October 2018 – Creativity in Mathematics

Jagodina, Serbia: 2018

This week, I write from Jagodina, Serbia! I am here with my friend Dr. Snezana Lawrence (author of www.mathisgoodforyou.com and Mathematicians and Their Gods) presenting at a History of Mathematics in Mathematics Education conference at the University of Kragujevac. Snezana secured funding for and helped organize this conference, which is intended to provide teachers and teacher educators resources for making mathematics more accessible to students by providing contexts for various mathematical ideas.

As mathematicians are well aware, mathematics is both a historical and cultural phenomenon. I know relatively little about the history of mathematics, but I did do some research into the relationship between Kepler and Tycho in preparation for a talk on Kepler’s Laws and the Music of the Spheres. The facts that Tycho was not only interested in astronomy, but was also a member of the Danish nobility (and hence had the money to design and create instruments which allowed for more accurate measurements than previously possible) were crucial in providing the data necessary for Kepler to deduce the laws which we know by his name today. The story is of course much more involved; the point is that providing some context for statements such as Kepler’s Laws makes them more accessible to students rather than just stating them as if they were abstractions devoid of cultural and historical context.

My contribution was running a workshop on polyhedra (the Platonic solids in particular). Participants were familiar with the Platonic solids at various levels, so I made sure there were nets available at varying levels of difficulty. For example, we were going to discuss the geometry of fullerene molecules, so I also had some truncated icosahedron nets printed out as well.

The weather was particularly warm this weekend; plans were altered so we could be outside for much of the workshop. So I moved everything I wanted to discuss using the computer to the first hour. For example, to put our work into some context, I discussed my correspondence with Magnus Wenninger (which I excerpted in a series of blog posts earlier this year).

I also wanted to make a reference to Kepler’s Harmonices Mundi, as Snezana had mentioned Kepler earlier in the conference. Among many other things, Kepler included a discussion of the Platonic solids.

Further, I wanted to discuss the fairly recent (certainly compared to Kepler) discovery of fullerenes, as this discovery led to an interesting, if relatively simple, result: in a fullerene molecule, where each carbon is adjacent to exactly three others, and the carbon atoms come only in rings of five and six, then there must be exactly twelve rings of five carbons, regardless of how many rings of six there are.

This is actually not very difficult to prove using Euler’s formula (and I have done so in a previous post), but it is somewhat surprising the first time you hear it. More difficult is to prove that there can be any number of hexagons except one; I didn’t go into this in any more detail, however. But I did want to go to the Wikipedia article and put the question in some historical context.

After the lunch break, we proceeded to an outdoor classroom which we populated with two movable whiteboards. First, I had participants build at least one of the Platonic solids using the double-tab method I learned from Magnus Wenninger. We then moved on to an algebraic enumeration of the Platonic solids as an application of Euler’s formula (which was also mentioned earlier). As some of the participants were involved with teaching younger students, I wanted to avoid spherical trigonometry.

Then, as mentioned above, we looked at fullerenes. Since the technique involved here was similar to that used to enumerate the Platonic solids, after we defined the problem, I gave the participants time to solve it on their own. Relatively quickly, one of the educators did prove that there were exactly twelve rings of pentagons, and presented her solution to the others. This was a great way to end the afternoon; afterwards, we took a walking tour of the busier section of Jagodina.

Of course one interesting aspect of attending any international conference is learning about the similarities and differences between education in different countries. In Serbia as well as the United States, teachers are not paid as well as other professions. But mathematicians have fewer opportunities for employment than they do in the US, so they are more likely to go into teaching as a career.

One striking difference revolves around school subjects and the corresponding teacher training. As early as fifth grade, the sciences are separated as subjects. All students take biology and informatics (computer science) beginning in the fifth grade, physics beginning in the sixth grade, and chemistry beginning in the seventh grade. Moreover, teachers of mathematics, biology, physics, and chemistry have to major in that subject to teach in grade five and beyond. (Up to the fourth grade, there is just one teacher for all subjects except foreign languages.)

A suggestion was once made to change this in mathematics: just allow mathematics teachers in middle school grades to take a few courses, similar to what we could call a mathematics endorsement here. This was strongly rejected by those responsible for training mathematics teachers. We can learn a lesson from this, as many teachers of middle school mathematics in the US are not adequately prepared for their classrooms. However, as the teaching of informatics is a fairly recent addition to the curriculum, a major in computer science is not necessary to teach informatics in the middle school grades.

Another significant difference is how mathematics is taught at university. Each major in the sciences has their own sequence of mathematics courses, taught by faculty in that department; physics typically has a three-semester sequence, and chemistry a two-semester sequence. So there is no “one size fits all” calculus sequence, as is commonplace in American universities.

One consequence of this is that there is no gateway proof-writing course in the mathematics major. Theory and proof are integrated into mathematics courses from the very beginning. Thus, to earn the highest marks in an introductory calculus course, a student must show an ability to think abstractly and write mathematical arguments. This would seem like such a luxury for the typical American mathematics professor – a Calculus I course consisting entirely of math majors! But this is just standard operating procedure in universities in Serbia.

So my first experience in Serbia was an enjoyable one, and I learned a lot about the culture and education. I look forward to a few days in Belgrade before returning to the US later on in the week. I leave you with this interesting piece of found art I saw walking down the main street of Jagodina — you truly can never tell what surprises await when visiting somewhere new!

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.