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).

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):

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:

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

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

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?

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:

1.  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;
2.  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;
3. 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;
4. 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:

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.

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.

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.

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….

Bay Area Mathematical Artists Seminars, XI

This past weekend marked the eleventh meeting of the Bay Area Mathematical Artists Seminars.  Our host this month was Scott Vorthmann, the mastermind behind vZome.  Scott lives in Saratoga, and so those participants who live in the San Jose area were glad of the short commute.

It seems that the content of our seminars is limited only by the creativity of the artists involved, meaning fairly limitless….  Scott invited anyone interested to come early — 1:00 instead of our usual 3:00 — and be involved in a Zome “build;” that is, the construction of a large and intricate model using Zome tools.  Today’s model?  The omnitruncated 24-cell!

This is not the place to have a lengthy discussion of polytopes in four dimensions.  In a nutshell, the 24-cell is a polytope in four dimensions with 24 octahedral facets.  This polytope is truncated in a particular way (called omintruncation), and then projected into three-dimensional space.

But there is just one problem with the projection Scott wanted to build.  You can’t build it with the standard Zome kit!  No matter.  Scott designed and 3D-printed his own struts — olive, maroon, and lavender.  If you’ve ever played around with ZomeTools, you’ll understand what a remarkable feat of design and engineering this is.

The building process is a modular one — six pieces like the one shown below needed to be built and painstakingly assembled together.

Scott built two of the modules before anyone arrived, so we had something to work from.  That left just four more to complete….

The modules were almost done, but we needed to move on.  In addition to the Zome build, we had two other short presentations.   Andrea and Andy were planning to present a workshop at Bridges 2018 in Stockholm, but at the last minute, were unable to attend.  So they brought their ideas to present to us.

The basic idea is to encode a two-dimensional image using two overlays, as shown here.

Your friend has an apparently random grid (pad) of black and white squares.  You want to send him a secret message; only you and he have the pad.  So you send him a second grid of black and white squares so that when correctly overlaid on the pad, an image is produced.

This is a great activity for younger students, too, since it can be done with premade templates and graph paper.  And even though Andrea and Andy were not able to attend Bridges, their workshop paper was accepted, and so it is in the Bridges archives.  So if you want to learn more about this method of encryption, you can read all the details about the process in their paper in the Bridges archives.

Our next short presentation was by pianist Hans Boepple, a colleague of Frank Farris at Santa Clara University.  Frank happened to have a very stimulating conversation with Hans about a mathematics/music phenomenon, and thought he might like to present his idea at our meeting.

The idea came from a time when Hans happened to look down a metal cylinder of tubing, like you would find at a hardware store.  It seemed that there was an interesting pattern of reflections along the sides of the tubing, and knowing about music and the overtone series, he wondered if there was any connection with music.

Here is part of a computer-generated image of what Hans produced using paper and pencil many years ago:

How was this picture generated?  Below is how you’d start making the image.

You can see that the red lines take two zigzags to move from one corner of the rectangle to another, the blue lines take three zigzags, the green four, and the gold lines take five.  If you keep adding more and more lines, you get rather complex and beautiful patterns like the one shown above.  Those familiar with the overtone series will see an immediate connection.

Of course, the mathematical question is about proving various properties of this pattern.  It turns out that the patterns are related to the Ford circles; BAMAS participant Jacob Rus has created an interactive version of this diagram.  Feel free to explore!

In any case, we were delighted that Hans could join us and share his fascination with the relationship between mathematics and music.  You can  learn more about Hans in this interview in The Santa Clara, which is Santa Clara University’s school newspaper.

When Hans finished his presentation, it was time to finish building the omnitruncated 24-cell.  It was quite amazing, as Scott is certainly one of the foremost experts on ZomeTools in the world.  Here is the finished sculpture, suspended from the ceiling in his home.  Just getting the model up there was an engineering feat in its own right!

It is difficult to describe the intricacy of this model from just a few pictures.

Here is an intriguing perspective of the model, highlighting the parallelism of the blue Zome struts.  It seems there is no end to the geometrical relationships you can find hidden within this model.

And, as usual, the afternoon didn’t end there.  Scott arranged to have Thai food — one of our favorites! — catered in, and we all chipped in our fair share.  We all were having such a great time, the last of us didn’t leave until about 8:30 in the evening.  Another successful seminar!

It is quite heartwarming to see so many so willing to take on hosting our Bay Area Mathematical Artists Seminars.  We have all enjoyed these meetings so much, and we are so glad they continue to happen.  I am confident there will be many, many more delightful Saturday afternoons to experience….

Teaching Three-Dimensional Geometry, I

I have recently had a rather unusual opportunity.  I’ve talked a bit over the last few months about my consulting work producing online videos for a flipped classroom; I’ve been working busily on the Geometry unit.

Now the last section of this unit is on three-dimensional geometry, and I’ve been given pretty free reign as to what to cover in this 20-lectures series of 5-7 minute videos.  And given my interest in polyhedra (which I could focus on exclusively with no shortage of things to discuss!), I felt I had a good start.

But the challenge was also to cover some traditional topics (cones, cylinders, spheres, etc.) — as well as more advanced topics — while not using mathematics beyond what I’ve used in the first several sections of the Geometry unit.

There is, of course, no “correct” answer to this problem.  But I thought I’d share how I’d approach this series of lectures, since geometry is such a passion of mine — and I know it is for many readers as well.  The process of reforming high school geometry courses is now well underway; I hope to contribute to this discussion with today’s post.

Where to start?  Cones and cylinders — a very traditional beginning.  But I thought I’d start with surface areas.  Now for cylinders, this is pretty straightforward.  It’s not much more difficult for cones, but the approach is less obvious than for cylinders.

Earlier in the unit, we derived the formula for the area of a sector of a circle, so finding the lateral surface area of a cone is a nice opportunity to revisit this topic.  And of course, finding the lateral surface area of a cylinder involves just finding the area of a rectangle.

Now what do both of these problems have in common?  Their solution implies that cones and cylinders are flat.  In other words, we reduce what is apparently a three-dimensional problem (the surface area of a three-dimensional object) to a two-dimensional problem.

This is in sharp contrast to finding the surface area of a sphere — you can’t flatten out a sphere.  In fact, the entire science of cartography has evolved specifically in response to this inability.

So this is a nice chance to introduce a little differential geometry!  And no, I don’t really intend to go into differential geometry in any detail — but why not take just a minute in a lecture involving spheres to comment on why the formulas for the surface areas of cones and cylinders are fairly easy to derive, and why — at this level — we’re just given the formula for the surface area of a sphere.

I try to mention such ideas as frequently as I can — pointing out contrasts and connections which go beyond the usual presentation.  Sure, it may be lost on many or most students, but it just may provide that small spark for another.

I think such comments also get at the idea that mathematics is not a series of problems with answers at the back of the book…on the face of it, there is no apparent reason for a student to think that finding the surface area of a cone would be simpler than finding the surface area of a sphere.  This discussion gets them thinking.

Next, I’m planning to discuss Archimedes’ inscription of a sphere in a cylinder (which involves the relative volumes).  This is a bit more straightforward, and it’s a nice way to bring in a little history.

I also plan to look at inscribing a sphere in a right circular cone whose slant height is the same as the diameter of the base, so that we can look at a two-dimensional cross-section to solve the problem.  In particular, this revisits the topic of incircles of triangles in a natural way — I find it more difficult to motivate why you’d want to find an incircle when looking at a strictly two-dimensional problem.

Now on to calculus!  Yes, calculus.  One great mystery for students is the presence of “1/3” in so many volume formulas.  There is always the glib response — the “3” is for “three” dimensions, like the “2” in “1/2 bh” is for “two” dimensions.

When deriving these formulas using integration, this is actually exactly a fairly solid explanation.  But for high school students who have yet to take calculus?

It is easy to approximate the volume of a right circular cone by stacking thin circular disks on top of each other.  If we let the disks get thinner and take more and more of them, we find the volume of the cone as limit of these approximations.  All you need is the sum

$\displaystyle\sum_{k=1}^n k^2=\dfrac{n(n+1)(2n+1)}6.$

I plan to prove that

$\displaystyle\sum_{k=1}^nk=\dfrac{n(n+1)}2,$

and then prove (or perhaps just suggest — I’m not sure yet) the formula for the sum of squares.

I think a fairly informal approach could be successful here.  But I do think such discussions are necessary — in calculus, I’ve routinely asked students why certain formulas they remember are true, and they struggle.  As a simple example, students can rarely tell me why the hypotenuse of a 30-60-90 triangle is twice as long as the shorter leg.

When teachers just give students formulas and ask them to plug numbers in to get answers to oversimplified word problems, of course there is a sense of mystery/confusion — where did these formulas come from?  I’m hoping that this discussion suggests that there is a lot more to mathematics than just a bunch of formulas to memorize.

As usual, I realize I have much more to say on this topic than I had originally supposed…I’ve only discussed up to the fifth lecture so far!  Since I have not had extensive experience teaching more traditional topics at the high school, it has been an interesting challenge to tackle the usual geometry topics in a way that grabs students’ attention.  It’s a challenge I enjoy, and of course I’ll have much more to say about it next week….