Title: Introduction to Abstract Algebra
Course Number: MAT 411
Semester: Fall 2013
Time: MWF at 9:10-10:00AM
Location: AMB 163
Course Information and Policies
MAT 320W with a grade greater than or equal to C.
Basic algebraic structures. Topics selected from groups, rings, and fields
MAT 411 introduces students to the basic ideas, definitions, examples, theorems and proof techniques of abstract algebra.
Upon successful completion of the course, students will be able to do the following within the topics of groups, rings and fields:
1. Read and write expository text on elementary aspects.
2. Distinguish truth from falsehood.
3. Provide examples and counterexamples of statements.
4. Perform needed computations.
5. Construct concise and correct proofs.
- Group Theory: axioms, examples of groups of numbers, matrices, and permutations; abelian groups, cyclic groups; order of an element, subgroups, cosets, normal subgroups, factor groups, homomorphisms, kernels; Cayley’s Theorem, LaGrange’s Theorem, First Isomorphism Theorem.
- Rings: axioms, examples of rings of numbers, matrices, and polynomials; unity, units, divisibility, zero divisors, integral domains, division rings, field of quotients, ideals, homomorphisms, factor rings, prime and maximal ideals.
- Fields: axioms, examples; polynomials, divisibility criteria, irreducible polynomial, construction of finite fields and their cyclic multiplication groups.
Learning Management System
We will be making limited use of Bb Learn this semester, which is Northern Arizona University’s default learning management system (LMS). Most course content (e.g., syllabus, course notes, homework, etc.) will be housed on our course webpage that lives outside of Bb Learn. I suggest you bookmark the course webpage. In addition, we will be utilizing an LMS called Canvas. We will be utilizing Canvas to manage grades and (possibly) the submission of Weekly Homework assignments. Also, we will use the forums within Canvas to facilitate out of class discussion. I will send the class an invite to our Canvas page and spend time in class discussing its use.
We will not be using a textbook this semester, but rather a task-sequence adopted for IBL. The task-sequence that we are using was written by me. Actually, I’ll be finishing up the notes as we go. Any errors in the notes are no one’s fault but my own. In this vein, if you think you see an error, please inform me, so that it can be remedied. The course notes are available here.
I will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions in class or posting questions to the course forum on Canvas. Here’s what Paul Halmos has to say about reading mathematics.
Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof
use the hypothesis?
An Inquiry-Based Approach
In a typical course, math or otherwise, you sit and listen to a lecture. (Hopefully) These lectures are polished and well-delivered. You may have often been lured into believing that the instructor has opened up your head and is pouring knowledge into it. I absolutely love lecturing and I do believe there is value in it, but I also believe that in reality most students do not learn by simply listening. You must be active in the learning you are doing. I’m sure each of you have said to yourselves, “Hmmm, I understood this concept when the professor was going over it, but now that I am alone, I am lost.” In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL).
Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work).
Perhaps this is sufficiently vague, but I believe that there are two essential elements to IBL. Students should as much as possible be responsible for:
- Guiding the acquisition of knowledge, and
- Validating the ideas presented. That is, students should not be looking to the instructor as the sole authority.
If you want to learn more about IBL, read my blog post titled What the heck is IBL?.
Much of the course will be devoted to students proving theorems on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the work “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!
Furthermore, it is important to understand that proving theorems is difficult and takes time. You shouldn’t expect to complete a single proof in 10 minutes. Sometimes, you might have to stare at the statement for an hour before even understanding how to get started. In fact, proving theorems can be a lot like the clip from the Big Bang Theory located here.
In this course, everyone will be required to
- read and interact with course notes on your own;
- write up quality proofs to assigned problems;
- present proofs on the board to the rest of the class;
- participate in discussions centered around a student’s presented proof;
- call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar on first glance.
As the semester progresses, it should become clear to you what the expectations are. This will be new to many of you and there may be some growing pains associated with it.
(Adopted from Chapter Zero Instructor Resource Manual) Aside from the obvious goal of wanting you to learn how to write rigorous mathematical proofs, one of my principal ambitions is to make you independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much “classroom knowledge” is fairly superﬁcial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are “getting it” comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills.
At the beginning of most class meetings, 1-3 students will be chosen at random to answer a single question related to the material we have covered so far in the course. The questions will be basic. In most cases, students may be asked to recite a definition, a key theorem, or to provide an example/counterexample that exhibits some given properties. Each response to a query will be graded using the following rubric.
|4||Completely correct and clear response. Yay!|
|3||Response has minor technical flaws, some unclear language, or lacking some details. Essentially correct.|
|2||A partial explanation is provided but a significant gap still exists.|
|0||A response is given that does not address the question.|
|-1||Student was chosen at random, but was absent.|
Your scores on the Daily Queries will impact your class presentation and participation score (see below).
Class Presentations and Participation
(Adopted from Chapter Zero Instructor Resource Manual) Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken very seriously since they spearhead the work of the class. Here are some of my expectations:
- In order to make the presentations go smoothly, the presenter needs to have written out the proof in detail and gone over the major ideas and transitions, so that he or she can make clear the path of the proof to others.
- The purpose of class presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the proof clear to the other students.
- Presenters are to write in complete sentences, using proper English and mathematical grammar.
- Presenters should explain their reasoning as they go along, not simply write everything down and then turn to explain.
- Fellow students are allowed to ask questions at any point and it is the responsibility of the person making the presentation to answer those questions to the best of his or her ability.
- Since the presentation is directed at the students, the presenter should frequently make eye-contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.
Presentations will be graded using the rubric below.
|4||Completely correct and clear proof or solution. Yay!|
|3||Proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct.|
|2||A partial explanation or proof is provided but a significant gap still exists to reach a full solution or proof.|
|1||Minimal progress has been made that includes relevant information & could lead to a proof or solution.|
|0||You were completely unprepared.|
However, you should not let the rubric deter you from presenting if you have an idea about a proof that you’d like to present, but you are worried that your proof is incomplete or you are not confident your proof is correct. You will be rewarded for being courageous and sharing your creative ideas! Yet, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.
I will always ask for volunteers to present proofs, but when no volunteers come forward, I will call on someone to present their proof. If more than one student volunteers, the student with the fewest number of presentations has priority. The problems chosen for presentation will come from the Daily Homework assignments. After a student has presented a proof that the class agrees is sufficient, I may call upon another student in the audience to come to the board to recap what happened in the proof and to emphasize the salient points. Each student in the audience is expected to be engaged during another student’s presentation.
In order to receive a passing grade on the presentation portion of your grade, you must present at least twice prior to each of the three midterm exams and at least once after the third midterm exam for a total of at least 7 times during semester.
Your overall performance during presentations and the Daily Query, as well as your level of interaction/participation during class, will be worth 20% of your overall grade.
Daily Homework: Homework will be assigned each class meeting, and students are expected to complete (or try their best to complete) each assignment before walking into the next class period. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation and spelling. You will almost always write a draft of a given solution before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment.
The Daily Homework will generally consist of proving theorems from the course notes. On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting some subset (maybe all) of the proofs of the theorems that are due that day. At the end of each class session, students should submit their write-ups for all of the proofs that were due that day. Daily Homework will be graded on a $\checkmark$-system.
Students are allowed (in fact, encouraged!) to modify their written proofs in light of presentations made in class; however, you are required to use the felt-tip pens provided in class. I will provide more guidance with respect to this during the first couple weeks of the semester.
Weekly Homework: In addition to the Daily Homework, we will also have Weekly Homework assignments. For most of these assignments, you will be required to submit two formally written proofs. In general, you may choose any two problems marked with $\star$ that were turned in during a given week to submit the following week (due dates to be announced). For example, you may choose any two problems marked with a $\star$ that were turned in during week 2 for the second Weekly Homework. These problems are due during week 3.
Beginning with the second Weekly Homework, you will be required to type your submission. You should type your Weekly Homework assignments using $\LaTeX$. I will walk you through how to do this.
The Weekly Homework assignments described above are subject to the following rubric:
|4||This is correct and well-written mathematics!|
|3||This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.|
|2||There is some good intuition here, but there is at least one serious flaw.|
|1||I don’t understand this, but I see that you have worked on it; come see me!|
|0||I believe that you have not worked on this problem enough or you didn’t submit any work.|
Please understand that the purpose of the written assignments is to teach you to prove theorems. It is not expected that you started the class with this skill; hence, some low grades are to be expected. However, I expect that everyone will improve dramatically. Improvement over the course of the semester will be taken into consideration when assigning grades.
As well as the type of Weekly Homework described above, we will also have a few assignments that require each student to write a “referee report” for a proof written by an anonymous student. I will provide details about these types of assignments later in the semester. The referee reports will be graded using a 4 point scale similar to the rubric above.
Unlike a traditional Moore method course, you are allowed and encouraged to work together on homework. However, each student is expected to turn in his or her own work. In general, late homework will not be accepted. However, you are allowed to turn in up to 5 homework assignments (daily or weekly) late with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. Your overall homework grade will be worth 15% of your final grade.
On each homework assignment, please write (i) your name, (ii) name of course, and (iii) Daily/Weekly Homework number. You can find the list of homework assignments here. I reserve the right to modify the homework assignments as I see necessary.
There will be three midterm exams and a cumulative final exam. Each midterm exam will be worth 15% of your overall grade and the final exam will be worth 20%. All of the exams will likely consist of both an in-class portion and a take-home portion. The in-class portions of the midterm exams are tentatively scheduled for Friday, September 27,
Friday, October 25 Monday, October 28, and Friday, November 22. The in-class portion of the final exam will be on Tuesday, December 17 at 7:30–9:30AM. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.
Rules of the Game
You should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course. On the other hand, you may use each other, the course notes, me, and your own intuition.
Basis for Evaluation
Your final grade will be determined by the scores of your homework, presentations/participation, and exams.
|Homework||15%||Mix of Daily & Weekly Homework|
|Daily Query, Presentations, & Participation||20%||Each student must present at least twice prior to each midterm exam and once after the third exam (that’s at least a total of at least 7 times during semester)|
|Exam 1||15%||Friday, September 27|
|Exam 3||15%||Friday, November 22|
|Final Exam||20%||in-class portion on Tuesday, December 17 at 7:30–9:30AM|
Regular attendance is expected and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your participation grade (see above). Students can find more information about NAU’s attendance policy on the Academic Policies page.
Department and University Policies
You are responsible for knowing and following the Department of Mathematics and Statistics Policies (PDF) and other University policies listed here. More policies can be found in other university documents, especially the NAU Student Handbook (see appendices) and the website of the Office of Student Life.
As per Department Policy, cell phones, mp3 players and portable electronic communication devices, including but not limited to smart phones, cameras and recording devices, must be turned off and inaccessible during in-class tests. Any violation of this policy will be treated as academic dishonesty.
Students are expected to treat each other with respect. Students are also expected to promote a healthy learning environment, as well as minimize distracting behaviors. In particular, you should be supportive of other students while they are making presentations. Moreover, every attempt should be made to arrive to class on time. If you must arrive late or leave early, please do not disrupt class.
Please turn off the ringer on your cell phone. I do not have a strict policy on the use of laptops, tablets, and cell phones. You are expected to be paying attention and engaging in class discussions. If your cell phone, etc. is interfering with your ability (or that of another student) to do this, then put it away, or I will ask you to put it away.
There are many resources available to get help. First, I recommend that you work on homework in groups as much as possible, and to come see me whenever necessary. Also, you are strongly encouraged to ask questions in the course forum on our Canvas page, as I will post comments there for all to benefit from. I am always happy to help you. If my office hours don’t work for you, then we can probably find another time to meet. It is your responsibility to be aware of how well you understand the material. Don’t wait until it is too late if you need help. Ask questions! Lastly, you can always email me.
(Adopted from pages 202-203 of The Moore Method: A Pathway to Learner-Centered Instruction by C.A Coppin, W.T. Mahavier, E.L. May, and G.E. Parker) There are two ways to approach this class. The first is to jump right in and start wrestling with the material. The second is to say, “I’ll wait and see how this works and then see if I like it and put some problems on the board later in the semester after I catch on.” The second approach isn’t such a good idea. If you try every night to do the problems, then either you will get a problem (Shazaam!) and be able to put it on the board with pride or you will struggle with the problem, learn a lot in your struggle, and then watch someone else put it on the board. When this person puts it up you will be able to ask questions that help you and the others understand it, as you say to yourself, “Ahhh, now I see where I went wrong and now I can do this one and a few more for the next class.” If you do not try problems each night, then you will watch the student put the problem on the board, but perhaps will not quite catch all the details and then when you study for the exams or try the next problems you will have only a loose idea of how to tackle such problems. And then the anxiety will build and build and build. So, take a guess what I recommend that you do.