SPEECH AND GESTURE IN PATTERN GENERALIZATION TASKS INVOLVING
GRAPHS: EVIDENCE THAT PERCEPTIONS INFLUENCE CONCEPTIONS
Kristen Bieda
Mitchell Nathan
University of Wisconsin – Madison
University of Wisconsin - Madison
## email not listed ##
## email not listed ##
This paper presents a study aimed at describing middle school students’ views of graphical
representations. From interviews of middle school students solving pattern generalization
tasks using graphs, both speech and gesture responses were analyzed to determine students’
beliefs about graphical representations. Results indicate that a bounded view of graphs, as
evident in gestures, may influence performance on pattern generalization tasks.
Background
There is limited empirically based research in the literature about student reasoning in the
context of early algebra graphical representations. Zacks and Tversky (1999), using a sample
of Stanford undergraduates, found that students view graphs as inherently limited by their
form. Stevens and Hall (1998) showed how a student, involved in a tutoring session using
Cartesian coordinate graphs, was influenced by the graph’s spatial relation to the grid edges
as he predicted how an entire function would change in appearance when an equation was
transformed. Videotaped interview data allows for the collection of multiple modes of
response that can be analyzed to provide a more accurate and complete account of middle
school student thinking. Our gesture analysis methodology draws from the established
theoretical work by Goldin-Meadow (2003) and Alibali and Goldin-Meadow (1993).
Methods
Thirteen students, in grades 6-8, participated in videotaped interviews. All students were
from a large urban middle school with a high percentage of non-Caucasian students (91.7%)
and students in the free/reduced lunch program (86%). Problem 1 asked students to interpret
a two-dimensional, Cartesian coordinate graph with four points plotted along a linear
function, y = 3x + 1. In Problem 1a, a low-complexity FP (far prediction) task, students were
asked to determine the cost to make 10 copies of a CD using the information shown in the
picture. Problem 1b, a high-complexity FP task, asked students to determine the cost to make
31 copies of a CD, and thereby extrapolate beyond the bounds shown in both the x- and y-
axes.
Results
Students in this study articulated both bounded and unbounded beliefs about the graph.
Bounded beliefs about a graph are articulated when a student’s speech or gesture response
demonstrates that the perceived information is constrained by the graph’s physical
boundaries. Unbounded beliefs about a graph manifest when a student’s speech or gesture
response indicates reasoning beyond the physical boundaries of the graph. Since FP tasks are
tasks with solutions outside of the numerical range shown on one or more axes of the graph,
only unbounded verbal responses are considered to reveal the novel information about
students’ interpretations presented through their gestures. Figures 1 and 2 show examples of
bounded and unbounded verbal and gesture responses from the study:
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Convention