Warren W. Esty and Anne R. Teppo
Students' performance in today's secondary school mathematics curriculum is measured to a large extent by unit examinations administered on a regular basis. The numerical grades earned on these tests are averaged to help determine a final course grade. This article examines the logical consequences of this process of test-score averaging on the arrangement of course material, the nature of the tests themselves, and the learning emphasized. We believe that in each area, these consequences are negative. Furthermore, we believe that in line with the evaluation standards, test scores can be used to assign course grades in an objective and valid way that does not rely on averaging. An example of this type of grading will be described.
Consider two examinations in a mathematics class, one given early on and the other given near the end of the term. Logically, the only way that they can be considered equal in assessing learning is if this learning is not regarded as cumulative. That is, each examination is truly a "unite" examination and the unit studied has no cumulative importance if it contributes nothing to subsequent concept development and if the procedures and concepts learned never reappear. These assumptions are false. The objections to averaging of test scores apply doubly to averaging of homework scores. Yet mathematics teachers continue to assign final course grades on the basis of averaged scores.
Grade averaging has an impact on the arrangement of course material and the types of tests that are used. Material tends to be compartmentalized into small, discrete "digestible chunks," with related short examination problems that test specific and isolated skills. This approach allows teachers to demand immediate mastery of objectives with an accompanying 90-80-70-60 percent performance scale, yielding the grades they want to give.
Compartmentalizing course material severs connections between one topic and another. Concepts, which by definition must be abstracted from numerous examples in various contexts, are neglected in instruction because they cannot be mastered in small chunks at the 90-80-70-60 percent level during the short intervals between tests. Averaging of unit test scores forces teachers to compromise and write tests according to what the students can learn between tests, not according to what teachers really want the students to know.
For example, few students will fully grasp the difference between a function, f, and its image, f(x), immediately after functions are introduced. The distinction between the rule itself and the number created by applying the rule to x is subtle but extremely important. Calculus students need to know what f(x+ h) is when f(x) = x^2, yet many do not distinguish it from f(x) + h because they have not learned to isolate the rule, f ("square it"), from the image, x^2. After the term function is introduced in algebra, a unit test is, of course, given that explores some aspects of particular functions. But the concept of function will not have been fully developed through instruction and certainly not yet fully understood by the students. Thus the teacher can ask questions only at a low level of comprehension if he or she is designing a unit test that students can pass. Success on such a test, although it improves a student's chances for an acceptable final course grade, gives little indication of the student's progress toward understanding the essential concept of function.
The evaluation standards recognize that tests "are one way of communicating what is important for students to know" (p. 189). Because students will not do well on conceptual questions, such as the distinction between f and f(x), such questions are usually not asked. Because conceptual questions are not asked, students fail to recognize the importance of concepts. Then whatever misconceptions the students develop on their own persist. The pernicious effects of averaging carry through to higher-level courses.
Averaging and compartmentalizing reinforce one another. Without compartmentalizing averaging is inappropriate. With compartmentalizing averaging works. With averaging compartmentalizing seems necessary to justify the grading system. Therefore these two have settled in together. The compartmentalizing-averaging method of teaching and evaluating mathematics has come to be taken for granted. The vast majority of textbooks and courses depend on it. Teachers use it and it works-for material organized in compartments. But this type of organization is precisely what must be changed if mathematics courses are seriously to teach mathematical connections.
The evaluation standards recognize that "students' grasp of mathematical concepts develops over time. Many concepts introduced in the early grades are later extended and studied in greater depth" (pp. 223-24). Clearly this type of development over time must happen throughout individual classes as well as from elementary to secondary grades. If we recognize that learning mathematics is cumulative and expect students to make connections among procedures and develop concepts over time, then we propose that it is acceptable-and completely expected-for students to be unable to demonstrate concept mastery until late in a course. The current practice of test-score averaging does not recognize these factors of time and complexity. The evaluation standards specifically assert that assessing what students do not know should receive decreased attention (p. 191), but averaging penalizes students for what they don't know during the progress of the course instead of rewarding them for what they do know at the end.
The evaluation standards call for us "to reassess the manner and methods by which we chart our students' progress" (p. 192). In terms of assigning students' grades, the evaluation standards assert that assessment should measure (a) how well the student has understood and integrated the material, (b) if the student can apply his or her learning in other contexts, and (c) if the student is prepared to proceed to the next grade or level (p. 200). These questions are best answered only at the end of a mathematics course. They imply that assessment for grade assignment should be based on long-term course goals. Using a scoring method other than averaging for assigning class grades can free existing instruction from its present constraints and make it possible to emphasize the learning of complex concepts and related multistage procedures.
At the beginning of this class, students are informed that although they will be examined with diagnostic quizzes and tests throughout the course, only their performance in the final weeks of the class will be counted for assigning grades. They are told that what they learn will be cumulative and that at the end of the course they will be held responsible for all the material.
Daily homework is checked for accuracy but not scored. Early quizzes and examinations are scored but not counted toward the final grade. Class participation is required and supplies useful feedback for both the instructor and student. Thus, assessment during the first two-thirds of the class is used to inform students and instructor of individual progress, not to generate course grades, that is, assessment is diagnostic and furnishes instructional feedback.
Students' final grades are not penalized by averaging in test scores that measure incorrect or incomplete understanding that occurs in the early stages of the course. Course grades are assigned on the basis of knowledge and skills displayed in the final few weeks of the class using quizzes and cumulative examinations, which are written tests, and, to a much lesser extent, class answers, which are spoken-yielding quite different information.
Because this approach is so nontraditional, we have had to deal with students' fears that we don't mean it and that in the end we really will average in some poor early score. We repeatedly reassure the students that we are looking for improvement and eventual mastery, not for a chance to take points off. For instance, quiz scores, instead of being rated on a 90-80-70-60 percent scale, are classified as "already at the C (or A or B) level" or "soon to be C or better."
Students' performance is expected to improve throughout the course. By omitting the requirement of immediate mastery, students can be held to a higher standard and posed more challenging problems. For example, on the second test administered halfway through the autumn 1990 course, a performance of 50 percent was judged by the instructor to be what was expected, at that time, of an eventual C student. The instructor's comment to the students on this "low" test score was, "You will have another chance soon to demonstrate that you have mastered the exam material." This recognition that examination scores were acceptable even if they did not fit a 90-80-70-60 percent scale is to be distinguished from curving, which is sometimes necessary under an averaging system according to which each score directly affects the student's final course grade.
Students' perceptions of the progressive grading system were obtained from interviews conducted as part of a qualitative study of the autumn 1990 course (Esty and Teppo 1991). Students remarked that this approach kept them working when they were not doing well on the diagnostic examinations and homework. "It keeps me feeling like I'm in the race," one student commented. Students knew they still had a chance to earn an acceptable final grade by continuing to improve throughout the course. As another student explained, "[You're given] a chance to make up the things you didn't understand [after] you've found a way to acquire the knowledge. You're rewarded for that, and I think it's really important."
This grading system acknowledges the reality that mathematics learning takes time. Instead of increasing anxiety early in the course for those who have not yet been able to grasp the material, this approach allows students the time actually required to put it all together and produces less negative reinforcement. Furthermore, less emphasis is placed on filing facts in short-term memory in cramming sessions just before unit examinations.
We have found that the students do not abuse this grading system-any more than they do any other grading system-by not working at the beginning of the course. As one interviewed student explained, "It isn't as though the teacher relieved us of the obligation or anything like that. He put all the students in a position where they could be confident in themselves." The students used the diagnostic quizzes to measure their performance as the course progressed. They understood that growth would occur and that they could realistically look forward to assimilating the material eventually. We have found that performance in the final weeks of the course accurately reflects this integration of knowledge throughout the course.
National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.
Warren Esty teaches at Montana State University, Bozeman, MT 59717. He does research in applied probability and is working on his second freshman-level textbook. Anne Teppo is a mathematics education consultant, 1611 Willow Way, Bozeman, MT 59715. She is involved in qualitative research investigating the development of mathematical concepts within the classroom environment.