## Foundations of Math

## General Information

**Title:** MAT 320: Foundations of Mathematics

**Semester:** Spring 2013

**Credits:** 3

**Section:** 2

**Time:** MWF at 11:30-12:20PM

**Location:** AMB 162

## Instructor Information

**Instructor:** Dr. Dana C. Ernst

**Office:** AMB 119

**Office Phone:** 928.523.6852

**Email:** dana.ernst@nau.edu

**Office Hours:** MWF at 10:00-11:30AM (or by appointment)

**Webpage:** http://danaernst.com

## Course Information and Policies

### Prerequisites

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

### Catalog Description

Symbolic logic, set theory, functions, and number systems. Focuses on techniques of proof and mathematical writing. This course fulfills NAU’s junior-level writing requirement. This course contains an assessment that must be successfully completed in order to register for student teaching. Letter grade only.

### 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.

### Purpose

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. Also, the purpose of any mathematics course is to challenge and train the mind. Learning mathematics enhances critical thinking and problem solving skills.

### Learning Management System

We will be making limited use of Bb Learn 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 on a course webpage (see link above) that lives outside of Bb Learn. I suggest you bookmark the course webpage. In addition, we will be utilizing an LMS called Canvas. We will use Canvas to manage grades and the submission of Weekly Homework assignments. Also, we will use the forums within Canvas to facilitate out of class discussion. I will send the class an invite to our Canvas page and spend time in class discussing its use.

### Course Notes

We will not be using a textbook this semester, but rather a task-sequence adopted for inquiry-based learning (IBL). The task-sequence that we are using was written by me, but the first half of the notes are an adaptation of notes written by Stan Yoshinobu (Cal Poly) and Matthew Jones (California State University, Dominguez Hills). 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.

In addition to working the problems in the notes, I expect you to be *reading* them. 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 in class or posting questions to the course forum on our Canvas page. Here’s what [Paul Halmos}(http://en.wikipedia.org/wiki/Paul_Halmos) has to say 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?

### Comments about this course and expectations

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. 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). Modifications of the Moore method are also referred to as inquiry-based learning (IBL) or discovery-based learning. If you want to learn more about IBL, go here.

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 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 proving theorems is difficult and takes time. You shouldn’t expect to complete a single proof in 10 minutes. Sometimes, you might have to stare at the statement for an hour before even understanding how to get started. In fact, proving theorems 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 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. This will be new to many of you and there may be some growing pains associated with it.

### Goals

(Adopted from *Chapter Zero Instructor Resource Manual*) 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 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.

### A little more propaganda

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.

### Class Presentations

(Adopted from *Chapter Zero Instructor Resource Manual*) 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.

Presentations will be graded using the rubric below.

Grade | Criteria |
---|---|

4 | Completely correct and clear proof or solution. Yay! |

3 | Proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct. |

2 | A partial explanation or proof is provided but a significant gap still exists to reach a full solution or proof. |

1 | Minimal progress has been made that includes relevant information & could lead to a proof or solution. |

0 | You were completely unprepared. |

However, you should not let the rubric deter you from presenting if you have an idea about a proof that you’d 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.

I will always ask for volunteers to present proofs, but when no volunteers come forward, I will call on someone to present their proof. Each student is expected to be engaged in this process. The problems chosen for presentation will come from the daily assignments. After a student has presented a proof that the class agrees is sufficient, I will often call upon another student in the audience to come to the board to recap what happened in the proof and to emphasize the salient points.

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 6 times during semester. Your grade on your presentations, as well as your level of interaction during student presentations, will be worth 30% of your overall grade.

### Homework

**Daily Homework:** Homework will be assigned each class meeting, 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 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 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. Daily Homework will be graded on 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 felt-tip pens provided in 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, 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 Tuesday by 5PM. For example, you may choose any two problems marked with a * that were turned in during week 2 for the second Weekly Homework. These problems are due by 5PM on Tuesday in week 3.

Beginning with the second Weekly Homework, you will be required to type your submission. You should type your Weekly Homework assignments using $\LaTeX$ (see below for more on this), MS Word, OpenOffice/LibreOffice, or Google Docs. However, **everyone is required to electronically submit a PDF file** via Canvas. I will walk you through how to do this. If you need help with converting your document to a PDF, please let me know.

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. |

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. |

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 Weekly Homework problems that you received a score of 1, 2, or 3 on 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 5 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 will be worth 25% 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 homework assignments here. I reserve the right to modify the homework assignments as I see necessary.

### Exams

There will be two midterm exams and a cumulative final exam. Each exam will be worth 15% of your overall grade and may consist of both an in-class portion and a take-home portion. The in-class portions of the midterm exams are *tentatively* scheduled for **Friday, February 22** and ~~Friday, April 12~~ **Friday, April 19**, and the in-class portion of the final exam will be on **Wednesday, May 8** 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.

## Basis for Evaluation

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

Category | Weight | Notes |
---|---|---|

Homework | 25% | a mix of Daily & Weekly Homework |

Presentations/Participation | 30% | each student must present at least twice prior to each exam (that’s at least a total of 6 times during semester) |

Exam 1 | 15% | Friday, February 22 |

Exam 2 | 15% | Friday, |

Final Exam | 15% | in-class portion on Wednesday, May 8 at 10:00AM–12:00PM |

## Additional Information

### 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.

### 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. 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.

### 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.

### Getting Help

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 on our Canvas page, as I will post comments there for all to benefit from. 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.

To effectively post to the course forum, you should 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 $\LaTeX$, post a question in the course forum.

## Closing Remarks

(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.