How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

Purav Patel, Sashank Varma

Research output: Contribution to journalArticlepeer-review

Abstract

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self-reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.

Original languageEnglish (US)
Pages (from-to)1642-1676
Number of pages35
JournalCognitive Science
Volume42
Issue number5
DOIs
StatePublished - Jul 2018

Bibliographical note

Publisher Copyright:
© 2018 Cognitive Science Society, Inc.

Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

Keywords

  • Arithmetic
  • Irrational number
  • Magnitude
  • Number line
  • Perfect square
  • Radicals
  • Referential processing
  • Strategy

Fingerprint Dive into the research topics of 'How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares'. Together they form a unique fingerprint.

Cite this