Grading is simple in Honors if we don’t complicate it any further than need be and instead focus on what we’re learning. At the end of the semester you will receive one of three letter grades that go on your UNM transcript: A, CR (credit), or NC (no credit). In very rare cases, a grade of incomplete may be appropriate. The grades represent the quality of your work in class and in assignments you hand in:

A– Your work was excellent. There may be one aspect that is merely sufficient, but all aspects must be sufficient, and some significant part must be excellent enough to carry the rest.
CR– You showed up and did the work in a sufficient and satisfactory manner. There is no shame in this grade, but no great pride either.
NC– You have significant gaps in some aspect of your work and/or participation.

I will stress here that an A is not out of reach for any student, but is not awarded for merely showing up and doing some stuff. I have found that careful course design makes it possible for everyone to find a way to shine, to make expectations and standards clear on these same qualitative terms, and to see performance as something that ultimately you are responsible for, not me.

If you truly care about your grade, all you need to do is dig into the work. If this ever seems confusing or not true, I will work with you to fix it. Lower down, I have prepared some remarks on excellence, specifically it pertains to our assignments and goals.

What is Excellence?

Excellence can mean many things, and I hope to consider it broadly, by providing the opportunity to excel at different things and in different ways. In short, an excellent student has done everything to a level of competence and responsibly, and has taken some major aspect of the work and worked to make it their own. This can mean taking risks to go far beyond your comfort and experiences, taking an assignment beyond its boundaries, forging a new path, or in cases of extreme competence, simply never missing a beat.

The syllabus and schedule detail a number of formally handed in assignments, proof presentations, and our class discussions which will form the basis of your grade. You will receive written feedback on each major assignment, to give you some ideas regarding expectations and how to improve in the future. Here, I will introduce each of these major areas and what it means to engage with them.

Problem Sets

To do well on written problem sets, yes you do need to get the right answer, but this is the beginning of success, not the end. What is important is that you make a real effort at communication in your writing, that you have gone to some pains to think about how you are helping another person to understand what’s going on. Care for how the piece looks and flows, and especially any connections that can be drawn further afield or extensions on the given subject, these are the unmistakable acts of quality.

As for the answers, please understand you are not on your own. I only require that the writing you produce is honestly your own. We have each other, and of course the internet, to help. For some reason, it is hard to ask for this help, especially ahead of time, to organize so that everyone knows the stuff. This is certainly another way to succeed in this class.

What is the opposite of this? When I feel like I have wasted my time reading an assignment. When there is no care or attempt to share meaning between fellow humans, only enough chicken scratch to hopefully get some points. That’s not what I’m here for and it’s just as much a waste of your time. I understand that quality takes time and that’s why the total number of assigned problems is low.

These homework assignments are different than other math problem sets you have likely done before. It may take some time before you pick up on an appropriate style. We will together consider examples of good work to help set the tone. You will have at least one chance to hand in a problem and receive feedback but no grade. I will pay close attention to how your work evolves in response to the feedback you receive.

Writing Mathematics

Our written work will be based on solving problems for the most part, answering questions that have at least one right answer and plenty of wrong ones. The difference will be in the role played by the answer itself. Instead of proving that you can follow a procedure, on your own without help, to get a certain kind of answer like you’ve seen before, here the answer will help us to focus on the act of communication itself. Our problem sets are writing assignments.

This is about more than “showing your work”. The idea is to take a hand at trying to explain to someone who doesn’t know, to recruit different strategies as you try to make ideas accessible and clear to another person. Everything from the words you choose, to the patterning of words, equations, and diagrams, even the whitespace you include are what we care about.

Why? Because we aim to embrace some of the values that have guided the development of math over the millennia, in this case that the facts uncovered are of less importance than understanding why. The overall system in which these questions and answers fit together is the principal creation, the thing we hope to understand. No one cares if you can find xx using the quadratic formula or compute a standard integral. Your suspicions are correct. These skills are not directly applicable to life, modern or not. Our work, though it involves some of these same computations, pertains to something of deeper meaning and relevant if a bit abstract. If we find a way to see ourselves in it too, looking behind the curtain with math will ultimately be far more meaningful than the sort of solace one gets from knowing how to multiply matrices. Just as a proper understanding of history is not the lists of names, dates, and places, learning math should mean more than memorizing formulas and practicing methods.

Sometimes the role of asking and answering “why” will be very obvious. Mathematicians even have a word for this kind of problem: “proof”. You are supplied with a true statement and your job in producing an answer is to show us why it is true. Often this game has even stronger rules, where your tools for showing “why” are limited in some fashion, say by the techniques of algebra or within the world of ruler-and-compass constructions (i.e. Euclidean Geometry). If you’ve done proofs in school, you may also be familiar with rules that we need to, at this point, discard. These now-unnecessary rules mostly revolve around overly formal approaches to writing proofs: two column formats, specifying every single detail, like in the PBJ sandwich instructions some of you will have done in other classes.

See, the definition of truth in mathematics is neither that of objectivity—what the universe conforms to—or computer science—that which follows from following logic to a T. Truth in math is what you can convince other mathematicians of. And while this doesn.’t share everything with other areas of rhetoric, it borrows the idea that in a good story you try to focus on the important details. Answering the question “why” ends up more like a story than a computer program.

Electronic Writing

Here too, we aim to make baby steps into a more realistic world, one shaped by what life calls for than the game of assessing student progress. For most of you, writing math electronically—not just entering answers into a homework system or a handheld calculator—will be quite new. We undertake it partly to make a habit of learning new things, we are after all in school to learn, yes?, and partly as a baby step into a world where our mathematical thinking is aided by modern technology and responsible to it, becoming a part of other faculties rather than separated from all other thinking by the digital wall.

Concretely, you will be writing these problem sets in Google Docs, turning them in by simply having them in our shared folder. This writing tool provides a lot to help you express ideas that slip from equation to sentence to diagram and maybe even beyond. It also places your work as a part of and easily connected to the web of knowledge which we live among. Since we are not on our own, citing your sources of inspiration, and even remixing them, is but a hyperlink away.

If any of you wish to take more than a baby step into this world, you are also welcome to use LaTeX, RSweave, your own blog, or other similar means—the tools of serious students and professionals—to do your writing. If you plan on majoring in math or physics, this would be a good jump start on things you will need to learn anyway. Please speak with us if you’re interested.

Once you demonstrate an ability to work with electronic media, perhaps for the final portfolio, we will also consider paper craft or other media as substrates for your mathematical ideas in their final form. There is a great history of math and paper craft, and YouTube videos are among our most useful sources to name but two possibilities. Inventiveness in format is not something we seek to preclude, only to postpone as a way of making sure we have time to get to the ideas.

Regardless of format though, when you write in this course, there has to be an intent towards communication. Your message won’t make any sense without knowing how to get the right answer, but the correctness of the answer is not the point; showing us the way there is.

Taking over the class for a day

(AKA Great Theorem Presentations)

The easiest shorthand for this major assignment is “presentations of the great theorems”. And like the answers in the problem sets, this names a central necessary part of what you’re doing but would be a mistake to confuse for the whole thing. You and a couple classmates will be responsible for us on these days, for what we learn, and that’s more than seeing a presentation. But let’s start there so there’s no confusion and its likely not as familiar as getting the right answer to a math problem. So yes, first and foremost, each “presentation” involves you proving for us the major theorem in a chapter of JTG. Each chapter is written to focus on these “great theorems” and our major goal is to understand them. Dunham proves them for us, but as you’ll see this doesn’t mean your only job is to copy him.

At a basic level, though Dunham is an accomplished writer, his approach may not be flawless or wholly appropriate for us. You will want to interpret his work more than reproduce it. This may mean making different judgements about order, nomenclature, which examples to use, how to label or illustrate the work, and so on. Even the artifacts of publication, like the fact that printed books tended to avoid whitespace and diagrams due to cost, are there for you to revise. Your job is to help us understand what Dunham is trying to do, to give us a better version of what he does.

And this extends beyond the “great theorem”. Surrounding each, in JTG, the rest of the chapter aims to provide context to help the reader get a feel for why the theorem is important, what was going on in math and the world that led to or followed from the theorem, and how proving this theorem was a part of the life of the person who first proved it. When it comes to context, it should be far easier to improve on Dunham, at least if you realize that this is your job.

Dunham wrote his book before Wikipedia existed, in an age and for an audience that is gone. Ideas about the themes and participants in math history have changed since he wrote, and there are multiple perspectives to consider. Many of the stories, times, and places only have their surfaces scratched by Dunham, due also to the need for brevity in a small book for laypeople. You may be able to easily find out more about these people, those who Dunham doesn’t mention, whether we know anything about them really at all or where he uncritically passes on myths. Even just digging up a few pictures or more original works would enlarge our view of these worlds.

Class Participation and Reading

The major assignments listed above only make sense within the environment of the larger course, most of which is produced through our shared reading of a central text and meeting times together. This shared endeavor is of critical importance even though there is no due date or percentage derived from them. Quite simply, if we can carry the responsibility we have to each other through making these meetings mean something to each other, then this class will mean something to us all. If we leave it up to me, Bibiana, or the “other students” too keep up, involved, and present, then there’s no point to any of it.

As with the problem sets, and presentations, there is a necessary minimum: showing up and being a part of this course. Attendance is very important, but on its own, not nearly enough. Absences are to be avoided at all costs. If life intervenes and makes attendance impossible, realize that you responsible to us for that time.

Likewise, because this time is ours, I welcome you to make it into something you do value. Ask questions, bring in your interests, and build them into our subjects. If there is no way to make these connections and to create value for yourself in these meetings, you should think seriously about dropping the course. There is no reason someone needs to be stuck here.

Being here, and having this amount to more than just a body in a seat depends on your ability to work through the reading. Especially with math, and especially especially with our deeper approach to math, this means more than passing your eyeballs over the words in a chapter. Reading math is hard and different from other reading. It’s more like reading a comic than a novel or textbook.

1. It’s not linear. Expect to read a bit, realize you haven’t picked up much, and go back. Expect to repeat this cycle.
2. You don’t lean it until you do it. Especially and critically for the mathematical arguments, they literally don’t mean anything until you get out a pen and paper and to to get through the argument.
3. You are unlikely to understand everything until later. The first thing to get good at is finding what you are confused by, enough to ask about. The second is being able to put a pin in it: note the thing that’s a hole and pick up the story on the other side, hoping to fill it in later.
4. You need to bring yourself to the material. How can you care about a book if you don’t have any feelings about what comes next? The development of math is pretty far from a lot of other interests. You will need to find something to attach to and care about to get through the material: parallel history outside math, psychology of genius, how this relates to math you do know, etc.
5. So those are the basics: actually read, and be there, ready to do more with each other. The reason discussing together is the other part of what we do is precisely because the reading, problems, and ideas are difficult taken alone.

Accommodations

If for any reason, elements of this class or the work it requires are difficult to participate in due to a recognized disability or similar difference, let me know how I can accommodate you. My aim is to provide a safe, inclusive environment for all learners. To begin, since this class is not lecture-based, and does not include timed-exams (or really any information storage and retrieval tasks), note takers and extra exam time should not be needed by most students who would otherwise require those in a typical class. However, this general observation may not fit all students. Again, please let me help to make this learning environment work for you.

According to UNM specifically:

Accommodation Statement

In accordance with University Policy 2310 and the Americans with Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. It is imperative that you take the initiative to bring such needs to the instructor’s attention through official channels, as he/she are not legally permitted to inquire. The Accessibility Resources Center (Mesa Vista Hall 2021, 277–3506) provides academic support to students who have disabilities. If you think you need alternative accessible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner.

Title IX Statement

All UNM faculty members are required to inform the Title IX Coordinator at the Office of Equal Opportunity (oeo.unm.edu) of any report they receive of gender discrimination which includes sexual harassment, sexual misconduct, and/or sexual violence. You can read the full campus policy regarding sexual misconduct at policy.unm.edu/university-policies/2000/2740.html. If you have experienced sexual violence or sexual misconduct, please ask a faculty or staff member for help or contact the LoboRESPECT Advocacy Center at loborespect.unm.edu.

Academic Integrity Statement

Each student is expected to maintain the highest standards of honesty and integrity in academic and professional matters. The University reserves the right to take disciplinary action, up to and including dismissal, against any student who is found guilty of academic dishonesty or otherwise fails to meet the standards. Any student judged to have engaged in academic dishonesty in course work may receive a reduced or failing grade for the work in question and/or for the course. Academic dishonesty includes, but is not limited to, dishonesty in quizzes, tests, or assignments; claiming credit for work not done or done by others; hindering the academic work of other students; misrepresenting academic or professional qualifications within or without the University; and nondisclosure or misrepresentation in filling out applications or other University records

Email Use Statement

Students often use email to inquire about protected and sensitive matters, including grades and class progress, and faculty often use email to individually report such protected and sensitive matters. Unless students opt out in writing to the Honors College, the Honors College and Honors Faculty will assume that all email sent individually to students via their official UNM email address (generally their @unm.edu address) is private and confidential and that the student assumes all risk of inappropriate interception of email transmissions. If students opt out of this policy, they are agreeing to only receive such information either in person (students may be required to show identification before information is shared with them) or through regular mail to the student’s official address on file with UNM.

Citizenship and/or Immigration Status

All students are welcome in this class regardless of citizenship, residency, or immigration status. Your professor will respect your privacy if you choose to disclose your status. As for all students in the class, family emergency-related absences are normally excused with reasonable notice to the professor, as noted in the attendance guidelines above. UNM as an institution has made a core commitment to the success of all our students, including members of our undocumented community. The Administration’s welcome is found on our website: http://undocumented.unm.edu/.

Background in Case It’s Helpful Though It Usually Isn’t

Grades in this class are simple enough for us to almost always agree without incident. But somehow this can be seen as suspicious. If you’re wondering, I don’t do this because it’s easier—it’s rather harder, even before I write sections like this my my syllabi. I do it because it’s the right thing to do and the way that I can feel I ask students to put significant work into their learning, to focus on their learning instead of the number of points I’ve taken off, maintain objectivity, and teach students to work with evaluation as it operates most places outside school.

What about points?

Even though it takes some work to get used to a grading system not based religiously on a large number of points, I believe it is worth the effort for a few reasons. First, I am well aware of the biases that plague grading, from structural differences between disciplines to unconscious prejudices that are reflected in unfair practices. We often mistake numbers for objectivity forgetting the numbers that result from uncritical grading practices result in the perpetuation of sexism and racism and the like. I believe it is far more objective to follow the lead that our three point system provides, where the idea is that there should be solid grounds for believing that I, you, my colleagues and any other interested observer would be likely to agree on the grade. In most cases, you will not need me to tell you the grade that you’ve merited.

Second, while numerical grading is the norm at this stage of education and life, overall qualitative assessments are a far more frequent and crucial aspect of evaluation as you get further in life. It is rarely enough to have checked all the boxes you were tasked with filling, and numerical grades set up the situation where we confuse rudimentary completion of delegated tasks with actual achievement.

We use numbers in places like this when we become afraid of each other’s humanity, find it inconvenient, or need something efficient.

Since we do not face the difficulties with numbers, time, and sorting that help numerical grades persist despite the obvious drawbacks, I believe it is my responsibility to you and those around me to engage in a more meaningful, objective, and serious form of evaluation.

How Do You Know How You’re Doing?

I will provide extensive preparation and feedback on individual assignments to arrive at shared understandings of quality as they apply to each specific area of work, and where you stand in relation to excellence and competency. I welcome further conversation on any of these assessments, whether it is a matter of the standards being sought, evaluation of your work, or what is possible and necessary in the way of improvement. At least twice in the semester, I will both ask you to evaluate your own summative performance and I will share my sense of how you’ve done and how you might improve your standing.

While in practice I have found that it has been easy to agree on grades when it comes down to a frank appraisal and conversation, I do not claim to be free from bias, or to always be able to see every sort of dedication. If there is a disagreement about what constitutes “enough for an A” or any other decision between these ultimate three qualities, I welcome you to provide your own assessment and to work out the details together. In the event that we cannot agree or where that conversation is not possible for some reason, I welcome the consultation of a colleague. As in mathematics, the ultimate notion of truth is what we can agree upon.

As part of this, you may find that without numbers, you have forgotten how to assess your performance. My reticence to provide the former should not be mistook for a lack of interest in the latter. While some discomfort is always part of learning, do not let this rise to the level of existential crisis. If you find you are worrying more about your grade than your work, I am always happy to spend the time and effort to alleviate any confusion about your progress, resolve any inconsistencies in what determines quality, and help you find ways to more fully engage in the work. I will begin by asking you about where you feel you stand, and you will need to take stock of the basics of your performance: presence and participation in class or on Slack, completion of assignments, the feedback you’ve received on those assignments, and any earlier conversations we’ve had.

Part of including you materially in this process is that I don’t feel it’s fair to treat me like a grade robot. I do not keep track of where you measure up at all times on the top of my head and in some spreadsheet. Deciding on a grade requires effort, time, and input. I cannot provide a number to you off the top of my head without warning. If you wish to have a progress form filled, or just want to know where you stand, that’s fine, but figure out what you would give you and why, let me have some time, and be ready for a conversation.

If you do need some help thinking about your performance, or are looking for ways to improve, the last week of class is usually too late. Only deciding to care when the grade is imminent and when you have no time left to try harder is usually a good sign that you have not put enough into the course to get an A.

Narrative Evaluation

One way to think about getting an A is if I have a good story to share about what you’ve accomplished and I do not need to temper those accolades with a caveat related to a major shortcoming.

In Honors (also in my High School), in addition to letter grades, we traditionally had a written evaluation system, including a blank space to fill with actual words about what students accomplished. I like telling stories about the good work students do, risks they have taken by trying something new and difficult, and ways they have taken learning into their own hands. Even though this is no longer pat of the official process, it is still at the center of how I think about evaluating your performance.

Thinking a bit more broadly, an A is not the best external gratification you can get from an Honors course. Even better is a recommendation letter to a scholarship committee, admission committee, or prospective employer. The narrative evaluation was like a warm up for these. The qualities I’d like to see you embody, the kinds of things that also relate to the A you would get, however they are expressed, are some of the biggest things people on the other end of recommendation letters want to hear about you.

Not enough A’s

Computing grades in this class again is simple, perhaps so much so that it is easy to miss. No one has yet missed an A who actually put in significant time and effort. But I end up giving fewer A’s than I would hope. If you are at any time—probably the last week of class is a bad time to start though—unsure where you stand or what you can do to do better, I’m only a PM on Slack or a walk down the hall away. If you could use help on any assignment or presentation, I and the other students are always here to help. If for some reason I am standing in the way of your top performance and am not approachable or responsive, please bring someone else to help me understand.