Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge

Abstract

Summary Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem‐solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation‐solving ability. An aptitude‐by‐treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.

Advances in designing instruction based on examples

“…As represented in the first line of Table 2, one of the first and still most popular issues concerns whether it is better to ask learners to view a demonstration of how to carry out a procedure through a worked‐out example or to ask learners to carry out a procedure through actually solving an exercise problem. In the present volume, there is evidence that observing the step‐by‐step solution of an example problem is more effective than solving an example problem (Barbieri & Booth, 2020; van Harsel, Hoogerheide, Verkoeijen, & van Gog, 2020) which is consistent with the larger literature on instruction based on examples (Renkl, 2014, 2017). Specifically, van Harsel et al (2020) found that college students performed better on a posttest involving solving calculus problems if they had learned by observing calculus problems being solved rather than being asked to solve the problems on their own, yielding a large effect size.…”

“…Jaeger, Marzano, and Shipley (2020) reported a small effect for college students solving spatial problems when prompts were added to explain erroneous examples. Barbieri and Booth (2020) also found a small effect on middle school students' solving and analyzing algebra equations when they observed erroneous examples rather than solving the same problems. It appears that more work is needed on how best to incorporate erroneous examples in example‐based instruction.…”

“…Specifically, van Harsel et al (2020) found that college students performed better on a posttest involving solving calculus problems if they had learned by observing calculus problems being solved rather than being asked to solve the problems on their own, yielding a large effect size. In another study, Barbieri and Booth (2020) found that middle school students performed better on a posttest involving solving algebra equations if they had learned by observing algebra problems being solved rather than being asked to solve problems on their own, yielding a small effect size. Overall, there appears to be support for asking students to observe demonstrations of how to solve mathematics problems rather than asking them to solve them on their own.…”

Embracing complexity in research on learning from examples and from problem solving

“…Renkl, 2014; van Gog, Rummel, & Renkl, 2019). The studies reported by Van Harsel, Hoogerheide, Verkoeijen, and van Gog (2020) and Barbieri and Booth (2020) attest to this. Jaeger, Marzano, and Shipley (2020) seem to obtain contradictory findings by showing that sketching, which they consider equivalent to problem solving, is equally effective compared with learning from (erroneous) examples, but I will turn to this finding later arguing that there is in fact no contradiction.…”

“…To conclude, sometimes in our research, we may induce certain ways of processing example information without intending so. For instance, Barbieri and Booth (2020)—without addressing this aspect in their research—choose an example design, where solutions are introduced as coming from a fictitious student, which may have positive (Bandura, 1986; cf. social agency theory, Mayer, Sobko, & Mautone, 2003) or negative effects (Riggs et al, 2015) on the way students process the examples.…”