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<p>Due to the cognitive obstacles students face when transitioning from arithmetic to algebra, the relationship between these two domains has long been emphasized. Although many studies suggest that students can build on their arithmetical experiences to support algebra learning, the cognitive connections between arithmetical and algebraic thinking remain underexplored. In this study, we examined the associations between multiplicative reasoning and functional thinking (core components of arithmetical and algebraic thinking, respectively) from a units coordination perspective. The sample consisted of 152 fourth-grade students from two elementary schools. We used a developmental framework to assess elementary students’ multiplicative reasoning, focusing on the first four schemes in the progression: Multiplicative Double Counting (mDC), Same-Unit Coordination (SUC), Unit Differentiation and Selection (UDS), and Mixed-Unit Coordination (MUC). We also designed tasks to assess functional thinking, focusing on correspondence and covariational relations in linear whole-number contexts. A psychometric analysis was conducted to validate the assessment instrument. Results from the chain mediation analysis indicate that MUC is the only multiplicative scheme that directly and significantly predicts functional thinking in whole-number contexts, whereas the mDC, SUC, and UDS schemes indirectly predict functional thinking via the MUC scheme. Furthermore, within the UDS scheme, students’ ability to employ the <i>Difference-First</i> strategy (rather than the <i>Total-First</i> strategy) is critical for the development of functional thinking. Finally, the theoretical and practical implications of these findings are examined.</p>

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Relating elementary students’ multiplicative reasoning and functional thinking: a units coordination perspective

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Xixi Deng,
  • Rui Ding,
  • Ron Tzur,
  • Rongjin Huang

摘要

Due to the cognitive obstacles students face when transitioning from arithmetic to algebra, the relationship between these two domains has long been emphasized. Although many studies suggest that students can build on their arithmetical experiences to support algebra learning, the cognitive connections between arithmetical and algebraic thinking remain underexplored. In this study, we examined the associations between multiplicative reasoning and functional thinking (core components of arithmetical and algebraic thinking, respectively) from a units coordination perspective. The sample consisted of 152 fourth-grade students from two elementary schools. We used a developmental framework to assess elementary students’ multiplicative reasoning, focusing on the first four schemes in the progression: Multiplicative Double Counting (mDC), Same-Unit Coordination (SUC), Unit Differentiation and Selection (UDS), and Mixed-Unit Coordination (MUC). We also designed tasks to assess functional thinking, focusing on correspondence and covariational relations in linear whole-number contexts. A psychometric analysis was conducted to validate the assessment instrument. Results from the chain mediation analysis indicate that MUC is the only multiplicative scheme that directly and significantly predicts functional thinking in whole-number contexts, whereas the mDC, SUC, and UDS schemes indirectly predict functional thinking via the MUC scheme. Furthermore, within the UDS scheme, students’ ability to employ the Difference-First strategy (rather than the Total-First strategy) is critical for the development of functional thinking. Finally, the theoretical and practical implications of these findings are examined.