Title: MA4510: Introduction to Analysis
Time: MWF 11:15am-12:05pm
Location: Hyde 317
Instructor: Dr. Dana C. Ernst
Office: Hyde 356
Office Phone: 603.535.2857
Office Hours: MWF at 1:00-2:00pm (or by appointment)
Course Information and Policies
A satisfactory grade in MA3110: Logic, Proof, & Axiomatic Systems and MA3120: Linear Algebra.
A rigorous treatment of classical topics in calculus including: Completeness Axiom, Heine-Borel Theorem, differentiability and/or integrability of functions.
This course is an introduction to real analysis, which is a subject area of mathematics that deals with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, and related properties of real-valued functions.
We will spend most of our time studying sequences and the continuity of functions. We will take an axiomatic approach (definition, theorem, and proof) to the subject, but along the way, you will develop intuition about the objects of real analysis and pick up more proof-writing skills. The emphasis of this course is on your ability to read, understand, and communicate mathematics in the context of real analysis.
(Adopted from Chapter Zero Instructor Resource Manual by Carol Schumacher) 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 the student 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.
This course will be different than most math classes that you have taken. You are used to being asked to do things like: "solve for $x$," "take the derivative of this function," "integrate that function," etc. Accomplishing tasks like these usually amounts to mimicking examples that you have seen in class or in your textbook. The steps you take to "solve" problems like these are always justified by mathematical facts (theorems), but rarely are you paying explicit attention to when you are actually using these facts. Furthermore, justifying (i.e., proving) the mathematical facts you use may have been omitted by the instructor. And, even if the instructor did prove a given theorem, you may not have taken the time or have been able to digest the content of the proof. This course is all about "proof." Mathematicians are in the business of proving theorems and this is exactly our endeavor. You will be exposed to what "doing" mathematics is really all about.
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 the reality is that 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 the Moore method (after R.L. Moore, a former professor of mathematics at the University of Texas, Austin). Modifications of the Moore method are also referred to as inquiry-based learning (IBL) or discovery-based learning.
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. I learned to ride a bike by getting on and then falling off, and in a similar fashion, you will learn mathematics in this course by attempting it and sometimes falling off.
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.
We will not be using a textbook this semester, but rather we will be using a theorem-sequence adopted for inquiry-based learning and the Moore method for teaching mathematics. The theorem-sequence that we are using is an adaptation of the analysis notes by W. Ted Mahavier published by The Journal of Inquiry Based Learning in Mathematics. The published original version of the notes can be found here. The author has been gracious enough to grant me access to the source of these notes, so that we can modify and tweak for our needs if necessary. Every attempt will be made on my part to maintain the integrity of these notes and any intentional modifications or additions to the notes will be clearly indicated. Any new errors introduced 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.
Regular attendance is expected and is vital to success in this course, but you will not be graded explicitly on attendance.
Homework will usually be assigned every lecture day and will usually be due at the beginning of the next lecture day. The 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. I expect the write-ups to be well-written using complete sentences and proper grammar. Students are allowed to modify their written proofs in light of presentations made in class. Typically the next homework assignment will begin where we left off in class on the previous day. I reserve the right to modify the homework assignments as I see necessary.
On each homework assignment, I will grade 1-2 problems (perhaps ones that were not presented in class), where each problem that is graded is worth 4 points and 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 for this problem.|
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.
Any problem that you received a score of 1, 2, or 3 on can be resubmitted up until 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 write "Resubmission" on top of any problem that you are resubmitting and keep separate from any other problems that you are turning in. Resubmissions on problems that received a 0 are not allowed.
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. Five (possibly more) of your lowest homework scores will be dropped. In general, late homework will not be accepted. However, you are allowed to turn in up to 5 homework assignments 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 25% of your final grade.
(Adopted from Chapter Zero Instructor Resource Manual by Carol Schumacher) 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.
I will be grading presentations both on the content of the proof and on the quality of the presentation. I will probably grade the quality more harshly as the semester wears on, since I expect your presentation skills to develop with practice. I will always ask for volunteers to present proofs, but when no volunteers come forward, I will call on someone to present their proof (perhaps by rolling a die or equivalent). Each student is expected to be engaged in this process. The problems chosen for presentation will come from the assigned homework. Your grade on your own presentations, as well as your level of interaction during other's presentations, will be worth 30% of your overall grade. A student must make at least 3 presentations during the semester to receive a passing presentation grade. At any point during the semester, you may ask me for your current presentation grade. Lastly, students may choose to present a problem on the board in my office that has not already been presented in class. However, presentations in my office cannot be the only presentations that a student makes.
There will be two midterm exams and a cumulative final exam. All exams will may consist of an in-class part and a take-home part. The midterm exams will take place on or around Wednesday, October 6 and Wednesday, November 5. The in-class portion of the final exam (if there is one) will take place on Friday, December 17 at 11:00–1:30PM, and the take-home portion of the final exam will be due by 5:00PM on Friday, December 17. Each exam will be worth 15% of your overall grade. 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 presentations, written homework, and exams.
|Homework||25%||Each graded problem is worth 4 points|
|Presentations||30%||Each student must present a minimum of 3 times|
|Midterm Exam 1||15%||Take-home: due October 8|
|Midterm Exam 2||15%||Take-home: due November 5 November 19|
|Final Exam||15%||December 17|
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, as I will post comments there for all to benefit from.
To effectively post to the course forum, you will need to learn the basics of LaTeX, the standard language for typesetting in the mathematics community. See the quick LaTeX guide for help with $\LaTeX$. If you need additional help with $\LaTeX$ or editing/posting content to the wiki, post a question in the course forum.
There are substantial resources available online to help you learn the Wikidot markup, which is quite simple. I regularly consult Wikidot's Wiki Syntax page and Wikidot's Quick Reference Guide. Lastly, you can always contact me.
The PSU Student Handbook addresses policies pertaining to students with disabilities, religious observation, honor code, general conduct, etc. The Handbook can be found here.
ACT for Growth
All teacher education majors are subject to the Areas of Concern/Targets for Growth policy, which is located here.
(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.
If you are struggling too much, then there are resources available for you. Work together and help each other learn. Use the course forum! 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!