in triplicate

Academic grades, as conventionally understood and used, have three distinct audiences, and they serve a different function for each audience. Grades are problematic largely due to the ways these functions are at odds with each other.

The three audiences are:

• The student. In the context of a single course, assessment and feedback are necessary parts of the educational process. They let the student know how they’re doing as they go along. Traditional letter grades are a crude form of feedback, and where they are used students rarely have a chance to meaningfully respond, other than to try to adapt for the next assessment. Alternative grading systems often treat the entire course duration as a traning period, in which the student can respond to feedback through revision, reassessment, or other forms of demonstrated proficiency following an initial evaluation.
• The institution. Let’s be as generous as possible and assume that the primary goal of the college or university in which a student is enrolled is to educate that student in such a way that they reach their fullest potential. This requires communication among the numerous instructors that student will encounter throughout their studies, and also coordination with the other offices and institutional bodies that are present to support the student. A grade can be a succinct summary to the next instructor in a sequence about how developed are the student’s prerequisite skills. It can also be a signal to the institution about how well the student is progressing in their overall academic career. (This is the reason for “mid-semester” grades that can be used to alert the school when a student needs assistance or intervention.)
• The outside world. Outside of the institution (and inside as well, to an extent), grades become currency that the student can trade for prestige or opportunities. This currency may be in the form of a GPA (to which all of the individual course grades contribute, despite being largely incommensurable with each other) or, for a more granular view, a transcript (which provides the opportunity to craft a narrative about the grades). Of course, not all persons or organizations outside academia value this currency in the same way. But “good grades” provide an inexhaustible supply of recommendations, and “bad grades” are a perpetual obstacle to be minimized or navigated around.

Thus grades are expected to operate at three different social scales, and also at three different time scales. The feedback to a student within a course is short-term, the communcation with the institution is medium-term, and the message to the outside world is long-term (or as they say, permanent). Much of the “objectivity theater” surrounding the assignment of grades is based on the pretence that these three purposes can be fulfilled by a single summative object.

The fact that the third use of grades has both the largest audience and the longest-lasting effects means that it becomes their dominant purpose, their telos. Student anxiety about grades, for the most part, is caused not by an intrinsic dislike of getting feedback about how they can improve their understanding and performance in a subject, but by the belief that in the end what has the greatest practical impact is the final letter or number they can show to the outside world when the course is done. Institutional concerns about “rigor” are based not on the needs of the student, but the needs of the school to present the final scores to the outside world as meaningful indicators of their students’ quality.

Those of us engaged in the practice of developing and implementing alternative grading systems are primarily focused on the first and smallest-scale purpose of grades, providing a useful feedback process to the student. Yet our systems must also interface with the institutional and outside world audiences. At those interfaces lie, in my view, the most difficult ethical questions of grading: how do we support students beyond our time as their instructor? how do we provide an honest evaluation that meets the needs of all three audiences? to whom are we primarily responsible? is the merging of the three functions into a single metric flawed in such a way that it needs to be overthrown?

Honestly, I think (at the time of this writing) that grades are most useful at the institutional level. If it were not for the outward-facing use of grades, they could serve as a quick, qualitative (not quantitative) shorthand in communicating among the internal parts of a college or university what are the needs or successes of an individual student. (To be supplemented by more personalized detail as necessary.) Within a course, as we’ve seen, any number of assessment/feedback systems can work, as long as they’re built on clear communication and building trust in the student-faculty relationship. As for the presentation of grades to the outside world, well, that’s where the dirty work happens.

initial thoughts on teaching in a time of crisis

For those reading in the future, this is the week that the reality of the COVID-19 (coronavirus) pandemic struck the U.S. Hundreds of schools announced that their campuses would be closing, and students would be expected to continue their studies from home, or wherever else they might find to stay. The president of our university and the dean of the college announced on Wednesday that face-to-face instruction would continue through Friday, students would have to leave campus by Sunday afternoon (unless they received an exception), and instruction would resume remotely the following Wednesday. That leaves Monday and Tuesday for redesigning courses and implementing them in a new format. Some instructors managed to get an earlier start before the week ended, but my mind and time were occupied with trying to wrap up well with students in person, and also home life obligations (I have a two-year-old daughter, and my wife is a graduate student with a full-time job), so I’m really just beginning to collect my thoughts.

I am entirely in the camp of those who describe this new mode of teaching as “remote” rather than ”online” learning (and I am grateful that is the language our administration has chosen to use). I do not plan to create an online class, and I do not have any pretensions that I could make a good one in the time available. Even in ordinary circumstances I forget to update course pages on our LMS half the time. I’m focused on what’s going to happen during the class meetings. I’m probably going to have to record a few lectures, but I’ll also look for other video resources that already exist, because me making a video will be either entirely off-the-cuff or hours of planning and scripting. I’m focused on the “remote” part, as a substitute for focusing on class meetings. Students are going to be studying on their own, in a wide variety of settings and living situations. The global upheaval and concomitant personal stresses make it likely that calculus or abstract algebra is not the primary concern in their lives. I want to give them the best chance to learn despite those conditions. The “online” element is present strictly to mitigate the isolation of self-study. Thus I plan to use online elements in ways that will maintain our community and help students feel like their study is meaningful. We’ll have online discussions, to the extent possible given the geographic diversity, and I plan to formulate new projects that will allow students to implement new knowledge in ways that directly affect their understanding of the world. Ideally, these projects themselves will enable assessment of the relevant skills, and I’ll be able to rely less on traditional test formats.

I am not in either camp regarding whether this switch to remote teaching is beneficial or detrimental. In fact, I would describe myself as wholeheartedly ambivalent on that matter. On the one hand, nothing about this situation is normal. Literally the entire world is focused on containing a disease that could kill tens or hundreds of millions of us. Many social institutions (schools, religious groups, local government) have shuttered their brick-and-mortar locations. Friends are instantly made distant, and any challenges to family life are made proximate. On the other hand, I believe my students truly value education. Continuing classes means maintaining some measure of normalcy. It is an element of life where one has some control, unlike almost everything else around us. And I have seen learning of complex mathematics happen in truly extraordinary circumstances, such as during the time that I taught in Peace Corps. (I do not support the argument, made by some in our community, that we can manage this “because we’ve done it before.” A year and a half ago, our college switched to remote instruction for three weeks due to wildfires; that was a more traumatic time, though a briefer one. This represents a fundamental shift in how we complete our courses, not just maintain progress in the short term, and its ramifications for higher education and our university in particular should not be minimized. The fact that, at the administrative level, so many decisions seem to be driven by the threat of future problems with accreditation exceeds my ability to worry.) For now, it is my job, and it is one way that I can contribute to bettering the world.

I am mindful of those for whom this time presents even greater challenges: people who already suffer from loneliness and isolation; people without homes, for whom the loss of public spaces and services will fray an already thin network of support; people in prison, who are neglected in the best of times, despite being in custody of the state; people in immigrant detention centers, who are constantly treated shamefully and forced to live in appalling conditions. My job and my position do not exonerate me from doing what I can to aid them, as well.

“Create Your Own” part 1

Today was the first day of calculus for the fall semester of 2018. As a first-day activity, I wanted to do something that didn’t require any calculus knowledge and could break students out of the mindset that doing math is always about solving particular problems that have been fed to you.

So I initiated a sequence of exercises I’m planning, which I’ve come to think of collectively as “Create Your Own…”. In this case, I gave the following prompt:

The number 1 can be written many different ways, for example 4 – 3 or 10/10.
Come up with ten different expressions that equal 1. Be creative!
Try to have at least four of your solutions involve some kind of algebraic expression, like a variable x.

After they had a few minutes to work individually, I had them share their answers in small groups, and each group picked out what expression by its members they thought was most creative. At the end of class, I collected all of their solutions to look at later in the day.

In having students do this exercise, I learned a lot, and I would definitely do it again, with a few tweaks. Here’s some of what I learned:

• Students judge creativity differently than I do. In looking over the collective work this evening, I saw some excellent examples of splitting 1 into a sum of fractions or decimals and some elaborate expressions involving absolute values or square roots. But the groups often picked examples with the fanciest functions as most creative. Each section had some students come up with cos²(x)+sin²(x) as an answer, and some used logarithms, as in ln(e) or log₁₀(10). And I’m glad those functions were there! It gave us a chance to talk a little about them and for me to give assurance that we would review them at an appropriate time. But 1/10+2/5+1/2 is much more personal, somehow, and I’d like it to have its due.
• This kind of exercise was surprising and unfamiliar. I’m not quite sure how much time I gave for the creative process; I started out in my head with the idea of 2–3 minutes, but that clearly wasn’t enough, so it was probably 4–5. In that time, not everyone came up with ten solutions. (Which is fine! We’d spent an earlier part of the class watching Jo Boaler’s “Four Key Messages” video, which emphasizes that speed isn’t essential in learning math; a couple of students added that comment to their work.) I saw a few get stuck for a while, however, and next time I’ll have some ideas for how to gently prod.
• The notion of “variable” is very strongly connected with “solving an equation”. The vast majority of students interpreted the direction “involve some kind of algebraic expression” to mean “write an equation whose solution is 1.” This led to answers like 2x=2 and x+3=4, and many others (one group gave log₅(x)=0 as an answer!). There was a remarkable amount of creativity in the creation of these equations; I’d like to figure out how to leverage that. But now I also know that the distinction between an “expression” and an “equation” has not yet been made clear, and when we start simplifying algebraic expressions (e.g., to compute limits), we’ll still need to inject some flexibility into our thinking.

The main adjustments I would make next time are:

• Rephrasing the instructions to say that the expressions simplify to 1, rather than equalling 1. Hopefully this will give clearer direction regarding algebraic expressions. Also, I would probably add an algebraic example like (x+1)–x.
• Preparing, nonetheless, for a discussion about what the term “expression” means.
• Giving a more definite and slightly longer period of time, and providing more useful interventions as the students do individual work.

That’s all for now. More updates as warranted.

summer activities

This summer I had a fair amount of travel and conference/workshop activities. I’ve also been working on several projects that need finishing, and of course I have an eight-month old daughter. So I haven’t been blogging, even though several ideas for posts have been kicking around in my head. In order to get something posted, here’s a summary of some major events of the summer.

In May I taught a four-week session of “Transition to Abstract Mathematics”, our introduction-to-proofs course. I had seven students who worked very hard throughout the session. I was pleased that we were able to reach a point where the students could present results they selected from Proofs from THE BOOK during the final exam period.

In June I attended a program on “Teichmüller dynamics, mapping class groups and applications” at the Institut Fourier in Grenoble. (If any of those topics are of interest to you, videos of all the talks are available on YouTube.) I did not give a lecture, but I had a chance to talk with several people about work I did last summer with a Pepperdine student on the topic of “homothety” or “dilation” surfaces. Got a couple of more projects started during this time, to mix into the three or four I was already working on. ¯\_(ツ)_/¯ It was also my first time visiting that part of France, so I traveled with family around the region. A couple of touristic highlights: tasting Chartreuse at the distillery in Voiron, and exploring the Citadel in Sisteron.

In July I participated in an IBL workshop run by the Academy of Inquiry-Based Learning. Four days with 25 enthusiastic teacher-learners and six fantastic facilitators. I started a set of IBL notes to use in the course on complex variables I’m teaching next spring and garnered several new tools and ideas for increasing student activity and engagement in the classroom.

Last weekend I joined the mastery grading session at MathFest. Tons more ideas here! It’s great to be part of so many communities of people who are generating and sharing ideas big and small.

I’ve promised to write at least one more blog post in the next week, for Sam Shah’s Virtual Conference on Mathematical Flavors. So it won’t be quite so long before I post again!

specifications in analysis

Earlier this week, I wrote about expectations for my analysis class this fall (which also apply broadly to upper-level math classes) and some things I learned about specs grading this summer. In this post, I’ll share the specifications I have created for analysis. (I have taught real analysis before, and last time I tried a standards-based approach. Frankly, that basically turned into a point system, albeit a simplified one, which is why I’m trying something completely different this time.)

The rest of the post is taken verbatim from (the current draft of) my syllabus.

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Effective learning requires effective methods of assessment. The assessments should relate as directly as possible to the expectations of the class, and they should provide both feedback on how to improve and opportunities to demonstrate improvement as the semester progresses. In my experience, “traditional” grading schemes based on assigning points or percentages to individual tasks do not serve these functions well. Therefore, this course adopts specifications grading*, in which grades are tied to specific outcomes. This is likely to be different from grading policies in other classes you have taken, so feel free to ask me questions or let me know if you have concerns. I hope that this system will make clear the connections between the expectations stated in the previous section and the ways you will be assessed.

Overall grading. At the end of the semester, I am required to submit to the university a letter grade reflecting your achievement in this class. That grade will be determined on the basis of a set of specifications in four areas: (1) class participation, (2) written proofs, (3) exams, and (4) synthesizing activities. Each of these areas will receive a simple grade of A, B, C, D, or F. The following sections describe how these grades will be determined. Your final grade will depend on your performance in all four areas, according to the following table.

Final grade	based on individual grades of
A	all As, or 3 As and 1 B
A–	two As and two Bs
B+	one A and three Bs
B	all Bs, or 3 Bs and 1 C
B–	two Bs and two Cs
C+	one B and three Cs
C	all Cs, or 3 Cs and 1 D
D–	two Cs and two Ds

I will use my discretion to assign a final letter grade to other combinations of individual letter grades.

Class participation. Attendance at every class meeting is required. Most weeks, we will alternate days between discussing reading assignments and presenting solutions to exercises. The end of this syllabus has a schedule of what we will be doing in class each day (with allowance for adjustments, as needed).

Reading. In order to participate effectively on discussion days, you will need to read the textbook before coming to class. Each reading assignment is about 10 pages. The textbook attempts to be very accessible, but that does not mean it is easy. We will be working with ideas that stretch reason and imagination. You should be prepared to spend at least 1–2 hours on each reading assignment; rereading pages, paragraphs, or sentences; working out examples; and writing questions or comments in the margins or on separate paper. You should be especially mindful of definitions. These are not always set apart from the text, so pay attention when new vocabulary is introduced. Start working on a list of definitions and theorems from the start of the semester. The chapter summaries can be an aid in this process.

Collaborating. On days with a reading assignment, you will work in small groups to discuss the material. I will assign these groups at the start of each week. You should bring your own questions and thoughts to these discussions. If there is extra time, you can also discuss the current set of exercises.

Presenting. On the remaining days, you will take turns presenting solutions to exercises distributed previously. The solution you present does not necessarily need to be entirely correct, but it should show evidence of a serious effort. You should also be prepared to answer questions from me or other students. To maintain balance, no one will be allowed to present more than once every two weeks, unless every student in the class has already presented during that time period. In exceptional cases, some of these verbal presentations may be made to me outside of class (no more than one per student).

To earn a	you must do the following
D	attend at least 75% of class meetings present at least one proof in class
C	attend at least 85% of class meetings and contribute to discussions present at least three proofs in class
B	attend at least 90% of class meetings and contribute to discussions present at least four proofs in class
A	attend all class meetings (2 unexcused absences allowed) and contribute to discussions present at least five proofs in class

Written proofs. Over the semester, you will develop a portfolio of work that you have submitted for formal assessment. Most of your contributions will be proofs. Each week I will indicate one or more exercises whose solutions could be submitted to your portfolio. You may discuss your work with other students in the class, to have them check whether it meets the standards of the class and give you feedback. A proof for the portfolio is due the Monday after it is assigned. These proofs must be typed using LaTeX, Google docs, Microsoft Word, or another system.

When you submit a written proof for your portfolio, I will judge whether it is Successful, Quasi-successful, or Unsuccessful (see the earlier section on “Proofs” under “Expectations” for details about these ratings), and mark it correspondingly with one of S/Q/U. Proofs marked Q or U will not be counted towards your grade. However, proofs can be resubmitted at the cost of one or two of your allotted tokens; see section on “Tokens” below.

To earn a	your portfolio must contain
D	at least four successful proofs
C	at least six successful proofs
B	at least eight successful proofs
A	at least ten successful proofs

Exams. There will be two midterm exams and a final exam. Each one will have a take-home portion and an in-class portion. [Dates and times, listed in syllabus, omitted here.]

The take-home portions will consist of two or three proofs that you are to complete on your own, without consulting other students. (You may discuss your work with me before turning in the exam, although I might not answer questions directly.) These will be judged as successful, partially successful, or unsuccessful, like the proofs in your portfolio. They cannot be resubmitted after grading, however.

The in-class portions will test your mastery of definitions and the statements of theorems. You will need to be able to state both definitions and theorems properly. You will also be asked to recognize and provide examples of situations or objects where a definition or theorem does or does not apply.

To earn a	you must do the following
D	correctly answer 60% of in-class test questions write at least two successful proofs on take-home exams
C	correctly answer 75% of in-class test questions write at least three successful proofs and one quasi-successful proof on take-home exams
B	correctly answer 85% of in-class test questions write at least four successful proofs and two quasi-successful proofs on take-home exams
A	correctly answer 95% of in-class test questions, write six successful proofs on take-home exams

Synthesis. To master the ideas of the class, you must spend time synthesizing the material for yourself. The items in this graded section will be added to your portfolio, to complement the proofs. All materials in this section must be typed using LaTeX, Google docs, Microsoft Word, or another system.

List of definitions and theorems. It should be clear at this point that being able to produce accurate statements of definitions and theorems is essential to success in this class. To encourage you to practice these, I am requiring you to create a list of these statements for the entire course. Your list should be organized in some way that makes sense to you—e.g., alphabetically or chronologically.

The textbook can be used as a reference, as can the internet, but how do you quickly recall what definitions we’ve used and how they're related? How do you find the phrasing of a theorem that’s become most familiar? This list should help you in these situations. More importantly, creating it will help you review and organize the material in your own mind.

I will verify your progress on these lists at each in-class exam.

Papers. Twice during the semester, once in the first half and once in the second half, I will provide a list of topics that we have been discussing, from which you can choose to base a paper on. These will be due approximately two weeks after the midterm exams.

There is a third paper that can be completed at any point in the semester on a topic of your choosing, but you must get the topic approved by me before Thanksgiving.

These papers will for the most part be expository, meaning they will present previously known mathematical results (not original research). Here are the requirements for a paper to be acceptable:

• It should have 1500–4500 words.
• It should use correct grammar, spelling, notation, and vocabulary.
• It should be organized into paragraphs and, if you wish, sections.
• It should cover the topic clearly and reasonably thoroughly, with an intended audience of other math students (who may be assumed to have studied as much analysis as you).
• It should contain a proof of at least one major result.
• The writing should be original to you. Of course, small pieces like definitions may be taken directly from another source, but apart from these the paper should be your own work.
• Citations are generally not necessary in expository mathematical writing, except for the following: a statement of theorem that you are not proving, a peculiar formulation of a concept/definition, or a creative idea (e.g., an uncommon metaphor or illustration) from another source.
• You may choose to follow the style of our textbook, or a more formally structured math textbook, or something more journalistic or creative, as long as the previous criteria are met.

Papers that do not meet these criteria will be considered unsatisfactory and will not count towards your grade. An unsatisfactory paper can be revised and resubmitted at the cost of three tokens.

To earn a	you must do the following
D	create a list of definitions and theorems to include in your portfolio
C	create a list of definitions and theorems to include in your portfolio write a paper on one of the topics provided
B	create a list of definitions and theorems to include in your portfolio write two papers on the topics provided, one during each half of the semester
A	create a list of definitions and theorems to include in your portfolio write two papers on the topics provided, one during each half of the semester write a third paper on a topic of your own choosing related to the class

Tokens. You start out the semester with seven (7) virtual “tokens,” which can be used in various ways:

• One token allows you to resubmit a written proof initially judged to be quasi-successful (must be used within one week of initial grading).
• Two tokens allow you to resubmit a written proof initially judged to be unsuccessful (must be used within one week of initial grading).
• Three tokens allow you to resubmit an unsatisfactory paper (must be used within one week of receiving paper back).
• One token gives you a 48 hour extension past the due date for a paper.

Unused tokens may be exchanged for a prize at the end of the semester. [maybe?!?]

*Based on Linda Nilson’s book Specifications Grading: Restoring Rigor, Motivating Students, and Saving Faculty Time.

taking specs seriously

I’ve been an advocate of standards-based grading since I started using it over three years ago. It has addressed many of the concerns I had about the dominant point-based grading system and encouraged students to move forward in their understanding rather than feeling trapped by past performance.

I’m not solely an SBG proponent when it comes to grading, however. For one thing, I find it hard to adapt SBG to upper-level math courses. For another, the time seems ripe for experimentation in grading practices as more of us realize the shortcomings of what we have inherited from decades past. Not that we should constantly reinvent the grading process, but we should be open to various thoughtful ways of providing authentic assessment.

So I was certainly interested a couple of years ago when several fellow instructors began talking about specifications grading, a method espoused by Linda Nilson in her book Specifications Grading: Restoring Rigor, Motivating Students, and Saving Faculty Time. I adopted some of the ideas I heard and appreciated the increased flexibility it offered.

However, it was not until this summer that I read through Nilson’s book. It was useful because it seems Nilson and I think differently in ways I can’t quite put my finger on, and so the book has lots of ideas I would not have intuited on my own. Here are a few of the things I garnered from reading the book that I hadn’t picked up from online discussions (not that these things weren’t said, but this time they stuck):

• Sometimes it’s OK to use percentages. I’ve been highly points- and percentages-averse since starting SBG. Percentages, my argument went, were essentially meaningless, because they’re constantly being curved (so they don’t really represent a “percentage” of anything) and the difference between 80% and 81% is essentially a coin toss (so they aren’t as linearly ordered as people like to think). But that argument isn’t uniformly true. In a course where precision is important, it is possible to measure, for instance, how many definitions a student can correctly state. For my upcoming analysis class, I expect “A” students to get definitions right 95% of the time, “B” students 85% of the time, “C” students 75% of the time. This really is quantifiable, and a definition is either correct (with respect to the established conventions of the subject) or not, so each one can be graded yes/no. As long as not everything is forced into a percentages model, this can be an effective way to give feedback.
• Make students work for an A, but give them some choice in how to get there. As instructors, we want an A to represent mastery, an indication that the student can think nimbly and complexly about the subject. Ideally, students who earn an A will be the ones most invested in the subject. To demonstrate all this, students should have ownership of their work. They should make meaningful choices that reflect their interests and their skills as well as the subject at hand.
• Not everyone has to do everything. This is closely tied to the previous point. Nilson uses the metaphor of “hurdles”: grade levels can be differentiated by having students clear either higher hurdles (more complex, better quality work) or more hurdles (more extensive work), or a mix of the two. I’m not generally a fan of having students earn higher grades by just proving they can do more—that takes more of my time, and more of theirs. But true mastery requires a measure of initiative. Having a small number of optional assignments that give students opportunities to distinguish themselves makes sense as part of a larger grading scheme.
• There are good reasons to limit reassessments. Of course, one of these reasons is the subtitular “saving faculty time.” In past upper-level classes where I’ve allowed essentially unlimited resubmission, I’ve been swamped/behind at several points in the semester as students frantically tried to get something accepted. But that’s not even the best reason. By limiting reassessments and grading work pass/fail (or pass/progressing/fail or some other variant), students are encouraged to submit their best work each time, and to spend extra time making sure they check its quality before asking me to do so. The onus is on me to establish clear expectations, and on students to meet them. We’re not negotiating what’s acceptable through repeated revision and grading.

I also found the chapter on cognitive models (Chapter 3, “Linking Grades to Outcomes”) helpful in considering what it means to have a higher level of mastery; previously I wasn’t really familiar with anything beyond Bloom’s Taxonomy.

If this post was of interest to you, I hope you’ll consider joining the Google+ Community on Standards-Based and Specifications Grading” (SBSG) Slack workspace on mastery grading, where teachers of diverse disciplines are meeting to discuss how to implement these two particular alternative forms of grading.

Tomorrow I’ll share my full set of specifications for real analysis.

expectations in analysis

I’m working on the syllabus for my (junior and senior level) analysis class this fall, and I’d like to share some parts of it, hopefully thereby eliciting feedback. The main thing I’m concerned about is the type of specifications grading I’m adopting for the class—I’ll share that later this week. This post is about establishing the expectations of the course, on which the specifications will be based. None of these are particular to analysis; they establish what I believe any student in an upper-level mathematics course should achieve.

The rest of the post is taken verbatim from (the current draft of) my syllabus.

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To learn mathematics, it is essential to engage actively with the material. This is especially true at this stage in your mathematical careers, as the focus of study shifts from developing computational tools to examining underlying concepts and practicing abstract reasoning. This shift may be described as a move from pre-rigorous thinking, which is informal and intuitive, to rigorous thinking, which is formal and precise. (This terminology has been suggested by mathematician Terence Tao; he also includes a post-rigorous stage, in which professional mathematicians work, where one is able to make intuitive arguments that are grounded by formal training.)

The content of this course resides in definitions, theorems, and proofs. You will be expected to state both definitions and theorems accurately and to illustrate them through examples. Mathematics is not merely a collection of disconnected facts, however, and so you will also develop your logical skills by proving mathematical truths, linking definitions to their profound consequences captured by theorems. All of this will happen in the context of a community—two really, our class and the larger mathematical community.

Definitions. In mathematics, as in other sciences, it is necessary to quantify what is being studied and to be able to identify what is of interest at each moment. This is done by carefully establishing and internalizing definitions. This is not to say that definitions do not involve creativity; as a subject develops, often definitions evolve to encompass more or fewer cases, to be more precise, or to reorganize ideas.

By the end of the course, you should be able to:

• state definitions accurately and explain any notation or previously-defined terms they contain;
• identify whether or not an object meets the conditions of a given definition;
• give examples that satisfy a given definition as well as examples that do not satisfy it;
• test an unfamiliar definition using examples;
• create new definitions when needed.

Theorems. A theorem has two parts: the antecedent (its assumptions) and the consequent (its conclusions). To take a familiar example, the equation \(a^2 + b^2 = c^2\) by itself is not a theorem; rather, the Pythagorean Theorem states that “If \(c\) is the length of the hypotenuse of a right triangle, and \(a\) and \(b\) are the lengths of its other two sides, then \(a^2 + b^2 = c^2\).” A theorem may not always include the words “if” and “then,” but you should always be able to determine what are the antecedent and the consequent. Sometimes rephrasing the theorem’s statement can help. For example, “Every differentiable function is continuous” can be rephrased as “If a function is differentiable, then it is continuous.” In most cases, the consequent does not imply the antecedent (e.g., not every continuous function is differentiable). A theorem that says one set of conditions holds “if and only if” another set of conditions holds is logically making two statements (the antecedent and consequent can be reversed), and both must be proved.

By the end of the course, you should be able to:

• state theorems accurately and identify what are their assumptions and their conclusions;
• determine whether the conditions of a theorem do or do not hold in a given situation, explain why, and determine what the theorem does or does not imply in that situation;
• recognize logically equivalent forms of a theorem;
• formulate and test conjectures.

Proofs. Proofs are how we as individuals and as a community determine the truth of mathematical statements, i.e., theorems. Here is one definition of a proof, due to David Henderson: A proof is “a convincing communication that answers -- Why?” The extent to which a proof succeeds, therefore, depends on how well it embodies these three properties: it should be logical (does it convince?), it should be comprehensible (does it communicate?), and it should be intentional (does it answer why?). Evidently, each of these properties depends somewhat on the others. It is thus reasonable to classify proofs into an S/Q/U system:

• (S) A successful proof makes an argument for the truth of a mathematical statement that is fully convincing to an informed reader or listener. It employs appropriate vocabulary and carefully chosen notation. It avoids sloppy reasoning. It makes clear use of the theorem’s assumptions and, when necessary, previously known results. The best examples provide motivation for the methods chosen. Minor revisions may be advisable, but they do not hinder the overall effectiveness.
• (Q) A quasi-successful proof contains most of the ideas necessary to make a complete argument. It may have slips in logic or notation, or it may neglect a special case, or it may be hard to read. It contains sufficient evidence, however, that the argument can be “salvaged” by filling in gaps or clarifying language. Serious revision is necessary. [Not in syllabus: thanks to Dan for suggesting “quasi-”.]
• (U) An unsuccessful proof does not convince an informed person of the truth of the purported theorem, for one or more of the following reasons: – It makes logical leaps or omits key ideas. – It demonstrates incomplete understanding of definitions or notation. – It fails to reference previous results when appropriate. Complete revision is generally necessary.

In other words, a successful proof is of sufficient quality that it could reasonably be accepted as part of a paper in a professional journal. A quasi-successful proof has some merit, but it requires revision, after which it might or might not be acceptable at a professional level. An unsuccessful proof is sufficiently flawed that it would not be acceptable as part of a professional publication.

By the end of the course, you should be able to:

• evaluate, on the basis of professional standards, whether a given proof is successful or not;
• write original, successful proofs.

Community. Our class time will be structured primarily around discussion rather than lecture. The idea is to have a space that promotes sharing ideas, making guesses, taking risks, and sharpening our reasoning abilities. I will guide and facilitate these conversations, but everyone is responsible for contributing to discussions, both in small groups and with the entire class. That is, in this course mathematical authority resides not just with me as the instructor, but with every class member. I will give short lectures (20 minutes) when the entire class agrees it would be beneficial, but not more often than once a week.

By the end of the course, you should be able to:

• engage in discussions about mathematics by sharing questions, proposals, and insights;
• evaluate others' contributions critically and respond constructively;
• present your own work in front of an audience and address their comments and questions.

using calculus to understand the world

In my last post, I wrote about how I returned to teaching related rates in my calculus class and ranted a bit about the inanity of most related rates problems. There I mainly discussed the difficulty in reading the statement of such problems and how to make the questions they raise seem more natural. I’d like to expand on this theme with some more examples.

One feature of mathematics that doesn’t get emphasized enough, IMHO, is that it is a science, and as such is based in observation. Often, either we lead students through abstract reasoning to a previously unanticipated result, or we prove things that are so self-evident that the notion they need proof is itself baffling. Now, in the world of professional mathematics, it is true that even apparently obvious facts need proving (remember that “to prove” just means “to test”), and we often do get excited when we are led to something unexpected and beautiful. That is because we have learned how to use and trust our logical skills to examine the truth of something, and we delight in the uncovering of new truth by means of those skills. But even when a result is surprising to an audience, and even if it was at first surprising to the speaker, it is no longer so. A mathematical researcher plays with ideas until she notices something interesting, and then she tries to understand why it is so. That’s the exciting part of math, and that is what I believe we can share with our students through the process of modeling.

My goal in teaching related rates has become to ground as many questions as possible in direct observation. When I ask about the sliding ladder, as I described in my last post, before setting up the math but after asking students what they think will happen, I demonstrate by leaning a ruler against a book and slowly pulling the bottom end away. What one notices in this experiment is that the top end of the ruler moves very slowly at first, and very quickly just before reaching the ground. The speed in the final moment is so great that one is tempted to think the person pulling the bottom end lost control, and gravity took over. (This is even more credible when using the much larger, heavier ladder in a demonstration.) But the math shows that even if the person keeps complete control and moves at precisely the same speed, the same effect will occur. Let’s see why.

The exact length of the ladder doesn’t matter, of course, so call it $L$. If $x$ measures the distance from the wall to the bottom end of the ladder and $y$ measures the distance from the floor to the top of the ladder, then we have $x^2 + y^2 = L^2$. Then we differentiate both sides with respect to time and get $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$, or \[ \frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt}. \] At this point most related rates problems would ask you about the size of $dy/dt$ for some particular values of $x$, $y$, and $dx/dt$, but look at how much we can determine just from this related rates equation: when $y$ is larger than $x$, the top end is moving more slowly than the bottom end, and conversely when $x$ is greater than $y$, the top end is moving more quickly than the bottom end. There is just one moment when the two ends are moving at the same speed, which is when $y = x$, or in other words, when the ladder is at a 45 degree angle. And as the distance between the top end and the floor approaches zero, the speed of the top end approaches infinity. (Not physically possible, of course, but it explains why there’s such a quick movement at the end of the process.) There; now I feel like I’ve learned something!

I have five more examples to illustrate how much more interesting I think related rates are when tied to direct observation. This will probably belabor the point, but unfortunately these examples are also stripped of any interest by focusing too much on a single moment in time, which is what every standard textbook does with them.

The next example involves inflating a balloon. This, again, is easy to demonstrate. I can’t take a deep enough breath to fill the whole balloon at once, but even with two puffs, exhaled at a near-constant rate, it’s obvious to students that the size (i.e., diameter, or radius) of the balloon grows more quickly at first, then more slowly. Anyone who’s worked with an air or helium tank has surely experienced this phenomenon. Why is this happening? And how much more slowly is the diameter increasing as time goes on? Here there’s very little modeling involved; essentially the entire model is provided by assuming the balloon is a sphere and using the formula for the volume of a sphere in terms of its radius, $V = \frac{4}{3}\pi r^3$. Differentiating with respect to time gives the relation $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$, so \[ \frac{dr}{dt} = \frac{1}{4\pi r^2}\frac{dV}{dt}. \] Some books reach this equation, but as with the ladder they again jump to plugging in values for a specific time, rather than noting the following: if $dV/dt$ is constant, then $dr/dt$ is inversely proportional to the square of the radius! And even more, $4\pi r^2$ is the surface area of a sphere with radius $r$, so this relationship between rates is directly related to the fact that the derivative of the volume of a sphere with respect to its radius is the surface area! That is, the size of the surface of the balloon is what determines, together with the rate the volume is increasing, how quickly the radius is increasing.

A similar phenomenon happens with the standard filling-an-inverted-cone problem. The demonstration here involves a martini glass and some colored water. (As I promise my students when doing this experiment, it’s just water.) My martini glass is about 12 cm across on top, and about 8 cm deep, giving it a volume of 300 milliliters. (That’s about 10 ounces, the size of two regular martinis—you don’t want to fill this glass with gin and drink it too quickly.) Having a nice big glass is useful for this demonstration: if I pour at a constant rate, the water level rises much more slowly near the top than near the bottom. The math shows just how much more slowly. The volume of a cone with height $h$ and base radius $r$ is $V = \frac{1}{3}\pi r^2 h$. From the geometry of this situation (using similar triangles, for instance), for my glass the radius of the surface of the water is always three-quarters of the water’s depth (here interpreted as height). We could use the relation $r = \frac{3}{4}h$ and substitute into the volume formula to get rid of the variable $r$, but there’s also no harm (as my students taught me) in differentiating first, using the product rule: \[ \frac{dV}{dt} = \frac{\pi}{3} \left( 2rh\frac{dr}{dt} + r^2\frac{dh}{dt}\right). \] Notice that this formula is valid for all cones varying in height, radius, and volume, whether or not the height and radius are linearly related at all times. The most obvious quantity of interest (assuming constant $dV/dt$) is $dh/dt$. From $r = \frac{3}{4}h$ we get $\frac{dr}{dt} = \frac{3}{4}\frac{dh}{dt}$, and also $h = \frac{4}{3}r$. The reason to solve for both of these quantities is that, by keeping both $dh/dt$ and $r$ in the equation and substituting out $h$ and $dr/dt$, we get $\frac{dV}{dt} = \frac{\pi}{3}\left(2r^2\frac{dh}{dt}+r^2\frac{dh}{dt}\right)$, or, after solving for $dh/dt$, \[ \frac{dh}{dt} = \frac{1}{\pi r^2} \frac{dV}{dt}. \] First of all, the ratio between the height and the radius has disappeared, so this formula now works for any inverted cone, not just my martini glass. And second of all, just as with the balloon, the rate at which the height increases depends on the “surface area” that is expanding, which in this case is just the base of the cone! Thus, again, the reason the water level rises more slowly near the top of the glass has a clear geometric interpretation. (Here’s a real-world application: I argue this works to the benefit of bartenders, who can pour into a martini glass fairly quickly without risk of overflowing, because the beverage level rises slowly near the top of the glass.)

I took the next two examples from Cornell’s Good Questions Project; come to think of it, it may be these questions that first planted in my head the idea of looking at related rates problems over time, without numbers. The situations are again standard for related rates problems, but the conclusions are much more interesting than a single rate at a single moment.

Consider an actor (say, Benedict) on a stage, illuminated by a light at the foot of the stage. Benedict casts a shadow on the back wall; how does the length of his shadow vary if he walks towards the light at a constant speed? The demonstration of this situation is particularly exciting, because you get to turn off the classroom lights, pull out a flashlight and a doll or figurine, and watch what happens to the shadow of the doll/figurine/actor on the wall as it moves towards the flashlight. Students observe that at first the shadow grows slowly (when the figure is close to the wall), then more quickly as he approaches the light. Modeling this situation generally provides the first major geometric hurdle for my students, because it involves the imagined line that emanates from the light, passes by Benedict’s head, and finally reaches the back wall, thereby determining the height of the shadow. (I wonder if many of them have never thought about the geometry of how shadows relate to the objects that cast them.) I’ll let the reader work out the fact that, if Benedict’s height is $h$, the distance from the light to the back wall is $D$, the distance from Benedict to the light is $x$, and the height of the shadow is $s$, then $\frac{s}{D} = \frac{h}{x}$. (Hint: use similar triangles.) Here the only variables are $x$ and $s$, so the related rates equation is \[ \frac{ds}{dt} = -\frac{hD}{x^2}\frac{dx}{dt}. \] Students are at first perplexed by the negative sign: shouldn’t the shadow be increasing? If so, why does its derivative appear to be negative? Then they realize: ah, if Benedict is walking towards the light, then $dx/dt$ is negative, so $ds/dt$ is in fact positive! And so it becomes clear that the height of the shadow increases much more rapidly when Benedict is near the light than when he is near the wall. (I generally give specific values for the height of the actor and the distance from the wall to the light in this question, so that it’s more obvious which values are constant.)

I don’t have a standard demonstration for this next problem, because I use it as a quiz question (although maybe not anymore, now that I’ve written about it here), but it’s easy enough to devise an experiment. This situation is similar enough to the previous one that its result is a bit surprising. Suppose a streetlight at height $L$ is the only source of illumination nearby, and a woman (say, Agatha) of height $h$ walks at a constant speed away from the light. As she gets farther away from the light, does her shadow grow more quickly, more slowly, or does it grow at a constant rate? If $x$ again denotes the distance to the light (well, really from Agatha’s feet to the base of the lamp, which is not the same as her distance to the source of illumination), and $s$ is the length of Agatha’s shadow, then similar triangles produce the relation $\frac{s}{h} = \frac{s + x}{L}$. We can rearrange this into a simple proportion between $s$ and $x$: $s = \frac{h}{L-h} x$. (Here’s an interesting feature of this equation already: it only makes sense if $h < L$, that is, if Agatha is shorter than the lamppost!) Now we differentiate to get \[ \frac{ds}{dt} = \frac{h}{L - h} \frac{dx}{dt}. \] So if Agatha’s speed is constant, then her shadow’s length is also increasing at a constant rate. This example shows especially well why it’s dumb to look at related rates at a single moment in time. Most book exercises of this sort ask how quickly the shadow is growing when Agatha is at a particular distance from the lamp. But it doesn’t matter how far away she is, and the math proves that it doesn’t matter.

There’s a risk in related rates exercises to always resort to problems that only involve differentiating polynomials, so here’s an example that uses trigonometric functions. The demonstration I use: I walk back in forth in front of the class and tell the students to be mindful of what their heads do as they follow my movement. After a couple of times, several of them observe that their heads must turn more quickly when I’m closer to them. I point out that this is something anyone who’s had to run a video camera at a race must be aware of. (It’s also apparent to someone riding in the passenger seat of a car, keeping their gaze fixed on a single tree or other immobile object: for a long time, your head turns little, but when you’re close to the object, you have to turn quickly to keep it in view.) I generally set up the problem on the board as though it is taking place at a racetrack. Suppose a runner is moving along a track (let’s assume it’s straight for simplicity) at $v$ feet per second. You’re watching from a position $D$ feet away from the track. How quickly does your head need to turn to keep following the runner? The answer depends on how far away the runner is. One has to introduce a reasonable coordinate system and some useful variables: good choices are the position $x$ of the runner relative to the point of the track closest to you, and the angle $\theta$ by which your head is turned from looking at this closest point. Then we get the relation $\tan\theta = \frac{x}{D}$, and differentiating with respect to time results in the equation $\sec^2\theta \frac{d\theta}{dt} = \frac{1}{D} \frac{dx}{dt}$, or \[ \frac{d\theta}{dt} = \frac{v}{D} \cos^2\theta \] (using the assumption that $dx/dt = v$). When $\theta = 0$, so that the runner is closest to you, the rate at which your head turns is $v/D$, which depends only on how fast the runner is going and how far away from the track you are. (Notice that the units work out: the radian measure of an angle is technically dimensionless, and so we expect its rate of change not to have any dimension other than 1/time. Since $v$ has dimension of distance/time and $D$ has the dimension of distance, $v/D$ has the dimension 1/time.) As $\theta$ increases (in this scenario, $\theta$ is never greater than a right angle), the change in the angle of your head to follow the runner happens more slowly, because $\cos^2\theta$ is closer to zero.

These are just a few examples of standard situations involving related rates that become much more interesting when the myopic attention to a single moment in time is removed. I’m sure most readers of this post can do the calculations I’ve shown on their own, but the tendency to hone in on a single rate at a single point in time is so entrenched that I wanted to show how much more interesting related rates become when that element is removed. I don’t know that my students are better at solving related rates problems than other students, but I have noticed that they’re much less likely to insert specific quantities into a relation before it’s necessary than when I taught the subject years ago. I haven’t had time to strip all such problems of the detritus that comes with wanting a numeric answer, but I believe our understanding (and our calculus students’ understanding) of the world will be much improved by making the effort to transform these problems into meaningful questions.

Here are two other examples that I won’t work out in detail. One scenario has a boat being pulled into a dock by a rope attached to a pulley elevated some distance above the boat. If the rope is pulled at a constant rate, the boat in fact speeds up as it approaches the dock! (I tried demonstrating this once with a string tied to a stuffed animal pulled across a desk, with moderate success.) Another common type of problem considers two boats moving in perpendicular directions (or cars moving along perpendicular roads), and asks at a certain point in time whether the distance between them is increasing or decreasing. That’s silly. Why not establish the relation between them, and ask at what times the distance is increasing, and at what times the distance is decreasing? If there’s a time when the rate of change in distance is zero, then the boats (or cars) are at their closest (or farthest) positions, which connects to the study of optimization, which has its own set of issues…

P.S. I should have known better than to look at Khan Academy’s treatment of related rates. His videos show all the marks of what is classically wrong with these problems: the irrelevant information of what variables equal at a single moment in time is presented up front along with everything that’s constant in the situation, and in the end the answer is a single, uninformative number. Even when an interesting equation is present on the screen, Khan rushes past it to get to the final number. How can we get our students to ask and answer more interesting questions than these, about the same situations?

formative assessment isn’t scary

I get a little jumpy around nomenclature. This probably comes from being a mathematician; we spend a lot of time coming up with names for complex ideas so that they’re easier to talk about. Naming a thing gives you power over it and all that. So when we come across a new name, it could take anywhere between a few minutes and a few months to unpack it. An abelian group, for instance, can be completely and formally defined very quickly, whereas a rigorous definition of Teichmüller space often takes several weeks in a course to reach. Some things are in between, easy to define but not-so-easy to figure out why the object has a special name (see dessin d’enfant). Very often a major step along the way to understanding something is grasping the simplicity—the inevitability, even—of its definition.

So it is with formative assessment. When I first learned about the formative/summative assessment distinction, I got nervous. I thought, “So, besides giving tests and quizzes, I need to be doing a whole bunch of other things in class to find out what students are thinking? How much more class time will this take? How much more preparation will it take? How will I ever incorporate this new feature into my class, and how bad will it be if I don’t manage to?” I think I got caught up in the impressiveness of the term assessment; that seemed like a big “thing”, and doing any kind of assessment must require a carefully crafted and substantial process.

So let’s back up a bit. In teaching, assessment means anything that provides an idea of students’ level of understanding. If it’s not graded, it’s formative.

That’s it.

As a teacher, unless you have literally never asked “Are there any questions?”, you have done formative assessment. Asking “Are there any questions?” is a crude and often ineffective means of formative assessment, but it is assessment nonetheless. You and I are already doing formative assessment, which means that we don’t have to start doing it; we can instead turn to ways of doing it better. Somehow I find that easier.

“Formative assessment” is more like “abelian group” than “Teichmüller space”. If you have ever added integers, you have worked with an abelian group. But having an easily-grasped definition doesn’t have to mean than a concept is limited. In fact, simple definitions can often encompass a broad range of ideas, which happen to share a few common features. There are entire theorems and theories built on abelian groups. Naming a thing gives you power over it. Now that we’ve named formative assessment, let’s see how we can build on it.

David Wees has a collection of 56 different examples of formative assessment, which range from the “Quick nod” (“You ask students if they understand, and they nod yes or no”—possibly virtually, which enables anonymity) to “Clickers” to “Extension projects” (“Such as: diorama, poster, fancy file folder, collage, abc books. Any creative ideas students can come up with to demonstrate additional understanding of a topic.”) John Scammell has a similar collection of Practical Formative Assessment Strategies (some overlap with Wees’s list), grouped into sections like “Whole Class Strategies”, “Individual Student Strategies”, “Peer Feedback Strategies”, “Engineering Classroom Discussion Strategies”, and so on.

Formative assessment doesn’t have to take much time or preparation. You’re probably already doing it without realizing it. Adding some variety to the methods of assessment, however, can provide a more complete picture of students’ understanding, to their benefit. Feel free to add more resources in the comments.

a reflection on course structure, and standards for calculus

Here’s what I’ve learned about writing standards: it’s hard to get them balanced properly. This challenge is inherent in developing any grading system. I used to fret about whether quizzes should count for 15% or 20% of the final grade; now I fret about whether the product, quotient, and chain rules should be assessed together or separately. (I’m happier trying to solve the latter.)

Another challenge is in setting up standards so that assessments have some coherence. I’ll explain. My first couple of times creating standards, I sat down and made a list of all the things I wanted my students to be able to do by the end of the semester, grouped into related sets, with an eye towards having each standard be of roughly equal importance (as I mentioned in the previous paragraph). After all, that’s what standards are, right? All the skills we want students to develop? That done, I told myself, “Okay, now every assessment—every homework, quiz, and test—will have to be graded on the basis of items in this list.” In principle, it’s nice to have this platonic vision of what students should do and know, including all the connections between related ideas (parametrization means imposing coordinates on an object; it doesn’t really matter what dimension it has, so parametrizing curves and surfaces should go together as a single standard). However, while this list said a lot about what I thought students should do, it didn’t say much about what I was going to do. It didn’t fit the structure of the course, just of the ideas (oh, wait, we’re parametrizing curves in week 2 and surfaces in week 10—why didn’t I notice that before?). Looking back, I can see that a lack of contiguousness within a standard does reflect a conceptual distinction between the concepts involved (hmmm, maybe the idea of drawing a curve through space is conceptually different from laying out a coordinate system on a curvy surface). I ended up assessing “partial” standards at various points in the semester, which is absurd on the face of it. It’s one thing to assert that a standard may be assessed at different points in the semester, based on how the skills are needed for the task at hand; it’s another to say, “Well, you’re learning part of a skill now, and I’ll test you on that, and you’ll learn the rest of this same skill later.”

I’ve had fewer slip-ups of this sort as time goes on, but I’ve never quite been happy with how the standards match up with the time spent in class. Both of the problems above keep rearing their heads. So for this fall, I decided to look at the schedule of the class and write standards based on what we do in 1–2 days of class. (Reading this blog post by Andy Rundquist earlier in the summer helped push me in this direction.) If it seemed like too little or too much was getting done in a day, well, that’s an indication that the schedule should be modified. In a semester with 38 class meetings, there should be sufficient time allotted for review, flexibility, and a few in-depth investigations, which leads me to having 25–30 content standards for the course. That’s a few more than I’ve had in the past, but not by many.

Here’s the conclusion I’m coming to: standards both shape and are shaped by the structure of the class. Part of what we as instructors bring to a class is a personal view of how the subject is organized and holds together. If you and I are both teaching calculus, there will be a great deal of overlap in what skills we believe should be assessed, but there will be differences, and we’ll find different dependencies. A fringe benefit of writing out standards is that we can see this structure clearly—even better, I believe, than just by looking at the order of topics. They force us to be honest about our expectations, thereby combatting a certain tendency, observed by Steven Krantz in How to Teach Mathematics, to give tests based on “questions that would amuse a mathematician—by which I mean questions about material that is secondary or tertiary. … In the students’ eyes, such a test is not about the main ideas in the course.” You may want students to use calculus mostly in applied settings where exact formulas for the functions involved are not known, whereas I may be primarily concerned with students’ ability to deal formally with closed-form expressions and to deeply understand classical functions. We can both be right. We should both let our students know what we expect of them, rather than making them guess. In short, standards are not completely standardized—they highlight the commonalities and the particularities among courses that treat basically the same material.

With all that said, here I will share my list of standards for Calculus 1 this semester. Because of the length of the list, I’ll just link to a Google document that contains them: Standards for MTH 111, Fall 2014. They are grouped into twenty-six “Content standards” and three “General standards”. Over time, I’ve settled on these last three as skills that I want to assess on every graded assignment: Presentation, Arithmetic and algebra, and Mathematical literacy and numeracy. These are essential skills for doing anything in calculus, and struggles in calculus can often be attributed to weaknesses in these areas. We’ve all had students who are fine at applying the quotient rule to a rational function, but are stymied when it comes to expanding and simplifying the numerator of the result. That can hamper solving certain kinds of problems, and I want to be able to point to “algebra”, not anything calculus-related, as the area that needs attention. The descriptions of the content standards are shaped in part by our textbook, Calculus: Single Variable by Deborah Hughes-Hallett et al. I like to introduce differential equations fairly early in the course—this follows a tradition at my college, too—so some standards related to that are sprinkled throughout. I should also confess an indebtedness to Theron Hitchman for the language of using verb clauses to complete the sentence “Student will be able to …”

In addition to the 29 standards in the document linked above, I have one more for this class: Homework. Oh, homework. The calls to treat homework purely formatively and to stop grading it (link goes to Shawn Cornally’s blog) have not quite reached the halls of post-secondary education. Many college and university instructors believe homework is so important that they make it worth a substantial fraction of the students’ grades. And it is important, but solely as a means for practicing, taking risks, developing understanding, and making mistakes. (See this video by Jo Boaler* on the importance of making mistakes: “Mistakes & Persistence”.) Grading homework almost always means that its usefulness as a place to take risks is undermined. Last semester I didn’t grade homework at all, although I did have a grader, who made comments on the homework that was submitted. On average, about 1/3 of the class turned anything in. At the end of the semester, I got two kinds of feedback on homework. A few students expressed appreciation that the pressure to make sure that everything in the homework was exactly right was relieved. Several, however, said they realized how important doing homework is to their understanding—often because they let it slip at some point—and urged me to again make it “required”. I want to honor both of these sentiments. I want to encourage students to do the homework and to feel like it is the safest of places to practice and make mistakes, and thereby improvements. So I will count both submissions and resubmissions of homework towards this standard. A student who turns in 20 homework assignments or thoughtfully revised assignments will earn a 4 on this standard, 15 will earn a 3, and so on. I hope this will have the desired effect of giving students maximum flexibility and responsibility in their own learning, while also acknowledging the work and practice they do.

All of the rest of the standards, general and content, will also be graded out of 4 points, with the following interpretations: 1 – novice ability, 2 – basic ability, 3 – proficiency, 4 – mastery. (I’ve adapted this language from that used by several other SBG instructors). At the end of the semester, to guarantee an “A” in the class, a student must have reached “mastery” in at least 90% of the standards (that is, have 4s in 27 out of 30 standards), and have no grades below “proficiency”. To guarantee a “B”, she must have reached “proficiency” in at least 90% of the standards, and “basic ability” in the rest. A final grade of at least “C” is guaranteed by reaching “basic ability” in at least 90% of the standards.

Two other blog posts about standards in college-level math classes went up yesterday:

• Bret Benesh wrote about his near-final list of standards for calculus 1, and again explained his idea to have students identify for which standards they have demonstrated aptitude when they complete a test or quiz. I really like this idea, as it essentially builds metacognition into the assessment system. I will have to consider this for future semesters.
• Kate Owens posted her list of standards for calculus 2, which she has organized around a set of “Big Questions” that highlight the main themes of the course. This is particularly important in calculus 2, which can sometimes seem like a collection of disconnected topics. In an ensuing discussion on Twitter, it was pointed out that these kinds of Big Ideas are what can really stick with students, far beyond the details of what was covered.

After reading Kate’s post, I looked at my monolithic list of standards, and attempted to organize them into groups based on three big questions: “What does it mean to study change?” (concepts of calculus), “What are some methods for calculating change?” (computational tools), and “What are some situations in which it’s useful to measure change?” (applications). I was not particularly successful at sorting my standards into these categories, but I like the questions. I may ask the students how they would use the various standards to answer these questions. There are trade-offs in any method of developing a set of standards. I am grateful for these other instructors who are also working on changing how we think about grading and sharing their ideas.

* Jo Boaler’s online courses on “How to Learn Math” are currently open:
For teachers and parents until October 15 ($125)
For students until December 15 (free)

standards for analysis

Writing standards for a proof-based class is a different beast than for introductory calculus, or even probability. In my last post, I described a bit of the structure of the analysis class I’m teaching this fall: inquiry-based, primarily structured around group work, running on a weekly cycle of tackling a problem, agreeing on an approach, and presenting a solution to the class for discussion. My usual way of compiling standards—looking through the course content and breaking it into 20–30 skill sets of roughly equal importance—sort of falls apart here. Do I want students to be able to prove that every Cauchy sequence in the set of real numbers is convergent, and to explain what this implies about the completeness of the reals? Yes, but what I really want is for them to be able to assimilate new concepts and make sense of them by creating examples and fitting the definitions into proofs. Do I want them to be able to compute integrals with respect to both Lebesgue measure and singular Dirac measures? Yes, but what I really want is for them to see how these represent the interplay of mathematics and other sciences—how the exigencies of other fields of science led to the development of both the Lebesgue integral and the Dirac delta—and to feel part of a scientific community, both in and out of the classroom.

While considering these questions, I determined that there are six standards I want students to actively develop during the semester, and on which I want to be giving targeted feedback. These skills will be grounded in the content of the course, but they will also provide the benchmarks of success in mastering the content. Here they are:

1. Correct use of vocabulary and notation: Using mathematical terminology and symbols, especially those particular to analysis, correctly and appropriately.
2. Correct and convincing argumentation: Creating and recognizing complete proofs, with their various pieces presented in a logical order.
3. Clear written exposition: Organizing a paper for the benefit of the reader, making it easy to read and using proper English grammar.
4. Broad vision of the subject: Providing context in papers, including statements of solved problems, a guide to the structure of proofs, and connections with other ideas in the class (previous work or larger themes).
5. Effective verbal presentation: Using good speaking habits (e.g., speaking confidently, talking to the class and not to the board, being sensitive to the audience, handling questions well) to present mathematical content.
6. Collaboration and participation in discussion: Attending class regularly, engaging in discussion through questions and critical feedback, seeking ways to serve the overall community.

(As usual, I’m grateful to Bret Benesh and Theron Hitchman for helping me think through these at an early stage.) As I will acknowledge to my students, some of these standards depend to a certain extent on others. For example, it’s hard to make an effective presentation without mastering the vocabulary of the topic. But I believe these are distinguishable skills, all of which are important for students’ development as mathematicians. And I believe the students should be reflecting on their mastery of these skills as much as their mastery of analysis, and have the chance to show when they’ve improved.

My grading scheme for this class is somewhat of a compromise. I am keeping as many of the features of standards-based grading as I can—including scoring individual assignments by standards and providing opportunities for reassessment—but in order to take into account how well the content has been mastered, at the end of the semester I will weight and total points to determine a final grade. This last step is a kludge made necessary by the continued use of letter grades. If I had my druthers, I would leave the final assessment in terms of the students’ demonstrated mastery of the standards on the individual assignments, so that their focus would always be on improving in those areas rather than reaching a particular grade. I have tried to set this up in a way that, to quote T.J., “if you tried to ‘game the system’ to improve your grade, you would be doing exactly the kinds of things I wanted you to do, and improving your abilities as a mathematician.” (This suggests that we’re having to work against the current grading system to encourage students to grow in the ways we want. I suppose it’s a bit idealistic to believe that we can create a grading and reporting method that will provide both useful feedback to students and a helpful summary to those outside, but I digress.)

Of the standards I’ve listed, 1–4 are basically about writing and 5–6 are basically about active involvement. They will be handled separately in the grading scheme. Each student will write, as part of a group, eleven papers that state and solve a particular problem. These papers will be graded on the basis of standards 1–4, with each standard receiving either a 0 or a 1. After a paper has been graded, the groups will have the benefit of feedback from me and from their classmates, and they will revise, if necessary, until the paper merits at least 3 of the possible 4 points. This final version will be included in a document for the whole class to share. There will be a midterm and a final exam, as required by the college. Both will be take-home, and the individual problems on the exams will be graded according to the same standards as the papers. Following the midterm, students will have the chance to revise their solutions, as they do with the group papers.

Standards 5 and 6 will be graded over the whole semester. Each student will have approximately four chances to present in front of the class; although they will be presenting as part of a group, I will give individual presentation grades, again out of 4 points. The baseline will be 2 points. Grades of 3 or 4 will be achieved based on the quality of the presentation and adherence to the principles stated in the description of the standard. I’ll only consider the highest presentation grade at the end of the semester. For the participation grade, the baseline will again be 2 points, for regular attendance. (This is my first time giving an attendance grade. I generally believe college students should be free to decide for themselves whether coming to class is useful or not. In this case, however, the presence and participation of individual members is essential for the class to work, so I think this grade is justified.) Grades of 3 or 4 will be achieved based on involvement in class discussion, either during meetings or online in the class forum (where each week’s papers will be posted), and in general contributing to a supportive, scientific atmosphere. Since this grade is not given on any particular assignment, I will meet with students individually a couple of times during the semester to gauge their progress and experiences, and to discuss their level of participation.

Now, at the end of the semester, I want students’ work on the group papers and the exams to count about equally towards their final grade, and I want each of those to count about four times as much as their presentation and participation grades. So I will convert everything to a 40-point scale (16 possible points for papers, 16 for exams, 4 for presentation, and 4 for participation Edit: I’ve clarified these numbers in the comments). A letter grade of A will require at least 38 points, with no grades lower than 3 on any assignment (paper or exam problem) or standard (presentation and participation). A B will require at least 28 points, with no grades lower than 3. A C will require at least 18 points. This is as close as I can get to my usual way of assigning final grades: a 4 on 80% of standards (or 90%, depending on the class), with no grade below 3, and so on. It also follows relatively closely the French grading system based on 20 points, with 10 required for passing.

It’s not perfect, but that’s my current grading plan for this inquiry-based Introduction to Analysis course. Thoughts?

low-threshold exercises for analysis

This fall, one of my courses will be Introduction to Analysis. At my school, this has been taught using a modified-Moore method for the last few years, and I will be largely adopting the structure and content of these previous years. In this IBL implementation, students work in groups on one problem per week. Each week has three assigned problems (so generally multiple groups are working on the same problem) that are loosely related. At the end of the week one class period is devoted to presentations: for each problem, one group is selected to present their solution in about 20 minutes, and the rest of the class is expected to be engaged in discussion with the presenters. Many of the problems were developed by David Cohen (now professor emeritus), who described the method in an article for the American Mathematical Monthly. Further developments were made by Christophe Golé, with whom I co-taught the course two years ago. From my first exposure to the materials for this class, I have been impressed by the clever way students are led through standard material by a non-standard path.

As with many introductory analysis courses, one goal of this class is to help students transition to more formal mathematics, giving them experience with absorbing definitions and writing proofs. The problems themselves guide students through much of this process. I felt, however, that at times students could benefit from having exercises that allow them to interact more rapidly and immediately with new definitions. So one aspect I’m adding this year is a collection of “Warm-up exercises”, one per week. These are intended to be “low-threshold” activities, in the sense that a student should be able to work on them and produce results even with just a superficial understanding of the definitions involved. My hope is that by interacting with the definitions in a meaningful and productive way, they will feel more prepared to grapple with the assigned problems.

Here is a list of the exercises I’ve written, together with a rough description of the corresponding week’s topic. In addition to being “low-threshold”, several of these are also “high-ceiling”, meaning that immediate extensions and generalizations are evident. (For most of the course, however, the “high ceiling” is provided by the main set of problems.)

• (Counting and cardinality) Prove that the sets {1,2,3} and {4,5,6} have the same cardinality. Prove that {1,2} and {1,2,3} do not.
• (Balls in metric spaces) Recall |x|=x if x≥0 and |x|=−x if x < 0. Prove |x+y|≤|x|+|y| for any real numbers x,y.
• (Topology of real numbers) Prove that if x is isolated from a set T ⊂ R, then x cannot be an accumulation point of T.
• (Topological properties) In R, is a set that contains just one point compact? (A bit of clarification here: in this course, the definition given for “compact” is a variant of sequential compactness, namely, that every infinite subset has an accumulation point.)
• (Continuity) Prove that x^n is continuous at zero for any n∈N.
• (Properties of functions) Prove that x^n is differentiable at zero for any n∈N.
• (Sequences of functions) Use the algebraic identity (1–r)(1+r+r^2+…+r^n) = 1–r^(n+1) to prove that the series 1+r+r^2+r^3+… converges to 1/(1–r) if |r| < 1. (I keep finding that students have forgotten the sum of a geometric series in classes after calculus, so I figured it made sense to remind them of this fact while also suggesting they prove it.)
• (Uniform convergence and degrees of differentiability) For any k∈N, give an example of a function that is C^k but not C^(k+1).
• (Borel sets) Suppose X is any set and A is the power set of X, i.e., the collection of all subsets of X (including ∅ and X itself). Show that A is a countably complete Boolean algebra.
• (Lebesgue integration) Show that a sum of simple functions is a simple function.

There’s one more week’s worth of problems—all focused on properties of the Cantor set—which don’t require any new definitions.

I’m not quite sure what role to give these in the course. I don’t want them to be required, and I definitely don’t want to make them “extra credit”. I do want them to provide a useful entry into playing around with definitions and not seem like extra work. Thoughts?

my one goal for teaching next year

Over the last few years, I’ve introduced several new aspects to my teaching: standards-based grading, student essays, prompting class discussion with questions on slips of paper, explorations with Desmos, assignments through Google docs, and so on. Some of these have had real positive effects, and I definitely believe in continuing to try new things. However, this year I’ve decided to focus on just one element of my teaching, which is to engage every student in every class. This means not worrying about all the potential newness, paying attention to what happens each time the class meets, and figuring out from those observations how to make every class meeting productive for everyone. This doesn’t mean I won’t try new things, but I want my focus to be on student engagement rather than experimentation.

Here are a few specific things I think this entails:

• More preparation before the semester begins. I’m doing more work ahead of time to prepare my classes than I have before. Usually I make sure my syllabus has an outline of the topics in rough chronological order, a description of when homework is due and exams will be given, and a litany of other policies and expectations. Then, during the semester, I choose homework assignments as we go along and follow the schedule with some fluidity, which means lots of time spent figuring out just what the next class can cover. I want my time outside of class to be more reflective. That is, instead of emerging from class and picking a homework assignment that goes with what we did, I want to have time to think about what each student did during the class and what might encourage them next time to be even more involved in the work. Instead of spending prep time picking topics, I want to look at the topics already before me and think about how each student might connect with them. (Writing standards is already a big help towards this: when I consider what skills I want the students to demonstrate by the end of the semester, it forces me to balance the material, on a global scale, in terms of importance and time invested.)
• More peer-instruction methods, like think-pair-share. In other words, I should talk less (but PI is the positive formulation of this principle). How many answers can the students generate on their own? While some might think having students come up with the answers rather than providing a nice clear explanation myself would take more time, I am thinking of the fact that even in my “good” classes any explanation I give usually has to be given multiple times, because not everyone is focused at the same time. The next level would to be see how many questions the students can generate on their own before they start coming up with the answers, and I have that goal in mind. Nothing like trying to answer your own question to keep you engaged!
• Effective use of silence. I have absolutely no problem with periods of silence in my class. If nothing else, stopping the flow of information for a few moments now and then underlines the message that “class is not an info dump”. But I want to be sensitive to what kind of silence is occurring. The best kind is when you know there’s cogitation going on: the students are faced with a new idea or a collision of ideas and are trying to sort it out in some way they can enunciate. But there’s also the kind where everyone is just so baffled and lost that they can’t come up with answers, questions, or anything else. And sometimes in the silence you sense that the students know the prompt they’ve been given is banal, and responding to it proves nothing other than that they’re not literally asleep. I want to be attuned enough to know which is happening. Even better, I want the students attuned enough that they can tell me which is happening and whether the period of silence is worthwhile.
• Finding and using “low-floor, high-ceiling” activities. These are the kind of things anyone can get excited about. A student who is floundering should have something to grasp on to. A student who has mastered the material so far should have somewhere to grow. One way to do this is to have a whole bunch of questions of increasing “difficulty”, and I’ve used that tactic, but it conflicts with some of these other goals. In particular, someone who has trouble getting started on the list might feel at the end of class like they’ve failed if they don’t get to all the questions, and someone who rushes through and gets to the end might get the sense that there’s nowhere further to go. Moreover, when I ask more questions it leaves less room for students to ask theirs. I guess what I’m saying is that tracking down these types of activities is hard, and defies the way in-class activities are often done in calculus. (Possibly the objection I have to many traditional types of calculus problems, like Optimization and Related Rates, is that they have such a high floor and low ceiling. They’re basically puzzles, aimed at a particular level of understanding, which means they’re fun for some but not really broadly useful for learning.)
• Being more deliberate about formative assessment. This might be the hardest one for me, and yet I think it’s key to the whole endeavor. It’s easy to have a sense of how a few particular students and the class as a whole are doing. It’s easy to grade a quiz or a test and look over the results to draw conclusions about students’ understanding (a.k.a., summative assessment). It’s harder to come up with ways that encourage students to work independently, take risks, and also produce something concrete I can assess and provide feedback on. So I’ll be mining the math-twitter-blogosphere for ideas on a variety of ways to make formative assessments!