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.
","abstract_html":"<p>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.</p>","abstract_has_math":false,"creators":["Altindis, Nigar"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Teaching and Leadership","degree_department":null,"school":null,"contributors":["Fonger , Nicole N. F."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-05-23T07:00:00Z","date_published":"2021-05-23T07:00:00Z","updated_at":"2026-07-24T04:56:14Z","subjects":["Functional Thinking","Networking Theories","Quadratic Functions","Quantitative Reasoning","Representational Fluency","Representations","Education","Science and Mathematics Education"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/1508","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Fonger , Nicole N. 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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.
"]},{"key":"dc:title","label":"Title","values":["Exploring the Nature of the Co-emergence of Students' Representational Fluency and Functional Thinking"]}]}],"canonical_facts":{"dc:contributor":["Fonger , Nicole N. F."],"dc:creator":["Altindis, Nigar"],"dc:description.abstract":["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.
"],"dc:identifier":["https://surface.syr.edu/etd/1508"],"dc:subject":["Functional Thinking","Networking Theories","Quadratic Functions","Quantitative Reasoning","Representational Fluency","Representations","Education","Science and Mathematics Education"],"dc:title":["Exploring the Nature of the Co-emergence of Students' Representational Fluency and Functional Thinking"],"thesis:degree_discipline":["Teaching and Leadership"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:56:14Z"}