TY - JOUR

T1 - Reversible reasoning in fractional situations

T2 - Theorems-in-action and constraints

AU - Ramful, Ajay

N1 - Includes bibliographical references.

PY - 2014/3

Y1 - 2014/3

N2 - The aim of this study was to investigate, at a fine-grained level of detail, the theorems-in-action deployed and the constraints encountered by middle-school students in reasoning reversibly in the multiplicative domain of fraction. A theorem-in-action (Vergnaud, 1988) is a conceptual construct to trace students’ reasoning in a problem solving situation. Two seventh grade students were interviewed in a rural middle-school in the southern part of the United States. The students’ strategies were examined with respect to the numerical features of the problem situations and the ways they viewed and operated on fractional units. The results show that reversible reasoning is sensitive to the numeric feature of problem parameters. Relatively prime numbers and fractional quantities acted as inhibitors preventing the cueing of the multiplication–division invariant, thereby constraining students from reasoning reversibly. Among others, two key resources were identified as being essential for reasoning reversibly in fractional contexts: firstly, interpreting fractions in terms of units, which enabled the students to access their whole number knowledge and secondly, the unit-rate theorem-in-action. Failure to conceptualize multiplicative relations in reverse constrained the students to use more primitive strategies, leading them to solve problems non-deterministically and at higher computational costs.

AB - The aim of this study was to investigate, at a fine-grained level of detail, the theorems-in-action deployed and the constraints encountered by middle-school students in reasoning reversibly in the multiplicative domain of fraction. A theorem-in-action (Vergnaud, 1988) is a conceptual construct to trace students’ reasoning in a problem solving situation. Two seventh grade students were interviewed in a rural middle-school in the southern part of the United States. The students’ strategies were examined with respect to the numerical features of the problem situations and the ways they viewed and operated on fractional units. The results show that reversible reasoning is sensitive to the numeric feature of problem parameters. Relatively prime numbers and fractional quantities acted as inhibitors preventing the cueing of the multiplication–division invariant, thereby constraining students from reasoning reversibly. Among others, two key resources were identified as being essential for reasoning reversibly in fractional contexts: firstly, interpreting fractions in terms of units, which enabled the students to access their whole number knowledge and secondly, the unit-rate theorem-in-action. Failure to conceptualize multiplicative relations in reverse constrained the students to use more primitive strategies, leading them to solve problems non-deterministically and at higher computational costs.

KW - Division

KW - Fraction

KW - Multiplicative reasoning

KW - Reversibility

KW - Vergnaud's theory

KW - Units

U2 - 10.1016/j.jmathb.2013.11.002

DO - 10.1016/j.jmathb.2013.11.002

M3 - Article

VL - 33

SP - 119

EP - 130

JO - Journal of Mathematical Behavior

JF - Journal of Mathematical Behavior

SN - 0732-3123

ER -