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

v

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

vi

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.

vii

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  

viii

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  

ix

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  

x

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

xi

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

2

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

3

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.

6

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.

7

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

8

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.

9

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

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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.

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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.

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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

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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

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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

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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

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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).

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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

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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.

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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

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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 ornicolefm@udel.edu. Please include your study title and reference number in all correspondence with thisoffice.