# Math 311: Real Analysis

## Course Information

• Institution: Dordt University
• Course: Math 311-01 (3 cr.)
• Term: Fall 2021
• Instructor: Dr. Mike Janssen, Associate Professor of Mathematics
• Classroom: CL 92
• Class time: 12:25-1:40pm TTh
• Office: SB 1612
• Student Hours: Make an appointment or drop by
• Course Notes: https://prof.mkjanssen.org/ra/notes/ | PreTeXt source
• Course website: https://prof.mkjanssen.org/ra/
• Catalog Course Description: An introduction to the content and methods of single-variable real analysis: infinite sets, the real number system, sequences, limits, series, continuity, differentiation, and integration. Prerequisite: grade of C- or higher in Mathematics 212; or permission of instructor.

## Learning Objectives

In this course, students will:

• be communicators through regular presentations to the Math 311 learning community and growing fluency in the writing of mathematical proofs. (CD)
• be explorers by engaging with in-class discussions, activities, and regular work outside the classroom. (CD)
• be learners by leveraging knowledge of logic, functions, and sets to explore foundational questions in the study of functions of one real variable. (CS)
• be ambassadors by exploring the ways in which humans have developed the mathematical aspects of creation and reflecting on our creative roles as image-bearers. (RO, CD)

## Course Liturgies

The best way to learn mathematics is to do mathematics. There are two main types of regular work in this course: daily work, and written work.

### Daily Work

You will be assigned 3–5 problems from the book to work on before coming to class. You may not use any outside resources to help you solve these problems–no books, no websites, no friends who have taken this course before. Using these resources will constitute plagiarism and will be reported to the Student Life Committee. You may work with others in the course, but you will need to ensure that you can completely understand and explain the proofs you come up with.

One of the main goals of this course is to improve your mathematical communication. Thus, the majority of each class period will be devoted to you presenting your work on these problems to the class. You should expect that approximately 90% of the typical class period will consist of presentations and discussion.

By 11:00am before each class, you will claim on Canvas problems whose solutions you are willing to present. In general, you will be allowed to present at most one problem per class meeting. You will earn 0.1 daily work points per problem you sign up to present, even if you are not ultimately the person to present it.

You will be notified by 12:00pm on Canvas of any problems you are assigned to present.

You will then write the problem and solution up on the board, highlight the main points of the proof/solution, and generally lead the class discussion. That is, merely writing the solution on the board will not be sufficient to earn full credit. The presenter will earn points as follows.

1 point. The solution/proof is correct and complete, and all questions are answered.

1/2 point. The solution/proof was prepared, but there are gaps and/or questions that are not satisfactorily answered.

$-0.1$ points. The solution/proof was either not prepared or is in completely the wrong direction.

Each presented problem will also have a scribe assigned on Canvas by 1:00pm. The first problems will be scribed in alphabetical order by last name, and subsequent scribes will be assigned in ascending order from least daily work points to most. The scribe will have the responsibility of taking notes on the presented proof and asking questions when something is not clear. They then will write up a formal version of the proof and discussion and post it to our Overleaf document. The scribe will earn 1/4 point when a correct proof is submitted.

This course structure effectively models the way professional mathematicians conduct and share their research. Thus, we will abide by the Policy Statement on Ethical Guidelines1 adopted by the American Mathematical Society, in particular Section I on mathematical research and its presentation. As this statement describes, “[t]he knowing presentation of another person’s mathematical discovery as one’s own constitutes plagiarism and is a serious violation of professional ethics. Plagiarism may occur for any type of work, whether written or oral and whether published or not.” When you present your work in this class, both orally and in writing, you must cite any conversations you have had about this problem with anyone in the class. Looking to any resource outside of the people in our class for information about the problems at hand constitutes plagiarism and will be reported to the Student Life Committee.

Daily work points will be monitored and factored into the final grade.

### Written Work

Roughly every other week (other than the weeks we have exams), you will be assigned three problems to solve, write up, and submit online by 5:00pm on Sep. 2, 16, 30; Oct. 14, 28; Nov. 11; Dec. 2. These will be written in LaTeX, and will generally not be problems that have been presented in class (though they may have been assigned as daily work). Each problem will be graded on a four-level scale (each explained more fully on the proof rubric distributed here) as:

Exceeds expectations. Dr. Janssen would be happy to post this as the official class solution.

Meets expectations. The logic is generally correct and it is reasonably well written, but there is room for improvement.

Revision needed. Some major gaps in logic, misuse of notation, or unclear communication requires revision.

Not assessable. This is difficult to read, abuses notation, or contains significant mathematical flaws.

Writing proofs is as much art as science, and initially it can seem daunting and confusing. In order to aid your growth, you will have the opportunity to revise your work once for free after Dr. Janssen returns the graded version provided the work was (a) submitted on time (or within the 24-hour grace period) and (b) received an initial assessment of R or higher.

In short, your submissions will go through the following workflow (with the number representing the number of days since the Wednesday submission):

• Day $0$: Initial submission due 5:00pm Thursday
• Day $7$: Initial assessment feedback returned by the end of the day Thursday
• Day $12$: Revised submission due at 5:00pm Tuesday
• Day $12+n$, $1\le n\le 7$: Graded revised work returned; grade is final

Work submitted late, work assessed at an N, or additional revisions requires a meeting with Dr. Janssen and short accompanying reflection on why the work was assessed at an N and how such assessments will be avoided in the future. If earning an N becomes a regular occurrence on written assignments due to perceived lack of effort, you may lose the grace afforded by the submission/feedback/resubmission process and only be allowed a single submission.

### Historical Project: What is a real number, anyway?

There are many fascinating historical vignettes in the development of real analysis which reveal both God’s wondrous design in creation and the creativity exercised by humans as image-bearers. An example of this is Dedekind’s work defining the real numbers themselves–how can one define a real number in such a way that the definition applies equally well to natural numbers, integers, rationals, and irrationals, all of which are reals? We will read some of Dedekind’s Continuity and Irrational Numbers and complete a sequence of mathematical tasks which will help us explore his work. You’ll submit your work on these tasks, along with an accompanying reflection, one week after we finish spending class time on it.

### Exams

There will be two exams, the first the week of October 18, and the second during the last week of class/finals week. The first will be cumulative up to the previous class, and worth 50 points, while the second will be cumulative over the whole semester, but with an emphasis on Chapters 3-5, and worth 75 points. Both exams will be oral exams. Your exam average will be a major factor in your final grade.

In general, your final grade will be the highest fully completed row in the following table.

Grade Daily Work Written Work (M or higher) Written Work (E) Exam Average Project
A 30 19/21 14 87% E
A- 28 18/21 12 84% E
B+ 26 17/21 10 80% E
B 24 16/21 9 77% M
B- 22 15/21 7 74% M
C+ 20 14/21 6 70% M
C 19 13/21 4 67% M
C- 18 12/21 2 64% M
D 15 11/21 0 55% -

• I am generally fairly accepting of late work, with a built-in 24-hour grace period for any non-classroom activities (e.g., written work submissions, projects, reflections). Additional time beyond the 24-hour grace period must be approved ahead of time.
• Student hours are your time to ask questions about all aspects of the class and college life. If you can’t find an appointment, send me an email! I will do my very best to accommodate your schedule.
• Email Policy: I check my email twice per school day: once in the morning, where I’ll deal with any emergencies, and once in the afternoon, when I’ll respond to other emails (including any that have come in since the morning). If you require a more immediate response, you’re welcome to come find me in my office.

#### Dordt University Student’s Right to Accommodations Policy

Any student who needs access to accommodations based on the impact of a documented disability should contact the Coordinator of Services for Students with Disabilities (CSSD): Marliss Van Der Zwaag, Academic Enrichment Center, (712) 722-6490, marliss.vanderzwaag@dordt.edu.

#### Dordt University Academic Dishonesty Policy

Dordt University is committed to developing a community of Christian scholars where all members accept the responsibility of practicing personal and academic integrity in obedience to biblical teaching. For students, this means not lying, cheating, or stealing others’ work to gain academic advantage; it also means opposing academic dishonesty. Students found to be academically dishonest will receive academic sanctions from their professor (from a failing grade on the particular academic task to a failing grade in the course) and will be reported to the Student Life Committee for possible institutional sanctions (from a warning to dismissal from the university). Appeals in such matters will be handled by the student disciplinary process. For more information, see the Student Handbook.

#### Classroom Attendance Protocol

As we begin the Fall 2021 semester, class attendance policies and procedures as outlined in the Student Handbook are in place. To paraphrase the Student Handbook, Dordt University as an institution remains committed to in person instruction for face-to-face courses. As a result, you are expected to be present for every class period and laboratory period. Should you need to miss class for any reason, contact your instructor as soon as possible (either prior to the absence or immediately following). Absences for Dordt-sponsored curricular or co-curricular activities will be communicated by the activity sponsor and are considered excused.

Methods of making arrangements for missed work are back to normal (pre-COVID). You are responsible to contact your instructor. Your instructor is not required to provide real time (synchronous) learning for you should you be absent for class for any reason (ex. Zooming into your real time class). Your instructor is also not required to provide asynchronous virtual learning materials for you (ex. recordings of missed classes, slide decks, other materials on Canvas). While some instructors might utilize some of the synchronous/asynchronous methods of making up work on occasion, you should not expect all instructors to provide these experiences automatically. Methods of making up missed work might include: contacting a fellow student to get notes from class, extensions on assignments or labs, or other methods as determined by your instructor. Making arrangements for missed class work is your responsibility!