When Confidence Masks Conceptual Gaps: Examining Teachers’ Knowledge of Graphs and Procedural Approaches to Teaching

Graphs are a conceptually demanding yet foundational component of school mathematics, yet their teaching is often shaped by procedural routines that can obscure weak conceptual understanding. A persistent concern is that teachers’ confidence in teaching graphs may be mistaken for secure topic-specific knowledge, even though graph teaching requires explanation, interpretation, and responsiveness to learner thinking. This study examined how teachers’ responses to graph-related tasks reflected their confidence, conceptual understanding, and perceptions of learner difficulties in teaching Grade 8 and Grade 9 graphs. The study was guided by the Mathematical Knowledge for Teaching framework, which distinguishes between general mathematical competence and the specialized knowledge needed for teaching, including the ability to explain concepts, interpret learner thinking, and respond to misconceptions. A qualitative exploratory case study design was employed during the baseline phase of a broader engaged scholarship project in a rural circuit in the Capricorn North District, South Africa. The targeted group comprised nine in-service mathematics teachers participating in the professional development initiative. Data were generated through written responses to five open-ended diagnostic questions focusing on teachers’ confidence, conceptual understanding of graphs, and perceptions of learner difficulties, and were analysed using qualitative content analysis. The findings showed that teachers’ reported confidence did not always align with equally strong conceptual understanding, that some graph-related explanations remained procedurally oriented, and that teachers’ recognition of learner difficulty did not always reflect strong diagnostic insight into its conceptual roots. The study contributes to the field by showing that confidence, specialized content knowledge, and knowledge of learner thinking need to be examined together when evaluating graph teaching. Although the findings are context-specific, they offer insights that may be transferable to similar under-resourced professional development contexts where routine familiarity may mask conceptual gaps in mathematics teaching.