RECKONING WITH INTANGIBLES
David Landy and the science of learning math
In 2007 PBS Professor David Landy was a PhD student at IU in computer science and cognitive science, studying the cognitive process by which people learn and perform mathematical problems. He was also the father of a newborn baby girl. Graduate research and new parenthood do not always happily coexist. But on one particular night, as Landy held and rocked his daughter into the late hours of the night, he had an idea that decisively shaped the course of his career as a scholar and scientist: a computer program that could dramatically change the way we learn and do math.
“Algebra is hard. It’s hard intrinsically to think abstractly,” Landy observes. Making it even harder is “the 500-year-old interface” with which mathematicians work out a problem. This ancient interface, otherwise known as paper, has certain advantages, but it also has “some unnecessary limitations.”
“Imagine,” he explains, “that you were going to learn to play chess, but you didn’t have a chessboard and had to recopy the whole board each time you make a move. After five moves, you’d make a mistake. There’s too much room for error. And most of your time would be spent in copying, rather than thinking.”
Such is the case with algebra
Paper is static. At each step of a problem, you need to rewrite your equation, increasing the likelihood of making a mistake. A more flexible, fluid medium, Landy suggests, would be less error prone and lessen the working memory load more generally which simultaneously includes remembering the general rules of algebra and the variables of a specific problem.
The programs he and his colleagues have created address these issues. They also have the advantage of tracking the users’ computational moves, thereby preserving reams of data for further research into the cognitive underpinnings of math and the efficacy of their own applications.
Tackling math literacy rates
Landy’s fascination with how we learn math comes out of a broader interest in how we make sense of things we can’t see and physically experience.
“We seem to be really good at dealing with things we can touch and see and work with,” he explains. “Once it gets beyond that, it gets very hard. Math, however, combines being both intangible and knowable. Or at least, it is knowable at times, while at others, utterly elusive.”
“We seem to be really good at dealing with things we can touch and see and work with. Once it gets beyond that, it gets hard.”
Math has proven especially elusive for the majority of U.S. schoolchildren, about half of whom, according to the National Center for Education Statistics, a federal entity that analyzes educational data, only make it to an eighth-grade level. Landy and his colleagues believe that the applications they are developing can help to tackle this national issue.
Currently Landy dedicates a major portion of his lab work to developing the project within the context of a company — Graspable Math, Inc. — which includes collaborators from four universities and several disciplines. The next step, he says, is to determine what he and his team can do to make their applications more robust, more usable and more appealing.
Cognition Unbound
Landy paraphrases the mathematician/philosopher Alfred North Whitehead, suggesting, “We often think civilization advances by increasing the number of things we can think about. But it actually advances by extending the number of things we don’t have to think about.”
From this perspective he and his colleagues are continuing a process that was set in motion hundreds of years ago by mathematicians from India, the Middle East and Europe who invented the language and symbols of algebra. As some of Landy’s own studies show, the visual symbols and conventions of algebra already act as a cognitive shorthand, relieving us of some of the burden placed on our working memory. Once we learn these visual forms, we can think less about the rules.
“Making algebra tangible is exactly what people in the 16th century saw themselves as doing, using pictures to represent the intangible.”
“Making algebra tangible is exactly what people in the 16th century saw themselves as doing, using pictures to represent the intangible,” Landy suggests. “If you go back to the history, it’s what they saw themselves as doing. The deep insight was already there. I see what we are doing as taking another small step in the process.”
“My dream,” he adds, “is that something in this ballpark is a better way to manipulate mathematical symbols. At some point we won’t want to use paper for math any more than we would want to use a sand board, the 12th-century predecessor to paper.”
