## Abstract

In 2 studies (Ns = 55 and 54), the authors examined a basic form of conceptual understanding of rational number arithmetic, the direction of effect of decimal arithmetic operations, at a level of detail useful for informing instruction. Middle school students were presented tasks examining knowledge of the direction of effects (e.g., "True or false: 0.77 * 0.63 > 0.77"), knowledge of decimal magnitudes, and knowledge of decimal arithmetic procedures. Their confidence in their direction of effect judgments was also assessed. The authors found (a) most students incorrectly predicted the direction of effect of multiplication and division with decimals below 1; (b) this pattern held for students who accurately compared the magnitudes of individual decimals and correctly executed decimal arithmetic operations; (c) explanations of direction of effect judgments that cited both the arithmetic operation and the numbers' magnitudes were strongly associated with accurate judgments; and (d) judgments were more accurate when multiplication problems involved a whole number and a decimal below 1 than with 2 decimals below 1. Implications of the findings for instruction are discussed.

Original language | English |
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Pages (from-to) | 374-386 |

Number of pages | 13 |

Journal | Journal of Educational Psychology |

Volume | 109 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Apr 2017 |

### Bibliographical note

This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details## Keywords

- Conceptual knowledge
- Decimal
- Mathematical cognition
- Mathematical development
- Rational number arithmetic