Developing knowledge about how students acquire mathematical understanding is a focus of mathematics curricula and research, including the ability of students to move from manipulating concrete materials to abstract number properties when solving problems. This study, informed by the Numeracy Development Projects (Ministry of Education 2007a, 2008b) and the work of Pirie and Kieren (1989, 1992, 1994a, 1994b), examines the role of ‘imaging’ in supporting the development of students’ mathematical thinking and understanding. Imaging is an important phase of the teaching model advocated by the Numeracy Development Project. The context of this study is a primary school mathematics programme, which involved the teaching of two mathematics units that focused on the addition and subtraction of decimal fractions and whole numbers. There is considerable research about what is effective in mathematics education for diverse learners, and how students learn. There is, however, very limited research about the role of imaging in mathematical learning.

This qualitative study adopted a case study approach and focused on a group of Year 6 students. Data collection methods included observation, interviews, field notes and document analysis. A thematic approach was used to analyse data and to develop and inform an emerging theoretical framework.

During this study I developed a model, entitled A Model for the Development of Students’ Mathematical Understanding, which illustrates six mathematical resources students use as they solve problems. These resources are: materials, mental picture images, drawn picture images, transformed mental images, transformed drawn images and number properties. Students’ engagement with these six resources illustrates how they develop understanding of mathematical concepts. The students identified a preference for using drawn rather than mental images when solving problems. This study also emphasizes the complexity of the imaging process, and the fluid and multifaceted nature of learning in mathematics. This study serves to highlight the complexities of the teaching and learning process in mathematics for both teachers and students.