Views of mathematical modeling in empirical, expository, and curricular references typically capture a relationship between real-world phenomena and mathematical ideas from the perspective that competence in mathematical modeling is a clear goal of the mathematics curriculum. However, we work within a curricular context in which mathematical modeling is treated more as a venue for learning other mathematics than as an instructional goal in its own right. From this perspective, we are compelled to ask how learning of mathematics beyond modeling may occur as students generate and validate mathematical models. We consider a diagrammatic model of mathematical modeling as a process that allows us to identify how mathematical understandings may develop or surface while learners engage in modeling tasks. Through examples from prospective teachers' mathematical modeling work, we illustrate how our diagrammatic model serves as a tool to unpack the intricacies of students' mathematical experience while engaging in modeling tasks.
Developing deep conceptual understanding of what Ma (1999) calls fundamental mathematics is a well-accepted goal of teacher education. This paper presents a microanalysis of an intriguing episode within a course designed to encourage such understanding. An adaptation of Krummheuer's (1995) elaboration of Toulmin's (1958Toulmin's ( /2003 diagrams is used to examine video recordings and transcripts of a group of graduate students in secondary mathematics education grappling with the idea of a three-dimensional line having negative slope. The graduate students' understandings of slope are examined using an expansion of Stump's (1999Stump's ( , 2001b categories of conceptions of slope. The episode ends in an interesting impasse, in which the graduate students agree to pursue the idea no further, purposely ignoring the question of negative slope, despite the clear intention of the task. The analysis explores the argumentation, factors of the learning environment, and conceptions of slope that may have contributed to this impasse.Slope is a universal topic in mathematics curricula that is usually introduced with lines or linear functions. Its importance for describing the behavior of a curve and essential role in the development of derivative are undeniable. Yet, as a secondary mathematics topic, it has the peculiar fate of being well-known but not well understood. Ironically this prominent concept has received scant scrutiny; however, the research literature that does address slope makes valuable contributions to understanding this complex concept. Walter and Gerson (2007) suggest that the emphasis of the mnemonic phrase "rise-overrun" has contributed to an instrumental understanding of slope to the extent that students are poorly equipped to make connections between slope and line position or slope and rate Educ Stud Math (2011) 76:3-21
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