When a high school math teacher saw a need for education reform, she ended up in the perfect position to make a difference.

Amber Simpson went back to school for her doctoral degree in curriculum and instruction in mathematics education at Clemson University. Now she’s an assistant professor of math education at Binghamton University.

“I had no idea what a PhD actually meant or required,” Simpson says. “But I started taking research classes and my world just flipped. I loved how you can contribute to the education field through research, and so that was a huge eye-opening experience for me, almost by happenstance.”

At Binghamton, Simpson says she has focused her research and coursework on changing the stigma associated with math education.

“Most students think of learning mathematics as, ‘Let me sit at my desk, listen to the teacher present procedures and do my homework,’” Simpson says. “But there’s a lot of research that supports the idea of inquiry, allowing kids to brainstorm ways to solve the problem through various approaches and having classroom discussions about their strategies.” 

In spring 2019, Simpson designed and taught a service-learning course called Making and Tinkering — an education class focused on design, physical creation and problem-solving. Students worked in groups to develop youth making programs. They implemented them in the community through a partnership with the Windsor Central School District and Your Home Public Library in Johnson City, Simpson says.

“Makerspaces provide a place for anyone to just go and make things, and tinker with things,” Simpson says. “Making can including high-tech tools such as 3D printers and laser cutters, low-tech tools such as makey-makeys and arduinos, or upcycled materials such as cardboard and yogurt containers. All of it has a huge presence within STEM education and as part of my research, I examine the role of making and tinkering for young children and educators.”

Jose A. Morales Collazo, a doctoral student in the department of teaching, learning and educational leadership, has worked with Simpson since she came to Binghamton in 2017. He attributes much of his success to her guidance.

“I was very intrigued by Dr. Simpson because she was this energetic person that was coming in with this fire,” Collazo says. “I waited a little bit, but then I just went to her office and knocked on her door and asked if I could help. … We started with just a smaller project that she was working on. And currently we are working on four projects involving STEM identity and meta-analysis.”

Simpson has published work in identity research, particularly STEM identity research, which is driven by her fascination of what people experience and their stories, she says. 

Aside from developing unique curricula, Simpson is involved in community research. She was awarded a three-year National Science Foundation grant to work with families and engineers on considering the engineering design process in their own home and/or community. Simpson says families are guided by local engineers and students on how to come up with solutions or prototypes for something they see as a problem.

“The family engagement piece is sort of a newer interest of mine,” Simpson says, “and I always try to build in identity research and making and tinkering into my projects. So this particular grant has all three of my research studies coinciding.”

Birthday conferences are a longstanding tradition in mathematics, but they’re not thrown for everybody, says Jim Davis, a mathematician from Indiana University. Farrell received this special honor, Davis says, because his work revolutionized the field of topology — an area of mathematics that explores the deformation of theoretical geometric objects.

Farrell is famous for formulating a theory with Stony Brook University mathematician Lowell Jones that came to be known as the Farrell-Jones Conjecture. The research began while Farrell studied at Yale in the 1960s. “My advisor suggested a couple of problems to work on, but then a few months later, he told me they’d been solved already by this person named Borel,” Farrell explains. “After this happened a few times, my advisor told me to go down to Princeton and ask Borel to suggest a problem for me to solve.” The problem he was assigned became known as the Borel Conjecture. “It’s what I’ve been working on for most of my life,” Farrell says.

The Borel Conjecture deals with geometric objects called manifolds, which often look like twisted ribbons or hollowed-out pretzels. Mathematicians who study manifolds have abstract ideas about shape. For example, you probably wouldn’t say that an egg and a sphere are the same shape, but in topology they’re thought of as homeomorphic: the points along their surfaces are arranged in a similar order. “There’s a saying that topologists don’t know the difference between a doughnut and a coffee mug,” Davis says. “The mug has a hole in the handle, so if it was made of clay, you could mold the mug into a doughnut.” But you can’t reshape an egg into a doughnut without tearing or gluing its surfaces, so those two objects are not homeomorphic.

What Armond Borel proposed was that if two manifolds have the same number of holes and a negatively curved surface (like the side of a nuclear power plant’s tower), then the manifolds would be homeomorphic — their points have a one-to-one correspondence. Some researchers disagreed with the conjecture. In 1988, after 21 years of puzzle-solving, Farrell and Jones were able to prove those dissenters wrong for most examples.

“Farrell, together with his advisor Wu-Chung Hsiang and later with Jones, did the most important work in proving this conjecture in many cases,” Davis says. “Farrell has said that in the beginning, some of the results were so surprising that even he didn’t believe it. It took a while for people to really understand it.”

Within the past few decades, the Farrell-Jones Conjecture has gained a following. The conjecture has been highlighted at the International Conference of Mathematicians, and a German mathematician named Wolfgang Leuck has founded an entire school of thought based on the Farrell-Jones Conjecture.

Davis says the Farrell-Jones Conjecture is a theoretical tool that has few real-world applications at this time.  But “when Einstein came along and needed mathematical tools for Theory of Relativity, he needed abstract differential geometry. The math developed 60 years earlier by (Bernhard) Riemann gave him a tool to use. So you never know! There’s a bunch of reasons why people like Tom and I can justify spending our lives doing mathematics. It can sometimes be influential, but the main thing is the sheer beauty in it.”