Mathematical Challenge in Connecting Advanced and Secondary Mathematics: Recognizing Binary Operations as Functions

Abstract

For secondary mathematics teachers, it is important that their mathematical coursework helps deepen their understanding of the school mathematics they will teach. That is, making connections between advanced and secondary mathematics is vital for practicing and prospective teachers (PPTs). However, forming these connections poses significant mathematical hurdles. In this chapter, I explore the mathematical challenges that arise when PPTs are asked to make connections by recognizing ideas in advanced mathematics as being an instance of an idea studied in secondary mathematics. In particular, I look at the mathematical challenges faced by two PPTs as they tried to reconcile the definition of a binary operation in abstract algebra (i.e., ∗ : A × A → A) in terms of it being a function – something studied in secondary school. In this example, mathematical challenge is evident through the conceptual stages and shifts these two PPTs went through as they came to understand a binary operation as a function itself. I use this example to ground the discussion of mathematical challenges faced, more broadly, as PPTs develop connections from their advanced mathematical coursework. I also elaborate on the purposes such connections might serve, and why, for PPTs, these connections merit the mathematical challenges encountered to develop them.