The multiplicative reasoning that students should develop in elementary school is a key area of research in contemporary mathematics education. Researchers employ various views including multiplication as arithmetic operation, multiplicative structures, and multiplicative relationships. They also propose various classifications of multiplicative structures to support students’ development of multiplicative reasoning and problem solving. Our work contributes to this conversation by focusing on simple multiplicative relationships. Drawing on Davydov’s Theory of Developmental Instruction, we employ the relational view to analyze some multiplicative structures identified by researchers and practitioners. We propose a typology of basic multiplicative relationships as well as their specific graphical representations as a set of mental tools allowing for a holistic flexible understanding of the multiplicative situations, simple and complex, primary school students usually encounter (in grades 1–6). We suggest that the new approach may better support students’ multiplicative and relational reasoning and problem solving.

中文翻译：

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基础教育中的一些乘法结构：从关系范式看

学生应在小学阶段发展的乘法推理是当代数学教育的一个重点研究领域。研究人员采用各种观点，包括作为算术运算的乘法、乘法结构和乘法关系。他们还提出了乘法结构的各种分类，以支持学生发展乘法推理和解决问题的能力。我们的工作通过关注简单的乘法关系来促进这种对话。借鉴达维多夫的发展教学理论，我们采用关系观点来分析研究人员和实践者确定的一些乘法结构。我们提出了基本乘法关系的类型学及其特定的图形表示，作为一组心理工具，允许对小学生通常遇到的乘法情况、简单和复杂的情况（1-6 年级）进行全面灵活的理解。我们建议新方法可以更好地支持学生的乘法和关系推理和问题解决。