2024 Andy Ash - Using semantic pathways to analyse multiple representations of ‘school maths’

There are a range of representations that are useful for teaching and learning such as diagrams, graphical images and analogies, however it can also be seen that using representations may potentially over-complicate things and confuse students. In mathematics, where the objects of knowledge are entirely abstract, representations present the only way for teachers to communicate meaning. For example, when mathematics is translated into ‘school maths’, teachers will often use representations such as shaded in shapes or number-lines to represent the abstract concept of fractions. It is the teacher’s role to carefully orchestrate the use of these so that pupils develop a shared understanding of concepts. In this Roundtable, Andy Ash demonstrates how he enacted the LCT dimension of Semantics to analyse how representations are used in the school maths classroom. His doctoral study analysed the practice of a single classroom teacher in England and used Semantics to better understand how she used representations to engage pupils in mathematical thinking. In the classroom, her movements between real-life examples, textbook diagrams and student representations of their thinking show how her pedagogy shifts in the lesson between simpler, context-dependent examples and more general and complex mathematical ideas. These moves can be thought of as signature semantic pathways, which make knowledge-building pedagogies explicit and reproducible. Understanding and tracking these pathways offer opportunities for teacher learning across different subjects.