Date of Award

Spring 5-22-2021

Degree Type


Degree Name

Doctor of Philosophy (PhD)


Teaching and Leadership


Nicole Fonger


Functional Thinking, Networking Theories, Quadratic Functions, Quantitative Reasoning, Representational Fluency, Representations

Subject Categories

Education | Science and Mathematics Education


Abstract In this dissertation, I explore ways to support secondary school students’ meaningful understanding of quadratic functions. Specifically, I investigate how students co-developed representational fluency (RF) and functional thinking (FT), when they gained meaningful understanding of quadratic functions. I also characterize students’ co-emergence of RF and FT on each representation (e.g., a graph, a symbolic equation, and a table) and across multiple representations. To accomplish these goals, I employed a design research methodology: a teaching experiment with eight Turkish-American secondary school students in an after-school context at a Turkish Community Center. I constructed the design principles and design elements for the study by networking two distinct domains of literature—representations and quantitative reasoning—to support students’ meaningful learning. I conducted ongoing and retrospective analyses on the enhanced transcriptions of small- and whole-group interactions. The analyses revealed a learning-ecology framework that supported secondary school students’ meaningful understanding of quadratic functions. The learning-ecology framework consisted of three components: enacted task characteristics, teacher pedagogical moves, and socio-mathematical norms. Furthermore, the findings showed that students employed two types of reasoning when they created and connected representations of quantities and the relationships between them: static thinking and lateral thinking. Static thinking is recalling a learned fact to represent a quantitative relationship with no attention to how quantities covary on a representation, while lateral thinking is a creative way of thinking wherein students conceive of concrete representations of functions as an emergent quantitative relationship. The findings also showed that students’ co-emergence of RF and FT can be operationalized into four levels starting from lesser sophisticated reasoning to greater sophisticated reasoning. Level 0 is a disconnection, level 1 is a partial connection, level 2 is a connection and level 3 is flexible a connection between students’ RF and FT. The dissertation informs teachers and the mathematics education community by (a) reporting and verifying the learning-ecology framework that supported students’ meaningful understanding of quadratic functions; and (b) characterizing students’ co-emergence of RF and FT within and across multiple representations.


Open Access