As teachers dig into implementing new standards in mathematics, a federal research project is working to make their curricular materials more effective at getting content across.
A five-year, $10 million project by the National Research and Development Center on Cognition and Mathematics Instruction is field-testing how to apply cognitive-science findings on how students learn. In randomized controlled experiments in 17 states, researchers are testing changes to one of the most popular middle school math-textbook series, Pearson’s Connected Mathematics.
Teams of researchers from WestEd, Carnegie Mellon University, Temple University, the University of Illinois at Chicago, the University of Wisconsin-Madison, and Worcester Polytechnic Institute are testing potential revisions to the Connected Mathematics curriculum materials for grades 6 to 8, based on the federal Institute of Education Sciences’ practice guide, “Organizing Instruction and Study to Improve Student Learning.”
Research teams are testing better text and illustration integration, use of sample problems, and homework and practice pacing. They are also looking at how well teachers are trained to implement the curriculum.
Researchers used eye-tracking software to monitor middle school students as they read original pages from a popular math text and pages that have been revised to reflect new research on how children learn.
The length of time that someone’s gaze pauses on a spot is related to the “cognitive load”—the mental difficulty in processing the information. For students with low and high background knowledge in math before the study, researchers found the revised illustrations reduced the mental effort needed to read the material, as measured by how long they looked at the text. But the cognitive load increased for students in the middle.
SOURCE: University of Wisconsin-Madison
Though full reports are not expected until next year, researchers presented some early findings at the American Educational Research Association conference here last month.
“This isn’t just change the materials, change the world,” said James W. Pellegrino, a co-director of the Learning Sciences Research Institute at the University of Illinois at Chicago and one of the researchers in the Math Center project. “How is the curriculum structured? What is being practiced and assessed?”
In one study, Mr. Pellegrino and his colleagues looked at how often students practiced needed skills, and what supports they had in doing so.
The researchers analyzed 150 skills and 70 relationships between concepts that map onto the Common Core State Standards in mathematics. They found the curricular materials asked students to practice the skills more than 8,200 times, and tested 105 skills 57 times.
For many of those skills, there was a long time between when students were introduced to and asked to practice skills through homework and when they were actually tested on them. “It turns out that a lot of things kids are expected to have learned earlier, they don’t retain,” Mr. Pellegrino said. “The effects of practice on prior skills were substantial—students who practiced and remastered relevant prior skills scored higher on the post-test for new skills."Moreover, for most of the skills, Mr. Pellegrino and his colleagues found there was little review of homework or class practice sessions to correct students’ misconceptions or errors.
As a result, “kids do homework, and they are practicing, but often they are practicing mistakes,” Mr. Pellegrino said.
His team is working to identify the best pace for practice, concept review, and testing to help students master concepts before moving on.
Learning by Example
Another pilot tested how students use the worked-example problems—samples in which each step is fully or partially completed that commonly appear at the start of the homework and chapter-review sections of math textbooks.
Carnegie-Mellon University researchers Eliane S. Wiese and Kenneth R. Koedinger and their colleagues asked 10 middle school students to think aloud as they worked through 25 math-problem sequences. Each sequence included one regular problem, one full or partially worked example, and then one additional problem that would require the same strategy as the worked example.
In theory, a student reads each math problem and notices the example, then reads each step of the work and uses the steps to develop a strategy to tackle other, similar problems.
In practice, it was a different story. In eight sequences out of 25, the student ignored the example problem entirely. Students in only 14 sequences read the example all the way through, and none of them developed strategies to solve related problems without being specifically prompted to do so.
“A lot of worked-example studies are done at the college level, where students have more metacognitive strategies,” Mr. Koedinger said, but middle school students may not be able to use resources in a textbook without explicit guidance. “We need to help students with not only the content, but with acquiring better learning strategies.”
For both teachers and curriculum publishers, Ms. Wiese said, it can help to use examples that are only partially completed, or that ask students to explain why each step was made, to encourage them to think more deeply about the strategies being used. In the study, students explained steps when explicitly asked to do so in 15 out of 16 problems, but they did not explain steps without prompting.
It’s important, Ms. Wiese said, to “make sure those prompts point students to the core of what the worked example is intended to illustrate and not assume students will intuitively know what to draw from it.”
Focus or Distraction?
In another study, researchers at the University of Wisconsin-Madison found that changing how text and illustrations were paired could improve students’ focus and reduce the mental burden of learning a new math concept.
Most math series, including Connected Mathematics, include pictures or charts to illustrate problems or simply to liven up a layout. The UW team revised a unit that removed any irrelevant illustrations—such as a girl on a bicycle next to a word problem on distance—and used color-coding, labels, and boxes to highlight visuals relevant to the math lesson. Both the original and revised lessons used the same text.
Researchers used eye-tracking software to monitor 57 6th and 7th graders as they read the original or revised lesson one page at a time. The length of time that someone’s gaze pauses on a particular spot is related to the “cognitive load"—the mental difficulty in processing the information.
For students with low and high background knowledge in math before the study, researchers found the revised illustrations reduced the mental effort needed to read the material, as measured by how long they looked at the lesson pages. No improvement occurred, however, for students with a medium level of background knowledge.
Overall, the students did not perform significantly better on a post-test after using the revised lesson, versus the original material. But students who showed lower levels of cognitive load performed better.
“One explanation is that revisions intended to guide learning may add redundant information that is helpful for lower-knowledge students,” Mitchell J. Nathan, an educational psychology professor and a co-author of the study.
