Monday, July 15, 2019

cardioid, deltoid, folium

The cardioid and the deltoid are two of my favorite curves. They arise in similar ways: one is an epicycloid, and the other is a hypocycloid. In a sense, each is the simplest non-trivial example of their respective type. They make excellent examples for calculus problems. But as I learned this week, they are actually the same curve.

This post is about the claim made in italics in the previous paragraph. Obviously I don’t mean that the classical constructions mentioned above (and described below) produce the same curves in the Euclidean plane. Rather, they are the same from the perspective of complex projective geometry. When I searched for this fact on Google after uncovering it for myself, I only found one mention of it, in a textbook from 1923 entitled An Introduction to Projective Geometry. I assume it was well-known at the time, and today is probably known to certain algebraic geometers, but it seems worth explicating for a larger audience.

First, the curves. Epicycloids and hypocycloids are both examples of roulettes, curves traced out by a point marked on one curve, which is free to move, as it rolls along another curve, which is fixed, without slipping. To generate an epicycloid or hypocycloid, both the fixed curve and the moving curve are circles; the difference is that for an epicycloid, the rolling circle is outside the fixed circle, and for a hypocycloid the rolling circle is on the inside. The shape of the epicycloid or hypocycloid is determined by the ratio of the circles’ radii. For an epicycloid, we can choose a 1:1 ratio, which means the marked point on the rolling circle makes contact with the fixed circle once as the outer circle completes a circuit. A hypocycloid cannot be constructed from circles whose radii have a 1:1 ratio, and a 2:1 ratio simply produces a line segment, so the simplest hypocycloid arises from a 3:1 ratio. The construction of these simplest examples is illustrated below. (These animations were created using a Desmos graph with the help of GIFsmos.) The first is called the cardioid (“heart-like”) and the second is the deltoid (“triangle-like”).

In both cases, the rolling circle is given a radius of 1, and in both cases the centers of the two circles remain at a distance of 2. By watching carefully, one can see that in both cases the marked point makes two revolutions around the center of the rolling circle. For the cardioid, these revolutions are counterclockwise, and so the cardioid can be parameterized by \[ (2\cos\theta + \cos2\theta, 2\sin\theta + \sin2\theta)\text. \] In the case of the deltoid, the marked point’s revolutions are made clockwise, and so the deltoid can be parameterized by \[ (2\cos\theta + \cos2\theta, 2\sin\theta - \sin2\theta)\text. \] These formulas are very similar, but certainly not the same, and the pictures they produce are quite different. So how can I claim that the curves are the same?

Our first step toward understanding the claim involves switching to complex numbers. If we collect the \(x\)- and \(y\)-coordinates of the plane \(\mathbb{R}^2\) into a single complex coordinate, then the parameterizations above become

\(2e^{i\theta} + e^{2i\theta} \qquad\) and \(\qquad 2e^{i\theta} + e^{-2i\theta}\).
Now we want to extend to the complex plane \(\mathbb{C}^2\) (note: I think of \(\mathbb{C}\) as the complex line because it is one-dimensional as a complex vector space). A standard trick is to add a second coordinate that is conjugate to the first, which makes the parameterizations
\(\big(2e^{i\theta} + e^{2i\theta},2e^{-i\theta} + e^{-2i\theta}\big) \qquad\) and \(\qquad \big(2e^{i\theta} + e^{-2i\theta},2e^{-i\theta} + e^{2i\theta}\big)\).
Now let’s set \(t = e^{i\theta}\) and allow \(t\) to take on all complex values (except \(0\), but we’ll take care of that later) instead of just values on the unit circle. At the same time, let’s label the parameterizations \(\gamma_C\) and \(\gamma_D\), with \(C\) standing for cardioid and \(D\) for deltoid. This gives us
\(\gamma_C(t) = \left(2t + t^2,\dfrac{2}{t} + \dfrac{1}{t^2}\right) \qquad\) and \(\qquad \gamma_D(t) = \left(2t + \dfrac{1}{t^2},\dfrac{2}{t} + t^2\right)\).
We still can see superficial similarities in these formulas, but not enough to conclude that they define equivalent curves. In order to see their equivalence, we need to see what’s happening at infinity, which means introducing some projective geometry.

The complex projective line \(\mathbb{P}^1\), also known as the Riemann sphere, is obtained by adding a single point, labeled \(\infty\), to the ordinary complex line \(\mathbb{C}\). The points of \(\mathbb{P}^1\) may be thought of as the “slopes” of lines through the origin in \(\mathbb{C}^2\). Indeed, it is often useful to assign coordinates to \(\mathbb{P}^1\) using non-zero vectors \((s,t)\) in \(\mathbb{C}^2\), where two vectors correspond to the same point of \(\mathbb{P}^1\) if they are scalar multiples of each other, \((s,t)\sim(\lambda s,\lambda t)\) if \(\lambda\in\mathbb{C}\setminus\{0\}\). We write the equivalence class of \((s,t)\) as \([s:t]\); these are called homogeneous coordinates on \(\mathbb{P}^1\). We can recover \(\mathbb{P}^1\) as \(\mathbb{C}\cup\{\infty\}\) by sending \([s:t]\) to the slope \(t/s\) if \(s \ne 0\); then \([0:1]\) is sent to \(\infty\).

In a similar way, we can extend \(\mathbb{C}^2\) to the complex projective plane \(\mathbb{P}^2\) by adding points at infinity, and the most convenient way to do so is by homogenous coordinates. We start with non-zero vectors \((u,v,w)\) in \(\mathbb{C}^3\) and consider \((\lambda u, \lambda v, \lambda w)\) to define the same point of \(\mathbb{P}^2\) as \((u,v,w)\) if \(\lambda\in\mathbb{C}\setminus\{0\}\). Then \([u:v:w]\) are homogeneous coordinates on \(\mathbb{P}^2\). The points with \(u\ne0\) correspond to points of the original complex plane \(\mathbb{C}^2\), by sending \([u:v:w]\) to \((v/u,w/u)\). The points with \(u=0\) constitute the new line at infinity, which is just a copy of \(\mathbb{P}^1\) with coordinates \([0:v:w]\).

Now we can extend the cardioid and the deltoid to curves in \(\mathbb{P}^2\), not just \(\mathbb{C}^2\). We start with the parameterizations \(\gamma_C\) and \(\gamma_D\), append an initial coordinate of 1, then clear denominators (we can do this because of the equivalence that defines homogeneous coordinates). Then we get

\(\gamma_C(t) = \big[t^2:2t^3 + t^4:2t + 1\big] \qquad\) and \(\qquad \gamma_D(t) = \big[t^2:2t^3 + 1:2t + t^4\big]\).
These allow for the possibility of \(t = 0\), but apparently leave out the point at infinity \(\infty\), so we make one more modification, replacing \(t\) with \(t/s\) and again clearing denominators to obtain
\(\gamma_C([s:t]) = \big[s^2 t^2:2st^3 + t^4:2s^3t + s^4\big] \qquad\) and
\(\qquad \gamma_D([s:t]) = \big[s^2t^2:2st^3 + s^4:2s^3t + t^4\big]\).
Here we see a feature characteristic of maps from one projective space to another, when homogeneous coordinates are used: each component of the map must be homogeneous of the same degree (in this case, four). By expressing the parameterizations of the cardioid and the deltoid in this way, we see that both curves touch the line at infinity at the two points \([0:1:0]\) and \([0:0:1]\), corresponding to \([0:1]\) and \([1:0]\), respectively, for the cardioid, and in the reverse order for the deltoid. Still this isn’t enough to show that the curves are the same! We need one more ingredient.

A projective transformation of \(\mathbb{P}^1\) or \(\mathbb{P}^2\) is induced by a linear transformation of the homogeneous coordinates. Readers who are already familiar with the Riemann sphere will recognize projective transformations of \(\mathbb{P}^1\) as Möbius transformations (also known as fractional linear transformations): given \(a,b,c,d\in\mathbb{C}\), we can convert \([s:t] \mapsto [as+bt:cs+dt]\) to a Möbius transformation in the coordinate \(z = s/t\), where it becomes \(z \mapsto \dfrac{az+b}{cz+d}\). The condition for this function to be invertible is \(ad - bc \ne 0\), which is the same as the condition for the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) to be invertible. In the same way, projective transformations of \(\mathbb{P}^2\) arise from invertible linear transformations of \(\mathbb{C}^2\). Two objects in \(\mathbb{P}^1\) or \(\mathbb{P}^2\) are called projectively equivalent if there is a projective transformation that carries one to the other. And now we can state precisely what was meant in the opening paragraph:

The cardioid and the deltoid are projectively equivalent in \(\mathbb{P}^2\).

But how do we find the projective equivalence? A clue may be found in one clear difference between the original curves drawn in the Euclidean plane, which niggled at me while I was trying to figure out their relationship. The deltoid clearly has three cusps, while the cardioid apparently only has one. If the curves are equivalent, where are the other cusps of the cardioid? The answer: on the line at infinity!

How can we tell? It’s time to apply some differential geometry and look at the tangent lines of these two curves. Returning to the parameterizations in terms of \(t\), we find

\(\gamma_C'(t) = \left(2 + 2t,-\dfrac{2}{t^2} - \dfrac{2}{t^3}\right) \qquad\) and \(\qquad \gamma_D'(t) = \left(2 - \dfrac{2}{t^3},-\dfrac{2}{t^2} + 2t\right)\).
Now a line in \(\mathbb{C}^2\), with coordinates \((v,w)\), passing through \((a,b)\) in the direction \((s,t)\) has the equation \(\begin{vmatrix} s & v - a \\ t & w - b \end{vmatrix} = 0\). Thus the tangent line to the cardioid at \(\gamma_C(t)\) has the equation \[ \begin{vmatrix} 2 + 2t & v - \big(2t + t^2\big) \\ -\frac{2}{t^2} - \frac{2}{t^3} & w - \big(\frac{2}{t} + \frac{1}{t^2}\big) \end{vmatrix} = 0 \] which, after some simplification, becomes \[ wt^3 - 3t^2 - 3t + v = 0\text{.} \] This is the line equation of the cardioid. In a similar fashion, we can find the line equation of the deltoid, which is \[ t^3 - vt^2 + wt - 1 = 0\text{.} \]

Having the line equation of a curve, in terms of a parameter \(t\), can be useful in several ways. As \(t\) varies over \(\mathbb{P}^1\), it produces all the tangent lines of the curve. (We’ll clarify what happens when \(t = \infty\) in a moment.) But we can also let \((v,w)\) vary over \(\mathbb{C}^2\) and find, for each point, which tangent lines of the curve pass through that point. Because the line equations of the cardioid and the deltoid are cubic polynomials in \(t\), most points of \(\mathbb{C}^2\) will lie on three tangent lines. Those points that lie on fewer than three tangent lines play a special role.

Let’s illustrate first with the deltoid. We’ll be looking at lots of cube roots, so let \(\omega = e^{i\,2\pi/3}\); this means that \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\). When \((v,w)=(0,0)\), the line equation becomes \(t^3 - 1 = 0\), so the tangent lines of the deltoid that pass through the origin correspond to the parameters \(1\), \(\omega\), and \(\omega^2\). Indeed, the three points \(\gamma_D(0) = (3,3)\), \(\gamma_D(\omega) = (3\omega,3\omega^2)\), and \(\gamma_D(\omega^2) = (3\omega^2,3\omega)\) are the three cusps of the deltoid. On the other hand, a point that belongs to the deltoid lies on tangent lines corresponding to at most two parameters (two of the points of tangency have “coalesced”). For example, when \((v,w)=(-1,-1)\), the line equation becomes \(t^3 + t^2 - t - 1 = 0\), or \((t+1)^2(t-1) = 0\). At a cusp, all three tangent lines coincide: for example, when \((v,w)=(3,3)\), the line equation is \(3t^3 - 3t^2 + 3t - 3 = 3(t-1)^3 = 0\). See the pictures below.

We can homogenize the line equation of the deltoid by replacing \(t\) with \(t/s\) and \((v,w)\) with \((v/u,w/u)\) and clearing denominators to obtain: \[ ut^3 - vst^2 + ws^2t - us^3 = 0\text. \] When \([s:t] = [1:0]\) or \([0:1]\) (remember, this second point in homogeneous coordinates corresponds to \(t=\infty\)), we get the same equation of the tangent line, \(u = 0\). This is the equation of the line at infinity, so the line at infinity is tangent to the deltoid at both \([0:0:1]\) and \([0:1:0]\)! A line that is tangent to a curve at two points is called a bitangent.

The cardioid also has a bitangent, which is easier to see: when \(t = \omega\) or \(t = \omega^2\), respectively, the line equation of the cardioid becomes \(w - 3\omega^2 - 3\omega + v = 0\) or \(w - 3\omega - 3\omega^2 + v = 0\), both of which are equivalent to \(v + w = 3\). The visible cusp occurs at \((-1,-1)\), where the line equation becomes \((t + 1)^3 = 0\). For an example of more generic behavior, look at \((-3,-3)\), where the line equation becomes \(3t^3 + 3t^2 + 3t + 3 = 0\), or \(3(t + 1)(t + i)(t - i) = 0\). See pictures below.

The homogeneous version of the cardioid’s line equation is \[ wt^3 - 3ust^2 - 3us^2t + vs^3 = 0\text. \] When \([s:t] = [1:0]\), this becomes the \(w\)-axis \(v = 0\), and when \([s:t] = [0:1]\), we get \(w = 0\). In each of these cases, we see that only one tangent line passes through the point, just as we saw for the cusps of the deltoid. So we have identified the three cusps of the cardioid—\([1:-1:-1]\), \([0:0:1]\), and \([0:1:0]\). The tangent lines through all three of these cusps pass through the origin in \(\mathbb{C}^2\), with homogeneous coordinates \([1:0:0]\).

We now have enough information to show the equivalence of the cardioid and the deltoid. To define a projective transformation from \(\mathbb{P}^1\) to itself, we need to specify where three points go; to define a projective transformation from \(\mathbb{P}^2\) to itself, we need to specify the images of four points, no three of which are collinear. We’ll show how to transform the line equation of the deltoid into the line equation of the cardioid via pullback.

We’re looking for projective transformations \(f : \mathbb{P}^1 \to \mathbb{P}^1\) and \(g : \mathbb{P}^2 \to \mathbb{P}^2\) such that \(\gamma_D \circ f = g \circ \gamma_C\). Starting with \(f\), we require

\(f\big([1:0]\big) = [1:\omega]\),   \(f\big([0:1]\big) = [\omega:1]\),   and   \(f\big([1:-1]\big) = [1:1]\),
so that the parameters of the cardioid’s cusps are sent to those of the deltoid’s cusps. This can be accomplished by defining \[ f\big([s:t]\big) = [s - \omega t : \omega s - t]\text. \] Meanwhile, \(g\) needs to satisfy
\(g\big([0:0:1]\big) = [1:3\omega:3\omega^2]\),  \(g\big([0:1:0]\big) = [1:3\omega^2:3\omega]\),
\(g\big([1:-1:-1]\big) = [1:3:3]\),   and   \(g\big([1:0:0]\big) = [1:0:0]\),
which is accomplished by \[ g\big([u:v:w]\big) = [ 3u+v+w : 3\omega^2 v + 3\omega w : 3\omega v + 3\omega^2 w ]\text. \] Now substitute the components of \(f\) and \(g\) into the variables of the deltoid’s line equation, expand, and simplify. The result is the line equation of the cardioid. You can calculate this by hand, or just let SageMath do it for you:

One of the curves mentioned in the title of this post has been conspicuously absent so far: the folium of Descartes. This is another favorite curve of mine, invariably given in my calculus classes as an exercise in implicit differentiation. Its equation is \(x^3 + y^3 = xy\).

So what’s the connection between this curve and the others? Well, if we extract the coefficients from the deltoid’s line equation and use them to define a new curve \(\gamma_F\), we get \[ \gamma_F\big([s:t]\big) = [ s^3 - t^3 : st^2 : -s^2 t ]\text, \] which parameterizes \[ v^3 + w^3 = uvw\text, \] the homogeneous version of the folium’s equation. This means that the folium is dual to the deltoid (and thus also to the cardioid)! The tangent lines of the cardioid/deltoid have been converted into points of the folium, and likewise points of the cardioid/deltoid become tangent lines of the folium. Just as each point of \(\mathbb{C}^2\) lies on three tangent lines of the cardioid/deltoid, counted with multiplicity, each line of \(\mathbb{C}^2\) intersects the folium at three points, counted with multiplicity. The bitangent of the deltoid and cardioid has been converted into a point of self-intersection. If we look at points of the form \([1:v:\bar{v}]\), then the threefold symmetry of the folium is revealed (the three asymptotic directions correspond to the three tangent lines that pass through the origin, which as we saw are the tangent lines at the cusps).

Saturday, December 29, 2018

an IBL preface

In just over a week, I will distribute to students the first piece of the complex variables notes I have been writing. Here is a preface to be included with the notes, to motivate the IBL structure. The details of the class will be spelled out in the syllabus; this is just to set the tone.

You are the creators. These notes are a guide.

The notes will not show you how to solve all the problems that are presented, but they should enable you to find solutions, on your own and working together. They will also provide historical and cultural background about the context in which some of these ideas were conceived and developed. You will see that the material you are about to study did not come together fully formed at a single moment in history. It was composed gradually over the course of centuries, with various mathematicians building on the work of others, improving the subject while increasing its breadth and depth.

Mathematics is essentially a human endeavor. Whatever you may believe about the true nature of mathematics—does it exist eternally in a transcendent Platonic realm, or is it contingent upon our shared human consciousness? is math “invented” or “discovered”?—our experience of mathematics is temporal, personal, and communal. Like music, mathematics that is encountered only on as symbols on a page remains inert. Like music, mathematics must be created in the moment, and it takes time and practice to master each piece. The creation of mathematics takes place in writing, in conversations, in explanations, and most profoundly in the mental construction of its edifices on the basis of reason and observation.

To continue the musical analogy, you might think of these notes like a performer’s score. Much is included to direct you towards particular ideas, but much is missing that can only be supplied by you: participation in the creative process that will make those ideas come alive. Moreover, the success of the class will depend on the pursuit of both individual excellence and collective achievement. Like a musician in an orchestra, you should bring your best work and be prepared to blend it with others’ contributions.

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

These notes are a guide. You are the creators.

Monday, September 03, 2018

2018 calculus syllabus

In my last post, I explained a bit about how I feel like my syllabus is a work-in-progress, even though the semester has started and I’m already using it. In this post I’ll give some more details and even more history. I’ll quote extensively from my syllabus verbatim; here is a link to the entire thing for anyone who is interested.

Revising my syllabus for this semester really began last fall. I wasn’t entirely blind to the faults that were starting to show. One major change was in restructuring the exam schedule. When I switched to standards-based grading in calculus 1, I also started weekly quizzes (which students took on their own time outside of class) and had three midterm exams plus a final. The quizzes functioned as a sort of preliminary assessment for most of the standards. Each test covered about eight standards. After the third test, there were a couple more standards we covered in class, which were only assessed on a quiz and the final exam. Even with three midterms, however, I had often felt like students were rushed in completing them. I also began to question the value of the out-of-class quizzes. So I turned the quizzes into “labs” that students were free to collaborate on, and I switched from three midterm exams to five, which would formally assess every standard before the final.

I really liked how having five midterms broke up the material. Each test became more coherent in the material it included. Exam 1 covered limits. Exam 2, definition and interpretation of derivatives. Exam 3, rules for differentiation. Exam 4, applications of derivatives. Exam 5, definition of integrals and the Fundamental Theorem of Calculus. Especially helpful was splitting up the applications of derivatives (l'Hospital's rule, optimization, related rates, and so on) from the introduction to integrals; these topics had usually been all jumbled together in the last midterm, compounding the difficulty already created by it being late in the sester. Also, by dedicating one test just to derivative rules, I was moving towards having a Differentiation Gateway Exam, as several of my colleagues at Pepperdine use. And paired with that move was an awareness that I was gravitating towards a specifications framework.

This fall, I decided to maintain the five-midterm structure and get rid of the quizzes/labs, to be replaced by an occasional more substantial homework exercise that will be used in class. I collected seven standards into the Gateway Exam, which will form the bulk of the third midterm. I split the remaining standards into 45 “tasks”, which is a term I hope will be clearer than “standards”; each standard split into approximately two tasks. The idea of tasks goes back in my mind to the list of problems Kate Owens shared from her Ph.D. advisor George McNulty. That is, a task is a specific type of problem that students will show they know how to solve. Here is the new introduction to the “Goals and Assessment” section of my syllabus:

Change is present all around us, and understanding it is an essential component of many fields of study. Calculus is fundamentally a set of tools for measuring, quantifying, estimating, and interpreting change in a variety of contexts. In this course, we will delve into some of the most profound ideas in mathematics, whose roots are from ancient times and which began to develop fully in the 17th century; they continue to form the basis for much of modern science. My hope is that this class will develop your analytical ability and deepen your appreciation for the power and elegance of mathematics.

The skills you should acquire are related to the Learning Outcomes stated on the first page of this syllabus. Your mastery of the course content will be assessed through your performance on a collection of definite tasks. A complete list of tasks is on the last two pages of this syllabus. These tasks, rather than points or percentages, will be the primary basis for grading. The following sections provide details on how the tasks will be assessed and what you should accomplish in order to earn your final grade.

My hope is that this method of assessment, called standards-based grading or mastery grading, will keep you clearly informed as to the expectations of the class and how well you are meeting them, while also removing the (often distracting) elements of linear grading that uses letters or total points. Learning is not always a straightforward process, and one of my goals is to give you as many opportunities as possible to demonstrate your understanding. I will be glad to do everything I can to help you towards your goal of mastery. If you have questions or concerns at any time, please feel free to discuss them with me.

Another potential source of confusion from my SBG system in the past was the levels of ranking. I really liked that we were using the vocabulary of mastery / proficiency / basic ability / novice to talk about students’ progress, but it was rare that a student could rate their own work with one of these levels. So this fall I opted, as many others have, for a simple pass/fail approach on tasks. I don’t like the pass/fail language, however, so I chose successful for a task completed satisfactorily and progressing or incomplete for work that has major mistakes or is absent. I also wanted to handle small mistakes through a faster revision process, an idea I picked up from MathFest; for these situations, I added a revisions needed category. Here is how I describe the rating system in my syllabus:

A task is a problem or a collection of similar problems that should be solved using calculus tools. Your progress in the class will be measured in terms of the number of tasks that you accomplish. Partial credit is not given; a task must be fully successful in order to count towards your final grade. Whenever a task is included on an exam, your work will receive one of four ratings:
Ssuccessful Solution is complete and correct.
Rminor revisions needed Solution is correct except for small errors.
Pprogressing Partial understanding is evident, but solution contains substantial errors.
Iincomplete Not enough evidence is available to provide an assessment.
A task marked “S” has been completed; you can check it off the list at the end of the syllabus.
A task marked “R” can have a small mistake such as an arithmetic error or a miscopied value. You will have 48 hours (or over the weekend if the work is returned on a Friday) to complete a Revision Form that explains how to correct the mistakes, and to submit the form along with your original work, in order to earn a successful rating.
A task marked “P” demonstrates progress in mastering the topic, but reassessment is necessary in order to successfully accomplish the task.
A task marked “I” shows little or no relevant work. Reassessment is necessary.
To show mastery of a task after it has received a rating of P or I, see the section entitled “Reassessment” on the next page.
Hopefully this simplified rating system will also make it easier for me to track student progress over the duration of the semester and analyze trends afterwards. (I agree with Kate that Drew’s “A tale of two students” chart was a moment of clear inspiration at MathFest.)

In order to help students know what is expected to prepare for reassessment, and to help me schedule them more effectively, I have introduced Reassessment Tickets:

After a task has been assessed on an exam, you may schedule a reassessment if you did not successfully complete the task. This is a two-step process:
  • First, pick up a Reassessment Ticket from my door or download and print one from the Courses site. Complete the form and return it to me at least 24 hours before you want a reassessment.
  • Second, once a meeting is scheduled, come to my office and I will give you a new opportunity to demonstrate mastery of the task. If possible, I will grade your work immediately; otherwise, I will let you know the result by the following day.
I will reassess up to two tasks per student per week. In addition, you can use exam days as opportunities for reassessment of up to three tasks, provided you let me know 48 hours in advance which ones.
I plan to use one or two class days at the end of the semester for reassessment alongside review, as well.

Another element I introduced was subcategories of tasks: “core”, “modeling”, and “additional“ (not a great name, I’ll try to find a better one in the future). Again, lots of other people are already doing this, and I like what many of them are doing, which is to require two demonstrations of mastery for core skills and only one for the rest. I couldn’t figure out how to make that work with my system, so I made the following distinction: core tasks are the ones that could appear on the final exam. There are 14 of them, and I will choose seven to go on the final. (The final exam will also include a reflection essay, and a period of time for additional reassessment.) I also set higher expectations for how many core vs. additional tasks needed to be successful at each grade level.

What I learned from creating a list of tasks is that, because I state exactly what types of questions I will include on the tests, there is less wiggle room than with standards, which could always be applied to new sorts of problems. (This is the distinction between activity and ability I talked about in my last post.) I don’t know if my list of tasks, or the categorizing thereof, is ideal, but it is certainly enough to guarantee that a student who succeeds at all of them will have mastered calculus 1. (I used Robert’s classification of “core” and “supplemental” learning targets as a reference while I was sorting, but our lists don’t match up exactly.)

After all the work that goes into getting away from letter grades in a standards-based system, it’s always a bit dispiriting to turn back to them. So I start my section on “Final letter grades” with a bit of reluctance (not to say snark).

At the end of the semester, I am required to submit to the university a letter grade reflecting your achievement in this class. Here is how that grade will be determined.

To earn an A: in addition to passing the Gateway Exam and completing the Final Reflection,
  • Submit 20 homework reports.
  • Complete all modeling tasks.
  • Complete all core tasks.
  • Complete 25 additional tasks.
  • Complete 6 core tasks on the final exam (minor errors are acceptable).
To earn an B: in addition to passing the Gateway Exam and completing the Final Reflection,
  • Submit 15 homework reports.
  • Complete 2 modeling tasks.
  • Complete 12 core tasks.
  • Complete 21 additional tasks.
  • Complete 5 core tasks on the final exam (minor errors are acceptable).
Passing the Gateway Exam is required to earn a final grade of B– or higher.

To earn an C: in addition to completing the Final Reflection,
  • Submit 10 homework reports.
  • Complete 1 modeling task.
  • Complete 10 core tasks.
  • Complete 17 additional tasks OR pass the Gateway Exam and complete 14 additional tasks.
  • Complete 4 core tasks on the final exam (minor errors are acceptable).
Failure to complete a Final Reflection will result in a grade of D or F.

To earn a D:
  • Submit 5 homework reports.
  • Complete any 30 tasks from C.1–C.14, A.1–A.28, M.1–M.3 OR pass the Gateway Exam and complete any 23 tasks.
  • Complete 3 core tasks on the final exam (minor errors are acceptable).
Plusses and minuses will be assigned as follows: if all criteria for a letter grade are met as well as two or three of those for a higher letter grade, then a plus will be added. If all but one or two criteria for a letter grade are met, and the remaining items meet the criterion for one letter grade lower, then the higher letter will be given with a minus added.

I will use my discretion to assign a final letter grade in cases where a different set of conditions is met.

So there it is. My syllabus for calculus this semester. I’m sure by December, or even October, I’ll have a much better notion of what changes I should have made. I’ll let you know how it goes.

(I should also have given more attribution in this post to the people I stole ideas from, especially at MathFest, but I don’t have those notes on hand right now. So a general word of thanks goes out to this very sharing community.)

Sunday, September 02, 2018

prologue to a syllabus

(This was originally supposed to be the post in which I describe my syllabus for the fall. I started writing some preliminary comments, and they got out of control. I’ll get back to the syllabus itself in my next post.)

First, I must express some gratitude. Thanks to parental leave provided by the state of California and my school, I did not have any teaching duties last spring. It was my first time not teaching first-semester calculus in four years. As I tell my students, calculus 1 is actually one of my favorite classes to teach, but I could tell by last fall that some parts of the course were getting stale. Having a semester break meant that, in addition to getting to know my newborn daughter, I could let my ideas on how to improve calculus instruction and assessment simmer for a bit.

Actually, it’s not entirely honest to refer to “my ideas” in this setting; what I really needed was a chance to reflect on ideas I’d been picking up (stealing) from others, and even better, to acquire (steal) some fresh ideas, which a workshop and conference provided over the summer. A fabulous community of college and university math teachers has formed around the question of how to improve our assessment practices, and the rate at which sharing/stealing/developing ideas is remarkably fast.

Over the past few weeks, as the fall semester has started up, several people have shared their syllabi along with extensive, thoughtful commentary on how they created them. I’ve been holding back, however, because while I believe my syllabus is better than it was last year, by the time the semester started I only felt like I had gotten it to “just good enough.” Some ideas aren’t fully developed yet, some feel out of balance, and some are plain risky. Nevertheless, in the spirit of community and maintaining a growth mindset, I’ve decided to go ahead and share my syllabus, too, warts and all.

Since I see this as a long-term work in progress, I’d like to begin with a few words about that progress. (These comments will parallel somewhat my talk from MathFest last month.) I started using standards-based grading in spring 2013, largely as a way to improve the feedback I was giving students. After a reasonably successful first attempt, I began using alternative assessment methods in all of my classes. Some worked better than others, but because I was teaching calculus 1 so often, my SBG system for that class developed into a collection of 25–30 standards that became fairly stable.

Around the same time, Robert Talbert was blogging about specifications grading, a well-developed and flexible framework whose goals, in the words Linda Nilson uses to subtitle her book on the topic, are “restoring rigor, motivating students, and saving faculty time.” For a while I remained skeptical about specs grading, because I couldn’t understand why anyone would turn to something besides my beloved standards. Eventually, however, I realized that SBG as I conceived it didn’t work in every situation, and so I delved more into specs. The Google+ community initiated by Robert goes by the name SBSG, to include both standards-based and specifications grading. Today the language of the community encompasses these and other alternative assessment systems under the broader term mastery grading, which hearkens back to Bloom’s terminology of mastery learning.

At MathFest, I talked a bit about this history of my classes and did some compare-and-contrast between SBG and specs grading. Possibly the most useful contribution I made to that session was the following six-word summary of how they relate:

Standards emphasize content.
Specifications emphasize activity.
Here’s another way to phrase the distinction in my mind: When we create standards, we are answering the question what do we want students to be able to do? When we create specifications, we are answering the question what do we want students to have done? More bluntly, standards are what we want to measure, while specifications are what we can actually measure; the latter is a proxy for the former.

I guess my claim is that standards and specifications support each other: they are two sides of the same coin. We need specifications in order to determine how standards will be assessed, and a clear list of standards keeps specifications from becoming arbitrary. (Or as Drew Lewis said on Twitter, “specs are how I assign letter grades, with the primary spec being mastery of standards.”) Whether I say that an assessment system is based on specifications or standards depends on whether the description of the system focuses on the proxy or the thing for which it proxies.

By last fall, some cracks in my SBG system for calculus had started to show. Every semester, I had a couple of students at the end of the course who still thought it wasn’t clear. The homogeneity of the list of standards was mushing the most important concepts together with secondary ones. Worst of all from a practical standpoint, I was finally getting overwhelmed by reassessments after years of claiming that SBG didn’t take any more of my time than traditional grading. I knew I needed to make some changes to clarify and streamline the assessment process.

What I have for now isn’t perfect, but it will get me through the semester. With this lengthy prologue complete, in my next post I’ll share parts of my syllabus and explain what I hope it achieves.

Monday, August 27, 2018

“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.

Monday, August 20, 2018

dialectics in mathematics

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns”. Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking. (intro by Michael Pershan)

I want to talk about how we respond to polarities. Here I mean “polarity” in the philosophical sense (a pair of concepts that are apparently in conflict) rather than in a mathematical sense. When we encounter a struggle or tension between goals or ideas, we tend to create one of two things:

  • dichotomy — a conclusion that the two ideas are irreconcilable and the choosing of sides, or
  • synthesis — a selection of desirable features from each and the attempt to make those features coexist.
While each approach is at times appropriate, both have their downsides. Establishing a dichotomy means that one side tends to be silenced and its contributions lost. Creating a synthesis can mean that neither side is fully honored; everything is compromise.

I propose a third option, an alternative to dichotomy or synthesis: this approach is dialectic — upholding both sides fully, maintaining the two ideas in tension so that a conversation may arise between them. Etymologically, “dialectic” comes from the roots “dia” (“across”) and “logos” (“word” or “reason”), so its underlying meaning may be read as “speaking across a divide”. Dialectics can simply refer to discussion or debate between two opposing sides, but I use it to denote a state that seeks not resolution, but rather the fruitfulness of an irreducible struggle. Doing so acknowledges the worth, validity, and potency of both sides. It can therefore be used in the classroom to foster the inclusion of diverse perspectives, even in mathematics.

Our group’s discussion began with an essay by Timothy Gowers entitled “The Two Cultures of Mathematics”. In this piece, Gowers makes the claim that most mathematicians are either “problem solvers”, who prefer to attack specific open problems that they believe are important, or “theory builders”, who prefer to develop a large, coherent body of understanding. The former are interested in general theory mainly insofar as it provides ways to solve their problems; the latter are interested in specific problems mainly insofar as they spur deeper insights or new directions for theory.

This subdivision is similar to the pure/applied separation we often talk about in mathematics, though it is not quite the same thing. Even the problems Gowers mentions fall well within the “pure” category. But these two polarities (pure/applied, theory/problems) share the feature that adherents of one side tends to be a bit snobbish towards those of the other.

Pure mathematicians tend to look on applied mathematics as, at best, a dirty form of math or, at worst, not truly math at all. G. H. Hardy, in his famous essay A Mathematician’s Apology, describes pure mathematics as more enduring, more exciting, and more “real” than applied mathematics. (He does make clear that what he considers “applied” mathematics limits itself to “elementary” tools, which more-or-less means grade-school arithmetic up through introductory calculus, and so his notion of applied mathematics might no longer suffice. I’ll get back to Hardy shortly.)

Gowers claims that, in a similar way, theory-building is currently “more fashionable” than problem-solving in the math world. (Rather than drawing the analogy with pure and applied mathematics, however, he compares this snobbishness with one, observed by C. P. Snow in “The Two Cultures”, held by humanities toward the sciences.) He laments that “this is not an entirely healthy state of affairs” and spends most of his essay defending problem-solving areas of math (combinatorics, in particular) against some perceived criticisms. His argument suggests to me that both the theory-building and the problem-solving camps should be upheld without one attempting to overcome the other; that is, a healthier state can be reached by sustaining a dialectic.

How can we think about theory-building vs. problem-solving in our classes?

For one thing, many of our students are trained problem-solvers. For them, learning mathematics means developing an appropriate response to any given stimulus. If a problem statement includes this-or-that word or phrase, then I should use such-and-such a technique to find a solution. For many of us instructors, however, it is the abstraction of ideas that drew us to mathematics. What is possible in this situation? To what extent can the possibilities be quantified and categorized? If theory-building is currently en vogue in mathematical culture, then I suspect we who teach are not immune to that trend. But here comes the question of motivation: what will draw students into doing mathematics? In many cases, the answer is… a problem. The problem may be “applied” (e.g., how does a population grow over time) or “pure” (e.g., how does the size of a square increase when its side length increases?), but a concrete connection provides an open door to considering broader mathematical truths. Such problems can lead into developing theory (e.g., what properties do exponential and polynomial functions share, and what distinguishes them?).

But developing theory for its own sake has been a part of mathematics since at least Euclid; we do our students a disservice if we neglect this aspect of doing math. A theory crystallizes into a single lattice ideas that might otherwise have been perceived as disconnected. Algebra in particular provides a unifying framework for solving individual problems. On the other hand, non-constructive statements are by turns inspiring and infuriating. It is no small movement from the (typically algebraic) claim that “A solution exists! And you can find it by following these steps…" to the (typically analytic) claim that “A solution exists! And you may never find it exactly…” This theory in turn motivates a slew of new problems: if nothing else, how shall we find solutions as close to the true answer as we desire?

In any case, it is useful to abide by a constructivist view of knowledge: students will understand best the structures that they form in their own minds, whether by induction (problem-solving) or deduction (theory-building), and they should be presented with ample opportunities for both forms of construction.

[Side note: in his keynote post for this series, Michael describes an occasion where he side-steps, or deconstructs, the theory-building/problem-solving divide by encouraging math-doers to create their own questions based on a simple prompt, questions which could easily veer in any direction, including problem-solving or theory-building.]

It is not hard to find other places in mathematics where polarities exist and a choice must be made: dichotomy, synthesis, or dialectic? A few weeks ago, I made a bit of a fuss on Twitter, claiming that everything Hardy wrote about mathematical culture should be read skeptically. The context for my criticism was an oft-shared quote from “A Mathematician’s Apology”: “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

A question immediately presents itself: who decides what is beautiful? Any claim to objectivity is nearly always tied up with privilege. The answer cannot be “all mathematicians” because we all have such different tastes and preferences. Nor can the answer be “a special subset of mathematicians” because the choice of that subset will inevitably be determined by power structures within the mathematical community. But neither is the answer that all mathematics is equally beautiful. The standard of beauty may be subjective, but that does not mean it is arbitrary. We value beauty, but it is not the sole or even the primary standard by which we judge mathematics.

Hardy argues that all mathematics considered “useful” is essentially “dull” or “trivial”. He seeks to create a dichotomy between the beautiful and the practical. Perhaps he didn’t foresee the computing revolution. He couldn’t predict that number theory would be used in encryption, or that general relativity would be used for GPS, or that differential equations would be used for movie animations. Perhaps he would not consider these applications to be built on the deepest, truest parts of those theories. (To be fair, at the time he wrote, Hardy was distressed by the ways in which science had been used in the cause of warfare, and wanted to establish some distance between pure mathematics and that particular set of applications.)

From an evangelistic perspective, potential converts (our students in particular) may be drawn in from either side: the aesthetic or the practical. I personally was first attracted to geometric form and the lovely, counterintuitive properties of mathematical relations. Some of my students have tastes similar to mine, but many more will be convinced of the predictive power of mathematics before they accept its inherent attractiveness.

Beyond this, however, neither beauty nor usefulness can or should be subjugated to the other. Mathematics is grounded in both. They can hone each other, but they can also proceed independently. Progress flows from the pursuit of either. It would be a mistake, I believe, to claim that either is the true purpose of mathematics; we should support both of them in our minds and in our classrooms.

Not every tension needs to be handled this way, but examples of dialectical pairings abound: precision and approximation, confidence and confusion, individual and community. I encourage us all to consider times when it can be productive not to resolve such conflicts but instead to foster a breadth of understanding from them.

Friday, August 10, 2018

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!