Catherine Pearn

How to Teach and Assess Whole Number Arithmetic: Some International Perspectives

This chapter focusses on the diverse theoretical and methodical frameworks that capture the complex relationship between whole number arithmetic (WNA) learning, teaching and assessment. Its aim is to bring these diverse perspectives into conversation. It comprises seven sections. The introduction is followed by a narrative of a Macao primary school lesson on addition calculations with two-digit numbers, and this sets the context for the subsequent three sections that focus on the development of students’ mathematical and metacognitive strategies during their learning of WNA. Apart from examining the impact of teachers’ knowledge of pedagogy, learning trajectories, mathematics and students on children’s learning of WNA, learning theories are also drawn on to interpret the lesson in the Macao Primary School. Two interpretations of the variation theory (VT), an indigenous one and a Western perspective, provide much needed lenses for readers to make sense of the lesson. In addition, the theory of didactical situations (TDS) is also applied to the lesson. The chapter also includes a reflection on possible classroom assessment and the role of textbooks, both of which were less apparent in the lesson, for the teaching and learning of WNA.

Generalizing Fractional Structures: A Critical Precursor to Algebraic Thinking

Our research focuses on how students find an unknown whole, when given a known fractional part of the whole, and its equivalent quantity. This chapter will show how Year 5 and Year 6 students, who have yet to meet formal algebraic notation, create algebraic meaning and syntax through their solutions of these fraction problems. Some students rely on diagrammatic representations using different mixes of multiplicative and additive strategies. Other students use fully multiplicative approaches to find the whole. Some students’ solutions show how they use “best available” symbols to move beyond arithmetic calculation and show evidence of algebraic thinking, especially when students are able to treat particular numerical and fractional values as quasi-variables. This chapter sets out to identify those precursors of algebraic thinking that allow students to move beyond particular fraction values to generalize their solutions.