I will report on two studies that involved the assessment and development
of Basotho (people of Lesotho, Southern Africa) elementary students' probabilistic
thinking. As an extension of framework research on probabilistic thinking
by Jones and his colleagues (Jones et al., 1997), we generated and validated
a framework that described and predicted the probabilistic thinking of
Basotho students' in grades 1, 3, 5, 7 and 8 across five constructs: sample
space, probability of an event, probability comparisons, conditional probability,
and independence (Polaki, Lefoka,& Jones, 2000). The framework described
students' probabilistic thinking in terms of four developmental levels
that were found to be consistent with the thinking levels in Case's (1996)
more general cognitive model, and in essence, demonstrated that Case's
model was applicable to probabilistic thinking in addition to the three
knowledge domains previously examined by Case: quantitative thinking,
spatial thinking, and narrative thinking.
As a follow- up to this study (Polaki et al., 2000), I designed and implemented
two versions of a teaching experiment that traced the development of Basotho
elementary (grades 4 and 5) students' growth in probabilistic thinking
(Polaki, 2000). The first version focused on small-sample experiments
and analysis of sample space composition. The second version incorporated
small-and large-sample experimentation (simulations) in addition to analyses
of sample space symmetry. Grounded in Cobb's (1999) developmental research
cycle, the main objectives of each version were to (a) to identify and
trace the evolution of mathematical practices associated with growth in
thinking about sample space and probability of an event, (b) identify
key episodes that were associated with students' growth in probabilistic
thinking, and (c) evaluate the effectiveness of each version of the teaching
experiment. Analyses of the data revealed existence of the following thinking
patterns: (a) a sample space misconception (Jones et al., 1999) that was
highly resistant to instruction, (b) a weak and often unstable part-part
schema that was minimally effective in enabling target students to identify
complete sample space and to order probabilities for 1-dimensional experiments,
and (c) a stronger and more stable part-part schema that enabled target
students to reason with greater consistency when listing the complete
set of outcomes and when ordering probabilities for 1-dimensional experiments.
Additionally, as evidenced by changes in thinking levels, each version
of the teaching experiment had a substantial impact on the target students'
Case, R. (1996). Reconceptualizing the nature of children's' conceptual
structures and their development in middle childhood. Monographs of the
Society for Research in Child Development, 61, (1-2, Serial N0. 246),
Cobb, P. (1999). Individual and collective mathematical development:
The case of statistical analysis. Mathematical Thinking and Learning,
1 (1), 5-43.
Jones, G. A., Langrall, C. W., Thornton, C. A., & Mogill, A. T. (1997).
A framework for assessing and nurturing young children's thinking in probability.
Educational Studies in Mathematics, 32, 101-125.
Jones, G. A., Langrall, C. W., Thornton, C. A., & Mogill, A. T. (1999).
Using students' probabilistic thinking in instruction. Journal for Research
in Mathematics Education, 30, 487-519.
Polaki M. V. (2000). Using instruction to trace the development of Basotho
elementary students' probabilistic thinking. Unpublished doctoral dissertation,
Illinois State University, Normal.
Polaki, M. V., Lefoka, P. J., & Jones, G. A (2000). Developing a cognitive
framework for describing and predicting Basotho students' probabilistic
thinking. BOLESWA Educational Research Journal, 17, 1-20.