Recent advances in science and engineering—combined with the prolific increase of available data and growth of data science as a field—have produced ample opportunities for early-career mathematicians. Undergraduates with mathematics degrees are well-qualified for well-paying positions at contemporary, fast-moving companies like Google, Apple, or Facebook. However, the math programs at most universities focus on 19th and 20th-century content. “What’s required for most math degrees doesn’t have a single concept that’s less than 100 years old, in terms of core classes,” Jeffrey Humphreys of Brigham Young University (BYU) said. “Many mathematicians are embarrassingly disconnected from other STEM disciples and the needs of industry. There’s a disconnect between what’s being used in the real world and what’s being taught in math departments.”
This raises an apropos question: how can institutions and their faculty make a degree in mathematics more applicable to the 21st century? During an invited presentation at the 2018 SIAM Annual Meeting, which took place last week in Portland, Ore., Humphreys spoke about BYU’s five-year-old undergraduate Applied and Computational Mathematics Emphasis (ACME) program. The major arms students with practical, relevant skills that make them both employable and valuable in a wide variety of settings.
“When you think about what your degree does for students, an interesting exercise might be to write a job description for which they would be well-qualified,” Humphreys said. He then proceeded to cheekily list the following “ideal” qualifications for those with a B.S. in math.
- Able to prove basic theorems from Moore method topology without getting shot down too often
- Able to identify when a ring is a Euclidean domain
- Able to solve integrals involving partial fraction decomposition and trigonometric substitution
- The ability to compute volumes with the shell method is a really big plus!
- Good row reduction skills; can solve three-by-three linear systems
- Able to solve power-series solutions of ordinary differential equations
- Knowledge of LaTeX and Maple a big plus.
While these skills are undoubtedly important, they are unfortunately not the type of qualifications that real-world employers seek. “They’re interested in people who can solve problems, work with teams, and make effective presentations,” Humphreys said. “It’s easier to teach a mathematician how to program than a computer scientist how to think.”
Jeffrey Humphreys of Brigham Young University (BYU) speaks about BYU's unique Applied and Computational Mathematics Emphasis (ACME) program, which prepares students for competitive industry jobs.
Humphreys acknowledged the challenge of structuring an effective math program around topics or theory that are “hot” or “in” right now, as those things can change. He also admitted that sustaining new advances is difficult, because innovations often die out due to burnout, administrative reassignment, or retirement of their facilitators. Furthermore, systemic changes typically require “off the shelf” solutions. “People probably won’t have the time to jump into something that’s in a constant state of revision,” Humphreys said.
BYU’s ACME program is structured from a ground-up perspective, and teaches both theory and the practical skills in mathematics, statistics, and computation necessary to tackle modern problems. “We wanted fresh, 21st-century content to make students competitive for 40 years,” Humphreys said. Participating students choose a specialty from a growing list of available concentrations, meant to attract those with broad interests and prepare them for graduate school — even if they are interested in a discipline other than mathematics. Economics, computer science, and statistics continue to be the most popular choices. The program culminates with a capstone experience.
ACME employs a lockstep curriculum for efficiency and a cohort model to breed teamwork and socialization. “This way we know what students have already done, and we reuse material they’re seen before,” Humphreys said. This leads to accelerated learning and helps with internship and career placement. 40 percent of BYU students who land an interview with Google are offered a position; for an extremely competitive employer, that is a high statistic.
When designing the program, Humphreys and his colleagues planned for tightly-integrated, rigorous math content with a focus on analysis and an emphasis on algorithmic thinking. They created the following four-course sequence complete with labs, which totals 32 credit hours spaced evenly over junior and senior year:
- Mathematical analysis
- Algorithms, approximation, and optimization
- Modeling with uncertainty and data
- Modeling with dynamics and control.
The labs offer students experience with cutting-edge computational training, data wrangling, and big data problems that pull broadly from applications and require algorithmic thinking. “An algorithm is a natural extension of a function,” Humphreys said. “Math analysis is about proving theorems about algorithms. When we’re proving a theory about an integral, we’re proving a theorem about the liming behavior of an algorithm.” First-year lab topics include Markov chains, facial recognition, PageRank, “Six Degrees of Kevin Bacon,” and the so-called “balloon pop problem.” Second-year students complete 12 data science labs and 16 machine learning labs.
ACME supplements these intensive labs with seminars focused on soft-skills training, including career essentials, data visualization, and competitive coding. Many projects emphasize teamwork. “We assign so much homework that very few students are able to do the assignments on their own,” Humphreys said. “This forces them to work together in a Darwinian sense.”
He then described some of the mathematical applications with which the students work. For example, they use the continuous linear extension theorem to define integration. “It’s a very slick way to derive the integral, and it doesn’t take nearly as much time,” Humphreys said. Tight integration is relevant to a myriad of situations, including inner product spaces, optimization, economics, operations research, engineering design, and almost any problem in machine learning. Students also have to earn the right to use any necessary packages; there are no black boxes, and they must code the core algorithms of the packages themselves.
Mathematical and statistical modeling feature heavily in the ACME curriculum. Humphreys emphasized the difference between models (the hypothesis) and modeling (a process much like the scientific method). “Your hypothesis is a mathematical relationship,” he said. He also spoke of the importance of understanding the different types of models: first-principle white box models, data-driven black box models, and the gray box models that combine features of both.
Students begin pursing the program track their junior year and continue through their senior year. “We’re not able to start it any earlier than that because students have to figure out what they want to do,” Humphreys said. He and his colleagues begin advertising the program freshman year and actively seek out qualified STEM majors; interested students must have taken a certain number of engineering and mathematics classes. Humphreys works hard to help place students as interns, and visits companies like Google, Amazon, Nike, Microsoft, and Boeing all over the country. He also uses his connections to help students negotiate salaries. The highest starting salary of a mathematics graduate of BYU was $150,000 for employment with Apple. And the median salary for a data scientist clocks in just under $117,000.
77 students are enrolled in ACME for the fall of 2018, a significant jump from the 32 enrolled the previous year. Humphreys concluded his presentation by emphasizing the value of a lockstep approach and cohort model, which helps students stick together and results in high retention, effective team-building, and a strong alumni base. Ultimately, Humphreys hopes that the program will continue to grow in popularity and equip young mathematicians with the skills, knowledge, and experience necessary to flourish in an ever-growing competitive and global workforce.
||Lina Sorg is the associate editor of SIAM News.