They Know the Math, but the Words Get in the Way
Hale, Patricia, Focus on Learning Problems in Mathematics
This study sought to better understand instructional models that could be expected to improve student understanding of graphs of kinematic variables (distance, velocity and acceleration). The effect of using CBL-instruments and cooperative group structure (alone and in concert) was examined for repairing students' misconceptions. Misconceptions were determined using Nemirovsky and Rubin's (1992) definitions for cues that indicate students' misconceptions. Laboratory activities utilized in the various instructional settings were created that incorporated strategies developed by Kykstra, D. I., Boyle, C. Fl, Monarch, I. A. (1992) to promote conceptual change. Results of the study support previous research that even though students understand the requisite mathematical concepts, they still have misconceptions concerning the interpretations concerning the interpretation of mathematical terms in a physical setting. The most problematic misconception was indicated by students' use of Linguistic Cues; students interpreted mathematical terms using "common language" interpretations, not mathematical interpretations. Students continued to use "common language" interpretations even when confronted with physical situations using CBL-tools that contradicted their (incorrect) conclusions. Students needed to not only be confronted by their misconception, but needed the confrontation to be confirmed by the teacher or they maintained that their interpretation was correct and that the physical evidence was wrong.
"You can not apply mathematics as long as words still becloud reality." (Herman Weyl 1885-1955)
Mathematicians and physicists believe that when people communicate mathematics using algebraic symbols, communication is precise and unambiguous. However, when applying the symbols of mathematics many students would agree with Wehl that there is a great deal of ambiguity. For example, many students have difficulty articulating their understanding of the relationship between a function, its derivative, and its graph. One of the principle applications of these concepts is with problems involving distance, velocity and acceleration of a moving object (kinematic variables).
Virtually all students come to the classroom with some personal experience with kinematics. The desire to build conceptual understanding of functions, graphs and the physical phenomena that they relate to, using knowledge that students already have, is consistent with widely accepted constructivist principles. Clement states, "We assume that it is desirable to be able to ground new material in that portions of the student's intuition which is in agreement with accepted theory. When this is possible, it should help students to understand and believe physical principles at a 'make sense' level instead of only at a more formal one (1989, p. 1)." Unfortunately, students' personal understanding of kinematic variables may be incomplete or erroneous. Students' difficulties are often grounded in knowledge based on their personal experiences (Monk, 1990; Nemirovsky, J.R., Monk, S., 1992) Further, many students continue to have difficulty interpreting graphs of kinematic variables even following instruction in mathematics and in physics courses (Beichner, 1994; McDermott, L.C., Rosequist, M. L., Van Zee, E. H., 1986). Students recognize that the slope of a velocity graph is acceleration, but fail to reflect on the physical interpretation of negative acceleration, and whether the interpretation is different when velocity is negative rather than positive.
The traditional model of instruction for mathematics and physics courses has been a lecture/homework format, with lectures concentrating on the algebraic interpretation of variables. This traditional format may not be effective for developing understanding of graphs of kinematic variables: "Teachers cannot simply tell students what the graphs' appearance should be. …