STUDENT INTUITIONS CONCERNING DESIGN PRINCIPLES FOR

REPRESENTATIONS OF MOTION

Jan LaTurno

University of California, Riverside

JLaTurno@aol.com

The NCTM Principles and Standards for School Mathematics (2000) emphasize the need for students to be able to construct and interpret data in various forms occurring over time. While an extensive number of studies have examined students’ difficulties in creating or interpreting an expert’s orthodox graphical representation, few research agendas have considered the design principles that guide and constrain the construction of representations. The process of examining representations as an object of thought has been referred to as metarepresentation. This paper examines the intuitive notions of metarepresentation that appear in students prior to instruction.

The researcher individually interviewed all 25 students from a combined 4th, 5th, 6th grade classroom, following a semi-structured protocol. The students (a) constructed 3 representations of motions enacted by the researcher, (b) described the motion they thought was depicted by 3 representations provided by the researcher, (c) critiqued all 6 representations, and (d) constructed a representation for a new motion.

Analysis focused on design principles reflected in the students’ construction, preference, and critique of representations. Indicators of design principles manifested by the students were identified and grouped into two categories: (a) normative principles (those used by an expert mathematician in creating a representation of motion), and (b) non-normative principles.

Few students adopted normative design principles, even after viewing and critiquing representations constructed with these design principles (such as appropriate abstraction, homogeneity, etc.). However, many of the subjects’ violations of normative design principles, as well as evidence of non-normative design principles from the interviews, point out an instructional challenge. In some contexts, the principles the subjects applied would make sense. It is only when viewed within the particular intellectual enterprise of the mathematical representation of motion, that they do not work. The challenge here is helping students gain knowledge of the goal structure that permeates the expert’s field of mathematics.