Lee S. Shulman
Stanford University

A large number of interesting and provocative ideas were presented in the con-
ference that led to this volume. Surely, there are many schemata or frames through
which we, as readers, can interpret the papers. This paper represents my own kind of
schematization or thematic summary. Since this volume contains 24 separate offerings, it
is not possible for me to comment on each paper individually. Rather, I will deal very
generally with five broad topics: (1) problem solving, (2) methodology, (3) curriculum,
(4) teaching and learning, and (5) teaching and teacher education.

One of the topics discussed in a number of the papers was, "What is problem solv-
ing?" I have a favorite epigram about what a problem is. I first encountered it in an essay
by Jerome Bruner ( 1961). Bruner cites a British philosopher named Weldon who asserted
that three basic things ought to be considered in defining what is a problem: troubles,
puzzles, and problems. A trouble is a circumstance, a situation, that leaves one upset
and at loose ends; one knows that things aren't going very well. It is a source of vague,
but persistent discomfort. A puzzle has a nice tight form, clear structure, and a neat
solution. A problem, then, is what you get if you can find a puzzle form to lay on top of
a trouble. This conception may serve as a framework for examining several topics discuss-
ed at the conference.

In his paper, Mayer's concern is with the translation process as it related to the ques-
tion of problem types. In fact all of the discussion about types seems to revolve around
the kind of cognitive processes and associated pedagogical processes that are involved in
finding the appropriate mathematical puzzle forms to lay on troubles to transform them
into problems that a learner knows something about. As Sowder pointed out in his reac-
tion, there are some researchers who do not even consider the solution of standard text-
book story problems to be problem solving.

In search for the definition of problem solving, Dewey's name has been invoked
repeatedly. In their papers, Nel Noddings and Patrick Thompson remind us how much
Dewey had contributed to this area. And of course, in this conclave of mathematicians,

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