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• W226453040 abstract "Several studies have shown the difficulties students encounter in making sense of situations involving rate of change. This study concerns how students discover errors and refine their knowledge when working with rate of change. The part of the study reported here concerns the responses of four precalculus students to a task which asked them to sketch a distance-time graph showing then These four students drew the same incorrect graph. This report is about how they discovered and corrected the error. Two general-conclusions from this study are that students use a variety of resources to address rate of change, and that slope and changes over intervals are both powerful ways for precalculus students to think about rate of change. An instructional implication of this study is that calculus and precalculus teachers should provide opportunities for students to use their knowledge of slope and changes over intervals to construct knowledge of rate of change. Teachers should notice the knowledge students use to make sense of situations and help students use that knowledge to construct new mathematical knowledge. Contains 41 references. Appendices contain statements of tasks and graphs. (Author/PVD) ******************************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ******************************************************************************** PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS BEEN GRANTED BY TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) This document has been reproduced as eived from the person or organization originating it. Minor changes have been made to improve reproduction quality. . Points of view or opinions stated in this document do not necessarily represent official OERI position or policy. Growth of knowledge of rate in four precalculus students Garnet Smith Hanger Department of Mathematics Spring Arbor College Spring Arbor, Michigan 49283 e-mail: ghaugerecougar.admin.arbor.edu AERA, Chicago, IL March, 1997 BEST COPY AVAILABLE 2 2 Abstract Several studies show difficulties students encounter in making sense of situations involving rate of change. There is less research on how students discover errors and refine their knowledge when working with rate of change. The study reported here concerns the latter. This study is part of a larger study about knowledge of rate of change of precalculus, calculus, and postcalculus students. The part of the study reported here concerns the responses of four precalculus students to a task which asked students to sketch a distance-time graph showing slowing down and then speeding up. These four students drew the same incorrect graph. This report is about how they discovered and corrected their error. The four precalculus students drew a graph that showed All of them discovered their error one before I asked her any questions; one when I asked him to explain how his graph showed and up; one after she explained how her graph showed and up, created a table of values for her table and tried to explain how it showed and up; and one when she began working on a different task which called for a graph that showed The first student came to see that slope and shape of the distance-time graph give information about speed and used this information to then draw a correct graph. The second student carefully examined and explained his graph to see that it represented steady pace. He then drew a correct graph and gave an explanation of how it showed and For both graphs he focused on change in the height of the graph over equal intervals of time to make decisions about the rate of change associated with each graph. The third student justified her initial graph by saying If they're steadily and they're steadily up, it should just be straight lines.. When she made a table of values for her graph and tried to explain how it showed and up, she saw that the distance was changing the same number of feet each second and concluded that it showed She then created a correct graph by carefully constructing appropriate changes in distance for each second to get a graph that showed and then The fourth student did not realize her error until she got to the second task, which called for a graph that showed She noticed then that her graph for the first task showed the same distance being traveled each second and proceeded to create a correct graph by thinking in terms of the time needed to cover a certain distance to show and then Of the four students, one used slope of graph, one used reference to changes on another graph, and two referred to relationships between distance and time to discover and correct their error. Two general conclusions from this study are that students use a variety of resources to address rate of change and that slope and changes over intervals are both powerful ways for precalculus students to think about rate of change. An instructional implication of this study it that calculus and precalculus teachers should provide opportunities for students to use their knowledge of slope and changes over intervals to construct knowledge of rate of change. More generally, teachers should notice knowledge students use to make sense of situations and help students use that knowledge to construct new mathematical knowledge." @default.
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• W226453040 title "Growth of Knowledge of Rate in Four Precalculus Students." @default.
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