Errorful Learning as Supports for Mathematics Learning & Motivation

• In Event: New Frameworks, Methods, and Teaching Strategies for Understanding and Addressing Emotions in Mathematics Classrooms

Abstract

Perspectives

“Mathematically proficient students are also able to…distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. (SMP 3).” This thinking can be supported in a variety of ways from turn-and-talks (e.g., “Do you agree with your partner’s thinking? Why or why not?”) or sentence starters to scaffold critique (e.g., “I disagree because…”). One useful approach is errorful learning. Students can study incorrect student solutions and identify and explain the mistake. These tasks can foster critical thinking and deepen procedural and conceptual understanding of mathematics (Barbieri & Booth, 2020).

Objective

The goal of this proposal is to provide recommendations on how best to leverage errorful learning approaches to support students’ mathematical learning and positive mathematics attitudes.
Methods and Data Sources. We rely on the growing body of literature illustrating the benefits of errorful learning practices on mathematical understanding and on motivation for and attitudes towards mathematics as well as perspectives shared by middle and high school teachers to provide recommendations on leveraging errors to support learning and motivation for mathematics.

Results

Errorful learning strategies can support math learning and positive math attitudes. Here, we highlight one errorful learning strategy: using incorrect worked examples. Incorrect worked examples directly target errors that reflect students’ mathematical misconceptions. Though incorrect worked examples have been studied for almost 20 years (e.g., Grobe & Renkl, 2007), many practitioners still hesitate to use these effective tools. Previously expressed concerns include: 1) “my students may start making the error they are being asked to study”, 2) “learning what NOT to do doesn’t really help my students learn what TO DO”, and 3) “My students may feel badly about themselves if they make the same error shown”.

However, recent work shows that studying incorrect examples can decrease errors students make (Barbieri et al., 2019; Barbieri & Silla, 2022). Further, increases in negative knowledge (Barbieri et al., 2021) or in knowledge about what doesn’t work in problem-solving, supports learning. Directly addressing the error through studying incorrect worked examples can weaken misconceptions, making it easier for students to learn new correct knowledge. Fostering a positive error climate in which errors are useful tools for learning supports students’ persistence in challenging mathematical tasks and positive achievement motivation (Steuer et al., 2013). We review this work in conjunction with recent reflections shared from secondary mathematics teachers on incorporating errorful learning into the classroom.

Significance

Incorrect examples are particularly beneficial for students with low prior knowledge (Barbieri & Booth, 2016, 2020; Barbieri et al., 2021), suggesting students most likely to make the errors displayed may reap the greatest benefits. Further, when students do not feel judged for making mistakes or having incomplete knowledge, they report a strong sense of belonging in the math class and identify as a math person (e.g., Joseph et al. 2019). These findings provide practical and actionable recommendations for teachers as well as considerations for theories of learning and motivation.

Authors

• Christina Areizaga Barbieri, University of Delaware

• Elena Silla, University of Delaware