The process of generalization is an important
component of mathematical ability, and to develop this ability is
an objective of mathematics teaching and learning (NCTM, 2000).
The research study documented how high school freshmen in an accelerated
algebra class developed generalization strategies in combinatorial
problem-solving situations. Students were asked to solve five non-routine
combinatorial problems in their journals, assigned at increasing
level of complexity. The generality that characterized the solutions
of the five problems was the pigeonhole (Dirichlet) principle.
The researcher documented the evolving strategies
of the students. Data gathered through journal writings, open-ended
interviews and classroom observations was analyzed using techniques
from grounded theory. In particular the constant comparative method
of Glaser & Strauss (1977) was used. The researcher expected
that student strategies would evolve with the complexity of the
problem and with time. Four students were successful in discovering,
verbalizing, and in one case successfully applying the generality
that characterized the solutions of the five problems, whereas five
students were unable to discover the hidden generality.
The research categorized and described student
behaviors that led to successful generalizations and those that
led to unsuccessful generalizations, as well as identified the variables
necessary for students to successfully arrive at mathematical generalizations.
The research study resulted in a modification of Lester's (1985)
problem-solving model, for the purpose of understanding the generalization
process in problem-solving situations. The modified model was an
adaptation and extension of Lester's (1985) model and elucidated
the properties of the categories in terms of student behaviors.
It included an explicit task component and an affective component.
The modified model could serve as a pedagogical tool in a mathematics
classroom.
