Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities

Abstract

Mathematical generalization can take on different forms and be built upon different types of reasoning. Having utilized data from a series of task-based interviews, this study examined connections between empirical and structural reasoning as preservice mathematics teachers solved problems designed to engage them in constructing and generalizing mathematical ideas aided by digital tools. The study revealed closer connections between naïve empiricism and result pattern generalization, between naïve empiricism and recognizing a structure in thought, between reasoning by generic example and process pattern generalization, and between reasoning by generic example and reasoning in terms of general structures. Results from this study imply that the ability to generalize based on perception and numerical pattern does not necessarily lead learners to generalize based on mathematical structure.

Characterizing Levels of Reasoning in Graph Theory

“…Students' knowledge of graph theory has been proven as a powerful tool for understanding other disciplines: chemistry (Bruckler & Stilinović, 2008), physics (Toscano, Stella, & Milotti, 2015), or computer science (Kasyanov, 2001). However, few studies have researched students' reasoning about graph theory, as noted by Hazzan and Hadar (2005), and Medová, Páleníková, Rybanský, and Naštická (2019), although numerous works in multiple fields of mathematics point out the importance of studying students' reasoning to design effective tasks sequences or explain the success or failure of different teaching strategies (Brito, Almeida, & Osório, 2020;Chaphalkar & Wu, 2020;Hokor, 2020;Yao & Elia, 2021). These facts demonstrate the need to develop theoretical frameworks that support this type of research.…”

The Development of The Flows of Educational Philosophy: Theoretical Concept and Implementation in 21st-Century Learning

“…The empirical influence obtained from the environmental effects on the development of the students. According to this flow, teachers are the external factor that holds an important role because the teachers provide the environment for the students and the students will accept education as an experience (Yao & Elia, 2021). These experiences will shape behavior, attitude, and students' character as learning goals.…”