Gender Differences on the American Mathematics Competition AMC 8 Contest

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Melissa A. Desjarlais
University of Nebraska at Lincoln, [email protected]
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GENDER DIFFERENCES ON THE AMERICAN MATHEMATICS
COMPETITION AMC 8 CONTEST
by
Melissa A. Desjarlais
A DISSERTATION
Presented to the Faculty of
The Graduate College at the University of Nebraska
In Partial Fulfilment of Requirements
For the Degree of Doctor of Philosophy
Major: Educational Studies
Under the Supervision of Professor David Fowler
Lincoln, Nebraska
May, 2009
GENDER DIFFERENCES ON THE AMERICAN MATHEMATICS
COMPETITION AMC 8 CONTEST
Melissa A. Desjarlais, Ph.D.
University of Nebraska, 2009
Adviser: David Fowler
This study examines gender differences on the American Mathematics Competition
AMC 8 contest between 2003 and 2007 by comparing the performances of male and
female United States eighth grade students after controlling for ability. During these
years 183,857 males and 178,857 females participated in the contest. Research on
gender differences frequently measures impact which is a difference in performance
between two groups that can often be explained by different ability distributions.
In contrast, differential item functioning (DIF) is a difference in performance after
controlling for ability. Three types of analyses were performed to compare the perfor-
mances. First, statistical analyses identified items with impact, DIF, and uniform or
nonuniform DIF. Differences in proportion correct were used to identify impact and
type of DIF while the Mantel-Haenszel procedure was used to identify items with gen-
der DIF. Second, substantive analyses placed the items into multiple categories based
onNCTM, Gierl, andHarnisch’s classifications of mathematics problems. Third, sub-
test analyses used these categories to look for gender differences in terms of impact
and DIF on subsets of the contest. While a majority of the items favored males in
terms of impact, after controlling for ability, few items demonstrated gender DIF.
None of the hypotheses of differing abilities in males and females suggested by earlier
studies were supported by the subtest analyses.
iii
ACKNOWLEDGMENTS
Writing my dissertation and finishing my doctorate while not living in Nebraska
creates many challenges, and successfully overcoming them during the past two years,
while also fulfilling the role of a faculty member, would have been much more difficult
if it were not for the encouragement, support, and aid of many people.
I would like to thank my supervisory committee for their guidance, support, and
flexibility. Thanks to my advisor, David Fowler. This journey began with a sugges-
tion by my advisor to analyze data from the American Mathematics Competition.
Finishing my degree at a distance can increase the responsibilities of an advisor, and
I want to express my appreciation to him for facilitating communication with my
committee and for obtaining signatures and submitting required forms.
I am grateful to Steve Dunbar for giving me permission to use the AMC data
and for providing electronic copies of the items included in this dissertation. He was
also very willing to answer my questions, about the data and the competition, and to
include me in an AMC 8 committee meeting. I also want to thank Del Harnisch for
his assistance in analyzing the data. Having conversations via his virtual office was
very beneficial both in terms of questions answered and encouragement given.
I want to thank the faculty and graduate students in the mathematics and ed-
ucation departments at UNL. I had many wonderful opportunities and experiences,
both in teaching and outreach, that helped me develop my philosophies about teach-
ing and learning and enhanced my interest in mathematics education. Teaching the
math content courses for pre-services teachers were especially meaningful experiences,
and I enjoyed working with Patience, Cheryl, Pari, and Raegan. The research meet-
ings with Rick, Josh, and Pari were also helpful. You all have helped me become a
better teacher and researcher.
iv
The Department of Mathematics and Computer Science at Valparaiso University
(VU) is a key reason why these past two years of working and writing have been
successful ones. The collegiality of the department is exemplified by a great sense of
humor, a willingness to help when needed, and a propensity for celebrating special
events with food. The gentle reminders to focus on my research and unflagging
support while writing and teaching were instrumental in helping me finish as planned.
I am blessed by the friends I have found in this department.
I would especially like to thank Rick Gillman for being protective of my time while
still encouraging me to pursue outreach opportunities related to my research area and
for always being willing to listen and offer advice. Thanks to Jerry Wagenblast for
his mentoring and for always being willing to talk about teaching and learning, and
to Michael Glass for his willingness to discuss my research and help with computer
issues. And I am grateful to Sara, Shane, and Daniel, who began their careers at
VU the same time I did; the transition to becoming a faculty member was easier and
more enjoyable because of sharing it with you.
A few words of appreciation are also due to those colleagues from math, CS, psy-
chologywithwhomIhavespent manyenjoyableFridayafternoons. Thecombinations
of backgrounds and personalities led to many interesting conversations, ranging from
insightful to “incite”ful, with both consensus and contention; no issue was too in-
significant to be discussed and debated. I have many fond memories of Fridays full
of camraderie and affectionate teasing. And special thanks to David, for being the
perennial instigator and always finding a way to make me laugh.
Finally, I would like to thank my family for their love, support, and patience
during the lengthy journey to finish my degree. I am grateful for growing up in a
home with two mathematics teachers, where mathematics and education were highly
valued, since it lead me to choose a career as a mathematics educator.
v
Contents
Contents v
List of Tables viii
1 Introduction 1
1.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Purpose statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Research questions or hypotheses . . . . . . . . . . . . . . . . . . . . 4
1.4 Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Literature Review 8
2.1 Review of the previous literature . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Differential Item Functioning . . . . . . . . . . . . . . . . . . 11
2.1.3 Differential Bundle Functioning . . . . . . . . . . . . . . . . . 12
2.1.4 Gender Similarities Hypothesis . . . . . . . . . . . . . . . . . 15
2.2 Summary of major themes . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 How present study will extend literature . . . . . . . . . . . . . . . . 19
3 Methods 21
vi
3.1 Sample and site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Access and permissions . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Instruments and their reliability and validity . . . . . . . . . . . . . . 22
3.3.1 American Mathematics Competition AMC 8 . . . . . . . . . . 22
3.3.2 Mantel-Haenszel Procedure . . . . . . . . . . . . . . . . . . . 27
3.4 Procedures of data collection . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Analysis of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.2 Substantive analysis . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.3 Subtest analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Results 36
4.1 Descriptive analysis of all data . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2 Substantive analysis . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.3 Subtest analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Analysis to address questions and hypotheses . . . . . . . . . . . . . 60
4.2.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 Research Hypotheses . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Tables and figures to display the data . . . . . . . . . . . . . . . . . . 66
5 Discussion 70
5.1 Summary of major results . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Relationship of results to existing studies . . . . . . . . . . . . . . . . 73
5.3 Limitations of the study . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Implications for future research . . . . . . . . . . . . . . . . . . . . . 78
5.5 Overall significance of the study . . . . . . . . . . . . . . . . . . . . . 79
vii
A Simpson’s Paradox 81
B Mantel-Haenszel Procedure 83
C Gierl et al. modified taxonomy 85
D Harnisch et al. attributes 89
E Items with Gender DIF 92
Bibliography 98
viii
List of Tables
1.1 Research hypotheses based on content and cognitive skills . . . . . . . . 5
3.1 Mean Scores on the AMC 8 Contest from 2003 to 2007 . . . . . . . . . . 22
3.2 Speededness on the AMC 8 Contest from 2003 to 2007 . . . . . . . . . . 23
3.3 Internal Consistency on the AMC 8 Contest from 2003 to 2007 . . . . . . 24
3.4 The 2×2 Contingency Table for ability level m. . . . . . . . . . . . . . . 28
3.5 Ability levels based on total score on the AMC 8 . . . . . . . . . . . . . 31
4.1 Impact and MH D-DIF for the 2003 AMC 8 Contest . . . . . . . . . . . 39
4.2 Impact and MH D-DIF for the 2004 AMC 8 Contest . . . . . . . . . . . 40
4.3 Impact and MH D-DIF for the 2005 AMC 8 Contest . . . . . . . . . . . 41
4.4 Impact and MH D-DIF for the 2006 AMC 8 Contest . . . . . . . . . . . 42
4.5 Impact and MH D-DIF for the 2007 AMC 8 Contest . . . . . . . . . . . 43
4.6 Summary of Classifications for the 2003 AMC 8 Contest . . . . . . . . . 45
4.7 Summary of Classifications for the 2004 AMC 8 Contest . . . . . . . . . 46
4.8 Summary of Classifications for the 2005 AMC 8 Contest . . . . . . . . . 47
4.9 Summary of Classifications for the 2006 AMC 8 Contest . . . . . . . . . 48
4.10 Summary of Classifications for the 2007 AMC 8 Contest . . . . . . . . . 49
4.11 Distribution of MH D-DIF Values . . . . . . . . . . . . . . . . . . . . . . 50
4.12 Statistical Analysis Classification . . . . . . . . . . . . . . . . . . . . . . 51
ix
4.13 NCTM Classifcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.14 Gierl et al. Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.15 Harnisch et al. Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.16 Classification by Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.17 Gender of Names included in Stem . . . . . . . . . . . . . . . . . . . . . 54
4.18 Impact by Classification Method . . . . . . . . . . . . . . . . . . . . . . 55
4.19 MH D-DIF by NCTM Standard . . . . . . . . . . . . . . . . . . . . . . . 56
4.20 MH D-DIF by Gierl et al.’s Modified Taxonomy . . . . . . . . . . . . . . 57
4.21 MH D-DIF by Harnisch et al.’s Attributes . . . . . . . . . . . . . . . . . 58
4.22 MH D-DIF by Length of Stem . . . . . . . . . . . . . . . . . . . . . . . . 59
4.23 MH D-DIF by Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.24 AMC 8 Contest Scores by Gender . . . . . . . . . . . . . . . . . . . . . . 67
4.25 Number of Items by Impact . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.26 Items with Non-negligible MH D-DIF . . . . . . . . . . . . . . . . . . . . 69
A.1 Summary of the Performance of Two Hypothetical Groups on an Imagi-
nary Item . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.1 Relationship of Gender to Item Response in the ith Stratum. . . . . . . . 83