Chinese Teachers’ Noticing in the Context of Lakatos-Style Proving Activity: A mixed-methods study to investigate its patterns and underlying mechanisms

Abstract

In the form of fictional classroom discussions, Lakatos portrayed the processes of how mathematicians construct and use mathematical knowledge. Some elements of his approach (e.g., the zig-zag path of mathematical discovery) have been suggested to be incorporated into school mathematical activities to enhance students’ engagement with authentic mathematics. However, there is still limited understanding of teachers’ perception and cognition in the context of the Lakatos-style proving activity. To address this research gap, this research investigated the patterns and underlying mechanisms of teachers’ noticing, focusing on how they attended to, interpreted, and decided how to respond to students’ arguments in the context of Lakatos-style proving activity. This research adopted a mixed-method research design, involving an interview study with 12 Chinese pre-service and in-service secondary mathematics teachers, followed by a survey study involving 169 Chinese pre-service and in-service secondary mathematics teachers. Based on the interview results, different ways that teachers used to notice students’ arguments were identified and characterised. In addition, three patterns of teacher noticing emerged: (1) the multiplicity of teacher noticing (when noticing a student’s argument, teachers tended to use more than one way of noticing); (2) the adaptability of teacher noticing (teachers tended to adapt their primary ways of noticing across different students’ arguments); and (3) the orientation of teacher noticing (teachers tended to organise their use of multiple ways of noticing throughout the activity, either consistently selecting a particular way or balancing the use of multiple ways). These three patterns appeared to be associated with multiple factors, including teachers’ teaching experience, teachers’ framings of their professional obligations, and the validity and purpose of students’ arguments. To quantitatively examine the identified patterns of teacher noticing and their underlying mechanisms with a larger sample size, a subsequent survey study was conducted with a new sample of 169 teachers. The survey results confirmed the multiplicity, adaptability, and orientation patterns of teacher noticing. Furthermore, a series of mechanisms were identified to provide additional insights into these three patterns. Specifically, regarding the multiplicity pattern of teacher noticing, teachers were more inclined to select multiple ways of noticing when the student’s argument was valid (versus invalid) or unrelated to the refutation of a conjecture. In-service teachers (ISTs) were more inclined to select multiple ways than pre-service teachers (PSTs). As to the adaptability pattern of teacher noticing, teachers tended to adapt their selection of primary ways of noticing according to the type of the student’s argument and their framings of their professional obligations for the given student’s argument. Compared to PSTs, ISTs were more likely to select the outcome (versus the essential process) of a student’s argument as their primary focus. Among the participants, PSTs with teaching internship experience (PSTs-TI) were most likely to select the essential (versus superficial) process of a student’s argument as their primary focus. Regarding the orientation pattern of teacher noticing, teachers’ framings of their professional obligations significantly predicted their orientation of noticing. The above findings empirically and theoretically contribute to both the field of proof-related instruction and the field of teacher noticing. These findings enrich both fields’ understanding of teachers’ perception and cognition in the context of Lakatos-style proving activity, and offer implications for supporting teachers’ noticing in this context. In addition, the identified teachers’ ways of noticing and teachers’ framings of their professional obligations can inform the design of teacher training resources. These resources can help teachers discuss and reflect on their practices, or introduce them to new ways of noticing and framing that may transform their teaching. In the future, researchers and teacher educators could consider the multiplicity, adaptability, and orientation of teacher noticing to more comprehensively assess or train teachers’ competencies in noticing students’ mathematical thinking.