Extra Credit Math talk by Kumar Murty

This was a really fascinating lecture to me. He talked about how the brain is more of a horizontally structured way of thinking instead of as vertical as mathematical thinking is. He brought up certain principles like the Heisenburg Uncertainty Principle where math is not really able to reach it. He brought up the fact that some group was offering money for a good mathematical model of the brain. Then he ended his lecture by saying that in reality, maybe by studying the brain we could come up with a new "field" of mathematics that would function more similarly to how the brain functions. This idea was so cool to me and I really enjoyed this lecture.

Final Blog, due Dec. 8

From looking at the review sheet I know that the concept of prime and maximal ideals is something that I would like to go over. Also, the concept of extension fields. I know that I need to work on better understanding quotient rings and quotient fields and some problems regarding these topics would be helpful. I think it would help to have a couple of examples of these quotient rings/fields and then describe them together (sort of like on our test).
Also, examples of different cyclic, non-integral domain etc. examples of rings/groups/fields would be helpful to me as this is a section on the tests that is always difficult for me. I also had a question: on the review sheet there are lots of theorems we should know well and I was wondering if that is the same list of theorems that we need to be able to prove.

Section 8.5, due on December 6

The most difficult part of this material for me was following the long proofs and connecting all of the ideas. In specific, Theorem 8.34 was slightly confusing to me. I also has a couple of questions regarding Theorem 8.33.

The most interesting part of this section was looking over the proof of theorems we had basically already done or applied in previous exercises and seeing how they proved it. I also round it really neat that so many groups have been classified as they show on page 281.

Section 8.4, due on December 3

The most difficult part of this section for me was pickinga part the different notation so that I got the full meaning of what it was trying to say. Also, In reading th proof of the Second Sylow Theorem, I found parts of it to be confusing. I also found it a little confusing when going over the proof of the First Sylow Theorem.

The most interesting part of this section to me was the fact that conjugacy is an equivalence relation. I was wondering if this means that given any equivalence relation, if that means that the equivalence classes partition whatever mega set you are talking about? I think this would be true, but for some reason it just dawned on me how cool that really is.

Section 8.3, due on December 1

I am confused as to whether Sylow refers to just subgroups or to groups as well. Every group is a subgroup of itself, so is that why it's called Sylow p-subgroup vs. just a Sylow p-group? Also, some of the examples were confusing to me especially following the Applications of the Sylow Theorems subtitle.

The most interesting part of this section was Corollary 8.18 because it seems as though it is a very helpful fact in figuring out what various groups are isomorphic to. Also I was wondering where the term Sylow came from. Is it named after a person?

Section 8.2, due on November 29

The most difficult part of this material for me was following Lemma 8.6. I got a little confused with the difference between a p-group and a cyclic group. Maybe it was just because it is a long theorem but there were some conclusions drawn that just didn't quite make sense to me.

The most interesting part of this material was the Fundamental Theorem of Finite Abelian Groups. When I think about all of the things that come from knowing a group is cyclic and has some characteristic associated with primes, it's very cool to think that every finite abelian group is the direct sum of cyclic groups and each of prime power order.

Section 8.1, due on November 23

The most difficult part of this material for me was understanding how you would write an element of a group that is a direct product of a bunch of different groups that don't have the same operation. Do you do it like a cross product like we had learned before?

The most interesting part of this material for me was the fact that just because G is a group that makes up some cross product, doesn't mean that G is a subgroup of that cross product.

Section 7.10, due on November 22

The most difficult part of this section was following all of the long proofs and trying to keep up with what was going on. I also don't really think I fully understand what the Alternating Group is and so some of the theorems made sense, but I didn't really understand the full implications behind them.

The most interesting part of this section was Lemma 7.53. I just thought it was interesting that even some An with n even has this property that every element is the product of 3-cycles. I also thought it was interesting to think about how all of these different theorems/lemmas of this section would apply to other areas and how they help.

Section 7.9, due on November 19

The most difficult part of this material for me was Theorem 7.5 and also just trying to keep the order of the cycles right. I don't think I quite understand why in Thrm. 7.5 they just remove the parantheses and put the element side by side because I thought the parentheses denoted cycles.

The most interesting part of this material for me was the Thrm. 7.47. I thought it was neat that every permutation in Sn can be written as the product of disjoing cycles. This seems like a really interesting concept and it just seems like a cool fact that has been proved.

Exam 3 review, due on November 17

Before the exam I need to better understand the concept of quotient groups and modding out by subgroups. I also need to go over again the various concepts of order and the theorems that apply to them. I feel like I also need to better understand some particular theorems such as the Second Isomorphism Theorem for groups.

I problem that I would like to see worked out in class would be one involving kernels/normal subgroups. I sometimes get confused in these types of proofs when transitioning between the different steps.

Section 7.8, due on November 15

The most difficult part of this material for me was understanding the Third Isomorphism Theorem and the theory that lead up to it. I still don't think I fully understand what subgroups of G/N look like and how they are related to G. I am also still a little confused about the section where they talk about how simple groups are the building blocks of all other groups and also the term composition factors.

The most interesting part of this section for me was theorem 7.45. I thought this was interesting considering that simple groups are the building blocks for all other groups and we know what the building blocks are isomorphic to, but when combined into a larger group we do not always know.

Section 7.7, due on November 12

The most difficult part of this material for me was thinking about what G/N actually does. The concept of modding out by a subgroup seems similar to modding out by ideals, but for some reason it still takes me a minute to really think about what is going on. Also some of their examples were a little confusing to follow.

The most interesting part of this material for me was noticing all of the similarities and differences to when we modded out by an ideal. I found it interesting that with groups we can talk about the order of G/N (thrm. 7.36). Why did the topic of order of a group just now come into play? Why didn't we talk about this when we were studying rings?

Section 7.6, due on November 10

The most difficult part of this material for me was following the all of the details that come along with Theorem 7.4. It was more difficult for me to understand in general why aN=N does not means that an=n for each n in N but rather that an=n1 for some n1 in N. The differences between these statements are a little confusing to me (ie. top of page 213).

The most interesting part of this material for me was the concept of a normal subgroup and how it is analagous to an ideal. It also made me wonder if it is at all related to a normal vector that we study in multivariable calculus and linear algebra. It is interesting, but still a little confusing, that normal subgroups aren't necessarily made up of elements that commute with all other elements of the group.

Section 7.5 (second part), due on November 8

The most difficult part of this material for me was completely following all of the logic and reasoning involved in Theorem 7.30 especially how they came up with the six elements from Ne and Nb that are elements of G.

The most interesting part of this material was that every group of order p (where p is a positive prime integer) is cyclic and isomorphic to Zp. I thought that this seemed to be a pretty powerful statement considering there are infinitely many primes. Considering the special behavior of Zp, it makes me wonder if a group of order p has special structure and behavior as well because it is isomorphic.

Section 7.5 (first part), due on November 5

The most difficult part of this material was understanding the concept of the index of H in G and also Lagrange's Theorem. The concept of congruence with subgroups doesn't quite make sense to me which is why the rest of these concepts aren't quite clicking. Also it talks about right cosets but not left cosets...are there left cosets and if there are do they have the same properties.

The most interesting part of this section for me was relating all of the similar properties to rings and integers. I also found it interesting that you can even define congruence when you mod out by a subgroup. This idea was really interesting to me, however it still doesn't quite make sense.

Section 7.4, due on November 3

The most difficult part of this material for me was trying to understand the proof of Cayley's Theorem. Also, I didn't really understand the second example of the section. I also don't remember what it means when they put two stars next to R, and whether that is just to denote something or if it is supposed to mean a certain something.

The most interesting part of this section for me was trying to figure out how isomorphisms of groups could be applied to the real world and if this is even a possibility. I was also wondering if certain things are preserved in a group isomorphism. Also, what is the difference between an isomorphism of rings and groups in a visual sense. Does it just sort of copy something more general whereas in the case of rings it is a copy of something more detailed? Or is the actual isomorphism different when dealing with rings?

Section 7.3, due on November 1

The hardest part of this material for me was understanding Theorem 7.17. I don't quite understand what is going on with the subgroup generated by S. I was also wondering in regards to the example under Theorem 7.13, why every unit mulitiplied by another unit must be a unit. It seems to make sense, I am just not connecting why this must be true.

The most interesting part of this section was the center. I found this interesting and was wondering if the center of a group is sort of like an ideal of a ring. They seem similar and so I was wondering if they are related in any way.

Section 7.2, due on October 29

The most difficult part of this material for me was comprehending Theorem 7.8 and its proof. I think I understand it, it just takes some time to sink in. I also found it a little bit confusing just because there are so many different types of examples that it is sometimes hard for me to wrap my thoughts around exactly what is being implied.

The most interesting part of this section was Corollary 7.9 because it closely paralleled what we did with degree in fields and factors with integers. It is interesting to see similar concepts apply while dealing with things that can be totally different from each other.

Section 7.1 (the rest of it), due October 27

The most difficult part of this material for me was understanding the dihedral group of degree n. I don't really what is going on to make this a group. I also, found Theorem 7.4 a bit confusing with the different symbols. I think one of the most difficult things for me is that there are so many different "operations" that could take place that it is confusing to me to figure out exactly what a group is.

The most interesting part of this section for me was also the most difficult. I found it really rather neat how a group it seems is more or less something with a pattern and that pattern/operation can be a bunch of different things. I also gound it interesting that symmetry groups have been used by physicists to predict the existence of certain elementary particles.

Section 7.1-pg.164, due on October 25

The most difficult part of this material for me was understanding the symmetric group on n symbols. I am not sure how they came to all of these generalizations about Sn by just having looked at S6. I also found it confusing to try and see how they came up with the inverse of f under composition. I also don't understand how S3 had order 6???? I thought if elements repeated then they were just counted as 1 element within a set. Is this just a process of trial and error or is there some method to this? I was also wondering exactly what a group is, is it is "larger" than a ring, and if all groups are comparable to permutations.

The most interesting part of this material for me was the properties of S3 under composition. In another one of my math classes we are showing that if f and g are continuous, then so is their composition. I was thinking about this and trying to relate it to the example in this section. I am not sure how it relates, but it made me wonder if being continuous had anything to do with the fact that the composition operation is a group when dealing with integers.

Exam 2 review, due on October 20

1. The topics that believe will be important to know are roots and irreduciblity as well as some basics facts about a ring mod by an ideal, or a Field mod by an irreducible. I think knowing the idea behind Thrm. 4.5 will be important as well as Thrm 4.11. I think Thrm 6.1 as well as cosets, quotient rings and kernels will all be necessary to know.

2. I need to work on understanding quotient rings, cosets and ideals. I also need to clarify when certain definitions and theorems apply- in other words if you must have a field, a commutative ring, etc. I need to better understand different properties regarding degrees of polynomials. I also need to try and understand the First Isomorphic Thrm. better and understand when to apply it.

3. In the proof of Cor. 4.16 I don't understand why (c-a) can't be equal to the zero in F. The whole strategy of this proof is also confusing for me. In regards to cosets I was wondering if a coset of I in R means any elements of I plus any element of R where the elements in R are related to the representative? If this is true, does this mean that an ideal is always just some set of multiples of something? How does taking an element and adding something from I make it congruent to to something else in R mod I?

Section 6.3, due on October 18

The most difficult part of this material for me was connecting our past versions of prime with this new version and then using this new version in different proofs and things. It just takes me some time to adjust to the new use of prime and visualize it. It was also a little difficult for me to follow the proof of Thrm. 6.15.

The most interesting part of this material to me was the fact that R/P may not always be a field when P is prime. It's interesting to me how when you get more general, certain properties don't always apply. Yet this principle can also be true as you get more specific.

Section 6.2, due on October 15

The most difficult part of this material for me was tying to follow and understand what exactly is meant by the First Isomorphism Theorem and trying to understand all of its implications. I am not sure I fully grasp all that is going on with the homorphic image and the kernel and such and how it is all so closely related to ideals and quotient rings. I guess I don't really understand the structure of R/K.

The most interesting par of this material was the relationship all of this has to the things learned in Linear Algebra. Although I don't think I fully understand the homomorphic image concept, I did think it was really interesting when they compared it to photography. I found this an interesting comparison. It makes me wonder about how accurate the way we see things really is and how much of the exact structure we miss by simply observing things with our eyes.

Section 6.1 and 6.2 through the middle of pg. 147, due October 13

The most difficult part of this material for me was keeping all of the new terminology straight while trying to relate it to things we have done in the past so that I understand it. I don't really understand how the ideal generated by c1, c2, c3, .... is in fact an ideal. The cosets are also a little confusing to me, as well as the quotient ring R/I because I am confused as to how you "mod out" in this ring.

The most interesting part of this material for me was just the ideal itself. I thought it was interesting that there is a name for something that "absorbs products". Originally I would only think of the zero elements to be able to be ideals, but there is obviously more than that. The concept of ideals reminded me of the type of stuff you learn in genetics with dominant and recessive traits. To me an ideal was like this dominant trait that when combined with other stuff, still dominates (ie. when combined with other stuff it still ends up in the ideal).

Section 5.3, due on October 8

The most difficult part of this material for me was conceptually understanding the extension field and trying to visualize it. I am also confused when they say the F[x]/(p(x)) contains a root of p(x), if that root is a class or just a number. And if it's a class, is it the same type of class as the classes in F. Also, I was wondering if every elements in F is a congruence class, or not necessarily.

The most interesting part of this material for me was trying to think about why it would be useful to have the congruence-class ring. I also thought it was interesting to see how many levels the properties of integral domain, irreducibility, etc. can be carried out and still apply. It seems like you can always just keep finding bigger and bigger areas to work in, which kind of boggles my mind. I wonder if these applications have anything to do with different dimensions in space.

Section 5.2, due on October 6

The hardest part of this material for me is distinguishing between the field F, F[x], and the ring F[x]/(p(x)). I have to keep reminding myself that elements of F[x]/(p(x)) are classes of polynomials and not just polynomials. Theorem 5.7 was also confusing to me because of my confusion with trying to distinguish between F and F[x]/(p(x)).

The most interesting part of this material was Theorem 5.8. It was a little confusing to me, but it seemed like a cool fact. I was a little confused about the difference between Theorem 5.7 and Theorem 5.8, but when I figured out how they were different I thought this was also interesting. This section also makes me wonder what types of things the ring F[x]/(p(x)) is used for and why its structure is so much more rich then the set of congruence classes modulo n.

Section 5.1, due on October 4

The most difficult part of this material was trying to understand congruence classes from this perspective. Trying to figure out possible remainders is a little more involved in some F[x]. I would like to know if there is some trick I am not quite understanding that would make me think about this stuff in a faster way. It seemed practically the same yet trying to figure out the equivalence classes for a particular polynomial actually helped me to better understand congruence classes in general.

The most interesting part of this material was observing just how much everything related to the class of integers modulo n. Almost every proof was similar to a past theorem we had already learned. I also thought it was interesting that there are n^k distinct congruence classes in the set of all congruence classes modulo p(x). This is a little different from what we learned before, but still very similar.

Section 4.5-4.6, due on Oct. 1

Some of the hardest parts of this material for me was the section with complex numbers. I feel as though I am less familiar with imaginary properties in general, so it was hard for me to visualize what was going on. Also, I didn't quite understand the proof of Thrm. 4.22. I think I just need to familiarize myself with degrees of polynomials to better follow what they are saying.

The coolest parts of this material were the Eisenstein Criterion that is a way of showing f(x) is irreducible in Q(x). I also liked the Rational Root Test. I remember the first time I learned that concept and I thought it was cool, but had no idea where it was coming from. To see the proof of this test was really interesting for me.

Section 4.4, due on September 29

The most difficult part of this section was the the idea of a polynomial inducing a function. I have never heard of this before and it is still not super concrete in my mind. I'm not really sure why the concept of a polynomial inducing a function is so important and why it needs to be so distinctly distinguished between polynomial. I also found Corollary 4.19 to be a little hard for me to understand.

The most interesting part of this section was the remainder theorem. I don't think I had ever realized that before and I thought it was really neat. I also thought it was cool that to show that a deg 2 or 3 polynomial is irreducible you need only verify that it has no roots in the field F. It reminded me a lot of high school algebra, but with a cooler twist than they teach you. I feel like I finally am starting to understand the origin behind some of those facts that I just took for granted.

Section 4.2-4.3, due on September 27

The most difficult part of these sections for me was remembering all of the implications of Theorem 4.2. It doesn't seem like that hard of a thing, but I had trouble remembering when a degree of a particular polynomial is less than or equal to another. In section 4.3 the hardest thing was figuring out and applying the definition of irreducible and associat element. This were use often in proofs and these concepts are still a little fuzzy for me. I had an especially hard time following the proof of Theorem 4.11.

The part of this material that I found to be most interesting was the fact that a nonzero polynomial may have infinitely many divisors where, in contrast, a nonzero integer has only a finite number of divisors. I also thought is was interesting that in an integral domain, a unit is a constant polynomial yet in a field a unit must be a nonzero constant polynomial. The conditions that apply to fields are a little more strict, which I thought was interesting.

Section 4.1, due on September 24

The most difficult part of this section for me was following the proof of the Division Algorithm in F[x]. The subscripts and different notation was a little hard for me to follow. The whole concept of P containing an element of x that is not in R, yet with properties related to R is a little confusing. I guess I don't really understand why this idea is necessary in order to define a polynomial.

The most interesting part of this material for me was now much it reminded me of my Differential Equations class with leading coefficients and degrees and such. I also thought it was interesting when they started talking about x being a specific element of the ring P, and not a variable that can be assigned values. It was also interesting to note how closely the Division Algorithm in F[x] relates to the previous Division Algorithm that we learned.

Exam 1 review, due on September 22

I think that some of the most important topics to study would be those concerning rings and modular arithmetic. Having clear how to show something is a ring, subring, field, integral domain, isomorphism, homomorphism, etc. seem like topics that will be important to know. So basically all of the theorems and definitions that relate to these topics are areas that I think will be important. I also think knowing modular arithmetic properties will be especially important because it applies to divisibility, primes, and rings. Modular arithmetic seems like something that connects together everything that we have learned.

On the exam I expect to see lots of definitions and about 2 theorems out of the list provided that we will have to prove. I also expect 1 or 2 other proofs that will be very similar to previous homework problems that we have had. I expect something about how to show that a particular set is a subring, isomorphism, and homomorphism whether it be a proof or just giving the definition.

Section 3.3, due on September 20

The most difficult part of this material for me was following the isomorphisms from a set a integers modulo n, to a Cartestian product. I also found it hard to comprehend exactly what was being described by isomorphism and homomorphism. I think that I understand, but this was a new idea for me and sort of difficult to follow. I am also finding that many of the past couple of theorems are very similar to past theorems as they both deal with certain properties that allows something to be defined as it is. It has been a little difficult to try and keep all of these straight in my mind.

The most interesting part of this material was the fact that certain properties are preserved after isomorphisms. For some reason I just thought that was a really cool concept. It serves as a good analogy for many other aspects in life. I also thought that structure being preserved from one ring to another was a cool concept in general. It reminded me of chemistry classes I have taken where sometimes certain reactions will preserve properties of elements, while other times things aren't preserved at all.

Section 3.2, due on September 17

The most difficult part of this section for me is remembering what can and can't be assumed for different rings. I also had difficulty following the proof for theorem 3.11. In this section in particular there is so much manipulating of equations, that at first it is hard to see where they are going or why they are doing what they do. This makes it a little confusing and hard to follow at times. The term "unit" is also new and confused me a little because I would always call a unit just a number with a multiplicative inverse.

The most interesting part of this material for me was Theorem 3.6. I found it interesting that just by having multiplication and subtraction that the process of showing that S is a subring of R is reduced down to 2 steps. I also found the relationships between integral domains and fields to be hard to follow, but really interesting at the same time.

Response to questions, due on September 15

The homework assignments have usually taken me between 2-3 hours. The lecture and reading definitely prepared me for them, it's just that the homework requires putting concepts together in ways that I usually haven't thought of quite yet.

I have really liked the fact that we are required to read before class. This has really helped me to understand the material better. I also like that we can still ask questions and turn the homework in by 4:30 pm of the day that it is due. This has helped me because I feel like I can get most of my questions answered before I finish the homework.

To help me learn more effectively I just like to see examples of the concepts being applied and how they connect to one another. Sometimes an overview sheet of all that we have learned is helpful, or even mini reviews from time to time. This class has built on past concepts for the most part though, which has definitely helped it all stick more than otherwise.

Section 3.1, due on September 13

The hardest part of this section for me was keeping track of all of the different axioms and definitions as they would explain the properties of various rings. It was harder to understand the more abstract types of rings because I was not as used to the symbols being used in the ways that they were. I am also not really quite sure I understand exactly what a ring is and why it is something useful to have defined.

The most interesting part of this material for me was to see how much it reminded me of vector spaces and subspaces from Linear Algebra. It was also interesting for me to see which axioms held for different types of rings, and which properties had to be proved in order to show that something is a subring. It makes me wonder what sort of applications various rings have in the real world and if they are widely used and put into practice during current times.

Section 2.3, due on September 10

The most difficult part of this section for me was following the steps of the proof to Theorem 2.8 as it showed step 1 implied 2 and step 3 implied 1. Following the switch of notation during the proof was a little difficult for me because it made me have to remember exactly what they were talking about as I followed the proof, whether it be numbers or congruence classes.

I found the second part of Theorem 2.8 to be especially interesting. I just had never realized that and so it was really cool to learn. I also thought that Theorem 2.11 was a handy pattern that has been discovered. Both Theorems 2.8 and 2.11 reminded me of definitions/theorems in Linear Algebra regarding invertible matrices which was an interesting connection. It made me wonder if there is any sort of use to putting the multiplication/addition tables of the different congruence classes modulo n, into a matrix.

Section 2.2, due on September 8

One of the most difficult parts of the material for me was trying to understand their definition of addition and multiplication in the set of integers modulo n. Theorem 2.6 makes sense to me until I start thinking more about the discussion preceding it. If 1 is in B and 7 is in C, then B+C=8, so the equivalence class containing 8. Yet if -3 is in B and 15 is in C, then B+C=12, so the equivalence class containing 12. So I don't understand why 8 and 12 aren't in the same congruence class? I also find it difficult to remember which modulo we are in and adjust my arithmetic accordingly.

The most interesting part of the material for me was Theorem 2.7. These properties that are valid for the set of integers modulo n are very similar (probably the exact same) as the axioms that must be proved in Linear Algebra to show that something is a vector space. It is interesting to me to try and think how the set of integers modulo n and the concept of congruence classes will be used in future sections.

Section 2.1,due on September 3

The most difficult parts of this material for me were understanding the definition of the congruence class of a modulo n. Congruence and modulus are fairly newer concepts to me (as far as using them in proofs and other mathematical problems) and so keeping track of what they actually mean seems to be a little tricky for me. Because of this, I also found the first part of Corollary 2.5 to be a little bit harder to follow because I had to do a couple of examples and convince myself that it actually is true as stated.

The most interesting part of this material to me was Theorem 2.3. Once I came to understand the congruence classes, I thought that this theorem and its proof were pretty neat. The patterns found in all of number theory related topics is really rather clever. The modulus is something that came in handy when I took a computer programming class. It was really useful in writing code when you wanted items to appear on the screen in a patterned way (like 4 items for every line). I have also heard, but am not sure, that modulus is used a lot in RNS cryptography and I find that to be really fascinating.

Section 1.1-1.3, due on September 1

Some of the more difficult parts of the material for me included the proof of the division algorithm. Towards the end of the proof when we are showing that q and r are unique (bottom of pg. 5) I am still a little confused as to why we can just add those two inequalities to further the proof.

The other difficult parts of the material for me were convincing myself that the theorems and lemmas actually worked, and understanding what they were actually saying at the same time. For example, Lemma 1.7 took me awhile to process what was actually going on. I also found it difficult to follow the Euclidean Algorithm and recognize the pattern, keep track of all the subscripts, and retain what it was saying. In general, as equations are written in different forms and we say x divides y and so on, it is difficult for me to keep track of what is the dividend and what is the divisor.

While this material was the most difficult for me, I also found parts of it to be the most interesting. I really think Lemma 1.7 is cool as it basically states that the g.c.d. of a dividend and a divisor is the same as the g.c.d. of the divisor and the remainder. This was a really interesting pattern to me. I also found it interesting how Theorem 1.5 and 1.8 related in that they were both provide specific cases which answer the question: If a divides bc under what conditions is it true that a divides b or a divides c?

Introduction, due on September 1

1. I am a junior in school and a mathematics major.

2. The post-calculus math courses I have taken include Linear Algebra, Differential Equations and Mathematical Proofs (Math 290).

3. I am taking Abstract Algebra because it is required for a bachelor's degree in mathematics, which is a subject I enjoy as it is my major.

4. The math professor I have had who was most effective was here at BYU. He did many things I like so I have a lot to say. He was very organized and spoke clear and loud. He wrote most everything on the board and went at a slow pace. This allowed us to write down what he was saying and let the concepts sink in so that we could think about what he was teaching us and ask questions. He was always making sure we were keeping up with what he was saying.

Every once in a while he would do in class worksheets with groups of 2-4 to get the class period going. These were like brain-starters for the material to be taught that day or from a previous day. This helped me to recall the material being taught and also allowed me to get to know other students in the class. He had periodic 2-3 question quizzes (about once a week) so that we would stay fresh with the material. I didn't like the quizzes at the time, but there is no doubt that it helped me when exam time came around because I was already working on retaining what he had been teaching and doing so on a regular basis.

Often times he would print out sheets that organized past material together which was very effective in seeing how past concepts lead/connected to the next. Important formula and theorem sheets, written in more understandable terms, were also given to us.

Our homework assignments were graded on completeness which is something I really liked. It was his philosophy that homework is still part of the learning experience and tests are what measured exactly what we knew and how well we knew it. This philosophy allowed me to be more open to the various ideas that flowed to my mind when trying to solve a problem because I wasn't solely focused on getting it textbook perfect that time in order to keep up. My creative thinking improved as a result. The next class period he would then proceed to answer any questions and problems we had with the homework before we had to turn it in. If we were really struggling it was not uncommon for him to extend the due date of that homework while still keeping the new/next homework assigned as normal.

His tests were a compilation of problems similar to the homework. We always took a class period to review before the test and always went over the tests afterwards. His tests were never timed, other than the final. He would print out a solution sheet to all of the test questions for us to look over. The final exam was cumulative and consisted of various questions similar to that in our past exams.

5. I like to play sports and be active. Volleyball, tennis, snowboarding, hiking, biking, basketball, frisbee and softball are among my favorite things to do. Something else unique about me is that I tried a piece of gum for the first time this summer!

6. When I am unable to come to your scheduled office hours the times that work best for me are Tuesday and Thursday at any time other than noon-1 and 6-7 pm.