Mathematical Argumentation in Pupils' Written Dialogues

In this chapter, we present some results from a project about mathematical argumentation and proving in the form of dialogues. Tasks were prepared in the form of written dialogues between two imaginary pupils discussing a mathematical problem, and pupils were invited to write their own dialogues continuing the mathematical discussion. An analysis of dialogues about fractions written by 33 pupils from two classrooms in Norway in Grades 5 and 6 (10-12-year-olds), working in small groups, revealed that many of the 5th grade pupils used forms of argumentation supported by visual representations of fractions, while the 6th graders used more rule-bound approaches based on conversion. The analysis showed that three of ten groups in 6th grade used both diagrammatic and narrative argumentation in contrast to 5th grade where half of the groups were able to use these two kinds of argumentation. Those groups who made use of both types of argumentation were most successful in their argumentation. We relate these findings to the theory of relational and instrumental understanding in mathematics.