UNDERGRADUATE STUDENTS

CONCEPTIONS OF INEQUALITIES

by

Elena Filofteia Halmaghi

Licence in Mathematics, University of Bucharest, Romania, 1985

Thesis Submitted in Partial Fulfilment of

the Requirements for the Degree of

Doctor of Philosophy

In the

Faculty of Education

Elena Halmaghi 2011

SIMON FRASER UNIVERSITY

Spring 2011

All rights reserved. However, in accordance with the Copyright Act of

Canada, this work may be reproduced, without authorization, under the

conditions for Fair Dealing. Therefore, limited reproduction of this work for

the purposes of private study, research, criticism, review and news reporting

is likely to be in accordance with the law, particularly if cited appropriately.

ii

Approval

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Degree: Insert your upcoming degree here

Title of Thesis: Insert your title here. Title page must be the same as this.

Examining Committee:

Chair: Name [Correct title Consult your Grad Secretary/Assistant

Name Senior Supervisor

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Date Defended/Approved:

iii

Abstract

Inequalities are vital in the production of mathematics. They are employed as

specialized tools in the study of functions, in proving equalities, and in approximation or

optimization studies, to enumerate only a few areas of mathematics where inequalities are

put to work. The concept of inequality, however, is problematic for high school and

university students alike. Moreover, the school curriculum seems disconnected from the

role of inequalities in mathematics and mostly presents inequalities as a subsection of

equations. The placement of inequalities in the school curriculum and the disconnect

between school mathematics inequalities and a mathematicians approach to inequalities

take the blame of research in mathematics education reporting on students

misconceptions when dealing with this concept. This study moves from the theory of

misconceptions to a framework of undergraduate students conceptions of inequalities. In

an effort to learn more about what students see when dealing with inequalities, three

research questions are pursued: What are undergraduate students conceptions of

inequalities? What influences the construction of the concept of inequalities? How can

undergraduate students conceptions of inequalities expand our insight into students

understanding of, and meaningful engagement with, inequalities?

Data for this study was produced mostly through learner-generated examples of

inequalities that satisfy certain conditions. The participants in the research were

undergraduate students enrolled in two mathematics courses a foundations of

mathematics course and a precalculus course. The results of this research are five

iv

conceptions of inequalities. It was also found that the undergraduate students

conceptions of inequalities mostly occupy the lower regions of Talls Three Mental

Worlds of Mathematics. The speculation is that what Tall calls the met-befores as well

as what I call the missed-befores influence the construction of the concept of

inequalities. Curriculum suggestions for preparing the ground for the work on and with

inequalities are presented. This study contributes to ongoing research on mathematics

concept formation.

Keywords: inequality; conception; understanding; learner-generated examples; met-

before; missed-before

v

Dedication

To the loving memory of my mom, Ana, my first partner in problem

solving.

To the person who declared me a mathematician in grade 5 and taught

me how to prove by reductio ad absurdum in grade 6, Professor Lazar

Samoila, my middle school mathematics specialist teacher.

Domnului Profesor Lazar Samoila, cel care m-a declarat matematician

in clasa a 5-a iar in clasa a 6-a m-a invatat sa demonstrez prin reducere la

absurd.

vi

Acknowledgements

My wholehearted thanks extend to my Senior Supervisor, Dr. Peter Liljedahl,

whose passion for teaching, support of learning, and devotion to research in mathematics

education were inspirational. Moreover, his guidance, encouragement, feedback, and

insight throughout the completion of my program and dissertation were tremendously

helpful.

A special thanks and my gratitude to Dr. Rina Zazkis, whos timely and valuable

feedback and suggestions, as well as her continuous support, helped me navigate my way

around the program and find the focus of my work. I am so grateful to her for pushing me

to take action and to make decisions in crucial moments.

I would also like to extend my earnest appreciation to Dr. Ami Mamolo for her

collegiality, support, and encouragement. A profound thank you goes to Dr. Tom

Archibald for helping me locate the most valuable references for the historical account on

inequalities event that pushed my research and thinking into new and productive

directions. The participants in this research also deserve my sincere recognition for their

contribution in the two studies that are part of my dissertation.

I am truly grateful to Dr. Stephen Campbell and Dr. David Pimm, my external

examiners, for their willingness to take me to a defence on a short notice, for their

insightful questions during the examination, and for their valuable feedback on my work.

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Table of Contents

Approval ............................................................................................................................. ii Abstract .............................................................................................................................. iii

Acknowledgements ............................................................................................................ vi Table of Contents .............................................................................................................. vii List of Figures ..................................................................................................................... x List of Tables .................................................................................................................... xii

Chapter 1: Beginnings ...................................................................................................... 1 1.1 Introduction ................................................................................................................... 1

1.1.1 The Protagonist .................................................................................................... 1

1.1.2 Background .......................................................................................................... 4 1.1.3 Finding a Niche .................................................................................................... 8

1.2 Narrowing Down the Topic ........................................................................................ 10 1.3 Outline of the Dissertation .......................................................................................... 14

Chapter 2: Inequalities in Mathematics, History of Mathematics, and

Mathematics Education Research.............................................................. 16 2.1 What Are Inequalities? ............................................................................................... 16

2.1.1 Definitions .......................................................................................................... 16

2.1.2 Inequality Manipulation ..................................................................................... 18

2.1.3 School Methods of Solving Inequalities ............................................................ 21 2.1.4 Famous Inequalities ........................................................................................... 26

2.1.5 A Mathematicians Approach to Inequalities: A Loved Inequality ................ 27 2.2 How Did Inequalities Come to Be? ............................................................................ 32

2.2.1 Why Educators and Researchers in Education Are Concerned with the

History of Mathematics ...................................................................................... 34 2.2.2 The Evolution of the Inequality Concept ........................................................... 37

2.3 Where Are Inequalities Located in the K-12 Curriculum? ......................................... 57

2.3.1 Inequality in British Columbia Curriculum from K to University .................... 57 2.3.2 Inequalities in Romanian Curriculum ................................................................ 60 2.3.3 School Curriculum and Inequalities ................................................................... 65

Chapter 3: Inequalities in Mathematics Education Research .................................... 67 3.1 Inequalities in Mathematics Education ....................................................................... 68

3.1.1 A Call for Research on Inequalities Was Issued ................................................ 69 3.1.2