematics relates to all other parts of mathematics as well. Crazy!
The most difficult part of the material was following how the proofs work for some of the corollaries and theorem sthat deal with homomorphism and isomorphism. For example, I have a hard time connecting Corollary 3.13 with the work we did with subrings from the previous sections. There are too many definitions, rules, and theorems to keep up with to understand all these weird properties for homomorphism and isomorphism.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
I think it is interesting how functions have popped back into this mathematics. I would not have expected functions to be included in working with rings or with our new concepts of homomorphism and isomorphism. It is interesting how most mathematics relates to all other parts of mathematics as well. Crazy!
Thursday, January 20, 2011
3.1 Part 2, due on Friday, January 20
1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material was knowing HOW to prove that all the axioms are true for something to be a subring. Most of the examples said this is, this isn't and because of what axiom is false. But nothing took you through the process of figuring out how to prove which axioms are true and false. I have a hard time coming up with proofs on my own and knowing if they are efficient or not, so that is the hardest part of this reading for me.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of this reading is how matrices are being used a rings and subrings. It is connecting to what I learned about matrices in linear algebra. I kind of thought I was not going to see matrices again, so this is interesting -- using them to learn and think about rings and subrings.
The most difficult part of the material was knowing HOW to prove that all the axioms are true for something to be a subring. Most of the examples said this is, this isn't and because of what axiom is false. But nothing took you through the process of figuring out how to prove which axioms are true and false. I have a hard time coming up with proofs on my own and knowing if they are efficient or not, so that is the hardest part of this reading for me.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of this reading is how matrices are being used a rings and subrings. It is connecting to what I learned about matrices in linear algebra. I kind of thought I was not going to see matrices again, so this is interesting -- using them to learn and think about rings and subrings.
Monday, January 17, 2011
3.1 Part 1, due on Wednesday, January 19
1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material for me was some of the technical details with the new definitions. For example, a couple of them talk about 1-sub-R not equaling 0-sub-R and other equations equaling 0-sub-R. I do not understand the 1 not equaling 0 or the other equations equaling 0 especially because I do not know what the sub-R's after all these mean. I am so confused and I have no idea where that technicality came from or what it means.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was how you could make a ring out of a set of letters and you could even redefine what addition and multiplication are, in particular stuff like homework problem 11. I had never thought about doing this type of math with letters and with new definitions of addition, multiplication, etc. It is cool, and I think it makes it applicable to then apply this stuff to other types of numbers if you can do it with letters.
The most difficult part of the material for me was some of the technical details with the new definitions. For example, a couple of them talk about 1-sub-R not equaling 0-sub-R and other equations equaling 0-sub-R. I do not understand the 1 not equaling 0 or the other equations equaling 0 especially because I do not know what the sub-R's after all these mean. I am so confused and I have no idea where that technicality came from or what it means.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was how you could make a ring out of a set of letters and you could even redefine what addition and multiplication are, in particular stuff like homework problem 11. I had never thought about doing this type of math with letters and with new definitions of addition, multiplication, etc. It is cool, and I think it makes it applicable to then apply this stuff to other types of numbers if you can do it with letters.
Thursday, January 13, 2011
2.3, due on January 14
1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material for me was trying to follow the proof the book gave for showing that point 3 in Theorem 2.8 was equivalent to point 1 in Theorem in 2.8. I got very lost when they assumed that when a*b=0 in Zp, then a=0 or b=0 and used it to show that p is prime. There was too much assumptions done throughout the proof for me to follow, so I am not yet convinced of this truth.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was Theorem 2.11. I think it is so interesting that if d/b then the equation has solutions, and furthermore, that the equation has d distinct solutions. I liked being able to use this theorem to do the homework. It was so simple. However, I am interested in seeing the proof for why the equation has d distinct solutions. I would be very interested to see how that works.
The most difficult part of the material for me was trying to follow the proof the book gave for showing that point 3 in Theorem 2.8 was equivalent to point 1 in Theorem in 2.8. I got very lost when they assumed that when a*b=0 in Zp, then a=0 or b=0 and used it to show that p is prime. There was too much assumptions done throughout the proof for me to follow, so I am not yet convinced of this truth.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was Theorem 2.11. I think it is so interesting that if d/b then the equation has solutions, and furthermore, that the equation has d distinct solutions. I liked being able to use this theorem to do the homework. It was so simple. However, I am interested in seeing the proof for why the equation has d distinct solutions. I would be very interested to see how that works.
Tuesday, January 11, 2011
2.2, due on January 12
1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material for me was how in the previous section we had been dealing with mods and now we are dealing with Z and my mind gets confused on transferring what I had learned about mods over to calling it all Z's. I mean, I think all the Z stuff is like using the idea of a mod or a change of base, but I'm still confused when approaching problems like the ones on the homework.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was Theorem 2.7 where there were like 10 properties that hold true for classes in Zn. I think that is fascinating how there are all these properties - very cool. Also, the proofs for them seem simple enough that it is interesting to see how they all simply work.
The most difficult part of the material for me was how in the previous section we had been dealing with mods and now we are dealing with Z and my mind gets confused on transferring what I had learned about mods over to calling it all Z's. I mean, I think all the Z stuff is like using the idea of a mod or a change of base, but I'm still confused when approaching problems like the ones on the homework.
2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was Theorem 2.7 where there were like 10 properties that hold true for classes in Zn. I think that is fascinating how there are all these properties - very cool. Also, the proofs for them seem simple enough that it is interesting to see how they all simply work.
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