MAT 401 Real Analysis

Welcome to the home page of Prof. Xiao Xiao’s Real Analysis at Utica University. You can find all the informtion and documents for this course on this page. Please check this page frequently for announcements and assignments.

Important Dates

• Add/Drop deadline: 9/1/23
• Fall break: 10/9/23 - 10/10/23
• Withdraw deadline: 11/6/23
• Thanksgiving break: 11/22/23 - 11/24/23
• SOOT: 12/1/23 - 12/8/23
• Final exam: 12/11/23 - 12/15/23

Instructor Information

• Instructor: Prof. Xiao Xiao
• Email: [email protected]
• Office: White Hall 255
• Office hour: Tuesdays and Thursdays 9:00am-9:45am, Wednesdays 10:00am - 11:30am.

An Important Course Policy

I pride myself on having a good environment for working and learning. It is very important to me that we all treat each other with care and respect, in equal measure. I know that I ask students to take risks in class almost every day, and this can be challenging for many. I ask that you help me keep our classroom a supportive place for each of the people in it. Each of us deserves the space to bring our full, authentic selves to class and be comfortable. (Adapated from T.J. Hitchman.)

General Course Information and Policies

• Course name: MAT 401 Real Analysis

• Course credit hours: 3-credit

• Course prerequisite: MAT 305, or permission of instructor

• Class time and location: Tuesdays and Thursdays 1:00pm - 2:15pm at Hubbard Hall 210

• Textbook: We will use The Number Line throught Guided Inquiry, by David Clark and myself. The ISBN of the book is 9781470465049.

• Course description: Foundations of the real number system, functions and sequences, limits and continuity.

• Program learning goals: In accordance to the learning goals of the Department of Mathematics of Utica University, MAT 401 will introduce and reinforce students’ ability of:
  • (PLG1) Reading and analyzing mathematical proofs.
  • (PLG2) Writing mathematical proofs.
  • (PLG5) Communicating mathematics in written form.
• Course learning objectives: Upon successful completion of this course, students will be able to:
  • understand and prove theorems about basic topological properties of the real number line using basic definitions including open and closed sets, limit points, converging and diverging sequences and monotonic sequences.
  • understand and prove the theorems about the relationships between rational numbers and infinite decimal representations of numbers.
  • understand and prove all the theorems that lead to the theorem that the real number field is the smallest complete field extension of the rational numbers.
  • understand and prove theorems about basic properties of continuous functions.

• 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 order for you to have a more thorough understanding of the knowledge, 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.

• Regular attendance is mandatory 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.

• MAT 401 is a writing intensive course. A writing intensive class has specific requirements on the style and the process of your writing. The first requirement is informal writing assignments which you will accomplish through your weekly journals and daily homework. The second requirement is a formal writing assignment that you will have the opportunity to revise after receiving feedback. Your work on the weekly homework and portfolio (which are based on your daily homework) include multiple submissions that support the writing process including an outline, rough draft, and revision of the mathematical writings describing your experiences.

Homework

• Daily Homework

  After each lesson, I will post a list of the problems that we are working from the textbook. You will be expected to read and solve (or make as much progress as possible) these problems before walking into the next class period. This will ensure that you are ready to take an active part in our class presentations and discussions. The research notebook is a place to keep and organize your notes on all of these problems. Submitted work 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. Also, keep your mistakes in your research notebook. The process of understanding and attempting to solve new problems will involve many dead ends. Do not erase your mistakes. Write a note next the mistake about why you believe it is wrong. These ideas might be useful in the future. Please use filler paper a 3-ring binder as your research notebook.

  Each Daily Preparation will be submitted twice. The first time you will submit in Google Classroom. I will review your draft and give suggestions and comments as soon as I can. You can read the comments and make necessary changes before you come to class on the next day. Your Daily Preparation work will usually be finished by hand and paper so the best way to do this is to take a picture of your writings. Please make sure that all pictures are properly oriented before submitting them. During the class, you will use felt tip color pens (provided by me) to take notes and make edits based on the presentations and class discussions on your Daily Preparation work. After the class, I will collect and grade them based on whether your pre-class work shows that you made a genuine effort before class. Incorrect but well-thought-out answers are fine because the goal is to be prepared, not be correct every single time!

• Weekly Homework

  In addition to the Daily Homework, you will also be required to submit two formally written problems each week. You may choose any two problems marked with * that were turned in during a given week to submit the following week. 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.

  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 15% of your final grade.

  All of your Weekly Homework must be typed using LaTeX. LaTeX is the industrial standard for typing scientific works in mathematics, physics, computer sciences, among others. The best way to learn how to use LaTeX is just like how you learn everything else: by using it! Fortunately, there is a website called Overleaf so you can use LaTeX online for free without having to install any software. I have also created a template for your Weekly Homework that should make things much easier for you. I will try to schedule a training session during week 1 to prepare you with some basics.

Presentations

• 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 problem 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 to prove to me that the presenter has done the problem. It is to make the ideas of the proof or the solution 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.

• Presentations will be graded using the rubric below.

  Grade	Criteria
  2	Completely correct and clear proof or solution.
  1	Proof has technical flaws, some unclear language, or is lacking some details.
  0	You were completely unprepared.

  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 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.

• A student can choose not to present on a day when he or she has a high rank in the sorted list. This is called a “pass”. No one shall “pass” for two consecutive classes. If you need help to prepare presentations, see me during office hours as soon as possible.

• In order to receive a passing grade on the presentation portion of your grade, you must present at least four times prior to each exam (2 midterms and 1 final) for a total of at least twelve 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.

Portfolio

• The object is to maintain a current account of the work we do. Every task that we encounter in the class needs to be typed up using LaTeX and 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 our 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 a task 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: Tuesday September 26, 2023, Tuesday October 31, 2023 and Tuesday, December 12, 2023, 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.

Examinations

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 Tuesday September 26, 2023 and Tuesday October 31, 2023 and the in-class portion of the final exam is Tuesday, December 12, 2023. 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.

Evaluation

Your final grade will be determined by the scores of your homework, presentations/participation, portfolio, journals, and exams.

Academic Integrity and Collaboration

Collaboration and cooperation are extremely helpful in the learning process and encouraged in most circumstances! However you may only collaborate with students currently enrolled in the same section of the course. When collaboration has occurred, you must acknowledge this clearly (state the name(s) of the person(s) you collaborated with on each problem.) You are permitted to collaborate on big ideas and hints with classmates, however, you must work independently when writing up solutions. All collaborations should occur when your collaborator is at essentially the same stage of the problem solution as yourself. In particular, if you have not yet started a problem and you ask a friend who has already completed it, how do you do this problem. This counts as plagiarism. The resulting work is not and cannot be considered your own. Regarding to outside resources: all work (including daily, weekly etc.), unless directly stated otherwise, the only resources you may use are our course textbook and your class notes. You are not permitted to go looking for completed solutions to problems in other texts or resources. In particular, use of internet resources is completely off limits for homework problems. If you see a solution, there is no way that you can claim to have an original solution. Evidence of using internet sources in your work will result in a minimum penalty of failing the assignment. Copying a solution, or any part of a solution, from any source (friend, internet, book, etc.) in any setting, constitutes plagiarism. You may not seek the help of an instructor or tutor (other than me) unless you first discuss this with me in advance. If you do not verify that this is acceptable before seeking help, this will be considered plagiarism as well. I am always willing to discuss any aspect of the course with you. Evidence of dishonest behavior on any assignment will be grounds for a minimum penalty of failing the entire assignment. In severe cases, the minimum penalty will be failure of the course. Peers who willingly assist others in acts of plagiarism are equally guilty, and will suffer similar penalties. All academic dishonesty will be reported to the Academic Standards Committee. There might be additional sanctions by the Academic Standards Committee such as dismissal from the university. See Utica University official page for Academic Honesty for more details.

Special Accommodation

The stuff just below is the University approved language, and is a bit… “legalese:. The point is, if you need accommodations to succeed in this course, talk to me and we can make sure you get what you need. And the social environment of this course is important to me, too. Let’s work together to make a welcoming and affirming space for everyone.

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 Judy Borner, Director of Learning Services in the Academic Support Services Center (315-792-3032 or [email protected] ) 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. See Utica University official page for Office of Learning Services.

Disclaimer

It is the students’ responsibility to keep informed of all announcements, syllabus adjustments, or policy changes during the semester via this web page or via school emails. The author of this syllabus reserves the right to change it with notice at any time during the semester.