William Yslas Vélez
Emeritus Professor, University of Arizona
This article presents four main points.
- Promote the undergraduate mathematics major not only as entry into the workforce but also to prepare students for graduate programs in STEM fields.
- Recognize the difference in the educational systems between the U.S. and other countries and the impact this has on graduate enrollment in the mathematical sciences.
- Suggest that the AMS take a leadership role in leading the mathematical community towards a comprehensive look into the evaluation of graduate school applications.
- Point out the lack of diversity in the graduate programs of our top research departments.
Just take one more math course
“Just take one more math course” was the lead sentence in the article [1] announcing that the Math Center at the University of Arizona (UA) had received the 2011 AMS Award for an Exemplary Program or Achievement in a Mathematics Department. The Math Center forms part of the department of mathematics and is dedicated to organizing the mathematics major program by providing advising, information about internships, research experiences and scholarships, and supporting recruitment efforts. The Center also records demographic information (citizenship, gender, ethnicity) in order to assess diversity efforts. I had taken over the directorship of the Math Center in 2003 with the goals of doubling the number of mathematics majors and increasing diversity. Rather than emphasizing outreach to increase the number of mathematics majors, I focused on reaching into enrollments in our mathematics classes, an activity I labeled as INREACH [2]. I sent emails to students enrolled in mathematics classes encouraging them to pursue further mathematical studies. At the UA, there were 30,000 undergraduate students enrolled, many of them in STEM fields. I advanced the view that the more mathematics a student took, the more opportunities would be available. I promoted the math major shamelessly. I learned early on not to suggest that a student change their major, but rather that they add mathematics as another major. This proved to be a successful strategy. When I retired in 2018, there were over 650 mathematics majors, 20% of them from minority backgrounds. Half of the graduating mathematics majors had another major or another degree. There were an equal number of mathematics minors with the percentage of minority students greater than 20%.
Most of the mathematics majors were not headed to graduate schools in the mathematical sciences, rather they were using their mathematics to make them more competitive in the job market or in applying for post-graduate studies. The mathematics major, combined with stellar grades in the students’ other fields of study, helped students gain admission into top graduate programs in STEM fields. This suggests that if mathematics departments encouraged undergraduates to increase the mathematical content of their curriculum and add the mathematics major or minor, this would increase the participation of U.S. students in STEM fields. As the data below shows, U.S. students do not constitute the majority of students in many STEM graduate programs. What should this percentage be? Certainly having graduate programs that would be 100% U.S. students would be unacceptable, as would 0%. A percentage of less than 50% is problematic. A reasonable percentage for U.S students would be between 70%-80%. After all, graduate programs in the U.S are part of the U.S. educational system. Graduate education is the economic engine in increasing the earning potential of its citizens.
Data on U.S. participation in STEM fields
Dependence on foreign oil, in the past, caused great concern in the U.S. because it endangered our national security. Yet, dependence on international students to power our graduate programs and our scientific enterprise appears to be acceptable. Exactly how dependent is the U.S. on international talent? In 2016-17 (the latest data available) international graduate students represent the majority (over 50%) in many STEM graduate programs, as Table 1 [3] shows.
Table 1 is an excerpt for Table 22 of [3] |
|||
Doctorate recipients, by subfield of study and citizenship status: 2017 |
|||
Area of doctorate |
total number of doctorates |
Temporary visa holders |
% temporary visa holders |
Computer engineering |
419 |
294 |
70% |
Structural engineering |
103 |
67 |
65% |
Electrical, electronics, and communications engr |
1900 |
1230 |
65% |
Industrial and manufacturing engineering |
253 |
161 |
64% |
Agricultural economics |
144 |
91 |
63% |
Computer and information sciences, general |
123 |
75 |
61% |
Civil engineering |
741 |
444 |
60% |
Computer science |
1587 |
931 |
59% |
Computer and information sciences |
1987 |
1101 |
55% |
Mechanical engineering |
1409 |
750 |
53% |
Econometrics, economics |
1184 |
626 |
53% |
Finance |
188 |
99 |
53% |
For the mathematical sciences profession, we can obtain data from [4] and it shows that 51% of doctoral degrees are awarded to international students. I strongly support the age-old view of “everything in moderation” as a good guideline for our graduate programs. We have long since passed the point of moderation.
Differences in the educational systems of the U.S. and other countries
In most other countries undergraduate students arrive at a university already having been accepted into a major. Some mathematics majors take analysis in their first year of study and three or four mathematics courses per semester as undergraduate students. Here is a link to the program of study at the University of Guanajuato in Guanajuato, Mexico (http://www.demat.ugto.mx/index.php/estudiantes2/consejos-para-nuevos-estudiantes-2/licenciatura-en-matematicas). International students have often completed academic training comparable to students with Master’s degrees in the U.S..
A liberal arts education is a requirement for most U.S. students. It is not unusual for U.S. students to declare a mathematics major in the second or third year of undergraduate study and take 1-2 mathematics courses per semester. The liberal arts education gives students the opportunity to explore other areas of interest before deciding on a major. Students interested in pursuing a graduate program in the mathematical sciences may not take analysis until their last year of study. In the meantime they have developed the mathematical maturity to understand this abstraction.
Is there any evidence that this later arrival at mathematical maturity correlates with creativity, or determination or any of the other factors that impact students’ abilities to become successful mathematicians?
Rethinking the admissions process for graduate school
Graduate programs must consider two criteria when accepting students into a graduate program. Can students pass entry-level courses and the requisite examinations, and are they creative enough to write a strong dissertation? Departments want the “best” graduate students. However, contrary to mathematical culture, “best” is not defined.
Perhaps “best” means what is best for the nation? Historically, minorities lack representation in the mathematical enterprise. By attracting students from under-served populations, mathematics departments could play a vital role in addressing STEM diversity. Certainly, increasing diversity is an important goal, one of increasing importance in a country undergoing such dramatic demographic changes. Mathematics departments could follow the lead of the Math Alliance [5] (https://www.mathalliance.org/), an organization whose stated purpose is to increase the number of minority students applying to graduate programs in the mathematical sciences.
Could “best” mean what is best for the department? International students have completed coursework equivalent to a U.S. Master’s degree and have demonstrated that they understand entry-level courses. This preparation enables them to move quickly on to their research interests and this demands less teaching from the faculty.
Perhaps “best” means selecting students with the best mathematical creativity? Mathematics departments choose prospective graduate students with the highest grades and GRE math subject scores. Instead of selecting students with mathematical creativity, admissions committees choose students who excel in test-taking and mathematical knowledge. In this process, mathematics departments fail to recognize the differences between domestic and international educational systems when comparing students’ GRE scores.
When I became director of the mathematics graduate program at the UA I was given no instruction. This is so typical of the mathematics profession. In graduate school we are prepared to become researchers. When we accept a faculty position we are then expected to become effective teachers, mentors, evaluators and administrators. The AMS should bring the community together to investigate new methods of evaluating selection criteria for graduate school and to develop training for faculty in these methods. Departments have relied on standardized testing platforms, like GRE, for decades. Its effectiveness in predicting success in graduate school could be one of the projects initiated.
Sadly, there is evidence of even further bias against U.S. students being accepted into STEM graduate programs. In a 2017 article, Mervis [6] stated that university administrators were concerned that Trump’s administrative policies would result in a 30% drop in the number of applications from international students, compared with the number of applicants from 2016. The article went on to say that a smaller applicant pool allows “administrators the option of admitting students who previously would not have made the cut, including more domestic students. But educators are loath to move the bar if it would lower the quality of the talent pool.” I find it quite surprising that university administrators are so willing to disparage U.S. students in public.
The U.S. educational system sets forth the requirements needed for a student to earn an undergraduate degree from the university. Presumably this preparation has prepared students to continue to post-graduate study. Students from outside of the U.S. must complete more advanced mathematics courses in order to earn this same degree and are therefore at a distinct advantage when applying to U.S. graduate schools. What is the rationale for changing the bar other than the fact that so many international students are applying? The bar should be set at a level commensurate with the requirements set forth by our educational system. This higher bar places U.S. students in an inferior position as far as the graduate admission process is concerned.
No taxation without representation
Universities are supported by U.S. tax dollars, both state and federal. Part of that support comes from the minority population. What is the minority population getting out of this investment [7]? For the academic year 2016-17, 1957 Ph.D.s were awarded by 299 departments. How many minorities were part of this group of doctoral recipients? Among the US citizens earning doctoral degrees, 4 were American Indian or Alaska Native, 30 were Black or African American, 33 were Hispanic or Latino, and 4 were Native Hawaiian or Other Pacific Islander [4]. This is a total of 71 minority doctoral recipients! For decades, billions of dollars have been spent on mathematics graduate programs with a negligible return for the minority community. If mathematics departments are not going to include minorities in their graduate programs, then don’t take their tax dollars!
The situation for minorities is even worse at the top research universities. The Doctoral Math Private Large grouping in the AMS survey [8] consists of 24 departments with the highest annual graduating rate of Ph.D.s (23 of the 24 departments responded to the survery). This grouping contains some of the best research departments in the country yet the percentage of female Ph.D.s is the lowest of all the groupings. Moreover, this grouping reported a total of 2 minority Ph.D.s! for the academic year 2016-2017. How is this even remotely acceptable! Given that NSF has an explicit goal of increasing diversity, why are these departments supported by NSF grants?
Where are the top research departments going to find minority faculty if they are not producing any minority mathematicians? Our faculty do not represent the population of the U.S. and this is a problem that needs attention. By reconsidering the admissions process for graduate schools, top research departments could begin to build pathways for faculty positions.
Everything in moderation went by the wayside in our graduate programs
In [4] we see that almost all of the percentages of U.S. citizen doctorates are around 50% in the mathematics groupings. I am surprised that percentages of U.S. students are not higher in biostatistics. The mathematical requirements for applying to graduate programs in biostatistics are three semesters of calculus, linear algebra and probability theory. (Harvard Biostatistics: https://www.hsph.harvard.edu/admissions/degree-programs/#research) and U of Washington Biostatistics: https://www.biostat.washington.edu/program/admissions/requirements). Biostatistics programs need to devote some energy towards recruitment of U.S. students.
In looking at graduate programs at different universities, I came across the mission statement from The Center of Mathematical Sciences and Applications (CMSA) at Harvard:
The Center for Mathematical Sciences and Applications will establish applied mathematics at Harvard as a first-class, interdisciplinary field of study, relating mathematics with many other important fields. The director of CMSA Yau states, “The center will not only carry out the most innovative research but also train young researchers from all over the world, especially those from China”. How is Harvard finding the funds to support an effort specifically aimed at training international postdocs? The U.S. minority community does not have such funds to encourage Harvard to create such a program for U.S minorities. The purpose of the Harvard program is to train international postdocs who would then out-compete U.S postdocs in the marketplace. Is this what is best for our nation?
Curiously, I gave a presentation at JMM in Denver in January 2020 pointing to the CMSA website. Curiously, this mission statement at CMSA disappeared soon thereafter.
The role of mathematics in producing U.S. STEM graduate students
Undergraduates with a strong background in mathematics are competitive for STEM graduate programs. The U.S would benefit if mathematics departments increased the number of mathematics majors. However, it is important to realize that the goal of an undergraduate degree in mathematics is not necessarily to pursue a graduate program in the mathematical sciences. We need to communicate to students that the mathematics major at the undergraduate level can lead to a wide variety of employment opportunities. For those interested in pursuing STEM graduate studies the mathematics major will strengthen their applications.
In my own work at the University of Arizona, the outstanding graduating senior in department X was often also a mathematics major or minor. This occurred year after year and the administration took notice. Mathematics majors were an integral part of STEM education at the UA. This model is worth replicating elsewhere. Mathematics does not just reside in a mathematics department. It needs to be incorporated into STEM and by increasing the number of mathematics majors across campus, we provide a well-prepared workforce for the country and a larger pool of eligible students for graduate schools.
Increasing diversity requires extra work and dedication. The UA model shows how a commitment and an infusion of resources can help faculty accomplish this goal and carry out this meaningful work. The importance of these efforts cannot be over-emphasized. The changing demographics of the U.S. requires a concerted effort to re-examine the application process of our graduate programs and respect the education of our undergraduates.
References
1. Arizona’s Math Center Wins AMS Award, Allyn Jackson, Notices of the AMS, Volume 58, Number 5, 2015, pages 718-721. http://www.ams.org/notices/201105/rtx110500718p.pdf
2. Inreach is the new outreach, William Yslas Vélez, MAA Focus, Volume 35, Number 4, August/September, 2015, pages 4-5. https://www.maa.org/sites/default/files/pdf/MAAFocus/Focus_AugustSeptember_2015.pdf
3. Data Tables, National Science Foundation, National Center for Science and Engineering Statistics (NCSES) Doctorate Recipients from U.S. Universities: 2017, Table 22. https://ncses.nsf.gov/pubs/nsf19301/data
4. Report on the 2016–2017 New Doctorate Recipients Amanda L. Golbeck, Thomas H. Barr, and Colleen A. Rose, Notices of the AMS, Volume 66, Number 7, August 2019, pages 1151-1160.
5. 2017 Award for Mathematics Programs That Make a Difference, Allyn Jackson, Notices of the AMS, Volume 64, Number 5, pages 476-478. .https://www.ams.org/journals/notices/201705/rnoti-p476.pdf
6. Drop in foreign applicants worries engineering schools, J. Mervis, Science, 17 February 2017, p. 676.
7. Broken Social Contract, Letter to the Editor, William Yslas Vélez, Notices of the AMS, September, 2020.
8. Departmental Groupings, The Mathematical and Statistical Sciences Annual AMS Survey. http://www.ams.org/profession/data/annual-survey/groups
Author notes. The author thanks Helen Grundman for thoughtful comments on this article and for suggesting a less combative tone and to the reviewers for many helpful suggestions. This article was originally submitted to the Notices of the AMS, but after several revisions, it was rejected. The referees were very helpful and accepting their comments improved this article. Regarding the section “Everything in moderation went by the wayside in our graduate programs”, one referee commented, “This entire section needs to be removed from the paper. It does not add value to the paper and it is not at the professional level of the rest of the paper or the Notices in general.” On the streets of this nation, people are demonstrating against injustices, but apparently, the Notices will not allow minority voices to complain about the injustices suffered. In the end, I could not accept the revisions suggested.