Quantifying and Documenting Inequity in PhD-granting Mathematical Sciences Departments in the United States

The kinds of problems mathematics and data science can be used to solve are extremely varied, running the gamut from theoretical with no foreseen applications to those that are immediately applicable to important real-world phenomena like climate change, epidemiology, and social networks. There is a rapidly growing body of work using tools from the mathematical sciences to analyze the mathematical sciences itself that has recently been described as the “mathematics of Mathematics” or “MetaMath” [5]. This term was chosen as an allusion to the broader field of “science of Science” [12] or “SciSci” that uses scientific and mathematical tools to analyze science as a whole as well as its individual disciplines.

In this paper, we present a contribution to the mathematics of Mathematics to introduce readers to the kinds of questions that can be asked and the types of tools and techniques that can be used in the emerging MetaMath area. We are inspired by the work of Wapman, Zhang, Larremore, and Clauset [37] published in Nature in 2022 that quantified hierarchy in faculty hiring and retention in a wide array of academic disciplines in the United States by analyzing a very large dataset of nearly 300,000 faculty members in over 10,000 departments at almost 400 PhD-granting institutions from 2011 to 2020. The research we present here builds on and leverages the ideas and analyses presented in Wapman et al. [37], but focuses on specific disciplines in the mathematical sciences that are available in their dataset [36], namely, mathematics, statistics, and operations research. This results in a dataset which includes nearly 10,000 faculty members in well over 200 departments that granted PhDs in the mathematical sciences from 2011 to 2020.

Using mathematical tools from network science [21], Wapman et al. produced a “prestige ranking” for every department at every PhD-granting institution in their dataset and quantified hierarchy in faculty hiring and retention in a large number of academic disciplines. In particular, Wapman et al. constructed a directed graph of PhD-granting departments with an edge from department A to department B corresponding to a faculty member who earned a Ph.D. at department A being hired into department B. Unlike other rankings like those published by U.S. News & World Report that purport to assign prestige based on an arbitrary selection of attributes, Wapman et al.’s prestige rankings are based upon the characteristics of the directed graph of departments. It is assumed that “prestigious” departments will prefer to hire faculty from other “prestigious” departments, and then the overall hierarchy of departments is inferred from the topology of the network using the SpringRank algorithm [9]. We will refer to the definition of prestige from Wapman et al. [37] as “Wapman-prestige.”

We utilize Wapman’s quantification of prestige to produce a definition of “elite,” which we will use to help us quantify and document inequity in PhD-granting departments in the United States. We define “elite” institutions as those that are within the top quartile as defined by Wapman-prestige. In this paper we will use Hasty et al.’s anthropological definition of inequity. They say inequity is “the unequal distribution of resources due to an unjust power imbalance. It is a type of inequality caused by this unequal distribution, often as a result of injustices against historically excluded groups of people” [18]. In the mathematical sciences, the underrepresentation of historically marginalized demographic groups at all levels results in an unjust power imbalance (see the brief literature review in Section 2 for further details). Because of the nature of the dataset [36], which contains gender data but not other demographic data (and in particular no data on race or ethnicity), we are only able to conduct analyses using gender and not other identity characteristics. However, by combining this dataset with publicly available data from the National Science Foundation (NSF) on awards made by the Division of Mathematical Sciences (DMS) [14], which is the primary funder of mathematical sciences research in the United States, we can investigate relationships among gender equity, Wapman-prestige, and NSF DMS funding.

Our thesis is that we can quantify and document inequity in mathematical sciences PhD-granting departments in the United States. Specifically, in this paper we will address the following research questions:

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  RQ1: What is the relationship between the percentage of women in a department and that department’s Wapman-prestige?

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  RQ2: What is the relationship between Wapman-prestige and funding received from the National Science Foundation Division of Mathematical Sciences?

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  RQ3: What is the relationship between the percentage of women in a Mathematics department and funding received from the National Science Foundation Division of Mathematical Sciences?

In this section, we provide a short survey of selected recent work that uses tools, topics, and techniques from mathematics and data science to describe, document, and discuss inequity in the mathematical sciences. We organize our summary of the literature in this area into three topics: 1) analysis of the (lack of) diversity in the mathematical sciences; 2) existence of hierarchy in mathematics and other academic disciplines; and 3) evidence of inequity in the mathematical sciences. For a longer survey of the areas discussed here, as well as the broader field of the mathematics of Mathematics, we refer the reader to the recent paper by Buckmire et al. [5].

In this section, we will describe the data and explain the methodology used to obtain our results. Our data and code is publicly available at [3]. We utilized two datasets, one sourced from Wapman et al. [36] and the second from awards made by the Division of Mathematical Sciences (DMS) at the United States National Science Foundation (NSF) between 2011 and 2020 [13].

The Wapman et al. dataset required some nuance to interpret. The dataset consisted of a census of tenured or tenure-track faculty employed at all PhD-granting institutions in the United States from the years 2011–2020. Faculty were only included in this sample if they were employed in the majority of the years under review. This dataset is centered on departments, rather than on faculty, and these departments are each assigned to a field such as “Mathematics.” In particular, a department may be accounted for in multiple fields; a “Department of Mathematics and Statistics” would have its faculty included twice, in the fields of “Mathematics” and “Statistics.”

Because our goal is to quantify and document inequity in the mathematics community as a whole, we choose to define the mathematical sciences as broadly as possible (see [4]). This choice results in a reduction of Wapman et al.’s original dataset of 295,089 faculty in all academic disciplines offering PhDs in the United States to 9,814 faculty that are distributed among the fields of Mathematics, Statistics, or Operations Research (Table 1). We adopt the convention of capitalizing these three terms when referring to these fields present in the data throughout the rest of this paper.

In our analysis, we incorporate the department prestige rankings from Wapman et al. [36] which we refer to as Wapman-prestige. Because of the way these are computed (a department has more Wapman-prestige if its PhD-graduates are hired by departments that also have Wapman-prestige), some departments do not have a Wapman-prestige ranking. This could happen if the department does not have a PhD program in one of the three fields (recall the Wapman dataset began with institutions that grant PhDs), or if none of its PhD-graduates were hired by departments with Wapman-prestige rankings. In Mathematics, for example, there are 223 departments listed, but only 161 of these have a Wapman-prestige ranking. Departments without Wapman-prestige rankings were not included in the data analyses involving Wapman-prestige below, but were included in the analysis of NSF funding later in the paper.

Since the Group I departments—in the groupings used by the AMS historically to divide departments by perceived quality (see Section 2.1 above)—constituted about 25% of departments, in our analysis we consider the upper quartile (in terms of Wapman-prestige rankings) as “elite” departments and compare this group to the remaining 75% of departments, which we refer to as “non-elite.”

As stated earlier in the paper, the Wapman et al. dataset does not have information on race or ethnicity but does include gender. However, gender is only included as a binary variable in their dataset. In fact, gender is self-reported for a small percentage (6%) of faculty in their initial dataset. They then attempted to infer the gender of the remaining faculty based on their names and the use of software that claims to assign gender based on names relatively accurately; ultimately binary gender was ascribed to a total of 85% of the faculty listed in their dataset. We include only these faculty who were ascribed a binary gender in our analyses,which eliminates roughly 15% of the total due to the inability to accurately ascribe a gender to these data entries.

We separately obtained data from the NSF’s publicly available data on awards made by the Division of Mathematical Sciences (DMS) in the Directorate of Mathematical and Physical Sciences (MPS) from 2011-2020 [13]. These data were aggregated by institution. Since the institution names in the Wapman et al. dataset typically did not match the formal organization names listed by the NSF, we manually adjusted these in order to compare the two datasets. On average, DMS awarded $235 million per year towards achieving its mission to support “a wide range of research in mathematics and statistics aimed at developing and exploring the properties and applications of mathematical structures” [14]. Of the awards to institutions, 80% were matched to the departments of interest in the following analysis. The NSF, and DMS in particular, is the primary source of funding for mathematics research in the United States [14]. While some mathematicians (such as several authors of this paper) receive funding from NSF divisions other than DMS and directorates other than MPS (e.g., the Division of Undergraduate Education and the Directorate for STEM Education) as well as from other federal agencies (e.g., the National Institutes of Health), we consider DMS funding a reasonable measure for overall financial support of mathematics by the federal government.

There are two major caveats to our use of the NSF DMS funding data. First, NSF awards are made to institutions rather than specifically to departments, which is the unit of analysis provided by the Wapman et al. data. Second, since faculty in Statistics and Operations Research typically have more varied sources of funding, we only considered the field of Mathematics in our analyses of funding discussed below.

Some of the institutions in the NSF data with the largest amount of funding were not included in our analysis because they were mathematics research institutes, not PhD-granting institutions. For example, the Mathematical Sciences Research Institute (MSRI, now known as the Simons Laufer Mathematical Sciences Institute or SLMath) and the Institute for Advanced Study (IAS) were awarded funds as separate entities from their associated universities, the University of California at Berkeley and Princeton University, respectively. We chose to treat MSRI and IAS as separate, non-PhD-granting institutions and therefore they are not included in our analyses.

In this section, we present the primary results of our research into the distribution of faculty and funding at mathematical sciences PhD-granting institutions in the United States. In Figure 1, the percentage of women in the fields of Mathematics, Statistics, and Operations Research from 2011 to 2020 is given. We note that the percentage of women in Mathematics lags behind Operations Research and Statistics throughout the time period of the dataset, mirroring the lower percentage of women that earn PhD’s in Mathematics versus the other two fields [17]. We further note the percentage of women in Mathematics, Statistics, and Operations Research is far below the percentage of women in Academia as a whole for the time period covered by the dataset.

Next, we computed the percentages of faculty in each department inferred to be women, and plotted these according to Wapman-prestige rank in the fields of Mathematics, Statistics, and Operations Research in Figure 2. To compute this percentage, we used as a denominator the total number of faculty in a department for which a gender was inferred, in effect removing from our sample any faculty members whose gender could not be inferred. In Figure 2, the blue circles are clustered in the lower-left corner of all three subfigures; this corresponds to the data demonstrating that elite departments (in the top quartile of Wapman-prestige) also have a low percentage of women (below 20% for Mathematics).

In order to address RQ1, we then considered the elite institutions as a group and calculated the percentage of faculty at these institutions that are women (Table 2); in each case, we see that the percentage of women among these elite institutions is lower than among non-elite institutions. A chi-squared test for each field was conducted, finding only the difference in Mathematics to be significant ($p<0.001$). We also conducted a Kendall tau test to determine if there is an association between the percentage of women and Wapman-prestige rank of Mathematics departments; we found a significant negative association between Wapman-prestige rank and rank by percentage of women ($\tau=-0.23$, $p<0.001$). In other words, higher Wapman-prestige of Mathematics department is associated with having a lower percentage of women.

To address RQ2, we then explored the distribution of DMS funding to Mathematics departments with Wapman-prestige rankings from 2011-2020. The elite institutions (i.e., the upper quartile by Wapman-prestige) were awarded in aggregate $119M per year in grant funding, while the non-elite institutions, of which there are three times as many, were awarded only $70M in aggregate of NSF money per year. We plotted this funding by Wapman-prestige of each department in Figure 3. Note that the two subfigures in Figure 3 depict total funding and per capita funding versus Wapman-prestige and yet they are remarkably similar in appearance. This demonstrates that we can use NSF funding to distinguish between different mathematical sciences PhD-granting institutions without worrying about individual outlier faculty members in Mathematics departments of those institutions. We also computed the total amount of funding received by elite (top quartile by Wapman-prestige) and non-elite (lower 3 quartiles by Wapman-prestige) departments over the time period in question. Elite departments received 64.7% of funding in our dataset, compared to 35.3% for the non-elite departments (despite these being thrice as numerous). Here we also conducted a Kendall tau test, finding a significant positive relationship between Wapman-prestige rank and DMS funding rank ($\tau=0.68$, $p<0.001$). In other words, higher Wapman-prestigeis associated with more DMS funding.

Another metric that can be used to quantify inequity in NSF DMS funding received by Mathematics departments in the United States is the Gini coefficient, a well-known measure of inequality often used to characterize wealth inequality on a scale from 0 to 1. A uniform distribution of wealth would result in a Gini coeffient value of 0 and a case where one individual holds all the wealth would result in a Gini coefficient of 1. The Gini coefficient of NSF DMS funding for the Mathematics departments in our dataset is 0.63. For more information on the Gini coefficient, refer to [27].

We also analyzed the full DMS-funded portfolio (excluding fellowships which are given to people instead of institutions) to avoid a possible sampling effect due to our focus on PhD-granting departments. In this more comprehensive dataset, the top 20% holds 86.1% of all DMS funding. The Gini coefficient of this distribution is 0.80. Thus, the larger set of all DMS funding recipients demonstrates greater inequality than the subset of PhD-granting institutions.

To address RQ3, we plotted the annual grant funding received by Mathematics departments against the percentage of women in those departments in Figure 4. We note an interesting effect in Figure 4, where none of the 29 departments (which are all non-elite with respect to Wapman-prestige) with at least 25% women received more than $1.1M in average annual funding from the DMS. We conducted a Kendall tau test to determine if there is an association between annual funding received from DMS and the percentage of women in Mathematics departments and found a significant negative relationship ($\tau=-0.22$, $p<0.001$). In other words, a higher percentage of women is associated with less annual DMS funding,

In addition to the statistical analyses above, we also ran a k-means clustering analysis on the dataset that contained NSF DMS funding and faculty gender percentage for mathematical sciences PhD-granting departments. We are particularly interested in the extent to which clusters produced by this analysis would reflect the Wapman-prestige rankings.

After normalizing the data, we ran an analysis to determine the optimal clustering number. Using the cluster [23] and factoextra [20] packages in the R programming language, we identified three as a consistent optimal number in the “bend-in-the-knee” analyses. These analyses identify the point at which increasing cluster number does not return significant advantages in lowering measures such as within-cluster variation. A visualization of the resulting three clusters is found in Figure 5. Cluster 1 in the bottom left, designated by red squares, identifies academic institutions with low representation of women and with low funding amounts. Cluster 2 in the bottom right, designated by blue diamonds, identifies institutions with high representation of women, but with low funding amounts. Cluster 3 in the top left, designated by green triangles, identifies institutions with low representation of women, but high funding amounts.

Note that there is no high gender diversity with high funding cluster, nor are there any institutions in the upper right area of Figure 5. In Table 3 the number of elite and non-elite institutions in each cluster are aggregated. We then conducted a chi-squared test to see if cluster membership and “eliteness” (being in the top 25% of Wapman-prestige rankings) were independent; they were not ($p<0.001$). In fact, Cluster 2 (blue diamonds) did not contain a single elite department, while 75% of Cluster 3 (green triangles) were elite departments. The red cluster of institutions with low gender diversity and low funding includes a mix of elite and non-elite institutions. Every institution identified in the gender diverse blue cluster was also ascribed to the lower 75% of the Wapman-prestige ranking list. The vast majority of institutions in the high funding green cluster were elite (in the top quartile by Wapman-prestige ranking). This cluster identified 68% of upper quartile (elite) institutions and only 7% of the lower three quartile (non-elite) institutions.

In this section we shall discuss the results presented above that demonstrate inequalities exist in faculty composition with respect to gender and the distribution of federal funding to mathematical sciences PhD-granting institutions in the United States.

The data shows that almost all PhD-granting institutions have Mathematics departments which are composed of faculty that are disproportionately male. In fact, not a single Mathematics department represented in this dataset was majority women. We found that the underrepresentation of women is more pronounced among elite Mathematics departments (recall that we defined “elite” departments as those in the upper quartile of departments in the prestige ranking generated by Wapman et al.).

We believe in the fundamental principle that mathematical talent is distributed equally among all groups of people who do mathematics. In the context of this paper, we therefore assume an equal distribution of mathematical talent among men and women. Under that assumption, the results presented here highlight the extreme level of underrepresentation of women in PhD-granting mathematical sciences departments.

Our analysis of NSF DMS funding identifies inequality in the amount of financial support for mathematics PhD-granting departments depending on Wapman-prestige. Pareto models, also popularized as the “20/80” economic model, predict that approximately 80 percent of assets are held, gained or earned by only 20 percent of the population being studied [30]. We found that in the elite institutions, the top 25% in our dataset by Wapman-prestige ranking, garnered 65% of the total funds given to the subset of PhD-granting institutions with a Wapman-prestige ranking. When we examine all NSF DMS funding, the top 20% of awardees receive 86% of all funds, with a Gini coefficient of 0.8. This result demonstrates a larger inequality than the classic “20/80” proportion.

The procedures and policies that NSF uses to determine which institutions receive funding may actually reinforce inequity in the mathematical sciences. NSF uses a process called merit review, where anonymous reviewers are asked to read, rate, and review funding proposals submitted to the agency. They are instructed to assess the intellectual merit and broader impacts of all proposals under five elements [16]. Two of these elements in particular may bolster inequity in the mathematical sciences. First, reviewers are asked “How well qualified is the individual, team, or organization?” This question is likely to skew reviewers towards considering institutional reputations since the information provided about qualifications typically includes individuals’ institutional affiliation(s). Second, reviewers are asked “Are there adequate resources available to the PI (either at the home organization or through collaborations) to carry out the proposed activities?” This second question is likely to skew reviewers to more positively rate proposals from well-resourced institutions, which contributes to a rich-get-richer phenomenon that could explain the extremely inequitable, “Pareto-like” distribution we found in our analysis of NSF DMS funding data.

Even more troubling are the systemic effects that are propagated over time by this inequality in the distribution of NSF DMS funding. The cliché “the rich-get-richer” is a colloquial distillation of how systems that disproportionately allocate resources then have an easier time justifying disproportionate funding. For example, well-resourced universities have significant funds internally for pilot projects. They have teams of grant departments that assist in the writing and administration of grants, funded by high indirect cost rates. They also have research support teams devoted to data gathering and processing as well as communications teams devoted to disseminating the results. We also note that expectations in obtaining grant funds vary widely between departments and institutions, and are often higher at the elite institutions (those in the top quartile using Wapman-prestige) in our dataset. This likely affects the number of proposals that are submitted by different institutions. In short, the effect we are seeing in NSF DMS funding is likely a result of a complicated collection of processes which reinforce and exacerbate the status quo that is a demonstrably unequal system, despite often being presented as a meritocracy.

To summarize, we have analyzed data on faculty in mathematical sciences PhD-granting departments and NSF DMS funding to institutions with these departments. We found that higher Wapman-prestige is associated with a lower percentage of women in Mathematics departments (RQ1). We also found, perhaps unsurprisingly, that NSF DMS funding was associated with higher Wapman-prestige (RQ2). Finally, we found that NSF DMS funding to departments was associated with a lower percentage of women (RQ3).

In this paper, we have shown that being an elite institution is an indicator of a disproportionately high allocation of financial resources from the federal government and a low proportion of women; this documents the existence of systemic inequality, i.e inequity, in the mathematical sciences community. We advocate for the redefinition of “prestige” (separate from Wapman-prestige) in mathematics in a way that better reflects equity for excellence [15]. For example, one could define institutions that are more prestigious to be institutions that “reflect the diversity of the US population” [15] among their faculty and doctoral graduates in mathematics. Using this re-envisioned idea of prestige, we present the ten most prestigious PhD-granting Mathematics departments based on representation of women in Table 4. Here, we intentionally choose to turn away from Wapman-prestige, but provide an alternative definition of prestige based on gender representation. One could also define “prestige” in the contribution that one makes not just to mathematics or academia, but in service to society. This is a cultural shift, but one that could have significant advantages to producing a vision for science and mathematics that improves all lives.

There are many other directions in which the research presented here could be extended. It is important to study the questions addressed in this paper with respect to other dimensions of diversity, particularly marginalized social identities including race/ethnicity, sexual orientation, national origin, and disability status, among others. This future work should be done in a way that allows analysis using intersections of multiple identity characteristics.

The addition of geographic location to the analysis of the gender diversity of PhD-granting institutions as well as of the distribution of federal funding is a possible direction of future research.

Another future direction is to expand this work to a wider range of institutions and faculty appointments. A study encompassing all types of institutions, and particularly community colleges, minority-serving institutions, and primarily undergraduate institutions, is necessary. Additionally, future work should investigate related questions about all types of faculty employed at these institutions, especially the increasing percentage of non-tenure track faculty.

This work’s primarily goal has been to document inequity in the mathematical sciences. Future work could involve developing mathematical models that are informed by the data we have that documents inequity in the mathematical sciences and lead to a better understanding of the mechanisms involved. Specifically, the data [2, 19, 17] showing the underrepresentation of women in new hires versus their underrepresentation in tenure-stream positions in PhD-granting institution raises some interesting questions that could also be addressed in future research.

We conclude by inviting interested researchers to join us in the ongoing MetaMath project to use mathematics and data science to analyze our discipline in order to promote social justice and enhance equity within all fields of the mathematical sciences.