A Rethinking of Mathematical Proficiency: Lessons Learned From Noticing Mathematical Missteps Situated In Science

This thesis investigates the recurring phenomenon I experienced as a teacher when students made seemingly trivial yet significant math mistakes in science class that they could not justify, despite demonstrating proficiency in the same skill during math lessons. This study utilizes a multiple-case study methodology with narrative methods, drawing on Mason’s (2002) noticing framework during data collection. I documented my experiences through field notes, focusing on instances where students made small, unexplainable math errors in science. I developed these observations into narrative-style vignettes, called Episodes, to represent the situations as I experienced them while preserving student anonymity. Using abductive reasoning, I analyzed each vignette individually, grounding my interpretations in existing literature. Following the multiple-case study approach, I then conducted cross-case analysis to identify broader themes and patterns. This iterative process involved forming hypotheses, testing them against relevant literature, and refining conclusions. The analysis reveals that the students I observed in my study appear to primarily rely on secondary intuitions, as defined by Fischbein (1987), as their main form of mathematics knowledge. These secondary intuitions emerge from prior mathematical instruction emphasizing procedural proficiency over conceptual understanding. While these intuitions can create the appearance of proficiency in math class, they break down in science contexts where flexible, nonlinear thinking is required. This finding suggests that procedural, intuitive knowledge alone is not transferrable to new learning environments, exposing gaps in conceptual understanding when students face unfamiliar contexts. These results raise critical concerns about how mathematical proficiency is defined and assessed. This study highlights the importance of prioritizing conceptual understanding in math education and fostering cross-curricular connections to improve students’ ability to apply their knowledge in diverse contexts. By addressing these issues, educators can better prepare students to engage with complex, interdisciplinary problem-solving and ensure a more comprehensive understanding of mathematics.