Can Erroneous Examples Help Middle-School Students Learn Decimals?

Abstract

This paper reports on a study of learning with erroneous examples, mathematical problems presented to students in which one or more of the steps are incorrect. It is hypothesized that such examples can deepen student understanding of mathematics content, yet very few empirical studies have tested this in classrooms settings. In a classroom study, 255 6th, 7th, and 8th graders learned about decimals using a web-based system under one of three conditions – erroneous examples, worked examples, and partially-supported problem solving. Although students’ performance improved significantly from pretest to posttest the learning effect for erroneous examples was not better than the other conditions, and unlike some earlier empirical work, the higher prior knowledge students did not benefit more from erroneous examples than from worked examples or problem solving. On the other hand, we were able to identify certain key decimal misconceptions that are held by a high percentage of students, confirming earlier mathematics education studies. Also, the incidence of misconceptions declined over the course of the lesson, especially for the worked example group. Overall, these results could indicate that erroneous examples are simply not as effective for learning as we (and other) researchers hypothesize. The results could also indicate that the manner in which erroneous examples were presented to the students in this study somehow missed the mark in promoting learning. It is also possible that erroneous examples, like some other e-learning techniques, do not work as well in classroom as they do in a laboratory setting. We discuss these possibilities and how we are redesigning the treatments to be more focused and appealing to learners for a subsequent study.