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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释放概念弹簧构建乘法推理

构建乘法推理对于学生的数学学习至关重要，尤其是在整个中年级及以后。Tzur、Xin、Si、Kenney 和 Guebert [美国教育研究协会，ERIC 编号 ED510991，（2010 年）] 得出结论，同化复合单元是乘法推理的概念弹簧。本研究根据学生对复合单元操作的流利程度，检查了在中等年级构建乘法推理的学生百分比模式。多项逻辑回归模型表明，六、七年级学生构建同化复合单元而非乘法推理的比率显着高于八、九年级。此外，六、七年级构建乘法推理的学生比例明显低于八、九年级构建乘法推理的学生比例。其含义之一是对 Tzur、Xin、Si、Kenney 和 Guebert (2010) 概念弹簧的定量验证。也就是说，在初中早期构建同化复合单元的学生很可能构建乘法推理；没有在初中早期构建同化复合单元的学生可能不会在中级构建乘法推理。Xin、Si、Kenney 和 Guebert (2010) 的概念弹簧。也就是说，在初中早期构建同化复合单元的学生很可能构建乘法推理；没有在初中早期构建同化复合单元的学生可能不会在中级构建乘法推理。Xin、Si、Kenney 和 Guebert (2010) 的概念弹簧。也就是说，在初中早期构建同化复合单元的学生很可能构建乘法推理；没有在初中早期构建同化复合单元的学生可能不会在中级构建乘法推理。