Restoring Confidence in the Value of Mathematics by Teaching Undergraduates Math They Will Use

1. Lost Confidence

Mathematics is deeply beautiful and extremely useful. But even when we mathematicians succeed at helping people see the beauty in mathematics, many don’t believe it has value beyond its beauty. Everyone knows that somehow, deep inside some computer somewhere, math is doing something that somebody thinks is useful. But most people don’t really believe that math is useful for them. They lack confidence in the value of mathematics.

Students don’t hesitate to tell teachers whenever we cover a difficult section or fail to engage them sufficiently in the math we teach: “When am I ever going to use this?!?”

1.1. Even math majors

But at least college math majors know math is useful, right? Not really. About ten years ago we surveyed the math majors at my university to get a better sense of how to recruit more students into mathematics. We asked them: Why did you major in math? and Why aren’t others majoring in math? Their answer was “I love it, but I know I can’t get a job if I don’t want to teach. I thought I’d just do something I enjoy now and worry about a job later.”

Their answers should not have surprised me, but they did. I thought math majors knew there were lots of applications of math relevant to them and to their future employment. We had alumni with good jobs that used math, but the students didn’t know that—they were just in it for the beauty. Even math majors had no confidence in the value of mathematics.

1.2. The cost of no confidence

That lack of confidence in the value of mathematics is a serious problem for at least four reasons:

1.

It diverts away resources. Funding, faculty lines, and other resources go where administrators and other decision makers think they will create the most value. When we aren’t seen as adding value, we lose resources. This is bad for mathematics, bad for us, and bad for students. Conversely, my university has provided our department with additional faculty and other resources as we have built confidence in the value of mathematics and attracted more and happier students.

2.

It damages learning. Students are more willing to put in the necessary effort to learn if they believe they will get something out of it. They don’t all find math as beautiful as we teachers do, and if they don’t see it as useful, how long can we keep saying “trust me, you’re going to like this” before they quit?

3.

It drives away “world changers.” Many students come to college wanting to change the world. They will not study math if they see it as only beautiful but not useful. Mathematicians have powerful tools to help overcome poverty, cure disease, and make the world more fair and just. Driving away these students means not only that we have fewer good students, but that these goals will be harder to achieve because the people working on them won’t have the tools they need.

4.

It excludes many. This lack of confidence excludes many students from our major because only a privileged few can afford to spend their college years doing what they like, ignoring their future careers.

1.3. Wrong solution: just advertise

Of course math is useful, even for undergraduates needing careers. At the time of our survey our students were getting jobs—some using math. And we knew lots of mathematicians in rewarding math jobs at interesting companies. Maybe, we thought, we just needed to tell students about these jobs. We told them about traditional math jobs in engineering, actuarial sciences, national security, and finance. We also told them about newer math jobs at places like Google, Amazon, LinkedIn, and Pixar.

This sort of worked. The number of math majors started to grow. But it was the wrong thing to do because many were still skeptical of the value of mathematics for them, and, more importantly, it wasn’t entirely honest. They were right to be skeptical.

Yes, our alumni got some of those jobs. But not really because of the math we taught them. Maybe they got them partly because of the problem-solving and critical thinking skills we taught. But much more often they got them because of some computer science classes they took on the side. They never got them because they could perform complicated matrix operations by hand or because of their knowledge of unique factorization domains and quadratic reciprocity.

We polled our alumni around this time and they were using math in their jobs—just not the math they learned in their degree. They told us: “I wish my major had prepared me better for my job.”

1.4. Better solution: deliver

This suggests a better solution: Actually deliver on the promise of mathematical value to our students. By this I mean that we should teach math that students can actually use, and enable them to use it by giving them the necessary skills and tools to work on problems that they truly care about. When we deliver on the promise that math has value, then we can advertise the careers, and then they believe us because then it’s true.

Let me be clear here. I am not talking about some sort of job training program. A rich, transformative education in rigorous mathematics and critical thinking (often called a liberal education) is a wonderful thing, and we should not give that up. I am talking about integrating the mathematics that is actually used in the real world, whether in computing, biology, engineering, physics, economics, or data science, into that traditional rigorous mathematical liberal education.

What I hope to convince you of is that it is possible to deliver on the promise that math has value, to teach math that students can really use, and to enable them to use it without losing what we love and what is most important about a traditional liberal education in mathematics. Moreover, as explained below, this can be done without a lot of effort from or retraining of faculty whose primary background is in pure mathematics.

2. Applied Mathematics is Good Mathematics

Some of you, like me, were raised by true believers in the religion of Bourbaki, Hardy, and Halmos. They taught me to be proud of the uselessness and “purity” of my math and to believe, incorrectly, that only pure math was beautiful, interesting, challenging, or rigorous.

But after 20 years of working in pure math, I finally looked a little closer at applied math, and, to my surprise, I found it to be just as beautiful, interesting, challenging and rigorous as pure math.

The first time I saw how the fast Fourier transform could be used to rapidly find a highly accurate low-degree Chebyshev polynomial approximation of an arbitrary smooth function, I was in awe. It’s a glorious combination of beautiful ideas, and it’s fabulously useful to boot. If you haven’t seen it yet, go look it up—it’s fantastic. See Tre20, Chapter 3 or HJ20, Section 9.5 for details. Some other things I find beautiful in applied mathematics include Noether’s theorem on symmetries and conservation laws; Thompson sampling to optimize the tradeoff between exploration and exploitation in multi-armed bandit problems; and the Metropolis–Hastings algorithm for Markov chain Monte Carlo.

In addition to being beautiful and useful, applied mathematics can also be taught in a way that is at least as rigorous and challenging as the traditional curriculum, and it can be taught in a way that develops mathematical ability and critical thinking skills as well as or better than the traditional major. As evidence for this claim, let me tell you about the undergraduate major at Brigham Young University we call the Applied and Computational Mathematics Emphasis (ACME) ACM22.

3. ACME

We started ACME as a way to deliver on the promise of mathematical value to our students. ACME is the brainchild of Jeff Humpherys and is the result of a lot of work by a lot of people, including Jeff Humpherys, Emily Evans, Jared Whitehead, and me, along with scores of other collaborators and generous support from the National Science Foundation. Since we started ACME, our university has provided our department with additional faculty and other resources because we have more, happier students.

I am describing ACME here not to say that everyone should do exactly what we are doing—every school is different, and you’ll have to make your own way. But I hope this is a useful proof of concept and that it gives you some ideas of things you could try. And I hope that some of the resources that we have developed can be useful to you.

3.1. 21st-century mathematics

The first question we faced was What mathematics should we teach? Applied math has traditionally been focused on problems of matter and energy—so much so that in 1998 V. I. Arnol$'$d insisted that math is a part of physics Arn98. But most of our alumni and industry contacts told us that the math they use is less about matter and energy and more about information, data, and computation.

I’ll let you decide for yourself whether that’s part of physics or not, but it’s clear that the mathematics of information and data needs to be a big part of any modern curriculum in applied mathematics. That doesn’t mean we abandon traditional applied math, but rather expand its scope to focus primarily on data-driven methods, modeling, and algorithms. These are also the three main components of applied mathematics identified by Weinan E in E21.

3.2. Mathematics, not data science

I need to emphasize that the ACME program is not a degree in data science. It is a rigorous education in the theory and practice of applied and computational mathematics.

What we do in ACME is relevant to data science, but it is not just data science. About a quarter of our students go into data science careers or data science graduate programs. But our students also go into many other careers, and to graduate school in many other disciplines, including pure math, applied math, economics, finance, biology, physics, engineering, computer science, and statistics. And they flourish in those programs because they have a deep and rigorous understanding of mathematics. An alumnus now enrolled in a PhD program in biology wrote about how ACME prepared him for that experience: “I work with machine learning every day, and cookie cutter methods don’t necessarily work for the problems I’m trying to solve. I need to be able to read scientific and mathematical papers and really understand how all of the parts of modeling with machine learning fit together. Having some feel for the mathematical foundations of it all really gives me confidence to try things and fail and not be afraid that there is some mysterious mathematics that I don’t understand or wouldn’t be able to understand if I tried.” —Karl Ringger ‘21

Some students initially think they want more training in data science and and less education in mathematics, but over time they come to appreciate the power of a rigorous education in mathematics. I recently received an email from one of our alumni, currently doing a PhD in network science at Northeastern, about this: “I can’t emphasize enough how grateful I am that my background is based in math theory, rather than just knowing how to plug and play with NumPy and scikit-learn. Thanks for requiring us to learn so much at such a high level, being patient while we complained about it, and encouraging us the whole time.” —Cory Glover ‘19

4. Key Features of ACME

From my perspective, the key features of the ACME program are

1.

A challenging and rigorous curriculum in mathematics.

2.

Lockstep cohorts for the junior and senior years.

3.

Computer labs for all advanced theory classes.

4.

A student-chosen concentration in an area of application.

The first half of the program is the same as our traditional math major, covering basic mathematical and computer programming prerequisites. But in the junior and senior years students enroll in a lockstep cohort through a rigorous and challenging mathematical core consisting of two theory classes and two lab classes, two hours a day, five days a week, every semester for four semesters. Students also choose a concentration of four to five courses in an application area. I’ll discuss each of these in more depth below.

4.1. Challenging curriculum

The ACME curriculum is at least as intense and demanding as our traditional major. It is built on a mathematically rigorous foundational core with daily (five days per week) homework sets or labs. Details on the curriculum for each course can be found at https://acme.byu.edu/.

I know some schools have been thinking about adding a data science program or an applied math program that is less challenging than the traditional major. I strongly recommend against this. First, mathematical tools are powerful, and those who wield them need to understand very well how they work and when and why they don’t work. Moreover, there are many benefits of a challenging curriculum to both the students and the program.

Benefits of a challenging curriculum

The most obvious benefit is that students learn more, but a challenging curriculum also attracts students, motivates collaboration, develops students’ ability to learn, develops ability and confidence to solve hard problems, and brings better job opportunities for graduates.

Students want a challenge. Students are attracted by the challenging nature of the ACME program—they don’t want a weak, watered-down experience. Here are two typical examples from our anonymous student ratings feedback about this. “I chose ACME because it challenges me.” “The most engaging and exhausting mental challenge of my life. I Love It.”

It encourages collaboration. Another big benefit of the challenging curriculum is that it motivates students to learn to collaborate with classmates. We do take many active steps to foster collaboration, and our cohort system is a big part of that effort (see Section 4.2 below), but the difficult curriculum itself also helps encourage collaboration. I’ll let one of the students explain: “I never enjoyed working with other students before ACME, but now I prefer it because I realize that I learn material better when I help others to understand it, and faster when they help me. The high expectations served as a catalyst for good habits that I would not have tried in their absence. With more to be done than I could accomplish on my own, I embraced working with others. The rigor of ACME taught me how to learn and gave me the opportunity to respond compassionately towards my peers. Because the load was challenging we learned—together—the value of working through difficult circumstances and the joy of rising to meet lofty expectations.” —Kolton Baldwin ‘21

Students learn to learn. Another benefit of the challenging curriculum is that students learn to learn more rapidly and effectively. Students and alumni often talk about how the ACME program has made them better learners, able to quickly learn new ideas, algorithms, and techniques that their coworkers struggle with. The following quote from a recent graduate is typical: “Mathematics was always a weakness of mine, and I’m now a lot stronger with it. But most of all my ability to soak in mass amounts of new information is what has improved most. It all prepped me for being a quick, efficient learner. I’ve been set up for life and I’m excited to keep learning.” —Lee Woodside ‘22

Students learn to solve hard problems. The challenging curriculum also helps build student ability and confidence to solve hard problems. Here is a comment from a student currently in the program: “Because of ACME, I am no longer afraid of math—math is afraid of me. I’m very grateful for the way that the program has built me into a reliable problem solver.” —Sam Goldrup ‘23

And the following is from an ACME alumna currently working on her PhD at Rice, studying applications of deep learning in medical imaging. “Consistently being challenged by the fast pace of ACME gave me the confidence to apply my deep learning research to imaging physics—something I had no prior background in.”—McKell Woodland ‘18

Employers want ACME students. The strong skill set of our graduates means that once someone has hired one ACME graduate, they usually want to hire more of them. Here’s an excerpt from an email I recently received from an employer trying to recruit more ACME graduates: “There are many programs out there which claim to prepare students for data science careers only to send them into the job market woefully underprepared…. But ACME students, on the other hand, have passed our technical interviews with flying colors and have shown they have the ability to solve hard problems.”

And here is the experience of one alumnus: “Technical leads that have known me now search out for ACME students to hire as a first preference.” —Wesley Stevens, ‘18

Problems of a challenging curriculum

I don’t want to imply that the challenging curriculum is without its problems. One of the difficulties includes the risk of students’ becoming intimidated or developing impostor syndrome. But our lockstep cohort (see Section 4.2) helps to mitigate that, as does the use of objective preparedness measures. It’s more likely a student will feel unprepared if we say the prerequisite is “good knowledge of analysis” than if we say the prerequisite is “a B or better in Math 341.” ACME faculty and TAs also explicitly coach students about impostor syndrome, why it happens, and how to overcome it.

The opposite problem also occurs, with a few students developing a big ego and a destructive attitude, thinking they are better than students not in the program or students not doing as well in the program. Again, explicit coaching is very powerful, and many students find that although they may be good at one thing (e.g., mathematical proofs), they are not necessarily so good at other things (e.g., computer programming). Needing and getting help from their classmates on the the things they struggle with tends to make them more humble and compassionate in those settings where they excel.

Students also sometimes struggle with time management and the trap of local optimization—focusing too much on one assignment and not enough on the big picture of their learning experience. This is partly helped by coaching from faculty and TAs about better learning strategies and by incentivizing good habits, but sometimes they have to learn it by experience.

Our ACME faculty and TAs regularly meet together to discuss how to best coach and otherwise support the students through these challenges. Managing these different challenges takes real work from the faculty and TAs. But the work is rewarding and brings significant benefits to our students.

“It’s HARD, but so powerful.” —Jesse Casillas ‘17

4.2. Lockstep cohort

The lockstep cohort starts in fall semester of the junior year and is a fundamental part of the ACME experience. Students take courses with the same classmates for two hours every day for two academic years and study together in common study rooms with those same classmates. They also organize social activities together.

Benefits of the cohorts

There are many benefits of the cohorts, both for the students and for the program. These include enabling us to to take advantage of interconnections between the parallel courses, building a sense of teamwork and group support, and building loyal and enthusiastic alumni.

Cohorts enable interconnections. Cohorts enable us to take advantage of interconnections between parallel “sister” courses. For example, in one course they learn about orthonormal bases and linear projections, and in the sister course they use that knowledge to understand Fourier series. As another example, in one course they learn about the uniform contraction mapping principle and in another they use that knowledge to prove the stable manifold theorem. Students appreciate the things they learn in one class much more when those things are used right away in another class.

Cohorts encourage teamwork. Learning to work together is essential but hard for many math majors. The cohorts help with that. One year a cohort entered themselves in the university intramural frisbee competition and won the championship for the entire university. That was partly because of some expert coaching by one member of the cohort, but it also shows how well they learned to work together.

Cohorts provide emotional support. The emotional and social support the cohorts give students is powerful. One student who was struggling with some mental health issues suddenly stopped coming to class and ditched his study groups. His peers recognized he needed help, went to his dorm, and banged on his door until he got out of bed. They told him to get dressed and come with them so they could all work on homework together and get him caught up. This wasn’t initiated or even noticed by the faculty until much later, but with the help of his peers he finished the semester strong and is now flourishing in a good graduate program working on his PhD in mathematics. Without the cohort, I don’t think he would have finished the semester.

Another student just this month came to consult with me about how to help a classmate struggling with some personal issues. This is a stark contrast to my traditional math classes, where the students mostly don’t even know each others’ names, despite my best efforts to get them to engage with each other.

Cohorts grow loyal alumni. Working together with classmates as a team transforms students into loyal and enthusiastic alumni who stay connected after graduation and generously give time and money to support the students currently in the program. Our university’s most recent senior survey indicated that more than $40\%$ of all math majors (both ACME and traditional) were mentored by alumni—significantly more than any other department in our college. ACME students often mention how helpful it was for them to talk to alumni, and this is all a result of alumni volunteering and taking initiative to make themselves available to students. In contrast, before we started ACME we saw almost no alumni mentoring, and alumni of our traditional major still do not mentor students very often.

Challenges of cohorts

Of course there are difficulties with a lockstep cohort, including reduced flexibility for both faculty and students. Faculty must coordinate what they teach and when they teach it to be able to build on what is taught in sister courses. And students must take the cohort courses at the time and in the semester that they are taught. This sometimes also requires us to coordinate with other departments to avoid scheduling conflicts. And the schedule doesn’t always work perfectly for everyone. Some students need to switch to another cohort, or even take one of the off-ramps we provide to switch back to the traditional major from ACME. Conversely, students who realize late that they want what ACME has to offer can still join a junior cohort for just one semester or one year and use ACME courses to count toward their traditional degree.

The cohort system can also be a challenge for introverts who prefer to work alone or find it difficult to form study groups. Learning to work with others is an important skill even for introverts, but we try to help them overcome some of these hurdles by assisting with study group formation and providing dedicated study spaces and online collaboration tools like Slack.

Managing these difficulties takes work, but it’s worth it, because of the benefit to the students. And for many students the cohort itself is a strong draw. As one alumnus says, “I chose ACME because of the cohort situation and the in-depth learning about the math for many algorithms used in the industry today.” —Wesley Stevens ‘18

We started the cohort system as an efficiency, to reduce the need for faculty and TA resources, but the benefits of the cohort system are so significant we can’t imagine doing ACME without cohorts. In fact, after seeing the power of the cohorts in ACME, our department started a cohort experience for freshmen and sophomores for all our majors (both the traditional major and ACME).

4.3. Computer labs

Throughout the junior and senior core, students do a lab every week for each of the two core theory courses they are taking each semester. The general approach we take to labs is that the students first code up an implementation in Python of the mathematical ideas we are treating, then they compare their implementation for speed, scalability, and correctness to the industry standard implementation. Sometimes their code is competitive with the polished industrial version, which they find very rewarding. Finally, they use the mathematical tool to solve an interesting problem.

As an example, one lab involves using the FFT to filter the loud, annoying buzz of the popular vuvuzela noisemaker out of a recording of a World Cup soccer game. Students also experiment with convolution, starting with a recording of a Chopin piano piece played in the studio (a low-echo environment) and then convolving that with a recording of a balloon pop in an echoey stairwell. They get a kick out of hearing how the convolved result sounds like the piano is being played in the echoey stairwell. They often take this to the next level by convolving many of their favorite audio clips with the balloon pop.

Other popular labs include Markov chains for text generation, Perron–Frobenius for PageRank and March Madness brackets, finding Bacon (Erdös) numbers, Monte Carlo integration, Multi-armed bandits, SIR epidemic models, Hidden Markov model speech recognition, Random forests, Kalman filter, HIV treatment, and color quantization with K-means. These labs were developed with financial support from an NSF TUES grant (DUE-1323785) and all of our labs are free and open source HJE22.

Benefits of computer labs

The labs help students learn the math better, improve students’ attention to detail, improve students’ employability, and motivate students to learn more mathematics.

Labs improve mathematical learning. The best way to learn is to teach, and the computer is the dumbest possible student—it does only and exactly what you tell it and never gets the idea, sees the pattern, or fills in the details. To teach the computer the programmer must describe every part of every algorithm and formula and be able to debug all the errors that arise. Doing all that improves the programmer’s understanding enormously. As J. Betteridge et al. say, “Learning to use computers well is a very effective way to learn mathematics well: by teaching programming, we can teach people to be better mathematicians” BCC$^{+}$22.

Labs improve attention to detail. The Python interpreter usually won’t run students’ programs at all unless they have been careful about every aspect of their code, including syntax, order of operations, and carefully defining variables and methods before using them. Getting immediate feedback on these things in computer labs helps them learn to think more carefully and clearly about similar things in their written mathematics, where feedback is much slower. Importantly, students often seem to respond better to an impersonal error message from a computer than they do to a TA or professor telling them that the their proof is wrong. This helps them learn that mistakes are normal and expected, and that identifying mistakes is essential to growth.

Labs boost employability. Computer labs directly build students’ programming skills and their ability to convert complex ideas into efficient code. Moreover, the labs help students learn industry-standard tools, improving their employability and giving them a chance to build a portfolio of interesting projects to demonstrate their abilities to prospective employers. The labs focus on using computers and mathematics together, which is not something they can gain just by taking computer courses alongside their math courses, but it is something that employers tell us they want.

Labs motivate mathematics. Students are motivated by the applications in labs to learn more mathematics. The theory of Markov chains or the singular value decomposition may feel dry to them, but when they can use the theory to build a cool application, they become more motivated to learn and understand the mathematics.

Challenges with computer labs

As with anything, there are challenges with the labs, but so far these have been manageable for us.

The first challenge is limited resources for teaching. Not all our faculty program well, not all that do program well know Python, and faculty are busy with other things. For these reasons we designed the labs to be taught by teaching assistants (both graduate and undergraduate), and that works pretty well for us.

Another challenge is that the lab materials need regular updating, requiring a team of faculty and TAs to review and revise the labs regularly. But the benefits of the labs are so powerful for student learning that they absolutely outweigh the cost to us of managing these relatively small issues.

Why Python?

We use Python almost exclusively for several reasons. First, students need to learn to program well, so they need to have enough experience in a full-blown programming language to learn it in some depth. Other mathematical and statistical computing tools like Mathematica, MatLab, Maple, and R are great for what they do, but as programming languages they are not as versatile nor as widely used outside of the academic community as Python.

Python is the primary language of modern data science and is currently the most popular programming language in use, according to the TIOBE index TIO. It is easy to learn, and it is free and open source. So it has been our exclusive tool. There are many packages within Python that can be used for specific applications, and we use many of these in our labs HJE22, but the underlying tool is always Python. It is possible that Julia will eventually take the place of Python, but Julia is not yet as mature as Python and is not yet widely used.

4.4. Concentration

Students are required to do a concentration of four to five courses in an application area of their choice, usually from another department. Because the students have a strong mathematical background, these concentration courses are usually more advanced than a typical minor. Some of the most popular choices include computer science, data science and machine learning, economics, business, biology, and physics.

Benefits of concentrations

The concentration helps students learn to communicate across disciplines and see how math is used in a subject they care about. And they use it to prepare for their specific chosen career path, whether that’s a job in machine learning, going to graduate school in economics, or starting their own business.

Also, many students are attracted to ACME because the concentration allows them to study both math and another subject they love and use them together, rather than choosing between them. The following quote from a recent graduate is typical of what students tell us about why they chose ACME: “ACME offered me the opportunity to explore biology and mathematics simultaneously…. The idea of having a concentration that was unique to me and my interests really appealed to me.” —Karl Ringger ‘21

Challenges with the concentrations

In some concentrations students must fill many prerequisites before they can get to the interesting courses that really use math. Some departments are good about working with us to find alternative paths into those courses, and others aren’t. And, of course, the students often need guidance as they choose and navigate their concentration, and that takes faculty time. But it’s worth it, because it really helps them.

5. Additional Challenges

One big challenge we faced when starting ACME was a lack of suitable curriculum materials. Being naïve, we decided to write our own. The National Science Foundation helped with a grant to support our work; but it really was a lot of work, and I don’t recommend it if you can avoid it. I hope that some of what we have done will be useful to you. I’ve already shared a little about the labs above, but we also wrote some textbooks, which are published by SIAM HJE17HJ20. We were pleased that SIAM produced beautiful hardbound books in full color for less than what it would cost an individual to photocopy the book.

One of the greatest challenges we faced when starting ACME was limited resources, which motivated the lockstep cohort model. The cohort model is efficient, using only two faculty lines to run eight required courses, with labs run by graduate students.

We also had few faculty who knew all the material. To address this we developed our textbooks with faculty in mind as well as students, allowing faculty to learn the material ahead of the students. Many of our faculty also were not proficient at computer programming. To address this we use student TAs to teach the labs, so that the faculty need not code.

Finally, some faculty were suspicious of applied math and were reluctant to support any program that might move resources from pure math to applied math. In fact, when the first draft of the program went to the department curriculum committee for approval, it was unanimously opposed. But eventually the committee and the rest of the department agreed to let us try it, especially when the university academic vice president gave us one faculty line on condition that within five years we meet a target of 40 students enrolled per cohort and 25 graduating per year.

6. Results

Enrollments

Our first cohort had only 15 students in it, but by the fifth year, we had 70 enrolled in each new cohort and over 60 graduating each year—far exceeding the vice president’s requirement for keeping the faculty line. The total number of math majors (traditional and ACME combined, but not math education, statistics, or computer science) has grown from the low 200s to well over 400 (approximately 1.3% of the BYU student body). Since 2014, a year after ACME started, the percentage of minority students in ACME, while still lower than we’d like, has grown by $73\%$, compared to an increase of only $6\%$ for minorities in the university as a whole.

Jobs and internships

We are not a jobs training program, but ACME students generally get much better jobs than our traditional math majors. Data is incomplete because not everyone reports job and salary information back to us, but for those who do report, the highest starting salary of our traditional math majors is roughly the median starting salary for our ACME majors.

We have also seen a large increase in the number of employers coming to recruit our students. And many faculty in other departments now try to recruit our students as research assistants.

Graduate programs

Our students have been very successful in top graduate programs in both pure and applied mathematics, but they have also been successful in top graduate programs in other disciplines, including biostatistics, computational biology, computer science, economics, electrical engineering, geology, machine learning, marketing, math education, petroleum engineering, and statistics.

“In my graduate degree [Biostatistics at Berkeley] I have classmates who graduated from Ivy League schools who are constantly baffled by the breadth of topics in computer science, mathematics, and statistics I’ve been exposed to and the understanding I’ve retained…. I really am super grateful for ACME. It prepared me better than I could have imagined for grad school.”—Tyler Mansfield ‘20

Faculty

Although some of our faculty didn’t know all the material the first time they taught an ACME course, which meant extra work for them, most of them have loved teaching it. Even most of those who haven’t yet taught ACME classes recognize that ACME attracts more good students to math, with a positive spillover into the traditional major and graduate programs. And the university has provided our department with additional faculty and other resources because we have more, happier students.

7. First Steps

If you want to start something like this for your students, what should be your first steps? And what is the lowest hanging fruit or most bang for the buck?

Math + programming. I feel strongly that the most important thing math majors need is more computer programming in a widely used programming language, ideally merged with their mathematics in a way that enables them to use computers to solve mathematics problems and to implement deep mathematical ideas in efficient code.

One way to start this is to integrate programming labs into linear algebra for math majors. This helps them develop their programming skills, it frees them from the drudgery of solving large linear systems and finding eigenvalues by hand, and it helps them see how tedious computations can be assigned to the computer to let them do interesting things with their mathematics. Students can easily access a powerful computing environment through free tools like Google Colab without any special help or expertise. For an example of how this can be done, see HSWS22.

Another thing to consider for every math major is a course in algorithms and optimization, where they really think about the mathematics of computation and learn about optimization—the fundamental tool of data science, machine learning, and statistics. That course should also have lots of programming labs. We teach this course using HJ20 and the labs in HJE22, but there are many other ways that you could do such a course.

Finally, whether you adopt these math-plus-programing courses or not, consider requiring at least one or two standard computer science courses of all your math majors. Although Python and C++ are some of the most useful languages for mathematicians, courses in other popular languages like Javascript and Java are also useful.

Concentrations: math + X. Students benefit from seeing how mathematical ideas are used in other disciplines. But not every student likes the same applications, and not everyone will respond to the specific applications you choose to show in your classes. Consider encouraging or even requiring your students to take classes outside of mathematics in a complementary subject where they can apply their skills to something that interests them. It need not be in a STEM field; the social sciences, business, and other disciplines benefit greatly from the skills that mathematicians bring.

Cohorts. Cohorts are extremely powerful for improving the learning experience and helping students learn important soft skills. Even if you find it difficult to form formal cohorts among your majors, consider doing things to approximate cohorts. That could be scheduling two classes that majors typically take concurrently to run in the same classroom, back to back. It could also mean helping them form study groups, and, if possible, dedicating space for them to study together—maybe even in that same back-to-back classroom in the hours before and after the two classes. Almost anything that gets students talking to each other and working together is helpful.

8. Conclusion

I don’t think that our approach is necessarily the perfect fit for every program. But I hope that I’ve been able to give you some ideas of both why and how to implement a program that teaches students mathematics that they can use, and prepares them to actually use it.

When the students see that what we have to offer is relevant to their goals, their life, and their ambitions, then they are willing to trust us when we then ask them to do hard things. In the words of one student on our anonymous student ratings form, the experience “globally optimized our learning, happiness, and personal growth. That’s all you really need to know, since this is an optimization class.”

We can restore confidence in the value of mathematics by delivering what students have been promised—useful math and the practical skills to use it. I hope I’ve been able to convince you that we can do this without losing what we love about the traditional math major. Applied math is beautiful, and it can and should be taught in a way that is rigorous and challenging. I ask you to consider how you can use these ideas to open the doors for more of your students to enjoy mathematics, to succeed in mathematics, and to use mathematics to make the world a better place.