| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 229
11
Developing Classroom Process Data
for the Improvement of Teaching
lames TY Stigler and Michelle Perry
Of the many factors that determine student academic achievement, class-
room instruction is but one. Yet it is surely an important one. Indeed, all
attempts to improve education must of necessity at some point be mediated
through the classroom. This is obvious because classroom practice represents the
most direct means for affecting student outcomes. However, there has been
surprisingly little research on this link in the chain in affecting student outcomes.
As a nation, we collect very little data on what happens inside classrooms.
As Mandel (1996:3-29) wrote, "The national conversation about teaching has
always been compromised by a dearth of information about the quality of prac-
tice and practitioners. . . . When dismal or promising results about student
performance are reported, a new chain reaction of suppositions is often set off
about the degree to which teachers are to be blamed or praised. But these
suppositions are just that hypotheses disconnected from much of a factual base
that might shed some light on what is occurring, including the extent to which the
observed results can be accurately attributed to teacher actions." This relative
dearth of data can be blamed, at least in part, on what Burstein et al. (1995) point
out as the inherent difficulty in measuring instructional practice.
Despite this inherent difficulty, we argue that the merits of these data out-
weigh the obstacles in collecting them. As an example of the importance of these
data, here it is argued that we cannot know which instructional strategies lead to
positive learning outcomes unless we know which instructional practices are
being used and we cannot know which are being used without somehow looking
directly at educational practices. In other words, achievement data may tell us a
lot, but those data cannot tell us what should be done differently inside the
229
OCR for page 230
230
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
classroom. We argue that for test data to be most informative, classroom pro-
cesses need to be examined. If change in student learning outcomes is observed
in the tests, we still need to know whether change is due to something going on in
the classroom or something independent of that.
In this paper, we make the assumption that classroom process data, especially
when collected in conjunction with student achievement data, can play a critical
role in efforts to improve education. We further assume, however, that such data
will not necessarily improve education and that it is therefore extremely important
to have an explicit idea of exactly how data will be used to improve education and
by whom. In particular, we argue that researchers, policy makers, and teachers
need different kinds of data and will use data in different ways to improve the
quality of teaching and learning in classrooms.
Five questions guide this paper: (1) What is the nature of classroom instruc-
tion, and what implications does this have for developing indicators of instruc-
tional quality? (2) What kind of data can be collected, and what are the advantages
and disadvantages of each? (3) What kind of data ought to be collected, and how
will the data be used to improve the quality of instruction? (4) What are the costs
of collecting data of various kinds? (5) How can new kinds of data collection be
integrated into the existing National Assessment of Educational Progress (NAEP)
program?
Given these issues and questions, the goal of this paper is to consider what
sorts of data can be collected on classroom processes. With this goal in mind, we
examine the kinds of data that are currently collected on classroom processes and
evaluate what can and cannot be learned from these data. We then look beyond
current research practices and make suggestions for future data collection on
classroom processes.
STUDYING CLASSROOM PROCESSES
Nature of the Classroom
Having established a broad interest in collecting data on classroom processes,
we consider what kind of data might be collected. Before launching into a
discussion of specific data collection techniques, we need to ponder the nature of
classroom instruction. The data collected and measures constructed are only
indicators. To assess the validity of these indicators, we must first think through
the nature of what it is they are intended to be indicators of. Indeed, a framework
for thinking about the constructs that define classroom instruction provides a
necessary theoretical context in which indicators can be interpreted.
Classroom instruction, first and foremost, is a complex, dynamic, goal-
directed system. One goal of the system is student learning, although there
certainly are other goals as well. For purposes of this paper we will assume that
achievement, as measured by the NAEP, is an important overall goal of the
OCR for page 231
JAMES W. STIGLER AND MICHELLE PERRY
231
system we describe. The system consists of several important elements, includ-
ing a teacher, students, curriculum, and materials. These elements interact with
each other in complex ways. Teachers orchestrate the sequence of activities that
comprise the classroom lesson. These activities represent organized behavioral
interactions between students, teachers, and curriculum/materials. In addition,
these lesson elements interact with key contextual factors that impinge on the
classroom.
To say that the classroom is a system implies that it is more than the sum of
individual features or independent dimensions. Although features might be mea-
sured to indicate indirectly the functioning of the system, it is difficult to imagine
features of instruction that are always good or dimensions on which lessons
should be uniformly high. For example, although in general it might be true that
lessons in which students are cognitively challenged are better than lessons in
which they are not, there are many instances in which repeated practice with less
challenging tasks is appropriate and necessary for students' learning. This pre-
sents the researcher with a significant challenge. To define quality of instruction,
one must do more than define a set of features; one must evaluate features of a
specific lesson with reference to how they function in the context of a goal-
directed system. Indeed, one must describe the system itself to understand the
meaning of indicators.
An example will serve to illustrate the practical implications of this point. In
the process-product research of the 1970s and 1980s, it was demonstrated, across
many studies, that student learning of mathematics was significantly associated
with rapid coverage of a large number of problems during the lesson: the more
problems the teacher led students through, and the faster the pace, the more
students learned as measured by achievement tests (Leighton, 1994; Leinhardt
and Putnam, 1987~. As often as this effect was found, however, it turned out not
to hold up in cross-cultural comparisons. Japanese students achieve in math-
ematics at far higher levels than U.S. students, yet Japanese teachers often are
found to cover only one or two problems in a single lesson, compared with 30 to
40 in an American lesson (Stigler and Perry, 1988~. Clearly, the indicator of how
many problems are covered has different meanings in the context of different
instructional systems. U.S. teachers were using problems for repeated practice,
and clearly there is something to be gained by such practice. Japanese teachers,
in contrast, were using problems as the focus of students' deep thinking and
reflection. Simply knowing how many problems were covered was not enough to
characterize the kind of instruction students experienced.
Another truth about classroom teaching is that it is a cultural activity
(Gallimore, 1996; Stigler and Hiebert, 1997~. What this means is that teaching,
like other cultural activities, is constructed largely out of widely shared routines
that are learned implicitly and are highly resistant to change. Although in our
culture we perceive variability across teachers in their approach to teaching,
cross-cultural comparison reveals that such variability may be relatively insig
OCR for page 232
232
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
nificant compared with the large differences across cultures in the ways that
teachers teach. U.S. teachers, for example, have varied ways of providing feed-
back to students who are working on math problems during seatwork. But these
variations pale in size when we realize that virtually all U.S. teachers tell students
how to solve the problem before they ask the students to solve it, whereas most
times Japanese teachers do not. We tend not to notice those aspects of cultural
activities that are shared, focusing instead on features that vary. But it may well
be that the aspects of teaching that are widely shared in a culture are the ones that
have the most impact on student learning.
One important implication of this fact about teaching is that it shifts our
focus somewhat from the study of teachers to the study of teaching. Because the
literature on classroom indicators has been largely an American one, it has tended
to focus on aspects of teaching that vary in our culture. But we need to focus as
well on identifying the shared cultural scripts that underlie most or all of what we
see inside American classrooms. The improvement of teaching over time may be
much greater if we focus on changing widely shared scripts than if we focus on
understanding variations in the competence with which teachers use the scripts.
Research Questions
Viewing classroom instruction as a complex system and as a cultural activity
leads us to identify several important research questions to guide our inquiry into
instructional quality.
· What kinds of instructional systems can we identify? How can we describe
these systems? This will involve, minimally, identifying the key elements of the
classroom lesson and describing the ways in which these elements interact.
· What kinds of quantitative indicators can we develop to assess the func-
tioning of different types of instructional systems? What are the processes that
affect these indicators? We must quantify the descriptions developed in response
to the first research question if we are going to validate them across large numbers
of classrooms.
· What is the role of the student in different instructional systems? What
are the processes by which students learn from classroom instruction, and what
characteristics of different instructional systems affect how much students learn?
These are key questions, as our interest in instruction rests on the assumption that
student learning is affected by instruction.
· What is the role of the teacher in different instructional systems? How
can teaching be improved? Again, we assume that teachers play a critical role in
shaping the nature and quality of instruction in the classroom.
Each of these general research questions can be approached through various
analytic frames. For example, classroom lessons can be described on a more
OCR for page 233
JAMES W. STIGLER AND MICHELLE PERRY
233
macrolevel in terms of activity structures (e.g., classwork or seatwork) or from a
more microanalytic level (e.g., detailed analysis of discourse patterns as they
unfold throughout the lesson).
Units and Methods of Analysis
Starting with the assumption that classroom instruction is a complex cultural
system, we have proposed a broad set of research questions. The complexity of
instruction also has implications for the units and methods of analysis we choose.
Classrooms must be studied using units that make sense and that preserve the
crucial aspects of the system. These units might be relatively large (e.g., units,
grade levels), but they are probably not smaller than the classroom lesson. Class-
room lessons have ecological validity from the teacher's point of view. Teachers
plan their days in terms of lessons: "First we'll do math, then social studies."
Lessons are goal directed and orchestrated by the teacher. The explicit goal of
the lesson might be a student learning goal, or it may simply be the completion of
some series of activities. Regardless of the goal, the lesson itself can only be
understood in relation to the goal. Although we can study the lesson through
different lenses (e.g., we can study the nature of classroom discourse or the
patterning of teacher-student interactions), we will need to collect information
about the context in which the processes operate.
It is also important to note at the outset that both qualitative and quantitative
analyses will be required in our efforts to understand and improve classroom
learning. The first research question we listed is one that must be answered
through qualitative analysis. Identifying parts of lessons and figuring out how
the parts interact to produce student learning require a qualitative analysis of the
instructional process. Once the process has been described, however, it is useful
to develop indicators that can be used to validate and refine the descriptive model
of instruction.
Not only do we need both qualitative and quantitative data, we also need a
way to link the two kinds of data together. As we will see, this has been a
problem with more traditional approaches to the study of classroom processes.
TRADITIONAL METHODS: SURVEYS AND
NARRATIVE DESCRIPTIONS
Most commonly we have relied on surveys to collect data on classrooms.
Additionally, narrative descriptions have been used as a method of collecting
classroom data. In this section, we review those methods. In particular, first we
provide descriptions and overviews of the data forms. Next, we examine what we
typically learn from data collected with each of these methods. Finally, we offer
an evaluation of each of these methods, with some attention to both the limita
OCR for page 234
234
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
lions that each method has in terms of producing data on classroom processes and
the potential of providing new insights about teaching and learning.
Survey Methods
Descriptions and Overviews
Surveys represent relatively straightforward ways to collect data on a host of
issues related to classroom processes; however, surveys can take several different
forms. For example, even if we are just surveying teachers, teachers can be
surveyed about their recollections or their opinions with questionnaires (whose
answers can take the form of a rating scale, forced multiple-choice responses, or
open-ended answers), interviews, or diaries. In this section, we also include
observational checklists, which in some ways resemble the other data forms in
this section but in other ways resemble narrative observational records. In the
remainder of the section, we provide a general description of the various types of
survey methods.
Questionnaires and rating scales Questionnaires and rating scales are often used
to tap classroom processes. Questionnaires and rating scales used for these
purposes typically request information from teachers about the activities taking
place in their classrooms. Others, including classroom observers and students,
also may participate in completing questionnaires about classroom processes.
This data source can provide information about what is taught, how the
teaching takes place, and how much time is spent on various topics and activities.
As an example, Burstein et al. (1995:xiii) asked teachers to judge the percentage
of class time spent instructing with various strategies (e.g., whole-class instruc-
tion, administering tests, performing administrative tasks). One of their major
findings was that, "although the picture of teaching that can be drawn from
survey data is quite general, it is probably valid, because . . . data clearly show
that there is little variation in teachers' instructional strategies. The majority of
teachers use a few instructional approaches and use them often." With these
methods we can obtain data from a large number of informants who have direct
access to the information we find of interest.
Diaries We use the term diary to represent teachers' records of their lessons,
including lesson plans, outcomes, and the like. Diaries have been used, relatively
successfully, to measure curriculum content. Given that we are concerned with
classroom processes, one might wonder why we specified that diaries are used to
measure curriculum content. The reason is that curriculum content has often
served as a proxy for classroom practices, although it is not itself a direct measure
of classroom practices. Barr and Dreeben (1983:107) defined content coverage
(also commonly referred to as instructional pace) as the amount of curricular
OCR for page 235
JAMES W. STIGLER AND MICHELLE PERRY
235
material that is covered over a period of time. They argued that although other
indexes of productivity designed for judging the effectiveness of instruction are
possible, "we have selected this one because, when treated at the level of indi-
vidual children, it represents an instructional condition integrally connected with
learning." Another reason for focusing on diaries to measure content when we
are concerned about the relationship between teaching and learning, according to
Brophy and Good (1986:360) is that "the most consistently replicated findings
link achievement to the quantity and pacing of instruction."
As an example, Perry (1988) surveyed nine fourth-grade teachers' math-
ematics lesson plan books over the course of one year and recorded which prob-
lems were assigned. She then coded each problem as belonging to one of several
mathematical topics. She also measured the students' mathematics problem-
solving performance, both at the beginning and the end of the school year.
Problems that most children solved incorrectly at the beginning of the year were
designated as representing difficult topics, and problems that most children solved
correctly were designated as representing easy topics. Generally, Perry found
that problem assignment was related to student learning; more specifically, she
found that spending a great deal of time on a few difficult problems led to better
student achievement than covering many problems, especially problems that most
students could solve before receiving instruction. In this study, a diary of what
instruction consisted of was used to make inferences about teaching practices that
were related to learning outcomes.
Interviews Interviews, conducted face to face or by telephone, allow us to get
teachers' and/or the students' views of classroom processes. We can ask what
happened and we can ask for evaluations about what was reported to have
happened.
Interview techniques are especially useful, compared to paper-and-pencil
methods (such as questionnaires and rating scales) when the potential responses
have not been determined in advance. Interviews, especially those conducted by
well-trained interviewers who know what sorts of issues are of interest and which
deserve lengthy commentary, are desirable when we expect complex responses
because interviewers can ask respondents different questions, depending on pre-
vious answers. If the potential responses are already known, less expensive
methods may be more desirable.
Checklists Checklists often have been used to document classroom processes.
When using checklists, all of the behaviors of interest must be defined in advance.
Additionally, observers (i.e., the ones responsible for checking off observed
behaviors on a checklist) need to agree about what constitutes the observed
behavior. Thus, categories must not only be defined in advance, but must also be
specified as clearly as possible so that the observers check the appropriate entry.
Typically, checklists are completed by outside observers, which makes this
OCR for page 236
236
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
method different from those already discussed. In this way, checklists resemble
the narrative descriptions of classroom observations, which we discuss later.
However, this data form resembles the other forms of survey data in that the
questions to be examined generally are already known before the data are collected.
To lay out more clearly the data that can be obtained with observational
checklists, we provide a brief description of two well-known investigations that
have relied on this method. As a first example, Brophy and Evertson (1976) had
observers note each time a specified behavior occurred, such as teacher praise for
a student' s good response. From their observations and analyses, they concluded
that teachers whose students had the highest achievement treated their students in
a businesslike and task-oriented manner. As a second example, Stigler et al.
(1987) had observers in three countries check when certain classroom behaviors
and certain features of classroom organization were present. Their conclusions
centered around the idea that whole-class instruction means that every student
received some instruction, and teachers who relied heavily on individualized
instruction had some students who, basically, were never taught. Both of these
examples illustrate that checklists can provide a general snapshot of classroom
life.
Uses of and Outcomes from These Methods
Survey methods are used to assess many variables related to instruction and
life in classrooms. One reason these methods are used so frequently is that they
are easy to use. With these methods it is easy to measure curriculum content. For
example, researchers can read through teacher plan books or diaries kept for the
purpose of noting what topics were covered and easily judge what was and was
not taught. It is also easy to measure the amount and pace of instruction. For
example, researchers can ask teachers in an interview which pages in the text
were covered and can use a questionnaire to ask how much time was spent in
instruction. It is also easy to measure the format of instruction. For example,
researchers can ask teachers to check each form that was used on each day of
instruction (lecture, small-group work, etc.~.
More significantly, given the concerns motivating the present paper, these
methods can even be used to measure classroom processes. For example, we can
ask teachers in a questionnaire whether the questions they asked their students
required short answers or reflection and abstraction; we can ask whether the
students responded only to the teachers' requests or whether the students pro-
vided substantive contributions without teacher prompts. In short, researchers
have used these methods successfully to document a wide array of classroom
features. These methods typically have been used and analyzed in the process-
product approach to classroom investigation (e.g., Brophy and Good, 1986~. In
general, the process-product approach assesses classroom processes or their
proxies and relates these to student outcomes.
OCR for page 237
JAMES W. STIGLER AND MICHELLE PERRY
237
In addition, we note that these methods are typically used to test theories.
Because survey methods must generate categories and items before the data are
collected, the categories and items necessarily reflect a theoretical bias. The data
collected in surveys can, for example, support or call into question a relationship
that a theory would predict. In this way survey data can tell us when a theory
cannot be supported and thus when a new theory is called for.
Evaluation
These methods of collecting data are used frequently, in part because they
can be used on a wide scale: they are easy to administer and easy to analyze
relative to other methods. The ease associated with collecting survey data makes
these methods the most widely used for gathering data on classrooms. The
difficulty and costliness of other methods have sometimes made them prohibitive
altogether or at least have limited the number of classrooms that could be included
for study (we document these more fully as these other methods are discussed).
Burstein et al. (1995:35) say that "there is still much that survey data can tell
us about instructional strategy. Survey data can describe the major dimensions of
classroom processes and how they vary across course levels and types of schools.
National survey data, collected periodically, can document trends in teachers' use
of generic instructional strategies. Such information is important for determining
whether or not teaching is changing in ways consistent with the expectations of
curriculum reformers and policymakers." For these reasons we imagine that the
NAEP could collect and productively use these sorts of data.
Of course, with any method there are drawbacks. We see three major draw-
backs to the methods just described: (1) These methods leave open many threats
to validity; (2) Most significant among these threats is a lack of shared language;
and (3) These methods rarely contribute to generation of new ideas and thereby
do not prominently contribute to national discussion. We discuss each method in
turn.
Problems of Validity Probably the most serious problem with survey methods is
that responses often are not accurate, thereby making them not valid. In many
instances, typical paper-and-pencil survey instruments are not to be trusted be-
cause teachers are fallible human beings and may easily forget what they have
done or unwittingly skew their responses based on their individual biases. We do
not mean to say that teachers are not to be trusted. What we mean is that it is
sometimes difficult to produce accurate responses.
In particular, it is difficult to be precise about certain behaviors. This prob-
lem was made clear by some careful work (Mayer, 1999) on the reliability of
these methods. Mayer (1999:43) writes: "We cannot rely on the individual
survey questions to assess the amount of time . . . teachers use specific practices
. . . because the teachers do not report their practices in a consistent manner.
OCR for page 238
238
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
Thus, the portrait of specific practices conveyed by the survey is unreliable and
therefore invalid." It is much more reasonable to ask teachers what they believe
than exactly what they do or how they have impacted their students with what
they have done. For example, imagine how hard it would be to be precise about
whether you had conveyed the concept of equivalent fractions primarily with
questions, explanations, or examples. Imagine the further difficulty of knowing
which of these three methods of instructional practice had the greatest positive
influence on students' understanding of equivalent fractions.
Mayer (1999:43) investigated this directly by comparing teachers' responses
on surveys to classroom observations of these teachers. He found that "low
reliability existed for most of the practice items [i.e., items intended to measure
teachers' practices] examined in this study." In short, surveys probably could
never give us reliable and detailed data about classroom practice. And without
reliability we cannot claim to have validly measured their behaviors.
A cousin to this problem is that those who respond to surveys are often
tempted to answer questions as they imagine the researchers would like them to
be answered, rather than with accuracy and honesty (e.g., Burstein et al., 1995;
Cohen, 1990), thus making these methods susceptible to problems of social desir-
ability. For example, with the recent implementation of reform-based standards,
teachers are increasingly aware that their practice should reflect these standards.
However, their practice may lag behind their knowledge of these standards, and
so they honestly respond about what they know about the standards, even though
their knowledge may not be reflected in their practice, thus making their responses
on surveys inaccurate (i.e., not valid).
Although reliability is clearly a problematic aspect of relying on survey
methods for documenting classroom processes, the reliability of constructs mea-
sured by surveys increases when multiple, rather than single, items are used to
measure constructs (e.g., Light et al., 1990; Mayer, 1998; Shavelson et al., 1986~.
As Mayer (1999:43) writes: "Individual indicators of limited reliability can be
grouped into a highly reliable indicator." The point here is that if we can get at
a potentially important behavior with multiple approaches (e.g., use observa-
tional checklists to determine which instructional strategies were used and follow
them up with interviews to learn more about how often they are used and under
what conditions) or multiple items on the same measure, we are more likely to
avoid problems with reliability and validity than if we rely on a single item or a
single measure. Thus, we would recommend that if the NAEP were to include
survey measures of teacher behavior, multiple measures should be used.
Lack of Shared Language Related to the problem of not obtaining a valid picture
of classroom practices with typical paper-and-pencil survey instruments is that
these instruments require an evaluation of whether teachers understand the items
in the way they were intended. However, for this we need a common language
that we really do not have. As Burstein et al. (1995:35) put it: "Surveys typically
OCR for page 239
JAMES W. STIGLER AND MICHELLE PERRY
239
cannot capture the subtle differences in how teachers define and use different
techniques." For example, what one teacher means when she agrees with the
item "we had a discussion" may be very different from what another teacher
means when he agrees with the same item. Even something as specific as "We
folded paper to demonstrate equivalent fractions" is open to multiple, potentially
inconsistent interpretations (Was the paper a square or a rectangular shape to
begin with? How many folds were used?), thus rendering responses invalid, even
to specific descriptions.
This notion is corroborated by Palincsar and her colleagues (1998), who
argue that teachers' professional development should be constructed as a "com-
munity of practice." They argue that this model deals head on with two pervasive
problems in the culture of American schoolteachers: "(a) the lack of consensus
regarding the goals and means of education . . . and (b) the private, personal, and
individualistic nature of teaching . . . which deprives teachers of collegial and
intellectual support (Little, 1992~." In other words, Palincsar et al. believe that if
examples are collected and used for discussion, a common language can be
developed for teaching. Besides the inherent problems associated with not hav-
ing a common language when teachers respond to survey items, we note that
having a common language is the first critical step toward improvement and
change. In this case, a common language would enable teachers to share ideas;
teachers cannot be expected to implement and evaluate new practices until this
takes place.
Failure to Contribute to New Ideas Third, and perhaps most importantly, these
sorts of data rarely if ever contribute to the discussion of improving practice and
outcomes. Why not? Because to improve practice concrete new ideas about
classroom practice are needed. Without these, we cannot expect the dialogue
about classroom practice to move forward productively. And, of course, all of
the methods we have discussed thus far have the questions, issues, and items
defined before any data are collected, thus limiting or excluding altogether the
possibility of producing new, heretofore unimagined ideas about classroom prac-
tice. In this way, survey data are much better suited to supporting or questioning
existing theory than developing new theory. However, this must be qualified:
when theories are not supported by data, researchers are placed in a position to
refine, revise, or generate new theory. In this way, survey data have the potential
to contribute to theory.
Currently, most data on classroom practice can only tell us if what we want
to see in teachers' practice is there or not because people (researchers, policy
makers, administrators, etc.) have predefined what should happen. Thus, these
data can tell us what is not working but cannot help generate new ideas for
improvement. To generate new ideas for improvement, we would need to obtain
data that permit the development of a shared language to refer to concrete
OCR for page 254
254
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
examples, a dialogue could be built about which of these lessons were good and
why. In this scenario we would not have to worry as much as we do with other
data sources that our language about these lessons is not understood by others: if
we all watch the same lesson, for example, using folded paper to show equivalent
fractions, we will know exactly how the paper is folded and how it is marked. In
sum, video data can provide a shared set of examples for building language and
theories for analyzing classroom practices.
Data Needed to Test and Validate Theoretical Models
The data most useful to policy makers are probably those that say whether or
not teachers have implemented the stated policy and, if so, what the impact of the
implementation has been on student achievement. This can then be related to
student achievement data: if students perform well, the policy should remain; if
students perform poorly, the policy should be revised. Thus, the first concern for
policy makers is to know whether policy is being implemented. If the stated
policy is indeed being implemented, it is also important to know how it was
implemented.
Here is an example of this issue: the National Council of Teachers of Mathe-
matics recommends that students participate in mathematical discussions. Among
the many reasons for making this suggestion is that research has told us that
students learn better when they participate actively than when they are passively
taking in what the teacher tells them. To see whether insisting on discussions is
indeed a good policy to be recommended to all teachers, we would want to know
how frequently and how well teachers engaged their students in mathematical
discussions, especially in relation to the amount of time teachers expect their
students to be more passive (e.g., when the teacher stands at the front of the room
and explains to the students what she wants them to know). When we know the
absolute and relative amounts of time spent in mathematical discussions versus
just listening to the teacher, we can relate these to student outcomes.
How do we get these data? We can imagine several scenarios, but for this
sort of question we suggest that none involve teacher self-reports because teachers
cannot possibly teach and note when they are using different instructional tech-
niques and also report how much time they spent in these episodes. Thus, we
recommend videotaped observations because they permit a careful and relatively
accurate measure of what teachers do and do not do in their classrooms.
We also acknowledge that different types of data may be necessary to test
theoretical models of teaching and learning than the types of data used to develop
the models. For example, we can use videotaped records of classroom instruction
to develop ideas about what might facilitate learning and then test these ideas
using experimental methods. As an example, Flevares and Perry (2000) dis-
covered that teachers vary their presentations of nonverbal information to accom-
pany the verbal content and activities in a lesson. From this discovery, they
OCR for page 255
JAMES W. STIGLER AND MICHELLE PERRY
255
hypothesized that the naturally occurring nonverbal information may be crucial
to learning the lesson content. At this point, Flevares and Perry (1999) are
systematically presenting the same lesson content in verbal form but varying the
nonverbal forms and then measuring learning outcomes. Eventually, they expect
to understand which nonverbal forms aid learning of different concepts.
We also wish to make the point that even when we have what we believe is
a good policy, video data can clarify the policy. This point is important because
policy, such as that reflected in standards, is typically vague. When policy is
vague, it leaves plenty of room for interpreting and misinterpreting. As Cohen
(1990:313) puts it, "The [California] framework's mathematical exhortations
were general; it offered few specifics about how teachers might respond, and left
room for many different [implied: some bad] responses." Thus, we suggest that
clear examples, especially those derived from videotaped observations, not only
allow the development of a shared language about what practices actually reflect
policy and which do not but can hone and clarify the policy. In sum, a wide
array of data forms may be necessary to test models of the effects of policy and to
test theories of teaching and learning.
Data Needed as Basis for Communicating to the Public
Finally, we raise the point that data are also needed to communicate what has
been learned to the public. What sorts of data are these? Of course, the answer
depends on the type of data that best illustrate what we have learned. Here is a
simple example: if we have learned that teachers who spend a great deal of time
learning about a new curriculum do a better job of teaching it than teachers who
spend little time learning about the new curriculum, we simply need to present
the average number of hours spent in training of the teachers whose students
learned the material well compared to the teachers whose students did not.
Let's turn to a more complex example. If we learn that stating the goal of a
lesson in a clear fashion at the beginning of a lesson facilitates students' under-
standing of the lesson's content, we may need demonstrations of different teachers
stating the goal of their lessons. Data of this sort would allow the public to get a
sense of how powerful these opening goal statements can be, especially when
these are compared to other teachers' opening statements, which do not include
goal statements. The general point we wish to make is that the data we share with
the public need to be accessible and the data need to communicate or demonstrate
clearly what can be learned.
Recommendations
Classroom process data relevant to the needs of researchers and policy makers
are scant. In general we need more data of all kinds that can feed information
from the classroom back into the research and policy process. Specifically,
OCR for page 256
256
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
however, we stress the need to expand our data collection efforts beyond tradi-
tional surveys. We recommend three new initiatives.
First, we desperately need to collect more data on how policies are imple-
mented and their effectiveness inside classrooms. We need to know whether
policies are implemented or not, and we need to understand the conditions under
which they succeed or fail. Student outcome data must be linked into this effort,
but outcome data alone will not be enough to understand how policies work. In
particular, we propose that video surveys be used, in conjunction with more
traditional surveys, to study classroom processes. Through questionnaires we
can find out, for example, about teachers' opportunities to learn about new poli-
cies or new curricula. Through video surveys we can see what the new policy or
curriculum looks like as it is implemented in classrooms. Clearly, both kinds of
information are needed if we want to understand the mechanisms by which policy
affects teaching and learning.
Second, apart from policy, we should conduct video studies to aid in the
development of theories of teaching and to validate survey instruments. Video
data are especially useful for theory generation. Recall the example we presented
earlier in which we discussed "describe/explain" questions. Japanese teachers
asked their students to describe complete problem solutions, whereas U.S. teachers
asked students to present and justify single steps in a solution. Given that Japanese
students outperform their U.S. peers, we could use this information to advance
our theories of learning. In particular, we could hypothesize that it is not enough
to retell one portion of a problem's solution and have others tell about other
portions. Instead, for deep learning to take place, students may need to put their
explanations in the context of whole-problem solutions. This hypothesis, gener-
ated from video data, could be tested experimentally. Video records also allow
for validation of other instruments (see, e.g., Mayer, 1998~.
Aside from general surveys, we can think of two kinds of data collection
efforts that would be especially valuable. One would be the establishment of a
national sample of "indicator" districts or schools that could serve as a testbed for
developing theories of teaching and new survey instruments. We would propose
to collect all sorts of data in these schools, including, but not limited to, achieve-
ment data, survey data (from teachers, students, parents, and administrators), and
videotaped observations of lessons. In these settings, quantitative data could be
linked with rich contextual data to yield important insights. Moreover, with the
availability of multiple indicators and videotaped records, new theoretical ideas
could be explored.
Another important use of video would be to study special classrooms: either
those in which students have been shown to learn a great deal or those in which
new or experimental teaching techniques are being used. Such data would not
only advance our understanding of what works in classrooms but also provide
guidance to teachers about what the process of changing teaching can look like.
Examples of teachers who are in the process of changing allow other teachers to
OCR for page 257
JAMES W. STIGLER AND MICHELLE PERRY
257
see what it is like to have mixed (i.e., new and old) practices (e.g., Cohen, 1990)
and can provide teachers with direct knowledge of what may be problematic in
adopting something new. In addition, examples of teachers who have accom-
plished a successful change can provide a model, replete with explicit tactics for
instructional success. Our point is simply that special cases may well be more
useful than random samples in advancing our knowledge of teaching and how to
. .
Improve it.
Our third recommendation is to conduct international studies in order to
increase our exposure to novel variations in teaching practices. New ideas are
essential if we are to improve teaching. Systems, and individuals, have a difficult
time learning without a steady diet of variability (Siegler, 1996~. Innovations,
alternative images, different ways of doing things, and new information are all
needed to create new experiences from which the system can learn (Stigler and
Hiebert, 1999~. Looking across cultures can be an especially useful source of
new ideas about what is possible in classrooms, but only if we use research
methods that can spot what is new. Questionnaires are not well suited to this goal
because on them teachers can only answer the questions the researchers were
clever enough to ask. Video data, especially those that are collected outside our
own country, can serve this function of generating new ideas and new hypotheses
about teaching.
DATA FOR CLASSROOM PRACTITIONERS
We have described the role that data can play in helping researchers and
policy makers understand the chain of influence that relates policy to classroom
practice to student learning. But what about classroom teachers? What role can
data play, if any, in teachers' efforts to improve their own practice?
The traditional view is that teachers can use the findings from research, and
the recommendations of policy makers, to improve their teaching. So, for ex-
ample, teachers are assumed to read documents such as the NCTM Professional
Standards for Teaching Mathematics and be able to use the recommendations
therein as a guide for improvement. Recent data and a lot of experience suggest,
however, that teaching is not easily changed by having teachers read such docu-
ments (e.g., Stigler and Hiebert, 1997~. The reason, we believe, is that general
research findings, because they are general, are not situated in the complexities of
classroom life. As we pointed out earlier, there are few features of instruction
that are always desirable or always undesirable; it depends on the lesson context.
We propose an alternative to the traditional view. Because teaching is so
complex, general research findings will have limited applicability to the improve-
ment of practice. Such findings can serve as a guide, but they will not be
sufficient. Teachers need a different kind of knowledge as well, knowledge we
might refer to as localized theories grounded in practice. Teachers themselves
will be the ones to develop this kind of knowledge.
OCR for page 258
258
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
What Teachers Need to Know to Improve Practice
Much has been written about what teachers need to know to perform their
craft (e.g., Shulman, 1986~. We will not review that literature here except to
point out that there is a marked difference between the kind of knowledge teachers
use, as indicated by post hoc analysis, and the kind of knowledge teachers have
available in their quest to become better teachers. Most attempts to improve
teaching through workshops, courses, and so forth, provide knowledge that is of
limited relevance in the classroom. On the one hand, teachers are exposed to
theories, generated by researchers, that are decontextualized and difficult to link
to classroom practice. On the other hand, teachers are given models or examples
of what they "should do" in their classrooms and asked to copy them. But in
these cases the examples are not grounded in theory and thus are not easily
adaptable in local classroom contexts.
Our view is that teachers, to improve their practice, need a kind of knowl-
edge that has been in short supply to this point: theories linked with examples.
This is what we mean by localized theories of teaching. To be useful, such
knowledge needs to be organized around curricular goals and needs to be pack-
aged in units that are shareable across teachers and classrooms. Currently we
have no means of generating this kind of knowledge, no means of accumulating
and storing this knowledge, and no mechanism for sharing this knowledge across
teachers. A major goal of data collection about teaching, therefore, should be to
produce data that can contribute to producing theories of teaching linked with
examples, and that can help in the accumulation and sharing of this knowledge.
Role of Data for Improving Teaching
We believe that teachers must play a central role in the generation of local-
ized theories of teaching and learning in classrooms. Teachers are the ones with
the best access to relevant information about classrooms, and they are in the best
position to evaluate the validity of localized theories. In addition, there are many
more teachers in the country than there are educational researchers. Unless
teachers are involved in a central way in this process, progress will be exceed-
ingly slow. Of course, it will take more than data to engage teachers in this
process, but data can play a central role.
Generating localized theories of teaching will require prolonged reflection
and discussion of examples of classroom practice. Video can play a central role
in these discussions because it allows what is normally a complex and transitory
phenomenon to be slowed down and replayed for study. The theoretical descrip-
tions of teaching that can result from analysis of classroom videos will naturally
be linked to actual examples of classroom practice. Thus, what teachers learn
from joint analysis of such examples will be easier to situate in terms of their own
classrooms. The collaboration is important, too, for it means that teachers will be
OCR for page 259
JAMES W. STIGLER AND MICHELLE PERRY
259
developing a shared language for describing the events and activities they see on
video. This shared language is critical as it becomes the foundation on which
localized theories of teaching can be stored, accessed, and communicated about
with other teachers.
In the process we envision by which teachers could use classroom videos, it
is interesting to ponder what kinds of examples ought to be collected. Some
might think that the most important videos to analyze would be those that teachers
collect in their own classrooms (see, e.g., Lampert and Ball, 1998~. Although
there certainly is a place for such examples in the teacher development process,
they are by no means the only or even the most important examples. Because
teaching is a cultural activity, and because variation in teaching methods might
therefore be limited in a single culture, it is probably most important that teachers
gain exposure to genuine alternatives, examples that depart significantly from
what they are accustomed to seeing. Even risking possible misinterpretation,
videos of lessons from other cultures, and videos of lessons in which serious
efforts to reform are evident, would be a high priority for teachers because these
present clear alternatives to typical and/or culture-bound lessons.
For teachers, contextual data about the lessons taped are even more critical
than for researchers and policy makers. Teachers need to know what happened
yesterday and what the students knew and understood before the lesson started.
Test data and interview data from students both before and after a lesson would
be highly relevant to teachers' analyses. Interviews with the teacher on the video
would also be important, especially questions that elicit from the teacher explana-
tions of what she or he was intending to accomplish with each part of the lesson.
For teachers, the key is not sampling: lessons need not be representative, and the
number of lessons need not be large. What is important is that the cases be
selected to expand and inform teachers' developing understandings of teaching
and learning in classrooms.
Finally, there is one more function that can be served by access to video
examples. As noted by Cohen and Hill (1998), analysis of the possibilities
exemplified by other teachers can provide a powerful incentive for teachers to
improve their own teaching. We are reminded of the beginning Japanese teacher
described by Lewis and Tsuchida (1997) who broke down in tears after watching
one of her senior colleagues teach a science lesson. She explained that she
thought the other teacher was so skilled that she felt badly for her own students,
who, through the luck of the draw, ended up in her class. The result was a strong
feeling of wanting to improve, coupled with concrete images of what improved
teaching might look like.
Recommendations
Teachers can videotape themselves at the local level, but the federal govern-
ment can play an important role in collecting, and then giving teachers access to,
OCR for page 260
260
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
variant examples of teaching in different cultures, different subject areas, and so
forth. The federal government also can document and collect examples from
teachers who are unique, either through some special talent or through participa-
tion in systematic programs of reform.
The National Center for Education Statistics also should consider accumulat-
ing examples into a national database of video cases that could be accessed by
teachers over the Internet. If rules were established to control quality, it would be
possible to build and maintain a database to which classroom teachers could add
their own examples. Nothing would do as much as such a database to facilitate
the development and sharing of curriculum-based localized theories of teaching.
VIDEO AND THE EXISTING NAEP
Having discussed new methods of studying classroom processes and having
thought through how data on classroom processes might be used by different
audiences to improve teaching, we return to the question of the NAEP. In
particular, we wish to address the issue of how new methods, particularly video,
might be used in conjunction with the existing NAEP.
The primary focus of NAEP has been on student achievement. For more
than a quarter of a century, NAEP has documented national trends in what
students know and are able to do in various academic subject areas. Yet there has
also been a growing interest in documenting changes in the context of achieve-
ment at a national level. Student and teacher questionnaires are now included in
the NAEP as a means of measuring everything from student demographics to
teacher preparation, instructional practices, school policies, and out-of-school
activities.
We believe that video surveys can be integrated into the NAEP framework
and that they can contribute greatly to the study of instructional practices over
time. Of course, it is not feasible to videotape in every classroom included in
NAEP, but collecting video records of lessons in a substantial subsample of
NAEP classrooms is both practical and useful. Using techniques similar to those
in the TIMSS video study, videotaping in national samples of classrooms can
provide the first reliable means of tracking changes in instructional practices over
time. Meanwhile, before data can be accumulated on instructional trends, video
surveys can provide a means of studying the classroom mediators of such vari-
ables as race and social class. For example, NAEP already provides a means of
tracking racial gaps in achievement over time. But are such gaps correlated with
gaps in teaching quality and instructional practices? Video records would clearly
be the best means of asking such a question, especially over time.
One way to implement such an effort would be to send videographers around
the country, much as was done in TIMSS. But another possibility is even more
intriguing: just as the Nielsen ratings measure television viewing by placing
continuous monitoring devices in a sample of homes, NAEP could place video
OCR for page 261
JAMES W. STIGLER AND MICHELLE PERRY
261
cameras in a sample of classrooms and conduct continuous monitoring of class-
room processes. This idea is not as farfetched as it sounds. Cameras are cheap,
and the technology for connecting them to the Internet also is cheap. It would not
be necessary to record all of the camera images. Instead, sampling plans could be
devised to get valid and reliable pictures of what goes on inside classrooms. If
NAEP assessments could be administered more frequently in this subsample of
classrooms for example, three times a year we would have the best data ever
available for studying the relation of instruction and learning inside real class-
rooms. This idea is feasible and should be considered seriously.
Another use of video surveys in NAEP should be to aid in the development
and validation of better traditional measures of classroom practices such as ques-
tionnaires. A well-designed sample of video data could serve both immediate
research purposes and instrument development purposes, provided the two are
integrated in their conception and design. It may be that some aspects of class-
room practice are well measured by questionnaires, but validity studies to docu-
ment this possibility are scant. Over time, using video in the development of
questionnaires will increase the power of both methods of studying classroom
practice. One way to approach this goal is to fund the development of a thesaurus
of teaching practices. The problem of developing a shared language for indexing
complex materials is a common one in library and information science. Library
scientists have resolved the problem by relying on thesauruses, the meanings of
which are painstakingly developed over time. Using similar techniques, we
propose a project in which researchers, subject-matter specialists, teachers, and
the public contribute to constructing a thesaurus of teaching practices linked with
video examples. We believe that such a thesaurus could provide a foundation for
developing new measures of instructional processes.
Yet another use of videos collected as part of NAEP would be in the commu-
nication of study results to the public. Although testing of student achievement is
a complex and difficult task, the public nevertheless has some intuitive sense of
what achievement tests measure. Moreover, achievement measures themselves
have been validated over many years. The study of instructional practices is
different on both counts. There is little agreement as to what the basic constructs
are, and, as noted earlier, we lack a public vocabulary for describing teaching
practices. Not only do teachers need to develop such a vocabulary if question-
naires are ever to be a useful means of studying classroom practice, but the public
must do so as well if it wants to understand the information collected about
classroom practices.
In terms of cost, we reiterate the fact that the cost of video data primarily
resides in the analysis phase, not in the collection. For this reason we encourage
the collection of larger quantities of video data, even if funds are insufficient to
support in-depth analyses. Our reasoning is that an archive of nationally repre-
sentative videos will become more and more valuable over time. Imagine if we
had video data of instructional practices over the past 100 years. It would not be
OCR for page 262
262
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
the analyses of 100 years ago that would interest us but the opportunity for
analysis now. Education is a field in which many "facts" are never really estab-
lished as such, most especially those that pertain to the way things "used to be."
Solid data from classrooms can play a key role in mediating and dampening the
polarization that characterizes most educational debate in this country.
CONCLUSION
Data on classroom processes are critical if we are to improve education,
either through policy channels, research, or teacher professional development.
All attempts to improve education must, if they are to work, pass through the final
common pathway that is the classroom. If we fall to collect information on what
is happening in classrooms, we risk missing the key processes that could effect
change. But simply collecting data is not enough. We must, before we collect
any data at all, develop an understanding of how the data will be used, and by
whom, to improve education. We have ruminated on how classroom process data
might be used by policy makers, researchers, and classroom practitioners, but this
is only the beginning. The way data are used is a subject of study in and of itself.
We need more empirical studies of this process. We also need to realize that
there are multiple models of data use, and so we must be flexible in collecting the
data we need for different purposes.
REFERENCES
Barr, R., and R. Dreeben
1983 How Schools Work. Chicago: University of Chicago Press.
Brophy, J., and C. Evertson
1976 Learning from Teaching: A Developmental Perspective. Boston: Allyn and Bacon.
Brophy, J., and T.L. Good
1986 Teacher behavior and student achievement. In Handbook of Research on Teaching, M.C.
Wittrock, ed. New York: MacMillan.
Burstein, L., L.M. McDonnell, J. Van Winkle, T. Ormseth, J. Mirocha, and G. Guitton
1995 Validating National Curriculum Indicators. Santa Monica: RAND Corp.
California State Department of Education
1985 Mathematics Framework for California Public Schools: Kindergarten Through Grade
12. Sacramento: California State Department of Education.
Cohen, D.K.
1990 A revolution in one classroom: The case of Mrs. Oublier. Educational Evaluation and
Policy Analysis 12:311-329.
1995 What is the system in systemic reform? Educational Researcher 24(9):11-17, 31.
Cohen, D.K., and H.C. Hill
1998 Instructional Policy and Classroom Performance: The Mathematics Reform in California.
Paper presented at the NCTM Research Presession, April, Washington, D.C.
Fernandez, C.
1994 Students' Comprehension Processes During Mathematics Instruction. Unpublished doc-
toral dissertation, University of Chicago.
OCR for page 263
JAMES W. STIGLER AND MICHELLE PERRY
263
Flevares, L.M., and M. Perry
1999 Seeing what place value means: Building students' understanding through nonverbal rep-
resentations. Poster presented at the biennial meeting of the Society for Research in Child
Development, April, Albuquerque.
2000 How many do you see? The use of nonspoken representations in first-grade mathematics
lessons. Manuscript under review for publication.
Gallimore, R.
1996 Classrooms are just another cultural activity. Pp. 229-250 in Research on Classroom
Ecologies: Implications for Inclusion of Children with Learning Disabilities, D.L. Speece
and B.K. Keogh, eds. Mahwah, N.J.: Lawrence Erlbaum Associates.
Lampert, M.L., and D.L. Ball
1998 Teaching, Multimedia, and Mathematics: Investigations of Real Practice. New York:
Teachers College Press.
Leighton, M.S.
1994 Measuring Instruction: The Status of Recent Work. Unpublished manuscript, Policy
Studies Associates, Inc., Washington, D C
Leinhardt, G., and R.T. Putnam
1987 The skill of learning from classroom lessons. American Educational Research Journal
24:372-387.
Lewis, C., and I. Tsuchida
1997 Planned educational change in Japan: The shift to student-centered elementary science.
Journal of Education Policy 12(5):313-331.
Light, R.J., J.D. Singer, and J.B. Willett
1990 By Design: Planning Research on Higher Education. Cambridge,MA: HarvardUniver-
sity Press.
Little, J.W.
1992 Opening the black box of professional community. Pp. 157- 178 in The Changing Contexts
of Teaching, A. Lieberman, ed. Chicago: University of Chicago Press.
Mandel, D.R.
1996 Teacher education, training, and staff development: Implications for national surveys.
Pp. 3-29 to 3-42 in From Data to Information: New Directions for the National Center
for Education Statistics, G. Hoachlander, J.E. Griffith, and J.H. Ralph, eds. Washington,
D.C.: U.S. Department of Education.
Mayer, D.P.
1999 Measuring instructional practice: Can policy makers trust survey data? Educational
Evaluation and Policy Analysis 21:29-45.
Palincsar, A.S., S.J. Magnusson, N. Marano, D. Ford, and N. Brown
1998 Designing a community of practice: Principles and practices of the GIsML community.
Teaching and Teacher Education 14(1):5-19.
Perry, M.
1988 Problem assignment and learning outcomes in nine fourth-grade mathematics classes.
Elementary School Journal 88:413-426.
Rosenshine, B., and N. Furst
1973 The use of direct observation to study teaching. In Second Handbook of Research on
Teaching, R.M.W. Travers, ed. Chicago: Rand McNally.
Shavelson, R.J., N.M. Webb, and L. Burstein
1986 Measurement of teaching. Pp. 50-91 in Handbook of Research on Teaching, Third
Edition, M.C. Wittrock, ed. New York: MacMillan.
Shulman, L.S.
1986 Paradigms and research programs in the study of teaching: A contemporary perspective.
Pp. 3-36 in Handbook of Research on Teaching, Third Edition, M.C. Wittrock, ed. New
York: MacMillan.
OCR for page 264
264
CLASSROOM PROCESS DATA AND THE IMPROVEMENT OF TEACHING
Siegler, R.S.
1996 Emerging Minds. New York: Oxford University Press.
Stigler, J.W.
1996 Large-scale video surveys for the study of classroom processes. Pp. 7.1 to 7.29 in From
Data to Information: New Directions for the National Center for Education Statistics, G.
Hoachlander, J.E. Griffith, andJ.H. Ralph, eds. Washington, D.C.: U.S. Department of
Education.
Stigler, J.W., and C. Fernandez
1995 Learning mathematics from classroom instruction: Cross-cultural and experimental per-
spectives. Pp. 103-130 in Basic and Applied Perspectives on Learning, Cognition, and
Development, C.A. Nelson, ed. Mahwah, N.J.: Lawerence Erlbaum Associates.
Stigler, J.W., and J. Hiebert
1997 Understanding and improving classroom mathematics instruction: An overview of the
TIMSS video study. Phi Delta Kappan 79(Sept.): 1, 14-21.
1999 The Teaching Gap: What Teachers Can Learn from the World's Best Educators. New
York: Free Press.
Stigler, J.W., S.Y. Lee, and H.W. Stevenson
1987 Mathematics classrooms in Japan, Taiwan, and the United States. Child Development
58: 1272-1285.
Stigler, J.W., and M. Perry
1988 Mathematics learning in Japanese, Chinese, and American classrooms. Pp. 27-54 in
Children's Mathematics, New Directions for Child Development, G.B. Saxe and M.
Gearhart, eds. San Francisco: Jossey-Bass.
Representative terms from entire chapter:
classroom process