UNDERSTANDING INTRODUCTORY STUDENTS‘ APPLICATION OF INTEGRALS IN PHYSICS FROM MULTIPLE PERSPECTIVES by DEHUI HU B.S., University of Science and Technology of China, 2009 AN ABSTRACT OF A DISSERTATION Submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Physics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2013 Abstract Calculus is used across many physics topics from introductory to upper-division level college courses. The concepts of differentiation and integration are important tools for solving real world problems. Using calculus or any mathematical tool in physics is much more complex than the straightforward application of the equations and algorithms that students often encounter in math classes. Research in physics education has reported students‘ lack of ability to transfer their calculus knowledge to physics problem solving. In the past, studies often focused on what students fail to do with less focus on their underlying cognition. However, when solving physics problems requiring the use of integration, their reasoning about mathematics and physics concepts has not yet been carefully and systematically studied. Hence the main purpose of this qualitative study is to investigate student thinking in-depth and provide deeper insights into student reasoning in physics problem solving from multiple perspectives. I propose a conceptual framework by integrating aspects of several theoretical constructs from the literature to help us understand our observations of student work as they solve physics problems that require the use of integration. I combined elements of three important theoretical constructs: mathematical resources or symbolic forms, which are the small pieces of knowledge elements associated with students‘ use of mathematical ideas; conceptual metaphors, which describe the systematic mapping of knowledge across multiple conceptual domains – typically from concrete source domain to abstract target domain; and conceptual blending, which describes the construction of new learning by integrating knowledge in different mental spaces. I collected data from group teaching/learning interviews as students solved physics problems requiring setting up integrals. Participants were recruited from a second-semester calculus-based physics course. I conducted qualitative analysis of the videotaped student conversations and their written work. The main contributions of this research include (1) providing evidence for the existence of symbolic forms in students‘ reasoning about differentials and integrals, (2) identifying conceptual metaphors involved in student reasoning about differentials and integrals, (3) categorizing the different ways in which students integrate their mathematics and physics knowledge in the context of solving physics integration problems, (4)exploring the use of hypothetical debate problems in shifting students‘ framing of physics problem solving requiring mathematics. UNDERSTANDING INTRODUCTORY STUDENTS‘ APPLICATION OF INTEGRALS IN PHYSICS FROM MULTIPLE PERSPECTIVES by DEHUI HU B.S., University of Science and Technology of China, 2009 A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Physics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2013 Approved by: Major Professor N. Sanjay Rebello Copyright DEHUI HU 2013 Abstract Calculus is used across many physics topics from introductory to upper-division level college courses. The concepts of differentiation and integration are important tools for solving real world problems. Using calculus or any mathematical tool in physics is much more complex than the straightforward application of the equations and algorithms that students often encounter in math classes. Research in physics education has reported students‘ lack of ability to transfer their calculus knowledge to physics problem solving. In the past, studies often focused on what students fail to do with less focus on their underlying cognition. However, when solving physics problems requiring the use of integration, their reasoning about mathematics and physics concepts has not yet been carefully and systematically studied. Hence the main purpose of this qualitative study is to investigate student thinking in-depth and provide deeper insights into student reasoning in physics problem solving from multiple perspectives. I propose a conceptual framework by integrating aspects of several theoretical constructs from the literature to help us understand our observations of student work as they solve physics problems that require the use of integration. I combined elements of three important theoretical constructs: mathematical resources or symbolic forms, which are the small pieces of knowledge elements associated with students‘ use of mathematical ideas; conceptual metaphors, which describe the systematic mapping of knowledge across multiple conceptual domains – typically from concrete source domain to abstract target domain; and conceptual blending, which describes the construction of new learning by integrating knowledge in different mental spaces. I collected data from group teaching/learning interviews as students solved physics problems requiring setting up integrals. Participants were recruited from a second-semester calculus-based physics course. I conducted qualitative analysis of the videotaped student conversations and their written work. The main contributions of this research include (1) providing evidence for the existence of symbolic forms in students‘ reasoning about differentials and integrals, (2) identifying conceptual metaphors involved in student reasoning about differentials and integrals, (3) categorizing the different ways in which students integrate their mathematics and physics knowledge in the context of solving physics integration problems, (4)exploring the use of hypothetical debate problems in shifting students‘ framing of physics problem solving requiring mathematics. Table of Contents List of Figures ................................................................................................................................ ix List of Tables ................................................................................................................................. xi Acknowledgements ....................................................................................................................... xii Dedication .................................................................................................................................... xiii Chapter 1 - Introduction .................................................................................................................. 1 1.1 Motivation ............................................................................................................................. 1 1.2 Brief Introduction to the Theoretical Framework ................................................................. 3 Mathematical Resources ......................................................................................................... 3 Metaphors in Mathematical Thinking ..................................................................................... 3 Conceptual Blending ............................................................................................................... 4 1.3 Research Questions ............................................................................................................... 5 1.4 Main Contributions of This Research ................................................................................... 5 1.5 Overview of Dissertation ...................................................................................................... 6 Chapter 2 - Literature Review on Students‘ Use of Calculus ......................................................... 8 2.1 Introduction ........................................................................................................................... 8 2.2 Student Understanding of Calculus Concepts ...................................................................... 8 2.2 Student Application of Calculus Concepts in Physics ........................................................ 11 2.4 Summary ............................................................................................................................. 14 Chapter 3 - Conceptual Framework .............................................................................................. 16 3.1 Introduction ......................................................................................................................... 16 3.2 Resources Framework ......................................................................................................... 16 General Description of Resources ........................................................................................ 16 Mathematical Resources ....................................................................................................... 19 3.3 Conceptual Metaphor Framework ...................................................................................... 22 Metaphors in Everyday Language ........................................................................................ 22 Metaphors in Mathematical and Physics Thinking ............................................................... 23 3.4 Conceptual Blending Framework ......................................................................................