Mike Clapp and Susan Kasparian

Department of Mathematics



Mathematics Appreciation and Understanding

For many people, mathematics is a subject inspiring dread rather than curiosity or fascination. Such negative feelings often come from experiences with school mathematics and the role in the mathematics learning process played by teachers or parents. Students frequently hold these views and lack confidence in their ability to do mathematics in two of the classes we teach. Mathematics 110, Liberal Arts Mathematics, satisfies the quantitative methods general education requirement and is taken by students majoring in the arts, humanities, communications, liberal studies, and child and adolescent studies. For many of them, this will be their last exposure to a course in mathematics. Mathematics 303A, Fundamental Concepts of Elementary Mathematics, is designed for prospective elementary teachers. So in both these courses, beyond the conventional outcomes associated with mathematics instruction, we seek to modify fundamental attitudes our students have about mathematics. We believe we are achieving some promising outcomes.

In our classes we attempt to present in a stimulating and non-threatening environment both the practical and the aesthetic aspects of mathematics. We also explore some useful ways of thinking about and using mathematics, and demonstrate why mathematics is both an enjoyable and entertaining discipline. We feel that the ability to appreciate mathematics and the challenges of problem solving are vital outcomes for our students. Many of them are destined to become teachers, who will provide children with their first exposure to the discipline. All of them as adults will have a role to play with mathematics, whether as teachers, parents, voters, and decision-makers. We want them to take away from our class a greater appreciation for mathematics as a living, exciting subject, many parts of which they can comprehend and enjoy. Even if they don't become enthusiasts, we hope that they will at least become friends with respect to the importance and accessibility of the study of mathematics.

Beyond teaching the content, we endeavor to influence students' views of mathematics and their own confidence in using it. Because of these objectives, we use a markedly different approach to instruction and assessment from that traditionally used in mathematics classes. Our belief is that students learn mathematics best when they are actively engaged with it. Consequently, we actively engage students in the learning process. To reduce anxiety, build confidence, and stimulate interest we focus classroom time on activities that involve students working in groups. The activities, in contrast to typical mathematical exercises, are designed to encourage students to communicate, explore, speculate, take risks, and to become personally engaged in the learning of the mathematical ideas being presented. Whenever possible, activities involve more concrete and visual experiences to bridge the transition from simple beginnings to more abstract understanding. Our techniques lead to lively classroom discussions in which students share discoveries and insights, successes and failures in a non-judgmental setting.

This approach to instruction and our own philosophy about how best to measure learning mean that conventional mathematics assessment is not an option for us. The one-hour, closed book examination, so prevalent in mathematics classrooms, contributes to student anxiety and lack of confidence, and yields limited insight into student understanding. We use portfolios as the primary means of student assessment. Portfolios offer students the opportunity to extend ideas beyond the classroom and to reflect upon the content and the experience of learning mathematics. Without the time constraints imposed by a traditional examination, students are better able to respond to a richer variety of assessment items and to present evidence of their best work.

Some Outcomes

Student reactions have been overwhelmingly positive to our methods of teaching and assessment. Of equal importance, we believe we are accomplishing our objectives. We base this on the information we have gathered over the past two years of teaching Math 110 and three semesters of Math 303A. In Math 110 we have been measuring changes in the students' confidence in their ability to do mathematics and their feelings about the subject of mathematics. We have only recently begun to gather analogous quantitative information for Math 303A, but for both classes we have a variety of qualitative feedback.

Since the summer of 1997, we have asked students in Math 110 to respond to two "before and after" prompts:

Each item requires a response on a scale of one to five. The choices for the first item are

(1) strong dislike (2) dislike (3) neutral (4) like (5) strong like.

For the second items the choices are

(1) very low (2) low (3) average (4) high (5) very high

Table 1 below displays information obtained from these items. The first row reflects the before and after scores for all 150 students in five sections who responded to the survey offered over the last two years. In the category of feelings about mathematics, the original average of 2.61 (a score midway between "dislike" and "neutral") increased 1.45 points to an average value of 3.92 (almost a score of "like"). For the category of confidence, the scores increased from an initial value of 2.92 by .89 to 3.81, a change from near "average" to near "high."

While these results from the entire group are encouraging, we recognize that many of the students may have entered the class already feeling positive about mathematics and their ability to do it. Consequently, we were particularly interested in how our approach to the course influenced the more than two-thirds of the students whose initial feelings about mathematics ranged from neutral to strong dislike or whose confidence in their ability to do mathematics ranged from average to very low. There were 116 students in the former category and 103 students in the latter. We examined the changes in average scores for these groups. The results here are very rewarding. The second row in Table 1 reflects the before and after scores for these selected students. For the group whose feelings were neutral to negative the initial score of 2.07 increased by almost two points to 3.92. In the confidence category, the score went from 2.33 to 3.59, an increase of 1.26.










All Students







Selected Students







Table 1

Improvement in Math 110 Students Feelings and Confidence

Based on Averages in All Sections

Beyond the increase in average score, we were interested in the number of the selected students whose attitude and confidence improved to scores of three or above. Table 2 contains the data for these students. One hundred four (almost 90%) of the one hundred sixteen recorded an increased score of three or better, while eighty-nine (76.7%) reported their feelings about mathematics as either "like" or "strongly like." Confidence, too, improved for the selected group of one hundred three, with seventy-four (71.8%) reporting an increased score of three or better. Remarkably, fifty-four (slightly more than half) of those students who originally had described their confidence in their ability to do mathematics as "very low," "low," or "average" now describe themselves as having "high" or "very high" confidence in their abilities.




> 3


> 4



> 3


> 4












Table 2

Improvement in Math 110 Students Feelings and Confidence

Based on Initial Student Scores Below Four



We do not yet have comparable data from our experiences in Math 303A. Nevertheless, students' written comments indicate that positive changes in feelings and confidence occur in this class as well. The examples of student reactions that follow are indicative of the responses that we receive from our students in Math 303A and closely parallel those from Math 110.

While we believe that the manner in which the classes are taught contributes in a significant way to these changes, we also find that the use of portfolios for assessment of student learning plays an important role. Our students tell us that they uniformly prefer portfolios to traditional in-class exams. Almost all say the portfolios are challenging, but that they learn and retain more from assembling them. Because of the opportunity for reflection, our students feel that they produce higher quality work and are better able to demonstrate what they have learned with less stress. Again the students' comments speak for themselves.



When we began to teach Math 110, we knew that many of our students would enter the course with views of our discipline that ranged from neutral to hostile, with corresponding levels of confidence in their ability to do it. We also knew that we wanted to try to influence the way our students thought about themselves and mathematics. We have designed our approach to the teaching and assessment of our classes using ideas based in our own experience and on the literature on what constitutes good practice. Our work thus far, supported in part by the Los Angeles Collaborative for Teacher Excellence, a five-year, multi-campus project funded by the National Science Foundation, indicates that we have been able to change student perceptions for the better in a significant number of cases. These changes are occurring in the mathematics courses for elementary teachers as well.

Two obvious next steps lie ahead. We want to try to determine how lasting these changes may be. If a few years from now, we measure how these same students view mathematics, will our class have had any lasting influence or will this turn out to be simply a temporary euphoria? Many of the students who enroll in liberal arts mathematics and never take another mathematics course will go on to hold opinions about mathematics education as parents, workers, voters, and opinion-shapers. Others, from that same course or from courses for prospective teachers, will live lives that may actively involve the use or teaching of mathematics. We know that early experiences with mathematics shape initial opinions; will positive experiences with the last courses taken restructure negative perceptions? We believe they will.

We also must determine if our success can be generalized to other instructors and other classes. Ours is a challenging way to teach and the preparation is demanding of time and creative energy. But if we are correct that this approach enables more students to learn the subject with confidence and enthusiasm, and to demonstrate successfully what they have learned, then there is impetus for change on the part of our colleagues. Will faculty be willing to shift from traditional methods? We believe they will.