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This study examined the role of student generated drawings to offload cognitive demands of a mathematical problem. We used Unit Transformation Graphs to compare students' thought processes when they had to solve the problem mentally, and when they were allowed to use pen and paper. The results indicated that the possibility to rely on drawings helped the participants to free up working memory resources and complete a cognitively demanding fractional task. [For the complete proceedings, see ED630210.]

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. [For the complete proceedings, see ED629884.]

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. [For the complete proceedings, see ED629884.]

The Complex Connections: Reimagining Units Construction and Coordination working group has been running since PMENA in 2018. Previous years have focused on developing a website to share research articles and share completed work and facilitating collaboration between researchers and educators regarding students' units construction and coordination. This goal aligned with Steffe's (2017) call for work focused collaboration between researchers and educators to advance units construction and coordination for all learners. The group has grown beyond the initial topic of individuals' units coordination in whole number and fraction settings to discuss units coordination in algebraic and covariational reasoning. [ABSTRACT FROM AUTHOR]

Units Coordination and Covariational Reasoning are powerful frameworks for modeling students' mathematics in arithmetic reasoning and construction of relationships between changing quantities, respectively. This case study of an advanced stage 2, 8th-grade algebra student, Daniel, investigated connections between his units coordination and covariational reasoning on non-graphical covariation tasks. Results show Daniel leveraged his units coordination structures to reason about how two quantities varied together in several distinct ways. From Daniel, new insight was gained into underlying mental structures and actions involved in Carlson and colleagues' (2002) covariational reasoning framework. Implication for engaging a diversity of learners is included. [ABSTRACT FROM AUTHOR]

Units coordination, defined by Steffe (1992) as the mental distribution of one composite unit (i.e., a unit of units) "over the elements of another composite unit" (p. 264) is a powerful tool for modeling students' mathematical thinking in the context of whole number and fractional reasoning. This paper proposes extending the idea of a numerical unit to an algebraic unit and a covariational unit. Evidence to support this extension is taken from two qualitative studies conducted with middle-grades students. Results suggest the coordination of two unknowns in algebraic contexts is limited for students who assimilate with composite units. Additionally, in the context of covariational reasoning, results describe a new partitioned unit that students who assimilate with composite units may apply. Implications for engaging all learners are discussed. [ABSTRACT FROM AUTHOR]

Units construction and coordination is a powerful tool for modeling mathematical thinking about whole numbers and fractions. This working group will support collaboration of mathematics educators to make connections between foundational contexts for units coordination, and emerging contexts such as combinatorial reasoning, mathematical knowledge for teaching, and communicating mathematical ideas in our daily lives. [ABSTRACT FROM AUTHOR]

The Complex Connections: Reimagining Units Construction and Coordination working group began at PMENA 2018, with the aim of facilitating collaboration amongst researchers and educators sharing Steffe's (2017) concerns about units construction and coordination for all learners. This working group made dramatic progress in their ability to conceptualize their work and impact the mathematics education field in the form of publications and task development. [ABSTRACT FROM AUTHOR]

This study examined the role of student generated drawings to offload cognitive demands of a mathematical problem. We used Unit Transformation Graphs to compare students' thought processes when they had to solve the problem mentally, and when they were allowed to use pen and paper. The results indicated that the possibility to rely on drawings helped the participants to free up working memory resources and complete a cognitively demanding fractional task. [ABSTRACT FROM AUTHOR]