2025 Theses Doctoral

Engaging Students in Mathematical Reasoning in School Calculus: A Case Study

Yong, Zhaojun

Since the early 2000’s, ongoing efforts for educational reform in the teaching of school mathematics have taken place to foster students’ mathematical reasoning in the classroom. Nevertheless, teachers have been hesitant to embrace the newfound role of reasoning in their instructional approach, as they continue to primarily focus on fact recall, symbolic manipulation, and procedural fluency. However, engaging students in reasoning is crucial because it allows students to analyze problems as they learn how to break them down into manageable parts and instills confidence so they can regard themselves as capable problem-solvers.

The study aims to identify mathematical practices that calculus students engage in to develop their reasoning ability as well as teacher moves employed to facilitate students’ learning of those practices and the reasons for them. Data collected for this case study were derived from class observations as well as semi-structured interviews individually with each of three experienced calculus teachers.

Findings suggest that the teachers often helped students to make connections. In particular, teachers frequently summarized students’ prior knowledge that was relevant to the completion of a task, in order to help students link new concepts with already familiar ideas. Such a move in making ideas public has evidently signaled to students their ability to solve a problem by applying what they already know. Asking students to discuss multiple approaches to a task help them think about how best to approach problems.

This move in eliciting student ideas causes students to look for connections or similarities between their ideas or processes with those of their peers. Making connections between current learning and prior knowledge can decrease the cognitive load, while making connections across different representations can address potential misconceptions, as visuals do not always lead to accurate conclusions.