Difficult:
I thought one of the most difficult parts of this section was the division algorithm in F[x]. I understood how it works, but I thought the proof was very difficult to follow. I think I got lost in all of the coefficients and exponents. But I thought it was very cool that the division algorithm also works for fields and polynomials. I also got confused about what x was supposed to represent in our polynomials. It talked about how x is not a number that we are trying to solve for, but that it is a specific element of P. I just don't understand what x is.
Connections:
I like that we are working with polynomials now. I understand how to do mathematics on polynomials since I have been dealing with them pretty much since 7th grade. I hope that my background with polynomials will make it easier for me to understand the new applications in this class. I am comfortable adding, multiplying, and dividing polynomials, so I am happy that there is something I am familiar with as I learn some new things that I am just seeing for the first time.
Saturday, January 29, 2011
Thursday, January 27, 2011
Double Bubble Theorem, Presented on January 25
I went to the presentation by Dr. Morgan on Tuesday, and I really enjoyed it. I felt that he was a very entertaining presenter and that I actually understood what he was talking about. The thing that I found most difficult about the presentation was trying to determine which form had the least amount of surface area for the double, triple, quadruple bubbles and so on, even up to 13 or something like that. I don't understand how they would calculate that to find which structure was the best. He had students go up and guess and pick which one they thought had the least surface area, but I just didn't really understand how to even begin to determine that.
I really enjoyed going to the presentation, especially I had heard of the double bubble theorem, but I honestly had no idea what it was. It was great to learn a little bit more about it, especially in such a fun way. Another thing that I really enjoyed was the displays he put on the overhead projector about finding the equilibrium state of the bubbles. He dipped the structure into the soapy water and showed that it found its balance when all three angles forming the connection were the same. (This is really hard for me to explain, but it was cool to see). It was interesting to see the lines of the bubbles move to this equilibrium state just as Dr. Morgan said they would. I learned that I am a terrible guesser since I got most of his questions wrong on the projector, but I also learned a little more about what the double bubble theorem is.
I really enjoyed going to the presentation, especially I had heard of the double bubble theorem, but I honestly had no idea what it was. It was great to learn a little bit more about it, especially in such a fun way. Another thing that I really enjoyed was the displays he put on the overhead projector about finding the equilibrium state of the bubbles. He dipped the structure into the soapy water and showed that it found its balance when all three angles forming the connection were the same. (This is really hard for me to explain, but it was cool to see). It was interesting to see the lines of the bubbles move to this equilibrium state just as Dr. Morgan said they would. I learned that I am a terrible guesser since I got most of his questions wrong on the projector, but I also learned a little more about what the double bubble theorem is.
Response Blog, Due on January 28
On average, the homework assignments take about 2-3 hours. I usually can't seem to answer all of the questions on my own, but by working with others we can talk about what works and what doesn't work and can usually come up with at least a general idea of how to attack the problems. I often feel very confused though. I think I am just not very good at knowing where to start with proofs and I have a hard time understanding what the questions seem to be looking for on my own.
I feel like I understand more from lecture than I do from the readings, but maybe that is because I have done the readings before I go to class, so I have a good idea about what will be covered, so I know what questions I am already trying to understand before lecture starts. I feel that the readings help me prepare for lecture and lecture helps me prepare for the homework. I think so far in this class, lecture and homework has contributed most to my learning. I can focus and learn what is covered in lectures. But I feel that the homework helps me really apply what I have learned to test whether or not I am really understanding.
As far as making my learning more effective, I feel like if I study more as I go I will be able to refer back to previous material and know what theorems, definitions, or other things will help me as I am attempting a proof instead of having to flip through each section of the book to see what will be useful. I think I need to be better about memorizing things as I go instead of cramming it all in before the test. I also learn very well from examples, so I really appreciate all of the examples that we can see in class that will be beneficial for our understanding of the concepts presented.
I feel like I understand more from lecture than I do from the readings, but maybe that is because I have done the readings before I go to class, so I have a good idea about what will be covered, so I know what questions I am already trying to understand before lecture starts. I feel that the readings help me prepare for lecture and lecture helps me prepare for the homework. I think so far in this class, lecture and homework has contributed most to my learning. I can focus and learn what is covered in lectures. But I feel that the homework helps me really apply what I have learned to test whether or not I am really understanding.
As far as making my learning more effective, I feel like if I study more as I go I will be able to refer back to previous material and know what theorems, definitions, or other things will help me as I am attempting a proof instead of having to flip through each section of the book to see what will be useful. I think I need to be better about memorizing things as I go instead of cramming it all in before the test. I also learn very well from examples, so I really appreciate all of the examples that we can see in class that will be beneficial for our understanding of the concepts presented.
Tuesday, January 25, 2011
3.3, Due on January 26
Difficult:
I thought the beginning of the chapter was very difficult. I got confused with all the relabeling and and new notation of the bars. I also have a hard time remembering what injective and surjective means and how to prove that a function is injective and surjective. I think the most difficult part of this section for me will be to use the notation and to show injectivity and surjectivity of functions.
Connections:
I remember proving that something is injective, surjective, and bijective in my 290 and 341 classes, but I never really understood what it means, especially how to prove it, and to know if you have actually proved it. I have seen the function notation and read the definitions of these things many times, but I always have a hard time remembering them. So hopefully I will be able to remember what I had been taught and be able to create proofs using some of my previous knowledge of injectivity and surjectivity.
I thought the beginning of the chapter was very difficult. I got confused with all the relabeling and and new notation of the bars. I also have a hard time remembering what injective and surjective means and how to prove that a function is injective and surjective. I think the most difficult part of this section for me will be to use the notation and to show injectivity and surjectivity of functions.
Connections:
I remember proving that something is injective, surjective, and bijective in my 290 and 341 classes, but I never really understood what it means, especially how to prove it, and to know if you have actually proved it. I have seen the function notation and read the definitions of these things many times, but I always have a hard time remembering them. So hopefully I will be able to remember what I had been taught and be able to create proofs using some of my previous knowledge of injectivity and surjectivity.
Saturday, January 22, 2011
3.2, Due on January 24
Difficult:
I had a hard time understanding the definition of a unit. I got confused because I sounded to me like every ring with identity may not always have a unity, but I thought that was the definition of being a ring with identity. If a ring has identity, then it has a multiplicative inverse. But the definition made it sound like a ring with identity only has a unit if there exists and element a such that ua=1=au. I just was confused if there is a unit for every ring with identity.
Connection:
I really liked this section. I like that subtraction in a ring is defined as most people think of subtraction (just adding a negative). I like that arithmetic in rings, for the most part, is something that I am familiar with. For example, theorem 3.5 made a lot of sense to me. I like when I am able to follow the logic of the proofs and understand the meaning of the theorems.
I had a hard time understanding the definition of a unit. I got confused because I sounded to me like every ring with identity may not always have a unity, but I thought that was the definition of being a ring with identity. If a ring has identity, then it has a multiplicative inverse. But the definition made it sound like a ring with identity only has a unit if there exists and element a such that ua=1=au. I just was confused if there is a unit for every ring with identity.
Connection:
I really liked this section. I like that subtraction in a ring is defined as most people think of subtraction (just adding a negative). I like that arithmetic in rings, for the most part, is something that I am familiar with. For example, theorem 3.5 made a lot of sense to me. I like when I am able to follow the logic of the proofs and understand the meaning of the theorems.
Thursday, January 20, 2011
3.1, Due on January 21
Difficult:
In this section of 3.1, I had a hard time understanding why you would have to have 4 new axioms to determine whether or not the subset of a ring would be a subset. If its a subset of a ring, I don't understand why these 4 axioms need to be re-fulfilled. And if these 4 do, why don't the other 4? I don't understand why closed under addition and multiplication, and a 0 element is in the subset, and that there is a solution to a + x = 0 ( the 0 element in the subset). I didn't understand why these are the 4 axioms that need to be fulfilled.
Reflection:
I really like using the Cartesian product of z modulo 6 and z modulo 4 to see if it is a ring. I like being able to write out addition and multiplication tables to organize the information to help determine if the Cartesian product is a ring. I also like that we can redefine addition and multiplication to find other rings and use the axioms to see if the set is a ring or if that ring has any subrings.
In this section of 3.1, I had a hard time understanding why you would have to have 4 new axioms to determine whether or not the subset of a ring would be a subset. If its a subset of a ring, I don't understand why these 4 axioms need to be re-fulfilled. And if these 4 do, why don't the other 4? I don't understand why closed under addition and multiplication, and a 0 element is in the subset, and that there is a solution to a + x = 0 ( the 0 element in the subset). I didn't understand why these are the 4 axioms that need to be fulfilled.
Reflection:
I really like using the Cartesian product of z modulo 6 and z modulo 4 to see if it is a ring. I like being able to write out addition and multiplication tables to organize the information to help determine if the Cartesian product is a ring. I also like that we can redefine addition and multiplication to find other rings and use the axioms to see if the set is a ring or if that ring has any subrings.
Tuesday, January 18, 2011
3.1, Due on January 19
Difficult:
I thought the most difficult part of this section was keeping track of all the different types of rings. I had a hard time distinguishing when a ring is commutative, is a ring with identity. And then once I figure out if its commutative, then it could be an integral domain or a field. I didn't really understand these definitions and how best to keep track of them. I also didn't really understand what the 0 with a subscript R meant, or a 1 with subscript R. I think in this section I mostly got lost in all the definitions and new notation.
Connection:
As with modular arithmetic, I like that rings follow properties of addition and multiplication like the integers. It makes it easier for me when I already understand the axioms that define a ring. So once I can better understand the new notation, I will feel more comfortable with recognizing rings because I already know about the axioms, since I have been using these axioms pretty much all of my life.
I thought the most difficult part of this section was keeping track of all the different types of rings. I had a hard time distinguishing when a ring is commutative, is a ring with identity. And then once I figure out if its commutative, then it could be an integral domain or a field. I didn't really understand these definitions and how best to keep track of them. I also didn't really understand what the 0 with a subscript R meant, or a 1 with subscript R. I think in this section I mostly got lost in all the definitions and new notation.
Connection:
As with modular arithmetic, I like that rings follow properties of addition and multiplication like the integers. It makes it easier for me when I already understand the axioms that define a ring. So once I can better understand the new notation, I will feel more comfortable with recognizing rings because I already know about the axioms, since I have been using these axioms pretty much all of my life.
Thursday, January 13, 2011
2.3, Due on January 14
Difficult:
For the most part, I felt like I pretty much understood this section, which is a pretty big surprise for me! However, the part that I found to be most difficult was in the proof of Corollary 2.9. In the last part of the proof when the authors are proving uniqueness, they have this great idea to multiply both ends of the equation by u. This made sense, but the reason I found it to be so difficult was because I would never be able to think of doing something like that. I see why it works, but when I am doing proofs, I never think of doing something like this. Multiplying through by the number I want to both sides of the equation is not something that I ever think to do. This step in the proof was difficult for me to understand only because it is not something that I ever think of doing.
Connection:
I really liked the method used to solve the equation ax=b in Z mod n using the Euclidean algorithm to find a linear combination. Then using the solutions in the linear combination, you can find the solution to ax=b. I like being able to use algebra skills that I am already familiar with to solve new mathematics that I encounter in this course. Even though modular algebra is a newer, more difficult concept to me, I feel like I better understand the material when I can apply concrete ideas that I already know how to use.
For the most part, I felt like I pretty much understood this section, which is a pretty big surprise for me! However, the part that I found to be most difficult was in the proof of Corollary 2.9. In the last part of the proof when the authors are proving uniqueness, they have this great idea to multiply both ends of the equation by u. This made sense, but the reason I found it to be so difficult was because I would never be able to think of doing something like that. I see why it works, but when I am doing proofs, I never think of doing something like this. Multiplying through by the number I want to both sides of the equation is not something that I ever think to do. This step in the proof was difficult for me to understand only because it is not something that I ever think of doing.
Connection:
I really liked the method used to solve the equation ax=b in Z mod n using the Euclidean algorithm to find a linear combination. Then using the solutions in the linear combination, you can find the solution to ax=b. I like being able to use algebra skills that I am already familiar with to solve new mathematics that I encounter in this course. Even though modular algebra is a newer, more difficult concept to me, I feel like I better understand the material when I can apply concrete ideas that I already know how to use.
Tuesday, January 11, 2011
2.2, Due on January 12
Difficult:
At the beginning of the section, it defined classes of integers, A, B, C, D, E, where the classes were all disjoint, and every integer was in one of the equivalence classes, but it showed that B+C=D and B+C=A. I was confused why this happened. Did they set up these equivalence classes in such a way just to prove this point? I was just really confused where these equivalence classes came from, and why the addition property didn't work in them. I had a difficult time understanding when this would happen in a problem, and why the addition and multiplication that were defined at the beginning of the chapter were just "tentative".
Connection:
The most interesting part of this section for me was that modular arithmetic also follows the same properties as arithmetic in the integers (with the exception that if ab=0, then a=0 or b=0). I thought that this section was really cool to see that modular arithmetic will follow the same rules we have been taught over and over in math. These rules are ones that we have practiced over and over, so it is interesting to be able to apply them to a new kind of arithmetic. I really like when I can use something that is familiar to me when I am trying to learn something new and more difficult.
At the beginning of the section, it defined classes of integers, A, B, C, D, E, where the classes were all disjoint, and every integer was in one of the equivalence classes, but it showed that B+C=D and B+C=A. I was confused why this happened. Did they set up these equivalence classes in such a way just to prove this point? I was just really confused where these equivalence classes came from, and why the addition property didn't work in them. I had a difficult time understanding when this would happen in a problem, and why the addition and multiplication that were defined at the beginning of the chapter were just "tentative".
Connection:
The most interesting part of this section for me was that modular arithmetic also follows the same properties as arithmetic in the integers (with the exception that if ab=0, then a=0 or b=0). I thought that this section was really cool to see that modular arithmetic will follow the same rules we have been taught over and over in math. These rules are ones that we have practiced over and over, so it is interesting to be able to apply them to a new kind of arithmetic. I really like when I can use something that is familiar to me when I am trying to learn something new and more difficult.
Saturday, January 8, 2011
2.1, Due on January 10
Difficult:
I understood the congruence classes, but I got confused about the second part of Corollary 2.5. It says that there are exactly n distinct congruence classes. I am confused about the meaning of distinct. Does this just mean they do not overlap with each other? I also found it difficult to understand that the set of all congruence classes has exactly n elements. I was confused because at first it points out that the congruence class of 2, 5, -1, and 14 are the same modulo 3, but that only the congruence class of 2 is in the set of all congruence classes modulo 3. I just didn't understand why all of the other ones wouldn't be considered to be in the set of congruence classes modulo 3 because they are the same as the congruence class of 2 modulo 3. I guess overall, the most difficult part of this section for me was understanding congruence classes and properties of them.
Reflection:
I really like that the reflexive, symmetric, and transitive properties also apply to congruence modulo n. I know that as a secondary mathematics teacher, I will encounter these three properties quite often when teaching algebra and other subjects. I like that these are properties that I am familiar with and understand so that I can apply the information that I already understand, and will one day teach, to something that I am less familiar with, congruence and congruence classes. I am excited to be able to teach my students about these properties and how they apply in algebra classes.
I understood the congruence classes, but I got confused about the second part of Corollary 2.5. It says that there are exactly n distinct congruence classes. I am confused about the meaning of distinct. Does this just mean they do not overlap with each other? I also found it difficult to understand that the set of all congruence classes has exactly n elements. I was confused because at first it points out that the congruence class of 2, 5, -1, and 14 are the same modulo 3, but that only the congruence class of 2 is in the set of all congruence classes modulo 3. I just didn't understand why all of the other ones wouldn't be considered to be in the set of congruence classes modulo 3 because they are the same as the congruence class of 2 modulo 3. I guess overall, the most difficult part of this section for me was understanding congruence classes and properties of them.
Reflection:
I really like that the reflexive, symmetric, and transitive properties also apply to congruence modulo n. I know that as a secondary mathematics teacher, I will encounter these three properties quite often when teaching algebra and other subjects. I like that these are properties that I am familiar with and understand so that I can apply the information that I already understand, and will one day teach, to something that I am less familiar with, congruence and congruence classes. I am excited to be able to teach my students about these properties and how they apply in algebra classes.
Thursday, January 6, 2011
1.1-1.3, Due on January 7
Difficult:
For me, the most difficult part of this section was in section 1.3. In the fundamental theorem of arithmetic, it states how every integer is unique. I was very confused by how p could be a plus or minus q. I just couldn't understand what this statement was saying, and how it made each prime factorization unique. I think I just got lost in all the plus or minus signs. I had a really hard time trying to keep track of them, and whether or not they were canceling out, or if the answers were negative or positive. I found the plus and minus signs in the Fundamental Theorem of Arithmetic and its proof to be very confusing and the most difficult part of this section of reading.
Reflection:
I really liked section 1.3. It reminded me of what prime numbers are and that all numbers can be factored into a product of primes. This is something that as a secondary math teacher, I will encounter a lot as I teach high school and middle school students. As I gain a better understanding of prime and prime factorization, I will be able to better articulate these ideas to my future students. Also, in section 1.2, I liked learning about relatively prime. I made a task in one or my previous education classes that used the concept of relatively prime to solve the problem, so it was nice to see this concept come up again in this class. I plan to use my task in my future algebra classes, so I will also teach my students about what it means for two numbers to be relatively prime which was discussed in this abstract algebra book.
For me, the most difficult part of this section was in section 1.3. In the fundamental theorem of arithmetic, it states how every integer is unique. I was very confused by how p could be a plus or minus q. I just couldn't understand what this statement was saying, and how it made each prime factorization unique. I think I just got lost in all the plus or minus signs. I had a really hard time trying to keep track of them, and whether or not they were canceling out, or if the answers were negative or positive. I found the plus and minus signs in the Fundamental Theorem of Arithmetic and its proof to be very confusing and the most difficult part of this section of reading.
Reflection:
I really liked section 1.3. It reminded me of what prime numbers are and that all numbers can be factored into a product of primes. This is something that as a secondary math teacher, I will encounter a lot as I teach high school and middle school students. As I gain a better understanding of prime and prime factorization, I will be able to better articulate these ideas to my future students. Also, in section 1.2, I liked learning about relatively prime. I made a task in one or my previous education classes that used the concept of relatively prime to solve the problem, so it was nice to see this concept come up again in this class. I plan to use my task in my future algebra classes, so I will also teach my students about what it means for two numbers to be relatively prime which was discussed in this abstract algebra book.
Wednesday, January 5, 2011
Introduction, Due on January 7
My name is Haley Stevenson. This is my 4th year here at BYU, so I am technically a senior. However, I still have one year left in school before I will graduate. Beyond calculus, I have taken Linear Algebra, Multivariable Calculus, Theory of Analysis, Math 290, and am currently enrolled in Differential Equations. I am taking this Abstract Algebra class not only because it is a requirement for my math education major, but I really like algebra so I hope this course will help me better understand the kinds of algebra that I will teach in my future classroom.
One math professor that I had that was a very effective teacher truly cared about the success of his students. I really appreciated that he took extra time when necessary to explain difficult concepts. I also appreciated that he would show examples in class of what to expect on homework and exams. I find it difficult in courses when I feel that the homework or exams are completely different than what is covered in a class. I also liked that he was willing to take extra time to help students through office hours and such to further explain and help solidify concepts or go over difficult homework problems.
The only thing that I find to be unique about myself is that I am semi-left handed. I write with my left hand, but I do all other things right handed (eating, painting my nails, playing sports, brushing my teeth, etc). I'm not sure why this is... but that is just how I am I guess!
One math professor that I had that was a very effective teacher truly cared about the success of his students. I really appreciated that he took extra time when necessary to explain difficult concepts. I also appreciated that he would show examples in class of what to expect on homework and exams. I find it difficult in courses when I feel that the homework or exams are completely different than what is covered in a class. I also liked that he was willing to take extra time to help students through office hours and such to further explain and help solidify concepts or go over difficult homework problems.
The only thing that I find to be unique about myself is that I am semi-left handed. I write with my left hand, but I do all other things right handed (eating, painting my nails, playing sports, brushing my teeth, etc). I'm not sure why this is... but that is just how I am I guess!
Subscribe to:
Posts (Atom)