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Proportional reasoning as the capacity to compare situations in relative (multiplicative) rather than absolute (additive) terms is an important outcome of primary school mathematics. Research suggests that students tend to see comparative situations in additive rather than multiplicative terms and this thinking can influence their capacity for proportional reasoning in later years. In this paper, excerpts from a classroom case study of a fourth-grade classroom (students aged 9) are presented as they address an inquiry problem that required proportional reasoning. As the inquiry unfolded, students' additive strategies were progressively seen to shift to proportional thinking to enable them to answer the question that guided their inquiry. In wrestling with the challenges they encountered, their emerging proportional reasoning was supported by the inquiry model used to provide a structure, a classroom culture of inquiry and argumentation, and the proportionality embedded in the problem context.

Inquiry-based learning (IBL) is a pedagogical approach in which students address complex, ill-structured problems set in authentic contexts. While IBL is gaining ground in Australia as an instructional practice, there has been little research that considers implications for student motivation and engagement. Expectancy-value theory (Eccles and Wigfield 2002) provides a framework through which children's beliefs about their mathematical competency and their expectation of success are able to be examined and interpreted, alongside students' perceptions of task value. In this paper, Eccles and Wigfield's expectancy-value model has been adopted as a lens to examine a complete unit of mathematical inquiry as undertaken with a class of 9-10-year-old students. Data were sourced from a unit (∼10 lessons) based on geometry and geometrical reasoning. The units were videotaped in full, transcribed, and along with field notes and student work samples, subjected to theoretical coding using the dimensions of Eccles and Wigfield's model. The findings provide insight into aspects of IBL that may impact student motivation and engagement. The study is limited to a single unit; however, the results provide a depth of insight into IBL in practice while identifying features of IBL that may be instrumental in bringing about increased motivation and engagement of students in mathematics. Identifying potentially motivating aspects of IBL enable these to be integrated and more closely studied in IBL practises.

Background The recommendation from national documents and reports to promote inquiry-related science activities has not been supported by recent studies, which have found the overall frequency of inquiry activities to be negatively associated with student learning outcomes. This study was inspired by such conflicting reports and aimed to clarify the associations of science-specific, inquiry-related activities and epistemological beliefs with students’ mathematical and scientific literacies. Results A secondary analysis of the database from the Programme for International Student Assessment 2015 of Australia (N1 = 14,530) and Taiwan (N2 = 7708) utilizing structural equation modelling revealed that these two countries exhibited similar data patterns. Results suggested that open-inquiry activities (such as debating and planning experiments) had a negative relationship with secondary students’ mathematical and scientific literacies. Structured inquiry learning (such as students explaining their ideas and teacher explaining how an idea can be applied to different phenomena) and epistemological beliefs about science were significant and positive predictors of student mathematical and scientific literacy performance. Conclusions The current study further highlights and provides empirical evidence that the teacher’s role in structured inquiry (especially pertaining to the relevance and applicability of these ideas) appears to be essential to the development of student literacy. Educational implications and recommendations are discussed.