Conversations about connections: how secondary mathematics teachers conceptualize and contend with mathematical connections

The importance of mathematical connections in learning and understanding mathematics is widely endorsed in both the research and the professional literature but teachers’ understanding of mathematical connections is underexplored. This study examined teachers’ conceptions of mathematical connections as knowledge at the interface of content knowledge and pedagogical content knowledge. I had individual conversations with nine secondary mathematics teachers in a three-stage process of progressively more structured interviews. Interviews focussed on teachers’ explicit connections related to particular mathematical topics, including a common task about quadratic functions and equations. I coded transcribed interviews according to a model I developed that identified five types of connections – different representations, implications, part-whole relationships, procedures, and instruction-oriented connections. Teachers’ thinking about connections was completely bound up with their thinking about teaching. They talked about real-world connections and connections to students’ prior knowledge, but only a few explicitly pointed out connections to their students. Most teachers were enthusiastic in their approval of considering mathematics as an interconnected web of concepts. While some teachers saw mathematical connections as integral to the way they taught, others were conflicted, and expressed a tension between teaching concepts and teaching algorithms. In the context of a structured task, teachers demonstrated knowledge of specific mathematical connections at a fine-grained level, but only with considerable effort. Teachers do have knowledge of specific mathematical connections but that knowledge is largely tacit. Teachers described specific mathematical connections in all five categories of the model. The model proved robust in classifying connections across a range of mathematical topics and grain-size. The mathematical connections that teachers articulated dealt with a narrow range of content, and favoured connections that were explicitly described in their textbooks. Nevertheless, teachers were also able to identify certain connections as crucial to students’ understanding of a topic. This systematic and detailed examination of the way that teachers view mathematical connections has laid a foundation for future research by demonstrating a methodology that facilitated the expression of teachers’ tacit knowledge, and by developing a model for classifying the explicit mathematical connections that teachers did express.