The Development of Arithmetic Sense and Its Predictive Relationship to Mathematical Achievement

Abstract

Cognitive psychologists and mathematics education researchers have discovered that people employ a range of diverse arithmetic strategies to efficiently (i.e., quickly and accurately) solve arithmetic problems. To date, psychological studies of arithmetic problem solving have emphasized arithmetic fluency—the use of direct strategies like fact retrieval when solving simple, conventional problems. Comparatively little attention has been focused on arithmetic sense, which I define as the adaptive use of direct and indirect strategies like decomposition when solving complex, novel problems. This dissertation aims to articulate the new construct of arithmetic sense, investigate the development of this construct in elementary school children, and evaluate its predictive relationship to mathematical achievement among elementary and college students. Experiment 1 was developmental in nature and utilized paper-and-pencil classroom assessments. This experiment focused on 4th and 6th graders to investigate the development of arithmetic fluency and arithmetic sense and to evaluate their predictive relationship to mathematical achievement. The results showed that (1) both arithmetic fluency and arithmetic sense improve with grade level and (2) arithmetic sense is the best single predictor of mathematics achievement and pre-algebra skills in older elementary school children (i.e., is better than arithmetic fluency). Experiment 2 was cognitive in nature and utilized computerized mathematical cognition tasks. This experiment focused on college students to identify the best predictor of mathematical achievement among indices of (1) number sense as measured by the distance effect in the symbolic number comparison task, (2) arithmetic fluency as measured by the problem size effect in the arithmetic fact verification task, and (3) arithmetic sense as measured by the algebraic principles effect in the arithmetic sentence verification task. The results showed (1) that arithmetic sense was a significant predictor of ACT mathematics scores and (2) that this predictive relationship held even after controlling for ACT English scores, number sense, and arithmetic fluency. Taken together, the current experiments provide insights into the various direct and indirect strategies underlying arithmetic problem solving and establish a predictive relationship between arithmetic sense and mathematical achievement in children and adults. The scientific result is a better understanding of the mental representations and cognitive processes underlying numerical, arithmetic, and algebraic thinking. The educational significance is opening a new pathway for improving school-aged students’ mathematical achievement and algebraic skills.