Upper Elementary Students' Connecting and Interpreting Function Representations
Abstract: This work is an exploration of upper elementary students' sense making around four conventional representations of function: equations with algebraic notation, Cartesian graphs, function tables, and natural language. The cornerstone to the empirical work is a task called the Function Puzzle, where students are given 16 cards representing four functions across these four representation ... read moretypes and asked to make sets of cards "that belong together." Without prior instruction on interpreting these representations, students successfully create sets where all four cards represent the same function. The three empirical studies examine students' reasoning around the Function Puzzle representations by analyzing one-on-one interviews, held after solving the puzzle, where students discuss their solutions. Study 1 is a case study which explores mediational influences of discourse and the representations themselves on a 4th grade student's developing understandings of algebraic notation. Study 2 examines how four 5th-grade students "discover" the semantic rules of algebraic notation and connects those discoveries with students' noticing of dependent and independent variables in the function representations. Finally, Study 3 uses discourse analysis to examine how students' patterns of discourse not only communicated their solutions of the Function Puzzle, but reinforced connections among representations in such a way as to potentially impact students' understandings of functions. Across the studies, I provide evidence that students employ sense making to negotiate connections and interpretations across the function representation types. Students reason dynamically about covarying relationships represented in the task, and several students make conjectures about interpreting algebraic notation, an unfamiliar representation type. These ways of reasoning are important to developing a sense of functions and demonstrate mathematical disciplinary engagement. In summary, elementary students can attend to covariational relationships between quantities -functions - when given opportunities to use sense making and their propensities for generalization. I argue that cross-referencing activities with multiple function representations like the Function Puzzle are important and generative mathematical experiences for elementary school children.
Thesis (Ph.D.)--Tufts University, 2018.
Submitted to the Dept. of Education.
Advisor: Bárbara Brizuela.
Committee: Maria Blanton, Moon Duchin, and Brian Gravel.
Keywords: Mathematics education, and Elementary education.read less