Pedagogic Scenarios: Where's the Metric?
Last year, The New York Times [13 January 2009,12:2] published an article that was a very supportive discussion of the creation and operation of the TEAL physics course. It quoted a staff member of the course saying that the student failure rate in the course TEAL replaced had reached 30 percent, signaling that a course change was needed. Doubtless that conclusion is bolstered by other factors, but taken by itself it is a rational opinion and all opinions are true.
The problem with rational opinions is that other rational opinions will exist, so reaching a conclusion is a matter of compromising opinions in some arbitrary fashion. On the other hand, a logical system would yield the same conclusion for all who consider the situation.
A logical system would have the characteristic metric and an arithmetic that contains the basic assumptions of the system that determines how the elements of the metric transform. In simplified form, this is Euclid's axiomatic form of plane geometry, a metric of theorems and the assumptions, that parallel lines do not intersect, etc., or the form Newton used to establish logical physics, with the laws of motion the metric and the calculus the arithmetic. These forms admit closure, so no matter by whom or where a conclusion is drawn the answer will be the same for metrics embodying the same assumptions. After one has established the basic assumptions about grades, it is too simple to think the logic based on those assumptions can be used to model the total education environment.
James A. Garfield, a professor of Classics at Hiram College who became the twentieth President of the United States for four months before he was assassinated, said [See Diane Zabel's occasional column at rusq.org for an essay listing what might be the original expression and its many plagiarisms], "The ideal college is Mark Hopkins on one end of a log and a student on the other."
In the terms of the present discussion this statement places the teacher in the determining position. The student on whom the lecture focuses is also in a determining position, for the student must be capable of being engaged by the lecture, by having made the preparation to possess the capability to provide a basis on which the lecture can build. And the log, it expresses the determining effect of the adequacy of the classroom to support this activity.
Any one of these elements can be judged in terms of rational opinion that can range from religious faith to something approaching pure logical results. It is a fact that a lecturer can hold the attention of an audience and transfer no information. Or for a lecturer to have much information to transfer to an audience yet be unable to do so because of a lack of the ability to hold their fascinated attention. Luther Burbank, the genius horticulturist, is said to have been one of the unfortunate latter types [Jane S. Smith in the review of her book by Janet Maslin in The New York Times, 04 May 2009, C4:1]. He was also Visiting Lecturer on Evolution at Stanford University, 1904-1906.
It appears then that all these factors, teacher, classroom, student selection, syllabus, examinations deserve metrics and arithmetics to be used to define a given teaching environment. The metrics will not be as simple as the one Newton presented to the world, force equals mass times acceleration, with the arithmetic, the calculus, to determine transformations of the metric.
But the basic assumptions, which compose the metric, can be assembled at least in part. I have given a start for a metric for examinations and grades [M. W. P. Strandberg, Design of Examinations and Interpretation of Grades, Am. J. of Phys. 26, 555 (1958)].
The arithmetic, the function that transforms examination elements is logical steps.
In the ‘30s, when K. T. Compton and J. C. Slater began their renovation of the MIT Physics Department, they used what could be called unarticulated metrics to fashion the physics course structures. After all, the structure and implementation of the courses in a physics department is intuitively obvious. For the first two years all students were required to take physics courses. N. Frank wrote an introductory physics book that was strong in mathematics and taught from it. F. Sears wrote an introductory physics text that was lighter in mathematics and more graphic and taught from it. At times a course and syllabus based on some field of application of physics but light on a professional level was added.
It is clear that the lecturer, syllabus, and student metrics were exercised to satisfy the need to try to approach an environment that was different for different students. Admittedly, a triage of the students does not get to the one student and Hopkins scenario, but it is an attempt that rational opinions made, and that logical analysis might have improved.
So what about the high failure rate in the course? Was the selection of students to take the course a poor choice? Were the examinations unintentionally difficult as evaluated by a grades metric? Or was the lecture and recitation section format a poor one for this course material? Obviously these are not easy questions to answer without using a logical process and a good deal of rigor.
This essay is meant to be a challenge and so it lacks the explicit elements of a course design system. But the elements of the metrics controlling various aspects of the course are known or readily discovered and need to be declared as assumptions.
The scheduling matrix is readily diagonalized for the college course, for there cannot be more courses taught than there is staff, a given, to present them. Even the addition of attributes does not make the scheduling much more difficult. Considering attributes does make staffing more difficult. The lecturer who charms audiences and leaves them with a slight grasp of the topic of the lecture, and the lecturer who can embed the topic of the lecture in the mind of the audience in spite of not charming them, are two different people. Or, in the future, the lecture room may be supplied with netbooks that the lecturer could use to interact with each student separately by storing the response of each student to a query the lecturer makes, in a modern multi-dimensional version of Mark Hopkins' log. Are these three different lecturers or three similar people adapting to course needs?
The classroom metric in a similar manner contains the indisputable properties of each class environment and how they are modified by the client taking the course. At one time student seating was assigned and roll was taken by an empty seat scan. How important is attendance?
The student metric has to do with preparation and interest of the student in the course material and its mode of presentation. Richard Feynman organized a physics course with three students in order to get the material he wanted presented at a level he wanted. The human factor is so dominant in this metric that it is certainly the most difficult to be satisfied.
As for grades and grading, I have presented elements of a metric in the Logical Steps Rule referenced earlier. My experience tells me that my colleagues are offended if one does not understand or accept their intuitive judgment as to the difficulty of a question. And they are puzzled by the degree of difficulty that students find a question to have that does not mirror my colleagues' views. If I had to guess, I would probably say that remedial work on an effort to broaden the population with a logical basis for an understanding of the design of examinations and the meaning of grades would yield the richest dividends when compared with work on any of the metrics discussed above.