**Article:**
Yackel, E., Cobb, P., & Wood, T. (1991) Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22 (5), 390-408.

**Quotes and Comments:**

"The salient features of the project for study are these. First, instruction was provided by the regular classroom teacher in the regular classroom setting. Second, the duration of the study was an entire school year. Third, a cooperative learning approach was used for all aspects of second-grade mathematics instruction including computational activities, usually thought of as skill and practice activities, and other traditional second-grade topics such as time, money, and measurement. This third feature is especially important, since small-group problem solving in mathematics education has come to mean that students work together on word problems or nonroutine problems. Our approach, however, was to develop instructional activities so they were likely to give rise to problematic situations for the children, rather than be treated as exercises for practice or drill... Finally, the classroom norms for cooperation were mutually constructed in the context of working on mathematics activities and in the absence of an
external reward system for either individuals or groups"
(390-1).

"We set out initially to form pairs of a variety of types to
gain information about influences that might be relevant in
developing productive working relationships. Accordingly, we
formed pairs that were homogeneous with respect to mathematical
development and pairs that were heterogeneous in this regard,
and we were careful to include male-male, female-female, and
male-female pairings" (392).

Further on in the article, the authors evaluate the results of
the homogeneous and heterogeneous ability groupings: "[I]n
groups where children were of different mathematical conceptual
levels the teacher sometimes modified the obligation that they
should reach a consensus about solution methods because she
realized that this was an impossibility, given the disparity in
their mathematical understandings. Instead, the children had to
agree only about an answer, and it was acceptable for the more
advanced child to construct a solution method that the other
could not understand provided the weaker child had also
constructed a solution. Although this appears to suggest that
children of different mathematical conceptual levels should not
be partners, there are other factors that make such
partnerships viable. For example, one such pair in the project
classroom formed a productive working relationship because the
weaker child continually drew his partner back into active
involvement... The partnership proved mutually beneficial
because of the social relationship that developed between the
two children" (400).

"...[T]eaching through problem solving acknowledges that
problems arise for students as they attempt to achieve their
goals in the classroom. In this approach students are seen as
the best judges of what they find problematic and they are
encouraged 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 attempt to make sense of a situation in terms of their
current concepts and procedures, accounting for a surprise
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, developing a framework that accommodates
alternative solution methods, and formulating an explanation to
clarify another child's solution attempt" (394-5).

Thus the authors believe that social situations are crucial in
mathematics instruction, especially with regard to allowing the
child to construct and modify his or her own mathematical
knowledge.

"In the project classroom, the teacher initiated and guided the
mutual construction of a variety of social norms as explained
below. These included the following: that students cooperate to
solve problems, that meaningful activity is valued over correct
answers, that persistence on a personally challenging problem
is more important than completing a large number of activities,
and that partners should reach consensus as they work on
activities... In addition to norms for social cooperation,
there were classroom norms for individual activity. These
included the following: that children figure out solutions that
are meaningful to them, that they explain their solution
methods to their partner, and that they try to make sense of
their partner's problem-solving attempts" (397-8).

On pp. 398-99, the authors discuss two instructional methods used
by the teacher to create and reconfigure these norms. In some cases
she would use an incident that had occurred in the classroom to
illustrate a general class expectation, in others she would explicitly
introduce a norm to the whole class - drawing the children into a
discussion of the particular obligation and the reasons for its existence.

"Children engage in two types of problem solving as they work
together in small groups. On the one hand, they attempt to
solve their mathematical problems, and on the other hand, they
have to solve the problem of working productively together, as
discussed above. Once social problems have been temporarily
resolved, the interactions that take place give rise to
opportunities for learning that result directly from the
interactions. As the children work together and strive to
communicate, opportunities arise naturally for them to
verbalize their thinking, explain or justify their solutions,
and ask for clarifications. Further, attempts to resolve
conflicts lead to both the opportunity to reconceptualize a
problem and thus construct a framework for another solution
method, and the opportunity to analyze an erroneous solution
method and provide a clarifying explanation. These types of
discussion-based learning opportunities do not typically occur
in traditional classroom settings and are qualitatively
different from the opportunities that do arise in traditional
settings where children typically do not engage in mathematical
discussions with each other but work individually to complete
many exercises by repeating a method demonstrated by the
teacher" (401-2).

- summary by Jane Ehrenfeld