Revista Brasileira de Informática na Educação – Número 1 – 1997
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Explorando Funções Matemáticas através de Micromundos Dinâmicos
Verônica Gitirana Gomes Ferreira
Email: [email protected]
Tese de grau de PhD em Educação Matemática defendida em 28/02/97
Instituto de Educação, Universidade de Londres
Orientadora: Prof. Celia Hoyles
Título original: Exploring Mathematical Function through Dynamic Microworlds
Abstract
The aim of this research was to investigate students' perceptions of function as they interacted
with the different dynamic representations of function made available through computer
environments. Microworlds were designed comprising sequences of activities around two
software: a multi-representational software, Function Probe (Confrey et al, 1991), which
allows students to transform a graph into another directly in the Cartesian representation; and
two adaptations of DynaGraph (Goldenberg et al, 1992), which allows students to vary x,
observing the variation of ƒ(x), DG Parallel (with parallel axes) and DG Cartesian (using
Cartesian axes). This microworld design considered its technical (representation of concepts
in a computer environment) and pedagogical sense (“a software designed to be adaptable to
pupils' initial conceptions together with carefully sequenced sets of activities on and off the
computer...” (Hoyles et al, 1991: 1)). A series of case studies of four pairs of students was
undertaken in Brazil in order to trace the evolution in students' perceptions of a selection of
function properties; namely turning point, variation, range, line symmetry and periodicity.
Perceptions were considered to be student’s understanding of these properties evidenced by
the way they characterised functions when represented in each microworld. This diversity of
properties was chosen to examine different ways students analyse functions: pointwise,
variational, global and pictorial.
Starting with an examination of the curriculum followed by the case study students as a means
to describe the origins of their perceptions, a longitudinal investigation was undertaken in
order to identify the main features of each of the microworlds that appeared to contribute to
students' progress. The student’s perceptions were analysed by drawing attention to their
origins, their usefulness and their potential limitations (from a mathematical point of view). A
methodology for this longitudinal study was devised which incorporated visual presentations
to capture the main characteristics of students' perceptions.
The results showed that DG Parallel, a ‘new’ representation, prompted the development of
perceptions free of previous limitations and sufficiently robust to allow revision. However,
Revista Brasileira de Informática na Educação – Número 1 – 1997
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properties previously perceived pictorially were rarely identified in DG Parallel. Together
with DG Cartesian, interactions with this microworld provoked the students to develop a
variational view of some of the function properties. In addition, DG Cartesian served as a twoway bridge between variational and pictorial views. By way of contrast, using the tools in FP
to transform graphs seemed not to shape perceptions, but to assist in the exploration of the
function properties.
References
Confrey, J.; Smith, E. & Carroll, F. (1991) Function Probe: Academic Version, Department of
Education, Cornell University, Ithaca - NY.
Goldenberg, E.P.; Lewis, P. & O'keefe, J. (1992) “Dynamic Representation and the
Development of a Process Understanding of Function”, in G. Harel and E. Dubinsky (Eds.)
The Concept of Function - Aspects of Epistemology and Pedagogy, MAA Notes 25,
pp.235-60.
Hoyles, C.; Noss, R. & Sutherland, R. (1991) Final Report of the Microworlds Project: 19861989, Institute of Education University of London.