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Despite the importance of multiplicative reasoning (MR) in mathematics, there is limited empirical study of the role of MR in multi-digit multiplication tasks. This study examined children’s multiplicative reasoning within their use of partial products. This mixed methods study of 16 fifth-grade students revealed that half of participants did not attend to partial products, and those that did tended to operate with the second or third multiplicative concept. 

Abstract Quantitative measures in mathematics education have informed policies and practices for over a century. Thus, it is critical that such measures in mathematics education have sufficient validity evidence to improve mathematics experiences for students. This article provides a systematic review of the validity evidence related to measures used in elementary mathematics education. The review includes measures that focus on elementary students as the unit of analyses and attends to validity as defined by current conceptions of measurement. Findings suggest that one in ten measures in mathematics education include rigorous evidence to support intended uses. Recommendations are made to support mathematics education researchers to continue to take steps to improve validity evidence in the design and use of quantitative measures. 

Abstract This study sought to explore math and science teacher educators' use of various media to represent practice within methods courses. There is little understanding of why certain media is used over other representations and the rationale for these choices. Specifically, the study focused on the prevalence and familiarity of teacher educators with comics and animations, standard videos, and 360 videos. This mixed methods study utilized a survey and interviews to ascertain math and science teacher educators' level of familiarity and perceived usefulness of representations of practice. Results indicate that standard video is by far the most used representation of practice in methods classes with three themes explaining this finding: access to representations of practice, dimensions of representation, and pertinacity of using representations. Familiarity with representations of practice relates to teacher educators' perceptions of access thereby indicating a need for teacher educators to have better access to representations. Implications of this study include supporting current literature about the relationship between the level of familiarity and perceived usefulness in media along with the potential need for a central platform that houses these representations of practice resources for teacher educators. 

Use of video as a representation of practice in teacher education is commonplace. The current study explored the use of a new format (360 video) in the context of preservice teachers’ professional noticing. Findings suggest that preservice teachers viewing 360 videos attended to more student actions than their peers viewing standard video. In addition, using a virtual reality headset to view the 360 videos led to different patterns in where preservice teachers looked in the recorded classroom, and to increased specificity of mathematics content from the scenario. Thus, findings and results support the use of 360 video in teacher education to facilitate teacher noticing. However, future research is needed to further explore this novel technology. 

Multiplication and division are vital topics in upper level elementary school. A teacher’s pedagogical content knowledge (PCK) influences both instruction and students’ learning. However, there is currently little research examining teachers’ PCK within this domain, particularly regarding professional education of future teachers. To help address this need, the present paper presents an initial validity argument for a survey of preservice teacher’s PCK for multiplication and division.