The research literature has made calls for greater coherence and consistency with regard to the meaning and use of the term advanced... Show moreThe research literature has made calls for greater coherence and consistency with regard to the meaning and use of the term advanced mathematical thinking (AMT) in mathematics education (Artigue, Batanero, & Kent, 2007; Selden & Selden, 2005). Educators and researchers agree that students should be engaged in AMT but it is unclear precisely what is meant by AMT or how these engagements may manifest themselves in student work. Moreover, many contributions are aimed at advancing theory with little effort to connect with classroom practice. The intent of this study was to examine individuals in their natural problem solving states and to empirically describe how AMT might unfold in the context of nonroutine calculus problems. The study was conducted during the spring semester of 2012. Thirteen students solved three nonroutine calculus problems, each loosely couched in theories of AMT—process-concept duality (Gray & Tall, 1994), epistemological obstacle (Bachelard, 1938; Harel & Sowder, 2005; Sierpińska, 1987), and the basic metaphor of infinity of mathematical idea analysis (Lakoff & Núñez, 2000). While empirical instances of these theories were present, students predominantly displayed idiosyncratic strategies by way of imposing mathematical/scientific objects, tools, or concepts on the local domain of the problem space. Efforts to first individualize the task were followed by completing the task through this individualization. Both successful and unsuccessful problem attempts were documented in the form of novel tool usage, visualization, and/or abstraction through communication. x The outcome of this study suggests a need to further examine students’ use of procedures and how students successfully anticipate the utility of these procedures. Additionally, further work is needed to explain both the elevated use of visualization and how/why discussions with others provide momentum for abstracting solutions to contradictory problems. With respect to classroom practice, students’ novel use of well-known mathematics mirrors the habits of research mathematicians in their professional work. Importing such uses into classroom discourse has the potential to deepen connections of mathematics content and to enrich mathematics experiences through nonstandard methods of problem solving. PH.D in Mathematics Education, July 2013 Show less
