Abstract
Prior research suggest that introductory physics students have difficulty with graphing and interpreting graphs. Here, we discuss an investigation of student difficulties in translating between mathematical and graphical representations for a problem in electrostatics and the effect of increasing levels of scaffolding on students' representational consistency. Students in calculus-based introductory physics were given a typical problem that can be solved using Gauss's law involving a spherically symmetric charge distribution in which they were asked to write a mathematical expression for the electric field in various regions and then plot the electric field. In study 1, we found that students had great difficulty in plotting the electric field as a function of the distance from the center of the sphere consistent with the mathematical expressions in various regions, and interviews with students suggested possible reasons which may account for this difficulty. Therefore, in study 2, we designed two scaffolding interventions with levels of support which built on each other (i.e., the second scaffolding level built on the first) in order to help students plot their expressions consistently and compared the performance of students provided with scaffolding with a comparison group which was not given any scaffolding support. Analysis of student performance with different levels of scaffolding reveals that scaffolding from an expert perspective beyond a certain level may sometimes hinder student performance and students may not even discern the relevance of the additional support. We provide possible interpretations for these findings based on in-depth, think-aloud interviews.
Highlights
Physics is a challenging subject to learn and it is difficult for introductory students to associate the abstract concepts they study in physics with more concrete representations that facilitate understanding without an explicit instructional strategy aimed to aid them in this regard
We found that calculus-based introductory physics students have great difficulty translating from a mathematical to a graphical representation—only about one-quarter of the students without support plotted electric fields that were consistent with their expressions in all regions for the problem investigated here
Via think-aloud interviews, we identified possible reasons for the poor student performance in translating the electric field from the mathematical to the corresponding graphical representation
Summary
Physics is a challenging subject to learn and it is difficult for introductory students to associate the abstract concepts they study in physics with more concrete representations that facilitate understanding without an explicit instructional strategy aimed to aid them in this regard. When numbers are not given or when students run into a situation with two equations and two unknowns, they have much more difficulty solving the problem As evidenced by these examples and others [41], while students’ concept images are often not consistent with concept definitions, for mathematics experts, the concept images become tuned over time so that they are consistent with the conventionally accepted concept definitions. Some researchers have emphasized that this process of translating between the graphical and algebraic representations of functions presents one of the central difficulties for students in constructing an appropriate mental image of a function [47]. We explore the facility of students in a calculusbased introductory physics course in transforming a problem solution involving the electric field for spherical charge symmetry from a mathematical to a graphical representation, and the effect of different scaffolding supports on students’ ability to carry out the transformation consistently. In study 2, we designed two scaffolding support levels that built on each other based on the findings of study 1, and investigated their impact on improving students’ representational consistency
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