Future teachers’ knowledge of real numbers and functions on computers
PhD defence
Doctoral candidate
Abstract
Many high school mathematics teachers struggle to relate the theoretical knowledge they gained in university to the practical mathematics teaching in high school. This PhD project, inspired by Klein’s second discontinuity, seeks to bridge this gap by investigating the knowledge of university students who have completed their university mathematics study and plan to become high school mathematics teachers, focusing on a specific mathematical domain - real numbers. The primary goal of the project is to aid students in developing an advanced understanding of real numbers. As computer software has become ever more prevalent in Danish secondary mathematics education, particularly when dealing with real number calculations, the thesis considers three main research objects: real numbers in university, real numbers in high school, and real numbers on computers. The thesis explores the teaching of infinite decimal models of real numbers to bridge the models of real numbers in high school and university. Additionally, the concept of computability is introduced to analyze the meaning of computergenerated decimal representations of real numbers. Understanding real numbers as infinite decimal representations through the lens of computability is crucial for linking these three research objects. The theoretical framework of the Anthropological Theory of the Didactics (ATD) is foundational to this thesis. The notions of institutional relations and praxeology from ATD are presented as key elements for modeling Klein’s second discontinuity and establishing a connection to address this gap. The exploration of students’ work with infinite decimal models of real numbers and computability is carried out within a “capstone” course called UvMat at the University of Copenhagen. The present study involved the design of two weekly assignments that incorporate these concepts, the analysis of students’ responses to the two assignments, and interviews conducted in relation to these tasks. The analysis of students’ work reveals that engaging with infinite decimals from the standpoint of computability can, to a certain extent, assist future mathematics teachers in applying their university-acquired knowledge to address practical tasks in high school and aids them in opening the “black box” of computers when handling real numbers. However, many students fail to establish a link between mathematics teaching with digital tools in high school and the two assignments. This aspect needs further investigation, particularly in exploring whether explicitly teaching the concept of computability as a mathematical object within the course can help students perceive the connection.
Download the thesis: Rongrong_Huo_Thesis
Assessment committee
Professor Morten Misfeldt (chair), Department of Science Education, University of Copenhagen
Professor Reinhard Hochmuth, Leibniz U. Hannover, Germany
Assoc. Professor Berta Barquero, U. of Barcelona, Spain
Chair of defense
Associate Professor Adrienne Traxler, Department of Science Education, University of Copenhagen
Supervisor
Professor Carl Winsløw, Department of Science Education, University of Copenhagen