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We report on and validate a system for ranking the cognitive demand of mathematical tasks. In our framework, task rankings are determined by the sequences of units and unit transformations students might use to solve each task. Using this framework, we ranked a set of 10 fractions tasks. We then interviewed 12 pre-service teachers to assess the validity of the ranking system. Results validate the task ranking system by demonstrating that increases in task ranking predict increases in the cognitive demand experienced by the pre-service teachers, as evidenced by their responses to the tasks. These results hold implications for instruction that maintains appropriate cognitive demand and future research that models students' mathematics. [ABSTRACT FROM AUTHOR]

We introduce a methodology for diagramming the ways students use sequences of mental actions to solve mathematical tasks. We studied 12 pre-service teachers as they solved a set of fractions tasks, ranked by cognitive demand. We present the unit transformation graphs for one of those pre-service teachers, to illustrate how she experienced and met cognitive demand across the fractions tasks. Specifically, the graphs illustrate how sequencing mental actions places demands on working memory and how units coordination structures can offload some of that demand. [ABSTRACT FROM AUTHOR]

Understanding student cognition has been a crucial part of mathematics educational research for decades. The construct of units coordination has proven to be central in students’ mathematical development. The psychological construct of working memory has also been shown to have an important role in cognitive development. This report combines units coordination and working memory to create a new theoretical construct called unit transformation capacity, aimed to capture nuances in modeling students’ mathematics. [ABSTRACT FROM AUTHOR]

Recent research has highlighted the developmental importance of units coordination for students' continued mathematical learning. In this study, we introduce a task that ostensibly involves four levels of units. We examine student responses to this task and how they relate to student responses from tasks that assess students' coordination of three levels of units. This investigation helps to (1) characterize students' coordination of four levels of units in activity and, (2) explore the potential to use such activity as a valid indicator of students' interiorized coordination of three levels of units. Results of the study provide preliminary evidence for the validity of the task because it positively relates to another measure of units coordination and positively predicts students' construction of an iterative fraction scheme. [ABSTRACT FROM AUTHOR]

The article discusses a study on written assessments of units coordination (UC) in elementary mathematics learning. The study examined UC that occurs in a variety of contexts, including fractional and whole number continuous contexts. It indicated the participation of 96 sixth-grade students in the first iteration of the assessment.

The article presents a study which investigates the association of complexity of additive reasoning of sixth-grade students to the complexity of their multiplicative reasoning. Students whose zones of potential constructions (ZPCs) included constructing multiplicative comparisons were chosen in activity but not using them to assimilate situations. It showed that the additive complexity within every pair's ZPC were either greater than or comparable to the multiplicative complexity in their ZPC.