“It’s an honor to be recognized for the work I’ve done advocating for women to be successful in mathematics—and, more broadly, for diversity,” Mast said.

Mast was among 18 U.S. scholars recognized for increasing the visibility and success of women in the mathematics field.

As a mathematician and a woman, Mast said she knows firsthand the challenges that women in STEM face. She was one of the few female graduate students in her mathematics Ph.D. program at the University of North Carolina in the 1990s—and the only one in her original class to finish the program, she said. She also recalled people who told her, “You’re not going to have a problem getting a job because you’re a woman. Universities are supposed to be hiring women now.”

But thanks to the work of Mast and her colleagues, women are getting closer to achieving parity with their male counterparts.

Over the past two decades, Mast has promoted the participation of women in math through her leadership in several organizations. As co-chair of the Joint Committee on Women in the Mathematical Sciences, she designed panels for women in math, including a panel on balancing professional and family life. As a member of the AWM executive committee, she helped mitigate implicit bias in the honors and awards processes in the mathematics community. With three colleagues, she co-edited the book *Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America* (Springer International Publishing, 2017), which celebrates the contributions of women in mathematics.

As the first female dean of Fordham College at Rose Hill, Mast has mentored women in STEM student groups. Last fall, her office secured funding to create the ASPIRES Scholarship program, which provides mentorship and monetary support to underrepresented students in STEM—including young women. A few months ago, she became the director of the Clare Boothe Luce Program at Fordham, which provides scholarships for outstanding women undergraduate students and graduate fellows in the sciences.

“I’m very excited to work with the other deans and with the faculty to strengthen the support that we give to the Clare Boothe Luce scholars and to create a stronger community for women in STEM at Fordham,” Mast said. “We’ve got some amazing scientists and mathematicians here, and I’m really excited about bringing them together so that we can be even stronger.”

Mast will be honored at the AWM Reception and Awards Presentation in Denver on Jan. 16, 2020.

]]>Hein, a Princeton-educated mathematician who moved to the U.S. from his native Germany more than a decade ago, is the first Kim B. and Stephen E. Bepler Chair in Mathematics at Fordham University. Since his appointment last fall, he’s had more time to pursue his research in differential geometry, or geometry in any dimension.

On a daily basis, he develops methods and equations that lead to new shapes. These shapes can surpass three-dimensional space. It’s impossible to sketch some of them. But the point of his papers, often with titles as abstract as “A Liouville Theorem for the Complex Monge-Ampere Equation on Product Manifolds,” is to explore uncharted territory in the realm of mathematics and develop new ways of thinking that can describe complex phenomena like black holes, though perhaps only decades or centuries from now.

“[Mathematicians] try to figure out patterns, describe certain things that they observe, purely within math,” Hein said. “These methods and equations have a life of their own. They exist abstractly, without any specific application. And then 20, 30, 50 years later, it may turn out that this is exactly the right kind of math that you need to describe something that actually exists in the real world—like gravitational waves or black holes.”

There used to be these TV programs in Germany for people who didn’t finish high school or wanted to brush up on high school material before they went to college. I started watching the trigonometry program, just out of curiosity, when I was 9 or 10. I liked the shapes. They were explaining how to graph sin and cosine. I sat down after the lesson and tried to recreate that on paper on my own. And I got a shape that looked like the thing that I saw on TV.

Simple ideas that solve problems, that, in the end, are correct. It doesn’t depend on anybody’s opinion. It’s some pattern or idea that’s going to be correct a thousand years from now, if humanity still exists.

There’s the more elementary stuff, like basic differential geometry that actually happens in three-dimensional space, that actually exists in the real world, that engineers and physicists use all the time. Then there’s my work—the rarified, cutting-edge stuff in theoretical math.

I lie on the couch all day. I imagine shapes and connections between shapes and quantities and try to figure out if some quantity is going to be large or small—how different quantities interact with each other. It’s a little bit like art in the sense that you create shapes and patterns. And then if I have the complete picture in my mind, I’m usually able to see the solution.

My wife is a mathematician, too, so at least it’s not weird for her. She knows what’s going on … that I’m actually working.

No. It’s just in my head. If I’ve really thought something through, I can just go to my laptop and write like 10 pages of equations and formulas and arguments and reasoning, based on what I have been imagining. Sometimes I have to do some calculations on paper, but that usually comes later.

That you’re stuck constantly. You don’t know what you’re doing most of the time. It’s not like you’re applying some method that you learned in grad school, and you’re trying to use that to create something new. I mean, sometimes it’s like that. But more often, you’re working on some problem that nobody’s really thought about before—that certainly no one has ever solved before. What that usually means is that the methods that exist aren’t sufficient enough to solve that problem.

Right. Usually it’s a tweak on some method that you learned in grad school or from somebody else’s paper. But, you know, once in a while, you have to create something completely new.

This kind of math that I do is incredibly abstract. Right now, nobody knows if it’s ever going to have an application to anything real. Much of the math is developed completely independently of any applications to physics [for example]. We often create new ideas for their own sake. And then [decades or even centuries later]it turns out to be exactly the right math that’s needed to make sense of things like quantum mechanics.

You discover these new beasts, specimens. You can see them in your head. Somehow, they’re out there.

*This interview has been edited and condensed for clarity. *

“Ed, aren’t you a math teacher?” he asked.

Yes, said Burger, president of Southwestern University, a mathematics professor, and the author of a new book called “Making Up Your Own Mind: Thinking Effectively through Creative Puzzle-Solving,” which was the subject of his presentation at Fordham College at Rose Hill on Oct. 3.

The boy asked him for help. Burger scanned the math problems. Try the question about donuts, he suggested. But Seamus was stumped.

“He was, basically, intellectually constipated,” Burger remembered. “He was pushing and pushing, and not one idea would come out.”

So they tried something different.

“When I say go, I want you to give me an answer that you’re confident is wrong,” said Burger.

“16!” Seamus immediately said.

The boy was wrong, but he was close. And that was the point. Instead of just sitting there, trying to solve a problem he couldn’t, he took action. He ignored the pressure of getting the right answer, got a wrong answer, and learned from his mistakes. And once he did, the right answer wasn’t far behind.

“Effective failure is how you respond to failure. So the failing itself—not that interesting or important,” he said. “It’s what you do next.”

This concept—intentionally failing, and then gaining a different perspective that allows you to grow and react in a meaningful way—was the foundation of Burger’s presentation, “Making Up Your Own Mind: Thinking Effectively through Creative Puzzle-Solving.”

“Now, to be clear, I’m not suggesting that you do this on your final exam,” he told the students at the lecture. “But it’s an intermediate step.”

The lecture, co-sponsored by the Rose Hill dean’s office, the math department, and the Graduate School of Education, was about how to think more effectively and creatively in our daily lives. Burger’s advice, he said, could apply to all aspects of life: work, academic, personal.

Failing effectively was just one part of a three-part formula that Burger developed and described to the audience gathered in Keating Hall.

**Understand simple things deeply.** Don’t try to understanding something that, at first glance, seems very complicated, Burger said. Focus on something simple that’s related to that complicated problem. Examine it at a “subatomic particle level,” until you’ve noticed some feature that didn’t register before. “You’ll begin to understand that simple thing deeper, and then the more complex thing that you were first struggling with becomes easier to think about.”

**Add the adjective.** When you approach a math question, your first instinct is to solve it, said Burger. That can be a bad approach if you’re not fluent in the material. “Back up. Force yourself to describe it. Add as many descriptors as you can to describe the thing that you’re looking at. And every word that you add is going to reveal something you otherwise wouldn’t have seen,” he said.

**Fail intentionally.** Failure is good for you, he emphasized—as long as you gain some new insight from that failure, and then respond in a thoughtful way.

The audience was riveted. “How did you come up with this stuff?” asked one man.

It was an idea that was more than 20 years in the making, said Burger. He integrated these three steps into a class he developed at Southwestern University, where he gave students puzzles on which to practice this mindset, and eventually encouraged them to apply it to their own everyday lives.

After explaining the three-part approach, Burger quizzed his audience with a problem from his new book.

On a projector screen, he presented pictures of three black-and-white chess boards. The first one was normal. The other two were truncated: the second board was missing its northwestern and northeastern corners; the third one was missing its northwestern and southeastern corners.

“Imagine that you have a whole bunch of dominos, and each domino will cover exactly two squares,” he said. “And so the question is, can you cover each chess board with dominos so that every domino covers two squares, every square is covered, and no square is covered by two dominos?”

For two minutes, he said, apply those three practices to a puzzle. Brainstorm with the strangers next to you. And don’t actually solve the puzzles. Practice this way of thinking, and see where it leads you.

Each chess board had an even number of squares. The first board had 64.

“Sixty-four is a big number. What if I thought about a smaller chess board?” Burger said. “So understand simple things deeply.” Using a black ballpoint pen, he sketched the simplest version of the normal board—a two-by-two version.

And then, after drawing mini versions of the other two boards, an audience member “added the adjective.” The chess boards, said the audience member, were “white and black.” In a regular chessboard, a Domino needs to cover a white square and a black square, Burger reasoned. Then the audience noticed something different: the colors of the tiles that were removed. In the second board, a black and white tile were removed. In the third board, *two* black tiles were removed. Thus, the third board would not work.

“Now you realize that the missing squares in the second case were a different color, and the missing squares in the last case were actually the same color,” Burger said. “Just within a matter of minutes, you’ve looked at something that you’ve seen your whole life in a different way.”

“Just using those three things, you now see a chess board differently. And in some sense, I think that’s a metaphor for what we have within ourselves—the power we have through our own thoughts, when harnessed effectively, to see everything differently,” he continued. “And more magically.”

]]>Peter M. Curran, Ph.D., FCRH ’47, a professor of mathematics at Fordham for nearly five decades, died on April 18. He was 92.

Curran, a native of Queens, began his studies at Fordham in 1943 and graduated summa cum laude four years later; he was first out of a class of 320 graduates. He began teaching immediately after graduation and went on to earn a master’s and Ph.D. at Columbia University.

He served as assistant chairman of Fordham’s math department from 1967 to 1972 and chair from 1979 to 1984. He was a member of the University Faculty Senate’s tenure review and appeals committee and served on numerous committees concerned with the University statutes, faculty memorials, and faculty hearings.

Curran was awarded a National Science Foundation Science Faculty Fellowship for 1961–1962 and was appointed a member of the Institute for Advanced Study at Princeton for the fall term of 1968. His research specialty was in infinite group theory and cohomology.

In 1987, the University honored him for 40 years of service with a Bene Merenti medal; he was cited for his unwavering loyalty to the University and its students.

“As a teacher, mathematician, humanist, friend, Pete Curran in 40 years here has earned the respect and affection of students and colleagues,” the citation read.

He retired in 1992 after 49 years at Fordham. In 2006, the department established the Peter M. Curran Visiting Assistant Professorship, a two-year appointment offered to recent Ph.D. graduates who demonstrate strong research potential and success in undergraduate teaching.

Janusz Golec, Ph.D., current chair of the department, said Curran, whom he pegged a “human being of high caliber,” and “a true gentleman,” was an clear choice for the honor at the time.

“It was just a quick decision; we didn’t have to deliberate on it. It was obvious to us,” he said.

Armand Brumer, Ph.D., a professor emeritus of mathematics who worked for nearly 20 years with Curran, said he’d miss most the spirited yet respectful political debates he had with Curran, who, he noted, had an uncanny knack for remembering students he’d taught decades before.

“I had trouble remembering the names of everyone in my class currently, but he would remember people from his classes from 20, 30 years ago. It almost felt as if he had a photographic memory,” he said.

“He certainly cared. It was awesome to hear him say ‘Oh, this is a student from 1955.’”

Curran’s dedication to the University was undeniable; he visited the campus as recently as 2015 to hear one of his former students, Frank Connolly, Ph.D., FCRH ’61, professor emeritus at Notre Dame, speak. Connolly called him a linchpin of the department whose reputation for helping undergraduates who were serious about math came with a “quiet and wicked sense of humor.”

“His political outlook was a bit to the starboard side, but it was far more gently enunciated than is the current fashion,” he said.

“In the late 50s he spent a lot of time in informal seminars with mathematics majors.

He was very generous with his time that way. I was one of the students in those seminars, and I am greatly indebted to him.”

Curran’s niece Patricia Chimento said he had a heart of gold.

“He was the primary caregiver for both his sickly mother and brother during the last years of their lives. He went out of his way to help family, friends, former students, and members of the church and community,” she said.

“He was always available to tutor, give advice, and provide service to those in need. He will be greatly missed.

Curran is survived by Chimento, her husband Jim; a nephew, Brian Hummel; and six grand nieces and nephews.

]]>Mathematical jargon is a huge turnoff to students, especially in New York City where English is a second language for many of them. If I tell you what a “slope” is, and I make you memorize its definition, that’s not engaging. But say I give you a task, and in trying to solve the task you need to interact with the concept of “slope?” If you’re standing in front of the class and you’re stumbling trying to describe this concept, that’s when I give you the vocabulary behind the concept. It’s a real turnoff to come into a classroom and get a bunch of vocabulary that you don’t understand, and can’t even conceive the purpose of.

Students already come to the classroom with a lot of mathematical knowledge and understanding of numbers. By putting them at the center of learning, and trying to elicit information and ideas from them, we get a lot more engagement. You can do this is by engaging in rich tasks that have applications in the real world, and having *them* generate questions about the situation.

If you’re working with younger kids, use something called a three-act task: You show an intriguing video that begs some questioning and ask students to generate a list of questions to investigate. It could be as simple as a video of a football player running with a football down the field. ‘How fast are they running?’ ‘How many yards did they cover?’ One of my favorite activities to do with little kids is to watch a video of Cookie Monster eating cookies from a box, and getting them to answer, ‘How many cookies did he eat?’ You want them to mathematize the world, instead of imposing mathematics onto their world.

Most good games have a lot of mathematics inherent in them. If you’re talking to students about probability and expected value, you could look critically at the game Risk, and talk about different situations in the game and when it makes sense to attack or not. If you’re working on multi-step problems, and you’re trying to get them to think ahead, play a game like Nim or even chess, in which thinking steps ahead is a strategy to help you win. Appeal to their competitive nature. Give them a challenge. Let their desire to win be the motivation behind their mathematics.

In math class, anxiety is very real. By allowing students to work in groups, we give them resources in each other. We give them models for people who already understand it, and we also give them a little bit of camaraderie because they’re inevitably going to work with people who don’t understand what’s going on. We know that people solve problems better in groups, and we know from research that they solve problems better on their own from having had a group-solving experience.

In the real world, people do math in teams. In the real world, programmers and scientists work in teams. The only time you really do mathematics by yourself is in these contrived educational settings.

Problem solving is central to the study of mathematics, and it’s a big focus of Common Core. In order to keep kids engaged and challenged, try posting a non-routine problem somewhere in the classroom at the start of each week. If students are ever feeling unengaged or just want a challenge, they can get this problem and work on it. The problem of the week can help at the end of an activity when some students are finished and others are not. It gives students who are excited by mathematics an avenue in which to express that excitement. That creates a classroom culture in which students see others engaging in public extracurricular mathematics—which is a very powerful thing.

]]>Six years later, however, McCauley has embraced a different calling. The Houston native is graduating with a major in mathematics and a minor in economics. This year, he realized he had a knack for computer coding—and the response he received at a recent interview at Google has convinced him to pursue it as a career.

He discovered an interest in coding after spending his junior year studying abroad at the London School of Economics. Even though he was able to line up several internship interviews for positions in management consulting, nothing came of them. He found himself with no plans for the summer.

So he turned to his thesis adviser, David Swinarski, Ph.D., assistant professor of mathematics, to see if he could assist in any research. With a summer research grant, McCauley created an analysis suite using raw data from motion capture technology that Columbia Presbyterian Hospital doctors are using to study breathing.

“Breathing looks very different in a person with emphysema compared with a normal person,” he said. His resulting research shows how those differences can be expressed mathematically.

“When I started this project, I realized I could take it as far as I wanted to,” he said. “It ended up being a much more valuable experience, from the standpoint of developing skills and having some experience, than a management consultancy would have been.”

As his interests have evolved from theater to finance to coding, McCauley also feels like he’s learned how to relax more. A 4.0 student in his first semester, he joked that he felt like he learned more during some of those subsequent “3.75” semesters. His four years at Fordham also afforded him the chance to form long term relationships outside his immediate family. This year, he shared an apartment in Harlem with two fellow honors students who he first met his freshman year, and spent a great deal of his free time at a poetry collective that one of them started.

The college’s small size and liberal arts focus made it the perfect fit, he said.

“If you put yourself out there, you have direct access to everybody—from the professors all the way up to [FCLC Dean] Father Grimes,” he said.

“People really express an interest in the individual student’s well-being.”

]]>In the beginning, he focused on the mathematical theory behind experiments that Fredric Cohen, Ph.D. and Robert Eisenberg, Ph.D., physiologists at the Rush University Medical Center in Chicago, had been working on— until they advised that he move from theory to “reality.”

“They said, ‘this is what the experiments say. This is what actually happens in the experiments when we change conditions,’” said Ryham of his collaborators. “’Your mathematical description must account for these effects.’”

Ryham kept their recommendation in mind when he began to calculate non-spontaneous events in membrane fusion, a process for intercellular communication that can be critical in the delivery of drugs to combat diseases. He was particularly interested in how a virus and a cell join together, and how much energy is required to stimulate that process (activation energy).According to Ryham, if researchers know precisely where and how a particular virus and its machinery expend energy, it may be possible to develop treatments that interrupt the delivery of genetic material by that virus and prevent infection.

Last year, he partnered with Cohen and other researchers from the National Institutes of Health to publish results of a funded study about the properties of influenza virus haemagglutinin. The study, which was published in *Nature Microbiology*, explores how a deeper understanding of the structural details of membrane fusion machinery can help to effectively combat influenza.

While there have been several mathematical models that estimate activation energies, Ryham helped to identify a transition pathway that previous physicists had missed in their explorations of viral fusion: a new structure called ‘lipidic junction.’ The researchers made further headway in the study by quantifying the activation energy for this new pathway of membrane fusion, which had never been done in this capacity, he said.

Ryham calculated the minimum-energy paths necessary for membrane fusion. He said an approach to calculation called the “string method” allowed him to provide a mathematical explanation for what the biologists were observing under the microscope.

“The thing that they were seeing, which was totally unexpected, was that when the virus attaches itself to the cell or fuses with the cell, it doesn’t follow the conjectured pattern,” he said. “It seems to break open and re-attach in an abrupt process. You wouldn’t suspect that nature would permit something so abrupt.”

The experiments also revealed that the activation energy for membrane fusion was significantly lower than what researchers previously predicted, which has biological significance. Another unexpected outcome was that the amount of cholesterol in the cell governed whether the event was an abrupt or smoothly transitioning process.

“It just reinforces the idea that cholesterol is a control parameter for things occurring inside the body,” he said. “In this case, it was for influenza.”

Related work on the subject deals with gaining entry into a cell—but without the use of viruses.

“The application of this study is more toward pharmacology,” he said. “The idea is that you want to load up minute spherical vesicles with a drug, launch them in the bloodstream, and then release the drug in a controlled fashion. Having an equation describing how and when the vesicle is going to burst is part of what makes these treatments possible.”

Ryham said mathematical models like the ones he develops, whether big or small, could contribute to breakthroughs in medicine in the future.

“Mathematicians are often motivated by questions in basic science and research something because it’s interesting,” he said. “But it may end up having an application down the road, and you never know what that application is going to be.”

]]>August marks one year since students from the Brazil Scientific Mobility Program (BSMP) arrived on campus. Run by the Institute of International Education and supported by the Brazilian government, BSMP places top-achieving junior and senior students pursuing STEM fields (science, technology, engineering, and mathematics) at U.S. colleges and universities to gain global experience, improve their language skills, and increase international dialogue in science and technology.

The Fordham cohort—Aryadne Guardieiro Pereira Rezende, Tulio Aimola, Caio Batista de Melo, and Dicksson Rammon Oliveira de Almeida—have spent the year studying and researching alongside Fordham students and faculty.

“Fordham is a wonderful university. It teaches you to grow not just as a professional, but also as a person. I loved my semesters there,” said Guardieiro, a computer science major from Uberlandia, Minas Gerais.

Guardieiro worked with Damian Lyons, PhD, professor of computer and information science, on the use of drones to hunt and kill *Aedes aegypti *mosquitos, which spread diseases such as dengue and Zika virus, both of which are significant problems in Brazil.

“Different fields were available to research here,” said Batista de Melo, a computer science major from Brazil’s capital, Brasília. Batista de Melo researched with Frank Hsu, PhD, the Clavius Distinguished Professor of Science and Professor of Computer and Information Science, in Fordham’s Laboratory of Informatics and Data Mining.

“Our project used IBM’s Watson, which might not have been possible to use in Brazil, since it is such a new technology.”

The program has benefitted both Fordham and Brazilian students alike, said Carla Romney, DSc, associate dean for STEM and pre-health education, who oversaw BSMP at Fordham. Because it’s difficult for science students to devote a full semester to travel, the experience served as a sort of “reverse study abroad” for Fordham students.

“Having international students in the classroom has been an amazing internationalization experience for Fordham students, too,” Romney said. “It brings a different atmosphere into the classroom when you have students with widely divergent viewpoints and experiences. You get to know other cultures, other worlds.”

BSMP students complete two semesters of academic study at an American institution, followed by a summer of experiential learning in the form of internships, research, volunteering, or other types of “academic training.”

Earlier this summer, the four were joined by an additional 17 BSMP students who had been at other American colleges and universities and who took up residence at Fordham to undertake internships and positions at various New York City companies and organizations.

The experience was challenging both academically as well as personally, said Oliveira, a computer science major from Recife, Pernambuco who researched smartwatch applications in the Wireless Sensor and Data Mining (WISDM) lab with Gary Weiss, PhD, associate professor of computer and information science.

“The cultural shock was really unexpected, and for several months it made me feel uneasy,” Oliveira said. “Over time, I learned to overcome it. Being from a predominantly tropical country, I considered the winter to be the greatest challenge of all.”

In addition to culture shock, there was the inevitable loneliness, which Guardieiro said she felt deeply at times. However, she felt supported by her academic adviser and fellow students, and eventually came to love her newfound independence.

“I learned to never lose an opportunity to do what I needed or wanted to just because I did not have company to do so,” she said. “I learned to expose myself to new—and not always comfortable—experiences, and I was amazed with the results I got. I took dancing classes with great teachers, visited places like Wall Street companies and all kinds of museums, and visited many states by myself.”

The Brazilian government recently put a one-year moratorium on the scholarship exchange program, but Romney said Fordham would continue its partnership with the program when it resumes.

When it does, Guardieiro has advice ready for future Fordham-BSMP students:

“Don’t be afraid to do everything you want to… This kind of experience is given to us to learn as much as we can.”

]]>Working in Swinarski’s lab, Fordham College at Rose Hill senior Jeremy Fague set out to help find the answer. For almost two years, Fague has been helping Swinarski and a group of Columbia University Medical Center researchers use math and computer programs, like Visual Basic and Excel, to analyze patients who are short of breath.

The researchers place 89 sensors in different patterns on patients’ chests, backs, and stomachs to measure changes in their chest volume while they exercise. They’re hoping this new data will offer better insights compared with data from breathing tests, and will eventually lead to better ways of managing emphysema and other pulmonary issues.

In his work as a research assistant, Fague has been trying to determine what sensor patterns work for different types of people. “The original pattern only works for one body type,” Fague explains. “We want to use this technology to serve a wider population of patients. I’ve spent a lot of time writing a program to try to unlock a lot of this data.”* *

Fague, a Massachusetts native, started out on the pre-law track at Fordham before realizing he could apply his passion for logic through a double major in math and economics and a minor in computer science. After taking a linear algebra class with Swinarski, he approached him about research opportunities and was invited to join this project.

“I was really attracted to the applied nature of it,” he says. “I’ve always been drawn to how I can use math and technology to directly impact the world around me. Through this work, I can really see how being able to analyze the data differently is going to make a difference in someone’s life.”

Working with Swinarski outside the classroom has been particularly rewarding for Fague. “He challenges me to think on my own and to find solutions,” Fague says. “He gives me a lot of feedback and also works alongside me. He’s everything I hoped I would have in a mentor.”

It’s part of the Jesuit tradition that initially drew Fague to Fordham. “I’ve found that the different perspectives here on campus have always challenged me and made me grow. People here think deeply about fundamental questions.”

Fague’s Fordham experience—his research, his courses, and his work as president of the Alternative Investments Club—landed him a full-time job with Goldman Sachs postgraduation; he also plans to pursue a graduate degree in the near future. Although he is starting a career in finance, he believes this research experience will prove to be invaluable.

“I’m just a mathematician who has really found a lot of different things I can do with math,” he says.

]]>“It gets complicated very quickly,” said Swinarski, PhD, who managed the immense number of equations by turning to computers for help.

It proved an enduring interest. Today, his research program sits at the intersection of mathematics and computer science, two fields that are increasingly linked in the study of abstract questions but also more concrete topics like human health. Fordham’s joint major offering in the two fields, still awaiting state approval, is the latest sign of how they’re coming together.

“It’s not that they’re converging, it’s just that maybe the division between them has always been a little artificial,” said Swinarski, an assistant professor of mathematics who helped draw up the plans for the proposed joint major.

Advances in computing power are opening up lots of new possibilities in mathematics, as Swinarski came to appreciate while earning his doctorate at Columbia University. For his dissertation, he was trying to answer a question about string theory, concerned with infinitesimally small particles that are impossible to observe,.

“My adviser and I found a way to answer some very abstract questions with some very concrete calculations, but they were too difficult to do by hand,” he said.

This was the first time he had applied computers to algebraic geometry, and he was soon using computer programming to get results to many other math problems. “It just sort of snowballed,” he said.

Computers are an important part of his research program that involves students in a variety of projects at the Lincoln Center campus. Some projects have focused on economics, mathematical finance, and other areas; one current project, for which Swinarski wrote software, could lead to faster and more accurate analysis of large data sets.

Another project is applying math and computer programs—Visual Basic, Excel—to the subtleties of human physiology for the benefit of people suffering from emphysema and other conditions.

Swinarski and an undergraduate student, Jeremy Fague, are working with Columbia University Medical Center researchers to analyze the large data set gathered by placing 89 sensors on people’s chests, backs, and stomachs to measure changes in their chest volume while they exercised. They’re hoping this new data will offer better insight compared with data obtained from breathing tests administered while someone is at rest and lead to better ways to manage emphysema.

“The big question in pulmonary medicine is, ‘Why is this particular patient short of breath?’” Swinarski said. “Is it their lungs, is it their chest muscles, is it actually that they have poor circulation and this feels to them like they’re short of breath when actually their lungs are functioning fine (and) it’s the heart that’s the problem?”

“Answering the question is complicated for all those reasons,” and both math and computer science can be used to identify potential causes of the breathing problems, such as out-of-sync expansion of the chest and abdomen, Swinarski said.

The new computer science/mathematics joint major will give students more leeway for electives at the intersection of the two fields and better prepare them for working in today’s tech sector, Swinarski said.

For instance, problems in parametric statistics call for an extremely large number of computations because fewer assumptions are made about the variables in question. Because today’s computers are far better able to handle this kind of problem, it no longer needs to be relegated to the margins in the classroom, Swinarski said.

“If you take sort of a classic one-semester statistics course, that’s something you would maybe hear about for a week or two at the end of a semester,” he said. “Nowadays, with the explosion of data science in New York, this is the kind of thing we’d like to teach a whole second semester of statistics on.”

]]>That’s the crux of research by Yi Ding, PhD, associate professor of school psychology in the Graduate School of Education.

Ding’s research focuses on strategies that teachers can use to help children who are struggling with math. She makes the case that memorization of basic math facts, such as multiplication tables, is key because it allows children to store information in their long-term memory, and frees up their working memory to tackle more complex problems. For example, adults don’t have to actively remember their own names or birthdays because those facts are readily available in their long-term memory, which works on auto-retrieve.

“If you have to decode every word you are reading, what happens? You don’t have reading comprehension at all because your working memory is occupied by saying each letter. Your attention can’t go to the who, the how, the where, and the what,” she said.

“Math facts are the same. We have to memorize or automatically retrieve all this mathematical vocabulary so kids have this kind of fluency. Then their brain–their working memory–frees up to understand more complicated problems.”

She said that a lot of progress has been made treating the memories of children with special needs in the last 30 years, thanks to new pharmaceutical treatments and behavioral therapy, paired with changes in family and environment. Improving a child’s long-term memory allows them to use working memory to tackle more challenging tasks.

“Behavioral therapy early on can change the connectivity of the brain. So when we change the environmental stimulation and we change the way we approach the kids, many of them learn new skills.”

Ding has detailed her research in two upcoming papers: “Involvement of Working Memory in Mental Multiplication in Chinese Elementary Students,“ and “Working Memory Load and Automaticity in Relation to Mental Multiplication, under review for publication in the *Journal of Educational Research*. For the studies, she and her co-authors recruited fourth and fifth graders of differing academic levels to complete a series of arithmetic problems.

The study found that even if a problem involves more steps, if it comes with automatic retrieval to students, they do better. When asked to complete a two-step equations that they do not have automatic retrieval, like(25×3)×4, students stumble more frequently compared to when they tackled three-step problems that they have automatic retrieval, like 25×10+25×2.

A common critique of the American education system is “we try to dig three hundred miles wide and only half a mile deep,” she said. When Ding moved from Beijing to Iowa, she said she was taken aback by the thickness of children’s textbooks, coupled with the multiple topics teachers must cover in a short period.

“Teachers are running around like crazy trying to cover one topic to another. But then we don’t have enough time reserved for practice, for rehearsal, to give kids the time to make sure we have developed fluency in this most basic stuff. Your memory needs time to practice,” she said.

One way to address this is with drills. Ding remembers hating drills when she was a student but now appreciates them because they helped her develop a basic math vocabulary in her long-term memory.

Ding is working with engineering teachers to identify and aid low-performing students. She said it’s common for low-achieving engineering students to exhibit slow processing speeds, which leads them to struggle to complete exams in a timely fashion. Most of time, the automatic retrieval of math/engineering facts is very weak in these students.

To make matters worse, Ding’s evidence shows they tend to pick just one strategy to solve all problems, which results from a “very weak strategy flexibility.” They probably just simply do not have a wide range of effective strategies banked in their long-term memory and are not proficient with “which strategy fits what problem situation”.

“Even if they find the right strategy, when you ask them to calculate, they calculate it wrong. When the execution is problematic as well, it’s almost a double deficit,” she said.

“When you have lots of strategies that are well learned and banked in your long-term memory, then you free up your working memory. If a task has steps 1 through 10, and you can automatically retrieve steps 1 through 5 from memory, then you have what you need to do steps 6-10.”

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