| It has been well-documented that few high school Calculus students understand the relationship between the derivative and integral and, consequently, their relationship to the accumulation function. Also, students have difficulty utilizing this relationship when solving non-routine problems. Some research studies have shown that understanding the nature of the accumulation function may help students develop comprehension of the relationship between the derivative and integral. The derivative and integral, the mathematical models of certain real world phenomena and processes, can be presented to learners via multiple representations including word descriptions, graphs, diagrams, and tables. The available research shows that students exhibit certain attitudes and preferences towards the different representations of structurally the same concept. This study was designed to gain insight into the nature of students' conceptual understanding of the accumulation function. Advanced Placement (AP) Calculus students were surveyed in order to capture their attitudes and preferences when using multiple representations in solving problems involving the accumulation function. In particular, this study investigated proficient learners' strategies while solving problems, and inquired into proficient learners' ability to translate between multiple representations of structurally the same concept, as well as their structural or operational understanding of the accumulation function. Data were collected from three schools in the Northeastern United States. Ninety-two Advance Placement Calculus students completed a survey developed exclusively for the study. Ten surveyed students were interviewed. Analysis of data from the surveys and interviews focused on trends in students' attitudes and preferences when solving accumulation function problems in multiple representations, recognizing structurally the same concept in multiple modes, and utilizing strategies when solving accumulation function problems.;It has been found that proficient learners preferred accumulation function problems to be presented in multiple modes at the same time. In this study, proficient learners showed a slight preference for problems presented in words, as compared to problems presented in symbols or tables. Also, graphical representations, which required a certain depth of knowledge and ability to recognize the different relationship between multiple concepts, were not favored by the participants. Interestingly enough, among problem-solving strategies, such as finding the area under the curve, Riemann sums, "velocity vs. time," the "product of the units," and a rule, the participating students preferred the rule "the integral of the rate equals the amount." It is important to note, the language offered in quotations is colloquial, however, unfortunately widely-used in classrooms. Participants identified as proficient learners, who could fluidly translate between representations, showed flexibility in using different strategies in solving accumulation problems, thus demonstrated an understanding of the mathematical relationships. It is concluded that to facilitate students' development of conceptual understanding and procedural skills with the accumulation function, the problems should be presented in multiple ways. In addition, the use of imprecise mathematical language may inhibit students' conceptual understanding of the accumulation function. |
