Pedagogy – Creativity in Mathematics

My Polyhedra Textbook, II

In my last post, I talked about a textbook I had written to use in a course which, among other things, introduced students to a non-Euclidean geometry — spherical geometry. And while the primary purpose of the text was for teaching a college-level course, the actual content of the text had a different origin.

As I have mentioned previously, I got interested in polyhedra during graduate school, and was very fond of taking books about polyhedron models out of our mathematics library at Carnegie Mellon. It was easy enough to photocopy the nets provided, or use the numerical data to construct nets on my own.

But what was missing for me, a mathematician-in-training, was a more rigorous discussion of these intriguing models. For example, Wenninger’s Polyhedron Models contains many nets with some idea of how to construct them from a more general geometric point of view, but few details. And his Spherical Models contains many tables of data of angular measures to three decimal places, but many derivations are missing. Other books included metrical data such as the measures of dihedral and edge angles and circumradii, but again, often only numerically.

I should point out that this is not necessarily a criticism of these wonderful books, since each book had its own purpose. What I found, though, was that there was really no book out there which took a more mathematically sophisticated viewpoint at an intermediate level.

So I decided that if I wanted to know more about polyhedra — that is, know the exact metrical data associated with polyhedra, not just numerical approximations — I needed to do some work on my own.

I used two approaches — coordinates and linear algebra, and spherical trigonometry. In both cases, I wanted precise results — so typically, this meant knowing exact values for the cosines of dihedral and edges angles, for example. When you know the cosine of an angle, you know the angle.

Why the cosine? The sine has the disadvantage of being ambiguous, since many dihedral angles of polyhedra are obtuse — so you need more information than just the sine of the angle. Using the tangent involved the troublesome case of $\dfrac\pi2$ in formulas — which occurred rarely, but was still a nuisance. Some sources use the tangent of the half-angle, but this often involves additional calculations.

For my geometry course, I focused on spherical trigonometry — recall that I did not want linear algebra as a prerequisite for the course so that it would be accessible to a broader student demographic. But I still wanted to keep the mathematical rigor — use spherical trigonometry to calculate exact values of the cosines of relevant angles.

To give you an idea of what’s involved, let’s calculate the dihedral angle of a dodecahedron — that is, the angle between any two pentagonal faces of a regular dodecahedron.

Fig25c

To use spherical trigonometry (see a previous post on spherical geometry for a refresher), we imagine a small sphere centered at a vertex of the dodecahedron — it will carve out a spherical triangle, as shown below.

sphdodecahedron

Here, a is a vertex of the dodecahedron, and points b, c, and d are the points where the small sphere intersects the edges of the dodecahedron meeting at a. This creates a spherical triangle whose sides all have measure 108°, since the interior angles of a regular pentagon all have measure 108°.

The angles between the sides of the spherical triangle with vertices a, b, and c are the dihedral angles of the dodecahedron, whose measure we call D. We may apply the law of cosines for spherical triangles in order to find D (see the relevant Wikipedia page; in the textbook, spherical trigonometric formulas are discussed in detail):

$\cos108^\circ=\cos108^\circ\cos108^\circ+\sin108^\circ\sin108^\circ\cos D.$

To find $\cos D,$ we need to know the following (also explained in the text):

$\cos108^\circ=\dfrac12(1-\tau),\quad\sin108^\circ=\dfrac12\sqrt{\tau+2},$

where $\tau$ is the golden ratio. (I use $\tau$ for the golden ratio as this is the notation Coxeter used in his well-known Regular Polytopes, which inspired generations of polyhedral enthusiasts.)

Solving yields

$\cos D=-\dfrac1{\sqrt5},$

so that D is approximately 116.6°.

Now this is not the only way to find D; it is possible to find Cartesian coordinates for the vertices of the dodecahedron and use some linear algebra, for example. But using spherical trigonometry is straightforward and elegant — and is surprisingly versatile.

When I teach my polyhedra course, we use spherical trigonometry to find edge and dihedral angles of all the Platonic and Archimedean solids. Further, we use spherical trigonometry to design geodesic models, like the 4-frequency icosahedron shown here.

So the emphasis is on applying spherical trigonometry to the construction of physical models. In a typical two-lecture-per-week course, one lecture is always a hands-on laboratory, where we build a polyhedron or geodesic model we studied previously. We never build a model without knowing precisely how it is designed — we first understand the mathematics of the model, and then we build it. Included in the text is a week-by-week outline of how I’ve used the text in the classroom.

Now the classroom textbook is just Part I of the book. Part II is a bit more technical, and intended for the true polyhedral enthusiast. It turns out that spherical trigonometry is a powerful tool for studying polyhedra. In fact, the dihedral angles of all the uniform polyhedra may be calculated using spherical trigonometry — even the most complex snub polyhedra. However, it is sometimes necessary to solve sixth-degree polynomials in order to do so!

I thought to add Part II to the volume as I had already done all the calculations some years ago. And, to my knowledge, the calculation of the dihedral angles of all the uniform polyhedra using spherical trigonometry has not been published before. So I hope to contribute something to the literature of classical polyhedral geometry by publishing this book, in a way that someone with a modest mathematical background can understand.

As I continue with the book project, I’ll post updates as I discover interesting geometrical tidbits along the way!

On Assessment, V

Last week, I ended with a sample exam I might give a calculus course which included both Skills problems and Conceptual problems. Before presenting the final installment of this series on assessment, I thought I’d take a few moments to discuss the genesis of this exam format.

Again, the assumption here is that we are working in a more traditional system, where students must be assigned grades, and these grades must in large part be based on performance on exams.

Given IMSA’s statements about the advanced nature of their curriculum, I had concerns about the fairly traditional exams we gave in mathematics. In my mind, there was little to distinguish our exams from those given in any other rigorous calculus course.

The reason given to me by other mathematics faculty was that there just wasn’t time in a roughly hour-long exam to assess conceptual understanding. I wasn’t convinced, and I started thinking of an alternative. I did agree, however, that it wouldn’t work to have a conceptual question on an exam which might take half the exam period for a significant fraction of the students to complete.

What I finally settled upon was including a range of conceptual problems for which students only needed to provide a reasonable approach to solving. If you chose a conceptual problem which happened to be centered on a student’s weakness, you wouldn’t be able to assess a broader conceptual understanding. And if you insisted the problem be worked completely through, you encountered significant time constraints.

I’d like to share one last anecdote. I recall a parent visitation day one Saturday, which happened to be the day after I gave a calculus exam. Two of the parents approached me after the session and told me how much their son or daughter enjoyed my exam. This indicated to me that for the student who can perform the routine procedures easily, they want to be challenged to think outside the box, and indeed they thrive on such challenge. Shouldn’t we, as educators, find ways to stimulate all of our students, rather than be content with having students in the middle earn their B’s, making sure the struggling students earn their C’s, and relegating the very capable students to a sustained boredom?

And now for the last installment….

“Where does this bring us? Here are some key points as I see them.

We should move away from assigning grades punitively.
We should reconsider the “point'”system of evaluating student performance. Referring to the TIMSS (Third International Mathematics and Science Study): “In our study, teachers were asked what ‘main thing’ they wanted students to learn from the lesson. Sixty-one percent of U.S. teachers described skills they wanted their students to learn. They wanted students to be able to perform a procedure, solve a particular kind of problem, and so on….On the same questionnaire, 73 percent of Japanese teachers said that the main thing they wanted their students to learn from the lesson was to think about things in a new way, such as to see new relationships between mathematical ideas.” (Stigler and Hiebert, The Teaching Gap, ISBN 0-684-85274-8, pp. 89-90.) A point system reflects the assessment of procedural knowledge.
“We can think of all assessment uses as falling into one of two general categories — assessments FOR learning and assessments OF learning.” (From an internal document distributed to mathematics teachers at IMSA.) But why? The distinction is artificial. There are many other ways to compartmentalize assessments, such as timed/untimed, individual/group, skill/conceptual, procedural/relational, short-term/long-term, etc. The main argument for focusing on the “for/or” distinction is its relationship to student motivation — but we are given no context for it. I suggest that our typical IMSA student is highly motivated — certainly in relation to the average student in a typical high school classroom.
We should consider the assignment of letter grades in general. Right now, it would be impractical to suggest that we have formal written evaluations of each student in each class. But is it desirable? And if so, what resources are necessary to support such a system?
We should discuss the assessment of problem-solving.

Will any of these suggestions help to illuminate the power of ideas? I’m not sure. With the current need to assign grades, and their current cultural meaning and importance — especially when it comes to applying to college — there will be the necessary compromises in the classroom. I realize that many suggestions are of the “move away” rather than the “move toward” type. But I suppose that if there is something I am moving toward, it’s giving students at all levels more of a BC Fast-Track experience regardless of the depth of content.

This means actively moving toward a classroom environment where earning good grades is subordinate to learning complex concepts. Of course the two are not mutually exclusive — but I’d rather have students earn good grades because they learned, rather than learn in order to get good grades.

Of course many issues brought up in these remarks have been left hanging or only tentatively developed. These brief comments are meant to suggest questions for discussion, not definitive answers.

I can’t resist ending with the following challenge from Maslow: “In order to be able to choose in accord with his own nature and develop it, the child must be permitted to retain the subjective experiences of delight and boredom, as the criteria of the correct choice for him. The alternative criterion is making the choice in terms of the wish of another person. The Self is lost when this happens. (Maslow, source cited earlier, p. 58.) Is it possible to create a mathematics curriculum which can survive this test of course selection?”

Thanks for staying with this series! No, there is no simple resolution to any of the issues described in this essay. But that doesn’t mean we shouldn’t be involved in a conversation about them….

On Assessment, IV

Last week, I had ended with an interpretation for an A–D grading scale, shown below (here is a link to last week’s post for reference).

Day166grades

I remind you that this scale is not ideal; the purpose was to come up with some system of assigning grades which wasn’t punitive, but rather which motivated students to learn concepts rather than to avoid losing points on exams.

We continue with a discussion of how to use such a system in practice.

“Now let’s consider this in the context of an exam. The first part of an exam is a skills portion, with, say, ten short problems of roughly equal length. Expectations for this part of the exam [meaning a grade of B] are seven problems “essentially” correct, and four problems completely correct. These expectations are written on the exam for students to see.

Are these expectations too low? Perhaps. But then an A [refer to the chart below] means eight problems “essentially” correct, with at least five completely correct. Of course we must ask what it means for a problem to be “essentially” correct — but when in doubt, err on behalf of the student. (Students rarely suggest that their scores be lowered.)

Then grading is actually somewhat easier, and grades can be assigned as follows, with the abbreviations EC and CC meaning essentially correct and completely correct, respectively (for simplicity, a grade of C is assigned for all other cases not accounted for):

Day167grades

Now this eliminates the need for partial credit — but does require a judgment as to what “essentially correct” means.

This also makes grading much easier. I would suggest that each problem be marked as “EC,” “CC,” or left blank. Few comments, if any are necessary. This is the approach I have taken in BC Fast-Track, and it encourages further learning as it leaves the student in the position of needing to work through their mistakes.

I would have students keep a section of their notebooks for exams and revisions, and there they can keep their reworked problems, should they choose to do so. Then — as I did in BC Fast-Track — students could visit me periodically with their notebooks and I can take a look at their ongoing progress. This “additional” work, if sufficiently well done, could boost their grade at the end of the semester.

I think this could have the same effect it did in BC Fast-Track — exams were easier and more enjoyable to grade. But there were more discussions in my office about reworked exams and sources of error that were initiated by the students themselves, and these discussions were not about points, but about concepts.

Now what about the part of the exam which is intended to be more conceptual? Let us suppose that there are three problems, roughly comparable in length, and of various difficulties. Then grades might be assigned as follows:

Day167grades2

More details about how this would fit in a classroom environment may be found in a later document [I cannot recall which document is being referred to here]. But this system allows for a more qualitative approach to grading. Performance expectations are also clearer, but such expectations depend critically upon the nature of the problems given. Moreover, grades are not assigned punitively, but the emphasis is on doing problems completely and correctly.

For an example, below could be a set of ten skills problems and three conceptual questions for a basic assessment on the rules of differentiation. This would a 70-minute assessment. Given expectations for completely correct problems, I think this is reasonable.

Skills questions:

1. Evaluate

$\lim_{x\to\pi/4}\dfrac{\tan x-1}{x-\pi/4}.$

2. If $f(x)=e^x\cos x,$ find $f^\prime(x).$

3. Find

$\dfrac d{dx}\dfrac 7{\sqrt{x^3}}.$

4. Using the quotient rule, find

$\dfrac d{dx}\dfrac{x^3}{\sin x},$

simplifying as much as possible.

5. Find the derivative of $f(x)=\left(\sin\sqrt x\right)^{\!2}.$

6. Find the equation of the tangent line to $h(x)=\sec(2x)$ at $x=\pi/6.$

7. Using a definition of the derivative, find the derivative of $p(x)=x^2-x.$

8. Assume that $f$ and $g$ are differentiable functions. Find

$\dfrac d{dx}f(g(x^2)).$

9. Find the derivative of $q(x)=x^2e^x\cot x.$

10. Let $f$ be the greatest integer function. Using the definition of the derivative, determine whether or not the derivative exists at $x=0.$

Conceptual questions:

1. Using the product rule, find

$\dfrac d{dx}f(x)(g(x))^{-1}.$

Explain your result.

2. Suppose that the line $y=6x+a$ is tangent to both $f(x)=x^2+b$ and $g(x)=x^3+3x.$ Find $a$ and $b.$

3. Suppose that $f$ is a differentiable function. Discuss the following limit:

$\lim_{h\to0}\dfrac{f(x+2h)-f(x-h)}{3h}.$

Stay tuned for next week’s final installment of this series on assessment….

On Assessment, III

Today, I’ll continue with the discussion of assessment I began a few weeks ago. Last week, I ended with the observation that “The performance of our students determines our expectations, rather than the other way around.” Before continuing, I just want to make clear the context for that remark. When I wrote this essay, IMSA was fully funded by the state of Illinois.

What would it look like to legislators — who approved our funding — if you brought in the top students (relative to their local regions), and then they regularly received low grades, or worse yet, flunked out? Not good. And because the admissions process was fairly involved, would this indicate a major flaw in this process?

Yes, some students couldn’t handle the high-pressure academic environment. But I know that in several of my classes, a student needed to work hard not to earn at least a C. In other words, showing up, handing in homework on time, and doing a reasonable amount of studying for quizzes and exams would usually guarantee a grade of at least a C.

Now is not the time to dive more deeply into this artificial adjustment of expectations, but I did want to mention that this issue is a significant one, even at presumably elite schools for mathematics and science students.

So to continue with the essay….

“Now there is a natural give-and-take between evaluating student performance and setting expectations. And, of course, the above remarks are nothing but generalizations. But they illustrate some of the important issues at hand, and may bear fruitful discussion.

Moving to more concrete issues, I believe that the assignment of letter grades on exams in BC Fast-Track [recall, this was the colloquial name for the Honors Calculus sequence] was, on the whole, successful. Without going into unnecessary detail, the classroom environment was such that the assigned grades were meaningful to the students. To give a few examples, I assigned a grade of A+ for truly outstanding work, perhaps only a half-dozen times throughout the entire semester. The students knew this, and so that accolade truly meant something.

Moreover, an A meant something as well. It was truly rewarding to see the real pride of a student who, used to earning grades in the B range, began to earn the occasional, or perhaps more frequent, A. Admittedly, students who made it to the second semester were essentially guaranteed a grade of no lower than a B-. But this seemed to make an A that much more meaningful.

So student exams had two letter grades on them — one for the skills portion of the exam, and one for the conceptual portion of the exam. No points were assigned, and few comments were made. Students were expected to rework problems on which they made errors.

I bring up this point because I think this system of assigning grades really did motivate students to learn calculus rather than accumulate points. This is the critical issue: I suggest that the way we assign grades does little to disabuse many students that taking a mathematics course is about accumulating sufficiently many — or losing sufficiently few — points.

Let’s take a particular example. The past few semesters, I stopped assigning half-points on assessments [the usual practice at IMSA]. I might forgive a sign error now and then, but too many on a single assessment would warrant a point or two off.

In the past, I simply considered a sign error as a half-point off. And so it was. But consider that without being able to occasionally perform fairly involved calculations, it is not possible to become a successful mathematician. Attention to detail is as important in mathematics as it is in any number of other disciplines, and we try to develop that skill punitively — you don’t attend to detail, and we will take off points.

Of course one might argue that points are given for work well done — but any of us could, I think, agree that when discussing the grading of an exam, it’s how many points off for a particular type of error that is discussed as much as, or even more than, how many points are given for work correctly done.

And so the idea of “partial credit” is born. Perhaps now is not the place to begin this discussion, but consider that a student might meet expectations (that is, earn a B) without ever having done an entire complex problem on an assessment completely correctly. (Some teachers have even gone so far as to give no partial credit. See On Partial Credit, Letter to the Editor, MAA Focus, February 2002, p. 17.)

Why this system of points and partial credit? One may speculate as to its origins, and there is controversy even now about its use on standardized exams. But I cannot help feeling that one function of partial credit is that it allows a teacher to defend the assignment of a particular grade. “Every sign error is a half-point off. That’s why you got a B+ instead of an A-. I have to use the exact same scale for everyone in order to be fair.”

But doesn’t this simply shift the responsibility for the grade onto a rubric? I suggest that many of us would feel competent to take a set of calculus exams — with names removed — and within five or ten minutes, separate out all the A papers. Of course this is subjective — but no less subjective than saying that this problem is worth six points while another is worth ten, or that sign errors are a half-point off, unless, of course, the derivative of the cosine is taken incorrectly, in which case it’s a whole point.

Thus the assigning of points is no more “objective” than giving a letter grade. As I’m sure that anyone who has graded a complex word problem based on an assignment of points can attest to. Consider the student who has the entire procedure correct, but because of a few algebra errors, has no intermediate calculation correct. The problem is worth ten points. How many points should the student receive?

Well, of course, you say you’d have to see the problem first. But I say, no. The student receives a C. Having no intermediate calculations correct demonstrates — regardless of what else — that the student has not met expectations.

So is it possible to avoid points altogether? Perhaps. Consider the following grading system:

The essay continues in the next installment of On Assessment with a discussion of how to implement this system in practice.

On Assessment, II

Last week, I had mentioned finding an essay I wrote about assessment in an Honors Calculus sequence I had designed. I took some time to set the stage for this essay — so now it’s time to dive right in!

A caveat: This essay was written in 2011. I will attempt to keep as true to the original essay as I can, though I’ll edit for clarity and updated information/links, and will also [add commentary in square brackets]. So here goes!

“To educate is to illuminate the power of ideas.

This, of course, is an ideal — certainly not the last to be articulated, nor perhaps the highest. But putting practicality aside for the moment, how might we take on such a view of education?

Consider the example of planetary orbits. Tying together history, technology, politics, physics, and mathematics through a discussion of Kepler’s laws is — yes — illuminating. Of course there is no single idea at play here, but consider this: Kepler was perhaps the first astrologer to become an astronomer. He, in contrast to so many of his contemporaries, asked not only where a particular planet should be on some future date, but why it should be there. In his mind, predicting the positions of celestial bodies for the purpose of casting horoscopes for royal personages was not enough — he wanted to know why. That, however, would have to wait until Newton and the application of calculus. And the power of that idea!

There is no need to present further historical examples. What is important, however, is to move beyond historical ideas and address the issue of the power of a student’s own ideas. Two semesters of teaching Advanced Problem Solving [a course I taught at IMSA which emphasized the writing of original mathematics problems] have shown me that having students write original problems motivates them to learn. They are timid at first — for how could they come up with an original problem? But after some success, students are excited to create — and as a result, learn about a particular topic in a more profoundly personal way than they might have otherwise. (Interestingly, the writing of original problems as a teaching tool is common in eighth-grade Japanese mathematics classrooms. See The Teaching Gap by Stigler and Hiebert, ISBN 0-684-85274-8, pp. 6–41.)

Other assessments are more routine: students are informed that they need to be able to solve linear and quadratic equations, as well as graph linear and quadratic functions. This is perhaps more manageable — the task is well-defined within narrow limits. But it is rather different in nature than creating an original piece of work.

Abraham Maslow articulates a similar difference in a discussion of growth theory (see Toward a Psychology of Being by Abraham Maslow, ISBN 0-442-03805-4, p. 47). Psychologically, why do children grow? “We grow forward when the delights of growth and anxieties of safety are greater than the anxieties of growth and the delights of safety.” From an educator’s point of view, this implies making the classroom a safe environment for creativity and exploration.

“The opposite of the subjective experience of delight (trusting himself), so far as the child is concerned, is the opinion of other people (love, respect, approval, admiration, reward from others, trusting others other than himself).” (Maslow, same source, p. 51.) Who comes to mind is the student afraid of enrolling in BC Fast-Track [the colloquial name of the Honors Calculus sequence at IMSA] because he or she might not earn an A. [I did have a very capable student who was so concerned about grades, he opted to take the more traditional calculus sequence. I actually met with his father, who wanted a guarantee that his son would earn an A in the course. Given the nature of the course, there was naturally no way I could give such a guarantee.] Or perhaps the student who shuns a difficult course and takes an easier one instead. Grades are just that — opinions of other people.

The assignment of letter grades is not necessary for learning, but is merely practical for other reasons. The assignment of letter grades does nothing to illuminate the power of ideas.

A quick Internet search reveals that the assignment of A–F letter grades is a fairly recent phenomenon, not making its way into high schools until the mid-twentieth century. In the early twentieth century, grades of E (Excellent), S (Satisfactory), N (Needs improvement), and U (Unsatisfactory) were also used.

The question of whether a letter-grade system of evaluation is the best option for a school like IMSA is perhaps a worthwhile one (some argue against grades at all [this was the link, but it is now a dead end: http://www.alfiekohn.org/teaching/fdtd-g.htm ]). That is a question which cannot be answered in the short-term, if the question is considered relevant. So the question is this: given that letter grades need to be assigned this semester, what approach should be taken?

What do letter grades mean?

The only consensus I have heard, so far, is that earning a grade of A, B, or C means that a student exceeds, meets, or does not meet expectations. Of course, this is reminiscent of E, S, and N. Moreover, a grade of C is passing, so that a student receives credit for a course even if they do not meet expectations.

But this is not what the grades currently mean. Essentially, points are assigned to hundreds of problems given throughout semester, whether on assignments, papers, quizzes, or exams — and an arbitrary weighted sum of these point assignments is converted to a letter. (Of course letters may need to be converted to numbers so that they may be used by an online grade calculator to compute a number which is then converted back to a letter.)

What does this mean about meeting expectations?

Well, nothing really. I would venture to suggest that “meeting expectations” currently means “enough A’s, but not too many C’s.” Perhaps this is politically necessary, but expedient. The performance of our students determines our expectations, rather than the other way around.”

On Assessment, I

While downsizing for a recent move, I decided to sort through dozens of folders (thousands of pages!) of old problems, notes, exams, essays, and other documents generated in the course of twenty-five years of teaching mathematics. I found an essay written in 2011 which I though might be of some interest.

This essay might be considered a more thorough follow-up to The Problem with Grades, a satire on assessment practices which I’ve shared with you already. The essential point in that essay was that how we typically assign grades is punitive: begin with a given number of points (say, 100), punish students for making mistakes (subtract a certain number of points from 100), and then — magically! — the resulting number reflects a student’s understanding. (If you think that this is in fact a really great way to assess a student’s learning, you probably should stop reading right here….)

The essay I’d like to share here was written in the context of teaching an Honors Calculus sequence I designed while at the Illinois Mathematics and Science Academy (IMSA). So before diving in, let me take a few moments to set the stage.

We had quite a broad range of students at IMSA — and so when it came to teaching calculus, some students were bored by the “average” pace of the class. Of course this happens in a typical classroom; the tendency is to teach to the middle, with the necessary consequence that some students fall behind, and others could benefit from a faster pace or more advanced problems.

Now mathematics educators are looking at ways to design classroom environments where students can work more independently, at their own pace, incorporating methodologies such as differentiated learning and/or competency-based learning. These ideas could be the subject for wide-ranging (future) discussions; but in my case, I was working in an environment where all students did essentially the same work in the classroom, and assessments were fairly traditional.

In addition to creating a faster-paced, more conceptually oriented course, I incorporated two features into this Honors Sequence. First, I had students write and solve their own original mathematics problems, as well as reflect on the problem-posing process. I won’t say more about that here, since I’ve written about this type of assessment in a previous blog post.

Second, I experimented with a different way of assessing student understanding on exams. Each exam had two components: a Skills portion and a Conceptual portion. The Skills portion was what you might expect — a set of problems which simply tested whether students could apply routine procedures. The Conceptual portion was somewhat more challenging, and included nonroutine problems which were different from problems students had seen before, but which did not require any additional/specialized knowledge to solve beyond what was needed to solve the Skills problems. (As part of the essay I will share, examples of both types of problems will be included. I also discuss these ideas in more detail in a previous post, On Grading.)

Moreover, I graded these problems in a nonstandard say: each problem was either Completely Correct (CC), meaning that perhaps aside from a simple arithmetic error or two, the problem was correctly solved; Essentially Correct (EC), meaning that the student had a viable approach to solving the problem, but was not able to use it to make significant progress on the problem; and Not Correct (NC), meaning that the approach taken by the student would not result in significant progress toward solving the problem.

Letter grades (a necessity where I taught, as they are in most schools) were assigned on the basis of how many problems were CC and EC on the Skills/Conceptual portions. While I won’t go into great detail here, the salient features of this system are, in my mind:

Earning an A required a fair number of problems CC; in other words, a student couldn’t get an A just by amassing enough partial credit on problems — there needed to be some mastery;
Earning an A required as least some progress on the Conceptual problems; typically, I would include three such problems, which would be assessed more leniently than the Skills problems. In order to earn an A, a student would need to make some progress on one or two of these. I felt that an A student should be able to demonstrate some conceptual understanding of main ideas;
A student could earn a B+ just by performing well on the Skills problems (or perhaps an A- for performing flawlessly), but an A was out of reach without some progress on the Conceptual problems;
The approach was not punitive: students were assigned CC/EC based on their progress toward the solution to a problem, not how far they fell short of a solution.

Such a system is not new; I have talked with colleagues who used a 0/1/2 system of grading, for example. Whatever the format, the approach is more holistic. And given that problems in mathematics typically admit more than one solution, the idea of creating a point-based rubric for all possible solution paths does, in some real sense, border on the insane.

I should also add that our instructional approach at IMSA was essentially inquiry-based. While such a classroom environment is conducive to having students write original problems and using alternative assessment strategies, it is not strictly necessary. I have incorporated both the elements described above in a more traditional classroom setting, but with varying degrees of success. That discussion is necessarily for another time.

So much for a brief introduction! I would also like to comment that teaching this Honors Sequence was perhaps my most enjoyable and successful teaching experience in the past few decades. Students who completed the two-semester sequence left thinking about mathematics in a fundamentally different way, and I stay connected with a few of these students many years later.

Next week, I’ll introduce the essay proper, adding commentary as necessary to flesh out details of the course not addressed here, as well as making remarks which reflect my teaching this Honors sequence in the years since the essay was written. Until then, I’ll leave you with the opening words:

“To educate is to illuminate the power of ideas…..”

The Problem with Calculus Textbooks (Reprise)

Simply put, most calculus textbooks are written in the wrong order.

Unfortunately, this includes the most popular textbooks used in colleges and universities today.

This problem has a long history, and will not be quickly solved for a variety of reasons. I think the solution lies ultimately with high quality, open source e-modules (that is, stand-alone tutorials on all calculus-related topics), but that discussion is for another time. Today, I want to address a more pressing issue: since many of us (including myself) must teach from such textbooks — now, long before the publishing revolution — how might we provide students a more engaging, productive calculus experience?

To be specific, I’ll describe some strategies I’ve used in calculus over the past several years. Once you get the idea, you’ll be able to look through your syllabus and find ways to make similar adaptations. There are so many different versions of calculus taught, there is no “one size fits all” solution. So here goes.

1. I now teach differentiation before limits. The reason is that very little intuition about limits is needed to differentiate quadratics, for example — but the idea of limits is naturally introduced in terms of slopes of secant lines. Once students have the general idea, I give them a list of the usual functions to differentiate. Now they generate the limits we need to study — completely opposite of introducing various limits out of context that “they will need later.”

Students routinely ask, “When am I ever going to use this?” At one time, I dismissed the question as irrelevant — surely students should know that the learning process is not one of immediate gratification. But when I really understood what they were asking — “How do I make sense of what you’re telling me when I have nothing to relate it to except the promise of some unknown future problem?” — I started to rethink how I presented concepts in calculus.

I also didn’t want to write my own calculus textbook from scratch — so I looked for ways to use the resources I already had. Simply doing the introductory section on differentiation before the chapter on limits takes no additional time in the classroom, and not much preparation on the part of the teacher. This point is crucial for the typical teacher — time is precious. What I’m advocating is just a reshuffling of the topics we (have to) teach anyway.

2. I no longer teach the chapter on techniques of integration as a “chapter.” In the typical textbook, nothing in this chapter is sufficiently motivated. So here’s what I do.

I teach the section on integration by parts when I discuss volumes. Finding volumes using cylindrical shells naturally gives rise to using integration by parts, so why wait? Incidentally, I also bring center of mass and Pappus’ theorem into play, as they also fit naturally here. The one-variable formulation of the center of mass gives rise to squares of functions, so I introduce integrating powers of trigonometric functions here. (Though I omit topics such as using integration by parts to integrate unfriendly powers of tangent and secant — I do not feel this is necessary given any mathematician I know would jump to Mathematica or similar software to evaluate such integrals.)

I teach trigonometric substitution (hyperbolic as well — that for another blog post) when I cover arc length and surface area — again, since integrals involving square roots arise naturally here.

Partial fractions can either be introduced when covering telescoping series, or when solving the logistic equation. (A colleague recommended doing series in the middle of the course rather then the end (where it would have naturally have fallen given the order of chapters in our text), since she found that students’ minds were fresher then — so I introduced partial fractions when doing telescoping series. I found this rearrangement to be a good suggestion, by the way. Thanks, Cornelia!)

3. I no longer begin Taylor series by introducing sequences and series in the conventional way. First, I motivate the idea by considering limits like

$\displaystyle\lim_{x\to0}\dfrac{\sin x-x}{x^3}=-\dfrac16.$

This essentially means that near 0, we can approximate $\sin(x)$ by the cubic polynomial

$\sin(x)\approx x-\dfrac{x^3}6.$

In other words, the limits we often encounter while studying L’Hopital’s rule provide a good motivation for polynomial approximations. Once the idea is introduced, higher-order — eventually “infinite-order” — approximations can be brought in. Some algorithms approximate transcendental functions with polynomials — this provides food for thought as well. Natural questions arise: How far do we need to go to get a given desired accuracy? Will the process always work?

I won’t say more about this approach here, since I’ve written up a complete set of Taylor series notes. They were written for an Honors-level class, so some sections won’t be appropriate for a typical calculus course. They were also intended for use in an inquiry-based learning environment, and so are not in the usual “text, examples, exercise” order. But I hope they at least convey an approach to the subject, which I have adapted to a more traditional university setting as well. For the interested instructor, I also have compiled a complete Solutions Manual.

I think this is enough to give you the idea of my approach to using a traditional textbook. Every calculus teacher has their own way of thinking about the subject — as it should be. There is no reason to think that every teacher should teach calculus in the same way — but there is every reason to think that calculus teachers should be contemplating how to make this beautiful subject more accessible to their students.

Enumerating the Platonic Solids

The past few weeks, I outlined my approach to a series of lectures on polyhedra. One of my constraints is that students will not have seen a lot of trigonometry yet, and will not have been exposed to three-dimensional Cartesian coordinates. But there is Euler’s Formula! I just finished a pair of lectures on the algebraic enumeration of the Platonic solids using Euler’s Formula, and I thought others might be interested as well.

As a reminder, Euler’s Formula states that if $V,$ $E,$ and $F$ are the number of vertices, edges, and faces, respectively, on a convex polyhedron, then

$V-E+F=2.$

How might we use this formula to enumerate the Platonic Solids? We need to make sure we agree on what a Platonic Solid is: a convex polyhedron with all the same regular polygon for faces, and with the same number meeting at each vertex.

To use this definition, we will define a few more variables: let $p$ denote the number of sides on the regular polygons, and let $q$ denote the number of polygons meeting at each vertex of the Platonic solid. (Those familiar with polyhedra will recognize these as the usual variables.)

The trick is to count the number of sides and vertices on all the polygons in two different ways. For example, since there are $F$ polygons on the Platonic solid, each having $p$ sides, there are a total of $pF$ sides on all of the polygons.

But notice that when we build a cube from six squares, two sides of the squares meet at each edge of the cube. This implies that $2E$ also counts all of the sides on the polygons. Since we are counting the same thing in two different ways, we have

$pF=2E.$

We may similarly count all the vertices on the polygons as well. Of course since a regular polygon with $p$ sides also has $p$ vertices, there are $pF$ vertices on all of the polygons.

But notice that when we put the squares together, three vertices from the squares meet at a vertex of the cube. Thus, if there are $V$ vertices on a Platonic Solid, and if $q$ vertices of the polygons come together at each one, then it must be that $qV$ is the total number of vertices on all of the polygons. Again, having counted the same thing in two different ways, we have

$pF=qV.$

Thus, so far we have

$V-E+F=2,\quad pF=2E,\quad pF=qV.$

Note that we have three equations in five variables here; in general, such a system has infinitely many solutions. But we have additional constraints here — note that all variables are counting some feature of a Platonic Solid, and so all must be positive integers.

Also, since a regular polygon has at least three sides, we must have $p\ge3,$ and since at least three polygons must come together at the vertex of a convex polyhedron, we must also have $q\ge3.$

These additional constraints will guarantee a finite (as we know!) number of solutions. So let’s go about solving this system. The simplest approach is to solve the last two equations above for $E$ and $V$ and substitute into Euler’s Formula, yielding

$\dfrac{pF}q-\dfrac{pF}2+F=2.$

Now divide through by $F$ and observe that $F>0,$ so that

$\dfrac pq-\dfrac p2+1>0.$

Multiply through by $2q$ and rearrange terms, giving

$pq-2p-2q<0.$

How should we go about solving this inequality? There’s a nice trick here: add $4$ to both sides so that the left-hand side factors nicely:

$(p-2)(q-2)<4.$

Now we are almost done! Since $p,q\ge3,$ then $p-2$ and $q-2$ must both be integers at least $1;$ but since their product must be less than $4,$ they can be at most $3.$

This directly implies that $p$ and $q$ must be $3, 4,$ or $5.$

This leaves only nine possibilities — but of course, not all options need be considered. For example, if $p=q=5,$ then

$(p-2)(q-2)=9>4,$

and so does not represent a valid solution. But when $p=3$ and $q=4,$ we have the octahedron, since $p=3$ means that the polygons on the Platonic Solid are equilateral triangles, and $q=4$ means that four triangles meet at each vertex.

So out of these nine possibilities to consider, there are just five options for $p$ and $q$ which satisfy the inequality $(p-2)(q-2)<4.$ And since each pair corresponds to a Platonic Solid, this implies that there are just five of them, as enumerated in the following table:

PlatonicChart

Actually, this implies that there are at most five Platonic Solids. How do we know that twelve pentagons actually fit together exactly to form a regular dodecahedron? A further argument is necessary here to be complete. But for the purposes of my lectures, I just show images of these Platonic Solids, with the presumption that they do, in fact, exist.

Now keep in mind that in an earlier lecture, I enumerated the Platonic Solids using a geometrical approach; that is, by looking at those with triangular faces, square faces, etc. I like the problem of enumerating the Platonic Solids since the geometric and algebraic methods are so different, and emphasize different aspects of the problem. Further, both methods are fairly accessible to good algebra students. The question of when to take an algebraic approach rather than a geometric approach to a geometry problem is frequently difficult for students to answer; hopefully, looking at this problem from both perspectives will give students more insight into this question.

Teaching Three-Dimensional Geometry, III

This is the last of a three-part series on teaching three-dimensional geometry. A few weeks ago, I had begun describing how I would go about putting together a series of about 20 online videos on 3D geometry, each lasting 5–7 minutes. I just finished a discussion of buckyballs, and why regardless of the number of hexagonal faces on a buckyball, there are always exactly 12 pentagonal faces.

Euler’s Formula was key. We’ll look at another application of Euler’s Formula, but before doing so, I’d like to point out that students at this level have not encountered Cartesian coordinates in three dimensions, and so I need to find things to talk about at an accessible level.

On to the truncation of polyhedra! Again, we can apply Euler’s Formula, but it helps to think about the process systematically. You can count the number of vertices, edges, and faces on a truncated cube, for example, one at a time — but little is gained from a brute force approach. By thinking more geometrically, we would notice that each edge of the original cube contributes two vertices to the truncated cube, giving a total of 24 vertices.

We can continue on in this fashion, counting as efficiently as possible. This sets the stage for a discussion of Archimedean solids in general. A proof of the enumeration of the Archimedean solids is beyond the scope of a single lecture, but the important geometrical ideas can still be addressed.

This concludes the set of lectures on polyhedra in three dimensions. Of course there is a lot more that can be said, but I need to make sure I get to some other topics.

Like spherical geometry, for instance, next on the slate. There are two approaches one typically takes, depending how you define a point in spherical geometry. There is a nice duality of theorems if you define a Point in this new geometry as a pair of antipodal points on a sphere, and a Line as a great circle on a sphere. Thus two distinct Lines uniquely determine a Point, and two distinct Points uniquely determine a line.

This is a bit abstract for a first go at spherical geometry, so I plan to define a Point as just an ordinary point on a sphere, and a Line as a great circle. Two points no longer uniquely determine a Line, since there are infinitely many Lines through two antipodal Points.

But still, there are lots of interesting things to discuss. For example, there is no such thing as a pair parallel lines on a sphere: two distinct Lines always intersect.

Triangles are also intriguing. On the sphere, the sides are also angles, measured by the angle subtended at the center of the sphere. So all together, there are six angular measures in any triangle.

Since students will not have had a lot of exposure to trigonometry at this point, I won’t discuss many of the neat spherical trigonometric formulas. But still, there is the fact the angle sum of a spherical triangle is always greater than $180^\circ.$ And the fact that similarity and congruence on the sphere are the same concept, unlike in Euclidean geometry. For example, if the angles in a Euclidean triangle are the same in pairs, the triangles are similar. But on a sphere, if the angles of two spherical triangles measured the same in pairs, they would necessarily have to be congruent.

In other words, students are getting further exposure to non-Euclidean geometries. (I did a lecture on inversive geometry in a previous section.) One nice and accessible proof in spherical geometry is the proof that the area of a spherical triangle is proportional to its spherical excess — that is, how much the angle sum is greater than $180^\circ.$ So there will be something I can talk about without needing to say the proof is too complicated to include….

The final topic I plan to address is higher-dimensional geometry. The first natural go-to here is the hypercube. Students are always intrigued by a fourth spatial dimension. Ask a typical student who hasn’t been exposed to these ideas what the fourth dimension is, and the answer you invariably get is “time.” So you have to do some work getting them to think outside of that box they’ve lived in for so long.

One thing I like about hypercubes is the different ways you can visualize them in two dimensions.

Hypercube1

Viewed this way, you can see the black cube being moved along a direction perpendicular to itself to obtain the blue cube. Of course the process is necessarily distorted since we’re looking at a static image.

Hypercube2

This perspective highlights a pair of opposite cubes — the green one in the middle, and the outer shell — and the six cubes adjacent to both.

Hypercube4

And this perspective is just aesthetically very pleasing, and also has the nice property that every one of the eight cubes looks exactly the same, except for a rotation. Again, there won’t be any four-dimensional Cartesian coordinates, but still, there will be plenty to talk about.

I plan to wrap up the series with a discussion of volumes in higher dimensions. As I mentioned last week, I’d like to discuss why you should avoid peeling a 100-dimensional potato….

Thinking by analogy, it is not difficult to motivate the fact that the volume of a sphere $n$ dimensions is of the form

$Kr^n.$

Now let’s look at peeling a potato in three dimensions, assuming it’s roughly spherical. If you were a practiced potato peeler, maybe you could get away with the thickness of your potato peels being, say, just 1% of the radius of your potato. This leaves the radius of your peeled potato as $0.99r,$ and calculating a simple ratio reveals that you’ve got $0.99^3\approx0.97$ of your potato left.

Extend this idea into higher dimensions. If your potato-peeling expertise is as good in higher dimensions, you’ll have $0.99^n$ of your potato left, where $n$ is the number of dimensions of your potato. Now $0.99^{100}\approx 0.366,$ so after you’ve peeled your potato, you’ve only got about one-third of it left!

What’s happening here is that as you go up in dimension, there is more volume near the surface of objects than there is near the center. This is difficult to intuit from two and three dimensions, where it seems the opposite is the case. Nonetheless, this discussion gives at least some intuition about volumes in higher dimensions.

And that’s it! I’m looking forward to making these videos; I actually made my first set of slides today. As usual, if I come across anything startling or unusual during the process, I’ll be sure to post about it!

Teaching Three-Dimensional Geometry, II

A few weeks ago, I began a discussion of what I’d be presenting in a series of twenty (or so) 5—7 minute videos on three-dimensional geometry. I didn’t get very far then, so it’s time to continue….

So to recap a bit, I’ll begin with the usual cones/cylinders/spheres, looking at surface areas and contrasting flat surfaces with the surface of a sphere. Then on to a prelude to calculus by looking at the volume of a cone as a limiting case of a stack of circular disks.

Next, it’s on to polyhedra! A favorite topic of mine, certainly. Polyhedra are interesting, even from the very beginning, since there is still no accepted definition of what a polyhedron actually is. The exception is for convex polyhedra; a perfectly good definition of a convex polyhedron is the convex hull of a finite set of points not all lying in a single plane. Easy enough.

But once you move on to nonconvexity, uncertainties abound. For example, from a historical perspective, sometimes the object below was a polyhedron, and sometimes it wasn’t. Sounds odd, but whether or not you consider this object a polyhedron depends on how you look at the top “face,” which is a square with a smaller square removed from the center. Now is this “face” a polygon, or not? Many definitions of a polygon would exclude this geometrical object – which is problematic if you want to say that a polyhedron has polygons as faces.

So this brings us to a definition of a polygon, which is problematic in its own way – to see why, you can look at a previous post of mine on the definition of a polygon. Now the point here is not to resolve the issue in an elementary lecture, but rather point out that mathematics is not “black-and-white,” as students tend to believe. Also, it provides a nice example of the importance of definitions in mathematics.

Now this would be discussed briefly in just one video. Next would be the (obligatory) Platonic solids – where else is there to begin? The simplest starting point is the geometric enumeration by looking at what types of polygons – and how many – can appear at any given vertex of a Platonic solid. This enumeration is straightforward enough.

Next, I plan on computing the volume of a regular tetrahedron using the usual $Bh/3$ formula. This is not really exciting in and of itself, but in the next lecture, I plan to find the volume of a regular tetrahedron by inscribing it in the usual way in the cube by joining alternate vertices.

Of course you get the same result. But for those of us who work a lot in three-dimensional space, we deeply understand the simple algebraic equation, $2 \times 4=8.$ What I’m referring to, specifically, is that the number of vertices on a three-dimensional simplex is half the number of vertices of a three-dimensional hypercube.

This simple fact is at the heart of any number of intriguing geometrical relationships between polyhedra in three dimensions. In particular, and quite importantly, the simplex and the cross-polytope together fill space. This relationship is at the heart of many architectural constructions in additional to generating other tilings of space with Archimedean solids. But most students have never seen this illustrated before, so I think it is important to include.

Then on to a geometry/algebra relationship: having enumerated the Platonic solids geometrically, how do we proceed to take an algebraic approach? A fairly direct way is to use Euler’s formula to find an algebraic enumeration.

No, I don’t intend to prove Euler’s formula; by far my favorite (and best!) is Legendre’s proof which involves projecting a polyhedron onto a sphere and looking at the areas of the spherical polygons created. This is a bit beyond the scope of this series of videos; there simply isn’t time for everything. But it is important to note the role that convexity plays here; yes, there are other formulas for polyhedra which are not essentially “spheres,” but this is not the place to discuss them.

Next, I want to talk about “buckyballs.” I still have somewhat of a pet peeve about the nomenclature – Buckminster Fuller did not invent the truncated icosahedron – and so the physicists who named this molecule were, in my opinion, polyhedrally rather naïve. But, sadly (as is the case so many times), they did not come to me first before making such a decision…

The polyhedrally interesting fact about buckyballs is this: if a polyhedron has just pentagonal and hexagonal faces, three meeting at every vertex, then there must be exactly twelve pentagons. Always.

Now I know that the polyhedrally savvy among you are well aware of this – but for those who aren’t, I’ll show you the beautiful and very short proof. Once you’ve seen the idea, I don’t think you’ll ever be able to forget it. It’s just remarkable – even with 123,456,789 hexagons, just 12 pentagons.

So let $P$ represent the number of pentagons on the buckyball, and $H$ represent the number of hexagons. Then the number of vertices $V$ is given by

$V=\dfrac{5P+6H}{3},$

since each pentagon contributes five vertices, each hexagon contributes six, and three vertices of the polygons meet at each vertex of the buckyball.

Moreover, the number of edges is given by

$E=\dfrac{5P+6H}{2},$

since the polygons on the buckyball meet edge-to-edge. Of course, $F=P+H,$ since the faces are just the pentagons and hexagons. Substitute these expressions into Euler’s formula

$V-E+F=2,$

and what happens? It turns out that $H$ cancels out, leaving $P=12!$

Amazes me every time. But what I like about this fact is that it is accessible just knowing Euler’s formula – no more advanced concepts are necessary.

And yes, there’s more! This is now Lecture #12 of my series, so I have a few more to describe to you. Until next time, when I caution you (rather strongly) against peeling a 100-dimensional potato….