gender and doctoral mathematics: impactful factors …
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GENDER AND DOCTORAL MATHEMATICS:
IMPACTFUL FACTORS FOR THE SUCCESS OF FEMALE STUDENTS
by
Emily Miller
A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Education
Summer 2015
© 2015 Emily Miller All Rights Reserved
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GENDER AND DOCTORAL MATHEMATICS:
IMPACTFUL FACTORS FOR THE SUCCESS OF FEMALE STUDENTS
by
Emily Miller
Approved: __________________________________________________________ Ralph P. Ferretti, Ph.D. Director of the School of Education Approved: __________________________________________________________ Carol Vukelich, Ph.D. Interim Dean of the College of Education and Human Development Approved: __________________________________________________________ James G. Richards, Ph.D. Vice Provost for Graduate and Professional Education
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Dawn Berk, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Pamela Cook, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ James Hiebert, Ph.D. Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Charles Hohensee, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Ximena Uribe-Zarain, Ph.D. Member of dissertation committee
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ACKNOWLEDGMENTS
The journey of the past four years would not have been the transformational
experience it was without the support of my advisor and committee members, my
professors and fellow students, and my mom. First and foremost, I would like to
express my gratitude to my advisor, Dr. Dawn Berk, without whom I may never have
learned the meaning of the word pointify, and may still, tragically, be submitting
documents without page numbers. Thank you for your constant encouragement and
thoughtful mentoring, for always believing in me, and for making even the most
tedious tasks extremely entertaining. You have set such a strong example for the
professor and researcher I hope to become – balancing an impressive research agenda,
a passion for teaching, and a family, while maintaining a sense of humor about the ups
and downs of it all. Even though it started as a joke, I am sure I will find myself in the
future asking, “WWDD?” (What would Dawn do?) in times of uncertainty. Since day
one, I have been and always will be a REESE-ling!
To my committee members, Drs. Pam Cook, Jim Hiebert, Charles Hohensee,
and Ximena Uribe-Zarain: thank you for providing thoughtful, valuable feedback
throughout the dissertation process. I truly appreciate your time and dedication in
helping me to produce this work, of which, I am so proud. Each of you provided a
distinct perspective and the final product is better for it. Thank you for your support
and encouragement.
I would also like to thank the mathematics education faculty members at the
University of Delaware. I cannot envision a better introduction to the field than
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having the opportunity to work alongside such dedicated, accomplished, and
passionate scholars during my four years as a doctoral student. I am grateful for your
feedback and support, and I have learned so much from each of you that I will carry
with me into my career.
Thank you to the mathematics education doctoral students at the University of
Delaware. Whether I was feeling “motivated,” “stressed,” “cosmopolitan,” or
“concerned in slow motion,” I always knew I would find an empathetic ear. I can’t
imagine what this experience would have been like had I not shared an office with all
of you. It has been my pleasure to learn and collaborate with some of the smartest,
funniest, and most generous people I know. To Heather Gallivan and Steve Silber: it
is impossible to sum up what you mean to me in a single word. Wait… No, it’s not.
To Rebecca Chambers, Nicole Hansen, and my “other me,” Ali Marzocchi: so
much has changed since we enrolled as first-year doctoral students four years ago, but
one constant has been the support I received from the three of you. I could not have
asked for more caring, intelligent, hilarious, delightfully wacky friends to have by my
side. Most of my fondest memories of this experience star the three of you, and I
wouldn’t have had it any other way!
Last, but, most assuredly, not least, thank you to my mom, whose unwavering
love and support has shaped me into the person I am today. Even when my
confidence was shaken, you believed in me enough for both of us. Everything I have
accomplished and everything I will accomplish is because of your support. This
dissertation is dedicated to you, Mama.
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TABLE OF CONTENTS
LIST OF TABLES ........................................................................................................ ix ABSTRACT ................................................................................................................... x
Chapter
1 FACTORS CONTRIBUTING TO THE SUCCESS OF FEMALE MATHEMATICS DOCTORAL STUDENTS: A REVIEW OF THE LITERATURE ................................................................................................... 1
(Unsuitable) Hypotheses about the Cause of the Gender Gap ........................... 2 The Case for Gender Diversity .......................................................................... 4 Is the Problem Unique to Mathematics? ............................................................ 7 Research Questions and Methodology ............................................................... 8 Review of the Relevant Literature ................................................................... 11
A Longitudinal Model of Doctoral Persistence ......................................... 11 Personal Factors ......................................................................................... 12
Personal Characteristics ....................................................................... 12 Personal Considerations ....................................................................... 15 Content Preparation .............................................................................. 17
Programmatic Factors ................................................................................ 17
Sense of Belonging .............................................................................. 17 Quality and Availability of Courses ..................................................... 18 Interactions with Others in the Department ......................................... 19 Academic Benefits of Institutional Support ......................................... 21 Fairness of Policies .............................................................................. 23 Academic Support from Advisor ......................................................... 24
Factors Related to Gender .......................................................................... 24
Professor Gender Ratios ....................................................................... 24 Student Gender Ratios .......................................................................... 26 Gendered Beliefs and Actions of Others .............................................. 27
Conclusion ........................................................................................................ 28
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2 GENDER AND DOCTORAL MATHEMATICS: IMPACTFUL FACTORS FOR THE SUCCESS OF FEMALE STUDENTS ....................... 31
Research Questions .......................................................................................... 33 Methods ............................................................................................................ 34
Survey Instrument ...................................................................................... 34 Sample ........................................................................................................ 37
Power Analysis ..................................................................................... 39 Survey Responses Received ................................................................. 40
Data Collection ........................................................................................... 43 Data Analysis ............................................................................................. 43
Creating an Overall Scale for Institutional Support Experiences ........ 43 Formulating Latent Constructs with Exploratory Factor Analysis ...... 44 Multiple Imputation for Missing Data ................................................. 50 Partial Least Squares Structural Equation Modeling ........................... 50
Evaluating the Reflective Measurement Model ............................. 52 Evaluating the Formative Measurement Models ........................... 53 Evaluating the Structural Model .................................................... 58
Methods Used to Investigate Research Question 1 .............................. 60 Methods Used to Investigate Research Question 2 .............................. 60
Results .............................................................................................................. 61
RQ 1: Impactful Factors in the Success of Female Participants ................ 61 RQ 2: Comparison of Significant Factors for Female and Male Participants ................................................................................................. 62
Discussion ........................................................................................................ 64 Conclusion ........................................................................................................ 69
REFERENCES ............................................................................................................. 70 Appendix
A SURVEY INSTRUMENT ............................................................................... 81 B PATH MODEL ................................................................................................ 91 C INSTITUTIONAL REVIEW BOARD APPROVAL LETTER ...................... 92
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LIST OF TABLES
Table 1 Number (Percent) of Institutions by Type in the Sampling Frame ............ 38
Table 2 Number (Percent) of Survey Invitations and Responses Received by Institution Type ....................................................................................... 41
Table 3 Number (Percent) of Survey Responses Received by Gender and Job Title ......................................................................................................... 41
Table 4 Number (Percent) of Survey Responses by Gender and Institution Type . 42
Table 5 Number of Survey Responses by Time Since Degree (Leaving) and Completion Status ................................................................................... 42
Table 6 Factors and Factor Loading Values ............................................................ 45
Table 7 Formative Indicator Outer Weights ........................................................... 55
Table 8 Effect Sizes for Latent Constructs .............................................................. 59
Table 9 Pooled Path Coefficients for Female and Male Participants ...................... 62
Table 10 Multi-group Analysis for Female and Male Participants ........................ 63
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ABSTRACT
Although the gender gap in participation in undergraduate mathematics has
narrowed, significant disparities still exist at the doctoral level. To better understand
issues of retention for female mathematics doctoral students, this study was designed
to identify factors most crucial to success for female students and to compare these
factors to those identified as most crucial for male students. A survey instrument was
designed and administered to a stratified, random sample of male and female
mathematics faculty members (n = 662) employed at post-secondary institutions in the
United States. Data were first analyzed with exploratory factor analysis to identify
underlying constructs. Sixteen latent constructs were identified, which were then
analyzed using partial least squares structural equation modeling (PLS-SEM) to
determine the relative effects of each identified factor on doctoral program success.
Analyses of the data indicate that different factors were influential in the success of
male and female mathematics doctoral students. For female participants, personal
characteristics, personal considerations, support from their advisor, academic benefits
of their assistantship, and the obstacles faced were critical to their success in their
doctoral program. For male participants, the critical factors were personal
characteristics, content preparation, support from their advisor, the quality of their
coursework, and the fairness of policies within the program. Of all the constructs
tested, only one – Obstacles Faced – was a significantly stronger predictor of doctoral
program success for women than for men. Much of the previous research in this area
has focused on issues of attrition of female doctoral students, utilizing small samples
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and qualitative methodologies. In contrast, this study used a large, representative
sample of mathematics faculty members and investigated the comparative impact of
factors related to doctoral program success. Thus, this study makes a unique
contribution to better understanding the mechanisms underlying success in doctoral-
level mathematics. Based on the findings, five key recommendations are proposed to
guide revisions to mathematics doctoral programs to increase the success of female
students.
1
Chapter 1
FACTORS CONTRIBUTING TO THE SUCCESS OF FEMALE MATHEMATICS DOCTORAL STUDENTS: A REVIEW OF THE
LITERATURE
Although the gender gap in participation in undergraduate mathematics has all
but closed, significant disparities still exist at the doctoral level. In 2013, only 27% of
doctoral degrees in mathematics were awarded to women (Vélez, Maxwell, & Rose,
2014). Furthermore, women attempting to obtain doctorate degrees in mathematics
are less successful in completing their degrees than their male counterparts (Berg &
Ferber, 1983; Nerad & Miller, 1996; Sowell, 2008). Even those women who do
complete their Ph.D. degrees take longer to do so, on average, than men (Herzig,
2004a). As a potential consequence of gender differences in doctoral study, only 21%
of tenured mathematics faculty members are women (Blair, Kirkman, & Maxwell,
2013), and a meager six percent of faculty positions in top-tier doctoral-granting
universities are held by women (Lutzer, Maxwell, & Rodi, 2002; as cited in the Report
of the BIRS Workshop on Women in Mathematics, 2006).
This decline in participation has been characterized as a “leaky pipeline,” in
which more women leak out than men (Blickenstaff, 2005; Herzig, 2004a). Even for
those women who do reach the end of the pipeline and receive doctorates in
mathematics, Etzkowitz, Kemelgor, Neuschatz, and Uzzi (1994) question “whether
[they] have a graduate experience that is of as high a quality as that of men in terms of
technical research training, mentorship, first job placement, and care taken for their
introduction into an academic career” (p. 158). Whatever the cause or causes, the
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problem is both progressive, in that it worsens at more advanced stages, and persistent,
in that little has changed over time, even with efforts to remedy the problem (Cronin
& Roger, 1999).
(Unsuitable) Hypotheses about the Cause of the Gender Gap
Many hypotheses have been proposed as the trigger for this gender gap. Each
of these hypotheses centers on a single factor, inherent to students, as the explanation
for the differential success of male and female students in mathematics. First, one of
the most frequently cited rationales is women are less capable in mathematics (directly
or indirectly) due to biological differences (Geary, 1996). However, studies
comparing five main cognitive abilities with strong ties to mathematics performance
for males and females have failed to produce consistent, significant differences for any
age group (Ceci, Williams, & Barnett, 2009; Spelke, 2005). In 35 studies of
hypothesized differences in mathematical abilities of male and female students of
various ages, 15 found no significant differences, 16 found differences favoring male
students, and four found differences favoring female students (Maccoby & Jacklin,
1974). Maccoby and Jacklin (1974) attribute any existing differences in mathematical
performance, not to biology, but to differences in “mathematical styles” (p. 91), citing
male students’ greater use of a “space factor” (spatial ability) in solving mathematics
problems. These findings related to differences in cognition are what Halpern (2000)
describes as “differences […] not deficiencies” (p. 8). A more recent meta-analysis
conducted by Hyde (1996) found that, while boys did perform slightly better on
mathematical performance and spatial perception than girls, the magnitude of the
differences was not large enough to explain the extent to which women are
underrepresented in scientific fields. Furthermore, Blickenstaff (2005) warns about
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the dangers of adopting the biological deficit perspective: if the underrepresentation of
women in mathematics stems from differences in biology, there is then no need to take
action to remedy the supposedly unchangeable status quo.
A second frequently cited rationale for the gender gap is that the mathematical
abilities of men are more variable, and therefore, men are more likely to study
advanced mathematics. The primary study claiming that male students demonstrate
greater variability when it comes to mathematics performance bases this conclusion on
SAT results (Benbow & Stanley, 1980, 1983). In a study of “mathematically
precocious youth,” Benbow and Stanley (1983) found that 12 times as many
mathematically gifted male students as female students were represented in the top
one percent of scores on the mathematics portion of the SAT. Concerns about the
experimental design of this study and its underlying assumptions were published as
early as 1981 (Schafer, 1981; Stage, 1981). Furthermore, subsequent studies of
classroom performance and degree obtainment revealed that girls in the sample
received better grades in high school mathematics courses than boys and graduated
with degrees in mathematics at similar rates, and that a nearly equal percentage of
male and female participants went on to receive doctorates in mathematics (2.2% and
2.1%, respectively; Benbow, Lubinski, Shea, & Eftekhari-Sanjani, 2000; Xie &
Shauman, 2003). These findings led the researchers to conclude that the SAT
“overestimate[s] the abilities of talented boys, relative to girls” (Spelke, 2005, p. 955).
Finally, it has been posited that the gender gap in participation in doctoral-level
mathematics is due to an inherent lack of interest by women. However, this supposed
biological disinterest cannot fully explain the gender disparity at the doctoral level.
Given that gender parity now exists in mathematics at the undergraduate level (Hill,
4
Corbett, & St. Rose, 2010), this same achievement should then be possible at the
doctoral level. Furthermore, with a greater proportion of women enrolling in doctoral
programs, but choosing to discontinue their doctoral studies than men, Kerlin (1997)
proposes either that many female students are under-qualified upon their admittance,
or that the decision to leave is a product of their experience within the program. So,
while disinterest may in fact be an explanation for the differing participation in
doctoral mathematics for men and women, this disinterest may not be inherent.
In contrast to these hypotheses, which focus on a single, personal factor as the
explanation for the gender gap, this review is based on the premise that two other
hypotheses, either acting alone or in concert, better explain the underrepresentation of
women in mathematics doctoral study. The first hypothesis is that women become
increasingly disinterested in mathematics due to a confluence of cultural and societal
factors that systematically develop and encourage this disinterest. The second
hypothesis is that women experience their doctoral programs in qualitatively different
ways than their male peers. These differences, likely negative, in their doctoral
mathematics courses or in other interactions with professors, peers, and their
environment, may influence female students’ ability to be successful.
The Case for Gender Diversity
In the past, attrition may have been seen as beneficial in doctoral programs,
with an “only the best will survive” mentality differentially impacting female students
(Sternberg, 1981, as cited in Kerlin, 1997). Others have countered this assumption:
“The source of graduate student attrition is not inadequate students but indifferent and
wasteful programs” (Lovitts & Nelson, 2000, p. 44). Lovitts and Nelson (2000) claim
doctoral program attrition is wasting human capital and continues to do so, due to the
5
lack of improvement in retaining female students. The National Science Foundation
(NSF) recognizes the need for diversity in STEM by funding programs like
“Increasing the Participation and Advancement of Women in Academic Science and
Engineering Careers” (ADVANCE; NSF, n.d.-a). However, “much remains to be
done” to achieve gender equity (Institute of Medicine, National Academy of Sciences,
& National Academy of Engineering, 2006, p. 1).
Why is gender equity important for doctoral-level study of mathematics?
There are four primary arguments for the importance of this issue. First, all students,
regardless of gender, should have equitable opportunities to pursue the study of
advanced mathematics. As Drew (2011) states: “Women […] are consistently
discouraged from studying science and mathematics, the very subjects that would give
them access to power, influence, and wealth” (p. 195). Personal preferences and
personalities certainly play a role; however, Hsu, Murphy, and Treisman (2008)
contend, “systemic factors […] also affect students’ choices” (p. 205).
Second, the gender gap in mathematics at the doctoral level reinforces
persistent gender stereotypes in mathematics. A common archetype is that of the
mathematician as “someone who solved the Rubik’s cube at eight, took calculus at
fourteen, and was tackling serious mathematics at sixteen. And he’s a guy” (Haas &
Henle, 2007, p. 957). A similar stereotype has been described as the “boy wonder”
syndrome (Alper, 1993, p. 411). Or, as Nosek and Banaji (2002) succinctly title their
article, “Math = Male, Me = Female, Therefore Math ≠ Me.” Over time, slight biases
against women can lead to what Valian (2007) terms the “accumulation of advantage”
(p. 32). She analogizes this accumulation to compound interest, as “small imbalances
in evaluation and perception add up to advantage men and disadvantage women” (p.
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34). Stereotypes may contribute to the continuation of gender disparity in doctoral
study, where perhaps gender differences are not a result of differing competencies, but
of the cumulative effects of persistent views of mathematics as a masculine discipline.
These stereotypes, then, may be self-perpetuating, which only serves to worsen their
impact.
A third reason to work toward gender balance in doctoral mathematics is that
not doing so is simply a waste of mathematical talent that could be put toward
innovation and advancements in the field. With mathematics being designated an
“area of national need,” any loss in participation exacerbates the problem (United
States Department of Education, 2015). As Drew (2011) argues, “We must end the
cycle of negative expectations and wasted talent in this country” (p. 196).
The final, and perhaps most compelling, reason for the importance of gender
equity in doctoral-level mathematics is that diversity in the discipline is beneficial for
the discipline itself. Blickenstaff (2005) describes the issue in this way:
If only one kind of person asks the questions and interprets the results, then the field of scientific inquiry will be narrow and inbred. Science can be improved by broadening the diversity of its practitioners across gender, ethnic and racial lines and science education can be improved by acknowledging the political nature of scientific enquiry. (p. 383)
Although Blickenstaff makes the preceding case for diversity in science education, the
same arguments apply for the field of mathematics. Studies have shown that diverse
groups exhibit better problem solving behaviors and hold more complex discussions
than homogeneous groups (Antonio et al., 2004; Nemeth & Wachtler, 1983).
Furthermore, the description for an NSF program focused on diversifying the
engineering workforce states, “by seeing problems in different ways, a diverse
workforce can encourage innovation and scientific breakthroughs” (NSF, n.d.-b, para.
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7). In a broader sense, a lack of gender parity affects society as a whole: “The vitality
of the scientific enterprise and the prosperity of the North American countries depend
on the broad development of the mathematical sciences and on full access to that
development by all members of society” (Report of the BIRS Workshop on Women in
Mathematics, 2006, p. 1). Innovation in mathematics is currently jeopardized by a
lack of diversity of thinking and underrepresentation of women in the field, which
Porush (2010) likens to “design[ing] scissors only for righties” (p. 19).
Is the Problem Unique to Mathematics?
It is important to note that the underrepresentation of women is also
characteristic of other STEM disciplines. For example, in recent years, women
received only 18% of engineering doctoral degrees (Engineering Workforce
Commission, 2005), 18% of physics doctoral degrees (Ivie & Ray, 2005), and around
one-third of chemistry doctoral degrees (Institute of Medicine, National Academy of
Sciences, and National Academy of Engineering, 2006).
Given similar levels of gender disparity at the doctoral level in multiple STEM
disciplines, there are likely to be some common factors at play that would be relevant
to the specific issue of retention of women in doctoral-level mathematics. However,
mathematics doctoral programs differ in important ways from the laboratory-based
science disciplines. For example, many doctoral programs in other STEM disciplines
are structured differently, with groups of doctoral students working in a laboratory
environment under a common advisor. This may create an automatic community with
shared experiences, and result in students being less likely to withdraw from their
studies (McAlpine & Norton, 2006). Mathematics programs, on the other hand, tend
to involve students working individually on their own research. As noted in the
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Report of the BIRS Workshop for Women in Mathematics (2006), “mathematics has a
distinct culture, and the issue of systemic barriers in the advancement of women in
mathematics is difficult and subtle” (p. 4). Thus, findings pertaining to the retention
of women in doctoral programs for other STEM disciplines may have limited
applicability to understanding women’s experiences in doctoral programs in
mathematics.
Not surprisingly, in non-STEM disciplines, such as the social sciences and the
humanities, the problem is actually reversed. According to a survey of nearly 20,000
doctoral students, the ten-year completion rates in engineering, the life sciences, and
mathematics and the physical sciences were higher for men than for women , while the
reverse was true for social sciences and humanities disciplines (Sowell, 2008).
Research Questions and Methodology
McAlpine and Norton (2006) assert that retention and attrition are influenced
by the “interaction of a constellation of dynamic factors” (p. 5). Furthermore, Kerlin
(1997) suggests that it is beneficial to view doctoral student attrition, not as “a solitary
event,” but as “the consequence of a dynamic process” (p. 21). One key improvement
strategy will be to systematically identify and review all of the potential factors that
may influence the success of women in mathematics doctoral programs. Identification
of all the potential factors would enable researchers to design and conduct studies to
investigate whether, and to what extent, each of the key factors contributes to
women’s success. Better understanding which factors matter most, and in what ways,
would enable institutions and programs to make informed changes to better support
women in doctoral-level mathematics.
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In this review, I will identify and synthesize the various factors that have been
posited as having an impact on the degree completion of female mathematics doctoral
students and detail available evidence for each. To that end, the research questions
guiding this review are: 1) What factors have been identified in the research literature
as relating to or causing the attrition or retention of female mathematics doctoral
students? and 2) Which of these factors are most likely to influence the success of
female mathematics doctoral students?
The literature search for this review began with a narrow focus, consisting only
of peer-reviewed, empirical studies focused on the experiences of female mathematics
doctoral students. It was subsequently broadened with literature from related areas
pertaining to the experiences of female doctoral students in other STEM areas,
undergraduate retention, or women in the STEM workforce. This included works in
the fields of education, psychology, sociology, economics of education, cognitive
science and others. Works in related areas were either located because they were cited
in an article already included within the review, or because they further illuminated a
proposed factor from a piece already within the review. In this way, at least 230
works were reviewed. Of these works, 96 are cited here. The search for additional
literature was concluded when saturation of proposed factors was reached. At the
point of saturation, meta-analysis articles (e.g., Bair & Haworth, 2004) and literature
review articles (e.g., Herzig, 2004a) were consulted for a second time to ensure that
the set of identified factors was complete.
While a case was made earlier for the differences between doctoral study in
mathematics and other STEM disciplines, the intent of including literature from other
disciplines was to generate as comprehensive a list as possible of potentially impactful
10
factors. Once empirically tested, factors identified from other disciplines may prove
to be inconsequential or may impact success in different ways for students of
mathematics; however, this will not be known until these factors are identified and
tested.
Literature related to issues of both attrition and retention is included in the
review. Many constructs, such as retention, completion, success, and persistence have
been used in prior research to characterize the outcome of a student’s doctoral
program experience (e.g., Ampaw & Jaeger, 2011; Miller, 2015a, 2015b; Nerad &
Miller, 1996; Tinto, 1993). Each of these constructs – retention, completion, success,
and persistence – has been used to represent doctoral degree attainment. In the
absence of degree attainment, attrition has been used to describe the trajectories of
students who discontinued their studies (e.g., Bair & Haworth, 2004; Golde, 1998).
This review includes articles pertaining to any of these constructs.
Finally, no restrictions were placed in terms of the methods used in the
literature reviewed. Included in this review are empirical studies, employing
quantitative, qualitative, or mixed methods, as well theoretical articles and reviews of
literature pertaining to this topic. This review is not intended to be an exhaustive
description of every published work in this area; it is instead meant to identify and
describe the gamut of hypothesized factors influencing the experiences of female
mathematics doctoral students and make judgments, based on the strength of the
evidence, about which factors are likely to hold the greatest potential for reducing the
gender gap.
11
Review of the Relevant Literature
This section begins with a description of a theoretical model of doctoral
persistence proposed by Tinto (1993). Although this model is not specific to female
students or to the field of mathematics, it outlines a chronology of the doctoral student
experience, which is used to organize the relevant literature presented in the remainder
of the review. After describing the model, 12 factors that have the potential to
influence the success of female mathematics doctoral students will be presented, along
with supporting evidence.
A Longitudinal Model of Doctoral Persistence
My review of the existing literature did not identify a comprehensive model of
doctoral student persistence specific to mathematics or to gender. However, Tinto
(1993) proposed a general framework to examine doctoral persistence. His model is
one of the only attempts to organize the complex structure of doctoral persistence both
across time and across the local and external environments. Tinto’s model organizes
graduate student experience into three chronological stages: (1) transition and
adjustment, (2) development of competence, and (3) dissertation completion.
The first stage, transition and adjustment, lasts for about the first year of
doctoral study and is the process of socialization and accommodation into the work
and norms of the student’s graduate department. The student judges her compatibility
with the graduate department and attempts to establish herself in the academic and
social spheres of her surroundings. This stage is also dependent on the compatibility
of the student’s career goals and the goals of the doctoral program itself.
The second stage, development of competence, usually begins at the second
year of the doctoral program and ends when the student passes candidacy
12
examinations. As Tinto (1993) states: “Successful completion of this stage mirrors
both individual abilities and skills and the character of personal interactions with
faculty within the academic domain of the institution” (p. 236). The termination of the
second stage and the initiation of the third stage signifies that the student has acquired
the knowledge and skills required to conduct doctoral-level research, in the form of a
dissertation.
The third and final stage, dissertation completion, spans the period of time
from gaining doctoral candidacy to a successful dissertation defense. During this
stage, the student’s advisor and dissertation committee become the central figures in
the experience. This represents a shift from the first two stages, in which the student
likely has interactions with many professors on a regular basis due to coursework.
Persistence at this stage is not easily defined or predicted, since it relies so heavily on
the behavior and interactions of such a small group.
This model was used to provide a structure with which to organize the 12
factors that will be presented in the following sections. Factors will be presented
chronologically, as is the organization of Tinto’s (1993) model, in accordance with
how a student would experience them. The factors are organized into three main
sections containing related factors: Personal Factors, Programmatic Factors, and
Factors Related to Gender.
Personal Factors
Personal Characteristics
Each student enters her doctoral program with a unique set of personal
characteristics that may influence her ability to be successful in doctoral study.
13
Personal characteristics that may influence doctoral success include natural talent in
mathematics (e.g., Becker, 1984), motivation (e.g., Preckel, Goetz, Pekrun, & Kleine,
2008), commitment to doctoral study (e.g., Baird, 1993), and confidence (e.g., Ülkü-
Steiner, Kurtz-Costes, & Kinlaw, 2000). The attribution of natural talent in
mathematics may hold a differential impact on the graduate studies of men and
women. Research has shown that men view talent in mathematics as a symbol of
intellectual status, furthered by attending graduate school, while women are “less sure
of their goals in attending graduate school, and much less certain of their abilities to
do well” (Becker, 1984, p. 45). Women are more likely to respond to validation of
their abilities from others (Becker, 1984). Furthermore, students’ beliefs about the
nature of learning and intelligence may be influential. Those who believe that
intelligence is not a fixed trait and that abilities can be developed through hard work,
what Dweck (2006) terms a growth mindset, are more resilient to obstacles, more
motivated to learn and succeed, and less impacted by negative stereotypes in
mathematics (Good, Rattan, & Dweck, 2007).
Leslie, Cimpian, Meyer, and Freeland (2015) found that faculty, post doctoral
researchers, and graduate students in science, technology, engineering, and
mathematics (STEM) disciplines expressed stronger “field-specific ability beliefs,” or
the belief that “fixed, innate talent” is a prerequisite for success in a certain field (p.
262). As these researchers had hypothesized, the more a field expected brilliance in
order to be successful, the lower the proportion of doctorate recipients that were
women. In a hierarchical multiple regression, the field-specific ability beliefs score
explained additional variance in predicting the proportion of female doctorate
recipients in a field, even after controlling for variables representing alternative
14
hypotheses, such as number of hours worked, selectivity of the discipline, and the
extent to which work in the discipline requires systematizing versus empathizing
behaviors. Therefore, while a student’s own self-judgments of intelligence are
important, the expectations of the field may amplify a student’s insecurity.
Motivation and commitment are also key personal characteristics for dealing
with the challenges of graduate study. Motivations are characterized by Middleton
and Spanias (1999) as “reasons individuals have for behaving in a given manner in a
given situation” (p. 66). Studies have documented differences in motivation for
studying mathematics between male and female students (e.g., Preckel, Goetz, Pekrun,
& Kleine, 2008; Skaalvik & Skaalvik, 2004); however, these differences may stem
from gendered perceptions of mathematics (Middleton & Spanias, 1999). Nagi (1974)
reported that “motivation or the lack of motivation was the dominant personal reason
for either completion or non-completion of the doctoral program” p. 61).
Commitment to degree attainment and commitment to the field of study have also
been suggested as potential factors related to doctoral retention (Baird, 1993). In
summary, students entering a doctoral program in mathematics bring with them
particular personal attributes that may influence their ability to be successful in the
program. These attributes include natural talent in mathematics, motivation, and
commitment, and may differentially impact the experiences of women as compared to
men.
More generally, confidence in one’s own ability to succeed may influence
retention. Numerous studies have found that women exhibit lower self-confidence in
male-dominated fields when compared to students of either gender in gender-balanced
programs (e.g., Becker, 1984; Mura, 1987; Ülkü-Steiner et al., 2000). Zeldin and
15
Pajares (2000) investigated the self-efficacy beliefs of women in STEM careers,
drawing on Bandura’s (1977) definition of self-efficacy as “people’s judgments of
their capabilities to produce designated levels of performance” (p. 216). Zeldin and
Pajares found that higher self-efficacy was associated with greater resilience in the
face of obstacles. Women, in particular, are susceptible to “imposter syndrome”
(Widnall, 1988, p. 1743). Even women with objectively comparable qualifications
often fear “being ‘found out’” when their own perceived shortcomings can no longer
be hidden from those around them (Widnall, 1988, p. 1744). Thus, doctoral students’
confidence in their own abilities may be a key factor affecting their ability to
successfully complete a doctorate in mathematics.
Some doctoral programs require students to take benchmark examinations,
such as preliminary, qualifying, or candidacy examinations. These examinations have
been shown to have an impact on students’ confidence (Hollenshead, Younce, &
Wenzel, 1994). For those students who pass on their first attempt, confidence in their
mathematical abilities increases (Earl-Novell, 2006). However, students who take
these exams multiple times without success report that the exams are stressors and
described them with “adjectives such as ‘brutal,’ ‘terrifying,’ ‘difficult,’ and ‘hard’”
(Earl-Novell, 2006, p. 49).
Personal Considerations
The student’s experience may also be impacted by personal considerations
external to the program, such as caring for children or managing other family
responsibilities. For some, the support provided by family members could promote
success, as found by Zeldin and Pajares (2000): “For some women, the social
persuasions they received from members of their family regarding the idea of women
16
going into male-dominated areas and of women doing what they wanted to do were
critical and integral to their later paths” (p. 229). These verbal persuasions are one of
four key components of self-efficacy (Bandura, 1977). Interestingly, research
suggests that women are more affected by the presence, or lack thereof, of these verbal
persuasions than men (Becker, 1984; Zeldin & Pajares, 2000). Thus, having a
supportive family may be a factor that is particularly influential for women’s success
in mathematics doctoral programs.
However, for some students, family responsibilities may be a deterrent to
success in graduate study, because of external time commitments detracting from their
work. Herzig (2010) notes, “the conflict between the two greedy institutions of
motherhood and graduate school can be substantial” (p. 198). Sears (2003) reiterates
this point: “The perception is often that there is time for child rearing or academia, but
not both, and the potential costs are a failed marriage or a failed career” (p. 179).
Interestingly, research indicates that women in male-dominated doctoral programs,
such as mathematics, are more likely to experience conflict because of family issues
than women in gender-balanced programs (Ülkü-Steiner, Kurtz-Costes, & Kinlaw,
2000).
Graduate students may also be impacted by the financial implications of full-
time doctoral study (Nerad & Miller, 1996). Students pursuing graduate degrees
immediately after completing their undergraduate degrees delay their earning potential
for five or more years, while students returning to school from the workforce often
accept significant decreases in income to pursue graduate degrees. These sacrifices
may take a toll in already stressful circumstances, leading some students to
discontinue their studies in favor of employment with less financial burden.
17
Content Preparation
Each student also enters doctoral study with prior mathematical knowledge,
developed throughout the course of her undergraduate program. This knowledge must
be of sufficient breadth and depth for graduate-level study. However, in the past,
incoming doctoral students were rated by more than half of responding departments as
lacking in quality in terms of content preparation (Conference Board of the
Mathematical Sciences, 1987; as cited in Madison & Hart, 1990). Furthermore, Miller
(2015) found that female professors who received their doctorate from an institution in
the United States were more than five times as likely to report that their undergraduate
preparation was insufficient for doctoral study as compared to their international
peers. However, this finding is not universal; some studies have also concluded that
students who complete their degrees and those who do not are equally qualified for
graduate study, using Graduate Record Examination (GRE) scores and undergraduate
grade point averages as the means of comparison (Bair & Haworth, 2004; Lovitts &
Nelson, 2000).
Programmatic Factors
Sense of Belonging
Several researchers have argued that a student’s sense of belonging, or
integration into the community of the doctoral program, influences doctoral retention
(Girves & Wemmerus, 1988; Herzig, 2002; Lovitts, 2001; Tinto, 1993). Furthermore,
specific forms of involvement, including participation in “departmental, institutional,
and professional activities” are associated with doctoral student retention (Herzig,
2004a, p. 175). In fact, Girves and Wemmerus (1988) found that student involvement
had the greatest impact on a student’s progress to degree completion. Sufficient
18
integration into the doctoral community may allow a student to become more resilient
in the face of additional challenges. Lovitts (2001) found that students who were less
integrated were also less tolerant of financial difficulties caused by their doctoral
studies. However, female doctoral students are more likely to report feelings of
isolation than their male counterparts (Etzkowitz, Kemelgor, & Uzzi, 2000).
Quality and Availability of Courses
Previous research studies have documented the deficiencies women perceive in
their graduate coursework in mathematics. Hall and Sandler (1982) describe the
classroom climate for women in male-dominated fields, such as mathematics, as
“chilly” (p. 3) in that it “subtly or overtly communicates different expectations for
women than for men [that] can interfere with the educational process itself” (p. 3).
Herzig (2004b) found that women felt invisible in their courses and desired higher
quality instruction. The women also complained about the lack of feedback, reporting
that not only was work rarely collected and graded, but also that, in some courses,
work was not even assigned. Finally, the women felt that “they either could not ask
questions, or felt that they were rebuffed or chastised when they did” (Herzig, 2004b,
p. 386).
For women, who tend to prefer less competitive and more collaborative
environments (Barker & Garvin-Doxas, 2004), the classroom culture of the program
can have implications for their success (Gilligan, 1982). Multiple studies report that
women view competition as detrimental to their learning (e.g., Etzkowitz, Kemelgor,
Neuschatz, & Uzzi, 1992; Hollenshead et al., 1994; Tully & Jacobs, 2010). In fact,
“male students tend to thrive in a competitive classroom atmosphere (Fennema, 2000),
whereas female students often benefit from higher levels of cooperative activities and
19
open encouragement (Streitmatter, 1997)” (as cited in Tully & Jacobs, 2010, p. 458).
Hollenshead et al. (1994) conducted focus group interviews with female graduate
students in mathematics and physics. The women in their study reported feeling that
“competition [was] alienating, unfeminine, uncomfortable, and often distressing” (p.
69) and felt that the environment of their doctoral program was aligned with a
masculine “sink or swim” conceptualization of competition (p. 83). While a
preference for a less competitive atmosphere is by no means universal for female
students, alignment between one’s preferences in terms of competitiveness and the
actual competitiveness of the program itself may have implications for students’
success.
Interactions with Others in the Department
In addition to their coursework, doctoral students interact with faculty for
various purposes, including research experiences and mentoring. Interactions such as
these “define knowledge and disciplinary values, model the roles of academics in the
discipline, and produce practical help and advice” for students (Baird, 1993, p. 5). In
addition to helping students develop formal knowledge, interactions with faculty help
students develop “tacit knowledge.” Gerholm (1990) discusses the importance of
“tacit knowledge,” defined as implicit knowledge of the workings of the department
that is passed from professors to students, but is never formally taught. The concept of
tacit knowledge is similar to the idea of “the hidden curriculum”: that which is
learned, but is not overtly planned to be taught (Kelly, 2004). This tacit knowledge is
so crucial, that “failure to acquire this implicit knowledge is often taken as a sign of
failure to have acquired the explicit [content] knowledge itself” (Gerholm, 1990, p.
263). Since tacit knowledge is, by definition, not a part of the formal curriculum, it
20
must be transmitted through informal interactions (Etzkowitz et al., 1992). The more
frequent these interactions, the greater the likelihood of development of this tacit
knowledge, and finally, the greater the chance of socialization into the doctoral
community that is so crucial to doctoral success (Gerholm, 1990). Formal
mathematical knowledge, in addition to tacit knowledge, can be transferred from
professor to student by way of a cognitive apprenticeship (Brown, Collins, & Duguid,
1989).
Herzig (2004b) emphasizes the importance of these interactions in retaining
women and lessening the gender imbalance in mathematics: “Mathematicians need to
provide opportunities for women graduate students to develop meaningful and
substantive relationships with faculty, both in and out of class, in ways that enhance
these students’ participation in mathematical practice – leading them to learn to think,
act, and feel like mathematicians” (p. 390). However, the female respondents in
multiple studies conducted by Herzig (e.g. Herzig, 2002; Herzig, 2004b) did not report
experiencing these collegial connections. Instead, they described the interactions as
“limited or negative” (Herzig, 2002, p. 12).
Are these interactions any different for male doctoral students? According to
some researchers, the answer is yes, and the consequences of these gender differences
could be enormous. In a study conducted by Berg and Ferber (1983), both male and
female graduate students in male-dominated fields, including mathematics, were asked
to report how many male faculty members “they had come to know quite well in the
course of their graduate studies” (p. 638). Seventy-eight percent of male students, and
only 54% of female students responded “one or more” (p. 638). Differences in the
amount and quality of interactions with faculty could have a large impact on creating
21
very different experiences for male and female students, which could contribute to the
gender gap in mathematics doctoral program success.
In addition to interactions with faculty, interactions with one’s peers, the other
students in the program, may also be impactful. As with interactions with professors,
the opportunity to informally discuss mathematics with other students and to develop
tacit knowledge of the discipline and of the doctoral program may aid students in
having a successful doctoral program experience. In the quantitative studies reviewed
by Bair and Haworth (2004), those who completed their doctoral degrees were more
likely to have formed relationships with their peers than those who did not. Although
there is limited research in this area, the available literature suggests that student-
student relationships have a less prominent role in retention and attrition than student-
faculty interactions (Lovitts, 1996).
Academic Benefits of Institutional Support
Most doctoral programs offer students some sort of supporting experience –
most often an assistantship or fellowship – for at least part of their time in the doctoral
program. In addition to providing financial support, these opportunities are often
intended to be educative experiences for students. Graduate assistantships, such as
teaching and research assignments, have been characterized as “a means of enhancing
the professional development of graduate students, […] providing financial support
[and] the opportunity for socialization into the academic profession” (Ethington &
Pisani, 1993).
The type of financial support a doctoral student receives, whether it be through
a teaching or research assistantship, a fellowship, non-university associated
employment, or loans, has been shown to have an impact on completion rates, time to
22
degree completion, professional development, and scholarly productivity. Multiple
studies have found that the form of support a student receives is associated with rate of
degree completion, as well as time-to-degree, although to a lesser extent (Bowen &
Rudenstine, 1992; Ehrenberg & Mavros, 1992). Students who receive fellowships or
research assistantships had the highest completion rates and shorter times-to-degree,
while those with teaching assistantships or who supported themselves through other
means had lower completion rates and longer times-to-degree (Ehrenberg & Mavros,
1992).
Ethington and Pisani (1993) surveyed 603 doctoral students, and classified
each into one of four groups: “students who [had] received only research
assistantships, (2) students who [had] received only teaching assistantships, (3)
students who [had] received both, and (4) those who [had] neither” (p. 345). Findings
revealed that students who had a research assistantship or both teaching and research
assistantships during their doctoral study perceived greater contributions to their
professional development (consisting of growth in professional skills and
competencies and accumulation of professional accomplishments) than students in the
other two categories. Furthermore, students who were supported by teaching
assistantships for the entirety of their doctoral study perceived the least growth in their
research capabilities of the four groups. In terms of scholarly productivity, students
who had held both teaching and research assistantships were the most productive,
while students who had neither type of assistantship were the least productive.
However, Ethington and Pisani (1993) caution that the impact of student
selection into assistantship types cannot be overlooked: perhaps differences exist in
“abilities, desires, and motivations” that make students more suitable for one support
23
type over another (p. 351). If these selections are made based on gendered
stereotypes, women could be disadvantaged as a result. In a study conducted by
Miller (2015), one respondent described consistently being assigned remedial
mathematics courses for her teaching assistantship, a pattern she attributed to the
notion that women are perceived as being more patient. Given the research just
described associating teaching assistantships and lessened professional development,
any biases in assigning female students to certain types of assistantships over others
may influence their success.
Fairness of Policies
In general, the policies and procedures of the doctoral program may be related
to student success in the explicitness, clarity, and consistency of their application. A
study conducted by Golde (1996) suggests that doctoral departments with greater
structure had lower rates of attrition. In a study of a higher education administration
graduate program, Roberts, Gentry, and Townsend (2011) found that students viewed
“inconsistent policies and practices” of the department as detrimental to their success
(p. 1). Students in this program also expressed dissatisfaction with the distribution of
course requirements and their alignment (or lack thereof) with their intended career
goals. Inconsistent application of departmental policies on a student-by-student basis
may also decrease the success of some students in their doctoral program (Bair &
Haworth, 2004). If these inconsistencies arise due to biased opinions of female
students, they may be partially to blame for the gender gap.
24
Academic Support from Advisor
According to Tinto (1993), during the last stage of doctoral persistence, a
student’s doctoral advisor becomes the central figure in her academic world. A
qualitative meta-synthesis conducted by Bair and Haworth (2004) revealed that the
quality of the student-advisor relationship is the most frequent factor related to
doctoral attrition and persistence. Multiple studies included in the meta-synthesis
concluded that, in terms of students’ relationships with their advisors, positive
interactions were associated with greater rates of degree completion and negative
interactions were associated with greater rates of attrition (Bair & Haworth, 2004).
Furthermore, advisors may serve as a buffer, allowing the student to be resistant to
other problems in the doctoral program. Establishing a supportive relationship with
their advisor allows female doctoral students to be more resilient to familial
complications and stress than students with less supportive advisors (Ülkü-Steiner et
al., 2000). Other aspects of the student-advisor relationship that may be influential in
student success include help in networking with colleagues and aid in transitioning
into a career post-graduation (Miller, 2013).
Factors Related to Gender
Professor Gender Ratios
Professor gender ratios may also play a role in student success. In a study of
over 9000 undergraduates at the U.S. Air Force Academy, Carrell, Page, and West
(2010) found that, while professor gender had little to no impact on the performance of
male students in undergraduate STEM courses, female students’ performance
improved substantially with the presence of a female professor. In contrast, professor
25
gender was not a significant predictor of student performance for either gender in
more gender-balanced fields, such as the humanities.
Robst, Keil, and Russo (1998) found that the effect of female faculty was
amplified when female students had few same-gender peers. That is, as the percentage
of female students in a class increased, the importance of female faculty in retention
decreased (Robst, Keil, & Russo, 1998). Furthermore, Schroeder and Mynatt (1993)
found that female students reported that they experienced student-faculty interactions
of a higher quality when the faculty member was female. Herzig (2010) argues that
female doctoral students learn how to be female mathematicians by observing
established female professors conduct their work. In essence, they learn by example.
However, because of the low number of female faculty members in most mathematics
departments, female students are at an “inescapable disadvantage in finding mentors”
(Berg & Ferber, 1983, p. 639).
In a study of female students in engineering and chemistry courses, Young,
Rudman, Buettner, and McLean (2013) found that when these students viewed their
female professors as role models, they were less likely to characterize science as
masculine. Zeldin and Pajares (2000) support this finding in claiming that the
vicarious experiences component of self-efficacy, which entails “watching and
learning from others” (p. 238), is especially important for women in male-dominated
fields. However, due to current gender imbalances in mathematics, female
mathematics professors are scarce or unavailable.
Ülkü-Steiner et al. (2000) report that the converse is also true: “When female
faculty were noticeably few or absent in a program, female students experienced lower
self-concept, less sensitivity to family issues, and lower career commitment” (p. 306).
26
Because of this, these researchers claim that, beyond the gender of the individual
student, the alignment (or misalignment) of student gender ratios and faculty gender
ratios has the greatest impact on the success of a student’s doctoral experience (Ülkü-
Steiner et al., 2000). Therefore, tacit knowledge, developed through interactions with
professors and peers, may be harder to develop for female students, who are limited
both by the number of same gender peers, and by the even scarcer number of female
professors.
Student Gender Ratios
Student gender ratios may also play a role in determining doctoral program
success. Having a sufficient number of female peers, a so-called “critical mass”
(usually defined as at least 15% of the group in question) (Blickenstaff, 2005;
Etzkowitz, Kemelgor, Neuschatz, Uzzi, & Alonzo, 1994), is important for the
development of confidence, self-concept, self-efficacy, commitment and motivation
(Stout, Dasgupta, Hunsinger, & McManus, 2011; Zeldin & Pajares, 2000). According
to the theory of critical mass, once this threshold has been reached, the rate at which
qualitative improvements in the environment occur increases (Etzkowitz et al., 2000).
Reaching a “critical mass” of female students within a given program is also necessary
to avoid marginalization: “[The minority group’s] continued presence and survival is
in constant jeopardy, requiring outside intervention and assistance to prevent
extinction” (Etzkowitz et al., 1994, p. 51). Measures taken to equalize the numbers of
male and female mathematics graduate students will help to avoid this potential for
marginalization.
27
Gendered Beliefs and Actions of Others
While it is a fact that women are graduating from doctoral programs in
mathematics at lower rates than men, students may perceive these differences and
attribute them to biased or discriminatory beliefs or practices. Although many
doctoral students benefit from strong, positive relationships with faculty members and
other students at their institutions, not all inter-departmental interactions are
supportive. In fact, some interactions can actually be harmful to students’ success.
For instance, it has been demonstrated that male doctoral students view their female
counterparts as different from them, and interact with them less than peers of the same
gender (Berg & Ferber, 1983; Colbeck, Cabrera, & Terenzini, 2001).
Even subtle differences in treatment by professors or other students, termed
microaggressions, can be impactful. Microaggressions are defined by Sue (2010) as
“the brief and commonplace daily verbal, behavioral, and environmental indignities,
whether intentional or unintentional, that communicate hostile, derogatory, or negative
[…] slights and insults to the target person or group” (p. 5). This differential treatment
can lead to the activation of stereotype threat, defined as “being at risk of confirming,
as self-characteristic, a negative stereotype about one’s group” (Steele & Aronson,
1995, p. 797). In mathematics, a field where women are stereotyped as being less
competent than men, stereotype threat could cause women’s performance in
mathematics to falter in situations where that stereotype is activated (Oswald &
Harvey, 2000; Spencer, Steele, & Quinn, 1999). Interactions with consequences such
as these could potentially create a sense of isolation, interfering with female students’
opportunities not only for social engagement, but also for learning.
28
Conclusion
The differential success of male students as compared to their female peers is a
persistent problem in doctoral mathematics – one that has been studied for decades
without substantial improvement (e.g., Becker, 1984; Berg & Ferber, 1983; Herzig,
2004a, 2004b; Hollenshead et al., 1994; Stage & Maple, 1996). Given that Kerlin
(1997) suggests framing doctoral student attrition as “the consequence of a dynamic
process,” instead of as “a solitary event” (p. 21), it is crucial to investigate the
mechanisms that underlie such a process. There is still much work to be done to
ensure that doctoral study for women is “inclusive rather than isolating, collegial
rather than individualistic, and collaborative rather than competitive” (Kerlin, 1997, p.
18).
In summarizing the evidence for factors associated with the success of female
mathematics doctoral students, several recommendations can be made for mathematics
doctoral programs. Although personal factors may be difficult for a program to
change directly, there may still ways for programs to address these factors. Student
confidence and self-efficacy appear at the center of a constellation of other factors and
may be increased through verbal persuasions, positive feedback, and constructive
criticism from a student’s advisor and other faculty. Furthermore, in order to equalize
available time for schoolwork for students with families, support or provisions for
childcare could be integrated into the institutional structure.
At the programmatic level, since the importance of the advisor-advisee
relationship has been consistently found to have a large impact on student retention,
departments could choose to include advising and mentoring as part of their tenure
review process to reward faculty for devoting time to these relationships (Bair &
Haworth, 2004) and increase their awareness of the importance of their role as an
29
advisor through training or professional development. Furthermore, a wealth of
literature pertains to the importance of integration and socialization for doctoral
students. Students could be informed of the benefits of forming these relationships,
such as decreased stress, in order to increase their sense of belonging within their
departments (Bair & Haworth, 2004).
One interpretation of the literature presented here could be that doctoral
programs should admit female students in greater numbers, in order to provide women
with more same-sex peers. However, programs are cautioned against drawing this
conclusion. Any students entering a mathematics doctoral program should be
admitted with a realistic expectation of success. That is, admitting additional female
students who may be underprepared for doctoral study, while on the surface appearing
to promote gender equity, may eventually misappropriate limited resources and
perpetuate existing stereotypes of women in mathematics when these students are
ultimately unable to be successful in attaining doctorates.
Conversely, doctoral programs may infer that the factors presented here
suggest limiting admission to female students based on particular personal
characteristics. For example, since family responsibilities have been found in some
studies to impede a student’s ability to be successful, programs may decide to limit or
reduce the number of female students admitted with families. Again, programs are
cautioned against such strategies. While this may increase the proportion of female
students who attempt to earn a doctorate and are successful, this will not narrow the
gender gap. Instead, programs should focus on designing and testing policies,
strategies, and structures to support students with families in successfully obtaining
their doctoral degree.
30
Most of the work reviewed and synthesized here focused only on a subset of
the factors hypothesized to influence retention of female mathematics doctoral
students. That is, no single study has attempted to investigate all of these potential
factors simultaneously, and few have used a non-binary definition of success. There is
more to be learned about a student’s experience than simply their receipt (or lack
thereof) of a diploma. Even for those who do obtain doctorates, did they thrive or did
they merely survive until graduation? Furthermore, research investigating the gender
imbalance in STEM fields generally may not be as relevant as work conducted at the
subfield level (Kanny, 2014). Different factors may be at work in different disciplines
and important effects may be masked when fields are considered as a group.
Therefore, future research should seek to identify how the factors identified in this
review operate and interact to impact the multi-faceted construct of success for female
mathematics doctoral students. Only then can claims be made about the factors with
the utmost importance in retaining female students and can recommendations be made
for programs with limited resources seeking to promote the success of their female
students. With a better understanding of influential factors, improvements can then be
implemented and the underrepresentation of women in mathematics may soon be a
problem of the past.
31
Chapter 2
GENDER AND DOCTORAL MATHEMATICS: IMPACTFUL FACTORS FOR THE SUCCESS OF FEMALE STUDENTS
Women have long been underrepresented in mathematics. While progress has
been made at the undergraduate level, with 44% of bachelor’s degrees in mathematics
being awarded to women in 2007 (Hill, Corbett, & St. Rose, 2010), similar
improvement at the doctoral level has not occurred. For example, in 2013, only 27%
of doctoral recipients were female (Vélez, Maxwell, & Rose, 2014). This imbalance is
even further evident when one looks at gender ratios for mathematics faculty members
at doctorate-granting institutions. For example, in 2006, only 12% of mathematics
faculty at doctorate-granting institutions were female (Phipps, Maxwell, & Rose,
2007).
The underrepresentation of female mathematics doctorate students is well
recognized, but the solution is not apparent. According to McAlpine and Norton
(2006), retention and attrition are influenced by the “interaction of a constellation of
dynamic factors” (p. 5). A multitude of factors, emerging from quantitative and
qualitative research studies and theoretical arguments, have been hypothesized as
contributing to the retention or attrition of female mathematics doctoral students.
Proposed factors include those pertaining to students’ background characteristics and
external commitments (e.g., Becker, 1984; Herzig, 2010; Preckel, Goetz, Pekrun, &
Kleine, 2008); relationships with their advisor, other professors, and students in their
department (e.g., Bair & Haworth, 2004; Baird, 1993; Tinto, 1993); the quality and
32
culture of the courses they take in the doctoral program (e.g., Hall & Sandler, 1982;
Herzig, 2004b); the support they receive through assistantships or fellowships (e.g.,
Ehrenberg & Mavros, 1992; Ethington & Pisani, 1993); the presence or absence of
other female students or role models (e.g., Blickenstaff, 2005; Robst, Keil, & Russo,
1998; Schroeder & Mynatt, 1993); and perceptions of biases or discrimination against
female students (e.g., Berg & Ferber, 1983; Spencer, Steele, & Quinn, 1999; Sue,
2010). For a more detailed description of the literature supporting the constructs
included in this study, see Miller (2015b).
Most prior research concerned with the underrepresentation of women in
advanced mathematics has focused on identifying factors that impact the attrition of
women from mathematics doctoral programs (e.g., Herzig, 2002; Herzig, 2004a;
Herzig, 2004b). Because the sample sizes in these studies have typically been small, it
is unclear how generalizable these factors are in accounting for women’s attrition.
Moreover, additional factors may be at play in influencing retention, beyond those that
contribute to attrition. Thus, it is important to identify factors associated with the
success of women in mathematics doctoral programs. However, very few studies have
examined the problem from this perspective.
Many constructs have been used to characterize the outcome of a student’s
experience in a doctoral program. Constructs such as retention, completion, and
persistence have been used to represent doctoral degree attainment (e.g., Ampaw &
Jaeger, 2011; Nerad & Miller, 1996; Tinto, 1993); attrition has been used to describe
the trajectories of students who discontinued their studies (e.g., Bair & Haworth, 2004;
Golde, 1998). These constructs are essentially binary in nature: either a doctoral
student completes her program, or she does not. Although binary constructs are more
33
easily defined and measured, the focus is then on the end result, and not on the
confluence of decisions and experiences that contribute to that end result. In this way,
“attrition has been conceptualized as a solitary event, rather than as the consequence
of a dynamic process” (Kerlin, 1997, p. 21). There is more to be learned about a
student’s experience than simply their receipt of a diploma. Even for those who do
obtain doctorates, did they thrive or did they merely survive until graduation?
Research Questions
Identifying those factors that play a significant role in doctoral program
success is a necessary first step in addressing the gender gap in mathematics doctoral
study. This study aims to identify factors that have the strongest association with
student success, as reported by female and male graduates of mathematics doctoral
programs currently employed at post-secondary institutions, using a large-sample
quantitative survey methodology. Male participants are used as a comparison group
for the responses from female participants. In particular, this study was designed to
investigate the following research questions:
1. What factors have the strongest influence on mathematics doctoral program success for women who have earned their doctorate?
2. How do the factors influencing mathematics doctoral program success compare for men and women who have earned their doctorate?
A primary goal of this research is to identify those factors that are critical to
women’s success in obtaining a Ph.D. in mathematics so that future research can
investigate how these factors interact to influence doctoral program success.
Moreover, identification of critical factors can inform the redesign of doctoral
programs to better facilitate women’s success. Data collected about these factors will
describe the experiences of successful female doctoral students, contrast these
34
experiences with the experiences of successful male doctoral students, and evaluate
the importance of each factor in participants’ success in obtaining a doctorate in
mathematics. In contrast to previous studies using a binary construct, this study draws
on a more descriptive outcome measure, doctoral program success, to attempt to
capture the complex nature of this dynamic process. In addition, data collected will be
used to conduct group comparisons of critical factors for women who did and did not
complete their doctoral programs.
Methods
This section begins with a description of the survey instrument used in the
study. Then, the sample selection and resulting sample demographics are discussed,
followed by a description of the steps taken to prepare the data for analysis using
partial least squares structural equation modeling (PLS-SEM). This preparation
included exploratory factor analysis to determine underlying latent constructs,
multiple imputation for handling missing data, and model diagnostics to formulate a
path model suitable for PLS-SEM analyses. The section concludes by describing the
analyses conducted to investigate each research question.
Survey Instrument
The instrument used to collect data was an electronic survey consisting of three
sections. The first section contained three items designed to ensure that participants
satisfied the selection criteria. First, participants were asked if they had obtained a
doctorate in mathematics or applied mathematics. If the response was “Yes,” the
participant continued with the survey. If the response was “No,” then participants
were asked if they had ever enrolled in a doctoral program with the intent to earn such
35
a degree. If the response was again “No,” the survey ended. Finally, participants were
asked if they were currently enrolled in a mathematics doctoral program. If the
response was “Yes,” the survey ended. Participants who had enrolled in a doctoral
program in pure or applied mathematics, but did not complete their degree, were
directed to a slightly different version of the survey to collect information about their
experiences with attrition from their doctoral program. This version of the survey was
conceptually identical, but included a “Not Applicable” option for some items. This
allowed participants to distinguish factors they did not experience because of lack of
opportunity in the program from factors they did not experience because of their
departure from the program.
The second section of the survey asked participants to indicate their level of
agreement with 62 statements using a five-point Likert scale, with 1 = Strongly
Disagree, 2 = Disagree, 3 = Neither Agree, Nor Disagree, 4 = Agree and 5 =
Strongly Agree. The statements were designed to represent the key factors identified
from a systematic review of the literature and were organized into ten blocks based on
hypothesized themes (Miller, 2015b). The ten blocks evaluated student attributes,
prior educational experiences, external (non-academic) commitments, institutional
support experiences, interactions with professors, interactions with peers, academic
relationship with advisor, programmatic structure, quality of coursework, and gender
ratios within the program. An additional block evaluated the outcome construct,
doctoral program success, and contained five items. Approximately half of the 62
items in this section were worded in the negative; data for these questions were
reverse coded to improve the reliability of the items. Items worded in the positive
were assigned Likert values such that the higher end of the scale aligned with greater
36
success. For items worded in the negative, the lower end of the scale aligned with
greater success. Since the participants were asked to recall experiences that may have
occurred decades prior, the items were ordered chronologically to aid in memory
recall. The questions within each block were randomized so that participants’
responses were less susceptible to order effects.
In the third and final section of the survey, participants were asked
demographic questions, such as their gender, their current job title, and the highest
mathematics degree their employing institution grants. This section concluded with
items about aspects of the participants’ training in mathematics. For example,
participants were asked to identify the length of time spent in their doctoral program,
the gender of their doctoral advisor, and when they earned their doctorate. For the full
set of demographic questions, see the survey in Appendix A.
Ten mathematics education doctoral students and faculty members piloted the
survey. These pilot participants did not have doctorates in pure or applied
mathematics and so did not detract from the desired sample. However, most held
advanced degrees in mathematics and were, therefore, able to provide knowledgeable
feedback on the survey. The pilot process had three aims: (1) to garner feedback on
the content validity and clarity of the survey items; (2) to estimate the time required
for participants to complete the survey; and (3) to ensure that the online data collection
proceeds as planned. Pilot participants reported taking approximately 15 minutes to
complete the survey, as expected, and no issues with the online data collection process
were discovered. Minor feedback on the wording of survey items was received and
incorporated to improve the survey. The survey instrument can be found in Appendix
A.
37
Sample
The target population for this study was mathematics faculty members
employed at tertiary institutions in the United States, who either hold a doctorate in
mathematics (pure or applied) or who had at one time enrolled in a doctoral program
with the intent to earn such a degree. Those currently enrolled in mathematics
doctoral programs were excluded from completing the survey. The experiences of
these participants are incomplete, and therefore, are not comparable to the target
population.
Although graduates of mathematics doctoral programs have several
employment options available to them, most assume positions in academia. For
instance, in the 2012–2013 academic year, 65.8% of doctoral recipients in the
mathematical sciences accepted academic positions (Vélez, Maxwell, & Rose, 2014).
Since these data include the career paths of statistics and biostatistics graduates, fields
in which over 70% of graduates accept positions in industry or government, this
percentage is likely higher when limited to those earning mathematics doctorates.
Moreover, less than six percent of all new doctorate recipients in mathematics reported
being unemployed, and only five percent of recent female graduates reported
unemployment. Therefore, although the sample selected for this study may not be
generalizable to all graduates of mathematics doctoral programs, specifically those
unemployed or employed in industry or governmental positions, it is generalizable to a
large majority of doctorate recipients in mathematics. Since unemployed graduates or
graduates employed in non-academic positions would be nearly impossible to recruit
in a systematic manner, participation in this study was limited to participants
employed in academia.
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To obtain a sample from this population, a sampling frame was used from the
report, “Statistical Abstract of Undergraduate Programs in the Mathematical Sciences
in the United States” (Blair, Kirkman, & Maxwell, 2013). The sampling frame
contains an exhaustive list of two- and four-year colleges and universities granting
degrees in mathematics in the United States, separated into four strata by institution
type: associate’s colleges, baccalaureate colleges, master’s colleges/universities, and
doctoral/research universities. Any institutions not listed in one of the preceding four
categories (e.g., tribal colleges) were not included in the institution sample. Table 1
presents the distribution of institutions of the four types.
Table 1 Number (Percent) of Institutions by Type in the Sampling Frame
Number (percent) of institutions Associate’s colleges 1031 (42.78%) Baccalaureate colleges 553 (22.95%) Master’s colleges/universities 565 (23.44%) Doctoral/research universities 261 (10.83%) Total 2410 (100.00%)
Based upon the distribution of institutions of each of the four types, a
corresponding proportion of faculty members within each stratum were sampled. For
instance, approximately 11% of the total number of institutions in the sample fall into
the doctoral/research university category. Therefore, a corresponding percentage of
invitations for the survey were sent to faculty at that type of institution. To achieve
this, institutions in each stratum were randomly ordered. According to the random
ordering, all available participants at the highest listed institutions were selected for
39
the sample, until the required sample size was obtained. Contact information for
mathematics faculty employed at the selected institutions was collected through an
Internet search.
Power Analysis
There are different recommendations regarding the necessary sample size for
PLS-SEM analyses to be sufficiently powered. According to Cohen (1992, as cited in
Hair, Jr., Hult, Ringle, & Sarstedt, 2014), the sample size is dependent upon the
statistical power, the significance level, the minimum value of R2 desired, and the
maximum number of indicators pointing to a single construct in the path diagram of
the structural equation model. Based on these considerations, and with the
hypothesized latent construct structure created by identifying themes from the
literature, each analysis would require at least 166 participants, or 332 participants
overall.1 Others recommend using the results of a priori power analyses for multiple
regression, which would recommend a sample size of at least 64 per analysis, or 128
overall, as computed with G*Power power analysis software2 (Faul, Erdfelder, Lang,
& Buchner, 2007; Hair, Sarstedt, Pieper, & Ringle, 2012). Still others advocate using
the “ten times rule” and obtaining 10 times the maximum number of indicators leading
1 Assuming a power of 80 percent, with a significance level of .05, a minimum R2 of .10, and a maximum of 7 items defining a single construct.
2 Assuming a power of 80 percent, with a significance level of .05, an effect size f2 of .10, and 7 predictors.
3 While the original factor analysis was conducted using a cutoff value for the factor loadings of .4, that cutoff was lowered slightly to .38 in the second factor analysis in order to retain an item of great theoretical importance (PersChar3).
4 Bootstrapping was conducted using 5000 samples. For each gender, bootstrapping
2 Assuming a power of 80 percent, with a significance level of .05, an effect size f2 of .10, and 7 predictors.
40
to any one construct (Barclay, Higgins, & Thompson, 1995; Nunnally, 1967). For the
hypothesized latent construct structure, which had a maximum of seven indicators per
construct, this would result in a required sample size of at least 70 for each model (or
140 overall). Using the most conservative of these guidelines, the minimum sample
size recruited needed to be at least 332, with at least 166 men and at least 166 women.
In order to obtain a sample of this size, a total of 6887 invitations were sent to solicit
responses to the survey.
Survey Responses Received
Of the 6887 invitations, 1084 responses were received. Of these, 988 were
complete. After discarding the responses of those who did not fit the criteria and those
with more than 15% missing data, the analytic sample size consisted of 662 responses.
Of these 662 responses, 163 were from female doctorate recipients and 417 were from
male doctorate recipients. The remaining responses were from 40 men and 42 women
who had enrolled in, but did not complete, a mathematics doctoral program. As will
be discussed later, an exploratory factor analysis revealed a different latent construct
structure than was hypothesized, in which there was a maximum of six indicators per
latent construct. Therefore, revising the a priori estimates, and using the most
conservative estimate for the required sample size, 157 participants were required for
each analysis, or 314 overall. Therefore, the obtained sample size is sufficient to
detect significant differences within the data. Table 2 presents the number and percent
of survey invitations, responses received, and the response rate for each type of
institution. Tables 3 and 4 present the number (and percent) of participants of each
gender by job title and institution type, respectively. Table 5 presents participants’
time since degree obtainment (or time since leaving) by degree completion status.
41
Table 2 Number (Percent) of Survey Invitations and Responses Received by Institution Type
Number (percent) of invitations
Number (percent) of responses
Response rate
Associate’s colleges 2867 (41.63%) 63 (9.52%) 2.20% Baccalaureate colleges 1537 (22.32%) 262 (39.56%) 17.05% Master’s colleges/universities 1660 (24.10%) 125 (18.88%) 7.53% Doctoral/research universities 823 (11.95%) 212 (32.02%) 25.76% Total 6887 662 --- Note. Overall response rate = 9.61%.
Table 3 Number (Percent) of Survey Responses Received by Gender and Job Title
Number of male participants
Number of female participants Total
Full professor 162 (73.30%) 59 (26.70%) 221 Associate professor 122 (72.62%) 46 (27.38%) 168 Assistant professor 97 (61.39%) 61 (38.61%) 158 Post doctorate 10 (83.33%) 2 (16.67%) 12 Adjunct professor 13 (76.47%) 4 (23.53%) 17 Lecturer or instructor 35 (56.45%) 27 (43.55%) 62 Other 17 (73.91%) 6 (26.09%) 23 Total 456 (68.99%) 205 (31.01%) 661 Note. One participant left this item blank.
42
Table 4 Number (Percent) of Survey Responses by Gender and Institution Type
Number of male participants
Number of female participants
Total
Associate’s colleges 36 (57.14%) 27 (42.86%) 63 Baccalaureate colleges 173 (66.03%) 89 (33.97%) 262 Master’s colleges/universities 99 (79.20%) 26 (20.80%) 125 Doctoral/research universities 149 (70.28%) 63 (29.72%) 212 Total 457 (69.03%) 205 (30.97%) 662
Table 5 Number of Survey Responses by Time Since Degree (Leaving) and Completion Status
Time since degree (leaving) Obtained doctorate Enrolled, but did not complete doctorate Total
0 to 9 years 216 (88.52%) 28 (11.48%) 244 10 to 19 years 157 (84.86%) 28 (15.14%) 185 20 to 29 years 89 (88.12%) 12 (11.88%) 101 30 to 39 years 65 (87.84%) 9 (12.16%) 74 40 to 49 years 44 (89.80%) 5 (10.20%) 49 50 or more years 9 (100.00%) --- 9 Total 580 (87.61%) 82 (12.39%) 662
Previous research reports that male students complete their doctorates in less
time than female students (Herzig, 2004a). In contrast, participants in this study did
not reflect this trend. While male participants (nM = 409, MM = 5.61 years, SDM =
1.59) completed their doctorates in slightly less time than female participants (nF =
161, MF = 5.65 years, SDF = 1.35) in this sample, the difference was not significant
43
(t(568) = -0.253, p = .800). Also of interest, over 92% of the sample reported having a
male advisor, which aligns with previously reported research (Miller, 2015a).
Data Collection
Participants were invited by e-mail to participate in the study. The e-mail
contained a link to the electronic survey on Qualtrics. Data was collected through the
Qualtrics website, a service for building surveys and collecting data electronically
(Qualtrics Labs Inc., Provo, UT). After one week, an e-mail reminder was sent to
encourage those who had not yet completed the survey to do so. Data collection was
conducted in three waves, with additional invitations to participate in the survey being
sent in each wave until the necessary minimum number of participants of each gender
was met.
Data Analysis
Creating an Overall Scale for Institutional Support Experiences
One of the 10 blocks on the survey assessed participants’ experiences with
institutional supports (e.g., teaching assistantships, fellowships), in terms of both the
academic benefits and the time demands. Since doctoral students may receive
different forms and durations of institutional support, participants were asked to
evaluate only those sources of support they had received. Consequently, a consistent
measure for the evaluation of the particular institutional support each participant
experienced was needed. In order to summarize each participant’s various sources of
support and their evaluation of each, two new items were calculated (one for academic
benefits and one for time demands). Each new item represents the weighted average
44
of the participant’s evaluations of each source from which they received funding,
weighted by the duration of the funding:
𝐼𝑛𝑠𝑡𝑆𝑢𝑝1 =1𝑇 𝐿!𝑡!
!
!!!
where n is the number of sources from which the participant received funding, T is the
total duration of the funding received while in a doctoral program, Li is the
participant’s Likert evaluation of the academic benefits (or time demands) of ith
funding source, and ti is the duration for the ith funding source in years.
Formulating Latent Constructs with Exploratory Factor Analysis
After data collection, relationships between the 62 Likert items on the survey
were tested with exploratory factor analysis to formulate latent constructs for the PLS-
SEM analyses. Exploratory factor analysis was conducted in SPSS (IBM Corp.)
through a principal components extraction with a varimax rotation. The Kaiser-
Meyer-Olkin (KMO) measure of sampling adequacy was .863, above the
recommended minimum value of .6, indicating that the sample size of 553 participants
with no missing data was adequate for factor analysis (Hutcheson & Sofroniou, 1999).
Additionally, Bartlett’s test of sphericity was significant (χ2 (1891) = 13988.41, p <
.01), indicating that correlations exist within the data, making it suitable for factor
analysis (Dziuban & Shirkey, 1974).
According to Kaiser’s criterion, 17 factors exist with eigenvalues greater than
1. Collectively, these 17 factors explain 63.91 percent of the variance in the data.
Considering loadings above .4 as significant (Stevens, 2002), the following six survey
items did not load above .4 on any of the 17 factors: PersChar4, PersCons4,
ContPrep3, InstSup4, Fairness4, and the weighted average of the participant’s
45
evaluations of the time demands of each funding source, weighted by the duration of
the funding (similar to InstSup1).
After removing these six items, the data were reanalyzed for the remaining 56
items. Again, the KMO measure of sampling adequacy (.850) and Bartlett’s test of
sphericity (χ2 (1540) = 12495.14, p < .01) indicated the data were suitable for factor
analysis. After removing the items with low loading values from the analysis3, 16
factors (15 predictor constructs and one outcome construct) were detected with
eigenvalues greater than 1. These factors, explaining 64.89 percent of the variance in
the data, can be found in Table 6. It is these 16 factors that constituted the latent
constructs for the PLS-SEM analyses.
Table 6 Factors and Factor Loading Values
Variable Survey Item Loading Factor 1: Personal Characteristics
PersChar1 During my doctoral study in mathematics, I was motivated to succeed. .756
PersChar2 I was committed to my work during my doctoral study in mathematics. .696
PersChar3 I am naturally talented in mathematics. .389 Factor 2: Personal Considerations
PersCons1 My family responsibilities detracted from my ability to be successful during my doctoral study in mathematics.
.763
PersCons2 Outside of academics, aspects of my personal life did not detract from my ability to be successful in my doctoral study in mathematics.
.730
3 While the original factor analysis was conducted using a cutoff value for the factor loadings of .4, that cutoff was lowered slightly to .38 in the second factor analysis in order to retain an item of great theoretical importance (PersChar3).
46
PersCons3 During my doctoral study in mathematics, concerns about financial issues detracted from my ability to be successful.
.598
Factor 3: Content Preparation
ContPrep1 The mathematics courses I took in my undergraduate program did not prepare me well to succeed in my doctoral study in mathematics.
.821
ContPrep2 My prior educational experiences in mathematics prepared me well to succeed in my doctoral program. .807
Factor 4: Sense of Belonging
Belonging1 Before beginning my doctoral program in mathematics, I did not participate in research in mathematics.
-.700
Belonging2 During my doctoral study in mathematics, I felt I was a valued member of the courses I took. .428
Factor 5: Academic Support from Advisor
SuppAdv1 My doctoral advisor was academically supportive of me. .785
SuppAdv2 My doctoral advisor did not provide valuable feedback on my work. .759
SuppAdv3 My doctoral advisor aided me in networking with colleagues. .703
SuppAdv4 My doctoral advisor did not provide me with assistance in transitioning to my career. .691
SuppAdv5 My doctoral advisor treated me as a colleague. .631
SuppAdv6 My doctoral advisor did not help me in selecting my dissertation/thesis topic. .574
Factor 6: Interactions with Others in the Department
IntOthers1 During my doctoral study in mathematics, I had the opportunity to participate in informal conversations about mathematics with professors.
.713
IntOthers2 During my doctoral study in mathematics, I had the opportunity to participate in informal conversations about mathematics with other students.
.702
IntOthers3 During my doctoral study in mathematics, I did not have the chance to interact with the professors in my department outside the classroom.
.650
IntOthers4 During my doctoral study in mathematics, I had the chance to interact with the other students in my department outside the classroom.
.564
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IntOthers5 During my doctoral study in mathematics, I worked closely with other professors in the department (other than my advisor).
.543
Factor 7: Quality and Availability of Courses
Courses1 During my doctoral study in mathematics, I was satisfied with the course offerings available in my department.
.773
Courses2 During my doctoral study in mathematics, I was not satisfied with the courses I took. .726
Courses3 The requirements of my doctoral program allowed for enough electives that I could specialize in my area of interest.
.637
Courses4 During my doctoral study in mathematics, I was satisfied with the quality of the teaching in my courses.
.542
Courses5 During my doctoral study in mathematics, I did not receive valuable feedback on my assignments. .485
Factor 8: Academic Benefits of Institutional Support
InstSup1
The weighted average of the participant’s evaluations of the academic benefits of each source from which they received funding, weighted by the duration of the funding.
.641
InstSup2 During my doctoral program in mathematics, there was a competitive culture among students. -.553
InstSup3 During my doctoral program in mathematics, my funding source(s) provided me with opportunities that were academically beneficial to me.
.552
Factor 9: Professor Gender Ratios
ProfGender1 The professors in my department were predominantly male. .824
ProfGender2 There were approximately equal numbers of male professors and female professors in the department. .822
ProfGender3 The professors and students in my doctoral program reflected sufficient gender diversity. .609
Factor 10: Student Gender Ratios
StudGender1 During my doctoral study in mathematics, there were approximately equal numbers of male and female students.
.878
StudGender2 During my doctoral study in mathematics, there were noticeably more male students than female students. .853
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Factor 11: Ratios for Student Success by Gender
StudSuccess1
Male students in my doctoral program were more successful than female students, in terms of the proportion of incoming students of each gender who completed their doctorate.
.854
StudSuccess2
Male students and female students in my doctoral program were equally successful, in terms of the proportion of incoming students of each gender who completed their doctorate.
.804
Factor 12: Fairness of Policies
Fairness1 During my doctoral study in mathematics, teaching and research assistantship assignments were made fairly.
.726
Fairness2 During my doctoral study in mathematics, the policies and procedures of the department were inconsistently applied for certain students.
.725
Fairness3 During my doctoral study in mathematics, the policies and procedures of the department were unclear or were not made explicit to students.
.709
Factor 13: Obstacles Faced
Obst1 During my doctoral study in mathematics, passing benchmark exams (e.g., preliminary exams, candidacy exams) was an obstacle.
.682
Obst2 My confidence that I would succeed in obtaining a doctoral degree in mathematics wavered. .635
Obst3 I experienced significant setbacks during my doctoral program in mathematics. .518
Obst4
During my doctoral study in mathematics, I was hesitant to ask questions in class because of how those questions may be received by the professor or other students in the class.
.438
Factor 14: Unwanted Attention Due to Gender
UnAtt1 During my doctoral study in mathematics, I was the subject of unwanted attention due to my gender from one or more students in my program.
.798
UnAtt2 During my doctoral study in mathematics, I was the subject of unwanted attention due to my gender from one or more professors.
.768
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UnAtt3
During my doctoral study in mathematics, I felt that I was negatively singled out for reasons related to my gender by one or more professors, even in very small ways.
.761
UnAtt4
During my doctoral study in mathematics, I felt that I was negatively singled out for reasons related to my gender by one or more students, even in very small ways.
.748
UnAtt5 During my doctoral study in mathematics, I was the subject of unwanted attention due to my gender from my advisor, even in very small ways.
.550
Factor 15: Opinions About Success Due to Gender
OpSuccess1 Professors in my doctoral program appeared to think that female students were equally likely to succeed as male students.
.758
OpSuccess2 Other students in my doctoral program appeared to think that female students were equally likely to succeed as male students.
.754
OpSuccess3 Other students in my doctoral program appeared to think that male students were more likely to succeed than female students.
.731
OpSuccess4 Professors in my doctoral program appeared to think that male students were more likely to succeed than female students.
.713
Factor 16: Doctoral Program Success (Outcome Construct)
Outcome1 After graduating from my doctoral program in mathematics, I was hired at the type of job I wanted. .703
Outcome2 My doctoral program in mathematics equipped me with the knowledge and skills needed to succeed in my intended career.
.616
Outcome3 Overall, my experience in my doctoral program in mathematics was successful. .513
Outcome 4 I was able to complete my doctoral degree in a reasonable amount of time. .489
Note. Factor loadings less than .38 are suppressed. The factor analysis resulted in simple structure; therefore, only one column of loadings is presented.
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Multiple Imputation for Missing Data
Before any additional analyses were conducted, any participants with greater
than 15 percent missing data (i.e., participants who did not respond to at least 10 items
on the survey) were discarded from the sample. For the remaining missing data,
multiple imputation with five iterations was employed to maximize the useable sample
size. Multiple imputation is the preferred method for dealing with missing data, since
it does not bias the resulting data set as severely as other methods, such as mean
imputation (Sinharay, Stern, & Russell, 2001) and does not drastically decrease the
useable sample size for analyses, as with casewise deletion (Hair et al., 2014). The
imputation was conducted using the “Fully conditional specification” option in SPSS
(IBM Corp.), also known as Markov Chain Monte Carlo (MCMC) imputation,
meaning that five separate data sets were created, each with different imputed missing
values based on predictions from the observed data. Then, each PLS-SEM analysis
was conducted five times, with the final results being pooled from the five sets of
results according to Rubin’s (1987) rules.
Partial Least Squares Structural Equation Modeling
Data analyses were conducted using partial least squares structural equation
modeling (PLS-SEM). For the purposes of this study, only data from participants with
doctorates was analyzed. PLS-SEM, which is conceptually similar to multiple
regression, allows for evaluation of causal relationships between latent constructs,
instead of only observable variables (Hair, Ringle, & Sarstedt, 2011). Although the
analysis technique is most widely used for business applications, its use in social
science research has become more commonplace in recent years (e.g., Monteiro,
Wilson, & Beyer, 2013; Velayutham, Aldridge, & Fraser, 2012).
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The use of PLS-SEM allowed for the investigation of the comparative effects
on success of various factors associated with doctoral study in mathematics. This
analysis attempts to “maximize explained variance in the dependent constructs [while
evaluating] the data quality on the basis of measurement model characteristics” (Hair
et al., 2011). As opposed to covariance-based structural equation modeling (CB-
SEM), PLS-SEM is more appropriate for this study for several reasons. First, it is a
more suitable choice for exploratory analyses and reduces some of the biases inherent
in CB-SEM (Hair et al., 2014). Second, PLS-SEM has been shown to have greater
statistical power than CB-SEM and thus, has the ability to detect significant
differences when utilizing a smaller sample than with CB-SEM (Hair et al., 2014).
Third, PLS-SEM is based on less restrictive assumptions for the distribution of the
data. For instance, normality is not assumed; PLS-SEM analyses have been shown to
be robust to skewed or kurtotic data with sufficiently large sample sizes (Hair et al.,
2012; Hair et al., 2014). In order to conduct the PLS-SEM analyses, the software
program SmartPLS was used (Ringle, Wende, & Becker, 2014). Although other
programs, such as LISREL (Jöreskog & Sörbom, 2006) are more commonly used to
conduct CB-SEM analyses, SmartPLS is uniquely suited for conducting PLS-SEM
analyses.
Analyzing model diagnostics involves evaluating the suitability of two
components of the structural equation model: the inner model, consisting of the
relationships between the latent constructs, and the outer model, consisting of the
measurement models for the latent constructs (Hair et al., 2011). For each construct in
the path diagram, the measurement model can be reflective, in which causality is from
the latent construct to its measures (also called indicators), or formative, in which
52
causality is from the measures to the latent construct (Hair et al., 2014). Relationships
between indicators and constructs are called weights for formative constructs and
loadings for reflective constructs (Hair et al., 2014). In the case of the measurement
models for this study, all were formative, except the successfulness of doctoral
experience measure, which was reflective.
Evaluating the Reflective Measurement Model
For the reflective measurement model for the outcome construct (doctoral
program success), reliability and validity were assessed to determine the suitability of
the model’s measurement of the construct. While Cronbach’s alpha is commonly used
to assess internal consistency reliability for measurement scales, this measure is
calculated under the assumption that all indicators in the model are equally reliable
(Hair et al., 2014). Thus, composite reliability was computed instead. This statistic
ranges from 0 to 1, with values greater than .6 considered acceptable in exploratory
studies such as this (Hair et al., 2014). When analyzing composite reliability with all
four indicators of the outcome construct included, two of the resulting indicator
reliability values fell below the accepted minimum. After removing the two items
with low indicator reliability values (Outcome1 and Outcome4), the mean composite
reliability values across the five imputed data sets were .835 for female participants
and .909 for male participants, which are well above the minimum acceptable values.
Convergent validity, which is demonstrated when two supposed measures of
the same construct are positively correlated (Allen & Yen, 1979), was measured by the
average value extracted statistic (AVE). This measure is based on the stipulation that
the formulation of a latent variable should explain at least 50% of the variance in each
of its indicators (Hair et al., 2014). Therefore, the AVE, which is the mean of the
53
squared loading values of all of the indicators associated with a certain latent
construct, should be greater than .50 for a latent variable to have convergent validity.
Although the AVE values were above .50 with Outcome1 and Outcome4 in the model
(average AVE values of .5214 and .5538 for the female and male participants,
respectively), the AVE values markedly improved after the removal of these
indicators. After removing the two ill-fitting items, the mean AVE values across the
five imputed data sets were .7170 and .8340 for the female and male participants,
respectively.
Finally, for the reflective measurement model, discriminant validity was
assessed. Discriminant validity, which can be seen as the inverse of convergent
validity, occurs when two measurement scales purportedly measuring different
constructs share a low correlation (Allen & Yen, 1979). Although the indicators’
cross-loadings are sometimes used as a measure for this, this criterion tends to be very
liberal. Instead, Hair et al. (2014) recommend using the Fornell-Larcker criterion,
which is more conservative and compares the AVE values with the correlations among
the latent variables in the model. For sufficient discriminant validity, the square root
of the latent construct’s AVE value should be greater than its highest correlation with
any of the other constructs in the model (Hair et al., 2014). Using both the indicators’
cross-loadings and the Fornell-Larcker criterion, each of the five imputed data sets for
both genders (10 data sets in total) exhibited sufficient discriminant validity, both with
and without Outcome1 and Outcome4 in the model.
Evaluating the Formative Measurement Models
For constructs employing a formative measurement model, three steps were
undertaken to assess each model. First, as with reflective measurement models, it is
54
traditional to assess convergent validity via redundancy analysis. Redundancy
analysis is conducted by correlating each formative construct in the model with a
global indicator that purports to measure that construct directly (Hair et al., 2014).
Global indicators were included on the survey for the a priori hypothesized latent
constructs. However, following the factor analysis, three global indicators did not
load on any of the factors and many latent constructs were left without global
indicators. This, unfortunately, made conducting redundancy analyses impossible.
The second step is to identify and attempt to address any potential issues with
collinearity (or multicollinearity) among the indicators that constitute the construct in
question. If high correlations exist between two or more indicators, both the
interpretation of the results and the use of the analysis method itself are called into
question. To identify collinearity, the variance inflation factor (VIF) was computed,
which represents “the amount of variance of one formative indicator not explained by
other indicators in the same block” (Hair et al., 2014, p. 124). Therefore, the VIF
statistic represents the factor by which the variance has been magnified due to
collinearity (or multicollinearity). A VIF value greater than five indicates that
collinearity exists, and multiple indicators overlap to measure the construct. The
maximum VIF value detected for the female participants was 3.977; the maximum
VIF value detected for the male participants was 2.572. Thus, no issues with
collinearity or multicollinearity were detected for any indicators in any of the ten
imputed data sets.
Finally, the third step in assessing a formative measurement model is an
analysis of the relative and absolute contribution of each indicator to the formulation
of the latent construct. As a result of the model analysis, each formative indicator is
55
assigned an outer weight. In order to test if the outer weight of an indicator was
significantly different from zero and worthy of remaining in the model, a
bootstrapping procedure was used4. The bootstrapping procedure allows for the
estimation of standard errors and t values for significance testing. If the significance
test resulted in a p-value less than .05, the indicator was unconditionally retained in
the model. However, several indicators with non-significant p-values were still
retained, provided that their absolute contribution, provided by the outer weight, was
.5 or greater (Hair et al., 2014). The results of the bootstrapping procedure can be
found in Table 7.
Table 7 Formative Indicator Outer Weights
Formative indicators
Female participants Male participants Indicator outer weight
Outer loading value
Sig. Indicator outer weight
Outer loading value
Sig.
PersChar1 0.522 .792 NS-A 0.606*** .830 S PersChar2 0.576* .809 S 0.530*** .785 S PersChar3 0.308 .389 NS 0.218 .369 NS PersCons1 -0.418 .279 NS 0.053 .497 NS PersCons2 0.572* .671 S 0.767*** .907 S PersCons3 0.848*** .864 S 0.419* .662 S ContPrep1 0.726 .980 NS-A 0.290* .785 S ContPrep2 0.324 .892 NS-A 0.793** .974 S
4 Bootstrapping was conducted using 5000 samples. For each gender, bootstrapping was conducted using the no sign change option, the individual sign change option, and the construct-level sign change option, with the final decision on significance being made according to Hair et al. (2014, p. 137). Because of the computing time required for the bootstrapping procedure, bootstrapping was conducted on only one of the five imputation files for each gender (the first iteration of the MCMC procedure).
56
Belonging1 -0.248 -.336 NS -0.042 -.063 NS Belonging2 0.946** .969 S 0.998*** .999 S SuppAdv1 0.247 .812 NS-A 0.715*** .928 S SuppAdv2 0.473 .856 NS-A 0.234 .702 A SuppAdv3 0.186 .666 NS-A 0.055 .580 A SuppAdv4 0.134 .613 NS-A 0.243 .653 A SuppAdv5 0.053 .683 NS-A 0.059 .518 A SuppAdv6 0.266 .574 NS-A -0.175 .281 NS IntOthers1 0.859* .915 S 0.501** .829 S IntOthers2 -0.337 .230 NS -0.008 .514 A IntOthers3 -0.013 .556 NS-A 0.178 .643 A IntOthers4 0.268 .404 NS 0.346* .590 S IntOthers5 0.307 .622 NS-A 0.396** .682 S Courses1 0.244 .506 NS-A 0.220 .736 A Courses2 0.365 .764 NS-A 0.340** .770 S Courses3 -0.108 .312 NS 0.217* .605 S Courses4 0.002 .566 NS-A 0.396*** .796 S Courses5 0.705** .895 S 0.198 .653 A InstSup1 -0.089 .463 NS 0.554*** .853 S InstSup2 -0.217 -.229 NS -0.186 -.307 NS InstSup3 1.015*** .976 S 0.554*** .849 S ProfGender1 -0.145 .520 NS-A -0.329 -.195 NS ProfGender2 0.518 .760 NS-A -0.483 -.213 NS ProfGender3 0.737 .925 NS-A 1.109*** .751 S StudGender1 1.026 1.000 NS-A 1.538** .799 S StudGender2 -0.031 .846 NS-A -0.952 .241 NS StudSuccess1 0.995 1.000 NS-A -0.187 .555 A StudSuccess2 0.006 .837 NS-A 1.115** .990 S Fairness1 0.728* .937 S 0.553*** .822 S Fairness2 0.336 .719 NS-A 0.025 .600 A Fairness3 0.122 .618 NS-A 0.617*** .861 S Obst1 0.187 .616 NS-A -0.160 .166 NS Obst2 0.179 .512 NS-A 0.033 .463 NS Obst3 0.658*** .875 S 0.864*** .905 S Obst4 0.347* .629 S 0.423** .541 S UnAtt1 0.529 .752 NS-A 0.093 .651 A
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UnAtt2 -0.411 .182 NS 0.667** .958 S UnAtt3 0.727* .781 S -0.001 .652 A UnAtt4 0.133 .726 NS-A 0.141 .697 A UnAtt5 0.105 .115 NS 0.256 .795 A OpSuccess1 0.754 .888 NS-A 0.277 .796 A OpSuccess2 0.091 .763 NS-A -0.001 .474 A OpSuccess3 0.523 .823 NS-A 0.141 .685 A OpSuccess4 -0.219 .772 NS-A 0.707* .967 S
Note 1. S = significant, NS = not significant, NS-A = not significant, but absolutely important Note 2. *p < .05, **p < .01, ***p < .001
In order to maintain consistency of the model across genders, only the four
indicators that were neither significant nor absolutely important for both genders were
considered for deletion: PersCons1, Belonging 1, InstSup2, and PersChar3. Since the
PersCons2 item is included in the same factor as PersCons1, PersCons1 is
theoretically redundant and can be removed from the model without impacting the
content coverage of the construct. The removal of Belonging1 from the model
improves the R2 value (the percent of variance in the data explained by the model) of
the resulting path model slightly, and thus, the indicator is empirically acceptable to
remove. The interpretation of the construct to which InstSup2 belongs is more
theoretically sound without this indicator, and thus, InstSup2 can be removed from the
model. However, PersChar3, which addresses natural talent in mathematics, is
important according to prior literature and to the theoretical basis of the construct, and
thus it was retained.
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Evaluating the Structural Model
Finally, the structural model, containing the relationships between the latent
constructs, must be evaluated for its suitability. First, as with the formative
measurement model, the structural model must be assessed to detect any issues with
multicollinearity amongst the latent constructs. This is done by using the latent
variable scores for each participant to conduct a multiple regression, with the latent
variable scores predicting the outcome construct scores. As before, VIF values greater
than 5 indicate multicollinearity of constructs. In this case, across the ten imputed
data sets, the maximum average VIF value was 1.847 for any of the 15 predictor
constructs, well below the maximum allowable value, indicating no issue with
multicollinearity in the structural model.
The structural model is also assessed for the amount of variance in the data that
is explained by the model, interpreted from the R2 statistic. The average R2 value
across the five imputed data sets for female participants was .5132, while the average
R2 value for the five imputed data sets for male participants was .4492. Hair et al.
(2014) recommend that R2 values of .75 be interpreted as substantial, .50 as moderate,
and .25 as weak. Therefore, the structural models for both the female and male
participants can be interpreted as explaining a moderate proportion of the variance in
the latent construct scores.
To assess the predictive relevance (Q2) of the structural model, blindfolding
was used to obtain cross-validated redundancy measures for the outcome construct,
with an omission distance D = 7. Predictive relevance “shows how well the data
collected empirically can be reconstructed with the help of the model and the PLS
parameters” (Fornell & Cha, 1994, as cited in Akter, D’Ambra, & Ray, 2011). Values
of Q2 greater than zero are acceptable. The average values of Q2 for the five imputed
59
data sets for each gender were .239 and .324, respectively for female and male
participants, indicating sufficient predictive relevance.
Effect sizes for latent constructs were calculated with Cohen’s f2, the same
effect size calculation used for multiple regression analyses (Cohen, 1988). This
statistic represents the amount of unexplained variance in the model that is explained
by the addition of a certain latent construct. According to Cohen, effect sizes of .02,
.15, and .35 can be considered small, medium, and large, respectively (Cohen, 1988).
For the 15 predictor constructs included in this study, effect sizes can be found in
Table 8.
Table 8 Effect Sizes for Latent Constructs
Construct Effect size (f2) Female participants
Male participants
Personal Characteristics .070 .027 Personal Considerations .033 .005 Content Preparation .001 .012 Sense of Belonging .004 .005 Academic Support from Advisor .050 .113 Interactions with Others in the Department .038 .005 Quality and Availability of Courses .024 .029 Academic Benefits of Institutional Support .059 .006 Professor Gender Ratios .001 .001 Student Gender Ratios .001 .004 Ratios for Student Success .013 .006 Fairness of Policies .001 .009 Obstacles Faced .101 .003 Unwanted Attention Due to Gender .010 .003 Opinions About Success Due to Gender .001 .001 Note. For f2 effect sizes, .02 is considered small, .15 is considered medium, and .35 is considered large (Cohen, 1988).
60
Finally, bootstrapping was used to calculate the standard errors for the path
coefficient estimates between each latent construct and the outcome construct. As this
analysis relates to the research questions for this study, the results will be discussed in
the following section. According to the previously discussed modifications, the final
path model can be found in Appendix B.
Methods Used to Investigate Research Question 1
In order to identify the factors with the strongest association with female
participants’ doctoral program success, a PLS-SEM analysis was conducted using the
data from only female participants who had obtained doctorates. Analysis of these
data allowed for a determination of the strength of the associations between the latent
constructs and the outcome, doctoral program success. These associations, reported as
pooled path coefficients, were then compared to determine which latent constructs had
the strongest impact on doctoral program success. This analysis was conducted using
the following conventional specifications: a path weighting scheme, which maximizes
the value of R2 for the latent variables; a raw data transformation to standardize the
input data; an initial value of +1 to initialize the analysis; a threshold stopping
criterion of 0.00001 to ensure stabilization of the results; and a maximum of 300
iterations for convergence (Hair et al., 2014).
Methods Used to Investigate Research Question 2
In order to compare factors associated with participants’ doctoral program
success for male and female participants with doctorates, multi-group analysis was
conducted (Hair et al., 2014). Multi-group analysis compares pairs of path
61
coefficients for latent variables for different samples: in this case, female participants
and male participants (Kock, 2014). Path coefficients were compared for significance
using two methods: the pooled standard error method, which assumes the standard
errors of the two samples are not significantly different; and the Satterthwaite method,
which does not make assumptions about the standard errors of the data (Kock, 2014).
Results
RQ 1: Impactful Factors in the Success of Female Participants
Research Question 1 aimed to identify the relative impact of the 15 latent
constructs on mathematics doctoral program success of female students. Five
constructs were significantly predictive of the outcome construct (i.e., the pooled path
coefficient was statistically significant). These constructs were Personal
Characteristics, Personal Considerations, Academic Support from Advisor, Academic
Benefits of Institutional Support, and Obstacles Faced. Interestingly, these factors
include a mix of personal factors and institutional or program-level factors.
Of the five significant constructs, Obstacles Faced was most impactful in the
successful of female participants. Its pooled path coefficient of 0.257 implies that a
one standard deviation increase in a participant’s evaluation of the obstacles faced
while enrolled in their doctoral program would result in over a quarter of a standard
deviation increase in their evaluation of their doctoral program success. In comparing
the path coefficients, the obstacles a participant faces in her doctoral program are
nearly twice as impactful as a student’s personal considerations, such as familial and
financial responsibilities. The remaining 10 constructs did not reach statistical
significance. The pooled path coefficients can be found in Table 9.
62
Table 9 Pooled Path Coefficients for Female and Male Participants
Pooled path coefficients
Construct Female participants
Male participants
Personal Characteristics 0.196** 0.138*** Personal Considerations 0.144* 0.061 Content Preparation -0.002 0.090* Sense of Belonging -0.059 0.052 Academic Support from Advisor 0.205** 0.283*** Interactions with Others in the Department 0.141 0.066 Quality and Availability of Courses 0.141 0.169** Academic Benefits of Institutional Support 0.176** 0.061 Professor Gender Ratios 0.013 -0.002 Student Gender Ratios 0.026 0.040 Ratios for Student Success -0.077 0.059 Fairness of Policies 0.014 0.082* Obstacles Faced 0.257*** 0.050 Unwanted Attention Due to Gender 0.091 0.052 Opinions About Success Due to Gender 0.056 0.006 Note. *p < .05, **p < .01, ***p < .001
RQ 2: Comparison of Significant Factors for Female and Male Participants
Research Question 2 sought to investigate differences in the importance of the
identified factors for female and male participants with doctorates. One way to
determine how factors associated with doctoral program success compare for female
and male participants is to compare those constructs whose path coefficients reached
statistical significance in the model for the female participants and the model for the
male participants. Academic Support from Advisor and Personal Characteristics were
significantly predictive of doctoral program success for both genders. Personal
Considerations, Academic Benefits from Institutional Support, and Obstacles Faced
were predictive only for female students, while Content Preparation, Quality and
63
Availability of Courses, and Fairness of Policies were predictive only for male
students.
Additionally, recall that two types of multi-group analyses were conducted to
compare the path coefficients for female participants to those of male participants.
Table 10 presents the results of the multi-group analysis for all 15 predictor constructs.
Only one comparison reached statistical significance and that was for the construct
Obstacles Faced. Thus, the construct Obstacles Faced had a significantly stronger
relationship with doctoral program success for female participants than for male
participants (p = .001).
Table 10 Multi-group Analysis for Female and Male Participants
Construct t-value (Pooled standard error method)
t-value (Satterthwaite method)
Personal Characteristics 0.7338 0.6828 Personal Considerations 1.0109 1.0501 Content Preparation -1.1633 -1.2018 Sense of Belonging -1.2414 -1.2701 Academic Support from Advisor -0.8984 -0.9150 Interactions with Others in the Department 0.7484 0.7397 Quality and Availability of Courses -0.2689 -0.2820 Academic Benefits from Institutional Support 1.2934 1.3946 Professor Gender Ratios 0.1707 0.1831 Student Gender Ratios -0.1745 -0.1815 Ratios for Student Success -1.6302 -1.5218 Fairness of Policies -0.8506 -0.8113 Obstacles Faced 3.2253** 3.2968** Unwanted Attention Due to Gender 0.4482 0.3952 Opinions About Success Due to Gender 0.5746 0.5212 Note. *p < .05, **p < .01, ***p < .001
64
Discussion
This study was conducted to investigate the experiences of successful female
mathematics doctoral students and to compare these experiences to that of male
doctoral students. Much of the previous research in this area has focused specifically
on issues of attrition of female students, utilizing small samples and qualitative
methodologies. In contrast, this study used a large, representative sample of
mathematics faculty members and focuses on factors associated with doctoral program
success. While previous studies provided detailed descriptions of individuals’
experiences, the findings were not generalizable. Moreover, it was unclear which
factors were most critical in explaining retention and attrition. For this study, the use
of structural equation modeling, combined with the inclusion of male participants as a
comparison group, allows for more nuanced claims to be made than in previous work.
For instance, previous studies made claims about the importance of the relationship
between a student and her advisor based on the participant’s qualitative self-report
(e.g., Herzig, 2004b; Herzig, 2010; Hollenshead et al., 1994). However, it was
unknown how influential this factor is in comparison to other factors reported by the
participant as important. In this study, factors influencing doctoral program success
were compared quantitatively to confirm that the quality of the advisor-advisee
relationship was, in fact, one of the most influential factors, regardless of gender.
Moreover, because of the representativeness of the sample, these results are
generalizable to the population of mathematics faculty members in the United States.
Analyses of the data collected reveal that female participants with doctorates
found aspects of their personal lives, the academic support they received from their
advisor, the academic benefits of the institutional support they received (in the form of
assistantships and fellowships), and the obstacles they faced on their path to their
65
doctorate to be most impactful on their doctoral program success. Obstacles included
both personal or individual obstacles (struggling with confidence) and programmatic
or institutional obstacles (passing benchmark exams).
Differences were also detected in the experiences of male and female doctoral
graduates. Only one of the 15 factors – Obstacles Faced – reached significance in
comparing the latent constructs in the multi-group analysis. This means that obstacles
faced were a significantly stronger predictor of doctoral program success for women
than for men. One hypothesis that arises from this finding is that women might be
more inclined to interpret obstacles faced as a detriment to their doctoral program
success because they tend to have lower mathematics self-efficacy than men. Another
hypothesis deals with different tendencies for the attributions of success and failure by
men and by women: a woman’s success is more often attributed to luck or effort,
while a man’s success is more often attributable to innate ability (Lott, 1985).
Conversely, women’s failures tend to be associated with personal shortcomings, such
as ability, while men’s failures are usually attributed to external circumstances, such
as bad luck (Lott, 1985). Furthermore, the constructs that reached significance in the
separate male and female PLS-SEM models differed. Personal considerations (such as
family responsibilities), opportunities to learn from teaching or research assistantships,
and overcoming obstacles were predictive of success for female participants only. The
significance of assistantship assignments for female participants, but not for male
participants, could be a problem of inequity or of the perception of inequity. Female
doctoral students may, in fact, be assigned assistantships with inferior opportunities to
learn due to biased or inequitable practices. Alternatively, female students may
perceive that they are not able to gain the same level of academic benefit from their
66
assistantships as male students because male students may be able to form stronger
bonds with lead course instructors or principal investigators, who are likely also male.
Female students should thus be prepared to advocate for their own learning in this area
by requesting a range of funding opportunities, including both teaching and research
assistantships, during their doctoral program. For male participants, content
preparation, coursework, and the fairness of policies within the department had a
stronger influence on doctoral program success than for female participants.
However, it is noteworthy that both a student’s relationship with his or her advisor and
personal characteristics were predictive in both models, suggesting that improvements
or additional supports in these areas would be beneficial for all students, regardless of
gender.
The results of this study suggest five key recommendations for doctoral
programs and for female students. Doctoral programs could use these findings to
empower their female students to become better advocates for their own learning.
First, this research provides additional support for the finding from previous research
that the role of the advisor has a strong influence on doctoral student success (e.g.,
Bair & Haworth, 2004; Fagen & Wells, 2004; Miller, 2015a; Tinto, 1993). The
importance of developing and maintaining a productive advisor-advisee relationship
resonated across both of the two main analyses. A supportive advisor was a prominent
factor in explaining the success of both female and male participants. Therefore,
departments could emphasize the importance of advising as part of their tenure review
process to reward faculty for devoting time to improving these relationships (Bair &
Haworth, 2004) and increase their awareness of the importance of their role as an
advisor through training or professional development. Female students should be
67
aware of the importance of choosing a suitable advisor and make this decision with
great care. This choice should likely be based on considerations including, but not
limited to, alignment of areas of research interest, compatibility of personality types,
and potentially even discussions with a professor’s former doctoral students. Second,
additional female faculty members could be hired in order to provide visible female
mentors and role models to female students. Third, a culture could be created within
the program where both doctoral advisors and faculty members sponsoring students as
teaching or research assistants are encouraged to mentor students and focus on their
students’ development as future faculty members and scholars. Although personal
factors are outside the scope of a program’s control, doctoral programs could provide
supports in order to give admitted students the greatest possible chance of success.
The fourth recommendation, to equalize available time for schoolwork for students
with and without families, is for support or provisions for childcare to be integrated
into the institutional structure. Finally, for students with financial concerns, programs
could refer students to free or low-cost financial advisement in their area to help with
budgeting, student loans, and programs available to assist students and low-income
individuals.
Many of the findings presented here provide additional support for claims
made in other studies. However, the comparisons of the relative importance of each
factor for female students and between female and male students provide a unique
contribution to understanding the mechanisms underlying success in doctoral
mathematics. In the future, institutions could administer this survey to their current
students (as part of a yearly review) or recent graduates (as part of an exit interview),
creating a feedback mechanism to guide changes within the mathematics doctoral
68
program. Furthermore, the survey could be modified and used by other researchers to
investigate similar issues of retention and attrition for doctoral students in other STEM
fields. Revisions that may need to be made include the addition of items pertaining to
availability of laboratory time, space, and resources, and the quality and productivity
of interactions between laboratory group members.
Now that factors influencing the success of female mathematics doctoral
students have been empirically investigated in a more generalizable manner than has
previously been done, small-scale interventions can begin to be implemented at
individual institutions to see if modifications to these factors result in greater success
for female students. Because of the length of time required to obtain a doctorate, these
studies would need to collect longitudinal data over for a minimum of five years
before assessments of efficacy could be made. If substantial improvements occur,
these interventions could then be scaled up to include more institutions over time.
Additionally, it is an open question as to how these results would differ if the
sample were expanded beyond those doctoral recipients employed in academia.
Different factors may be important for success for those whose career goals will lead
to employment in government or in industry. However, obtaining a representative
sample from that portion of the population of mathematics doctorate recipients would
be challenging. A potential starting point would be to track the career trajectories of
recent graduates in order to determine if any trends exist in where these doctoral
recipients accept employment. Then, once this is known, it may be easier to recruit a
somewhat representative sample of those employed outside academia. Results from
this population could then be compared to the results presented here to make
69
recommendations to promote the success of all women, regardless of their career
aspirations.
Conclusion
The research described here is a key step in formulating a set of best practices
for retaining female mathematics doctoral students. This has the potential to make a
significant impact in narrowing the gender gap both in participation and in success for
mathematics doctoral students. An increased number of female graduates from
mathematics doctoral programs should eventually lead to a more balanced gender ratio
for mathematics faculty members. This, in turn, could have the effect of encouraging
more women to become interested in and study mathematics, diversifying the
discipline to the benefit of all involved (Hill et al., 2010). Over time, with increased
participation and a greater number of female mathematicians, mathematics educators,
and mathematics teachers as role models, gendered stereotypes of mathematical
competency may become a thing of the past.
70
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Appendix A
SURVEY INSTRUMENT
1. Have you obtained a doctorate in mathematics or applied mathematics?
• Yes
• No
If the participant replied, “Yes,” the survey skipped to Question 4.
If the participant replied, “No,” the survey continued to Question 2.
2. Have you ever enrolled in a doctoral program in pure mathematics or applied mathematics?
• Yes
• No
If the participant replied, “Yes,” the survey continued to Question 3.
If the participant replied, “No,” the survey ended and the participant’s data was
discarded.
3. Are you currently enrolled in a doctoral program in pure mathematics or applied mathematics?
• Yes
• No
If the participant replied, “Yes,” the survey ended and the participant’s data was
discarded.
If the participant replied, “No,” the survey continued to Question 4.
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4. From which of the following funding sources were you financially supported during your doctoral program? (Please select all that apply.)
• Teaching assistantship
• Research assistantship
• Fellowship, scholarship, or grant
• Government or private loans
• Non-university employment: (Please describe)
• Other: (Please describe)
5. For approximately how many years were you supported from each funding source?
Participants entered durations into a list containing only the funding sources they
selected in Question 4.
All of the following items were evaluated on a 5-point Likert scale, ranging from
“Strongly Disagree” to “Strongly Agree.” Participants who did not complete their
doctorate also had a “Not Applicable” option for certain items. Items followed by
“(R)” were worded in the negative for reliability purposes and were recoded for
analyses. Blocks of questions on the survey were ordered as seen below, but questions
within each block were randomized for each participant to minimize the possibility of
order effects.
Variable Name Items/Indicators
PersChar4 My personal characteristics were not well suited for success in my doctoral program in mathematics. (R)
PersChar2 I was committed to my work during my doctoral study in mathematics.
PersChar3 I am naturally talented in mathematics.
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PersChar1 During my doctoral study in mathematics, I was motivated to succeed.
Obst2 My confidence that I would succeed in obtaining a doctoral degree in mathematics often wavered. (R)
InstSup2
This item was coded for each participant for its alignment with Q23. If the participant preferred a competitive environment, the item remained as is. If the participant preferred a collaborative environment, the item was reverse coded. During my doctoral program in mathematics, there was a competitive culture among students.
ContPrep2 My prior educational experiences in mathematics prepared me well to succeed in my doctoral program in mathematics.
ContPrep1 The mathematics courses I took in my undergraduate program did not prepare me well to succeed in my doctoral study in mathematics. (R)
Belonging1 Before beginning my doctoral program in mathematics, I did not have the opportunity to participate in research in mathematics. (R)
ContPrep3 Before beginning my doctoral program in mathematics, I had realistic expectations of what would be required of me during my doctoral study.
PersCons2 Outside of academics, aspects of my personal life did not detract from my ability to be successful in my doctoral program in mathematics.
PersCons4 I received support and encouragement from my family during my doctoral study in mathematics.
PersCons1 My family responsibilities detracted from my ability to be successful during my doctoral study in mathematics. (R)
PersCons3 During my doctoral study in mathematics, concerns about financial issues detracted from my ability to be successful. (R)
Participants answered only the Global item and any items pertaining to their responses in Question 4.
InstSup4 During my doctoral program in mathematics, my responsibilities to my funding source(s) still left me with ample time to devote to my own work.
InstSup5 The amount of time I was required to devote to my teaching assistantship responsibilities was unmanageable and detracted from my ability to be successful. (R)
InstSup6 The amount of time I was required to devote to my research assistantship responsibilities was manageable and did not detract from my ability to be successful.
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InstSup7 My fellowship benefitted me in that it did not demand additional time that detracted from my own work.
Participants answered only the global item and any items pertaining to their responses in Question 4.
InstSup3 During my doctoral program in mathematics, my funding source(s) provided me with opportunities that were academically beneficial to me.
InstSup8 My teaching assistantship provided me with opportunities that were academically beneficial to me.
InstSup9 My research assistantship did not provide me with opportunities that were academically beneficial to me. (R)
InstSup10 While on fellowship, I missed out on opportunities that could have been academically beneficial to me. (R)
RelOthers3 During my doctoral study in mathematics, I did not have the chance to interact with the professors in my department outside of the classroom. (R)
RelOthers1 During my doctoral study in mathematics, I had the opportunity to participate in informal conversations about mathematics with professors.
RelOthers5 During my doctoral study in mathematics, I worked closely with other professors in the department (other than my advisor).
UnAtt2 During my doctoral study in mathematics, I was the subject of unwanted attention due to my gender from one or more professors. (R)
UnAtt3 During my doctoral study in mathematics, I felt that I was negatively singled out for reasons related to my gender by one or more professors, even in very small ways. (R)
OpSuccess4 Professors in my doctoral program appeared to think that male students were more likely to succeed than female students. (R)
OpSuccess1 Professors in my doctoral program appeared to think that female students were equally likely to succeed as male students.
RelOthers4 During my doctoral study in mathematics, I had the chance to interact with the other students in my department outside of the classroom.
UnAtt1 During my doctoral study in mathematics, I was the subject of unwanted attention due to my gender from one or more students in my program. (R)
UnAtt4 During my doctoral study in mathematics, I felt that I was negatively singled out for reasons related to my gender by from one or more students, even in very small ways. (R)
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RelOthers2 During my doctoral study in mathematics, I had the opportunity to participate in informal conversations about mathematics with other students.
OpSuccess3 Other students in my doctoral program appeared to think that male students were more likely to succeed than female students. (R)
OpSuccess2 Other students in my doctoral program appeared to think that female students were equally likely to succeed as male students.
SuppAdv1 My doctoral advisor was academically supportive of me.
SuppAdv2 My doctoral advisor did not provide valuable feedback on my work. (R)
SuppAdv6 My doctoral advisor did not help me in selecting my dissertation/thesis topic. (R)
SuppAdv3 My doctoral advisor aided me in networking with colleagues. SuppAdv5 My doctoral advisor treated me as a colleague.
SuppAdv4 My doctoral advisor did not provide me with assistance in transitioning to my career. (R)
UnAtt5 During my doctoral study in mathematics, I was the subject of unwanted attention due to my gender from my doctoral advisor. (R)
Fairness4 During my doctoral study in mathematics, the program’s structure and policies were conducive to my success.
Obst1 During my doctoral study in mathematics, passing benchmark exams (e.g., preliminary exams, candidacy exams) was an obstacle. (R)
Fairness3 During my doctoral study in mathematics, the policies and procedures of the department were unclear or were not made explicit to students. (R)
Fairness2 During my doctoral study in mathematics, the policies and procedures of the department were inconsistently applied for certain students. (R)
Fairness1 During my doctoral study in mathematics, teaching and research assistantship assignments were made fairly.
Courses3 The requirements of my doctoral program allowed for enough electives that I could specialize in my area of interest.
Courses1 During my doctoral study in mathematics, I was satisfied with the course offerings available in my department.
Courses2 During my doctoral study in mathematics, I was not satisfied with the courses I took. (R)
Courses4 During my doctoral study in mathematics, I was satisfied with the quality of the teaching in my courses.
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6. What is your gender?
• Male
• Female
Belonging2 During my doctoral study in mathematics, I felt I was a valuable member of the courses I took.
Courses5 During my doctoral study in mathematics, I did not receive valuable feedback on my assignments.
Obst4 During my doctoral study in mathematics, I was hesitant to ask questions in class because of how those questions may be received by the professor or other students in the class. (R)
ProfGender3 The professors and students in my doctoral program reflected sufficient gender diversity.
ProfGender1 The professors in the department were predominantly male. (R)
ProfGender2 There were approximately equal numbers of male professors and female professors in the department.
StudSuccess1 Male students in my doctoral program were more successful than female students, in terms of the proportion of incoming students of each gender who completed their doctorate. (R)
StudSuccess2 Male students and female students in my doctoral program were equally successful, in terms of the proportion of incoming students of each gender who completed their doctorate.
StudGender2 During my doctoral study in mathematics, there were noticeably more male students than female students. (R)
StudGender1 During my doctoral study in mathematics, there were approximately equal numbers of male and female students.
Outcome3 Overall, my experience in my doctoral program in mathematics was successful.
Outcome4 I was able to complete my doctoral degree in a reasonable amount of time.
Obst3 I experienced significant setbacks during my doctoral program in mathematics.
Outcome2 My doctoral program in mathematics equipped me with the knowledge and skills needed to succeed in my intended career.
Outcome1 After graduating from my doctoral program in mathematics, I was hired at the type of job I wanted.
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• Other
7. What is your current job title?
• Full professor
• Associate professor
• Assistant professor
• Post doctorate
• Adjunct professor
• Lecturer or instructor
• Other: (Please describe)
8. What is the most advanced degree in mathematics that is granted by your employing institution?
• Associate’s
• Bachelor’s
• Master’s
• Doctorate
9. Two versions of this question were used. The pertinent version was determined from Questions 1, 2, and 3.
For participants who were successful in obtaining a doctorate:
In what country did you earn your doctoral degree?
For participants who enrolled, but did not complete a doctorate:
In what country did you attempt to earn your doctoral degree?
10. In what country did you earn your undergraduate degree?
11. In what country did you complete the majority of your secondary education?
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12. Two versions of this question were used. The pertinent version was determined from Questions 1, 2, and 3.
For participants who were successful in obtaining a doctorate:
For how many years were you enrolled in your doctoral program before
graduating?
For participants who enrolled, but did not complete a doctorate:
For how many years were you enrolled in your doctoral program before leaving?
13. During your doctoral study in mathematics, did you:
• Work with only one dissertation advisor
• Work with more than one dissertation advisor (I had co-advisors)
• Work with more than one dissertation advisor (I changed advisors)
If the participant replied that they had only one advisor, the survey skipped to Q15.
If the participant replies that they changed advisors at some point, the survey will
proceed to Q14.
14. Please briefly describe your reason for changing dissertation advisors.
15. What was the gender of your doctoral advisor? (If you had more than one advisor, please respond for the advisor under whom you completed your doctorate.)
• Male
• Female
• Other
16. How many years have passed since you graduated from your doctoral program?
• 0-9 years
• 10-19 years
• 20-29 years
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• 30-39 years
• 40-49 years
• 50 or more years
17. Did your doctoral program require that students pass preliminary exams? Preliminary exams are traditionally taken during or shortly after the completion of coursework, usually within the first two years.
• Yes
• No
If the participant replied, “Yes,” the survey continued to Question 18.
If the participant replied, “No,” the survey skipped to Question 19.
18. How many times did you attempt your preliminary exams? (Please include the attempt on which you passed in the count.)
19. Did your doctoral program require that students pass candidacy exams? Candidacy exams are traditionally administered in order to allow a student to demonstrate expert knowledge in a specific subdomain related to the area of their dissertation, and are usually given around a student’s third year.
• Yes
• No
If the participant replied, “Yes,” the survey continued to Question 20.
If the participant replied, “No,” the survey skipped to Question 21.
20. How many times did you attempt your candidacy exams? (Please include the attempt on which you passed in the count.)
21. During your doctoral study in mathematics, did your marital status change?
• Yes, I was married during my doctoral study
• Yes, I was divorced during my doctoral study
• No, I was single throughout my doctoral study
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• No, I was married throughout my doctoral study
22. During your doctoral study in mathematics, did your parental status change?
• Yes, I had children during my doctoral study
• No, I had children before enrolling
• No, I did not have children during my doctoral study
23. In situations pertaining to mathematics, do you prefer an environment that is:
• Mostly collaborative
• Mostly competitive
• No preference
24. Is there anything else you would like to share about your experience in a mathematics doctoral program?
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Appendix B
PATH MODEL
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Appendix C
INSTITUTIONAL REVIEW BOARD APPROVAL LETTER
- 1 - Generated on IRBNet
RESEARCH OFFICE
210 Hullihen HallUniversity of Delaware
Newark, Delaware 19716-1551Ph: 302/831-2136Fax: 302/831-2828
DATE: November 7, 2014 TO: Emily MillerFROM: University of Delaware IRB STUDY TITLE: [677728-1] Factors Contributing to the Retention of Female Mathematics
Doctoral Students: Testing and Refining a Model of Graduate StudentPersistence
SUBMISSION TYPE: New Project ACTION: DETERMINATION OF EXEMPT STATUSDECISION DATE: November 7, 2014 REVIEW CATEGORY: Exemption category # (2)
Thank you for your submission of New Project materials for this research study. The University ofDelaware IRB has determined this project is EXEMPT FROM IRB REVIEW according to federalregulations.
We will put a copy of this correspondence on file in our office. Please remember to notify us if you makeany substantial changes to the project.
If you have any questions, please contact Nicole Farnese-McFarlane at (302) 831-1119 [email protected]. Please include your study title and reference number in all correspondence with thisoffice.