### Course Info

Title: MAT 320: Foundations of Mathematics
Semester: Spring 2017
Credits: 3
Section: 1
Time: MWF at 11:30AM-12:20PM
Location: AMB 225

### Instructor Info

Dana C. Ernst, PhD
AMB 176
MWF 10:15AM-11:15AM, T 11:00AM-12:00PM
dana.ernst@nau.edu
928.523.6852
danaernst.com

### Prerequisites

MAT 137 with a grade greater than or equal to C.

### Course Description

The course trains students on methods and techniques of mathematical communication, focusing on proofs but also covering expository writing and problem-solving explanations.

### Learning Outcomes

Upon successful completion of the course, students will be able to:

1. Write a readable and mathematically rigorous proof.
2. Express in writing, knowledge of the terminology, concepts, basic properties and methodology of symbolic logic, set theory, relations and functions, mathematical induction, cardinality, and number systems.
3. Propose useful definitions and make correct deductions from definitions.
4. Identify correct proof structures and criticize incorrect proof structures.

### What is This Course Really About?

The primary objective of this course is to develop skills necessary for effective proof writing. Students will improve their ability to read and write mathematics. Successful completion of MAT 320 provides students with the background necessary for upper division mathematics courses.

Aside from the obvious goal of wanting you to learn how to write rigorous mathematical proofs, one of my principle 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.

This course will likely be different than any other math class that you have taken before for two main reasons. First, you are used to being asked to do things like: “solve for $x$,” “take the derivative of this function,” “integrate this 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.

Unlike previous courses, this course is all about “proof.” Mathematicians are in the business of proving theorems and this is exactly our endeavor. For the first time, you will be exposed to what “doing” mathematics is really all about. This will most likely be a shock to your system. Considering the number of math courses that you have taken before you arrived here, one would think that you have some idea what mathematics is all about. You must be prepared to modify your paradigm. The second reason why this course will be different for you is that the method by which the class will run and the expectations I have of you will be different. 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.

All of the secondary skills you will develop in this course are highly valued by society. Whether you become a teacher, a lawyer, an engineer, or an artist, what differentiates you from your competition is your ability to think critically at a high level, collaborate professionally, and communicate effectively.

### An Inquiry-Based Approach

This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand with the readily available help of me and your classmates. Many of the concepts you learn and problems you work on will be new to you and ask you to stretch your thinking. You will experience frustration and failure before you experience understanding. This is part of the normal learning process. If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately. Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.

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, and 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 presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “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 solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems 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 and textbook on your own;
• write up quality solutions/proofs to assigned problems;
• present solutions/proofs on the board to the rest of the class;
• participate in discussions centered around a student’s presented solution/proof;
• call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar at 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.

### Course Structure and Approach

Class meetings will consist of discussion of problems, student-led presentations, and group work focused on problems selected by the instructor. A typical class session may include:

• Informal student presentations of progress on previously assigned homework problems;
• Summary of major steps and techniques of the solution of a finished problem;
• Exploration of alternative approaches, possible generalizations, consequences, special cases, converse;
• Discussion of relationships to previously assigned or solved problems;
• Assignment of new problems;
• Explanation of unfamiliar mathematical concepts as needed.

### Course Notes

We will not use a textbook this semester, but rather a task-sequence adopted for IBL. The task-sequence that we are using was written by me. 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. Here’s one of my favorite quotes 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?

### Learning Management System

We will make limited use of BbLearn 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 here on our course webpage that lives outside of BbLearn. I suggest you bookmark this page. The only thing I will use BbLearn for is to communicate grades.

### Homework

You are allowed and encouraged to work together on homework. However, each student is expected to turn in their own work. In general, late homework will not be accepted. However, you are allowed to turn in three late homework assignments with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. When doing your homework, I encourage you to use the Elements of Style for Proofs (see Appendix B of the course notes as a reference. Your overall homework grade will be worth 20% 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 assignments on the homework page. I reserve the right to modify the homework assignments as I see necessary.

Daily Homework: Homework will be assigned most class meetings, and students are expected to complete (or try their best to complete) each assignment before walking into the next class session. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation, and spelling. You should 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 completing exercises and proving theorems from the course notes. In addition to completing the assigned problems, I also want you to assess your own work. Next to each problem, I want you to write down a score between 0 and 4 that represents your perception of the validity and quality of your proposed solution/proof. Consider using the rubric given below in the description of the Weekly Homework assignments. Not completing the self-assessment step may impact the score on your homework.

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/solutions 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 using 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 colored marker pens provided in class. This will allow me to differentiate the work done in class versus the work you completed before class. The grade you receive on an assignment will be determined by the work you completed prior to 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 2 formally written proofs. Typically, these problems will come directly from the Daily Homework assigned the previous week. You will be required to type your submission using $\LaTeX$. I will walk you through how to do this. You can either submit a hardcopy of your assignment or email me the PDF of your completed work. If you email me the PDF, please name your file as WeeklyX-Lastname.pdf, where X is the number of the assignment and Lastname is your last name. Notice there are no spaces in the filename. Each problem on the Weekly Homework assignments is 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.

### Presentations and Participation

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 seriously since they spearhead the work of the class. Here are some of my expectations:

• 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 solution clear to the other students.
• 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 their 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 solution/proof. Yay!
3 Solution/proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct.
2 A partial explanation or solution is provided but a significant gap still exists.
1 Minimal progress has been made that includes relevant information & could lead to a solution/proof.
0 You were completely unprepared.

However, you should not let the rubric deter you from presenting if you have an idea about a solution/proof that you’d like to present, but you are worried that your solution/proof is incomplete or you are not confident your solution/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 general, I will curate the list of student presenters each class meeting. However, students are welcome to volunteer. This type of behavior will be rewarded. 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. Each student in the audience is expected to be engaged and respectful 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 midterm exams and at least once after the third midterm exam for a total of at least 7 times during the semester. Notice that I’ve described a minimum. To ensure a good grade, you should present as often as you can. Your overall performance during presentations, as well as your level of interaction/participation during class, will be worth 15% of your overall grade.

### Exams

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 Wednesday, February 15, Wednesday, March 22, and Wednesday, April 19. The final exam will be on Wednesday, May 10 at 10:00AM-12:00PM. 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.

### Attendance

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.

### Basis for Evaluation

In summary, your final grade will be determined by your scores in the following categories.

Category Weight Notes
Homework 20% A combination of Daily & Weekly Homework
Presentations & Participation 15% See above for requirements
Exam 1 15% Wednesday, February 15
Exam 2 15% Wednesday, March 22 March 29
Exam 3 15% Wednesday, April 19 April 26
Final Exam 20% Wednesday, May 10 at 10:00AM-12:00PM

In general, you should expect the grades to adhere to the standard letter-grade cutoffs: A 100-90%, B 80-89%, C 70-79%, D 60-69%, F 0-59%.

### 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 (PDF). 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.

### Class Etiquette

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.

### Important Dates

Here are some important dates:

• Thursday, January 26: Last day to drop/add (no W appears on transcript)
• Wednesday, February 15: Exam 1
• Monday, March 13-Friday, March 17: Spring Break (no classes)
• Wednesday, March 29: Exam 2 (rescheduled from March 22)
• Friday, March 24: Last day to withdraw from a course (W appears on transcript)
• Wednesday, April 26: Exam 3 (rescheduled from April 19)
• Wednesday, May 10: Final Exam

### Getting Help

There are many resources available to get help. First, I recommend that you work on homework in small groups as much as possible, and to come see me whenever necessary. 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.

### Changes to the Syllabus

Any changes to this syllabus made during the term will be properly communicated to the class.

# Dana C. Ernst

Mathematics & Teaching

Northern Arizona University
Flagstaff, AZ
Website
928.523.6852
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