Most educational research on argumentation comes from science, with argumentation in mathematics tending to focus on proof. We contend that argumentation can be used productively in learning mathematics even at the primary level. A research study was designed to explore children's development of argumentation in an Australian primary mathematics classroom. The classroom of 23 children (aged 9-10) had regularly used an inquiry-based approach to address extended, complex, ill-structured problems. The children's discussions and use of evidence is reported as they considered contentious media claims. The results of the design-based research study suggest that the children became proficient with Toulmin's argument framework (simplified). They were able to use this framework to plan, implement and defend the outcomes of a mathematical investigation they designed to provide evidence for or against the media claims. The paper highlights benefits and challenges with which student grappled while making and substantiating their final claims.

Mathematical inquiry is a process in which students respond to ill-structured, open-ended questions that reflect the authentic problems we encounter in 'real' life. This is unlike most problems we teach in mathematics, which are well-structured and close-ended. An open-ended ill-structured question has no single correct answer. It contains ambiguities in the problem or in the process of solving the problem that require students to make a number of decisions. This means that the emphasis is on the reasoning, judgements and evidence students provide rather than just on the answer (see Developing good inquiry questions on p 15).

Growth mindsets are vital for effective lifelong learning. Students with growth mindsets are more willing to learn new things, take risks, and embrace challenges. Students with fixed mindsets have limiting beliefs about their abilities, and will attribute success in learning to factors beyond their control. Inquiry in mathematics classrooms may have the potential to facilitate growth mindsets. This paper provides an analysis of inquiry mathematics in a primary classroom and reflects upon its potential to foster growth mindsets in classrooms.

The study we report on explores how primary children (aged 8-9) working on an inquiry-based problem draw on Game 1 and Game 2 reasoning about samples and processes (populations or mechanisms) in developing statistical arguments.

As this study is in an exploratory phase, our immediate aim is to build a foundation from which we can identify potential pathways for future research in inquiry-based statistical argumentation. In light of the theme of SRTL, we focus on three key questions:

1. To what extent does Makar & Rubin's (2009) inferential framework assist in identifying which game(s) students are playing as they conduct data-based inquiry?

2. What opportunities emerge for supporting students to stay in the [appropriate] game, when a particular pedagogical emphasis is placed on evidence in inquiry (Fielding-Wells, 2010)?

3. What role does the problem purpose play (Allmond & Makar, 2010) to assist or distract students from working in the appropriate game?

Mathematical inquiry is a process in which students respond to ill-structured, open-ended questions that reflect the authentic problems we encounter in 'real' life. This is unlike most problems we teach in mathematics, which are well-structured and close-ended. An open-ended ill-structured question has no single correct answer. It contains ambiguities in the problem or in the process of solving the problem that require students to make a number of decisions. This means that the emphasis is on the reasoning, judgements and evidence students provide rather than just on the answer (see Developing good inquiry questions on p 15).

Despite its importance for the discipline, the statistical investigation cycle is given little attention in schools. Teachers face unique challenges in teaching statistical inquiry, with elements unfamiliar to many mathematics classrooms: Coping with uncertainty, encouraging debate and competing interpretations, and supporting student collaboration. This chapter highlights ways for teacher educators to support teachers' learning to teach statistical inquiry. Results of two longitudinal studies are used to formulate recommendations to develop teachers' proficiency in this area.

Children's informal reasoning about uncertainty can be considered a product of their beliefs, language, and experiences, much of which is formed outside of formal schooling. As a result, students can adopt informal intuitions that are incompatible with formal reasoning. Although the creation of cognitive conflict has been considered as one means of challenging students' understandings, prior research in probability suggests that students may simultaneously hold multiple, incompatible understandings without conflict arising. Design-based methodology was adopted to investigate young (7–8 years old) students' inferential reasoning under uncertainty, using an inquiry-based unit developed around addition bingo. This paper selectively reports on students' inferences that initially suggested they were tacitly working from a uniform distribution (equiprobability bias), but shifted as students collected empirical data (from a discrete symmetric triangular distribution). Their inferences were challenged using an argumentation framework, with particular emphasis on the need for defensible evidence. Initial findings suggest potential for argumentation and inferential approaches that make students’ conceptions explicit through 'visibilizing' their knowledge.

Mathematical inquiry is a process in which students respond to ill-structured, open-ended questions that reflect the authentic problems we encounter in 'real' life. This is unlike most problems we teach in mathematics, which are well-structured and close-ended. An open-ended ill-structured question has no single correct answer. It contains ambiguities in the problem or in the process of solving the problem that require students to make a number of decisions. This means that the emphasis is on the reasoning, judgements and evidence students provide rather than just on the answer (see Developing good inquiry questions on p 15).

Mathematics is often perceived as a stand-alone subject in the school curriculum. When used as a tool to examine cross-curricular content, mathematics can enable deeper understanding of the context under investigation (Makar & Confrey, 2007). A study was designed to investigate opportunities and challenges that emerged when students addressed authentic interdisciplinary problems using an inquiry-based approach. This paper aims to identify aspects of students' engagement across two cohorts of a year 5 (age 9-10) classroom in Australia. Using a framework developed by Kong, Wong & Lam (2003), the paper discusses students' affective, behavioural, and cognitive engagement with mathematics during four integrated curriculum units over the course of a year. Results suggest that in both cohorts, students initially struggled with the shift from teacher-directed to student-driven learning within an inquiry-based, interdisciplinary environment. By the end of each year, however, the students had developed observable improvement in their ability to engage on multiple dimensions within the framework with ill-structured mathematical problems encountered across content areas. Implications for mathematics education research are addressed.