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…..”