When using a problem-centered teaching approach, the instructor needs new methods of assessment. This article explores one such approach.
Background and Purpose
In this article, we describe an assessment procedure the second author used at Florida State University in an upper level mathematics course on problem solving. Analysis of this course formed part of the doctoral dissertation of the first author . Florida State University is a comprehensive institution with a research mission. It is part of the state university system and has 30,000 students. The 16 colleges and schools offer 91 baccalaureate degrees spanning 190 fields. There is an extensive graduate program in each of the schools and colleges. Florida State University is located in Tallahassee and has strong research programs in both the sciences and humanities. The main goal of the course, taken by both undergraduate and graduate students, was to engender mathematics problem solving through the development of heuristics. A second goal was to provide opportunities for students to reconstruct their mathematics in an integrated way so that it could be utilized. The second author has taught this course several times using the methods described in this paper. We view mathematics as a personal activity one in which each individual constructs his/her own mathematics. It is our belief and experience that a problem-centered instructional strategy is more effective than explain-practice which tends to emphasize procedures . For assessment to be consistent with this view of mathematics, it should occur as students are engaged in problem solving/learning mathematics rather than as a separate activity.
In order to promote reflection on their mathematical activity, it is important to negotiate a classroom environment which focuses on the students' problem solving and explanations of their methods rather than prescribed procedures. In this problem solving course, nonroutine mathematical problems such as the following constituted the curriculum.
Fraction of Singles The fraction of men in a population who are married is 2/3. The fraction of women in the population who are married is 3/5. What fraction of the population is single?
Angle Bisector Problem Write the equation of the bisector of one of the angles formed by the lines 3x + 4y = -1 and 5x + 12y = 2.
The assessment plan involves considering each students' problem solutions and providing written feedback in the form of comments, without points or grades. Overall, assessment is based on classwork, homework, a midterm examination, and a final examination. However, it differs from traditional policies in that the grade is not simply a weighted average of numbers for these components but reflects the students' progress in becoming competent mathematical problem solvers. The assessment plan and the goals of the course are discussed with the students on the first day of class and elaborated in subsequent sessions.
Reading the homework on a regular basis is the richest source of assessment data as the student's mathematical competence and reasoning become clear. For example, one student usually wrote very little on each problem, unlike other students who used numerous pages explaining their methods. Yet he frequently developed quite elegant solutions. Elegance of solutions came to be valued by all students as the course progressed. In contrast, another student, while getting correct answers, would often use less sophisticated methods, nongeneralizable methods such as trial and error or an exhaustive search. On another occasion, a graduate student who thought he knew mathematics turned in simplistic solutions which did not withstand scrutiny. He was sure his answers were correct and was surprised to see the methods that were explained by other students. He initially underestimated the course demands but adjusted and became a strong problem solver. A few students were not successful in the course because their mathematical knowledge was inadequate. They had been successful in procedural courses but had difficulty when required to construct solutions without prescribed methods. By carefully tracking homework, the instructor is able to chart progress and assess quality. Some students who have little success on problems during the first few weeks of the course come alive and become successful problem solvers by the end the course. In previous years, the second author tried assigning letter or number grades on homework, but found that this practice encouraged students to do homework for the grade rather than focusing on learning and doing mathematics. The less students are thinking about their grade the more they can be doing mathematics. An average of test scores will not show this progression as clearly. Thus homework papers are an essential source of assessment data.
While generalizable and elegant solutions are valued, students are free to use any solution methods they wish. They are encouraged to explain their solution process and good ideas are valued. Comments made on papers include "very nice," "how did you get this?" "I do not follow," "this does not follow from what you did before," or "how did you think to do this?" Quality work is expected and the instructor will write "unacceptable" on some of the first homework papers if necessary. Students are not expected to be expert problem solvers when they enter the course; so weaknesses on early assignments are not unexpected. Furthermore, a student is not penalized by poor work initially.
In class, students present their solutions to the class for validation. There is no one method expected by the instructor. Some students are frequently eager to present their solutions while others are reluctant. By making notes on class activities after each session, these data can be a significant component of the assessment plan. On some days the students work in groups of three or four solving assigned problems. During this time, the instructor is actively engaged in thinking about the group dynamics: who initiated ideas, who is just listening; and assessing each student's involvement. In an early group session, the instructor mentioned that during the small group sessions he had seen several different approaches. He followed this by saying that "we are always interested in other ways." He was pointing out that there was not necessarily one correct way to solve a problem and stressing the belief that problem solving is a personal activity.
The midterm and final examination questions are similar to the nonroutine problems assigned on homework and classwork. Each examination consists of a take-home part and an in-class part. Early in the course it is announced that an A on the final examination will result in an A in the course, since it would be evidence that students had become excellent problem solvers. The comprehensive final examination consists of approximately 10 challenging problems. The goal is a high level of competence and many students reach this by the end of the course. In determining the final grades for students in this course, all the information about the student is considered, not just numerical entries in a gradebook. Particular attention is given to the level of competence at the end of the course rather than using an average of grades throughout the semester. Early in the course, some students are not able to solve many of the problems assigned but as they come to make sense of mathematics previously studied and develop effective heuristics, their level of competence rises dramatically. The students are also encouraged to assess their own work, and near the end of the course they turn in to the instructor the grade they think they should receive. This information can serve as a basis for discussion, but in the end the instructor determines the course grade. The final grade reflects the knowledge at the end of the course without penalizing students for lack of success early in the course. Thus, students may receive a higher course grade than would result from an average of recorded numbers.
Students became more intellectually autonomous, more responsible for their own learning. This was particularly evident in the whole class discussions. Students began to initiate sharing solutions, suggestions, and ideas. By the third class session, some students would walk to the board without waiting for the instructor's prompt in order to elaborate upon another idea during the discussion of a problem. Their additions to the class discussion added to the richness of their mathematics. In addition, students would suggest problems that were challenging and also voluntarily hand in solutions to nonassigned challenging problems.
This assessment plan, which encourages self-assessment, reflects student competence at the end of the course rather than being an average of grades taken at various points throughout the course. This practice encourages students who might get discouraged and drop the course when they actually have the ability to succeed. Students tend to like this grading method because there is always hope. Four students of varying levels of mathematical sophistication were interviewed throughout this Problem Solving course by the first author. The students generally liked the assessment plan. One student said that this plan was very appropriate as mathematical problem solving should be evolving and that the comments on their problem solving were helpful. When each of these four students was asked during the last week of the course what grade they believed they would receive for this course, all four reported grades which were identical to those later given by the instructor, one A, two Bs, and one C. Even without numerical scores, these students were able to assess their own work and this assessment was consistent with the instructor's final assessment.
Use of Findings
The results of this study suggests that the holistic plan described in this paper has merit and serves education better than traditional systems [1, 2]. Furthermore, by a careful reading of the homework, the instructor is better able to choose and design mathematics tasks to challenge and enrich the students' mathematical problem solving. For example, if students are seen as being weak in a particular area, e.g. geometric problems or proportional reasoning, the instructor can focus on these areas rather than spending time on less challenging problems.
Students enter a course expecting practices similar to those they have experienced in other mathematics courses. Thus, in implementing an assessment method which encourages students to focus upon doing mathematics rather than getting points, it is essential to help students understand and appreciate this unconventional assessment plan.
 Mathematical Sciences Education Board. Measuring what counts. National Academy Press, Washington, 1993.
 National Council of Teachers of Mathematics. Assessment standards for school mathematics, NCTM, Reston, VA, 1995.
 Trowell, S. The negotiation of social norms in a university mathematics problem solving class. Unpublished doctoral dissertation, Florida State University, Tallahassee, FL, 1994.
 Wheatley, G.H. "Constructivist Perspectives on
Science and Mathematics Learning," Science
Education 75 (1), 1991, pp. 9-21.