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

Relational Thinking and Fractions 28
We have further argued that Relational Thinking is a critical precursor – perhaps
the most critical – to learning algebra with understanding, because if children understand
the arithmetic that they learn, then they are better prepared to solve problems and
generate new ideas in the domain of algebra. However, Relational Thinking is almost
entirely neglected in typical U.S. classrooms with the unfortunate result that children
experience all types of learning difficulties as they move beyond arithmetic into learning
algebra. Some proposed solutions focus on a renewed emphasis on prerequisite skills
(e.g., U.S. Department of Education), while others emphasize the use of concrete
materials and models (e.g., Lesh, Post, & Behr, 1987). We have presented an alternative
view of how to address these difficulties that centers on cultivating children’s implicit use
of fundamental properties of the real-number system to solve arithmetic problems, to
better align the concepts and skills learned in arithmetic and algebra. At the heart of this
view is the reciprocal relationship between arithmetic and algebra as it is revealed in
children’s reasoning about quantity.