VISUALIZING RATES OF CHANGE FOR WATER TRAJECTORY: A
CASE STUDY WITH PRE-CALCULUS AND UNDERGRADUATE STUDENTS*
Patricia E. Balderas Cañas
The National Autonomous
University of Mexico
This is an interpretative
study of two current issues in the psychology of mathematics education. A first issue was to obtain further
understanding on visualization that undergraduate students manifest when asked
to imagine and draw the water trajectory that constantly flows from the end of
a hose pipe. The second issue concerned
how students’ conceptions of vertical and horizontal changes in addition to the
rate of both changes were influenced by their visualizations of water
trajectory. Both issues rose from
previous research (Balderas, 1992). This research guided instructional designs
for beginning and advanced calculus courses (Balderas, 2000; Balderas &
Schäfer, in preparation). The study was
based on visualization, problem solving and rate-of-change learning theory
(Presmeg, 1997; Zimmerman and Cunningham, 1991; Schoenfeld, 1992; Speiser and
Walter, 1994; Thompson, 1994 and 1999; between others). Students’ visualizations of the water
trajectory as a global and completed phenomenon were centered on numerical data
provided by the problem statement.
Their answers of rate of change were restricted to those visualizations.
Note
*Note. This study was developed as part of Balderas,
P. Visualization in Rate of Change Problem Solving. Research project supported
by CONACYT, Mexico, scholarship #145500, for a sabbatical leave at Florida
State University, 1999/2000.
References
Balderas, P. (2000). Variations
of Water Trajectory Problem for a Didactic Approach of Calculus. [Online]
Available: http://mailer.fsu.edu/~pbaldera/pbalderas.html
Presmeg, N. (1997).
Generalization using imagery in mathematics. In L. D. English (Ed.), Mathematical reasoning. Analogies, metaphors
and images (pp. 299-312). NJ:
Lawrence Erlbaum.
Speiser, B & Walter, C.
(1994). Catwalk: First-Semester
Calculus. Journal of Mathematical Behavior, 13(2), 135-152.
Thompson, P. (1999). Some Remarks on representation, conventions,
and common meanings. Prepared for
PME-NAXXI. Working Group on Representations, Cuernavaca, Mexico. [On line] Available: http://members.home.net/pwthompson/PMEXXI/ThompsonRep.html