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Constructing multiplicative reasoning is critical for students’ learning of mathematics, particularly throughout the middle grades and beyond. Tzur, Xin, Si, Kenney, and Guebert [American Educational Research Association, ERIC No. ED510991, (2010)] conclude that an assimilatory composite unit is a conceptual spring to multiplicative reasoning. This study examines patterns in the percentages of students who construct multiplicative reasoning across the middle grades based on their fluency in operating with composite units. Multinomial logistic regression models indicate that students’ rate of constructing an assimilatory composite unit but not multiplicative reasoning in sixth and seventh grades is significantly greater than that in eighth and ninth grades. Furthermore, the proportion of students who have constructed multiplicative reasoning in sixth and seventh grades is significantly less than the proportion of those who have constructed multiplicative reasoning in eighth and ninth grades. One implication of this is the quantitative verification of Tzur, Xin, Si, Kenney, and Guebert’s (2010) conceptual spring. That is, students who construct assimilatory composite units early in the middle grades are likely to construct multiplicative reasoning; students who do not construct assimilatory composite units early in the middle grades likely do not construct multiplicative reasoning in the middle grades.

Many undergraduate interior design students are unprepared for the required mathematical complexity (e.g., measurement, scale factor, and spatial reasoning). Holistically, the exercise of design is one of six universal activities through which mathematics is traditionally developed; however, researchers found no studies specifically related to interior design students and math. A small qualitative case study, therefore, was conducted to see if the mathematical theory of units construction, and coordination could provide a framework to measure interior design students’ mathematical abilities. This framework was used both to interpret sophomore interior design students’ understanding of measurement and scale factor and to identify their level of mathematical application relative to the interior design profession. Students measured a furnished lab and drafted floor plans, and semi-structured clinical interviews evaluated the students’ stages (1–3) of units construction and coordination, ruler fluency, and scale factor reasoning. Results indicate three of the students were stage two and two of the students were stage three. Stage two students applied whole number scale factors to linear measurements but could not accurately apply scale factors involving fractional linear units or square units. In contrast, stage three fluently applied whole number and fractional linear units and square units in the context of scale factor. The authors suggest that early assessment of interior design students’ units coordination structures is one method to evaluate their mathematical ability levels so that specific interventions tailored to individual student needs can be administered. Ongoing research is expanding the number of students evaluated with the instrument; future research will evaluate potential interventions.