Benefits and Challenges
A Challenging curriculum helps students learn more, but it also 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.
The title of the textbooks is very apt—Foundations of Applied Mathematics. The first year especially is very heavy on proofs and theory, but you leave with a deep understanding of so many mathematical tools.
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."
One of the major benefits of the ACME program is that it is very intentional in its build-up of essential topics in analysis, optimization, data science, and control theory. Within the text, the topics are well-explained and are demonstrated with very helpful examples and notes. The motivation for each topic is given and further justified as topics are integrated throughout the progression of the curriculum. I find myself often going back to my ACME textbooks for references and examples as I run across topics in grad school that I need a refresher on.
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, but the difficult curriculum itself also helps encourage collaboration.
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.
Of course we have to work with them on clarifying the difference between appropriate collaboration and cheating, but most students are really there for the learning, not the grade or the diploma, because there are many easier ways to get those.
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.
Students learn to solve hard problems
The challenging curriculum also helps build student ability and confidence to solve hard problems.
The program has increased my confidence in my problem-solving abilities. This major has been absolutely backbreaking at times,...but in the process, my skills have grown exponentially.
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.
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.
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.
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....
The biggest thing I see as someone who has conducted hundreds of interviews over the past few years is that students have knowledge of models and can sometimes talk about how they work in a technical sounding way, but rarely do they understand when and how to apply them. Secondly, they often come with very little preparation in terms of coding abilities and so it all falls apart for them in technical interviews.
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.
Technical leads that have known me now search out for ACME students to hire as a first preference.
Problems of a Challenging Curriculum
Of courser there are some difficulties with a challenging curriculum.
One of the difficulties includes the risk of students' becoming intimidated or developing impostor syndrome. But our lockstep cohort 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."
Our 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 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.
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.
Benefits of 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 a PhD in mathematics. Without the cohort, I don't think he would have finished the semester.
Students in the cohort notice and reach out to help classmates struggling personal issues.
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.
Challenges of Cohorts
Difficulties with a lockstep cohort include 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.
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.
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.
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
The first challenge with labs 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.
Students who did the labs last year are often excited to work as TAs the next year, and they do a very good job.
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.
Question: 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. 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, 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.
We do also encourage, but don't require, our students to learn a compiled language like C++, usually by taking more classes in computer science. This adds an important dimension to their skills and maturity as programmers, if they can make the time in their schedule to learn it.
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.
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.