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trial by ordeal: In the tutoring study, teachers' ideas for improvement were
"field tested" in a tutoring session with a real student and the ideas either did
or did not enhance learning. In the problem-design study, suggested
problems measured up to a set of guiding principles or failed to do so (Lesh
et al., 1993). In the assessment design study, proposed scoring rubrics
measured up to a set of guiding principles or failed to do so; (c) trial by
jury: In each of the three studies, each group of teachers compared and
contrasted their suggestions against those in consensus-forming discussions.
In these discussions, arguments were forwarded for which suggestions to
keep, which to weed out, how they should be organized and prioritized, and
how they might form a coherent conceptual system.
Overall, the driving force in selection appeared to be the result of the
resolution of various cognitive conflicts: task-interpretation mismatches,
interpretation-representation mismatches, environment-model mismatches,
model-tool mismatches, specific-general mismatches, and procedure-logic
mismatches.
2.5 Adaptation
As a positive consequence of mutation, demands from the environment
(challenges to proposed models) cause problem solvers, collectively and
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ACTION THEORY AND PHENOMENOLOGY
individually, to modify or extend existing ideas in one or more of the fol-
lowing ways: reorganization, coordination, differentiation, and integration.
Reorganization. We wanted to provide experiences that would encourage
teachers and students to switch to some completely new ways to think about
their suggestions. Sometimes these reorganizations occurred when cognitive
conflicts were presented. We noted some of the pertinent mismatches above.
Other reorganizations occurred when "wild ideas" (e.g., metaphors and
analogies drawn from brainstorming sessions) were considered that sug-
gested a reinterpretation of a direction or approach.
Coordination, or the "building up" process. Over time, teachers and stu-
dents gradually constructed more flexible and stable conceptual systems for
interpreting their suggestions. Sometimes we encouraged alternating be-
tween situations in which attention was focused on the constituent parts of
complex acts, other times on situations in which the focus was on the flexi-
bility and coordination of the systems-as-a-whole.
To help teachers gradually coordinate and refine their tutoring systems-
as-a-whole, we gradually increased the complexity of the contexts in which
the learner was to perform – while preserving the basic structure of the task.
For example, the complexity of tutoring sessions increased naturally as
teachers gradually noticed new types of relevant factors ranging from math-
ematical issues, to psychological issues, to pedagogical issues; and, tutoring
activities also became more complex as we introduced ways to use graphics
(other computer-based tools) as parts of hints, feedbacks, or follow-up
questions.
In the problem-design and assessment-design projects, we raised
concerns such as how well a given problem statement would draw upon the
students' experiences, or how well it documented students' work.
Alternatively, we asked if a scoring rubric that appeared satisfactory for
teachers was of equal value for parents or for the students themselves.
Differentiation: The "splitting" process. Conceptual systems do not sim-
ply get "built up" (or constructed) in a bottom-up manner; models also get
"sorted out." Teachers and students discriminate among alternative models:
those they have constructed and those that they have been given. The differ-
entiation process sometimes means that students and teachers temporarily
lose sight of the "large picture" when they pay attention to details of a
model. Alternatively, when the focus is on a single model, they lose sight of
others. We have found it to be a useful intervention with teachers and stu-
dents to redirect their attention to larger issues or components of their mod-
els or alternative models that they are neglecting.
2.6 Accumulation
When models are developing, the problem solver does not start from scratch
each time. The parts of the models that have served well in the past are re-
tained and become part of a larger and more comprehensive solution. The
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accumulating model draws upon representations and notation systems that
were helpful in the past. The emerging model helps guide the teacher in tu-
toring decisions, in designing tasks, and evaluating responses; it helps guide
students in applying articulated models to new problem situations.
Arising out of the design of action-theoretic studies of emerging expertise
among students and teachers, we have noted the following: In the tutoring
study, teachers moved progressively away from rule-oriented mathematics
and toward model-centered mathematics. Their interventions were reduced
in number and changed in character. They became less concerned, for ex-
ample, with procedural errors, and more interested in students' thinking:
Were students using more than one representational system? Were the
models they were constructing equal to the tasks that were set?
In the problem-design study, teachers moved away from designing prob-
lems that had a single correct answer (for which students' thinking and rea-
soning processes were not documented) toward ones in which the central
goal of the task was to promote model building and model documentation.
In the response-evaluation study, teachers moved away from assigning
holistic scores that covered many types of responses toward assessment pro-
cedures that considered the conditions of testing, student-related factors,
task-related factors, and their curricular goals in mathematics. They then
considered how to produce rich descriptions of the students' work. Finally,
considering both the conditions and descriptions, they evaluated the stu-
dents' work.
3. CONSEQUENCES OF CONSTRUCTIVISM
FOR A RESEARCH METHODOLOGY
When researchers adopt a constructivist orientation toward thinking and
learning, they must adapt their research methodology accordingly. Given
the assumptions of constructivism, can researchers predict with confidence
the state or level of construction of a concept that a student will reveal? If
not, attempts to prescribe what constitutes an "expert" state, and what con-
stitutes a "novice" state are open to question. As a corollary, pre- and
posttests that reify these a priori codifications of expertise are also open to
question. Further, research and teaching agendas whose goal is to bridge
this hypothesized "gap" with prescriptions may be misguided. Detailed
observations of children's thinking make clear that students' thinking is
often inadequately described by either the novice or expert prescriptions of
researchers (e.g., Carpenter, Fennema, & Romberg, 1993; Maher, Davis, &
Alston, 1991). Some children's thinking is haphazard, showing some
"expert" characteristics and some "novice" characteristics. The thinking of
other children frequently goes beyond the expectations of "expertise" that
were assumed for them (Lesh, Post, & Behr, 1989).
Further, since children's thinking evolves in complex ways over pro-
tracted periods of time, the a priori timing of a prescribed instrument to be
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used as a "posttest" may be quite arbitrary and may or may not succeed in
recording the changes in learning that it was designed to record. If children
construct ideas complexly over a long period of time, then researchers must
be willing to make continuous, rich, longitudinal observations of children.
Researchers must also focus on authentic tasks. Researchers in mathemat-
ics education should be primarily concerned about students' construction of
real mathematics, not about drawing remote inferences about mathematical
problem-solving based on scores from indices such as multiple-choice as-
sessments of procedural knowledge. We should be concerned with mathe-
matical problem-solving, not with surrogates of this process.
Finally, the constructivist approach suggests that researchers should pay