The development of prerequisite notions for an introductory conception of a functional limit
Nagle, Courtney Rose
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The limit concept plays a foundational role in calculus, appearing in the definitions of the two main ideas of introductory calculus, derivatives and integrals. Previous research has focused on three stages of students' development of limit ideas: the premathematical stage, the introductory calculus stage, and the transition from introductory calculus to the formal definition of the limit concept. Fairly little is known about the gulf between the premathematical stage and the introductory calculus stage. This study fills the void in research by investigating the knowledge components for limits developed during precalculus mathematics. One-hundred fifty high school and college precalculus students were engaged in a series of tasks designed to investigate students' ability to reason about limits. These tasks were designed based on the notion of predicting function values and required no prior knowledge of limits or calculus. The tasks included both graphical and numerical representations of various functional behaviors, including linear, non-linear, unbounded, piecewise, and oscillating functions. Students' solutions to the limit tasks were coded using the qualitative technique of grounded theory in order to develop a list of knowledge components demonstrated in students' solutions to the tasks. Five knowledge themes emerged from the coding of knowledge components: finding patterns, functions, reading and extrapolating graphs, notions of closeness and approaching, and unexpected function behaviors. Results showed that students scored better on functional tasks represented in graphical form compared with the same functional tasks represented in numerical form. A learning trajectory for the limit concept should provide a description of the desired outcome (a formal understanding of the limit concept), the stages of students' cognitive development, and instructional activities intended to advance students through the various stages. Additional statistical analyses contributed to informing a learning trajectory for limits. In particular, factor analysis showed that the numerical tasks and graphical tasks each measured a single underlying construct. Correlations between students' use of the various numerical and graphical knowledge components were used to identify simplex models and Guttman scalograms which supported developmental progressions. In particular, the developmental progressions supported knowledge of finding patterns, defining functions, reading and extrapolating graphs, and notions of the mathematical meaning of closeness and approaching as important prerequisites for developing additional, more advanced knowledge components. Results of the regression analysis showed that students' knowledge of approaching a value as well as their ability to recognize unexpected function behaviors predicted their achievement on the limit tasks. The notion of approaching was distinct from the other knowledge components in terms of its importance as both a prerequisite for other concepts and its power for predicting student achievement. The findings of this study support precalculus instruction that builds students' understanding of the mathematical meaning of closeness as a building block for introductory calculus instruction, which extends the idea to approaching via arbitrary closeness. The results also support an emphasis on multiple representations of functions so that students can build covariational reasoning that supports recognizing various graphical behaviors as illustrations of a relationship between two covarying quantities. The results corroborate previous findings that students struggle to interpret non-linear relationships, particularly when considering numerical representations. The findings are synthesized into a theory for how students develop notions of limit over time as well as into pedagogical suggestions for how precalculus and introductory calculus instruction can improve students' development of rich concept images.