For the final exam, there is so much to choose from as far as topics and theorems go. It is comprehensive, so we could really see anything that we've studied all semester. I think that the most important topics are the following, in no particular order:
1. Sets and subsets
2. Set operations
3. Proofs: direct, contrapositive, and contradiction
4. Equivalence classes
5. Relations: reflexive, symmetric, and transitive
6. Functions: injective, surjective, bijective
7. Sets: denumerable, countable, uncountable
8. Linear combinations and the division algorithm
I feel like the class was well-organized, with principles learned at the end of the semester having been effectively built on a foundation of what was learned earlier. When I look back at the semester, I think that I need to work on many things to better understand the material. I have read all of the readings and am familiar with the material, but I just don't know it enough to perform well on tests. I think that I really need to practice, no matter how time-consuming that may be, many sample problems. Especially the proofs. I need a variety of sample problems and I need to be able to work them out on my own without study aids.
I know there are study guides for the first three exams, but they don't include the last chapter we studied. Perhaps I'd like to see a sample problem of a typical calculus proof that might appear on a final exam, if possible.
In this course, I have learned a lot. The material was interesting, for the most part, and the professor was really great. Rather than list specific mathematical topics that I learned, I can say that, overall, I learned that I am not as strong at mathematics as I thought or I hoped I would be. For some strange reason, the material never made a permanent dwelling place in my mind. It would visit, stay awhile, and then leave. There are many bright mathematical minds and I really look up to the teacher and some of my classmates for how they just "get it" so well. So, I guess what I'm trying to say is that I learned that I didn't easily "get it" in this class and I've come to be okay with it. This will help me as I modify my future academic and career paths.
Sincere thanks for the good class!
Saturday, April 13, 2013
Thursday, April 11, 2013
* Make Up For Missed Reading Assignment * - Talk by Daniel Willingham
I attended the talk "What every student and teacher should know about human memory" by Daniel Willingham at 4:00 PM in 1170 TMCB on Tuesday March 12.
This was a topic that really interested me because - and this might sound silly - I feel like my memory isn't as good as it used to be. I was interested to see if he talked about why that could be. I already feel like an older undergraduate student at 32 years old. Particularly in Math 290 I haven't been able to memorize as well or remember things on tests like I feel like I could as a younger student. I was wondering if that had to do with "old age" (ha ha) or why it might be.
Anyway, the talk was very interesting. I thought that Daniel Willingham was a very good speaker, presenter, and teacher. He had a lot of applicable advice for his target audience. Three of the main points he made were:
1. Memory is the residue of thought.
2. Repetition, but repetition of the deep-thinking.
3. Testing yourself is good.
I've applied the "testing yourself" idea ever since I heard his talk and I think that is effective. It seems to really have prepared me better for my exams. Another suggestion I've tried is explaining the material to someone. I've tried this with my wife and, although she couldn't care less about the subject material, she listens and it's helped me know the material better through explaining to someone else.
Another thing he mentioned is that students stop studying when they get the feeling of knowing, but that this is not always reliable. I've definitely fallen victim to this false sense of security. I think I know something because I am familiar with it and then I go into the test and have forgotten, realizing that I might not have known it that well after all. I am only familiar with the material, not a master of it.
The final thing that I'll mention is that he talked about "overlearning," meaning he suggested still studying even after you know it. I agree that the advice is very good and would indeed help in earning A's on exams; however, I kind of had to laugh because I feel lucky when I'm able to put in just enough time to know the material, not "overlearn" it. I feel like I'm far too busy. So, if life ever slows down for me enough to "overlearn" something, I'll definitely do it - that is, if I can remember.
This was a topic that really interested me because - and this might sound silly - I feel like my memory isn't as good as it used to be. I was interested to see if he talked about why that could be. I already feel like an older undergraduate student at 32 years old. Particularly in Math 290 I haven't been able to memorize as well or remember things on tests like I feel like I could as a younger student. I was wondering if that had to do with "old age" (ha ha) or why it might be.
Anyway, the talk was very interesting. I thought that Daniel Willingham was a very good speaker, presenter, and teacher. He had a lot of applicable advice for his target audience. Three of the main points he made were:
1. Memory is the residue of thought.
2. Repetition, but repetition of the deep-thinking.
3. Testing yourself is good.
I've applied the "testing yourself" idea ever since I heard his talk and I think that is effective. It seems to really have prepared me better for my exams. Another suggestion I've tried is explaining the material to someone. I've tried this with my wife and, although she couldn't care less about the subject material, she listens and it's helped me know the material better through explaining to someone else.
Another thing he mentioned is that students stop studying when they get the feeling of knowing, but that this is not always reliable. I've definitely fallen victim to this false sense of security. I think I know something because I am familiar with it and then I go into the test and have forgotten, realizing that I might not have known it that well after all. I am only familiar with the material, not a master of it.
The final thing that I'll mention is that he talked about "overlearning," meaning he suggested still studying even after you know it. I agree that the advice is very good and would indeed help in earning A's on exams; however, I kind of had to laugh because I feel lucky when I'm able to put in just enough time to know the material, not "overlearn" it. I feel like I'm far too busy. So, if life ever slows down for me enough to "overlearn" something, I'll definitely do it - that is, if I can remember.
12.4 and 12.5, due on April 12
I find that these particular proofs in calculus are difficult to read speedily. I often have to reread the proofs in order to understand them fully. Also, I find myself rereading individual sentences several times just to comprehend what it is saying. A lot of the proofs are equations and inequalities that take time to see how it ties into what is trying to be proven. I'm glad that it is one of the later chapters because, to me, it takes some time to get them. Having been taught all that we've learned in this course, it goes from nearly impossible to understand (like it was for me in Math 112/113) to only time-consuming. The theorems are of particular difficulty, in my estimation. It seems easier and more practical if the proofs involve something in addition to the theorems. Proving the theorems themselves seems so hard - as was the case for all of the theorems presented in the text throughout the year.
I remember seeing the limit laws before - the ones involving their sums and products, for example. That was rather familiar and I readily made the connection. Knowing how to apply the theorems when solving problems in calculus makes it easier to understand the proofs, I think. It is nice when you know that what your learning has an application in another math class. Overall, I think that the text and the course tied into a lot of math I've learned and built upon it or deepened my understanding. The course and text were well-organized in the order in which things were taught - I think it was the ideal order. I learned a lot and it was continually surprising how many things were introduced that I hadn't learned before.
I remember seeing the limit laws before - the ones involving their sums and products, for example. That was rather familiar and I readily made the connection. Knowing how to apply the theorems when solving problems in calculus makes it easier to understand the proofs, I think. It is nice when you know that what your learning has an application in another math class. Overall, I think that the text and the course tied into a lot of math I've learned and built upon it or deepened my understanding. The course and text were well-organized in the order in which things were taught - I think it was the ideal order. I learned a lot and it was continually surprising how many things were introduced that I hadn't learned before.
Tuesday, April 9, 2013
12.3, due on April 10
Once again, I don't know what it is about the greek alphabet, but when I see greek letters in math my mind suffers from some sort of disconnect. Delta was introduced. Then delta schemed with my old nemesis epsilon to try and create superconfusion for me. I only half-joke. This material was familiar in that I've studied calculus and limits; however, I feel like these proofs would make more sense to me if they were explained again in class. Even the graphs and diagrams in the text do not really help me understand the proof better. I think that I just need a teacher to explain them to me in "English" and not "Greekmath." Another thing: in these proofs, how do you choose the values accurately so that the proof comes out nicely? How do you know what to choose?
I did find it interesting, though - the proofs of limits, I mean. The proofs that were proven were fairly easy limits, as far as calculation goes. Still, the proofs were complicated, in my eyes. The proof about the limit of 1/x as x approaches infinity was interesting. I remember studying in calculus about 1/x not converging, but how 1/x^2 does converge. I don't remember seeing a proof, so that was cool. I see that there are so many things in mathematics that I've studied - from the recent study of limits to the past study (way past) of division - and how there are proofs and mathematical theory behind all of it. I always took that for granted. Like the division algorithm, for example, and even now regarding the limit of 1/x. Interesting to read about the proofs.
I did find it interesting, though - the proofs of limits, I mean. The proofs that were proven were fairly easy limits, as far as calculation goes. Still, the proofs were complicated, in my eyes. The proof about the limit of 1/x as x approaches infinity was interesting. I remember studying in calculus about 1/x not converging, but how 1/x^2 does converge. I don't remember seeing a proof, so that was cool. I see that there are so many things in mathematics that I've studied - from the recent study of limits to the past study (way past) of division - and how there are proofs and mathematical theory behind all of it. I always took that for granted. Like the division algorithm, for example, and even now regarding the limit of 1/x. Interesting to read about the proofs.
Saturday, April 6, 2013
12.1, due on April 8
I have read about epsilon twice - once in Math 112 and once in Math 113. Both times, I had difficulty with it. I remember missing questions on tests about epsilon as I didn't really get how to calculate it. I saw it again in this reading and it involves a proof and I instantly became confused. Epsilon has become my least favorite Greek letter. I may just chalk it up as a lost cause, but if we do go over any proofs involving epsilon in class, then maybe it might click for me. Based on the reading, all of the convergence proofs seemed to involve epsilon in order to prove convergence. It would probably be a good idea to get a better handle on it.
In my Math 113 class that I'm taking right now, we just got through the section about sequences. This tied into the reading about sequences and convergence and divergence. I didn't gain a full understanding in my Math 113 study about this topic. There is still time in the semester to gain a better understanding. Perhaps my study in this Math 290 class right now will help me in my other math class. I remember reading in my calculus book about these proofs, but kind of reading through them quickly. Now I have the opportunity to seriously revisit the proofs and I do really hope that it will help me get a richer, deeper understanding of calculus and, of course, help with this class. It may end up being a blessing.
In my Math 113 class that I'm taking right now, we just got through the section about sequences. This tied into the reading about sequences and convergence and divergence. I didn't gain a full understanding in my Math 113 study about this topic. There is still time in the semester to gain a better understanding. Perhaps my study in this Math 290 class right now will help me in my other math class. I remember reading in my calculus book about these proofs, but kind of reading through them quickly. Now I have the opportunity to seriously revisit the proofs and I do really hope that it will help me get a richer, deeper understanding of calculus and, of course, help with this class. It may end up being a blessing.
Thursday, April 4, 2013
Preparation for Exam 3, due on April 5
I'm sure that the cardinality of sets will be very important. The proofs are difficult for me, so I plan to study hard to understand how to prove that a set is denumerable or uncountable. I would expect to see this topic on the exam.
Also, I plan to see the Division algorithm and the Euclidean algorithm. Fortunately, this seemed to make a little more sense than Chapter 10 and the cardinality of sets.
An important theorem is the Schroder-Bernstein Theorem. I would think that we would have to be able to prove that two sets are numerically equivalent. That appeared in the homework and I kind of struggled with it with regards to when to use it and quote it exactly.
It seems that the questions on the test, at least the proofs and free response, are similar in some ways to homework questions, which is nice. I feel that if you can do your homework well, then you should know it well enough to perform on the test without the textbook or notes.
As for the definitions, we went over many, as usual. Denumerable, countable, uncountable - it will be good to know these on the test. I need to memorize the definitions exactly as I wrote them down from class notes. I've found that I lose points on exams when I don't provide the exact definition from text/notes, even though what I've written on the test for a definition is correct. I suppose it wasn't the "correct" correct answer. Definition memorization will be a part of my preparation.
I would please like to see a problem showing how one set has a larger cardinality than another.
Also, I plan to see the Division algorithm and the Euclidean algorithm. Fortunately, this seemed to make a little more sense than Chapter 10 and the cardinality of sets.
An important theorem is the Schroder-Bernstein Theorem. I would think that we would have to be able to prove that two sets are numerically equivalent. That appeared in the homework and I kind of struggled with it with regards to when to use it and quote it exactly.
It seems that the questions on the test, at least the proofs and free response, are similar in some ways to homework questions, which is nice. I feel that if you can do your homework well, then you should know it well enough to perform on the test without the textbook or notes.
As for the definitions, we went over many, as usual. Denumerable, countable, uncountable - it will be good to know these on the test. I need to memorize the definitions exactly as I wrote them down from class notes. I've found that I lose points on exams when I don't provide the exact definition from text/notes, even though what I've written on the test for a definition is correct. I suppose it wasn't the "correct" correct answer. Definition memorization will be a part of my preparation.
I would please like to see a problem showing how one set has a larger cardinality than another.
Saturday, March 30, 2013
11.5, due on April 1
The difficulty I had with this section is that there were no examples of application of relatively prime integers. It was stated in the text that we would see that they are very useful. However, I only read about theorems and lemmas and proofs thereof. Perhaps we will see their usefulness later on. I suppose, actually, that there was one reference to its usefulness. At the end of the section it stated:
"By Theorem 11.16 then, if we wish to show that 12, say, divides some integer c, we need only show that 3|c and 4|c since 12 = 3 · 4 and 3 and 4 are relatively prime."
It didn't actually show how to do this, but it did mention it. I typically retain the knowledge better if I see some examples.
Looking back on previously learned mathematics, I don't recall ever hearing about relatively prime numbers. I am grateful for the opportunity in this class to learn about a lot of new topics, definitions, and mathematical ideas. When I come across something new in this class, I always think - I should have learned this, or at least heard about this, before during one of my mathematics classes or textbooks. It is surprising to me that I haven't. It leads me to believe that there is more to learn than I could even imagine. After all, many mathematicians let scholastic discovery be their life's pursuit and I'm sure there is no way to ever know it all.
"By Theorem 11.16 then, if we wish to show that 12, say, divides some integer c, we need only show that 3|c and 4|c since 12 = 3 · 4 and 3 and 4 are relatively prime."
It didn't actually show how to do this, but it did mention it. I typically retain the knowledge better if I see some examples.
Looking back on previously learned mathematics, I don't recall ever hearing about relatively prime numbers. I am grateful for the opportunity in this class to learn about a lot of new topics, definitions, and mathematical ideas. When I come across something new in this class, I always think - I should have learned this, or at least heard about this, before during one of my mathematics classes or textbooks. It is surprising to me that I haven't. It leads me to believe that there is more to learn than I could even imagine. After all, many mathematicians let scholastic discovery be their life's pursuit and I'm sure there is no way to ever know it all.
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