2. Teaching
3. MAT526
4. Syllabus

Prerequisites

MAT 226, MAT 316, and MAT 411 with grades of C or better.

Catalog Description

Topics in enumerative, algebraic, and geometric combinatorics, chosen at instructor’s discretion; may include advanced counting techniques, graph theory, combinatorial designs, matroids, and error-correcting codes.

What is This Course All About?

This course will be an introduction to enumerative combinatorics with an introduction to generating function techniques.

An ounce of practice is worth more than tons of preaching.

An Inquiry-Based Approach

This course will use an inquiry-based learning (IBL) to learning combinatorics. Rather than spending most of class listening to lectures, you will spend substantial time exploring problems, developing arguments, presenting solutions, and discussing the mathematics with your classmates. 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. 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). I have been a firm believer in this approach to teaching and learning for over a decade, but in this AI era, I am even more convinced of its merit.

Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.

In any act of creation, there must be room for experimentation, and thus allowance for mistakes, even failure. A key goal of our community is that we support each other—sharpening each other’s thinking but also bolstering each other’s confidence—so that we can make failure a productive experience. Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

You will become clever through your mistakes.

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;
• 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.

If you want to learn more about IBL, read my Inquiry-Based Learning Resource page.

Tell me and I forget, teach me and I may remember, involve me and I learn.

Course Structure

Class meetings will consist of discussion of problems, student-led presentations, and group work. 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.

The impediment to action advances action. What stands in the way becomes the way.

Rights of the Learner

As a student in this class, you have the right:

1. to be confused,
2. to make a mistake and to revise your thinking,
3. to speak, listen, and be heard, and
4. to enjoy doing mathematics.

You may encounter many defeats, but you must not be defeated.

Commitment to the Learning Community

In our classroom, diversity and individual differences are respected, appreciated, and recognized as a source of strength. Students in this class are encouraged and expected to speak up and participate during class and to carefully and respectfully listen to each other. Every member of this class must show respect for every other member of this class. Any attitudes or actions that are destructive to the sense of community that we strive to create are not welcome and will not be tolerated. In summary: Be good to each other. I would appreciate private responses to the following question: Are there aspects of your identity that you would like me to attend to when forming groups, and if so, how?

Students are also expected to minimize distracting behaviors. In particular, 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.

The secret to education is in respecting the student.

Course Materials

There is no textbook for this course. Instead, I incrementally will post an evolving sequence of tasks to the Course Materials page. The only way to achieve a sufficient understanding is to be digesting the material in a meaningful way. You should be seeking clarification about the content 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?

Rules of the Game

The goal is for you to solve problems on your own. Try this first! However, outside of class, you may use whatever resources help you learn—including books, websites, classmates, and AI tools such as ChatGPT. The important distinction is between finding a solution and understanding the mathematics. When you come to class, you should be prepared to explain, defend, critique, and adapt the ideas you encountered. However, you are required to provide appropriate citation and attribution for ideas that are not your own! Reviewing material from previous courses and looking up definitions and theorems you may have forgotten is fair game.

Because outside resources are readily available, homework and preparation will be relatively low-stakes. Your understanding will be assessed primarily through your mathematical contributions in class and regular short, individual, no-resource assessments. In short:

Use whatever helps you learn. Be prepared to demonstrate that you learned it.

There are some pitfalls to be aware of. First, not everything you encounter on the Internet or via AI is accurate. Second, it is paramount to recognize that there is a logical progression to the concepts in mathematics. Some ideas may only be introduced after others. I’m not suggesting there is only one right flow to various subjects, but there are definitely infinitely many wrong ways! When we haphazardly bounce around the Internet or engage in “discussions” with generative AI while trying to learn various mathematical concepts, the logical progression gets jumbled. One of the reasons for this is that there can be several equally-valid ways to approach a subject. But when we parachute into the middle of someone else’s writing or video, we usually lose this context. I want you to cultivate a flexible and robust mindset when it comes to solving problems. I want you to be able to tackle problems you don’t know how to do yet. I want you to be able to wrestle with future problems no one has yet encountered. This is quite challenging if all the various concepts, techniques, algorithms, and formulas are a jumbled mess in your mind. The goal is to have an understanding of how all the various concepts fit together and depend on one another. This way you do not have to memorize very much and later in life you can quickly look up the proper tool. In addition, you’ll have the foundation to build new tools to solve unfamiliar, possibly new, problems.

The ultimate goal is for each individual student to learn and to be successful. So, if you feel you need additional resources or support, please come talk to me and we will come up with an appropriate plan of action.

Time spent thinking about a problem is always time well spent. Even if you seem to make no progress at all.

Homework

You are allowed and encouraged to work together on homework. However, each student is expected to turn in their own work. On each homework assignment, please write (i) your name and (ii) 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.

Homework will be assigned most class meetings, and students are expected to complete each assignment before walking into the next class period. Homework will generally consist of completing exercises and proving theorems from the task sequence. 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. You are allowed (in fact, encouraged!) to modify your written work in light of presentations made in class; however, you are required to use a different color than what you used to complete your homework. This will allow me to differentiate the work done in class versus the work you completed before class. I would prefer you use your own writing utensil for this purpose, but I will also have a few colored pens you can borrow. 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. If you are completing your homework using a tablet (e.g., iPad), please annotate your work in a different color and then email me your homework with the annotations at then end of class.

Some homework problems will be straightforward while others are intended to be challenging. You should anticipate not knowing what to do on some of the problems at first glance. You may have several false starts. Some frustration, maybe even a lot of frustration, should be expected. This is part of the natural learning process. On the other hand, it is not my intention to leave you to fend for yourselves. I am here to help and I want to help. You are encouraged to seek assistance while adhering to the Rules of the Game. Please visit office hours and ask questions! I am always willing to give hints/nudges, so please ask.

Homework will be assessed using the following rubric.

You are allowed to turn in three late homework assignments. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. Your overall homework grade is worth ??% of your final grade.

I write one page of masterpiece to ninety-one pages of shit.

Presentations and Participation

General Comments

The problems chosen for presentations will come from the homework assignments. 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 for the presenter:

• 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 be prepared to address questions when they arise.

Alone we can do so little; together we can do so much.

Assessing Presentations

Presentations will be assessed using the following criteria.

You should aim to avoid unsatisfactory (U) presentations. An in progress (I) presentation should not be viewed as a bad thing as each of us will occasionally get stuck. However, you should strive for the majority of your presentations to meet (M) or exceed (E) expectations. Most presentations will meet expectations (M) while presentations that exceed expectations (E) will be rare. 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 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! In my view, an interestingly wrong solution or proof makes for the best presentation since it generates the best discussion. This is really what we are after. On the other hand, 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 alone cannot change the world, but I can cast a stone across the waters to create many ripples.

Selecting Presenters

In general, I will curate the list of student presenters each class meeting. A presenter is a student that either volunteered (V) or was chosen (C) by me. Volunteering is encouraged, but being chosen without volunteering isn’t bad. If more than one student volunteers for a specific problem, the student with the fewest number of presentations has priority. I reserve the right to decline your offer to present. This may happen if you are volunteering too often (and hence removing another student’s opportunity to present) or if I know in advance that another student’s presentation will lead to a fruitful discussion.

If you are chosen to present but would prefer not to present that particular problem, you can either negotiate presenting a different problem or take a pass (P). You may elect to pass at most two times during the semester, after which a presentation will be deemed unsatisfactory (U). By default, if you have an unexcused absence on a day when you have been chosen to present, then your presentation will be recorded as a pass (P) unless you have already exhausted your two passes, in which case the presentation will be recorded as unsatisfactory (U).

In summary, for each student presentation, I will record one of V, C, or P. In the case of V or C, I will also record one of U, I, M, or E based on the rubric given above. The most common pair will likely be CM (i.e., student was chosen to present and presentation met expectations).

Participation

You are expected to respectfully participate and contribute to class discussions. This includes asking relevant and meaningful questions to both the instructor and your peers. Moreover, you are expected to be engaged and respectful during another student’s presentation. Your class participation will be assessed as follows.

Determining Presentation and Participation Grades

Your Presentation and Participation grade is determined by your frequency and ability to foster productive class discussions through presentations and audience participation. The greatest determining factor in your Presentation and Participation grade is your willingness to present often. You should aim to present at least twice prior to each midterm exam. The table below provides a summary of how your Presentation and Participation grade will be determined.

I anticipate that most students will fall in the 80-89% range. Your Presentation and Participation grade is worth ??% of your overall grade.

Quizzes

We will have 6 quizzes this semester. Tentatively, the quizzes are scheduled for the following Fridays:

• Quiz 1: (week 3)
• Quiz 2: (week 5)
• Quiz 3: (week 7)
• Quiz 4: (week 9)
• Quiz 5: (week 11)
• Quiz 6: (week 13)

Each quiz will consist of a mixture of homework problems you encountered during the previous two weeks and new problems not previously attempted. During a quiz, you may have the opportunity to collaborate with your peers for a few minutes. Each problem will be graded using the following rubric:

Make-up quizzes will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time. Each Quiz is worth 10% of your overall grade (Quizzes constitute 60% of your overall grade).

Nandor Sieben, Monika Keindl, and I received a TRAIL AI grant to experiment with using AI to assess student work and to provide feedback. My current plan is to use ChatGPT and Codex to assess and provide feedback on the quizzes. This will be in addition to me grading the quizzes. In addition, I am required to not have your name associated with your work when ChatGPT does its assessment. You can opt out of this if you wish.

If you want to sharpen a sword, you have to remove a little metal.

Final Exam

There will be a cumulative Final Exam at the end of the semester, which may consist of both an in-class portion and a take-home portion. The in-class portion of the final exam will be on Wednesday, December 9 at 7:30-9:30AM. The Final Exam is worth ??% of your final course grade.

The best way to learn is to do; the worst way to teach is to talk.

I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past I will turn the inner eye to see its path.
Where the fear has gone there will be nothing.
Only I will remain.

Extra Credit

There are two ways you can earn extra credit. The first way is finding typos in course materials (e.g., syllabus, homework, website, Canvas). This includes broken links. However, it does not include the placement of commas and such. If you find a typo, I will add one percentage point to your next quiz. The second way is by finding legitimate errors in AI-generated feedback on quizzes. If you find an error in the AI-generated feedback on one of your quizzes, I will add two percentage points to your next quiz. You can earn at most four percentage points per quiz. They’re is a typo right here.

Basis for Evaluation

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

It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.

Department and University Policies

You are responsible for knowing and following the Department of Mathematics and Statistics Policies and other University policies listed here (PDF). More policies can be found in other university documents, especially the NAU Student Handbook (see appendices).

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 assessments. Any violation of this policy will be treated as academic dishonesty.

Career Readiness Skills

In every class you take at NAU, you learn professional skills that can support your future career. There are several ways that this course can help you meet and excel at your job goals and life desires. Below is a list of in-demand skills from National Association of Colleges and Employers (NACE) you will practice in this class:

1. Communication: Demonstrate the ability to articulate mathematical concepts clearly and concisely, whether through written explanations, oral presentations, or visual representations, ensuring comprehension by peers.
2. Critical Thinking: Demonstrate the ability to solve mathematical problems by considering the context in which they arise, ensuring that solutions are relevant and applicable to real-world situations.
3. Professionalism: Uphold academic integrity and accountability in mathematical assignments, demonstrating honesty and ethical behavior in the completion of individual and group tasks.
4. Teamwork: Collaborate actively with classmates to achieve common mathematical goals, working collectively on assignments, projects, or problem-solving exercises to enhance the overall learning experience.

Some Career Readiness resources:

• LinkedIn: CEFNS Career Development, NAU Career Development
• Handshake
• Udemy: Online courses and career searching advice. Log in with your NAU email account and search ‘NAU Career Steps’
• O*net Online: Occupation exploration reports
• Labor Market Insights

We cannot teach people anything; we can only help them discover it within themselves.

Changes to the Syllabus

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