Learning Algebra by Example in Real-World Classrooms

Abstract

Math and science textbook chapters invariably supply students with sets of problems to solve, but this widely used approach is not optimal for learning; instead, more effective learning can be achieved when many problems to solve are replaced with correct and incorrect worked examples for students to study and explain. In the present study, the worked example approach is implemented and rigorously tested in the natural context of a functioning course. In Experiment 1, a randomized controlled study in ethnically diverse Algebra classrooms demonstrates that embedded worked examples can improve student achievement. In Experiment 2, a larger randomized controlled study demonstrated that improvement in posttest scores as a result of the assignments varies based on students’ prior knowledge; students with low prior knowledge tend to improve more than higher knowledge peers.

Keywords:

• Worked examples
• prior knowledge
• algebra learning

ACKNOWLEDGMENTS

Preliminary versions of this work were presented at meetings of the Eastern Psychological Association and the Cognitive Development Society. Thanks are due to the Arlington Public Schools, the Madison Metropolitan School District, the Evanston-Skokie School District, the Evanston Township School District, and the Shaker Heights School District for allowing us to collect data in their classrooms.

Notes

Chi-squared analyses revealed no significant differences in demographics between the students who did not take the posttest and the rest of the sample. However, there was a trend toward a difference between pretest scores (t[54] = −1.72, p =.09) with excluded students (M =.7857) demonstrating higher prior knowledge than the rest of the sample (M =.6849). Thus, the final sample may have lower prior knowledge than the average for the population.

Overall attrition was 8%; differential attrition was 8% (control) – 6% (experimental) = 2%, which is lower than the conservative level required for maintaining minimal attrition (What Works Clearinghouse, 2014).

There was no difference in standardized pretest scores for students who did (M = −.006) and did not (M =.08) complete the posttest (t[423] =.483, p =.63). However, Chi-squared analyses revealed that a higher proportion of excluded students than expected were classified as low-SES, χ² (N = 425) = 4.00, p =.05. The distribution of excluded students also differed across classroom, with a higher proportion than expected coming from three of the classrooms, χ² (N = 425) = 257.93, p <.001. This may suggest that the teachers in those classrooms were less flexible with posttest administration and may not have allowed absent students to make it up on another day.

Standardized test scores were not collected for the study. However, one of the participating schools provided the students’ normed percentile scores from the math portion of their most recent EXPLORE test (ACT, 2014). Individual students’ normed math scores correlated positively with their scores on the linear equations pretest (r[64] =.25, p =.05), indicating concurrent validity for the linear equations test.

Figure 3 Sample content assessment items for Experiment 2.