Spontaneous Focusing on What and Why? What Children's Imprecise Responses Reveal about Their Mathematical Thinking and Relational Reasoning

In this study, we analyzed the imprecise (i.e., less mathematically precise) responses that 148 third- to fifth-grade Chinese students made on selected-response problems that were part of a spontaneous mathematical focusing task, the Quantitative Relations Test for Chinese Children (QRTC[superscript 2]). The purpose for this analysis was to ascertain whether the students' spontaneous responses represented random selections or followed discernible patterns. Those patterns were based on the specific attributes (i.e., featural, mathematical, numerical, and multiplicative) of the imprecise-response options for selected-response items on the QRTC[superscript 2]. Nonrandom selection patterns were identified by means of chi-square analysis and compared to patterns for the constructed-response portion of the QRTC[superscript 2]. For the chi-square analysis, the category involving imprecisions in both numerical and multiplicative attributes, and the category simultaneously involving numerical, multiplicative, and mathematical were significant. The analysis for the constructed problems also showed that children had difficulty describing the changes in the numerical and multiplicative attributes. Overall, the task of fully and accurately writing what they observed in the constructed-response problem on the QRTC[superscript 2] proved challenging for the children. Finally, imprecise response patterns the children exhibited on the QRTC[superscript 2] and their relational reasoning ability, as measured by the Test of Relational Reasoning-Junior (TORRjr) were significantly correlated. That significant correlation was attributable largely to the association between the children's analogical reasoning abilities and their QRTC[superscript 2] performance. We consider the theoretical, empirical, and practical implications of this work for mathematical learning and development.