A Journey to Understanding: Developing Computational Fluency in Multi-digit Multiplication

Abstract

This thesis presents a domain-specific instructional theory for multi-digit multiplication, focused on building students’ conceptual understanding and computational fluency. Multiplicative thinking is crucial to students’ mathematical development. Without a sound understanding of multiplicative structure students’ capacity to develop deep understanding and fluency in fractions, decimals, proportional reasoning and ratios is severely limited. Understanding multiplicative structure enables students to move beyond additive strategies such as skip counting and repeated addition, to efficient and sophisticated strategies that grow from their ability to think simultaneously about composite units. While there has been substantial research into early stages of students’ understanding in multiplication, there has been limited work exploring the more complex domain of multi-digit multiplication. The domain-specific instructional theory developed through the research described in this thesis includes a number of key features that build on and expand existing research. It draws on social constructivist perspectives to document the social and cognitive development of learners by uncovering, examining and analysing the mathematical practices that emerge through the collective learning in the classroom. Students’ invented strategies for multi-digit multiplication problems are connected to key developmental understandings of commutativity, associativity and distributivity. The crucial role of the array is explored in the process of students’ sense-making and reasoning. The theory developed in the research proposes an instructional sequence from a model of specific, contextualised situations, through to a model for more generalised mathematical reasoning in the domain of multi-digit multiplication. Design Research methods were used to inform the development of the domain-specific instructional theory. A hypothetical learning trajectory was constructed based on a review of relevant research-based literature into students’ use of the array and on curriculum documentation guiding teachers’ practices. The learning trajectory was tested in two separate teaching experiments, each involving ten teaching episodes conducted over a two-week period. Work samples, video and interview responses from a total of 55 Year 5 students from two classes were analysed and used to inform evidence-based refinement and modification of the learning trajectory. Ways in which learning could be supported through the implementation of the learning trajectory were also documented. Several key findings emerged through the design research. Students used a variety of invented strategies that drew on additive and multiplicative thinking, in some cases exclusively and in others, in combination. Students’ reasoning and justification relied predominately on the array, enabling them to make sense of the multiplicative structure in a way that symbolic recording alone did not. A number of different forms of the array were used in the study, with students electing to use different forms of the array based on the function they needed the array to serve. As students’ appreciation of and confidence with the multiplicative structure increased, their reliance on the array decreased, allowing them to move to more numerical notation underpinned by the sense-making developed through use of the array. Four mathematical practices relating to the social development of students’ understanding of the multiplicative structure were identified. Two of these: partitioning based on place value, and; using factors to manipulate the array, were based on students’ use of the array as a tool for sense-making. The other two, thinking multiplicatively, and; looking for friendly numbers, were based on ways that the students worked mathematically. Additionally, a set of five mathematical norms was identified as central to each students’ development of these mathematical practices. These were: looking for similarity and difference; making inferences; using representations; justification, and: forming generalisations. The research highlights some crucial aspects of teaching practice that are essential if students are to develop a sound understanding of multi-digit multiplication. First, instruction needs to focus on multiplicative structure built on representations that highlight fundamental mathematical properties. Second, computational fluency grows from an understanding of structure but needs to be explicitly developed through focused discussion of the mathematical features of particular strategies and representations. Third, mathematics classrooms must be focused on sense-making through carefully orchestrated discussion of students’ invented strategies and representations. Finally, the affordances and constraints of different forms of representation must be recognised in order to make clear the function that each of the possible forms might perform. The research therefore adds to the existing literature relating to multi-digit multiplication and to that relating to the development of sociomathematical norms. It brings cognitive and social perspectives of learning together in a new way, proposing a focus on form and function in multiplicative thinking and a set of transcendent mathematical norms that underpin students’ reasoning in mathematics.