The distributive property plays a pivotal role in advancing students' understanding of multiplication, enabling the decomposition of problems and the acquisition of new facts. However, this property of multiplication is difficult for students to understand. We used two unique data sets to explore middle school students' use of the distributive property. Study 1 involved data from 1:1 structured interviews of students (N = 24) discussing worked examples and solving associated practice problems. We examined whether or not students used the distributive property to solve the problems and whether or not interviewers followed the recommended distributive property prompts or defaulted to more conventional methods. Despite exposure to worked examples using the distributive property and a protocol calling for attention to it, students and interviewers favored methods like PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction) or long multiplication. Study 2 used a data set with middle school students' (N = 131) item -level responses on Kirkland's (2022; doctoral dissertation, University of Notre Dame) Brief Assessment of Mature Number Sense along with several related measures of domain -general and domain -specific skills. We extracted problems involving the distributive property for analysis. Surprisingly, there was no evidence that students' use of the distributive property improved from sixth grade to eighth grade. However, both gradelevel mathematics achievement and cognitive reflection uniquely predicted the correct use of the distributive property. Results suggest that middle school students who exhibit stronger reflective thinking tend to perform better on distributive property problems. Findings highlight cognitive reflection as a potentially important construct involved in the understanding and use of the distributive property. (c) 2024 Elsevier Inc. All rights reserved.

Students with mature number sense make sense of numbers and operations, use reasoning to notice patterns, and flexibly select the most effective and efficient problem-solving strategies (McIntosh et al., 1997; R. Reys et al., 1999; Yang, 2005). Although national standards and policy documents (e.g., CCSS, 2010; NCTM, 2000, 2014) emphasize the importance of number sense, limited research exists on the association between students' mature number sense and other important psychological constructs in mathematics education, such as grade-level mathematics achievement. The present study addressed this gap through a longitudinal design, advancing fundamental knowledge of number sense as a construct. At the start of the school year, middle school students (N = 129 at Time Point 1) completed measures of mature number sense and several related constructs, including grade-level mathematics achievement, executive functioning, and fraction and decimal computation. Students returned in the spring (N = 115 at Time Point 2) to complete an end-of-year mathematics achievement assessment as a measure of grade-level content learned during the year. We found mature number sense to be measurably distinct from related constructs and uniquely predictive of students' grade-level mathematics achievement at the end of the school year, controlling for their beginning-of-year mathematics achievement.

This Brief Report presents an example of assessment validation using an argument -based approach. The instrument we developed is a Brief Assessment of Students' Mature Number Sense, which measures a central goal in mathematics education. We chose to develop this assessment to provide an efficient way to measure the effect of instructional practices designed to improve students' number sense. Using an argument -based framework, we first identify our proposed interpretations and uses of student scores. We then outline our argument with three claims that provide evidence connecting students' responses on the assessment with its intended uses. Finally, we highlight why using argument -based validation benefits measure developers as well as the broader mathematics education community.

Students who exhibit mature number sense make sense of numbers and operations, use reasoning to notice patterns, and flexibly choose effective problem-solving strategies (McIntosh et al., 1997, https://ro.ecu.edu.au/ecuworks/6819). Due to its dispositional nature, mature number sense is typically measured through in-depth interviews or tests of strategy usage. Yet, the lack of an efficient, rigorously developed measure has made it difficult to collect systematic, replicable evidence on students' mature number sense. To address this, we developed a brief assessment of mature number sense. The present study provides additional convergent evidence of validity for this measure with US students in grades 3-8 (8-14 years old). We compared middle school (N = 40) and upper elementary school (N = 41) scores from the brief assessment with an established, time-intensive measure (Yang, 2019, https://doi.org/10.1007/ s10649-018-9874-8) and an in-depth interview of student strategy usage (Markovits & Sowder, 1994, https://doi.org/10.2307/749290). We found strong correlations (r > 0.7) across all three measures, and this held even when controlling for students' arithmetic scores (pr > 0.6). Researchers and educators can now use the brief assessment to investigate students' mathematical thinking and advance knowledge of a key aspect of mathematical cognition.