A CONSTRUCTIVIST APPROACH TO SECOND
GRADE MATHEMATICS
PAUL COBB, TERRY WOOD, AND ERNA YACKEL
Purdue University
Our overall objective in this paper is to share a few observations made and
insights gained while conducting a recently completed teaching experiment.
The experiment had a strong pragmatic emphasis in that we were responsible
for the mathematics instruction of a second grade class (7 year-olds) for the
entire school year.
Thus, we had to accommodate a variety of institutionalized
constraints. As an example, we agreed to address all of the school corporation’s
objectives for second grade mathematics instruction. In addition, we were well
aware that the school corporation administrators evaluated the project primarily
in terms of mean gains on standardized achievement tests
. Further, we had to be
sensitive to parents’ concerns, particularly as their children’s participation in the
project was entirely voluntary. Not surprising, these constraints profoundly
influenced the ways in which we attempted to translate constructivism as a
theory of knowing into practice. We were fortunate in that the classroom
teacher, who had taught second grade mathematics “straight by the book” for
the previous sixteen years, was a member of the project staff. Her practical
wisdom and insights proved to be invaluable.
It appears that we have had some success in satisfying the institutional
constraints. The achievement test scores did rise satisfactorily, the parents were
all universally supportive by the middle of the school year, and the administrators
developed a positive opinion of what they saw. As a consequence,
we are currently working with 18 teachers from the same school system.
In general, we hope that our on-going work constitutes the beginnings of a
response to Brophy’s (1986) challenge that “to demonstrate the relevance and
practical value of this point of view for improving school mathematics instruction,
they [constructivists] will need to undertake programmatic development
and research – the development of specific instructional guidelines (and
materials if necessary) for accomplishing specific instructional objectives in
typical classroom settings” (p. 366). Thus, we concur with Carpenter’s (1983)
observation that “If we are unable or unwilling to provide more direction for
instruction, we are in danger of conceding the curriculum to those whose basic
epistemology allows them to be more directive” (p. 109). Constructivism as an
epistemology is, for us, a general way of interpreting and making sense of a
variety of phenomena. It constitutes a framework within which to address
situations of complexity, uniqueness, and uncertainty that Schon (1985) calls
E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education, 157-176.
© 199 1 Kluwer Academic Publishers. Printed in the Netherlands.
158 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
“messes,” and to transform them into potentially solvable problems. Thus, like
any epistemology, constructivism influences both the questions posed and the
criteria for what counts as an adequate solution. Its value to mathematics
education will, in the long run, depend on whether this way of sense making, of
problem posing and solving, contributes to the improvement of mathematics
teaching and learning in typical classrooms with characteristic teachers. If it
eventually fails to do so, then it will become irrelevant to mathematics
educators.
I. OVERVIEW OF THE PROJECT
of learning mathematics. It should be clear that, for the constructivist, substantive
mathematical learning is a problem solving process (Cobb, 1986; Confrey,
1987; Thompson, 1985; von Glasersfeld, 1983). In this context, substantive
learning refers to cognitive restructuring as opposed to accretion or tuning
(Rumelhart & Norman, 1981). Consequently, our primary focus, as we
developed, implemented, and refined instructional activities, was on that aspect
of cognitive development that is both the most significant and the most difficult
to explain and influence.
At the risk of over-simplification, an immediate implication of constructivism
is that mathematics, including the so-called basics such as arithmetical
computation, should be taught through problem solving. This does not mean
that the instructional activities necessarily emphasize what are traditionally
considered to be problems – stereotypical textbook word problems. In fact, the
general notion that problems can be given ready-made to students is highly
questionable. Instead, teaching through problem solving acknowledges that
problems arise for students as they attempt to achieve their goals in the
classroom. The approach respects that students are the best judges of what they
find problematic and encourages them to construct solutions that they find
acceptable given their current ways of knowing. The situations that children
find problematic take a variety of forms and can include resolving obstacles or
contradictions that arise when they use their current concepts and procedures,
accounting for a surprising outcome (particularly when two alternative procedures
lead to the same result), verbalizing their mathematical thinking, explaining
or justifying a solution, resolving conflicting points of view, and constructing
a consensual domain in which to talk about mathematics with others. As
these examples make clear, genuine mathematical problems can arise from
classroom social interactions as well as from solo attempts to complete the
instructional activities.
In general, the instructional activities, classroom organization, and flow of
1.1 Rationale for Instructional Activities
Other contributors to this volume present constructivist analyses of the process
APPROACH TO SECOND GRADE MATHEMATICS 159
the lessons were designed to facilitate the occurrence of mathematical problems
that the children could attempt to resolve in conducive settings. Detailed models
of early number learning (Steffe et. al., 1983; Steffe, Cobb, and von
Glasersfeld, 1988) proved particularly relevant to the development of the
activities. The models’ specification of both young children’s meaning-making
capabilities and the sensory-motor and conceptual activities upon which they
can reflect guided the analysis of the problematic situations that might arise for
children at different conceptual levels. The instructional activities are appropriate
to the extent that they engender reflective, problem solving activity on
the part of the children.
1.2 Classroom Organization
The instructional activities are of two general types: Teacher-directed whole
class activities and small group activities. To the extent that any lesson can be
considered typical (Erickson, 1985), the teacher might first spend at most five
minutes introducing the small group activities to the children. Her rationale for
doing so is to clarify the intent of the activities. She might, for example, ask the
children what they think a particular symbol means or ask them how they
interpret the first activity. In doing so, she does not attempt to steer the children
towards an official solution method but instead tries to ensure that the
children’s understanding of what they are to do is compatible with the activity
as intended by its developers. Any suggested interpretation or solution, however
immature, is acceptable provided it indicates the child has made the appropriate
suppositions.
Once the teacher is satisfied that conventions used to present the instructional
activities have been established, she tells them to start work. One child
hands out an activity sheet to each group of two or, occasionally, three children.
As the children work in groups for perhaps 25 minutes, one notices that the
teacher spends the entire time moving from one group to the next, observing
and frequently interacting with them as they engage in mathematical activity. In
this setting she is free of managerial concerns and can focus attention on
children’s thinking and interactions. Because the children are encouraged to
take responsibility for their own learning and behavior in the classroom, she is
rarely interrupted by a child from another group asking for help or for permission
to, say, use particular manipulative materials. Further, the teacher does not
feel the need to monitor the children’s conduct – she is comfortable giving her
attention entirely to the group she is currently observing.
The noise level is somewhat higher than most classrooms because the
children are talking about the activities with their partners. However, the
teacher does not have to remind the children to lower their voices because the
noise level always stays within reasonable bounds and she has learned that their
talking is generally about mathematics and is not distracting to others. One soon
notices the purposeful way that children move around the classroom on their
160 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
own initiative. Some go to a table to take one of the available manipulatives that
they have decided is needed. Others get additional activity sheets or perhaps a
piece of scrap paper. It soon becomes apparent that some groups have completed
four or five activity sheets while others might complete just one, and
then only with the teacher’s assistance. This disparity in the number of activities
completed seems, for the most part, to be inconsequential to both the teacher
and the children. Finally, the teacher tells the children when there is only one
minute of work time remaining. Most of the children begin to put away the
manipulatives and prepare for the discussion of their solutions.
The teacher starts the discussion by asking the children to explain how they
solved the first activity. Typically, there is no shortage of volunteers. When
called upon, children spontaneously come to the front of the class to give their
explanations, often accompanied by their partner. If a child gives only an
answer, the teacher asks for an explanation of the solution process. Sometimes
she asks follow-up questions to clarify the explanation or to help the child
reconstruct and verbalize a solution. Occasionally, a child will become aware of
a problem with his or her solution while explaining it to the class. Because of
the accepting classroom atmosphere, the child does not become embarrassed or
defensive but might simply say, “I messed up” and sit down. In the course of
the dialogue, other children spontaneously explain why they now disagree with
their own solutions or why they thought their partner was wrong but now think
he or she is right. It is immediately apparent that the teacher accepts all answers
and solutions in a completely nonevaluative way. If, as frequently happens,
children propose two or more conflicting answers, she will frame this as a
problem for the children and ask them how they think the conflict can be
resolved. Children volunteer to justify particular answers and, almost invariably,
the class arrives at a consensus. On the rare occasions when they fail
to do so, the teacher writes the activity statement on a chalk board so that the
children can think about it during the following few days. Although the
discussion might continue for 15 to 20 minutes, the time is sufficient to
consider only a small proportion of the activities completed by some groups -
the children have much to say about their mathematics. Eventually, the teacher
terminates the discussion due to time constraints. She collects the children’s
activity sheets and might glance through them before making them ready for
distribution to parents. However, she does not grade their work or in any way
indicate whether or not their answers are correct.
In the remaining ten minutes or so of the hour long lesson, the teacher
introduces a whole class activity and poses one or more questions to the
children. She is again nonevaluative when the children offer their solutions and,
as before, attempts to orchestrate a discussion among the children.
1.3 Development Process
The teaching experiment bears certain resemblances to a type of Soviet
APPROACH TO SECOND GRADE MATHEMATICS 161
experiment that Menchinskaya (1969) called a macroscheme: “Changes are
studied in a pupil’s school activity and development as he [or she] makes the
transition from one age level to another, from one level of instruction to
and that of the Soviet researchers. Typically, Soviet investigators construct the
instructional materials before the experiment begins (e.g., Davydov, 1975). We,
in contrast, developed samples of a wide range of possible instructional
activities in the year preceding the experiment, but the specific activities used in
the classroom were developed, modified, and, in some cases, abandoned while
the experiment was in progress. To aid this process, two video-cameras were
used to record each mathematics lesson for the school year. Initial analyses of
both whole class dialogues and small group problem solving interactions
focused on the quality of the children’s mathematical activity and learning as
they tackled specific instructional activities. These analyses, together with the
classroom teacher’s observations, guided the development of instructional
activities and, on occasion, changes in classroom organization for subsequent
lessons. Thus the processes of developing materials, conducting a formative
assessment, and developing an initial explanation of classroom life were one
and the same.
Research Emphases
For the sake of explication, we have separated the development and research
aspects of the project. In practice, however, we found that the two frequently
blended together to such an extent that the distinction became irrelevant. As we
have noted, we struggled to achieve an initial understanding of classroom
events as part of the development process. In doing so, we attempted to clarify
situations that were problematic to us and, in the process, developed research
questions. The speculative solutions and working hypotheses proposed in turn
fed back to inform the development process. Thus, just like the children, we
encountered problematic situations that constituted opportunities to learn as we
attempted to achieve the goals of the project in the classroom.
Before describing our research emphases, it is perhaps well to state categorically
that our intent has not been, is not, and cannot possibly be to prove that
constructivism is right or even the way that all researchers should frame the
problems of mathematics education. It is simply a way of knowing which, we
believe, might open up potentially fruitful avenues of investigation. Further, we
have tried to be consciously aware of the danger of forcing our observations
into the conceptual boxes provided by constructivism as we understood it at the
beginning of the project. As a consequence, we have had to revise and relinquish
some basic assumptions that initially seemed beyond question. Thus, part
of the research agenda requires that we become increasingly aware of and probe
some of the weaknesses of constructivism. This, in our view, is the soundest
way of contributing to the vigor of this or any paradigm.
another” (p. 5). However, there is a crucial difference between our approach
1 .4
162 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
The issues of current interest include the children’s construction of increasingly
abstract arithmetical objects, their emotional/moral development, the
teacher’s learning in the classroom, and the changing nature of the children’s
beliefs about the activity of doing mathematics, their role as students, and the
teacher’s role. There is, however, one overriding problem that touches each of
these issues. Constructivism, at least as it has been applied to mathematics
education, has focused almost exclusively on the processes by which individual
students actively construct their own mathematical realities. Much progress has
been made on this front in recent years. However, far less attention has been
given to the interpersonal or social aspects of mathematics learning and
teaching. Given the emphasis of the evolving subjective realities of individuals,
how (to state the question in observer language) does mathematics as cultural
knowledge become “interwoven” with individual children’s cognitive achievements
(Saxe, Guberman, & Gearhart, 1985)? In other words, how is it that the
teacher and the children manage to achieve at least temporary states of intersubjectivity
when they talk about mathematics? More simply still, how do children
learn in such instructional situations? In our view, these are critical questions
for constructivism. We will be unable to talk about the specifics of instruction
in a theoretically grounded way unless we place analyses of learning within the
context of classroom social interactions. Brophy and other adherents of the
process-product approach are clearly trapped on one side of the chasm that
currently separates research on learning from research on teaching. Constructivists
are in danger of becoming trapped on the opposite side of the same
divide.
As a first step in coming to grips with the problem of learning in social
settings, we are currently analyzing the evolving regularities or patterns
identified in both whole class and small group interactions. These patterns are,
for the most part, outside the conscious awareness of both the teacher and the
children and are repeatedly reconstructed in the course of interactions (Voigt,
1985). In other words, the patterns consist of coordinated sequences of actions
and, at each occurrence, the development of a pattern begins anew – the
enactment and the construction of a pattern are synonymous. Thus, although the
teacher and children do not have a “blueprint” of the interaction patterns, each
knows how to act appropriately in particular situations. The patterns reveal the
largely implicit social norms negotiated by the teacher and children, the norms
that constitute the social reality within which they teach and learn mathematics.
Following Voigt (1985), we are analyzing the patterns and associated norms in
terms of both the implicit, taken-for-granted obligations that the teacher and
children accept in particular situations and the expectations they have for each
other. The investigation of how they mutually construct the social reality of the
classroom will therefore elucidate their beliefs about their own and each others’
roles and document how they created a “problem-solving atmosphere” (Silver,
1985).
At the same time, we are interested in how the evolving network of obligaAPPROACH
TO SECOND GRADE MATHEMATICS 163
tions and expectations influences individual children’s construction of mathematical
knowledge. The relevance of this question is apparent when one
observes with Balacheff (1986) that “most of the time, the pupil does not act as
a theoretican but as a practical man. His job is to give a solution to the problem
the teacher has given him, a solution that will be acceptable with respect to the
classroom situation” (p. 12). Thus, the social norms constrain what is
problematic for children and what might count as an acceptable solution.
However, it is while the children attempt to complete the mathematical
activities in groups and discuss their solutions in the whole class setting that the
social norms are renegotiated. This leads us to the contention that neither the
cognitions of individuals nor the mutual constructed network of obligations and
expectations are primary; we find it impossible to give an adequate explanation
of one without considering the other.
On the other hand, the social norms and consensually sanctioned mathematical
knowledge are created, regenerated, and modified by the coordinated actions
of the teacher and children as individuals. The norms and consensually
sanctioned knowledge do not exist independently of these actions (except in the
mind of the analyst). On the other hand, each individual’s understanding of the
norms and knowledge contrains his or her activity in the classroom. The norms
(and consensually constructed mathematics knowledge) constrain the activity
that creates the norms. Conversely, individuals’ activity creates the norms that
constrain that activity.
In this formulation, the individual and the social are interdependent, in that
one does not exist without the other. We thus acknowledge that social context is
an integral aspect of an individual’s cognitions without reifying mathematics as
a ready-made body of cultural knowledge that is somehow internalized from
without by individuals. Neither (the observer’s) physical, mathematical, or
social realities are taken as solid, immutable bedrock upon which to anchor an
analysis of learning and teaching.
EXAMPLES FROM THE CLASSROOM
We present our current, admittedly fragmentary understanding of classroom life
in the following sections. At the same time, the protocols and narratives of
specific episodes that support the analysis exemplify the kinds of interactions
that typified the activities of learning and teaching mathematics in the classroom.
2.1 Whole Class Interactions
The teacher’s overall intention as she led whole class discussions was to
encourage the children to verbalize their solution attempts. Such dialogues give
rise to learning opportunities for the children as they attempt to reconstruct their
II.
164 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
solutions, understand alternative points of view, resolve conflicts between
incompatible solution methods, and so forth. The teacher also had learning
opportunities in that she had to decenter and “see” things from the children’s
perspectives when she helped them to say what they wanted to say. However,
the teacher’s expectation that the children should verbalize how they had
actually interpreted and attempted to solve the instructional activities did not fit
with the expectations the children had developed on the basis of their prior
experiences of class discussions in school. During both first grade and second
grade with the exception of mathematics, such dialogues were typically initiated
with the intention of steering or funnelling (Voigt, 1985) the children towards
an officially sanctioned intepretation or solution (Wood, Cobb, & Yackel,
1988). As a consequence, class discussions in subject matter areas other than
mathematics were situations in which the children felt obliged to try and infer
what the teacher had in mind rather than to articulate their own understandings.
The project teacher therefore had to exert her authority in order to help the
children reconceptualize both their own and her role during mathematics
instruction. In effect, she had to actively teach the children that she had
different expectations for them when they did mathematics. To this end, she
initiated the mutual construction of expectations and obligations in the classroom.
In doing so, she simultaneously had to accept certain obligations for her
own actions. If she expected the children to honestly express their current
understandings of mathematics, then she was obliged to accept their explanations
rather than to evaluate them with respect to an officially sanctioned
solution method. Thus, the teacher had obligations to the children, just as they
did to her. This evolving, interlocking network of obligations and expectations
was beyond the conscious awareness of both the teacher and the children, but
nevertheless was mutually constituted. The teacher and the children initially
negotiated obligations and expectations at the beginning of the school year,
which subsequently made possible the smooth functioning of the classroom for
the remainder of the school year. Once established, this mutually constructed
network of obligations constrained classroom social interactions in the course
of which the children constructed mathematical meanings as they attempted to
achieve their goals (Blumer, 1969). Thus the patterns of discourse served, not to
transmit knowledge (Mehan, 1979; Voigt, 1985), but to provide opportunities
for children to articulate and reflect on their mathematical activities.
The teacher’s and students’ mutual construction of social as well as mathematical
realities is reflected in the dual structure of classroom dialogues. In one
conversation, they talked about mathematics whereas in the other they talked
about talking about mathematics. As these two conversations were conducted at
distinct logical levels (Bateson, 1973), one in effect setting the framework for
the other, seemingly contradictory statements such as “the teacher exerted her
authority to enable the children to express their viewpoints” make sense. As in a
traditional classroom, the teacher was very much an authority in the classroom
who attempts to realize an agenda. The difference resides in the way that she
165
translated her authority into action (Bishop, 1985).
discussion centers on word problems that are shown on an overhead projector.
T:
Peter: The tiger.
T:
Peter:
In the following episode that occurred during the first day of school, the
Take a look at this problem. The clown is first in line.
Which animal is fourth? Peter.
How did you decide the tiger? . . .. Would you show us how you got the fourth?
(Goes to the screen at the front of the room.) I saw the clown and then ... (He
counts the animals.) Oh, the dog [is fourth]. (Hesitates) Well, I couldn’t see from
my seat. (He looks down at the floor.)
OK. What did you come up with?
I didn’t see it. (He goes back to his seat quickly.)
T:
Peter:
The teacher realized that in making Peter obliged to explain his solution, she
put him in the position of having to admit that his answer was wrong in front of
the entire class. Peter’s response to the situation was to offer an excuse for his
error, “I couldn’t see it . . .” Peter’s concern with social comparison became
manifest by his construal of the situation as warranting embarrassment (Armon-
Jones, 1986). This construal confounded the teacher’s intention that the children
should feel free to publicly express their own solutions to problems. For her
purposes, it was vital that children feel no shame or embarrassment when they
present erroneous solutions in front of others. Crucially, the teacher immediately
initiated a second conversation by talking about talking about
mathematics as she suggested an alternative construal of the situation.
T: That’s okay Peter. It’s all right. Boys and girls even if your answer is not correct,
I am most interested in having you think. That’s the important part. We are not
always going to get answers right, but we want to try.
The teacher was directive in her comments. She expressed her expectations
by telling the children how she as an authority interpreted the situation. She
emphasized that Peter’s attempts to solve the problem were appropriate in every
way, and simultaneously expressed to the other children her belief that what is
important in this class is thinking about mathematics, not just getting right
answers. She then terminated this conversation and returned to the first.
T:
In this episode, the project teacher practiced what Schon (1 985) called
“knowledge-in-action.” She did not follow a set of prescribed rules as she
conducted this and other dialogues. Instead, as a practitioner, she made sense of
a problematic situation by reframing the situation.
The example makes clear that constructivist teaching does not mean that
“anything goes” or that the teacher gives up her authority and abrogates her
wider societal obligations. Because mathematical meaning is inherently
dependent on the construction of consensual domains, the activities of teaching
and learning must necessarily be guided by obligations that are created and
regenerated through social interaction. Critically, the teacher attempted to
All right, did anybody else come up with a different way to do the problem?
APPROACH TO SECOND GRADE MATHEMATICS
166 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
achieve her agenda by initiating and actively guiding the mutual construction of
obligations and expectations in the classroom. In doing so, she attempted to
invite actions that she considered appropriate while hindering others. These
invitations and her overall agenda reflected her beliefs that mathematical
learning is a problem-solving activity and that children’s understandings should
be respected. As a consequence of the teacher’s interventions, the realized
dialogue patterns contrast sharply with those of a typical classroom (Mehan,
1979; Stake & Easley, 1978; Voigt, 1985). The teacher exercised her authority
in different ways in each of the two conversations that intertwined in the
dialogues. When she and the children talked about talking about mathematics,
the teacher typically initiated and attempted to control the conversation.
However, when they talked about mathematics, she limited her role to that of
orchestrating the conversation. The following episode taken from a dialogue
that occurred later in the year again illustrates her authoritarian role with respect
to the classroom social norms.
T: Now another thing I noticed was happening and it is something I don’t like and I
don’t want to hear ... and it’s these two words. (She writes “That’s Easy on the
board and draws a circle around the phrase.) These words are no-no’s starting
today. What are these two words Mark?
... I’ve had kids come up to me and say, “Oh, that’s easy!” Maybe I look at it
and say, “I don’t think that’s very easy.” How do you think that’s going to make
me feel?
It is evident as she listened to several children’s suggestions that she was
looking for a specific answer. Finally she stated, “You are not saying the word I
am looking for.” She continued to fish for a specific interpretation of the
situation without success, and ultimately directly told the children.
T: . . . it hurts my feelings when someone says. “Oh that’s easy!” (She points to the
words on the board.) When I am struggling and trying so hard, it makes me feel
kind of dumb or stupid.
Because I am thinking, gosh, if it’s so easy, why am I having so much trouble
with it?
Once she had explicitly told the children the officially sanctioned interpretation
of the situation, she terminated the social conversation and turned to talking
about mathematics.
T: That’s easy is a real put down. It’s like if I think it’s easy, then you must not be
very smart, because it is not easy for you . . . OK. The first activity is balances.
The teacher’s highly directive intervention is consistent with her agenda for the
classroom. By outlawing actions that might make others feel stupid, the teacher
nurtured the sense of trust that was essential if the children were to talk publicly
about their mathematics. She demonstrated that if the children accepted the
obligation of expressing their mathematical understandings then she, as an
authority, was obliged to protect them. Her efforts bore fruit in that even the
Mark: That’s easy.
T
I
APPROACH TO SECOND GRADE MATHEMATICS 167
most conceptually immature child seemed to feel that his thinking was valued
and respected, and as a consequence continued to participate actively in whole
class discussion throughout the year.
When a conflict existed between the two conversations, the teacher always
gave priority to the social (Stake and Easley, 1978). She was right to do so in
that the norms established in the course of the social dialogue made possible the
conversations in which children felt free to take the initiative as they talked
about mathematics. The teacher was then able to adopt a nonauthoritarian,
nonevaluative role when the children explained their solutions. For the most
part, she orchestrated their contributions to the class discussions. The manner in
which the children felt free to take the initiative is illustrated in the following
episode which occurred near the end of the school year. The instructional
activities for the previous two days provided a setting in which to discuss the
meanings of fraction symbols such as 1/4, 1/2, 1/6, etc. The discussion prior to
this episode was about the meaning of 7/8ths. The episode begins when John,
who had previously stated that he has a problem he has been thinking about,
came to the front of the room where the teacher, the overhead projector, and the
screen were located.
John:
T:
John:
T:
John:
But um . . . what about if it was one and one? What would that be?
Like this? (She writes l/l). Good question. Or what if you had this over this?
(She writes 4/4 and 6/6.) What does that mean?
(Starts talking to the teacher while tracing two circles on the screen with his
hand.) Put a circle down.
O.K. Put a circle down (she draws a circle on the overhead, then looks at John).
Um like . . . what I’m thinking is one and one. (He points to l/l.) How could you
make like . . .? (He looks at the teacher.)
The teacher’s response at this point deviated distinctly from typical patterns of
classroom interaction in that she did not offer John an official explanation but
instead she continued the turn-taking dialogue by asking him a question that
placed him under the obligation of thinking through a solution.
T: How could you show one and one, one over one?
John: (Looks at the teacher and says nothing.)
Ann: (Interjects) Just fill it all in.
Mark: Yeah! That’s one piece.
T: OK. If we remember what a fraction tells us, John .. .
John: (Interrupts the teacher excitedly.) But it’s the whole thing (he gestures with his
hands, indicating a larger item), not just one piece (holds up one finger).
T: So, how many can we fill in?
John: The whole thing!
The conversations in which children discussed mathematics on their terms
gave rise to problematic situations for them which constituted opportunities for
reflection and the construction of mathematical knowledge. These interactions
were characterized by a genuine commitment to communicate. All the participants
came to assume that any contribution to the dialogue made sense to the
168 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
speaker (Rommetveit, 1985). Participation in the discussions therefore involved
a genuine attempt to exchange points of view and encouraged the children to
distance themselves from on-going activity (Sigel, 1981).
The manner in which the patterns of interactions evolved during the school
year exemplifies the development of what Silver (1985) called a “problemsolving
atmosphere.” The teacher’s priority was to establish a nonevaluative,
cooperative setting in which children could publicly verbalize and reflect on
their mathematical ideas without risk of social comparison or public embarrassment.
The obligations and expectations mutually constructed during whole class
discussions also provided a framework for the children’s activity as they
worked in small groups, in that they were expected to solve problems in a
cooperative manner and to respect each others’ efforts. However, the children
still had to negotiate obligations and expectations in the small group setting that
would constitute a productive working relationship.
2.2 Small Group Interactions
The children attempted to complete instructional activities in groups of two, or
occasionally three, prior to the whole class discussions. From the beginning of
the year the teacher used her authority to emphasize two major responsibilities
for each small group, namely that they cooperate to complete the activities and
that they reach a consensus. Ideally, consensus is attained when the children
develop a mutually acceptable method of solution. In a more limited sense,
consensus is reached when the children agree on a common answer, albeit via
different methods of solution. Thus, the children had two distinct types of
problem to solve. The first concerned the mathematical problems that arose as
they attempted to complete the instructional activities, whereas the second was
that of negotiating a viable relationship that would make it possible jointly to
solve their mathematical problems.
As with the whole class setting, the teacher was very explicit about what she
expected of the children as they worked to solve both types of problem. For
example, she attempted to place the children under the obligation of persisting
and thinking their problems through for themselves rather than trying to
complete as many activities as possible. Thus, she assured the children “it’s OK
to complete one problem or five problems. If you don’t get one finished, don’t
worry.” The children not only demonstrated by both word and deed that they
understood and accepted the teacher’s expectations (gaining personal satisfaction
by solving difficult problems, persistence, and task-involvement came to
characterize small group work), but felt free to extend and further emphasize
them. This is illustrated by the following dialogue which occurred in the whole
class setting at the beginning of a mathematics lesson. This episode occurred on
the first day following two weeks of winter holiday.
APPROACH TO SECOND GRADE MATHEMATICS 169
Lois:
Teacher:
Students: (in unison): Yeah.
Teacher:
Adam:
By contrasting her expectations with the typical classroom obligation of
completing a prescribed number of tasks, the teacher attempted to place the
children under the obligation of persisting to resolve what they found
problematic. Adam’s elaboration provides clear evidence that children understood
the teacher’s expectations and accepted the consequences. Even the most
extreme case, completing none, was acceptable provided the children met the
obligation of striving to solve their problems.
The above discussion illustrates that the teacher was very much an authority
when she told the children what she expected of them as they tried to solve their
mathematical problems. She was even more explicit about what she expected
socially, as is illustrated in the following continuation of the episode. Here, the
teacher reminded the children of their social obligations as they worked in small
groups.
Teacher: Any questions before we get started? OK. You’re going to cooperate. You’re
going to work with your partner. And if you figure out the answer and your
partner is looking at you like, “How in the world did you get that answer?” it’s
going to be up to you and your partner to work it out and understand it together.
Then you can get another [problem] card, but not until then.
In this directive the teacher attempted to place the children under a dual
obligation. They were expected to work cooperatively and, at a minimum, to
develop solutions that produced the same result. Ideally, they should also
develop mutually acceptable methods of solution. The two obligations are at
very different levels of complexity. At the basic level, children must negotiate a
way to work with each other and must share materials including the activity
pages (each small group received only one copy.) Cooperation at this level
includes listening to and attempting to understand each other’s comments and,
more generally facilitating rather than hindering each other’s mathematical
activity. The second level of cooperation, working together to construct a
mutually acceptable solution, is much more complex. It requires that students
communicate about mathematics, verbally or otherwise.
The social problem of cooperating at the basic level always has priority in
the sense that it makes the construction of mutually acceptable solutions
possible. The teacher played an important role in facilitating cooperation at this
level (Wood & Yackel, 1988). As the teacher observed small group activity, she
was able to identify those groups which were having difficulty at this basic
level. Comments such as, “if you put the paper here in the middle then both of
you can see it” and “Listen to what your partner is saying’’ exemplify the
How many problems are there?
There are eight. So the most you can do is eight. Some of them are very hard
though, so you may only get two done. Is that OK if you only get two done?
You bet it is . . . if you only get one done because it was so hard and you worked
so hard on it that it was the only one you got, that’s OK too.
If you get none done, but you’re still working on it, it’s OK too.
170 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
interventions the teacher made to encourage cooperation. If children were
unable to resolve the problem of achieving basic cooperation over several
weeks, even with the teacher’s intervention, they were assigned new partners. In
the few cases where basic cooperation was not achieved, there were underlying
reasons for failure such as extremely immature social communication skills,
high ego-involvement, or large discrepancies in mathematical conceptual level.
As the year progressed, the children’s social development alleviated the
difficulties of establishing cooperative relationships in the small group setting.
By the end of the year, instances of failure to cooperate at the basic level were
isolated and were typically resolved by the children themselves within the
group.
The problem of cooperating at the level of developing a mutually acceptable
solution is much more demanding. Nevertheless such cooperation became the
norm in the classroom. At this level of cooperation, the children have to
appreciate what each finds problematic and coordinate their mathematical
problem solving activities. Such interactions naturally give rise to opportunities
for reflection and cognitive reorganization. As an illustration, extracts of
a dialogue between two children as they solved a sequence of multiplication
tasks are presented. The children had just successfully completed the task
5 x 4 = -. (The children’s use of the term “sets” in talking about multiplica -
tion is a result of the teacher’s use of the term when she first introduced “X” as
the mathematical symbol for multiplication.)
John:
Andy:
John: Yeah!
Andy:
John: 40.
Andy: 40.
John:
Andy:
John:
Andy:
John:
Andy:
John: Just switch them around.
Notice that Andy constructed his solution by elaborating John’s initial comment
that “It’s 5 more sets [of four].” However, in his final explanation John
developed an alternative solution method. Andy’s initial statement that 40 is the
answer was followed by John’s exclamation “Yeah!” John’s subsequent
verbalizations suggest that once he heard “20 plus 20 is 40” he reconceptualized
the problem as four tens, rather than as 20 and 5 more fours, and then successfully
justified that conceptualization. This inference is supported by the
frequency and enthusiasm with which he repeated “Just turn it (or switch them)
It’s five more sets [of 4]. Look. Five more sets than 20.
Oh! 20 plus 20 is 40. So its gotta be 40. No?
No. 4, 8, 12, 16, 20,24,28 (keeps track on his fingers).
Yeah, I know . . . cause ten fours make 40.
Like five fours make 20.
Four sets of ten makes 40. Just turn it around.
Five sets of fours make 20 and so five more than that.
Yeah, just turn it around. Just turn it around.
5 times 4 is 20, so 20 more than that makes 40.
10 x 4 =- .
APPROACH TO SECOND GRADE MATHEMATICS 17 1
around.” Thus, even though each child eventually conceptualized the problem
differently their initial verbalizations were instrumental in the development of
the other’s solution. This illustrates an extremely productive level of cooperation
in which each partner facilitated the conceptual activity of the other.
9 x 4 =-
John:
Andy: 36
John: (pause) Yeah!
In this brief dialogue we see a different form of cooperation. One partner, John,
suggested a solution process and the other, Andy, carried it out. The pause
before John’s final agreement indicates that he verified Andy’s answer. In the
following dialogue the children agreed on an answer but constructed different
methods of solution.
Just take away 4 from that [10 x 4].
8 x 4 =-
John: Look! Look! Just take away 4 from that [9 x 4] to get that [8 x 4] See! Just take
away 4 from there [9 x 4].
Andy: Just take 8 away from that [10 x 4].
John: No. Take away 4 from there [9 x 4].
Andy: Take 8 away from that [10 x 4]. That makes 32.
Observer: Did you do it the same way as Andy?
John: Yeah, but I used that one [9 x 4]. Take away 4. It makes 32.
John’s final comment makes it evident that he was consciously aware of the
similarities and differences between his and Andy’s methods. Although they
constructed different methods, each was satisfied that his was viable. This was
sufficient to meet the obligation of reaching consensus. As the dialogue
continued, the children encountered a situation where they disagreed on the
answer.
8 x 5 =
Andy:
John:
Andy: No. 9,39, I think.
(Both children pause to reflect for a few moments.)
John: (very excitedly) It’s 40.
Andy: It is?
John:
Andy: No, it’s 39.
John:
Andy:
John: Huh?
Andy: Wait. 40. Yeah, 40.
On this task, Andy first attempted to modify his previously successful method
of solution and found eight fours plus five. Apparently John’s remark, “Eight
sets of 4. Eight sets of 5,” led Andy to reflect on this activity. His next answer,
Five more than that [8 x 4] is 37.
Eight sets of 4. Eight sets of 5.
Yeah, its 40! Yeah, look!
5, 10, 15, 20, 25, 30, 35, 40, 45, oops. Take away that last one. It’s 40.
5, 10, 15, 20, 25, 30, 35, 40, 45.
-
172 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
39, indicates that he modified his initial conceptualization. Andy then challenged
John’s answer of 40, and John was obliged to justify his solution. Andy
immediately adopted John’s method but his conceptualization of it is open to
question as he simply repeated exactly what John had said, including the error
of going up to 45. But John’s “Huh?” prompted further reflection and they
finally agreed on the answer. In this instance the problem of achieving a
consensus provided the basis for their mathematical activity.
Andy: Seven sets of 5.
John:
Andy and John: (simultaneously) 35.
John:
John appeared amused and pleased that they said the answer simultaneously, as
though that were the height of cooperation.
The above excerpts illustrate that the coordination of mathematical activity
can take a variety of forms. Notice that nowhere in the dialogue did either
partner make explicit reference to their obligation to cooperate. (Typically such
comments are made only when cooperation breaks down.) Yet it is apparent
that in every instance they persisted until each had constructed a viable method
that produced a common answer. The smooth flow of their discussions indicates
that they had mutually constructed a viable network of obligations and expectations.
For example, each knew when it was appropriate to explain or to justify a
solution without being explicitly asked to do so. The two children’s mathematical
discussions, which are characteristic of this level of cooperation, gave rise to
a variety of learning opportunities that are unique to the smaII group instructional
strategy.
The level of cooperation illustrated by John and Andy was made possible by
their ability to make sense of each other’s mathematical activity. Cases where a
child, for one reason or another, is unable to make sense of his or her partner’s
mathematical activity make it impossible for such cooperation to occur. This in
turn makes it more difficult for the children to cooperate at the basic social
level. In the episode presented below the method used by one partner was
incomprehensible to the other. Consequently, cooperation at the level of
developing a mutually acceptable problem solution was impossible. Each
worked independently to complete the instructional activities because they were
unable to talk to each other about their mathematics. What dialogue there was,
necessarily focused on whether or not they had the same answer and on the
more basic level of cooperation. In this episode the partners were working on a
multiplication activity. One child was using an (incorrect) algorithm to solve
the problems. Her partner, unable to make sense of her approach, chose to solve
the problems by drawing circles of tally marks and then counting. As a
consequence, it took him longer to generate an answer to each task.
5 x- = 35.
The final task in this excerpt is 7 x 5 = -. The dialogue is very brief.
Oh, it’s just 5 lower than that. It’s ...
We both said it at the same time.
APPROACH TO SECOND GRADE MATHEMATICS 173
Will you wait for me? Five times something equals 35. I already know the
answer to that. Five times five, I think. Let me check it again. (At this point
Adam draws five circles of five.) Seven.
Adam:
4 x 7=-
Adam:
Ann:
Adam:
Ann:
Adam:
Adam checks the answer 28 again and they agreed to give that as the answer.
Four times seven equals something. Let me draw these here. You can go ahead.
(He draws four groups of seven.) Oh, one more and then I’ll be caught up.
(Writes 22 as her answer on the activity page.) Do you agree?
Hold on. You’re going ahead.
I hate waiting for you.
I got 28 on that one but hold on.
Notice in the above dialogue that the children did not discuss their mathematics.
At best, they could compare answers and share the activity page and any
other materials they had. As a consequence, they continually had to resolve the
social problem of synchronizing their independent activities. Adam, for
example, continued to insist that Ann should wait for him and that they should
at least work on the same activity. Ann reluctantly obliged; she too understood
that they were expected to work cooperatively and to agree on answers. Despite
their different mathematical understandings, they attempted to maintain a basic
level of social cooperation. To do so, they had to continually negotiate cooperation
at this level verbally. This dialogue contrasts sharply with that between
John and Andy, who cooperated fully but made no verbal reference to it.
In the project classroom the social relationships the children negotiated were
influenced by, but not determined by, their level of conceptual understanding of
mathematics. For example, partners who were at approximately the same
conceptual levels could readily make sense of each other’s mathematical
thinking. Thus, cooperation at the level of developing mutually acceptable
problem solutions was facilitated. John and Andy exemplify this type of
relationship. Partners at widely different conceptual levels could cooperate only
if the more advanced partner operated far below his or her capabilities and
discussed mathematics at the level of the less advanced partner, or if partners
agreed to work independently on the instructional activities and compare
answers. Although the first option presents opportunities for the more advanced
child to serve as a tutor and can give rise to learning opportunities, in general it
denies that child opportunities for conceptual advancement. In the project
classroom partners who evidenced such a relationship were reassigned.
Some children who were at very different conceptual levels did develop very
productive working relationships. For example, in one pair the child who was
less advanced conceptually was very task-involved and continually requested
assistance from his partner. It was common for these children to work individually
on the mathematics tasks and then compare their answers. Dialogue
in this group, as in the Adam/Ann episode presented earlier, frequently focused
on negotiating a basic level of cooperation.
The previous discussion and examples illustrate that the social interactions
174 PAUL COBB, TERRY WOOD, AND ERNA YACKEL
between partners influence their mathematical activity and give rise to learning
opportunities. The interactions are in turn influenced by their conceptual
understanding and by the network of obligations and expectations mutually
constructed both within the group and in the wider whole class setting. Our
observations in the project classroom indicate that, as early as second grade,
children are remarkably successful at working together to complete instructional
activities. In doing so, they solve not only their mathematics problems
but also the problem of how to work together.
CONCLUSIONS
The teaching experiment was reasonably successful in that we were able to
satisfy the school district administrators’ expectations while allowing the
children to solve their mathematical problems in ways that were acceptable to
them. It also became apparent that the children’s abilities to establish productive
social relationships and to verbalize their own thinking improved dramatically
as the year progressed. However, most observers seemed to attribute the
greatest significance to the emotional tone of the classroom. In general, the
children were enthusiastic, persistent, did not become frustrated, and experienced
joy when they solved a personally challenging problem. We have, for
example, been unable to identify a single instance when a child evidenced
frustration while problem solving during the second semester of the school
year. Further, the children’s persistence was such that we found it necessary to
allocate three consecutive one hour class periods to general problem solving
whenever these instructional activities were used. The children requested
additional time because they wanted to continue working on an activity they
had been unable to complete during an entire class period.
We have tried to make the point that the teacher was very much an authority
in the project classroom, albeit a benevolent one. Her success in placing the
children under certain obligations while readily accepting compatible obligations
for herself was crucial to her effectiveness. As a constructivist teacher, she
did not merely refrain from carrying out certain activities characteristic of
traditional teachers and relinquish her authority. Rather, she expressed that
authority in action by initiating the mutual construction of certain obligations
and expectations. In doing so, she influenced the children’s beliefs about both
the nature of the activity of doing mathematics and their own and the teacher’s
roles in the classroom. Above all else, the obligations and expectations that
were established constituted a trusting relationship. The teacher trusted the
children to resolve their problems and they trusted her to respect their efforts.
This trust is, in our opinion, the most important feature of constructivist
teaching.
III.
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