Out-of-field teaching of mathematics is a reality in many secondary schools in the world. The incidence of out-of-field teaching generally occurs in low socio-economic communities and, in Australia, in schools located in rural and remote locations. The theory of boundary crossing enables positive perspectives of teaching out-of-field to be explored. In this paper, we explore changes in out-of-field mathematics teachers’ beliefs and practices about teaching mathematics over 3 years and the way in which their in-field teaching influenced and was influenced by their out-of-field mathematics teaching. Out-of-field teachers from schools in three Australian states participated in the study. The findings show that initially the majority of these teachers held instrumentalist beliefs about the mathematics discipline and its teaching and learning. Those who continued to teach mathematics out-of-field beyond the first year of the study presented evidence of some shifts in their beliefs about the teaching and learning of mathematics by including more student-centred or problem-solving approaches. Exploring the factors that enable continuity of discourse across subject boundaries is important not only for supporting and retaining teachers of mathematics in schools with a high incidence of out-of-field teaching but also for fostering interdisciplinary approaches to teaching and learning.

数学课外教学是世界上许多中学的现实。场外教学的发生通常发生在低社会经济社区，在澳大利亚，发生在位于农村和偏远地区的学校。跨越边界的理论使人们能够探索非现场教学的积极视角。在本文中，我们探讨了3年多来数学外场教师在数学教学信念和实践上的变化，以及他们的外场教学对其外场数学教学的影响和受其影响的方式。来自澳大利亚三个州学校的外地教师参与了这项研究。调查结果表明，最初这些教师中的大多数对数学学科及其教学和学习持有工具主义的信念。那些在研究的第一年之后继续在外教数学的人证明了他们对数学教学和学习的信念发生了一些转变的证据，这些方法包括更多以学生为中心或解决问题的方法。探索能够跨学科边界进行话语连续性的因素不仅对于支持和留住校外教学发生率高的学校的数学教师很重要，而且对于培养跨学科的教学方法也很重要。