All posts by: Sarah Hansen, M.S. '15

A new course in UMBC’s Department of Mathematics and Statistics is having a positive impact on student success in a notoriously difficult course for math majors everywhere. Two new papers by UMBC mathematicians and members of UMBC’s Faculty Development Center strongly suggest that MATH 300: Introduction to Mathematical Reasoning (IMR) is helping students succeed in MATH 301: Real Analysis, the first course math majors take that relies on one’s ability to construct and analyze proofs, rather than just do calculations. 

In Real Analysis, “We’re switching gears of how students think. They go from calculational things to proof-based work,” says Kathleen Hoffman, professor of mathematics and lead author on the new papers. “Now your solution is a paragraph that you have to write in full sentences. It has to have logical structure. It has to start with a hypothesis and end with a conclusion. It’s a big hump for students to get over.”

Real Analysis is easily one of the most challenging courses for math majors nationwide, Hoffman says. Over the years, many institutions have introduced a preparatory course that teaches students how to develop proofs without requiring them to learn new math content at the same time. The conventional wisdom is that these courses help, but almost no one had conducted a rigorous study to find out. 

“There was a huge gap in the literature,” Hoffman says. 

Forming the team

UMBC math faculty had seen the need and been talking about adding a dedicated proof-writing course for years, but it hadn’t quite come together. Hoffman jump-started the process by applying for a Hrabowski Innovation Fund Grant in the Scholarship of Teaching and Learning category. These awards support faculty who want to do ambitious projects that they might not otherwise have bandwidth for. 

When the award was funded, Hoffman formed a team with Justin Webster, associate professor of mathematics, and Kal Nanes, associate teaching professor of mathematics, to design the course for UMBC. Webster had re-designed and updated one of these proof-writing courses at his previous institution, the College of Charleston. The math team also worked with staff in UMBC’s Faculty Development Center to design a rigorous study to evaluate the course’s effectiveness over time. The team knows of only one other such study, from the 1980s, despite the rising incidence of proof-writing courses at universities nationwide.

Hoffman, Webster, and others believed that IMR would help UMBC math students, and informal observations supported their hunch once the course launched. These publications provide statistical analyses to back their intuition, and now Retrievers and students from other institutions can benefit from their successful formula.

Thinking about thinking 

The first paper, published in a special issue of Educational Sciences, focused on written reflections the students completed every week along with their proofs. Prior research suggests that students tend to struggle in specific skills related to proof-writing, so the students were required to address how well they thought they did on each of four skills in their reflections. 

“Students who did very thoughtful responses did much better in this course, but they also did much better in Real Analysis, where they didn’t do any reflections,” Hoffman says. The reflections “give the students a framework for understanding what they know and what they don’t know. It gives them the words to use.”

The study analyzed the quality of the reflections, but not necessarily the content. It didn’t seem to matter exactly what aspects of proof-writing the students addressed in their writing—simply the act of metacognition, or “thinking about thinking,” seemed beneficial.

The correlation was strong, but Hoffman admits the study does not prove causation. Strengthening the evidence, though, is that thoughtful reflections in IMR were not correlated with success in its prerequisite course, MATH 221: Introduction to Linear Algebra. That suggests the reflections, and potentially other elements of IMR, were the difference-maker for students moving forward.  

A solid foundation

A second paper, published in the International Journal of Mathematical Education in Science and Technology, compared students’ grades in Real Analysis depending on whether or not they had taken the proof-writing course. The findings showed that IMR did not much affect the outcomes for students who earned As in the prerequisite linear algebra course—they were also likely to do well in Real Analysis whether they took IMR or not. However, students who earned a B or C in the prerequisite course were much more likely to successfully complete Real Analysis if they had taken IMR. 

The researchers also received overwhelmingly positive feedback from students who had taken IMR about its benefits. One student said, 

“I feel like [IMR] gave me a solid foundation in understanding how to write proofs, which allowed me to come into [Real Analysis] with a bit more confidence. Without it, I probably would have struggled through [Real Analysis] since I would have been learning how to write proofs and the [Real Analysis] material at the same time.”

Based on the results of these studies, the UMBC mathematics and statistics department has decided to make IMR a required part of the curriculum for math majors and minors. Minors used to take Real Analysis as their terminal course, but now they take IMR. For majors, IMR provides the foundation needed to support success in Real Analysis.

More than pushing symbols around

When he took it as an undergraduate, an IMR-type course “was the thing that made me want to be a math major,” Webster says, so designing this course for UMBC was an exciting prospect. Becoming proficient in writing and analyzing proofs, rather than doing calculations, is like “writing versus writing literature,” he says—you have to spend a lot of time thinking about what you’re trying to accomplish and how to structure your arguments. You can’t just “plug-and-chug,” applying various theorems and techniques to instances of a given type of problem. “Math isn’t just the act of pushing symbols around,” Webster says.

To prepare students to write mathematical literature, Hoffman says that in IMR, “In one sense I’m teaching math, but in another sense I’m not. I’m teaching them how to think—how to structure their argument and express it clearly.” Almost never can a student simply sit down and write a proof in one sitting, like completing a problem set in prior math courses. It’s more like writing a paper.  

When you work on a proof, “You think, you don’t get it, you go do something else, you think, ‘Oh, I think I know what to do,’ you come back, and that is normal,” Hoffman says. “They have to understand, this is not instant gratification—you will struggle with this and I’m expecting you to. It’s inevitable that they will struggle—I’m teaching them to persist through the struggle.”

A collective commitment

The studies would not have been possible without support from the Faculty Development Center. While many faculty might like to conduct more rigorous analysis of their teaching methods, it’s not their area of expertise. “If you want people like me who do disciplinary research to engage in pedagogical research, you have to give me some help,” as Hoffman put it.

That’s where the FDC came in. Several staff members became involved in the project, including Tory Williams, Jennifer Harrison, Kerrie Kephart, and Linda Hodges, adding their individual areas of expertise.

“The math faculty have deep expertise in math pedagogy, but needed our support to help plan the intervention, design the research study, and analyze the data. The project required all hands on deck.” shares Kephart, co-author on the written reflections study and interim FDC director. “As a qualitative researcher with expertise in the teaching of academic writing, I enjoyed the challenge of figuring out how to study the effects of incorporating reflective writing into a math class.”

The project is also a demonstration of the math department’s commitment to supporting student success, even if that required some culture change. Since IMR’s initial offering in 2019, several additional math faculty have taken on teaching the course. Each time someone new takes it on, they work closely with experienced instructors, and all sections of the course are closely coordinated to ensure quality and consistency for students. 

The project is a masterclass in recognizing a challenge (a high failure rate in Real Analysis) and taking creative, concerted, and collective action to address it, with very positive results. “After realizing there was a gap in students’ preparation, a group worked together to fill it, and in the process learned a lot about how to measure progress in math education and pedagogy,” Webster says. “We leaped at this opportunity to effect change and measure outcomes in a novel and modern way.”

And because they carefully evaluated the project’s effects and published the results, now other math departments can benefit from their findings. That possibility was a highlight for Kephart. “Since our work in the FDC generally supports faculty, teaching, and learning here at UMBC,” she says, “it’s exciting to make a contribution toward the development of math pedagogy beyond our campus.”

Some mathematical and statistical challenges are so fundamental that the solutions can be applied to an array of real-world applications—which we all benefit from. But it’s not necessarily the applications that excite researchers. They’re on the hunt to develop an elegant set of equations or a more accurate model because of the work’s inherent beauty and satisfaction, and, simply, because it’s fun.

This past fall, members of the UMBC Department of Mathematics and Statistics received at least three major grants to pursue work with significant real-world potential, from self-driving cars to treatments for Alzheimer’s. And while the mathematical work underpinning these potential uses can seem opaque, abstract, and just plain hard to many people, these mathematicians and statisticians see the process of thinking about the challenges as exciting and stimulating in and of themselves.   

Optimizing optimization

Andrei Draganescu, associate professor of mathematics, has received a grant from the National Science Foundation to pursue a creative new approach for solving optimization problems. Anytime a system performs a basic functionality, but you would like to make it better or more efficient, that’s an optimization problem, Draganescu explains. 

Optimization means maximizing or minimizing something, such as cost, time, or power output. Think about today’s large language models, such as ChatGPT: Before it can respond effectively to any prompts, the model must rely on an algorithm to “learn” an optimal set of parameters from a vast amount of training data. Only once those parameters are fixed can it use them as a basis to determine the best response to your prompt. The time, electricity, and human work hours involved in training can cost billions of dollars, not including the capital investment in the hardware itself—so even a very small increase in efficiency makes a huge difference.  

All of the possible combinations of parameters can be represented as points along a curved surface with hills and valleys. An algorithm developed by a mathematician can search for the lowest or highest point on that surface to solve the optimization problem, depending if you are looking for a minimum or a maximum. One risk with these algorithms is that they may end up at a local minimum or maximum—the peak of a single hill or the base of a single valley—that isn’t actually the global maximum or minimum across the entire surface.

In his grant application, Draganescu proposed a novel way of solving this kind of problem. “Because this type of problem is so old, it’s hard to find something new. But I think I did,” Draganescu says. Rather than searching for the minimum or maximum along a series of straight lines that approximate its curved surface, Draganescu has proposed a way to search directly along a set of well defined curves. The idea for the new approach “is a completely new angle,” he says, and in correspondence with colleagues to date, “nobody has contradicted me so far.”

Sticking with the hard problems

In math, there are many problems that are easy to express but extremely difficult to solve. Some have puzzled mathematicians for centuries. Draganescu’s Ph.D. advisor encouraged his students to think about these problems from time to time, in case they found a solution, of course, but also to keep their creative juices flowing. Draganescu continues the practice today.

“Sometimes I’ve spent days in a row on some of these very hard problems that nobody knows how to solve,” he says. “And of course I would love the glory, but honestly, that’s not where the fun is. I’m old enough to know that the fun is actually thinking about it.”

Draganescu’s sticktoitiveness sometimes pays off in his everyday work. In the case of the idea that underpins his recent proposal, “I was surprised that it actually worked out. But ‘working out’ means many many months of work,” he says. “From having the initial idea to trusting that it could go somewhere, there were a lot of mistakes in the calculations and all that.” So why did he persist, knowing there was a good chance it wouldn’t “work out”? “I stuck with it because it was fun. I enjoy thinking about these problems.” 

Draganescu is continuing to develop his new technique for solving optimization problems, which could improve efforts to optimize all kinds of systems, from AI to airline schedules to agriculture—all because he found joy from sitting with a sticky problem.

Replicating the real world

Ansu Chatterjee, professor of statistics, is another UMBC math mind whose work could have a major impact on a wide range of fields. He and UMBC colleagues Animikh Biswas, professor of mathematics, and Karuna Joshi, professor of information systems, have received an NSF grant to expand the use of “digital twin” technology. The grant proposal highlighted the potential to advance diagnosis and treatment for neurodegenerative diseases such as Alzheimer’s or Parkinson’s, but digital twin technology has a huge range of potential applications.

A digital twin is “essentially a clone of something in the physical world replicated inside a computer,” Chatterjee says. Additional applications could include forest management or weather forecasting. These twins can offer up predictions about which direction a hurricane will turn, which treatment protocol might be most effective for a particular patient, or whether it’s the right time for a prescribed burn in a forest—but there will always be uncertainties in the models. 

“Measuring those uncertainties is a huge task,” Chatterjee says. How to deal with those uncertainties is one of the things the team wants to focus on with the new grant, which is very statistics-heavy, Chatterjee notes. Understanding and reducing the uncertainties requires understanding the physical processes at play in the real-world version of the system, and then working them into the model—and that’s hard. 

Beyond the black box

Typical AI models are frequently “black box models,” which means even the people who created them don’t always understand exactly what they’re doing. For one thing, they are not constrained by objective truth in the physical world, Chatterjee explains. “It’s not even attempting to get the biology or the chemistry correct,” he says, and that’s not acceptable for a digital twin that’s modeling the human brain and may be used to determine a real human’s treatment plan. 

“You don’t get the model right unless you get the underlying math of the molecules that constitute it right,” Chatterjee says. “So mathematics and statistics have to be the underpinning of anything related to digital twins.”

The digital twins Chatterjee and colleagues hope to further with their new research need not only replicate the outcomes of a system, but also the system’s mechanisms and internal processes. Again, that’s hard—especially when humans don’t yet fully understand many of these systems, such as the human brain, how cancer develops, or how various environmental factors influence carbon sequestration, for example.

“We are at the foundation of digital twins,” Chaterjee says. Once they are developed to a point where they are usable, digital twins and AI more generally would “open up a vast amount of opportunities for making fairly rapid progress with the actual science. This is where AI can help in scientific progress. It can suggest opportunities and possibilities which then can be verified,” Chatterjee says. “It would rule out certain possibilities. There would of course be false positives and false negatives, but as digital twins keep improving, it would be more and more convenient to use those first to find out a lot more interesting things.”

Fundamental building blocks

Chatterjee and colleagues are aware of the risks associated with digital twins. “There’s quite a lot of ethical issues that are related to digital twins in general, and it goes up several notches when it’s digital twins related to humans.” For example, there are regulatory concerns around who owns the data put into a digital twin (such as someone’s genomic data) and who is compensated when the twin is used for profit. “It’s exciting, and it’s also kind of a nightmare,” Chatterjee says. 

That’s why he and colleagues will be working directly with federal regulatory agencies to make recommendations about how to handle these AI models. “This grant is sort of laying down the first fundamental steps—saying this is what should be done, this is how the first steps should be done,” Chatterjee says. 

While that work is critical, it’s the math that truly excites Chatterjee. “For me, it’s about building the fundamental building blocks, whether it’s cancer or neurodegenerative diseases or forests” he says. “The essential ingredients for the digital component of it rely on the same math, the same statistics.”

Defining possible

Matthew Kvalheim, assistant professor of mathematics, is also working on developing tools that can help people in a wide range of situations. His ultimate goal? To be able to tell researchers whether whatever they are trying to do with their system is possible or impossible. Sound vague? That’s the point. 

“I’m never thinking about a specific application,” Kvalheim says. “I’m always working at the fundamental level, with a class of mathematical models that can describe all of them.” For Kvalheim, it’s the underlying math that is exciting—which is fortunate for all the people working on applications who will eventually use his multi-purpose tools. 

“For any real world system, you might want it to do some behavior, or not do some behavior,” Kvalheim says. This might be making sure two self-driving cars don’t collide, preventing a humanoid robot from falling on its face, or keeping an electronic device at a safe temperature. 

“Lots of people work on the set of models that I study,” he says, and the vast majority of them are investigating a particular application. These other researchers are very good at coming up with solutions that result in stability, or maintaining a certain state in a system, and safety, or preventing dangerous states in the system, “but there’s one thing they can’t do,” Kvalheim says. “They can’t determine if the thing they are trying to do, to make their system stable and safe in the way they want, is just fundamentally impossible. There could be a fundamental law of nature that says too bad, you can’t do it. You could try forever, but the heat death of the universe will occur first.”

A license to look

Kvalheim would like to discover new fundamental limitations that hold true for any application you could come up with, and also new fundamental capabilities. The U.S. Air Force recently awarded him a grant to further this work. He’s trying to generate a sort of mathematical litmus test that any practitioner can use to find out whether the thing they are trying to do with their system (whatever it is!) is fundamentally possible or impossible. If it’s impossible, it saves them a lot of time, energy, and money. And if it’s possible? “It’s like giving people a license to go look for it,” Kvalheim says.

The best part is that the end user of one of Kvalheim’s tools wouldn’t need to understand all the pure math behind it. “The idea is to produce useful tools that they can use even if they don’t have the time or interest in diving into all the details,” he says.

Math undergirds just about everything, Kvalhiem says, so it’s important to study the general case, rather than focus on specific applications. He wants people to see that it’s important “to know when there are fundamental laws of nature telling them that it is impossible or possible to do things that they want, and that math allows you to discover such laws of nature—and not just for one real-world system, but for many systems all at once.”

Somewhere a math or stat researcher is helping prevent your car from crashing, providing life-saving medical tools for yourself or a loved one, reducing wildfire risk where you live, or improving your experience with an AI tutor—and you might never even know it. There are humans behind each of these innovations who find joy and value in the math puzzles and problems that are all around us. And some of them work at UMBC.