# The College School

# Improving Students’ Performance in Mathematics

## CEHD research on the benefits of using worked examples in mathematics used to train The College School teachers

Christina Areizaga Barbieri, University of Delaware assistant professor, Dana Miller-Cotto of Kent State University and UD doctoral students Sarah N. Clerjuste and Kamal Chawla conducted a meta-analysis of 55 experimental studies and found that studying worked examples—or worked-out solutions to math problems—were moderately effective at improving learners’ mathematics performance when compared to more standard practices such as solving practice problems. In “A Meta-analysis of the Worked Examples Effect on Mathematics Performance,” published in *Educational Psychology Review*, the authors found that learners in elementary school through adulthood who study worked examples improve their overall math performance by an average of 18 percentile points compared to those who do not. The authors further identified variations of the worked examples effect, finding that worked problem studies using correct solutions yield larger improvements overall than those using incorrect solutions. They also found that studies that pair worked examples with self-explanation prompts—which ask students to explain a strategy or concept displayed in a worked example—lead to lower student performance compared to those using worked examples without self-explanation prompts. This study thus confirms and extends an influential 2007 Institute of Education Sciences (IES) report that claimed, based on a small sample of studies, that there was moderate evidence showing that interleaving worked examples and problem-solving was effective for learning.

Despite the widespread study of worked examples since their origin in 1985, a meta-analysis of the immense amount of research on this topic was not available prior to this study. In their work to quantify the average effect of worked examples on mathematics performance, Barbieri and her team first reviewed 8,033 abstracts to narrow in on a sample of causal studies with random assignment. This process resulted in a sample of 43 articles (41 experimental, 2 quasi-experimental) reporting on 55 studies and 181 effect sizes (with only four articles published before the seminal 2007 IES report). These studies included varying study environments (e.g., laboratory and classroom studies), varying mathematics performance measures (e.g., percent accuracy on a test, number of errors made during problem-solving) and varying modalities (e.g., paper-and-pencil tasks, computerized tutors). While the sample included students from elementary school through post-secondary education, most articles focused on middle school students in algebra or geometry. Barbieri and her team then used robust variance estimation to account for clustered effect sizes and identify how worked examples broadly affected student performance in mathematics.

In their work to assess what moderated the work examples’ effect on student performance, Barbieri and her team chose to focus on the most common modifications of worked problems. They assessed correct vs. incorrect examples alone and in combination with correct examples as well as the pairing of worked examples with self-explanation prompts. In addition, the team assessed the timing of administration, noting whether the worked examples were given to help a student acquire new knowledge or practice previously learned skills. In addition to the results noted above, Barbieri and her team found that worked examples yielded a medium effect on mathematics outcomes whether they were used for practice or initial skill acquisition.

Given these improvements in math performance, Barbieri and her team encourage instructional leaders and teachers to consider using worked examples in addition to more open-ended and exploratory strategies. While some math curricula have moved away from direct instruction, this study consolidates a large amount of research on worked examples in order to help teachers make informed decisions about the benefits of this strategy.

“I think what’s most notable about our findings is that, despite the immense variations in study design and administration—from specific math content area targeted, to time spent using worked examples, to grade, and even country of study—we still see this very meaningful effect of worked examples on mathematics performance,” Barbieri said. “Many students experience difficulties with learning mathematics. Most of the studies we reviewed were conducted in classroom settings. With consideration to these two points, I am hopeful that our findings can be used to inform further integration of worked examples-based instruction and materials into mathematics classrooms as one tool to support students.”

Read this spotlight on CEHD News.