A Reconceptualization of Teacher Thinking in Terms of Mediating
Resources1
Susan B. Empson
The University of Texas at Austin
empson@mail.utexas.edu
Abstract: This paper defines and illustrates resources
as a kind of cultural/psychological tool that mediates teacher thinking. The
focus is on resources that accord primacy to learning over teaching as the
activity around which to organize instruction. Four major feature are posited:
resources can exist anywhere in the material and social enactment of a task;
resources are dynamic rather than fixed or static; resources do not exist independently of teacher
action; and resources are a means to integrate the activities of teaching and
learning. In closely coupled activity, teachers use resources to appropriate several
aspects of students’ strategies to conceptual frameworks integral to elementary
math. When student productions become resources for further instruction, the
result is growth in understanding.
Although
learning can and does take place without teaching, teachers do not exist
without students. Lave (1996) proposed that learners should “constitute the
working conditions for teaching rather than the other way around” (p. 159).
Research on teaching often treats teaching and learning as conceptually distinct
activities. Lave’s claim is a call to reconsider the interrelationships between
the two, and to accord primacy to learning over teaching as the activity around
which to organize instruction.
Schools are not currently organized to
support learner-centered instruction in math, although there is a call for new
teaching practices centered on students’ understanding (National Council of
Teachers of Mathematics [NCTM], 2000). Teachers are faced with the challenge of
fostering and understanding student-generated strategies, and integrating those
strategies with instructional goals. This challenge has created a need for
schools and teachers to draw on and create new sets of knowledge resources to
elicit, make sense of, and act on children’s mathematical activity. In this
paper, I consider the features of resources that support teachers’ use of
children’s activity to advance mathematical thinking.
I
use resources to refer to a kind of
cultural/psychological tool (Wertsch, 1998) used to accomplish valued goals. Resources
are the means by which teachers engage in tasks designed to move classroom
activity towards those goals. As tools, their meanings and uses are continually
in flux (Wenger, 1998). Potential resources include: the knowledge frameworks
teachers use to interpret what children say and do; children’s strategies; the
materials, curriculum programs, and instructional goals set by policy; and the
physical set up of the classroom.
A focus on the role of resources in the
facilitation of instructional interactions shifts the burden of explanation for
student learning (or lack of it) from a major emphasis on teachers’ knowledge
to “[teachers]-operating-with-mediational-means” (Wertsch, 1998, p. 26). The
significance of this shift is to situate the teacher and his or her thinking
(i.e., his or her “operating”) in the contexts of the classroom and schooling,
and to begin to understand how what teachers do is the result of the confluence
and coordination of a variety of resources -- some generated through practical
inquiry (Franke, et al., in press), some provided through policy decisions,
some appropriated from professional development, and some contingent on the
activity and purposes of specific locales.
There are several implications of this shift in
perspective on teacher thinking. They
are organized around one main claim, founded on the premise that children
learning should be a major constituent of the context of teaching: To organize
classrooms in which children learn math with understanding, teachers need resources
that facilitate children’s participation in legitimate mathematical activity
and teachers’ responses to that activity.
The central idea is that teaching consists of a series of interactions
in which resources are continually deployed, invoked or collectively produced
to make sense of and direct activity. An important subclaim is that the more closely coupled teachers’ use of
resources is with students’ mathematical activity, the more likely students are
to learn and understand math (Empson & Junk, in preparation).
Example: Closely Coupled Activity
Closely coupled activity refers to teachers’ use of resources to
incorporate several aspects of students’ thinking activity into
instruction. In this example from a
third-grade classroom, Shatysh has solved a problem that asked how much cake
each person gets, if 12 children are sharing 20 little cakes. When the teacher2 approaches, she does not understand what
Shatysh has done. At the conclusion of the interaction, Shatysh’s strategy has
been refined as one where the sharing situation was transformed from 12 sharing
20 to 3 sharing 5. It will be used as a resource in later group discussion, as
an example of a ratio-based strategy. To reach this point, the teacher
appropriates several aspects of what Shatysh has done, including her initial
answer of 5, her grouping of the children and cakes, and finally her
distribution of groups of cakes to groups of children (which is not clear
initially).
<<Insert figure 1 here>>
Figure 1. Shatysh’s written work
for 12 people sharing 20 cakes. (1a on left; 1b on right)
T: (looking at Shatysh’s paper) OK, you
got an answer of 5? 5 what? (pause) Tell me what you did here (gesturing to
Fig. 1b)
S: 3.
T: 3 kids share 5 cakes?
S: Yeah. See, (points to children in
group of three) 1, 2, 3.
T: Those 3 right there? (pointing to
first 3 children in Fig. 1a)
S: Yeah.
T: So. (pause) So you have 3 kids
sharing 5 cakes?
S: (nods)…. I could split ‘em up
(i.e., share the 5 cakes among the 3 children).
T: (brief interaction where T
underlines each group of 3 children, and numbers groups 1 to 4)…. I see how you
have the 3 circled now. I understand what you’re doing. You went 5 (underlining
group of 5 cakes)-- and you tried to make it so this would be split up. So this
group (of 3 children) got 5 cakes…
S: (shows how she would share the 5
cakes among 3 children by giving out whole cakes to each person, then splitting
last 2 cakes in 3 pieces.)
T: This child right here (points to
first in group), how much cake does she get?
S: A whole.
T: A whole, plus what else? …. (S is
not sure) how much cake is that little piece right there? …. How many of
these little pieces does it take to make the whole cake?
S: 3.
T: OK, so what do we call it?
S: 1 third? (S writes “1/3 + 1/3 + 1”
with some assistance.)
Shatysh’s
ratio strategy is a resource that was collectively produced; note however that
Shatysh’s activity motivated the defining logic of the strategy. Resources the
teacher used in this production include her knowledge of ratios in
equal-sharing situations, the questions that helped Shatysh name the fractional
quantity, the task itself to motivate a range of student strategies.
Features
of the Framework
There are four main implications that follow
from the perspective that teachers’ thinking is best characterized in terms of
the use of mediating resources. First, rather than existing solely as
representations internal to a teacher’s thinking, resources can exist anywhere in the material and social enactment of a
task (Newman, Griffin, & Cole, 1989). They may be internalized to
teachers’ ways of operating, or they may exist or emerge externally. There is
no theoretical reason to separate these two categories of classroom activity,
since both have a bearing on what students take with them from instruction.
Second, resources
are dynamic rather than fixed or static. What they mean, how they are used,
and the form they take are in constant negotiation (Wenger, 1998), which
results in refinements and other kinds of changes. Thus, not only should
teachers be provided certain resources -- the “products of the inquiry of
others” (Lampert, 1998, p. 57) -- but more significantly their work should be
organized to create, test, and improve resources. This implication is in line
with other recommendations concerning teaching and inquiry (Cochran-Smith &
Lytle, 1999). Further, an inherent ambiguity in what resources are and how they
can be used means that the possibility of individual interpretation and
improvisation always exists.
Third, resources
“do not really exist independently of [teachers’] action” (Wertsch, 1998,
p. 25). The full significance of resources emerges in interaction.
Feiman-Nemser and Remillard (1995) pointed out that many perspectives on
teacher knowledge “leave open the question of what it means to know and use
such knowledge in teaching … misrepresent[ing] the interactive character of
teachers’ knowledge and sidestep[ping] the issue of knowledge in use” (cited in
Cochran-Smith & Lytle, 1999, p. 257). A focus on what teachers do and the
means by which it is accomplished, rather than on what teachers say about what
they do, ameliorates research problems posed by using measures that are proxies
for knowledge-in-action.
Fourth, resources
are a means to integrate the activities of teaching and learning, for purposes
of professional development and research. The emphases on the use of
children’s activity to inform teachers’ practice and on how teaching actions
can elevate and enfold children’s thinking has the potential to support an
inquiry-oriented feedback loop in instruction. There has been recent interest
in helping teachers engage in practical inquiry in ways that lead to generative
understanding of student thinking and mathematical content (Franke, et al., in
press), where generative refers to
the ability to continue to expand knowledge and deepen understanding. Possible
explanations for how some teachers become generative learners may be found
through examining teachers’ use of resources in practice, especially how
teachers appropriate aspects of students’ activity to ongoing instruction. In
closely coupled activity, teachers use resources to appropriate several aspects
of students’ strategies to conceptual frameworks integral to elementary math
(e.g., knowledge of equivalence or development of children’s equal-sharing strategies).
When student productions become resources for further instruction, the result
is in an iterative process of forward movement.
Conclusions and Future Questions
Understanding
teacher thinking in terms of mediating resources can shed new light on how we
address the problem of educational change, by allowing us to ask: What uses of
resources depend on teacher
learning? What uses of resources result in teacher learning? Further, what kinds of inquiry cycles or
feedback loops couple teachers’ practice with student outcomes? How do policy decisions regarding
instruction position children as certain kinds of learners? Only a small fraction of research has been
devoted to analyzing domain-specific aspects of teachers’ thinking (beyond
content knowledge) in a theoretically informed way (e.g., Carpenter, Fennema
& Franke, 1996). Just as Glaser (1984) argued for domain-specific analyses
of problem solving, one can argue for similar analyses of teachers’ thinking as
it is situated in the work of teaching fractions, geometry, whole-number
operations, and so on. I believe that
all of these questions are best answered by concentrating on a single, albeit
complex, aspect of teacher activity -- the use of learners as context for
teaching -- in a single content domain, such as fractions. This narrowing of
the questions provides a descriptive scale that is in accord with the problem
space(s) within which teachers operate on a day-to-day basis, but admits
analysis of contextual influences from several realms.
Notes
1. Acknowledgements: This research was supported in part by NSF grant
HER-9816023. Thanks to my research
assistants Debra L. Junk and Erin Turner for insights and feedback.
2. Debra L. Junk, a member of the research
team. We co-taught a unit on fractions
as part of pilot work.
References
Cochran-Smith,
M. & Lytle, S. (1999). Relationships of knowledge and practice: Teacher
learning in communities. Review of
Research in Education, 24, 249-305.
Carpenter,
T. P., Fennema, E. & Franke, M. (1996). Cognitively guided instruction: A
knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1),
3-20.
Empson, S.
B. & Junk, D. (In preparation).
Teachers' thinking about students' nonstandard strategies for
whole-number operations.
Feiman-Nemser,
S. & Remillard, J. (1995).
Perspectives on learning to teach.
In F. Murray (Ed.), The teacher
educators' handbook (pp. 63-91).
San Francisco: Jossey-Bass.
Franke,
M., Carpenter, T. P., Levi, L. & Fennema, E. (in press). Capturing teachers’
generative change: A follow-up study of teachers’ professional development in
mathematics. American Educational
Research Journal.
Glaser, N.
(1984). Education and thinking: The role of knowledge. American Psychologist, 39(2), 93-104.
Lampert,
M. (1998) Studying teaching as a thinking practice. In J. G. Greeno & S.
Goldman (Eds.), Thinking practices in
mathematics and science learning (pp. 53-78). Mahwah, NJ: Lawrence Erlbaum
Associates.
Lave, J.
(1996). Teaching, as learning, in practice. Mind,
Culture, and Activity: An International Journal, 3(3), 149-164.
Newman, D., Griffin, P. & Cole, M. (1989). The construction zone: Working for cognitive change in school. New
York: Cambridge.
Wenger, E.
(1998). Communities of practice:
Learning, meaning, and identity. New York: Cambridge.
Wertsch, J. (1998). Mind as action. New York: Oxford
University Press.