Constructing unitizing: the critical strategies and models that build this essential mathematical concept
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The focus of this longitudinal case study was to investigate the progressive development of unitizing in a cohort of students receiving reform-oriented mathematics instruction. One-on-one videotaped mathematics interviews were conducted twice annually for 4 years from Grade 1 to Grade 4 with a varying number of participants from 61 to 45 respectively. Multiplication and quotative division questions were analyzed for correctness, the physical model used to solve the problem, and the computation strategy used to solve the problem. Multiplication and adding up for division models that required the development of more sophisticated unitizing included modelling only one group, modelling just the groups, modelling the groups with the composite numeral, and modelling the new whole. Multiplication and adding up for division strategies that indicated the development of a unitizing structure included rhythmic counting, starting with a doublet, skip counting, regrouping to form a new composite, and splitting the composite and then iterating the sub-composites to find the total. The varying levels of simultaneity demonstrated by students were also noted as they pertain to the development of a unitizing structure. A theoretical landscape of the development unitizing is proposed.