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particular attention to the environment in which children are learning. Some
environmental factors (which include teachers and technology) will encour-
age and prompt children's thinking, others will constrain it. From this obser-
vation, we draw the lesson that the researchers themselves are part of the
environment that is both studying children's thinking and eliciting it.
3.1 Children, Teachers, and Researchers
We wish to make the more general assertion that constructivism is not sim-
ply a statement about how children think; rather, it is a statement about the
nature of thinking. A corollary of this premise is the principle that whatever
characteristics we ascribe to children's thinking, we should be willing to as-
cribe both to teachers' thinking and to our own thinking as researchers. If we
claim that children construct internal representations, or models, of the
world, and that these models are incomplete, flawed, subject to revision, and
evolve over a long period of time, then we must apply these principles
equally to our study of the models of the world held by teachers and re-
searchers.
We should not adopt uncritically the premise that mathematics teachers
are "expert" at mathematics, teaching, or tutoring. As researchers, we
equally should not entertain the conceit that we have an error-free
metavision of the thinking of all our "subjects." Teachers and researchers
construct models and often revise them. To use the language of Gadamer
(1975, 1976), the horizons of children, teachers, and researchers are limited
and contain what Heidegger would call many blindnesses (Heidegger,
1962). While each group has knowledge that the other does not yet possess,
all of our models of the world are historical, incomplete, fractious, contain
misconceptions and biases, and continually evolve.
3.2 Factors in a Research Methodology
Authentic performance: Tasks for students. We wish to elicit and develop
the mathematical intuitions of students using authentic tasks. An authentic
task for a student includes constructing mathematical models to gain lever-
age over general problems (the stage of model construction), explorations of
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the qualities of mathematical objects (the stage of model exploration), and
application of mathematical models to new situations (the stage of model
application). The type of authentic tasks that we have been developing an
d
using are called model-eliciting problems. Model-eliciting problems are de-
signed in accordance with the following principles (see Lesh et al., 1993):
1. The Model Construction Principle: Does the task create the need for a
model to be constructed, or modified, or extended, or refined? Does the task
involve constructing, explaining, manipulating, predicting, or controlling a
structurally significant system? Is attention focused on underlying patterns
and regularities rather than on surface-level characteristics?
2. The Simple Prototype Principle: Is the situation as simple as possible,
while still creating the need for a significant model? Will the solution pro-
vide a useful prototype (or metaphor) for interpreting a variety of other
structurally similar situations?
3. The Model Documentation Principle: Will the response require stu-
dents to explicitly reveal how they are thinking about the situation (givens,
goals, possible solution paths)? What kind of system (mathematical objects,
relations, operations, patterns, regularities) are they thinking about?
4. The Self-Evaluation Principle: Are the criteria clear for assessing the
usefulness of alternative responses? Will students be able to judge for them-
selves when their responses are good enough? For what purposes are the re-
sults needed? By whom? When?
5. The Model Generalization Principle: Does the model that is con-
structed apply to only a particular situation, or can it be applied to a broader
range of situations?
6. The Reality Principle: Is the scenario of the problem contrived so that
it would contradict students' knowledge of the scenario in a "real-life" situa-
tion? Will students be encouraged to make sense of the situation based on
extensions of their own personal knowledge and experiences?
Model-eliciting tasks allow and encourage students to display and docu-
ment their mathematical problem-solving.