The prestige and status of research fields within mathematics
TThe prestige and status of research fields within mathematics
Jean-Marc Schlenker ∗ Department of mathematics, University of Luxembourg ∗ To whom correspondence should be addressed; E-mail: [email protected] v1, September 1, 2020
Abstract
While the “hierarchy of science” has been widely analysed, there is no corresponding study ofthe status of subfields within a given scientific field. We use bibliometric data to show that sub-fields of mathematics have a different “standing” within the mathematics community. Highly rankeddepartments tend to specialize in some subfields more than in others, and the same subfields arealso over-represented in the most selective mathematics journals or among recipients of top prizes.Moreover this status of subfields evolves markedly over the period of observation (1984–2016), withsome subfields gaining and others losing in standing. The status of subfields is related to differentpublishing habits, but some of those differences are opposite to those observed when considering thehierarchy of scientific fields.We examine possible explanations for the “status” of different subfields. Some natural explana-tions – availability of funding, importance of applications – do not appear to function, suggestingthat factors internal to the discipline are at work. We propose a different type of explanation, basedon a notion of “focus” of a subfield, that might or might not be specific to mathematics.
Contents a r X i v : . [ c s . D L ] A ug Journals 15 focus of a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.3 Departmental and institutional strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A List of journals used 25
Certain fields of science benefit from a higher degree of prestige and status than others. This “status” offields or subfields can vary in time and between country, but it has deep consequences on the advancementof science, for instance: • Through the allocation of talented researchers, since talented students tend to disproportely choosesome fields over others. • Through its impact on institutional policy, and the choice by universities or research institutionsto invest in some fields more than others, depending on their profile and objectives. • Through its impact on science policy, and the allocation by governement or research funding agenciesof priorities in research.The prestige of a field should be distinguished from how fashionable it is. Some fields sometimesundergo rapid progress and promise to have a huge impact in the short or medium term (a recent exampleis for instance found in deep learning) and therefore attract both researchers and funding. Other fieldscan attract talented students because they are considered as stepping stones towards successful careers.We will see that, in mathematics, a clear distinction should be made between fields which are consideredas “high status” and those offering the best career perspective – in fact, a case could be made that thereis a negative relation between the two notions, see Section 8.1.We investigate whether and why some directions of research appear to be valued by top departmentsor top journals more than others, using quantitative and bibliometric tools and methods. It is difficultto compare different fields, since they tend to have different publishing practices, venues, etc. It is easier,at least when using bibliometric tools, to consider differences between subfields of a given field. Here wefocus on mathematics, and on the dynamics of its different subfields.The analysis of status between subfields of mathematics should be compared to the wider debateand ample bibliography on a “hierarchy of sciences” going back at least to Auguste Comte, see e.g.[6, 5]. The main indicator of a higher “status” of a field is generally considered to be the consensus evel , which is also related to the level of complexity of the object of study. The reader can find in [8]an analysis of the relations between the hierarchy/level of consensus of a field and different types ofbibliometric characteristics, such as the average number of authors, number of pages, or number of citedreferences. For instance, it appears that at the level of fields, shorter papers or less old references shouldbe related to higher consensus level and “status”, which might contrast with the situation within subfieldsof mathematics (see below).Among scientific fields, mathematics is generally considered as having a high level of consensus. In aslightly different direction of analysis, proposed by Hargens [11], see also [2], mathematics is characterizedby a high level of normative integration – the sharing of common beliefs and values – but a low level of functional integration – the activity of mathematicians does not depend directly much on those of hercolleagues. Mathematicians often have strong preferences for some fields over others, and preferences vary widely.However they also often express the idea that some fields of mathematics are “more difficult”, “morecentral”, or “more important” than others, and some convergence seems to appear in those opinions.While the assessment of each subfield varies from one author to the other, it is interesting to ask whetherthere is a “general pattern” in the way different mathematicians gauge the “difficulty”, “centrality” or“importance” of a each field, and to explain it. One might expect that the “importance” attributedto subfields is directly related to their relevance to applications in science and technology – we will seehowever that this does not seem to be the case.This observation leads to several questions. • Can a “hierarchy” of mathematical fields be observed, and how? • Do different observations provide similar rankings of fields? • Is this ranking stable over time? If not, how does it evolve? • How can it be explained?One way to approach those questions is through existing and established hierarchies: • of mathematics journals (as measured here by their MCQ, a kind of IF adapted to mathematics), • of mathematics departments.We can assume that if some fields were more valued than others, the “best” journals and the “best”departments would focus more on those fields than on others, while less prestigious departments andjournals would have to do with less prestigious fields. Those observations, confirmed by a set of biblio-metric observations, will serve as the basis for the analysis conducted here.Some of the key findings presented below are:1. There is a clear “ranking” of subfields of mathematics, with some more prestigious fields having amuch larger relative share of the mathematics departments of the most prestigious universities, ofthe papers published in the most respected journals (Section 3.2), of the top prizes (Section 3.1).2. Different “measures” of the rank of fields within mathematics tend to give very similar results,compare Section 3, 4 and 5. 3. The subfields with the highest apparent “status” are typically the most abstract ones and those withlittle direct applications, such as Algebraic Geometry, while applied fields like numerical modellingor statistics have low status – this in spite of efforts by some funding agencies to promote preciselythose low-status fields.4. The “status” of subfields can certainly not be explained by commonly used notions of impact, suchas citation count, since the most prestigious fields tend to be those with lowest citations numbers,and conversely.5. The prestige of some fields has changed quite dramatically over the period of observation consideredhere (1984–2016) in one direction or in the other. For instance the prestige of Differential geometryor of Analysis seems to have declined considerably, while those of Probability or Partial DifferentialEquations has increased markedly.6. The status of a subfield is related to differences in publishing habits: higher status is rather stronglyrelated to less authors/article, and to longer articles.7. The recruitment strategy of departments varies greatly with their “status”, measured here by theirranking according to an indicator of production. Top departments tend to favor disproportionatelythe “noble” subfields, while low-ranking departments hire more experts of subfields with a lowerstatus, see Section 4.3.8. There is a significant mobility of mathematicians between subfields. This mobility is higher in highlyranked universities, and tends to follow the evolution of the “status” of fields – mathematicians atbetter departments tend to move out of subfields of decreasing status, and into subfields of improvingstatus, see Section 4.4.9. The scientific productivity of authors is different between subfields. In some high-status fields ittends to be highly concentrated on a small number of highly productive authors, while in otherfields it decreases more slowly, see Section 7.2. This fact might explain why the optimal strategyof different departments might to focus on some fields more than others, see Section 8.3.Along the way, we consider in Section 7.1 what indicator of scientific productivity should be used forauthors. We use data on grants from the European Research Council to calibrate different indicatorsof production based on the MCQ of journals (see Section 2 below) and come to the conclusion thatindicators that best fit with the opinions of panels of mathematicians tend to give a much higher weightto a small number of journals with a high MCQ.Section 8 is dedicated to possible explanations of the “value” attributed to different fields. We considerseveral characteristics of subfields of mathematics that could be related to how “prestigious” an area ofresearch is. We then concentrate on the focus of a subfield: to what extent researchers in this subfieldtend share an interest for a small number of questions (or conjectures) which therefore become important.The notion of focus considered here is related to, but distinct from, the notion of level of consensus ina field. It is closer to the notion of normative integration considered by Hargens [11], but with a twist:what is shared is not only beliefs and values, but more specifically the interest in a small set of questionsor conjectures which are considered as particularly important.We will see that this notion can explain at least to some extend why some subfields of mathematicsseem to enjoy a higher degree of prestige than others. Section 8.2 shows how a keyword analysis – theuse of the word “conjecture” in the title of articles or their Math Reviews entry – correlates well withother measures of “status” identified here. 4 .3 Why study mathematics?
Mathematics is a suitable field of study for the questions considered here for a number of reasons. It is a“international” scientific field, present in one form or another in almost all universities (since mathematicsteaching is always present). It has well-defined fields and subfields, with relatively clear boundaries. Thenumber of authors in each article is usually relatively low, allowing for better identification of author’sroles. Another interesting feature is that the level of funding is probably less important for research inmathematics than in experimental fields, so that decision on topics to study might be less influenced bythe availability of funding.Mathematics as a field has other significant feature that make it an interesting object of study. • It is fully international, in the sense that while some nations tend to specialize more in somesubfields, there is no main difference either in the main questions being considered or in the methodsapplied by mathematicians across the world. This differs from some fields (e.g. economics) wherecultural and political factors can play a major role. • Mathematics is a large field, measured by the number of active mathematicians. For instance, thenumber of mathematicians with a faculty position in US post-secondary institutions was estimatedin 2017 as 25,632 (see [10]), to be compared to a total of 822,513 across all disciplines [12, Table315.20] (so that mathematicians represents close to 3% of faculty in the US). By comparison, thenumber of physicists with a full-time faculty position in US post-secondary institution was estimatedat 10100, so 2.5 times less than mathematicians. The size comparison between fields might varyfrom one country to the other, but one can expect that mathematics remains one of the largeracademic fields. • Mathematics is also an ancient field of study, with continuous development in the last centuries.As a consequence of this development, mathematics has branched into a variety of subfields, eachwith its own problems, methods and traditions. • Mathematics is also well determined as a field, with a clear definition: mathematicians insist on proving the statements that they consider as results. This distinguishes them quite clearly fromscientists from other scientists, who might be interested in the same questions and the same objectsbut with different methods. This clear line can be seen for instance in mathematical physics, with onone side mathematicians intending to prove results, on the other physicists who do not need a formalproof once they have reached a high degree of confidence in the truth of a statement. Similarly,computational engineers might be satisfied by an program providing numerical approximationsof solutions of a partial differential equations that are close to observed solutions, while appliedmathematicians tend to search for a proof that the numerical solutions are close to the real ones.Mathematics is also a convenient object of study thanks to the carefully curated and very completebibliographic information that is available, in particular through
Mathematical Reviews , a database pro-duced by the
American Mathematical Society . It has a number of very interesting features that don’thave analogs in most other fields, for instance: • It provides a clear identification of each institution and even each department within a given instition(each department is assigned a code, composed of a first block describing the country and a secondblock the institution). • It assigns to each individual author a number, so that even authors with the same first and lastname can be clearly distinguished. A similar picture is obtained through the U.S. Bureau of Labor Statistics data on the somewhat larger group of“Postsecondary teachers”, which includes professors at post-secondary institutions. In 2018 they estimated the total numberat 1,350,700, including 51,250 in mathematics, 13,780 in physics, and 13,270 in economics. See Each article is given a primary and one or several secondary
Mathematics Subject Classification (M.S.C.) code, so that one can easily determine to what extend a given article is related to acertain subfield of mathematics.We believe that a hierarchy of subfields might also be observed also in other fields of science. Onecan for instance find a hint in the description in [3] of the attitude of high-energy physicists towardscondensed-matter physics.
We use data collected selectively from
Math Reviews , a journal published by the
American MathematicalSociety containing synopsis of most articles published in the area of mathematics as well as in somerelated fields. The data available through
Math Reviews has a number of important qualities that makeit particularly useful for bibliometric studies on mathematics. For instance, it attributes a unique codeto each author – even when two authors have the same first and last name. It also attributes to eacharticles a series of precises disciplinary classification codes. Finally the
Math Reviews database attributesstable codes to each institution, so that affiliations of authors can be easily followed.Within the
Math Reviews database, we selected a list of approximatively 140 journals, which can beconsidered to be leading journals in pure and applied mathematics. The choice of the journals was madein two steps: • A first list of journals was selected for a previous study [7]. At the time the journals selectedwere those with the highest impact factor among those having a cited half-life above a threshold,according to the
Journal Citation Report • More recently, this list was completed by adding the journals ranked as A ∗ in the journal list ofthe Australian Math. Society for mathematical sciences, one of the very few journal rankings inmathematics produced by consulting with experts (rather than using only bibliometric data). Thislist can be considered as a reasonable proxy for a “consensus” among mathematicians of whichjournals are most selective and important.For each journal, we used the full list of articles published between 1984 and 2016, recording for eacharticle the most relevant data only (journal name, publication year, number of pages, number of authors,MSC (Mathematics Subject Classification) codes, and for each author, the MR code of the author andhis/her affiliation). The total number of articles in the database that we used is 247 677. We assumethat although this list of papers represents only a small fraction of the whole mathematics literaturepublished during the period of study, it does contain most of the articles considered as really significantby mathematicians.The list of journals considered here can be considered as somewhat arbitrary – while most of thosejournals should be in any study intending to study the most relevant papers in mathematics, some ofthe less central journals could be replaced by others with a similar standing. Moreover this collection ofjournals gives a certain weight for each subfield, and different choices of journals would lead to differentweighting of the subfields. However this weight of each field is not so important for the considerationsmade here, since our analysis of the status of subfields is based on the differences in weights of subfieldsin different departments (resp. journals, etc) rather than on the weight themselves. See .2 A database of authors and PhDs The data from
Mathematical Reviews was then merged with data from
Mathematical Genealogy , a re-markably complete and apparently quite accurate freely accessible database of mathematics PhD thesis.It appears that most “active” mathematicians appearing in our data from
Mathematical Reviews (thosewith at least 2 papers) also appear in
Mathematical Genealogy . This second source contains additionalinformation which is used in some of the results presented below, including the date of the defense andthe PhD-granting institution for many PhD theses in mathematics.Most of the data presented below are followed over the time frame of the study, from 1984 to 2016.To avoid too much random noise, we consider 8 periods each of 4 years (except the first one, 1984-1988,which is 5 years long).
For the purpose of this article, we defined a small group of subfields of mathematics, each correspondingto a small set of 2-digit MSC codes: • Algebra , corresponding to codes 06, 08, 20, 18, 15, 16, 17, • AlgGeom (for Algebraic Geometry) for codes 11, 12, 13, 14, • DiffGeom (for Differential Geometry) for codes 51, 52, 53, 32, 58, • Topology for codes 19, 54, 55, 57, 22, • Analysis for codes 26, 28, 30, 41, 42, 43, 46, 47, 33, 34, 39, 40, • PDE (for Partial Differential Equations) for codes 31, 35, 44, 45, 49, • DynSys (for Dynamical Systems) for code 37 (appearing only 2000), • Physics (for mathematical physics) for codes 70, 74, 76, 78, 80, 81, 82, 83, 85, 86, • Numerics for code 65, • Probability for code 60, • Statistics for code 62, • Other for codes 00, 01, 04, 97, 03, 05, 68, 73, 90, 91, 92, 93, 94.Note that all subfields do not have the same weight, and in fact the weight of the different fields, measuredin terms of total number of articles in the database, varies quite a bit over time, see Table 1. Note alsothat DynSys (Dynamical Systems) only appears in the period 1997-2000, since it did not have a specific2-digit code before the 2000 revision of MSC.
Table 1: Proportion of papers in different subfields7 .4 The MCQ as proxy for the “impact” of journals
Part of the analysis of the status of subfields of mathematics uses, or is based on, a stratification of thejournals appearing in our database, in terms of their prestige or level of selectivity. The choice of theproper indicator, to be used as a proxy for the prestige of a journal, requests some care, especially in thearea of mathematics.The use of any type of bibliometric indicators has been the object of heated debates within the mathe-matics community. Its use for individual evaluations is often considered quite negatively. Moreover, someof the bibliometric indicators widely used, such as the Journal Impact Factor, are sometimes consideredas crude and not well adapted to mathematics [1, 9]. In addition to general methodological objections,one main limitation of the use of IF for mathematics is that it is based on couting citations in a 2-yearwindow which is too short for mathematics, where the cite half-life of articles tends to be much higher.This short window creates biases between subfields of mathematics depending on their citation habits.Perhaps for this reason,
Math. Reviews also provides a measure of impact of journals, the
MathematicsCitation Quotient (MCQ). It measures the mean number of citations to articles published by each journalover a 5-year period, in a selected list of journals. The longer citation window, and the selection ofcitation sources, make it better suited than other impact measures for mathematics. This MCQ appearsmuch better suited to mathematics than the IF. Mathematicians tend to have a very precise idea ofthe ranking of journals, and anecdotical evidence (coming from e.g. evaluation panels or recruitmentcommittees) indicates that different mathematicians usually have relatively similar assessment of theprestige of different journals, even when they specialize in different subfields. It appears – although moreresearch on this topic would be welcome – that this subjective “prestige” of journals is much bettercaptured by the MCQ than by the IF.Here we use the MCQ 2016 as an (imperfect) proxy for the “prestige” of journals. This view issupported by a analysis presented in Section 7.1, where it is shown that weighing each paper by a certainpower of the MCQ provides an indicator that works well to identify mathematicians obtaining the highlycompetitive grants from the European Research Council.
The first indications of the “value” attributed to different fields can be found by following two sources ofprestige or status that are widely accepted in the mathematics community. • From the main specialties of recipients of the Fields medal, the best known and most revered prizein mathematics. • From the share of different fields in papers published by the “top 5” mathematics journals.
A first indication of the “status” of subfields can be obtained by observing the specialties of the math-ematicians who received the top prizes in mathematics. This is done in Table 1 of the
Supplementarymaterial for the Fields medal, by far the best known and most prestigious prize in mathematics.A number of observations follow from the list of subfields of Fields medalists. • Algebraic and Differential Geometry, as well as Topology, tend to dominate. • Probability, Dynamical systems, pPartial differential equations appear in the 1990s. • Statistics or Numerical mathematics do not appear at all (yet).8igure 1: Share of main fields in top 3 journalsThere are other well-known prizes in mathematics, such as the Wolf prize, or more recently theBreakthrough Prize , the Abel prize or the Chern Medal . It appears that similar conclusions wouldfollow from considering the recipient of those prizes. Of course a different picture would emerge from specialized prizes focusing only on one or several subfields, but none of those specialized awards has yetreached the same reputation as the Fields medal. Other indications can be obtained from the primary and secondary MSC classifications of the paperspublished in the “top 3” journals in mathematics. There is a relatively large consensus among mathemati-cians on what the most selective journals are, with a standard list:
Annals of Mathematics , InventionesMathematicae , and
Journal of the American Mathematical Society . Those journals are for instance thoseconsidered as “Top” journals in mathematics by the ARWU ranking and therefore used for their rankingof universities in the area of mathematics. Some mathematicians would tend to add a very small numberof journals to this list, in particular Acta mathematica and
Publications Math´ematiques de l’IHES , butthose two journals are publishing a small number of articles (typically less than 10 per year each) andadding them to the list would not change the results of any bibliometric study much. The share of articleswith primary MSC codes in the main subfields is shown in Figure 1Those three journals are highly selective, and typically publish less than 200 articles/year together ,for instance they published 139 articles in 2019. (This contrasts strongly with other scientific fields, suchas Physics, where one top journal such as
Physical Review Letters publishes around 4000 articles/year,while
Nature published around 4000 items in 2019.) For this reason, publishing in one of the top 3journals is seen as a significant achievements among mathematicians and can have a real impact on amathematician’s career.The main conclusions are similar to those from the main areas of Fields medalists, although in aquantitatively more precise manner. Algebraic and Differential geometry dominate, and their weight ismuch larger than in the database of articles as a whole, as seen in Table 1. But some fields – in particularDifferential Geometry an Topology – see a clear decrease in their share, while others – Dynamical Systemsand PDEs – have an increasing share of the number of papers. The list of recipients of the Wolf prize can be found at https://en.wikipedia.org/wiki/Wolf_Prize_in_Mathematics . See https://breakthroughprize.org/Prize/3 . See https://en.wikipedia.org/wiki/Abel_Prize . See https://en.wikipedia.org/wiki/Chern_Medal . See Departments
This section is focused on the disciplinary profiles of different groups of universities. We base this analysison the well-accepted idea that departments tend to have well-established and relatively stable levels of“status”, see [4] for a more refined analysis of department hierarchies in computer science, business andhistory.We consider different groups of universities, and different ways of asserting their disciplinary special-ization. • By their scientific production (articles published). • By the main fields of publication of their active mathematicians. • By their recruitments, that is, the disciplinary focus of mathematicians moving to those depart-ments.To obtain a synthetic view of the scientific focus of different universities, we defined several groups ofuniversities, aiming towards some homogeneity within each group. • Ivy-league type universities: Columbia, Cornell, Dartmouth, Harvard, Princeton, U. of Pennsylva-nia, Yale. • Technology Institutes: Caltech, GeorgiaTech, ETH, EPFL, MIT, Stanford. • Among the remaining universities we determined 4 groups depending on their total scientific pro-duction over the whole period: those ranked 1–10, 11–30, 31–100 and 101–300.Note that more refined analysis are possible, for instance differentiating US and non-US institutions, orby country, etc. We have not included such analysis here for lack of space.The motivation for defining those groups of universities might be clear to the reader. • Ivy-league universities are generally considered to be among the top universities in the US. Theyalso share a certain number of characteristics, for instance they are all (by definition) relativelyold and well-established institutions. They are not chosen here as the “best” universities, but as arather homogeneous group of universities which are presumably aiming at academic excellence. • The small group of Technology Institutes considered here are also considered as “excellent” insti-tutions, but with a somewhat different outlook then the Ivy-league universities. The student ofthose institutions tend to be mostly oriented towards engineering, and one can assume that theirresearch activity is oriented at least in parts towards innovation and real-life applications, ratherthan academic research. This would lead to a specialization in subfields of mathematics somewhatdifferent from other, more academically-oriented universities. • The other universities (in fact those publishing a sufficient number of papers, since we limit ourstudy to the top 300 institutions) are broken down into four groups of decreasing total productionaccording to an indicator that puts a relatively heavy weight on article in the top journals: eachpaper is weighted by its number of pages times the square of the MCQ of the journal. This weightingensures that the institutions with a better rank tend to be those producing more papers in highlyselective journals (see Section 2.4). 10igure 2: Share of some subfields in different groups of universities
The weight of different subfields varies widely depending on the type of department. Some subfields havea much larger role in “elite” departments, while others tend to be much more present in less ambitiousinstitutions. Figure 2 shows the evolution over time of the share of publication (weighted by number ofpages and MCQ of the journal) in a limited group of subfields, depending on the type of department –the full data, for all subfields, can be found in Figure 1 of the Supplementary material.Algebraic geometry provides an prime example of a high-status field: in spite of a somewhat decreasingshare, it remains dominant in Ivy league departments (approximately 17% of papers) while it plays a muchmore reduced role in less prestigious departments (approximately 8% in departments ranked 101–300).Its share is much more limited in Technology Institutes, which is to be expected since its applications toengineering are relatively limited.Differential Geometry offers a similar “elite” profile at the beginning of the observation period, butthe most striking phenomenon is that its share in Ivy league universities, and in fact in all groups ofuniversities except ITs, decreases very markedly. The opposite can be observed for Probability theory,which is almost absent from Ivy league universities and ITs at the beginning of the observation period,but becomes important by the end.On the opposite end of the spectrum, Analysis and Numerics appear to be low-status fields at theend of the period, with a very limited role in Ivy league departments and a much larger role in the lessprestigious institutions (those ranked 101–300). Partial differential equations (PDEs) shows a similarprofile, with a notable increase in status by the end of the observation period.It is quite apparent in the graphs that the Technological Institute have, as could be expected, asomewhat different focus than the other universities. They have less focus on Algebraic geometry andmore on Numerics, as could be expected from institutions with a strong focus on engineering programsand students, but the share of Algebraic geoemtry is still larger in those institutions than in lower-rankeduniversities. However they do have a strong focus on Differential geometry, which cannot really beconsidered as an applied field. 11he evolution over time of the status and share of some subfields is quite striking. For instanceAnalysis already appeared to be a low-status area in the 1980s, with a larger role in department with alower ranking. By the 2010s, it had almost disappeared from the Ivy-league departments, while retaininga significant place in the departments ranked 101–300. On the opposite, Partial Differential Equationswas almost absent from Ivy league institution in the 1980s, but had acquired a significant position thereby the 2010s.
Another way to look at the specialization of departments is through the specialties of their researchers.One obstacle in this direction is that articles appearing in our database are not necessarily writtenby members of mathematics departments. This is particularly true in some applied fields like statis-tics or numerical modelling, where authors might not be affiliated to a department of Mathematics (orMathematics and statistics, or even of Statistics).We intend here to measure the number of mathematicians, in the sense of scientists who devotea significant portion of their activity to mathematics research over a long period of time, and thereforeprefer excluding PhD students who leave academia after their thesis, or scientists of other fields producingonly occasionally a paper that is related to mathematics and included in the data we use. For this reason,we only consider authors who have over their lifetime produced a total of 100 pages in the journals weconsider, each page being weighted by the MCQ of the journal in which it is published and divided bythe number of authors (reflecting the relative contribution of the author).For each period, each mathematician is attributed a primary subfield, determined as the most com-mon field of their paper (weighted by MCQ and number of pages). Note that the primary field ofmathematicians can change from one period to another.Figure 3 shows the relative weights of a selection of subfields in different types of universites, overtime. Here a relative weight of 1.2 for Algebra in ITs, for instance, would mean that there are 120mathematicians with main focus on Algebra in ITs when the expected number (knowing the total numberof mathematicians in ITs and the proportion of those specializing in Algebra overall) is 100.Figure 3 shows the resulting data for the same subfields as in Figure 2. Again, the full data for allsubfields is available in the Supplementary material.The results are somewhat different from that presented in Section 4.1 since different subfields tend topublish more papers than others, on average (see below). Comparing numbers of active mathematiciansgives a better view of the focus put on different subfields by departments. The results are also quite clear,for instance Algebraic geometry appears again as a high-status fields, with a strong over-representation inIvy league departments and under-representation in departments ranked 101-300. Differential geometryis also a “high-status” field, but with a clear decrease in its position at the end of the period, whileProbability went from low status to high status over the period. Analysis, Numerics and PDEs appear aslow-status fields, with a strong under-representation in top departments and a strong over-representationin 100+ departments.
The graphs shown in the previous two sections show that the weight of some subfields in some departments— for instance Differential geometry, or Probability theory — varies very quickly. In this section andthe next we analyze the mechanism for those rapid variations, first by considering in what subfieldsrecruitments are made, and in the next section to what extend mathematicians change from one subfieldto the other. One key results is that both departments and individual mathematicians tend to be quitedynamic, but that the departments with a higher “status” (and their members) move more quickly.To extend the analysis and obtain a more dynamics view, one can look at the proportion of differentfields among mathematicians recruited in different groups of universities, see Figure 4.12igure 3: Relative weights of mathematicians in selected fields depending on groups of universitiesFigure 4: Share of recruitments in different fields for different groups of universities13igure 5:
Relative share of recruitments in different fields for different groups of universitiesFigure 4 shows the proportion or recruitments (defined as arrivals of new authors in an institution,at least 2 years after the PhD). It displays interesting differences between the main fields where differentgroups of institutions choose their new recruits. Top departments (Ivy league and top 10 “other” de-partments) continue to give a strong preference to Algebraic Geometry, and a decreasing place is givento Differential Geometry. Partial Differential Equations and Probability see their role increasing in basi-cally all groups of universities, while Analysis and Algebra remain important in lower-ranked institution(ranked 301 and below).Figure 5 is somewhat simpler to interpret as it shows the relative shares of different fields in therecruitments of different groups of institutions . For each group, only the 5 subfields with the largestrelative share of recruitments at the end of the period appear. There is again a very clear distinctionbetween top institutions (Ivy league, ITs, institutions ranked 1–10 in our list) where “high-status” fieldssuch as Topology and Algebraic Geometry dominate, with a relative weight close or larger than 2, togetherwith “new” highly regarded fields such as Probability, while Differential Geometry keeps a significant butslightly decreasing place. On the opposite, in less regarded institutions (ranked below 301 and even morebelow 901) the recruitments disproportionately favor Numerics, Analysis or and PDEs. It can be noted that the recruitments made in different fields are not sufficient to explain the importantvariation in the weight of different fields seen for instance in Figure 2. Another possible explanation canbe found in differing attrition rates : at all stages of their careers, a certain proportion of mathematicianstend to stop publishing, see [7], and this rate could vary from one field to another.Another important element, however, is the mobility of mathematicians between fields. Figure 6shows the mobility in and out of some “core” fields of mathematics (excluding Statistics, Physics andOther, where an important part of publications is not from members of mathematics departments). Morecomplete data can be found in tables in the Supplementary material. Here we record only the number Note that the subfields Statistics, Physics and Others were not considered here, again because they appear to reflect inan important manner recruitments made in non-mathematics departments.
14f mathematicians who changed their main field of publication from one period to the next (withoutconsidering incomers who did not publish before, or outgoers who stop publishing), where the “mainfield” is determined as the most common field defined by primary classification of papers published in agiven period. Some remarks that can be made are: • The in and out mobility is quite large for most fields and most periods – this can be explained bythe fact that a significant proportion of mathematicians are at the interface between two fields, andcan lean towards one side or the other from one period to the next. • The level of mobility tends however to decrease with decreasing rank of universities – mathemati-cians at “better” universities tend to change their topics more than those at “lower” institutions. • There are very significant imbalances between incoming and outgoing flow in some periods for somefields, for instance out of Differential Geometry in the periods 1997–2000 and 2001–2004 (out flowin red) and towards Probability in 2009–2012 (in flow in orange).Figure 6 hints at the role of individual strategies of researchers, who tend to leave some subfieldsof decreasing status and move towards those of increasing prestige. It is conceivable that publishing inthe most selective journals is perceived as easier in some subfields than in others. However, moving toa new field can be very costly in terms of time and energy, especially when moving to subfields wherean important background is needed, which tends to be the case of most high-status subfields (such asAlgebraic geometry or Topology).However the transfers of authors from one subfield to another, as seen in Figure 6, might not besufficient to explain the variations in the number of publications in some subfields in some groups of uni-versities. The priority given to some subfields in recruitments can also play a role, see Section 4.3. Otherexplanations can be found in the attrition rate (the proportion of mathematicians who stop publishing),which also appears to differ markedly between subfields and over time. We do not include an analysis ofthis phenomenon here since it would need to be relatively complex and various factors need to be takeninto consideration (e.g. the different numbers of PhD students in different subfields, relations to industry,risk of “missing” some publications in some fields more than others, etc).
As mentioned in the introduction, another way to look at the “status” of subfields of mathematics is toconsider whether top-ranked journals tend to publish more often papers in some subfields than in others.To perform this analysis, we use the MCQ provided by
Math Reviews as a proxy for journal “quality”(specifically, we use the 2016 value of the MCQ). One should note however that this analysis is somewhatcomplicated by the difference in publishing and in citing behaviors between subfields of mathematics.Typically, papers in some subfields (for instance Partial differential equations or Statistics) tend to becited faster, and therefore more often in the 5-year timeframe of the MCQ, than in others (for instanceAlgebraic Geometry). As a consequence, journals which are specialized in some fields or just tend topublish more papers in papers in certain subfields can have a higher MCQ than would be estimated fromtheir “quality” as perceived by mathematicians.To avoid this bias, we provide two figures: Figure 7 shows the weight of different subfields in groupsof journals depending on their MCQ, while Figure 8 provide the same data but excluding journals witha high degree of specialization. (The degree of specialization used here is computed as the sum of thesquares of the relative weight of each subfield among papers published by the journal, where the relativeweight in a given field is 1 if the proportion of papers in that field is the same as in the whole database).Although the two figures are quite comparable, some specific artifacts tend to disappear in Figure 8.15igure 6: Transfers between subfields. Top: Ivy, IT and top 10. Bottom: 101–300.16igure 7: Share of some fields according to journal MCQThose figures quite precisely support the idea of a “ranking” of subfields as already seen in Section 3.2and in Section 4, as well as the dynamics of this ranking. Algebraic Geometry is heavily over-representedin journals with a high MCQ, while Differential Geometry is over-represented, but with a declining sharein top journals. On the other hand, Analysis for instance is under-represented in top journals, with adeclining share, while the “status” of Probability is clearly increasing.The figures presented here only concern the same six subfields as above, more complete data is availablein the Supplementary materials.The general picture is again that Algebraic Geometry appears as a high-status field, with a muchhigher weight in high-MCQ journals, while Differential Geometry has a high but decreasing status – itsweight in high-MCQ journal decreases markedly, while its weight in lower-impact journals decreases moreslowly – and the status of Probability theory increases. On the lower side of the pictures we see thatAnalysis, Numerics and PDEs tend to be of lower status, with a clear increase for PDEs. Removinghighly specialized journals changes the picture to some (limited) extend, for instance it appears that therelatively high weight of Numerics in journals with MCQ between 1 and 2 is largely due to specializedjournals.
The “subfields” introduced above can be analysed further by considering separately the main 2-digit MSCclassification codes associated to each article by
Mathematical Reviews . This more detailed analysis isinteresting for instance for the field
DiffGeom considered above, which is defined by merging 5 different2-digit MSC codes: 32 (Several Complex Variables And Analytic Spaces), 51 (Geometry), 52 (ConvexAnd Discrete Geometry), 53 (Differential Geometry) and 58 (Global Analysis, Analysis On Manifolds).Figure 9 shows a finer analysis of the share over time of those 2-digit MSC codes in high-impactjournals. It shows quite cleary that the decline of Differential Geometry is due entirely to two 2-digitcodes: 32 (Several Complex Variables And Analytic Spaces) and 58 (Global Analysis, Analysis OnManifolds). A similar detailed analysis could be performed for other subfields, to get a better view andunderstanding of the evolving share of different subfields.17igure 8: Share of some fields according to journal MCQ, excluding specialized journalsFigure 9: Share of 2-digit MSC codes in
Differential Geometry in top 5 journals and in journals withMCQ between 2 and 4 18igure 10: Mean number of pages/article depending on the field
We have considered so far data that can reasonably be related to the “status” of a subfield, such as itsrelative weight in different departments or in different journals. In this section, we consider differences inpublishing habits between different fields, first in terms of lengths of papers, second in terms of numberof co-authors. It is quite striking that in both cases there appears to be a strong relation between the“status” of a subfield as seen above and the publishing habits of its authors. The results presented herecan be compared to, and contrasted with, those in [2, 8] concerning comparisons between different fieldsof science.In the next section we will see how those differences in publishing habits can hint at explanations ofthe differences in “status” between subfields.
The first parameter that we can consider is simply the average length of papers, see Figure 10. Tworemarks should be quite clear from the graphs.1. There is a general increase in the average length of papers, as already documented for mathematicsin [7].2. There is a remarkably direct relation between the “status” of subfields as seen in Section 4 and inSection 5, on one hand, and the average lengths of papers. Papers tend to be significantly longerin “high-status” fields like Algebraic geometry, Differential Geometry – where the length of paperscompared to Algebraic Geometry decreases over time – or Topology, and shorter in “low-status”fields such as Numerics or Analysis.It should be noted that the second point is in strong contrast with the inverse relation betweenconsensus level of a field and the average length of papers, observed when comparing different fields ofscience, see e.g. [8]. This difference could be explained by the nature of papers in mathematics, whichmight play a different role than in other fields of science. While in many fields a paper reports on theresults of research, for instance of a series of experiments, a paper in mathematics is expected to containa full proof of its main results – the paper is the research. Longer papers could therefore correlate witha higher level of complexity in the proofs. The second parameter is the mean number of co-authors of papers, as see in Figure 11. Here too, there isa very significant rise in the mean number of co-authors over time (as also documented for mathematics19igure 11: Mean number of authors/article depending on the fieldin [7]. Moreover, the higher the “status” of a field, the less co-authors a papers has on average: at theend of the period of study, papers had on average less than 2 co-authors in Algebraic Geometry, Algebra,Differential Geometry and Topology, but more than 2,5 in Numerics, Others, Physics and Statistics.Note that the interpretation of those data might require some care. The number of co-authors tendsto be higher in more applied areas, and one possible source of difference might be differences in habitstowards co-authorship, for instance it is more common in some subfield than in others for a PhD advisorto co-author papers with her/his PhD students.
It should come as no surprise, given the varying weight of different subfields as seen in Section 4 and inSection 5, that the “productivity” of mathematicians in different fields vary, with the amount of variationdepending on the indicator of production which is considered.
Before comparing scientific production between subfields of mathematics, it is necessary to define a properindicator of “quality” of scientific production. Clearly many indicators can be imagined: one could forinstance count the number of pages published, or weight this number in many different ways. Onetherefore needs a measure of how “reasonable” an indicator is.For the purpose of this study, the most relevant indicators are those which best fit with the assessmentsof experts in the field of mathematics. To assess the quality of possible indicators, we use a large set ofprojects positively evaluated by experts: the projects funded by the
European Research Council (ERC)in the area of mathematics (panel PE1). This is a total of 389 projects, in three categories (“Starting”,“Consolidator” and “Advanced” grants) corresponding to different age groups, always with only oneprincipal investigator. Since the main data on the funded projects is freely available (including theidentity of the PI), it is possible to estimate to what extend a given bibliometric indicator correlates wellwith the obtention of ERC grants.We considered what type of bibliometric indicator to use in this light. Specifically, we tried to estimateto what extend different indicators based on the number of pages, co-authors and MCQ of the journal. Weparticularly focused on the best way to take into account the MCQ of journals. For different indicators,we computed how many ERC grantees would be within the top 50, 100, 200, 500, 1000 and 5000 authorsin our database (over all years). The results are shown in Table 2 using as indicator the sum, for eachauthor, over all the papers in our database of the number of pages times a power of the MCQ 2016 of thejournal where it was published: from power 0 (first column) to power 3 (rightmost column). As can beseen, the best results are obtained, depending on the line, by a power 2 or 2 . MCQ MCQ MCQ . MCQ top 50 5 7 11 10 10top 100 13 16 16 16 15top 200 29 34 33 36 32top 500 54 62 73 69 63top 1000 93 109 110 113 113top 5000 241 257 267 266 261Table 2: Number of ERC grantees among top producers according to different indicatorsFigure 12: Mean production (pages times MCQ shared between authors) of different groups of authorsdepending on the field • Taking into account the number of pages leads to clearly better results than just counting thenumber of papers. • Taking into account the number of co-authors (for instance by dividing the weight attributed of apaper by the number of co-authors) leads to worse results.In the following section we use one of the indicators that gives the best fit – each paper is weightedby the product of the number of pages by the square of the MCQ of the journal – to give an estimate ofthe “production” of mathematicians in different fields.
Using the indicator above, it is possible to measure the “productivity” of mathematicians in differentsubfields. The results are presented in Figure 12. For each subfield, the graphs present the average“production” of the mathematicians in a group defined through the “ranking” of all authors by productionin a given period, for instance the first two graphs are for the top 50 authors in each field, in each period,the graphs number 3 and 4 for the authors ranked between rank 51 and 200, etc. In each case we splitthe subfields in two groups for clarity. 21o avoid biases related to specialized journals having a relatively high MCQ due to faster citation ratesin some fields, the data here was limited to journals having a specialization index (as defined in Section5.1) at most 70. This excludes a relatively small number of journals with a higher level of specialization.The data is quite reminiscent to the results presented above concerning the relative weights of differentfields in top 5 journals or with journals with higher MCQ (as could be expected) but also with recruitmentsin higher status departments. The mean production is highest in fields that were identified as “high-status”, in particular Algebraic and Differential geometry, lower in statistics.Two relatively striking phenomena however can be noted in the second part of the time frame consid-ered here. • Authors in Partial differential equations have a high production – as high as those in “high-status”fields for highly productive authors (top 50 of each field) and much larger among less productiveauthors (more than twice higher, in the last period, for authors ranked 501–2000). • Some authors in Numerics also have a relatively high “productivity”, as high in the last period, inthe group of most productive authors, as for Algebra or Analysis.Those phenomena should be considered with some care, since they could to some extend be explained bydifferences in the publication and citation practices in different fields.
The data presented above leads to a simple question: what can explain the preference of top depart-ments/journals for certain fields? We can propose some possible answers. • Some fields are more useful or important for applications than others.
This does not seem to bethe correct explanation since top departments tend to have preferences which are opposite to thisorientation. Funding agencies tend to prefer those application-oriented fields, which should providedepartments another strong reason to develop them, but that’s not what’s happening. • Maximizing external funding.
For the same reason, this explanation does not seem to resist scruti-nity, since funding agency tend to prefer more applied fields, which also tend to have lower “status”as determined above. • The “impact” of a field , as measured by standard indicators such as the number of citations that apaper can be expected to attract. Here again the data seems to clearly invalidate this explanation,since the “high-status” fields, such as Algebraic Geometry, are typically those where the impactof papers (at least in the 2 years after publication considered by the Impact factor) is the lowest ,while fields with much higher citation impact, such as Statistics of Numerics, appear as having amuch lower “status”. • Departments tend to maximize another type of impact, closer to the indicator used in the previousSection 7.1, and to the assessment that other mathematicians have of the relevance and importanceof results. This explanation seems broadly validated by comparing the results of Sections 4.2 and4.3 and Section 7.2, but it is somewhat circular and leads to another very close question: why dothe results in some field appear more relevant or important to mathematicians than in others? As an example, in 2011, EPSERC decided to fund PhD scholarships only in the areas of statistics and applied probability,see http://blogs.nature.com/news/2011/09/uk_mathematicians_protest_fell.html . A simple experiment confirming this view can be performed by checking the number of citations of say the 10th mostcited author in google scholar with a given “label”. On June 21, 2020, this yields for instance 10 156 citations for the 10thmost cited author with the label “algebraic geometry”, vs 166 049 citations for the 10th most cited author with the label“statistics”. Consensus level.
One can be tempted to extend to mathematics a main explanation of the “hierarchyof science”, namely, by a difference in level of consensus between different subfields. This explanationhowever runs into an obvious difficulty: the consensus level tends to be extremely and uniformlyhigh across mathematics, since all published papers are expected to contain full and complete proofsand therefore to only present results that are unquestionably true. • Focus.
This is the main explanation which we would like to put forward. Preferred fields tend tocorrespond to those where there is a strong and shared focus on well-identified problems. This canbe reflected in some bibliometric data: – Longer papers, because more efforts and technical developments are often necessary to makeprogress on a problem which is well identified and on which other experts already tried tomake progress. – Less co-authors/paper, for a similar reason – depth and technical difficulties tend to increasethe cost/benefit ratio of collaboration (too much time is spent explaining new ideas to collab-orators).We present in the next section some (limited) data supporting this assumption, showing thatsubfields where the word conjecture is used more often tend to be higher-status. • Relation to teaching.
Another possible hypothesis is that some subfields are more relevant to teach-ing, in the sense that their researchers tend to be more involved in or dedicated to the teachingactivities of their departments. We don’t have any data here to support or contradict this hypoth-esis. • Some fields offer better job perspectives.
This is another simple explanation that does not appearto resist scrutinity. In fact, a quick search by keywords on the available positions on mathjobs.org ,which is perhaps the main source of job advertisements in mathematics (whether academic or non-academic) seems to indicate that more jobs are available in lower-status subfields such as Statisticsor Numerical analysis than in “high-status” fields such as Algebraic geometry.In the rest of this section we discuss evidence supporting the idea that the level of focus of a subfieldof mathematics is related to its position in the “hierarchy of subfields”. focus of a field
The notion of focus of a field, as defined above, cannot be directly measured or even rigorously definedhere. A field should be considered as more focused if • it is structured around a limited number of important questions (or conjectures), • the active researchers in the field are aware of most of those questions, • they agree that progress on one of those questions would be highly valuable for the field.Algebraic geometry is an example of a strongly focused field, with a number of well-known conjecture(including for instance the Riemann Hypothesis, Fermat’s last theorem until it was proved in 1994, and anumber of others) shaping the field. There remain a number of old, well-known and more or less centralconjectures in the field. Differential geometry was probably more focused in the 1980s than it is now, since a number ofkey conjectures (the Calabi and Yamabe conjecture, the Geometrization conjecture, etc) were proved One can consult for instance the somewhat random list on https://en.wikipedia.org/wiki/List_of_conjectures tosee that Algebraic geometry, and in particular Number theory, appear prominently.
Math Reviews entrybetween the late 1970s and the early 2000s. On the opposite, One could consider that Probability theoryalso became much more focused on some key conjectures from the 1980s on, thanks in particular to newconnections to statistical physics ( e.g. the conformal invariance of the Ising model at critical temperature)or through internal motivations ( e.g. the self-intersection properties of the Brownian motion).Other fields, such as Statistics or Numerical modelling, appear much less focused on a small numberof important problems, and much more on finding new methods to solve problems that are important inapplications.The focus of a field is related directly to its place in the awards of Fields medals, as seen in Section3.1, since Fields medals tend to be awarded to individuals who have solved (or made a key progress) ina well-known problem. The same applies to a lower extend to publications in top 5 journals, as seen inSection 3.2, since the high level of selectivity of those journals means that papers that are accepted alsooften provide an important progress on a well-established problem.One indirect way to assess the focus of a subfield of mathematics is by measuring how often the word“conjecture” appears in the
Math Reviews entry of papers. Some results are shown in Figure 13. (Theresults are obtained by directly using the mathscinet web interface to
Math Reviews and counting thenumber of papers in a given period with a given primary MSC code, and then the number of those forwhich the word “conjecture” appears in the entry.)Although partial only, this data appears to confirm a relation between the focus of a field, as measuredin this manner, and the status of subfields as seen above. Algebraic geometry makes a considerable useof the word, which appears in the entries of more than 15% of articles. On the opposite, Statistics barelyuses it, while Probability theory is in an intermediate situation and the use of the word is increasingmarkedly over the period of study.
The data presented in Section 4.3 provides a glimpse into the recruitments choices of mathematics depart-ments. Can the “status” of subfields explain the thematic choices made by different types of departments,as seen in Section 4.3?Clearly, different types of departments have different needs. For instance, mathematics departmentsin Technology Institutes have a responsibility towards teaching mathematics to future engineers, andthis could explain a stronger activity in areas like Partial differential equations or in Probability (since astrong background in probability theory is necessary for curricula in data science or quantitative finance).This might explain some specificities in Figures 4 and 5.However the “productivity” data in Section 7.2 can also explain the differences in the recruitmentpolicy of different types of departments, if we assume (as an obviously simplistic model) that departmentsaim at maximizing their output, in the sense of the indicator used in Section 7.2. Indeed, under thishypothesis: 24
Top departments would tend to give priority to fields with high output for top researchers, andtherefore to “high-status” fields such as Algebraic geometry – as seems to be the case, see Figures4 and 5. • Lower-ranking departments might not be able to compete for the most productive researchers inthe “high-status” fields. Figure 12 shows that the output of authors drops quite dramaticallybetween say the group of authors ranked 1–50 (according to their output) and those ranked 200–500. As a consequence, a better strategy for less competitive departments might be to attracthighly productive researchers in a somewhat lower-status fields, such as Analysis or Algebra, or onnewly fashionable fields, such as Probability.Clearly, more research is needed into understanding how institutions or departments choose to orienttheir resources towards one field or research direction or another.
Acknowledgements
The author is grateful to Pierre-Michel Menger and Yann Renisio for many discussions, remarks andhelpful comments on this work and related results, and to Fr´ed´erique Sachwald useful remarks on apreliminary version of the text.
A List of journals used
Tables 3, 4 and 5 show the list of journals considered here, with, for each journal, its short code, thenumber of papers published, and the MCQ 2016 of the journal. References [1] Robert Adler, John Ewing, and Peter Taylor,
Citation statistics: a report from the internationalmathematical union (imu) in cooperation with the international council of industrial and appliedmathematics (iciam) and the institute of mathematical statistics (ims) , Statistical Science (2009),no. 1, 1–14.[2] John M Braxton and Lowell L Hargens, Variation among academic disciplines: Analytical frame-works and research , Higher education – New York – Agathon press incorporated (1996), 1–46.[3] Davide Castelvecchi and Barry Simon, The mathematician who helped to reshape physics , Nature (2020), no. 7819, 20–20.[4] Aaron Clauset, Samuel Arbesman, and Daniel B Larremore,
Systematic inequality and hierarchy infaculty hiring networks , Science advances (2015), no. 1, e1400005.[5] Stephen Cole, The hierarchy of the sciences? , American Journal of Sociology (1983), no. 1,111–139.[6] Auguste Comte, Cours de philosophie positive: La philosophie astronomique et la philosophie de laphysique , vol. 2, Bachelier, 1835.[7] Pierre Dubois, Jean-Charles Rochet, and Jean-Marc Schlenker,
Productivity and mobility in academicresearch: Evidence from mathematicians , Scientometrics (2014), no. 3, 1669–1701, Working PaperIDEI 606 and TSE 10-160, May 2010. Errors in the table, to be corrected, in particular MCQs added not taken into account. ournal name Code Table 3: List of journals considered[8] Daniele Fanelli and Wolfgang Gl¨anzel,
Bibliometric evidence for a hierarchy of the sciences , PLoSone (2013), no. 6.[9] Antonia Ferrer-Sapena, Enrique A S´anchez-P´erez, Fernanda Peset, Luis-Mill´an Gonz´alez, and RafaelAleixandre-Benavent, The impact factor as a measuring tool of the prestige of the journals in researchassessment in mathematics , Research evaluation (2016), no. 3, 306–314.[10] Amanda L Golbeck, Colleen A Rose, and Thomas H Barr, Fall 2017 departmental profile report ,Notices of the American Mathematical Society (2019), no. 10, 1721–1730.[11] Lowell L Hargens, Patterns of scientific research , Washington, DC: American Sociological Associa-tion (1975).[12] Thomas D Snyder, Cristobal De Brey, and Sally A Dillow,
Digest of education statistics 2017, nces2018-070. , National Center for Education Statistics (2019).26 ournal name Code
Table 4: List of journals considered, cont’d27 ournal name Code ournal name Code