Given the central role of explanations and justifications within mathematics, many important questions deserve further investigation, including:

• The role(s) of language and special notation within young children’s learning of mathematics.
• How to discern generality in students’ informal utterances.
• The interplay between generalisation and justification.
• Gender differences in argumentation.

Research, involving teacher collaboration, is needed in classroom situations to explore how we might best create learning environments that foster the development of children’s explicit development of and verification of conjectures. Specifically, we need to further explore the capabilities of young children’s reasoning abilities in relation to abstract thinking about number properties. Alongside this, further research is needed to understand effective ways to develop algebraic thinking by building on students’ informal knowledge and naturally occurring linguistic and cognitive powers.

Another area of research central to teaching children to make sense of and understand the structure of mathematics is the role of teachers’ mathematical knowledge. In what ways does the extent and depth of such knowledge influence the ability of teachers to:

• pose questions that go beyond asking children to describe their solution strategies;
• understand children’s mathematical thinking that differs from what might be expected based on the research-based information on children’s thinking;
• critically examine children’s thinking to determine if it is mathematically valid; and
• use what they learn about their children’s thinking to create tasks that enable children to extend their thinking.

Specifically, in the New Zealand context of numeracy development, we need to understand the ways in which mathematical knowledge influences teachers’ ability to transform their practice – to consider how teachers’ content knowledge changes in the context of their practice as they interact with students’ mathematical thinking.

Directives for improving the teaching and learning of mathematics in New Zealand schools have been so widely publicised and so actively supported that the reforms are often perceived as having large-scale impact. Yet this perception may be far from the truth. Evidence from this research suggests that mathematics teaching which places students’ thinking at the centre of instruction and places the development of conceptual understanding as its primary goal, is not commonplace. If students are to come to believe that learning and doing mathematics involves the solution of problems in ways that are meaningful, classroom instruction must encourage and support discussion and reflection on mathematical structure of number from a young age.

If school lessons are to involve “communities” of learners doing this kind of work rather than individuals acquiring skills and remembering rules, classroom will not be silent places where each learner is privately engaged with ideas. If students are to employ logic and mathematical evidence, they will need to be able to compose speech acts or written artefacts that expose their reasoning. If they are to conjecture and connect, they will need to communicate. (Lampert, 1999, p. 10)

More specifically, in relation to the assessment tasks analysed in this project, if students genuinely understand arithmetic at a level, and can explain and justify the properties that they are using as they carry out calculations, they have learned the foundations of much of algebra.