another experiment

(This post is an experiment in two senses. First, to test embedding graphs from the Desmos calculator into the post. Second, to show the results of a mathematical experiment carried out on Desmos and Twitter last night.)

In Spivak’s classic textbook Calculus, one exercise asks the reader to show that each of the following (complex) power series has radius of convergence 1: \[ \sum_{n=1}^{\infty} \frac{z^n}{n^2}, \hspace{0.5in} \sum_{n=1}^{\infty} \frac{z^n}{n}, \hspace{0.5in} \sum_{n=1}^{\infty} z^n. \] (I’ll leave that task to you. Hint: ratio test.) Another exercise then says, “Prove that the first series converges everywhere on the unit circle; that the third series converges nowhere on the unit circle; and that the second series converges for at least one point on the unit circle and diverges for at least one point on the unit circle.” Points where a series converges always raise a new problem: can we tell what value it converges to? Generally, that problem is hard. But at a point where a series is known to diverge, the story’s over, right? Well, no. There are many ways for a series to diverge.

I want to focus here on the behavior of $\sum z^n$ when $|z| = 1$. The series diverges, of course, because the size of every term is 1. But what do the partial sums look like? What do their real and imaginary parts look like? My thoughts on this began last night when I plotted the graph of $\sum_{n=1}^{50} \sin nx$:

(Click on the graph to go to an interactive version.) To my surprise, there appeared to be well-defined curves bounding the top and bottom of this graph. To be more precise, the points corresponding to critical values (or local extreme values) of the function lie on a pair of analytic curves. After some playing around, I found these curves to be the graphs of ${-\frac{1}{2}} \tan \frac{x}{4}$ and $\frac{1}{2} \cot \frac{x}{4}$ (shown in blue and green, respectively, below).

I sent out a tweet about my discovery:

This certainly *looks* like it fits, but I have no idea why. desmos.com/calculator/e67… @desmos #math #mathchat

— Joshua Bowman (@Thalesdisciple) June 11, 2013

I went investigating and found some hints that this behavior might be related to the Fourier series of the cotangent. Meanwhile, my tweet generated some interest, including this response:

@thalesdisciple @desmos Looks like using formula for a partial geometric sum on e^(inx) should explain it?Thanks for posting.Very cool.

— Paul Johnson (@ptwiddle) June 11, 2013

Also, later in the evening, Desmos took my initial graph and augmented it:

@ptwiddle @thalesdisciple What a fun exploration! I added a few things to your original graph. desmos.com/calculator/wz2… #SinesOfMadness

— Desmos.com (@Desmos) June 12, 2013

Paul’s on the right track, which brings us back to the Spivak exercise I mentioned earlier.

To get a point of the unit circle, write $z = \mathrm{e}^{ix}$, with $x \in \mathbb{R}$. Then the summation formula for partial geometric series yields \[ 1 + \mathrm{e}^{ix} + \cdots + \mathrm{e}^{imx} = \frac{1 - \mathrm{e}^{i(m+1)x}}{1 - \mathrm{e}^{ix}}. \] We can take imaginary parts of both sides and use some trig identities to get \[ \sin x + \cdots + \sin mx = \frac{\cos \frac{x}{2} - \cos \big(m+\frac{1}{2}\big)x}{2 \sin \frac{x}{2}}. \] (Note that Desmos included this latter formula in their augmented form of the graph. See also this nice derivation.) On the other hand, the tangent half-angle formulas give us \[ -\tan \frac{x}{4} = \frac{\cos \frac{x}{2} - 1}{\sin \frac{x}{2}} \qquad\text{and}\qquad \cot \frac{x}{4} = \frac{\cos \frac{x}{2} + 1}{\sin \frac{x}{2}}. \] When $\sin\frac{x}{2}$ is positive (for example, when $0 < x < 2\pi$), we have \[ -\frac{1}{2} \tan \frac{x}{4} \le \sin x + \cdots + \sin mx \le \frac{1}{2} \cot \frac{x}{4}, \] with equality on the left whenever $\cos\big(m+\frac{1}{2}\big)x = 1$ and equality on the right whenever $\cos\big(m+\frac{1}{2}\big)x = -1$. The direction of the inequalities is reversed when $\sin\frac{x}{2} < 0$, but the rest of the analysis remains the same. This is the desired result.

Thus the imaginary parts of the partial sums of $\sum \mathrm{e}^{inx}$ are always contained between $-\frac{1}{2}\tan\frac{x}{4}$ and $\frac{1}{2}\cot\frac{x}{4}$. To complete the picture, let's look at the real parts. Here is the graph of $\sum_{n=0}^{50} \cos nx$:

Using similar arguments as before, we can show that the value of $\sum_{n=0}^m \cos nx$ always lies between $\frac{1}{2}-\frac{1}{2}\csc\frac{x}{2}$ and $\frac{1}{2}+\frac{1}{2}\csc\frac{x}{2}$.

Therefore, even though the series $\sum z^n$ diverges whenever $|z| = 1$, the real and imaginary parts of its partial sums remain tightly constrained by values that depend analytically on the argument of $z$ (unless $z = 1$, i.e., its argument is a multiple of $2\pi$, in which case the series is just $1 + 1 + 1 + \cdots$).

Coda: The real and imaginary parts of $\sum \frac{z^n}{n^2}$ and $\sum \frac{z^n}{n}$ also look like Fourier series, no? Here, for instance, are the graphs of $\sum_{n=1}^{100}\frac{\cos nx}{n}$ (left) and $\sum_{n=1}^{100}\frac{\sin nx}{n}$ (right):

In particular, it looks like $\sum \frac{z^n}{n}$ diverges only when $z = 1$ (where it becomes the harmonic series). Can you find the functions to which its real and imaginary parts converge away from multiples of $2\pi$? Click on the graphs and try!

an experiment…

I’ve been wanting to get LaTeX working on this blog for at least a few months now. When I first tried it, I had the impression that it would be all sorts of convoluted. Fortunately, I just found another blog post with a variety of links on how to make it simple, so I’m going to try some of them out. (See the final remark at the bottom, plus the comments, for some observations on how well this works.)

Here is what prompted my initial crise de foi regarding the marriage of LaTeX and Blogger. This past spring, I taught an “advanced calculus” course, which amounted to an introduction to (embedded) manifolds. I was a TA for this class, twice, as a graduate student, and it has some of my favorite calculus material—the kind that makes you realize what a breathtaking endeavor it is. One of the first big revelations is the nature of the derivative. I love this part of the class, and I wanted to share it.

When we first teach the derivative, we teach it as a number. We have to. It’s hard to imagine conveying anything more abstract about it when the definition already involves tangents, limits, possibly infinitesimals, and we just want to instill some level of understanding. But the purpose of the derivative—indeed, the philosophy behind all of differential calculus—is to take a curvy object and make it straight. Since we apply it to functions, the result should be a straightened function, i.e., a linear function. A linear function from $\mathbb{R}^m$ to $\mathbb{R}^n$ may be encoded by an $n \times m$ matrix. That's often convenient, but not always. Here’s a simple example that shows that sometimes it’s best to avoid matrices.

First, the matrix-free definition of the derivative. Let $U \subseteq \mathbb{R}^m$ be an open set, and let $f : U \to \mathbb{R}^n$ be a function. Then $f$ is differentiable at $\mathbf{x} \in U$ if there exists a linear function $L : \mathbb{R}^m \to \mathbb{R}^n$ such that \[ \lim_{|\mathbf{h}|\to0} \frac{|f(\mathbf{x}+\mathbf{h}) - f(\mathbf{x}) - L(\mathbf{h})|}{|\mathbf{h}|} = 0. \] If such a function $L$ exists, then it is unique, and we write $Df(\mathbf{x}) = L$.

Now consider the set of $n \times n$ matrices, and identify this set with $\mathbb{R}^{n^2}$. Define $S : \mathbb{R}^{n^2} \to \mathbb{R}^{n^2}$ by $S(A) = A^2$. If we were to write this function out in coordinates, we would see that all of the entries are polynomials, and so it is differentiable. What is its derivative at a point $A$? Note that the derivative must be a linear map from $\mathbb{R}^{n^2}$ to itself, so writing out a matrix would be fairly taxing.

Instead, we can get an idea of what the derivative should be by adding a variable matrix $H$ (presumed to be small) to the matrix $A$ and seeing the result of the function: $S(A+H) = (A+H)^2 = A^2 + AH + HA + H^2$. The part of this expression that “looks linear in $H$” is the middle two terms. Indeed, if we set $L(H) = AH + HA$, then we find \[ \lim_{|H|\to0} \frac{|(A+H)^2 - A^2 - (AH + HA)|}{|H|} = \lim_{|H|\to0} \frac{|H|^2}{|H|} = 0. \] Ah-ha! The derivative of the squaring function $S$ at $A$ is $DS(A) : H \mapsto AH + HA$! I still find this computation incredibly insightful.

What happens when $n = 1$? Then our matrix $A$ is $1 \times 1$, so it’s just a number, say $a$. The squaring map is $S(a) = a^2$, and the derivative of this map sends $h$ to $ah + ha = ah + ah = 2ah$. In the one-variable setting, we find $S'(a) = 2a$, and so this perspective on the derivative in terms of linear maps has reaffirmed the geometric meaning of the ordinary derivative: it describes the amount by which the range variable changes, infinitesimally, when the domain variable is altered infinitesimally.

Remarks:

• I learned this way of thinking about derivatives from John Hubbard through his textbook Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. He was the instructor the first time I was a TA for this class.
• This experiment with including LaTeX seems to have been a success, provided I include the MathJax code in the post itself. Other methods which rely on altering the template have not worked for me.

SBG: what could have gone better

A couple of weeks ago, I posted about many great things that came out of my experiment with standards-based grading in multivariable calculus. I think the experiment was a success in that there was a lot more good than bad, and the bad things weren’t so bad. Nonetheless, things could have gone better, even in ways I don’t think the students realized, and I would be remiss not to mention them. Whence this post.

Arrangement of standards: There is a tough balance to achieve here, and I didn’t quite make it. The twenty-four standards included

• 7 “common standards” that could relate to any college-level math class
• 3 standards related to vectors and their geometry,
• 3 standards related to graphing and parametrization,
• 5 standards related to differentiation and its applications,
• 4 standards related to integration and its applications,
• 2 standards related to the classical theorems of vector calculus.

I wanted each standard to present a unified concept to be mastered. However, I didn’t want the number of standards to proliferate. I think the total number of standards is about right (anywhere in the range 20–30 would have worked), but the distribution of topics was a little off.

First, seven common standards is too many. Four or five would be more appropriate. Algebra and presentation are essential to treat separately. The others, while distinct in my mind before the class began, became somewhat muddled in their distinction during the semester, and some even overlapped a fair amount with the content-specific standards. The first six or seven standards scored on each homework assignment (often only six, because one was essentially “use of technology”, which rarely figured directly into the homework) became a blurry wash, occasionally used to try to indicate some general, but ill-defined, skill needed attention.

Second, I was too clever in collecting related topics, to the point that certain standards were only partially covered for several weeks at a time. For instance, “line integrals” included both arc length computation—covered in the first chapter—as well as integrals of vector fields—covered in chapter 4. This was perhaps the most egregious example. The topics covered by each standard should have been collected not only by commonality, but also chronologically.

Third, the relative importance of the standards, or at least the relative emphasis that was given to each during class, was not as balanced as I would have like. Setting up and evaluating double and triple integrals—a single standard on the syllabus—takes over a week of class time (although part of that time includes changing coordinates, which was a separate standard). Visualizing vector fields—the only standard that was only tested once, on the final exam—was dealt with sporadically in class. Is it as important to be able to interpret the visual information carried by a graphical vector field as it is to find integrals of several variables? Arguably, hence the separate and equal standards. Was that equality reflected in the amount of attention it was given during the semester? Again, not as much as I would have liked. Not sure this is a challenge of standards per se, but more of course design. Having the standards just highlights the inequity.

Finally, on this topic, even some of the content-specific standards overlapped more than I had intended. I mostly managed to avoid the obvious pitfalls: for instance, computing integrals and finding parametrizations were handled separately, so if a question asked students to find the surface area of a figure, say, and someone set up the wrong integral but computed it correctly from that point, they could get credit for integration but not for parametrization. But what exactly are the skills that go in to setting up an integral? There were standards for describing objects in 2 or 3 dimensions, as well as an “analysis” standard that, in part, required finding the domain of a function. When a double or triple integral is needed, one has to draw on one or more of these skills to find appropriate limits of integration. When it seemed to me like a problem could have been solved by several different approaches, does a failure to find any solution reflect a lack of mastery of all those skills? Hard call. No one’s scores suffered seriously from this ambiguity, but occasionally I found myself judging a surprising number of standards on the basis of one or two exercises.

Grading scale: Having a four-point scale for each standard worked well, on the whole. It was sufficiently refined to target both areas of success and areas needing work. However, the overall quality of work was so good that I had trouble distinguishing among the highest levels of performance. What should I do with a solution that reflects a clear understanding of the skills involved, but has one or two minor errors? Do those reflect some genuine misunderstanding, or simply a slip? How can I judge between a score of 3 (“generally good accuracy”) and 4 (“complete mastery”) in that case? I think I may have to move to a five-point scale, as described here, where 4 and 5 both indicate mastery, but 4 allows for small mistakes.

On the other hand, I think I could have been more exigent in what level of mastery should be reached across the spectrum. On the syllabus, I stated that attaining 4 on 80% of the standards with no scores lower than 3 was sufficient for an A, and I believe I could have raised that percentage to 90% to better reflect complete mastery of the course material. The end result of this is that perhaps a few final grades were more elevated than they might otherwise have been. But seriously—these students worked extremely hard, I am extremely proud of them and have full confidence in their calculus skills, and they deserve some recognition for working with me on this grading experiment and making it a success.

Assessments: I was exceedingly grateful to have a grader with whom I had worked before, and who I trusted to help me implement this SBG system as effectively as possible. I could not have made this first attempt work without her aid. Each week, she would mark the homework, making particular note of places that raised concern or showed exceptional mastery, and then we would meet together to assign scores. I don’t think this method is sustainable across terms. I need to shift to a model that depends less on explaining the grading system to a new assistant each semester, but also that will not vastly increase the amount of time I have to spend grading. (I don’t think having the only graded assignments be two midterm exams and the final is sufficient for me to trust those assessments, nor does it communicate with the students in the way I always hope SBG will.) This is perhaps the area I have to think most about revising as I move forward with SBG in future classes.

Re-assessments: About a fourth of the class took advantage of the opportunity to re-assess any standards. Not such a bad number, especially considering how well they were doing on the whole. But I feel more could have benefitted from this feature of SBG, had the process of reassessment been clearer, and had some of the above obstacles been removed. I do need to find a way to cut down on the time required for reassessment, however: I always tried to claim it would take 10–15 minutes, but often it was much longer than that. No student ever complained about the length of time, which arose both because I gave multiple chances to explain themselves and because of the relative complexity of the material. Nonetheless, I think retesting will have to be made more efficient for it to work in other classes.

Compiling scores: The students received regular updates on their scores in the form of score sheets attached to their homework and exams, but there was no established system by which they could see what their current scores on all the standards was. (Having had some troubles using our LMS in a much simpler grade book setting, I’m averse to the idea of using that or any other online reporting system.) Fortunately, I think this problem is easily solved. Most likely, I’ll handle it in the future by passing out sheets on which students can record their own scores, so that they don’t have to consult with me to find out their current standing. (This is a suggestion I got from Bret Benesh. I suspect some students were already doing this on their own.)

That’s probably not everything that needs improvement, but it’s what came to the forefront of my attention. I have some ideas, some listed above, on how to make my system better next time around. As I’m working on future syllabi, I’ll jot these ideas down and post them here.

SBG: what worked well

I’ve been lax all spring in blogging (other things, too, but that’s beside the point here), and now that the end of the semester has arrived, it’s time I settled down with a cup of coffee to share some of my thoughts on how standards-based grading went. I keep reading blog posts by other teachers and acquiring such cool ideas thereby (this blog brims with excitement about the possibilities for improvement SBG brings to both instruction and assessment; hat tip to Dan Meyer, who recently linked to it), but it is still the case that few teachers at the college/university level are writing about SBG in that context, so hopefully this will be a productive exercise. It’s a good thing qualifications aren’t a prerequisite for blogging, ’cause I ain’t got ’em. Which means this is at least as much about benefitting from the community as trying to contribute to it.

To make this a little more manageable, I’m going to deal with three topics in three (or more) separate posts: what worked well, what worked not so well, and how I plan to move forward.

The Backstory: When last we saw our intrepid blogger, he was heading off into the spring semester to teach multivariable calculus at a small liberal arts college in New England. Having spent several weeks thinking about how he might structure SBG in this class, he had settled on a 4-point system (where 0 represents “complete unfamiliarity” and 4 represents “complete mastery”) with 24 standards: 7 common standards and 17 content-specific standards (an early version of this list was posted here).

What came next: When I met with my class, I explained the system and why I was using it. Assessment should be about giving students the chance to demonstrate what they’ve learned, I said, and providing sufficient opportunity for them to show they’ve mastered the material during the course, even if it doesn’t happen in time for the first test on the material. A point-based system confounds this process. How many students really know what they got each “point” for? (How many of us teachers do?) And a point, once lost, cannot be regained, unless some system of “extra credit” is established, which just creates more work for everyone. I explained that the homework and tests would both create opportunities for them to demonstrate their understanding, and that there was no “weighting” of grades, just regularly-updated scores for the standards. A few expressed surprise, but overall they were accepting that this was how things would work. I explained that the process required honesty from all involved. For my part, I would give scores that I believed accurately reflected each student’s prowess with the various skills they were to learn. For theirs, since I was going to be assessing homework using the same system as the tests, they needed to present their own work each week. (From what I saw, this worked. Students worked together to tackle the problems, but they did not turn in assignments copied from each other. Had I not been at a private liberal arts college with a stringent honor code, I would definitely have had to find another way to handle this. Fortunately, my academic setting allowed me to try SBG this way without worrying about cheating.)

During the semester: We had weekly homework sets and two mid-semester exams. The students have just taken the final exam, and I’ll grade it over the weekend. The homework exercises were primarily taken from the textbook, Michael Corral’s Vector Calculus—available for free download here—and I also wrote some additional exercises to cover other material. (Side note, tangentially related: I chose this textbook because it seemed ridiculous to me to pay $150 for a book that covers material which is available for free almost everywhere. This book basically has the outline I wanted to use, and it has the additional benefit that the exercises are on the whole quite straightforward. I’m realizing that lots of books, and lots of instructors, like “clever” exercises that seem to students only distantly related to the material they’re learning. I’m often tempted that way myself. But if I’m going to assess standards rather than cleverness, a collection of direct applications is invaluable. More on this another time.) The tests were open-book and open-notes. While memorizing definitions, formulas, and theorems is an important step towards forming a coherent picture of the subject, I wanted to emphasize that in the Information Age one can use myriad tools to recall these facts, so that what’s really important is using them intelligently. (Tip: students are afraid of open-book tests, because they assume they’ll be harder. Does “more conceptual” equal “harder”? Possibly in their minds. They did well on the tests, however.)

In addition to the seven “common” standards, each homework covered between three and six other standards, so that many were assessed multiple times. None of the content-specific standards appeared on every assignment; most showed up 2–5 times, although some only once, and some only on the exams. Once a standard had been tested (not just appeared on homework), students could schedule appointments with me to reassess specific standards, up to two per week. To emphasize the importance of mastery, I told the students that I would guarantee an A for anyone who reached (and maintained) 4s in 80% of the standards, with no scores below 3; a B for anyone who reached 3s in 80% of the standards, with no scores below 2, and so on. Scores could be revised up or down, but to alleviate concerns that a fluke of a bad performance at the end of the semester would ruin their scores, I would average their highest and their latest score at the end of the semester.

Student response: When elicited, this was generally positive, which is the most important measure from my perspective. Several students said SBG reduced the stress of test-taking. Others liked how it affirmed their understanding in certain areas while pointing to areas that needed work. A handful took it as a personal challenge to reach all 4s by the end, even though having a couple of 3s wouldn’t change their grade. In the middle of the semester I used an online poll to get anonymous feedback. A couple complained that they didn’t know how their performance was compared with the rest of the class; I view this as part of the purpose of SBG (albeit a minor part)—the striving is against self, not in competition. One said she worked harder to master the material, but appreciated not having to worry about a single bad performance wrecking her grade. The consensus of more than half the students who responded was that SBG reflected their progress and communicated my expectations very well; other responses were at worst neutral. (I wish I had a comparison poll from my non-SBG classes to see if my expectations were being clearly communicated. But if I had done that, I probably would have been using standards anyway.) Even at this level (third-semester calculus), when one might think students’ feelings towards mathematics are firmly set, several students told me that they either had thought they were bad at math or didn’t like it, and now they’ve changed their minds.

My impressions: Mostly I have the sense that standards-based grading was freeing for the students. Far fewer worried about their grades than seems typical (though a few still did), knowing that the way to improve their final grade was the only sensible way: improving their understanding. I was glad to target my feedback, which was the main reason I started considering SBG to begin with. For example, most of the students were adept with algebra, but not all. Some had trouble moving between formulas and visual representations of graphs or objects. Some couldn’t quite grasp how to come up with parametrizations. No student, however, could come out saying “I’m not good at calculus.” They almost always knew which areas they struggled with, and by separating out the different skills, this method of assessment provided confirmation and encouragement at the same time. Each student could look at her scores and say, “Hey, I’m pretty good at a lot of this. I see an area where I’m having trouble, so I guess I’ll work on that.”

In the end, I have tried to be guided by the principle that it is not what I do, but what the students do that contributes the most to their learning. (I picked this up from somewhere, probably several places, and I’ll try at some point to elaborate on how else I applied it.) From that perspective, I would call SBG a success in this class. The participation and performance throughout the class was more uniform across all topics than I have ever seen before. By which I mean, each student knew she was responsible for a certain collection of skills, not just for an accumulation of points or a certain average letter grade, and so they all stepped up to learn all the skills. (Of course, this work ethic is characteristic of students at my school.)

Those are the upsides. In my next post (probably next week, after I’m done grading), I’ll discuss what didn’t go quite so well and why.

circles, tangents, and conceptual art

The math department at Smith College recently acquired a new art installation: Sol LeWitt’s Wall Drawing #139 (Grid and arcs from the midpoints of four sides). This piece was a gift to the Smith museum, and was first installed there in 2008. It is an example of “conceptual art,” of which LeWitt was a major exponent during the 20th century. While conceptual art was/is a large movement, of which I am almost completely ignorant, in this case (like many others of LeWitt’s wall drawings) it means that the art resides in a concept—more precisely, a set of instructions—which is created by the artist and carried out by a team in each physical location. This is analogous to the creation of music, with the artist playing the role of the composer and the installation team acting like the musicians, who must take the artist’s instructions and interpret them in their particular setting.

(You can click on each image below for a full-sized version.)

 In this case, the directions (paraphrased) are as follows:

• Draw a grid of lines evenly spaced 1 inch apart over the dedicated wall space.
• Draw circles centered at the midpoint of each of the four sides, with radii increasing by 1 inch, all the way across the wall.

Here are the four midpoints:



You can learn more about the original installation at the museum from a video. I just wanted to make these pictures available and to highlight the possibility of asking innumerable mathematical questions about this piece. For instance, the grid and circles produce varying patterns and densities throughout the space:


Can you tell where each of these pictures was taken? In the center of the piece, many coincidences appear and tangencies among the circles and the grid lines become evident:


The installation was done by three Smith students in art and math, directed by a professional installer from the LeWitt studio over the course of nine days in January. At a presentation last week, the students described the exactness and concentration that this project required, as well as certain accommodations that had to be made—for example, not all of the wall edges are perfectly straight, and so they had to determine how to adjust the grid, and what points to use as the midpoints. Apparently one circle has a radius that is slightly too large, because of slackness in the compass they were using. (I haven’t yet found where this circle is.) Clearly there is an interesting interplay between form and accident (in the Aristotelian sense), leading to all sorts of philosophical questions that I’m not up to expounding at the moment.

This is the first of LeWitt’s works that I have encountered. I’m sure others have plenty of mathematical material to explore, as well.

assessing standards

As promised, today I want to describe my plan for assessing the standards in my multivariable calculus class. I’ve pretty much settled on the “common standards” that I think would be appropriate for any intermediate college math class, and thanks to some feedback I’ve received since yesterday, I’m refining the list of “content-specific standards” for this class. (For some of the reasons I’m using standards-based grading in this class, see this post, or these slides by T. J. Hitchman from last week’s Joint Math Meetings.) As I see it, there are 4 issues to deal with in scoring standards:

• what scale to use;
• how to assess;
• how to re-assess;
• how to convert to a letter grade at the end of the semester.

I’m almost scared to bring up the last one, because it’s the issue that could unravel the whole process, but I’m certain my (highly driven and motivated students) will panic without it being addressed. If there are suggestions for other issues that should be ranked with these, please let me know. I’ll cover each of these briefly.

What scale I will use

I’ve seen several proposals, including the very simplest, a 2-point system for each standard. (To be fair, I think that works when the list of standards is more refined, so that very specific skills are treated separately and not clustered.) After thinking about what I believe will be the most useful to students, and based on my experience using a 3-point system, I’ve decided to score each standard out of a possible range of 0–4, with 0 indicating “complete unfamiliarity” and 4 indicating “complete mastery”. To aid the students in seeing what I expect at each level, I’ve written sentences they should be able to read and agree with when assessed at the various levels. This is another idea that I’ve borrowed from somewhere, but am having trouble finding at the moment. In my syllabus, I’m describing a standard as a set of closely related skills that represent a piece of knowledge towards mastering the class material, which should explain some of the language below.

1. “I have some idea of what this skill set and its vocabulary mean, but I don't really know how to use it.”
2. “I can complete basic exercises that involve these skills as long as I have some guidance.”
3. “I can use these skills in familiar situations with generally good accuracy.”
4. “I can use this skill set effectively and explain its significance. I can recognize when the skills are useful and apply them to both familiar and new situations.”

(I did not write a sentence for 0-level, as it would be hard for someone completely unfamiliar with a topic to muse on her understanding of it.)

How I will assess

In brief, there will be homework, two midterm exams, and a final exam. All of these will be assessed on the basis of individual standards, and each time a standard appears, its new score replaces the previous score.

I know the debate rages on about whether or not to grade homework, but because the learning time is compressed in a college class, and I do not get to see my students everyday, I think it’s important to have some way to encourage and recognize work done outside of class. That said, the homework grades will not be based on “completion”. Instead, they will provide an opportunity for students to set a “base-level” for their understanding. The report from each homework assignment will list the relevant standards and how the student’s work rates on those standards. This gives them immediate feedback, as well as a chance to see how prepared they are in advance of the exams. I suppose a student could just copy someone else’s work to inflate their scores, but I will explain that in that case their Presentation score (which is part of every assignment) will suffer; their work should be original.

Exams are larger collections of standards, integrated into a broader context. By the time a student gets to a test, she should have a good sense of which areas she will do well in, thanks to homework and earlier self-assessments. Part of the review for each test will include a list of the standards that have been covered to date and may be expected to appear. (This is another good reason for my standards to be a bit coarse, rather than drilling down to specific types of computations—it’s easier to guarantee that a test covers “parametrized curves” than “parametrizing lines”, “parametrizing circles”, “parametrizing spirals”, “checking for smooth points of a curve”, etc.) Again, I suppose a student could not have done any homework before the test and demonstrate total mastery of the material, but that outcome is not, in principle, outside of my goals for SBG.

How I will re-assess

This will be tricky to explain. For many students, tests have always been about how much they contribute to the final grade, rather than how much they say about the current level of understanding. I want to make clear that tests are important and useful only insofar as they create a rich opportunity for learning (through synthesizing the material) and showcasing one’s abilities. Whereas homework assessment is intended to establish a base level of understanding about a student’s ability from week to week, an exam provides a snapshot of her ability, and often a stressful one, at that. After the test, I want to give every student a chance to prove herself in the areas where she may have previously struggled. The experience of other teachers using SBG suggests that this not be done indiscriminately.

Thus, my policy (initially) will be to have students contact me to schedule reassessments for specific standards (during or outside of my usual office hours), at any point in the semester after a standard has been tested. This reassessment could take the form of either an oral examination or an expository presentation by the student. It is unlikely that another written assessment will be given, since I believe the obstacle is often precisely that written tests provoke anxiety. No standard can be reassessed more than once a week, and no more than three two standards can be reassessed in a week. The main point among these practical considerations is that if a student proves she has mastered a course standard, then she receives credit for doing so.

How I will convert to a final grade

This is the least important of the four issues, and yet it is the one that leaves the most lasting record. (In contrast, I hope that what leaves the most lasting overall effect is the knowledge and confidence the students gain.) I don’t want to encourage students to fiddle with a fixed formula, especially since this is my first time using SBG, but I do want to make it clear that mastery of standards is directly correlated with the final letter grade. So here’s what I’m starting with:

• In order to guarantee an A in the class, a student should attain 4s on at least 80% of the course standards and have no scores below 3.
• In order to guarantee a B in the class, a student should attain 3s on at least 80% of the course standards and have no scores below 2.
• In order to guarantee a C in the class, a student should attain 2s on at least 80% of the course standards.

This emphasizes that the goal is mastery. It is also commensurate with what one might expect of the scoring levels in any case: “mostly 4s” should look like an A/A-, “mostly 3s” should look like some form of B, etc.

The score that will be counted for each standard towards the final grade will be the average of the latest score and the highest score. That way earlier gains will not be wiped out by later retreats, but it is still important to keep up each set of skills. Because there are no opportunities for reassessment after the final exam, any prior standards that reappear on the final can only be raised by the scores on that test, not lowered.

And that’s it! That’s my plan for assessing the 20–25 standards that will finally form the basis for grading multivariable calculus this spring. Thoughts and advice are welcome.

standards for multivariable calculus, first pass

OK, it’s time to get real with this. In my last post, I explained some of my reasons for attempting to use SBG this spring and listed seven general standards for college-level mathematics classes. Now I’m listing the standards I have created specifically for multivariable calculus. There’s still time to tweak these, so I would certainly appreciate any feedback over the next few days (or, indeed, at any time!).

I have grouped the standards into five larger categories: “geometry of vectors”, “functions, curves, and surfaces”, “differentiation”, “integration”, and “classical theorems”. In some SBG implementations, the competencies listed below might be more finely sorted out, but I have come to believe that giving college students a somewhat broader classification will encourage them to guide their own education and to think holistically about the material. I have tried to sort these according to three basic principles:

• they cover roughly the same amount of course material;
• they are roughly of the same importance towards mastering the content;
• they can be more-or-less independently measured (although there are indisputably dependences among them).

During the semester, the content on which a standard is based may be introduced gradually over time. For this reason as well as the general expectation that skills should remain honed, many of the standards will be assessed several times. This is also one of my main sources of concern for confusion—what does it mean to have “mastered differentiation operators” at the level of computing partial derivatives and gradients, but not curl and divergence? If I were to distinguish these standards further simply because some parts are separated temporally, however, the number would increase two- or three-fold, making grading an intractable problem for me.

Despite these misgivings about the list itself, I feel it’s important to go ahead and publish it so that I and others can reflect on it. So here goes: (Update 1/18: Following some feedback and discussion, I have revised this list from its original form. Please ignore the parts that have been struck out.)

Geometry of vectors

• Operations – compute and interpret sum, scalar multiples, dot product, determinant, and cross product of vectors in the plane or in space
• Objects – describe points, lines, planes, spheres, and other surfaces using equations, vectors, set notation, or geometric objects of a different kind

Functions, curves, and surfaces

• Visualization – sketch or predict appearance of the graph of a function or curve based on a formula or other description; sketch or describe level sets of a function; use computer software to examine shapes of graphs
• Parametrization (added) – find parametrizations of lines, circles and other curves, as well as planes, spheres, and other surfaces (e.g., tori and graphs of functions)
• Analysis – find domain and range of functions of 1, 2, and 3 variables, and describe these using set notation or geometric terminology; determine continuity

Derivatives

• Operators – apply and interpret partial derivative, Jacobian, and gradient operators to functions; divergence and curl to vector fields
• Operations on functions (added) – compute and interpret partial derivatives and gradient of a function of 2 or 3 variables; apply and explain equality of mixed partial derivatives, including sufficient conditions for such equality to hold
• Operations on vector fields (added) – compute and interpret divergence and curl of a vector field
• Linearization – find tangent vectors, tangent lines, and tangent planes; use these to approximate curves and surfaces near a point
• Higher derivatives – apply and explain equality of mixed partial derivatives; use higher derivatives to collect data about shape of the graph of a function, including classifying critical points
• Optimization (added) – use higher derivatives to collect data about shape of the graph of a function; classify critical points; Lagrange multipliers
• Differential equations – interpret partial differential equations (such as the wave equation, the heat equation, and the Laplace equation) in terms of their solutions; verify that a given function is a solution to a PDE; use a computer to solve or approximately solve PDEs

Integrals

• Multiple integrals – accurately describe regions over which double or triple integrals are computed; perform calculations of double or triple integrals; apply and justify change-of-coordinate formulas; use computers to find integrals
• Line and surface integrals – effectively parametrize curves and surfaces, and use these use parametrizations of curves and surfaces to compute length, area, work, and flux integrals
• Applications – use double and triple integrals to solve problems in geometry and probability; use line and surface integrals to investigate physical phenomena

Classical theorems

• Integrability conditions – check conditions for a vector field to be irrotational or divergence-free (“closed” with respect to curl or divergence operators), and explain the meaning of these conditions; find potential functions
• Generalizations of FTC – explain the meaning and significance of Green’s Theorem, Divergence Theorem, and Stokes’ Theorem
• Applications – use Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem to convert integrals between various forms
• Generalizations of FTC – explain the meaning and significance of Green’s Theorem, Divergence Theorem, and Stokes’ Theorem; use these to convert integrals between various forms

In my next post (probably tomorrow), I’ll explain how I plan to grade these, including what scale I’ll use, how assignments will be broken into their component standards, and how students may improve their score on an individual standard.