Out-of-field" teaching refers to the practice of assigning secondary school teachers to teach subjects that do not match their training or education. This practice is an issue of concern in many countries around the world, and seems particularly prevalent in the teaching of mathematics. The aim of this paper is to analyse the design principles underpinning the development and delivery of a blended learning program of professional development for out-of-field teachers of secondary school mathematics in Ireland. Three theoretical frameworks inform our analysis of the blended learning design. The first identifies critical dimensions of blended learning environments as a boundary object facilitating coordination of face-to-face and computer-mediated instruction. The second framework conceptualises out-of-field teaching as a boundary-crossing event, and identifies contextual factors, support mechanisms and personal resources that influence identity formation in out-of-field teachers as they move between different disciplinary fields. The third framework identifies the structural and core features of effective teacher professional development: the form, duration and coherence of activities; nature of teacher participation; focus on (mathematical) content knowledge; and opportunities to engage in active learning. The original contribution made by our analysis is to integrate these frameworks within a blended learning context, with the aim of identifying the distinctive features of the Professional Diploma in Mathematics for Teaching afforded by this delivery mode.

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This paper reports on a research project which aims to improve prospective mathematics teachers' relational understanding and pedagogical beliefs for teaching in second-level Irish classrooms. Prospective mathematics teachers complete their teacher education training with varying pedagogical beliefs, and often little relational understanding of the mathematics they are required to teach at second level. This paper describes a course designed by the authors to challenge such beliefs and encourage students to confront and possibly transform their ideas about teaching, while simultaneously improving their subject knowledge and relational understanding. Both content and pedagogical considerations for teaching second-level mathematics are integrated at all times. The course was originally optional and was piloted and implemented in a third-level Irish university. Apart from offering an insight into the design considerations when creating a course of this type, this paper also addresses some of the challenges faced when evaluating such a course. Overall participant feedback on the course is positive and both qualitative and quantitative results are provided to support this and also highlight the efficacy of the programme.