Towards a Unified Theory of Rational Number Arithmetic

• In Event: Poster Session 05
  In Poster Session: PS 05 Topic - Cognitive Processes

Abstract

Rational numbers are among the most important topics children encounter in early math education. Rational number knowledge predicts algebra skill and high school math achievement (Booth & Newton, 2012; Siegler et al., 2012). However, many children struggle with rational numbers (Torbeyns, Schneider, Xin, & Siegler, 2015). These difficulties interfere with subsequent math learning and development (Hoffer, Venkataraman, Hedberg, & Shagle, 2007).
The present study aimed to advance our understanding of a particular aspect of rational number knowledge: decimal arithmetic. The study built on a theory of children’s procedural knowledge of fraction arithmetic (Authors, 2017). One goal was to test whether and how well key predictions of that theory generalize to decimal arithmetic. Another goal was to clarify, in the context of decimal arithmetic, how procedural knowledge interacts with conceptual knowledge, which was not included in the previous theory.
To achieve these goals, participants (N = 92 sixth and eighth graders) performed a problem-solving task and an explanation task. In the problem-solving task, children solved 12 decimal arithmetic problems while thinking aloud. In the explanation task, children were asked to justify decimal arithmetic procedures and explain why incorrect solutions were incorrect.
Three results confirmed predictions based on the previous theory. (1) Accuracies on different problem types paralleled problem distributions in textbooks, consistent with the assumption that the likelihood of selecting an appropriate procedure for a problem depends on how often similar problems have been encountered previously. (2) Most (70%) errors resulted from using strategies that would be appropriate for some problems to solve problems for which the strategies were not appropriate, consistent with the assumption that strategy over-generalization is the primary source of errors. (3) Children displayed four patterns of strategy use analogous to ones observed in fraction arithmetic (Authors, 2019) and predicted by the theory based on the assumption that continuous parametric variation among children generates qualitatively distinct strategy use patterns.
Three other analyses investigated spontaneous and prompted uses of conceptual knowledge. (4) A majority (79%) of children used conceptual knowledge when justifying procedures or explaining why incorrect procedures were incorrect, whereas a minority (36%) overtly used conceptual knowledge when solving problems. (5) These two types of conceptual knowledge use were positively associated with each other and with accuracy on the problem-solving task. (6) Children who frequently displayed doubt were more likely to use conceptual knowledge when solving problems.
The findings provide evidence that key assumptions of the previous theory of fraction arithmetic apply to decimal arithmetic as well as to fraction arithmetic. The results also demonstrate overt reliance on conceptual knowledge during routine problem-solving by a substantial minority of children, and suggest that lack of confidence with procedures (combined with having relevant conceptual knowledge) may increase the likelihood of such spontaneous conceptual knowledge use. Finally, the findings provide an empirical basis for a unified theory of rational number arithmetic that accounts for both fraction and decimal arithmetic and the role of conceptual knowledge in learning and using rational arithmetic procedures.

Authors

• David Braithwaite, Florida State University
  Presenting Author

• Robert S Siegler, Columbia University
  Non-Presenting Author