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.

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This game teaches algebraic generalizations through differentiated play in pairs, small groups, or as a whole class and uses manipulatives to bridge numerical and algebraic thinking.

This study presents the preliminary qualitative results of a larger mixed methods study. The qualitative phase utilized task‐based clinical interviews to examine the non‐symbolic and symbolic linear generalizations of middle‐grades students. This investigation identified similarities and differences in the students’ generalizations, and interpreted these distinctions within the number sequences framework, which describes a hierarchy of numerical reasoning stages. Participants included 14 students in grades 6 through 9 who operated at 3 distinct stages of numerical reasoning, as framed by the number sequences. Findings indicate that the number sequences were related to students’ generalizations, in that students with more sophisticated numerical reasoning produced more sophisticated generalizations. The analysis relates students’ generalizing behaviors to the mental operations that define each of the three number sequences under examination. This study demonstrates the importance of supporting students’ numerical reasoning as a means of preparing them to reason algebraically in the middle grades, contributes theoretically to an understanding of the relationship between numerical reasoning and generalizing, and suggests instructional strategies for engaging students in generalizing.