Proximity and precedence in arithmetic

Abstract

How does the physical structure of an arithmetic expression affect the computational processes engaged in by reasoners? In handwritten arithmetic expressions containing both multiplications and additions, terms that are multiplied are often placed physically closer together than terms that are added. Three experiments evaluate the role such physical factors play in how reasoners construct solutions to simple compound arithmetic expressions (such as “2 + 3 × 4”). Two kinds of influence are found: First, reasoners incorporate the physical size of the expression into numerical responses, tending to give larger responses to more widely spaced problems. Second, reasoners use spatial information as a cue to hierarchical expression structure: More narrowly spaced subproblems within an expression tend to be solved first and tend to be multiplied. Although spatial relationships besides order are entirely formally irrelevant to expression semantics, reasoners systematically use these relationships to support their success with various formal properties.

Keywords:

• Symbolic reasoning
• Mathematical cognition
• Embodied cognition

Acknowledgments

This research was funded by Department of Education, Institute of Education Sciences Grant R305H050116 and National Science Foundation REESE Grant DRL-0910218. We thank Lydia Nichols and three anonymous reviewers for comments and criticism.