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tic performance activities for teachers: The materials we develop for our
teaching experiments provide prototypes for some of the most important
components of a field-tested, on-the-job, teacher education program in
which teachers can simultaneously develop and document their mathemati-
cal/psychological/instructional knowledge and abilities.
3.4 Documentation of Change
In our teaching experiments, we wish to move away from state-focused
documentation and toward progress-focused documentation of learning.
State-focused documentation aims mainly at: (a) evaluating (or assigning
values to) states of development, (b) identifying deficiencies with respect to
some (preconceived and static) standard, and (c) inferring that progress has
been made by comparing one evaluation with another (using a subtraction-
based model to describe differences).
Progress-focused documentation aims at monitoring progress in a direct
manner by focusing on activities in which it is possible simultaneously to
encourage and document development in directions that are increasingly
“better” (without using pretests and posttests, which embody a fixed and fi-
nal definition of “best”). In progress-focused documentation, relevant activ-
ities tend to contribute to both learning and the documentation of learning.
Therefore, distinctions between instruction and assessment tend to blur and
the roles of teachers become particularly important. Further, the quality of
teachers' contributions tends to be strongly influenced by their own under-
standings about the nature of mathematics, mathematics learning, and math-
ematics problem-solving. Finally, the role of support and administrative
personnel (e.g., parents and supervisors) is neither ignored nor diminished.
In practice, teaching experiments for students provide an ideal context for
a teaching experiment for teachers. Teachers study the development of
mathematical thinking of students; the development of mathematical think-
ing and instructional models of the teachers is studied, in turn, by our pro-
ject staff.
3.5 The Role of the Researcher
The constructivist paradigm demands that we recognize that the models that
we use to make sense of the performances of both teachers and students are
themselves subject to revision and restructuring. For that reason, we find it
difficult to maintain the belief that we can act as disinterested, objective ob-
servers of our "subjects." Our "subjects" are actual teachers and actual stu-
dents engaged in meaningful learning. In a very real sense, we learn from
them; they are co-collaborators in the search for knowledge about how to
improve instruction, learning, and assessment. Our "data" are our models,
our understandings, of the models that the teachers and students are using to
make sense of the tasks that we have set them. Because we do not claim
RICHARD LESH AND ANTHONY E. KELLY
285

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