MAT 334 Introduction to Abstract Algebra
Welcome to the home page of Introduction to Abstract Algebra at Utica College. You can find all the documents for this course on this page. Please check this page frequently for homework assignments and announcements.
Course Information and Policies
- Instructor: Xiao Xiao
- Email: email@example.com
- Office: 109 DePerno Hall
- Office hour:
- Course notes: You can download them in Canvas.
- Class time:
- Useful Links: Canvas, Homework
- Textbook: We will not use a textbook, but rather a task-sequence adopted for inquiry-based learning. The task-sequence is written by me, adapted from notes of Margaret Morrow and David Clark, and it is available on Canvas. You are expected to work out all the tasks as the semester progresses.
- Course description: Introduction to Abstract Algebra covers basic group theory. We will discuss the following concepts in this course: groups, subgroups, abelian groups, normal subgroups, product groups, quotient groups, and group isomorphisms. Standard examples such as cyclic groups, dihedral groups, permutation groups and classical theorems such as Lagrange's Theorem will be discussed.
- Learning goals: In accordance to the learning goals of the Department of Mathematics of Utica College, MAT 334 will reinforce the following abilities and by the end the course, students will demonstrate their proficiency of these abilities:
- Reading and analyzing mathematical proofs.
- Writing mathematical proofs.
- Communicating mathematics in written form.
- Class organization: This course will likely be different from any other math course you have taken before. As an instructor, I will not be lecturing most of the time although I love lecturing very much. Scientific research shows that most people do not learn mathematics by listening, instead, they learn by doing it! I am sure you have said to yourself before "It looked so easy when the professor was doing it, but now I am confused when I have to do it by myself." Why? Because the knowledge belongs to your professor and does not belong to you. You do not learn the knowledge simply by hearing it once or twice from somebody else. In fact, good lectures will sometimes fool you by making you think you understand the materials when in fact you don't really understand it. In order for you to have a more thorough understanding of this course, we will incorporate ideas from an educational philosophy called the Moore method (after R. L. Moore). More precisely, we will use the modified Moore method, also known as inquiry-based learning. Most of the time during the class, students will be presenting proofs of theorems that they have produced by themselves, and not by other people or textbooks. A significant portion of your grade will be determined by how much mathematics you produce.
- 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. See the Homework section for more details on when and how you can use each other in class.
- 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.
- Each student will be asked to join a course Facebook page that is closed to everybody else outside of the class.
- Daily Homework: It will be assigned each class period, 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 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. You can work with each other for the daily homework and you can also come to see me for hints if you want. But you must write up your solution/proof by yourself.
When you write up your Daily Homework, I strongly
suggest that you use filler paper, such as this
. Write one
problem per page. There is a very good reason for doing this, which I shall explain later in the Portfolio section.
Each Daily Homework will be submitted twice. You will submit at Canvas before 12:00pm on the day before we meet in class. For example, a Daily Homework assigned on Monday will be discussed on Wednesday, so you should submit your draft by Tuesday 12:00pm at Canvas. I will review your draft and give suggestions and comments within 10 hours (namely before 10:00pm of the same day). You can read the comments and make necessary changes before you come to class on the next day. Your Daily Homework will be finished by hand and paper so the best way to do this is to take a picture of your writings and submit it as attachment on Canvas. Students must upload the draft in PDF format. You can use this
website to convert JPG format to PDF format. Or you can install CutePDF
(in Windows) to convert any format to PDF format.
The Daily Homework will generally consist of proving theorems or solving exercises from the task-sequence. On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting a subset (maybe all) of the proofs of the theorems that are due that day. Students are allowed to take pictures of board work and upload them to Facebook group so that everyone can see them. This means that you do not have to copy anything down from the board as you will have them available online. At the end of each class period, students should submit their write-ups for all of the proofs that are due that day. Daily Homework will be graded on a check-system. Students are allowed (in fact, encouraged!) to modify (not copy directly from the board as you will have the pictures in Facebook group) their written proofs in light of presentations made in class; however, you are required to use the felt-tip pens provided in class.
- Weekly Homework: In addition to the Daily Homework, you will also be required to submit two formally written proofs each week. You may choose any two problems marked with * that were turned in during a given week to submit the following week. For example, you may choose any two problems marked with a * that were turned in during week 2 for the second Weekly Homework assignment. These problems are due in week 3. Beginning with the second Weekly Homework, you will be required to type up your submission. You should type your Weekly Homework assignments using LaTeX. Look below for more information on LaTeX. The Weekly Homework assignments are subject to the following rubric:
|Grade ||Criteria |
|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 and/or there are too many grammatical mistakes. |
|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 or the work is not original and came from the internet or some other external source. |
Any Weekly Homework problems that receive a score of 1, 2, or 3 can be resubmitted within one week after the corresponding problem was returned to the class. The final grade on the problem will be the average of the original grade and the grade on the resubmission. Please label the assignment as “Resubmission” on top of any problem that you are resubmitting and keep separate from any other problems that you are turning in.
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 3 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 is worth 20% of your final grade.
- 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.
- The problems chosen for presentations will come from the Daily Homework. After a student has presented a proof that the class agrees is sufficient, I will often call upon another student in the audience to recap what has happened in the proof and to emphasize the salient points.
- In order to make presentations go smoothly, presenters need to write out the proof in detail and go over the major ideas and transitions, so that he or she can make the proof clear to others.
- The purpose of presentations is not just prove to me that the presenter has done the problem. The most important part is to make the ideas of the proof clear to the other students.
- Presenters need to write in complete sentences, using proper English and mathematical grammar. Here are some suggestions on how to write a proper proof.
- Fellow students are allowed to ask questions at any point and it is the responsibility of the presenter 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.
- When a presenter is stuck with a question from the audience or from the instructor or when the presented proof is not generally accepted by the audience, the presenter has the following choices:
- They can think on the fly and try to solve the problem although this should not take more than one minute as to save class time.
- They can sit down and think about the questions. After all the presentations have been completed, they can come back and address the questions.
- They can go home and think about the questions and present the same problem again in the next class. In this case, the presenter must make an appointment and meet with the instructor before the next class.
- Presentations will be graded using the rubric below.
|Grade || |
|2 || |
Completely correct and clear proof or solution.
|1 ||Proof has technical flaws, some unclear language, or is lacking some details. Or it is completed in more than one class. |
|0 ||Minimal progress has been made. |
However, you should not let the rubrics deter you from presenting if you have an idea about a proof that you would like to present, but you are worried that your proof is incomplete or you are not confident your proof is correct. You can only gain points by coming to the board and 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.
- In each class, a sorted class list produced by a computer program will be shown before presentations. Students whose ranks are high in the list have higher priority to choose problems. The sorted list is not produced randomly. It takes three factors into consideration:
- The number of past presentations: the more one presented in the past, the lower one is on the list.
- The quality of past presentations: the better one presented in the past, the lower one is on the list.
- Recentness of past presentations: the more recently one has presented, the lower one is on the list.
- Before each class starts, each student will be asked to submit a list of problems that he or she can present in written form. Based on that and the priority list, the instructor will decide who will present what on that day. If a student does not claim any problem on a day, then this is called a "pass". A student uses a "pass" in one class must make an appointment and meet with the instructor before the next class. No one shall "pass" for two consecutive classes.
- In order to receive a passing grade on the presentation portion of your grade, you must present at least twice prior to each exam (2 midterms and 1 final) for a total of at least six times during the semester. Your grade on your presentations, as well as your level of interaction during student presentations, are worth 20% of your overall grade.
- The object is to maintain a current account of the work we do. Every exercise, proposition, lemma, theorem and corollary that we encounter is to be included in your portfolio. Each entry in the portfolio is intended to be complete and polished. Do not include scratch work.
- Each of us will develop her or his own mathematical voice in this class. Not every solution will look the same. However, the form of the portfolio should be fairly standardized. It will include a cover sheet with your name on it. Begin each write-up with the statement of an exercise, a proposition, a lemma, a theorem or a corollary, followed by your solution or proof. Some write-ups will be two lines long, others may be several pages. You can use the filler paper (the same you use for Daily Homework) to write up your solution. If you have done a perfect job in your Daily Homework, then you can just insert the page into your portfolio. Though in most cases, you will have to improve your original work.
- The portfolio will be collected three times: Wednesday, February 17, Monday, March 28, and Wednesday, May 4, 2016, at the final exam.
- Because you will have already know whether your solutions or proofs are correct or not by discussing them during presentations and by having them graded as Daily and Weekly Homework, portfolios will be graded solely on completeness and clarity. Clear and complete portfolios will earn a check mark, all others will be asked to resubmit within a week. Keep your portfolio current as you work, it will be too much effort to get it all organized and collated the night before it is due.
- At the end of the semester, portfolios with three check marks will earn the full 15% possible. Two check marks will earn 10%, one check mark will earn 5% and no check marks will earn 0%.
- In the end, you will walk away with an organized and complete collection of your work on which you can look back with pride.
There will be two midterm exams and one cumulative final exam. Each exam is worth 15% of your overall grade and may consist of an in-class portion and a take-home portion. The in-class portions of the two mid-term exams are tentatively scheduled for Wednesday, February 17, 2016 and Monday, March 28, 2016 and the in-class portion of the final exam is Wednesday, May 4, 2016. 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.
We are all expected to make mistakes during the semester (including me) because we are learning. We should not let the anxiety of making mistakes prevent us from trying something. Learning from one's mistake is a very important skill as part of the general learning skills. Therefore, part of your grade is based on whether you can learn from your mistakes and make them productive. In order to get the full 5% grade for productive failure, you have to share a productive failure of yours during five separate presentations in class. Your sharing need to include the following points: what mistakes did you make, why do you think you made such mistakes, how did you realize that you make such mistakes, what lessons did you learn from this mistake. You can share your productive failure after your presentation, or after someone else's presentation.
Almost every week, you will be asked to write a journal on various topics. Details will be given in Canvas when they are assigned.
We will be using OverLeaf to edit and compile LaTeX files. OverLeaf is a free online program so you don't have to pay anything to use it but you need to have an internet connection. I will run a special session on how to use LaTeX in the first or second week of class. If you have experiences with LaTeX, then you do not have to attend the training sessions.
- Your final grade will be determined by the scores of your homework, presentations/participation, portfolio, journals, and exams.
|Category ||Weight |
|Homework ||15% |
|Presentations/Participation ||15% |
|Portfolio ||15% |
|Journals ||5% |
|Productive Failure ||5% |
|Midterm Exam 1 ||15% |
|Midterm Exam 2 ||15% |
|Final Exam ||15% |
- The correspondences between percentage and letter grades are explained in the following table:
|Percentage ||Letter Grade |
|>92% ||A |
|90%-92% ||A- |
|88%-90% ||B+ |
|82%-88% ||B |
|80%-82% ||B- |
|78%-80% ||C+ |
|72%-78% ||C |
|70%-72% ||C- |
|68%-70% ||D+ |
|60%-68% ||D |
|<60% ||F |
I have zero tolerance on dishonesty. Any forms of dishonesty such as copying homework or cheating on quizzes and examinations, will result in zero credit for that particular assignment, and will be reported to the Academic Standards Committee. The highest penalty a student can receive is "F for cheating" for the course. There might be additional sanctions by the Academic Standards Committee such as dismissal from the college.
Any student who has need of special accommodations in this class due to a documented disability should speak with me as soon as possible, preferably within the first two weeks of class. You should also contact Kateri Henkel, Director of Learning Services in the Academic Support Services Center (315-792-3032 or firstname.lastname@example.org) in order to determine eligibility for services and to receive an accommodation letter. We will work with you to help you in your efforts to master the course content in an effective and appropriate way.