The Algebraic Nature of Fractions: Developing Relational Thinking ...

Relational Thinking and Fractions 22
structure in the execution of standard algorithms as they are typically learned which can
be summarized as “do next.”
Further, the thinking displayed by these children resembles the “competent
reasoning” that proficient mathematical thinkers use to compare rational numbers, as
reported by Smith (1995). Based on an analysis of 30 students’ solutions to order and
equivalence problems, Smith argued that competent reasoning is characterized by the use
of strategies that exploit the specific numerical features of a problem and often apply only
to a restricted class of fractions. These strategies were reliable and efficient. He
contrasted this reasoning with the use of generalized, all purpose strategies – such as
conversion to a common denominator to compare fractions – which the students in his
study tended to use as a last resort. 5 These findings led Smith to conclude “the analysis of
skilled reasoning with rational numbers should … move beyond a focus on particular
strategies to examine the character of the broader knowledge system that has those
strategies as components” (p. 38). We are proposing that this system consists of
children’s informal algebra of fractional quantities and that it is expressed in children’s
Relational Thinking.
From this perspective, Jill and Keenan represent students who are in the process
of developing a knowledge base for reasoning about fractions in ways that can be
characterized as proficient and competent. Moreover, this knowledge base is integrated
with properties of whole-number operations and relations and anticipates the algebra of
generalized quantities, typically taught in the eighth or ninth grade. We are not
5 This approach to numerical reasoning is not unique to children. Dowker (1992) reported
that professional mathematicians prefer to approach computational estimation in the same
ways, that is, by exploiting specific numerical features of a problem rather than using a
generalized algorithm that works in all cases.