The Development of Multiplicative Calculation Flexibility by 3rd Grade Students

Abstract This article aims to understand how the students develop the multiplicative calculation flexibility in a context of exploratory teaching, answering three research questions: (i) What are the strategies used by students when solving multiplication tasks?; (ii) How do the strategies used by the students evolve?; and (iii) How do the students use the multiplicative calculation in a flexible manner? The investigation adopts a qualitative methodology within an interpretive paradigm having been held a teaching experience in a 3rd grade classroom. The data collection took place through the following techniques: participant observation and documentary collection. The article focuses only one pair of students. The results show that students used multiple strategies to solve tasks involving multiplication, evolving from additive strategies, associated with sequences of multiples, to multiplicative strategies, establishing numerical relationships, and using known basic multiplication facts and properties. This evolution is associated to the development of multiplicative calculation flexibility by the students, which is initially anchored in organizing the objects in equal parts, evidencing, at the end of the teaching experience, a clear connection between the idea of the same product being symbolized by a high number of multiplicative decompositions, the operation of division and the identification of the proportional relationship. The study shows the lack of parallelism between the development of multiplicative strategies in calculation and proportional situations.

摘要 本文旨在探究在探究式教学（exploratory teaching）情境下，学生如何发展乘法计算灵活性（multiplicative calculation flexibility），并回答三个研究问题：
（一）学生在解决乘法任务时会采用哪些策略？
（二）学生所使用的策略是如何演变的？
（三）学生如何灵活运用乘法计算？
本研究采用阐释主义范式（interpretive paradigm）下的质性研究方法（qualitative methodology），于三年级课堂开展了教学实践。数据收集采用以下方法：参与式观察（participant observation）与文献资料收集（documentary collection）。本文仅以一组学生为研究对象。
研究结果显示，学生可采用多种策略解决乘法类任务：其策略发展路径从基于倍数序列的加法策略（additive strategies），逐步转向能够建立数值关系、运用已知基础乘法事实与运算性质的乘法策略（multiplicative strategies）。该演变与学生乘法计算灵活性的发展高度相关：灵活性最初以将物体按等份分组为基础；在教学实践结束时，学生已能清晰建立起以下关联：同一乘积可通过大量乘法分解式（multiplicative decompositions）表征、除法运算，以及比例关系（proportional relationship）识别三者之间的内在联系。本研究表明，学生在计算情境中发展的乘法策略与比例情境中的乘法策略发展并不存在同步性。