How Can Students Better Apply Math Learning? New Studies Hold Answers
Even within STEM, transfer is tough
Mathematics is the language of science, the foundation of engineering, the power switch for new technology—but students often struggle to transfer their understanding of math concepts to practical application in other STEM subjects.
Researchers at the Association of Psychological Science conference here last month discussed new findings on ways to help students link their math learning to science, technology, and engineering.
"Looking at the longer history of transfer of knowledge, the research shows that if you have students pull a general concept out of a combination of specific examples and give multiple different examples, it increases transfer of that concept to new examples," said Holly A. Taylor, a psychology professor at Tufts University, in Medford, Mass.
"One reason that STEM concepts are difficult to transfer is because they are siloed. Although I believe that there is change afoot in this regard," she said, because new mathematics and science standards in most states are focused more on underlying processes than on learning just facts.
Words and Numbers
For example, another researcher at the conference, David J. Purpura, an assistant professor in clinical psychology at Purdue University, in West Lafayette, Ind., suggested young students' ability to apply early math skills can be hampered or propelled by their language development, independent of their math knowledge.
Many tests of early numeracy focus on children's "approximate number system," the ability to estimate size differences between two groups without counting.
However, in a study of the early math performance of 114 children ages 3 to 5, Mr. Purpura found that while preschoolers' ability to estimate the number in a group predicted the math achievement of students with low overall math skills, math-language ability was a better predictor of math achievement among middle- and high-performing math students.
Elementary school students in rural Vermont created paper sculptures (shown below) as part of a curriculum aimed at helping them bridge math and engineering concepts.
As part of the program, students in grades 3-6 learned to fold origami and build the paper structures by using diagrams and by reverse engineering from models. A pilot study of that effort was one of three studies on promoting transfer of math knowledge that were presented last month at the Association of Psychological Science’s annual conference in New York.
In the Vermont study, students in most of the grades tested were found to have improved their spatial reasoning and their ability to mentally fold objects. Fourth and 5th graders also improved their performance on a standardized math test.
"You have to have these basic language abilities—words like 'plus' or 'take away'—to do almost any math," Mr. Purpura said.
"Different skills predict early numeracy at different ability levels," Mr. Purpura said. "You might be able to compare sets but not to express that you can compare them. It might be a gateway."
The way a concept is presented can also affect how easily students understand when and how to apply it in other situations.
"The type of practice matters," said Charles W. Kalish, an education psychology professor at the University of Wisconsin- Madison.
"Even for familiar content, even for students who have had a lot of experience with these things, 10 or 15 minutes of practice that encourages them to map the underlying finding can really change the type of memory models that are activated," he added.
In two sets of experiments, first with college students and then with 2nd graders, Mr. Kalish and his colleagues had participants solve math problems focused on differences of ratios and magnitude. In each group, half the participants practiced using standard numbers and symbols. The other half solved problems within a simulation that highlighted an underlying relationship among the numbers.
For example, the 2nd graders were asked to add different levels of blue and yellow flavoring to an ice cream machine to make different shades of ice cream requested by various cartoon monsters.
"You can see there's a continuous underlying structure to these elements," Mr. Kalish explained. "In contrast, if we give you purely symbolic training, you are just learning arbitrary conjunctions of these features."
While both groups of adults and students performed equally well on similar problems, those who had practiced using symbols alone were not as good at applying the concept of changes in magnitude to a new set of problems in a different context.
Those who had practiced within the simulation were equally good at familiar and novel problems.
Better on Paper
Ms. Taylor, the director of the Spatial Cognition Laboratory at Tufts University, agreed. She is studying how elementary students in rural Vermont bridge math and engineering concepts through trial and error efforts with paper models.
The program, Think3d!, consists of six units in which students in grades 3-6 learn to fold origami and build paper structures, both from diagrams and by reverse- engineering from models. In the process, they learn to develop their own algorithms to explore and track how changes in the angle of a fold, for example, or in the number of cuts in a folded paper change the final sculpture.
While the curriculum at first differed by grade, Allyson Hutton, an architect and the president of Think3d!, the public-benefit corporation created to develop the program, said it was changed to the same sequence for all students after 6th graders proved no better than students in lower grades at understanding the directions in diagrams and charts.
"The kids wouldn't make the distinction between a line directing them to fold paper in half to make two rectangles and one showing a fold along the diagonal to make two triangles," she said. "Many did not connect the 2-D diagram to the piece of paper they were holding in their hands."
In a pilot study of the curriculum for grades 3, 4, and 5, Ms. Taylor found that students who took part in the curriculum improved their spatial reasoning and ability to mentally fold objects, compared with a demographically similar control group.
Fourth and 5th graders who went through the program also showed significantly better accuracy on a standardized math test and more frequent use of diagrams to solve problems. The 3rd graders did not show such a benefit—Ms. Taylor said the curriculum seemed to be difficult for them—and Ms. Hutton said she is now overhauling the curriculum for that grade.
"The sequences are designed to be catalysts, so students can just run with it," Ms. Hutton said. "The more time the kids are sitting down folding, mentally manipulating, visualizing, the more they are developing their spatial thinking."
"What we're working to do is train a skill that can be used across disciplines," she added.
Vol. 34, Issue 32, Pages 14-15Published in Print: June 3, 2015, as Studies Probe How Students Can Apply Math More Widely