Heuristics Employed by Problem Solvers Engaged in a Robotics-Based Task Restricted; Files Only

Vela, Patricia (2017)

Permanent URL: https://etd.library.emory.edu/concern/etds/9s161661z?locale=en
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Abstract

Although Polya(1945) provided a list of heuristics that have proven fruitful in solving nonroutine problems, the mathematics education community has not reached agreement on how to apply this knowledge in mathematics classrooms. The purpose of this dissertation was to identify what heuristics were employed and how by six high school and six undergraduate students, who enjoyed solving mathematics problems, during their attempt to solve a robotic arm task designed to stimulate the applications of trigonometry in its solution. Structured, task-based interviews, audio and video recorded, were used for documenting the problem-solving episodes and were coded using a modified version of Kilpatrick's (1967) problem-solving coding scheme. The results suggest that access to the robotic arm together with heuristics identified by Polya enabled the mathematically proficient students to engage with the mathematical task. In particular, the heuristics Changes Condition and Successive Approximations promoted scientific engagement with mathematics - enabling students to experience mathematics as a discipline that includes exploration and reformulation of the original problem. The findings also suggest that when using the heuristic Drawing a Figure, identifying the unknown in the diagram is not an easy endeavour. This has implications for the teaching of mathematics because it suggests problem solvers should be given ample thinking time to recognize the salient features of the problem and, thus, be able to distinguish the unknowns from the knowns. Lastly, the study also finds that using the search engine Google (i.e. "Googling"), expands Polya's Recalling a Related Problem heuristic to now include Finding a Related Problem. Problem solvers are no longer limited to the information they know or can recall. Students, today, may need to be presented with more problems that challenge them to identify what information they need to help solve the problem at hand and then, be encouraged to search the web to find the specific missing knowledge needed for a successful resolution to the problem.

Table of Contents

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION ………………………………………….....…… 1

Statement of the Problem ………………...……..…....................................3

Purpose of the Study ……………………...………………..……………….4

CHAPTER 2. REVIEW OF THE LITERATURE ……..…….……..……………...5

Definitions and Theoretical Framework …………………..……………….5

Nonroutine Problem Solving ………….…..................5

Heuristics and Their Role in Nonroutine Problem Solving ......7

Heuristics in Mathematics Education ……………………………………...9

Robotics-Based Tasks and Heuristics ……………………...……………… 13

Robotics-Based Tasks and Mathematical Nonroutine Problem Solving …...17

Summary …………………………………………………………………...18

Research Questions ……………………………………….………………..20

CHAPTER 3. RESEARCH METHODOLOGY ………………………..………… 21

Pilot Studies ……………………………...…………………………..…...… 24

First Pilot Study ……….…………………………………………... 24

Second Pilot Study ……….………………………………………... 25

Sampling and Participants ………………………………….……………... 27

Data Collection ………………………………………………………………28

Initial Background Interview ...………………….………………… 29

Robotics-Based Task …………………………………….………... 30

Final Interview ……………………………….…………………….. 31

Materials …………………………………………………………... 33

Procedure ………………………………………………………….. 34

Qualitative Analysis ……………………………………………………….37

Reliability and Validity …………………………………………....38

Role of the researcher ………….…………………………... 39

Chain of evidence ……………………………….…………. 41

Internal and inter-coder reliability ………………….…….… 41

Data Storage …………………………………………………….....41

CHAPTER 4. RESULTS …...……….……..……………………………………… 42

The 12 Problem Solvers ……….……………………………………...42

Research Question 1 ……………………………………………………….66

Research Question 2 ………………………………………………….……67

Draws a Figure Heuristic ……………………………………………68

Changes Condition Heuristic ………………………………………..69

Recalls a Related Problem Heuristic ……………………………….70

Uses Related Problem in Solution Heuristic ………………………...71

Uses Successive Approximation Heuristic ……………………….…72

Sets up Equations Heuristic …………………………………………72

Uses Auxiliary Elements/Problems Heuristic ……………………….73

Works Backwards Heuristic ………………………………………..73

Decomposition Heuristic ……………………………………………74

Checks Solution Heuristic ………………………………………….74

Derives Solution by Another Method Heuristic …………………….75

Research Question 3 ………………………………………………….…….76

CHAPTER 5. DISCUSSION ………...……………………………………………78

Observation #1 …………………………………………………….……… 81

Implications for Theory ……………………………………………..83

Implications for Educators …………………………………………...84

Observation #2 …………………………………………………….……… 85

Implications for Theory………………………………………………87

Implications for Educators ………..……………………….………… 88

Observation #3 …….……………………………………………….……… 89

Implications for Theory ……………………………………………...90

Implications for Educators …………………………………………...91

Needed Future Research ………...………………..………………………….92

Final Remarks .………………………………………………………………. 96

REFERENCES ………………..…………………………………………………… 98

LIST OF TABLES AND FIGURES

Table 1. List of Heuristics Used by the 12 Participants ……………………………..108

Figure 1. John's Sample Work ……………………………………………………...109

Figure 2. Adam's Sample Work …………………………………………………….110

Figure 3. Sarah's Sample Work ……………………………………………………..111

Figure 4. Sarah Describing the Angle of Rotation of the Elbow ……………………112

Figure 5. Juana's Sample Work ……………………………………………………..113

Figure 6. Mark's Sample Work ……………………………………………………..114

Figure 7. Ebony's Sample Work …………………………………………………….115

Figure 8. Mena's Sample Work ……………………………………………………..116

Figure 9. Imani's Sample Work ……………………………………………………..117

Figure 10. Maisha's Sample Work ………………………………………………….118

Figure 11. Lola's Sample Work ……………………………………………………..119

Figure 12. Aniya's Sample Work …………………………………………………...120

Figure 13. Pablo's Sample Work ……………………………………………………121

APPENDICES

A. Instrument for the Initial Interview ………………………………………….122

B. Instrument for the Robotics-Based Task ……………………………………124

C. Instrument for Robotics-Based Task Activity Script ……………………….125

D. Instrument for the Final Interview ………………………...………………...127

E. Robotic Arm Prototype Used in the Study ………………………………….128

F. Echo pen used in the study …………………………………………………..129

G. Flash Cards Used in the Study ………………………………………………130

H. Instrument - Coding Sheet with Heuristic Definitions Used in the study …..135

I. Matrix Display Used to Keep Track of How Each Participant Employed Each Heuristic …..141

J. Technology Used for Recording and Analyzing the Problem-Solving Episode …………………142

K. Matrix Used to Keep Track of Information from the Initial and Final Interviews …………………143

L. Participant 5: Mark's Work and Sample of Counting of Figures Drawn …...144

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