### If You Give a Student (a Fraction of) a Cookie . . .

Eliane Stampfer Wiese has just received a Ph.D. from Carnegie Mellon University, where her research turned up interesting revelations about how young students learn math—and what adults might get wrong.

As an undergraduate at Columbia University, Eliane Stampfer Wiese thought she might want to become a teacher. The computer science major enrolled in a teacher-training program at Barnard, where she quickly found she was more fascinated by studying how students learn than by teaching them.

“I’d be working with a student trying to explain something and thinking that someone must have looked at the best way to explain this concept,” Wiese said. “Or *has* someone looked at the best way to explain this concept? I found I was drawn to those questions.” [pullquote]“I think teachers would probably be surprised that a lot of the reasoning that adults and instructional designers take for granted is actually very difficult for students.”[/pullquote]

Wiese has just received a Ph.D. from Carnegie Mellon University, where her research turned up interesting revelations about how young students learn math—and what adults might get wrong.

Math learning looks different than it did when Wiese was growing up in the 1990s. Today, there is a stronger emphasis on reasoning and logic—teaching kids why a solution makes sense rather than having them memorize the process. To prepare kids for 21^{st} century learning, math educators want them to be able to tackle complex problems. Common Core math instruction places more emphasis on the conceptual understanding of mathematics than on procedural memorization.

“But actually creating instruction that does that for students in a way that they can understand is really difficult to do—and also interesting,” Wiese said.

Take fractions, the focus of her doctoral research and a common line of inquiry at CMU. Fraction addition, she said, demands a “kind of reasoning that seems very obvious to adults” but is not so clear-cut to kids.

While the steps to fraction addition are not necessarily logical, the doctoral student thought it should at least be clear to the student when an answer was wildly incorrect. Because you are adding two positive sums when you combine fractions, the resulting quantity should clearly be bigger than each original piece. So as long as students know the size of the fractions and the size of the sum, they should be able to determine whether the solution is even possible, right? Wrong, Wiese found out.

Through a series of studies with Pittsburgh-area public school students, Wiese’s research found that while the concept of adding positive magnitudes might make sense to kids when the equation involves whole numbers, the lesson is not always transferable to fractions. On top of that, our traditional graphical representations of fractions (a brightly colored rectangle divided into four pieces, say) do not necessarily make sense to students.

“I think teachers would probably be surprised that a lot of the reasoning that adults and instructional designers take for granted is actually very difficult for students,” Wiese said. She cautions that her research does not provide conclusive advice to educators, but said it speaks to the power of scaffolding.

Her best results came when she broke the process into smaller steps, starting with the graphical representations and adding numerical symbols only after the kids understood the images.

Wiese is still researching learning, only now she is a postdoc at UC Berkeley. She has mostly left her fractions in Pittsburgh, but continues to study visual representations. This time, she is looking at integrating graphs into science instruction.

Wiese said she will always have fond memories of Pittsburgh. For a doctoral student interested in learning science, she said, Pittsburgh was the place to be. CMU’s multidepartmental approach to learning science, specifically through the Program in Interdisciplinary Education Research, led to a rich cross-pollination of ideas between fields. And high-tech research facilities helped her collect data.

Even across the country, she hopes to continue debunking adult assumptions about learning and making the fundamental concepts accessible to kids.

There may be easier ways to offer visual representations. After all, Wiese said, if you tell a fifth grader you’re going to give him two cookies of a certain size, and instead give him one single cookie that isn’t twice as big, he would know there was something wrong with that picture.