Martin Hall
In an article that was published a few years ago in the Annual Review of Psychology, Professor Janet Metcalfe argued that learning from errors has specific and significant benefits in education:
Considerable research now indicates that engagement with errors fosters the secondary benefits of deep discussion of thought processes and exploratory active learning and that the view that the commission of errors hurts learning of the correct response is incorrect. Indeed, many tightly controlled experimental investigations have now shown that, in comparison with error-free study, the generation of errors, as long as it is followed by corrective feedback, results in better memory for the correct response.
This is supported by comparisons between teaching methods in Japan and the United States. In the US tradition of teaching Mathematics in schools, set procedures are followed for teaching specific categories of problems, with an emphasis on avoiding mistakes. In contrast, rather than starting with an account of the correct approach, teachers in Japan first require their students to attempt the problem on their own. Inevitably, learners get into difficulties and their errors become the focus of the lesson:
The time spent struggling on their own to work out a solution is considered a crucial part of the learning process, as is the discussion with the class when it reconvenes to share the methods, to describe the difficulties and pitfalls as well as the insights, and to provide feedback on the principles at stake as well as the solutions.
This is significant because, year-on-year, schools in Japan achieve significantly better outcomes in Mathematics than school in the United States.
More recently Aarifah Gardee and Karin Brodie, Wits University School of Education, have researched the benefits of learning from errors in the context of teaching Mathematics in South African schools. Their perspective comes from theories of knowledge construction and identity development, augmenting the conclusions that Metcalfe derives from experimental psychology.
Gardee and Brodie start from the position that making errors is normal; what matters is the ways in which teachers work with learners’ mistakes. For some teachers, avoiding errors is taken as an indicator of ability and intelligence while making errors is a source of shame, resulting in anxiety and self-doubt. Other teachers see opportunities for learning from errors, introducing the possibility of building learners’ self-confidence – the approach in Japan. This direction of research has particular importance in the context of South Africa’s marked educational inequities and the achievement gap in educational outcomes.
For this research, Aarifah Gardee spent two years following teachers in a South African school who taught Mathematics to Grade 9 and 10 classes. She filmed lessons and interviewed both students and teachers, resulting is a granular and detailed analysis that complements the general patterns emerging from transnational enquiries such as the TIMMS reports.
One of the specific lines of enquiry in this research was the ways in which teachers’ approaches to errors influenced learners’ “mathematical identities” : the embodied and reflexive sense of self that is shaped by interests, actions and a person’s social context. The hypothesis was that an individual learner’s mathematical identity would determine the extent that they learner would embrace the teacher’s objectives, formally set out in the curriculum. Mathematical identities ranged from a strong affiliation with the “classroom community” (“affiliating learners”) through to a self-perception of being on the margins of conformity.
Affiliating learners exercise agency by participating as full members of the community and by utilizing the tools and resources of the mathematics classroom community to develop their social identities as full participants. These learners tend to be emotionally invested in learning mathematics for their lives and futures, as indicative of their personal identities. Concomitantly, learners who develop their identities in marginalization, when offered social identities of marginalization by teachers, may not be emotionally invested in or motivated to learn mathematics for their lives and futures, as indicative of their personal identities. These learners exercise agency by engaging with the tools and resources of the mathematics classroom community in limited ways, or not at all, to develop their social identities as marginal members.
The two teachers who took part in this study took different approaches in their classes.
One offered lengthy explanations and multiple examples. For him “errors had negative connotations, being able to ‘grow’ and ‘breed’. He organized his lessons by limiting learner opportunities to explore and devise their own methods and solutions to problems to avoid the occurrence of errors, and usually corrected errors when they occurred using lengthy explanations”.
The other teacher preferred group discussions in which learners devised their own solutions:
Errors were discussed positively as lessons, and he used errors as a means of encouraging learners to learn from their mistakes. He structured his lessons to support learner reasoning and he often probed learner errors to access their reasoning before correcting errors.
As is so often the case in this kind of close qualitative research, the patterns that emerged were far from straightforward. Some learners liked the first teacher’s approach, identifying with the implication that making errors is a sign of weakness. Some learners in the second teacher’s class felt uncomfortable with his approach, preferring direct instruction. But whatever their attitude, all learners in the study formed a discernible “mathematical identity” over the two years of the study.
The outcomes of this research open up tantalising questions for future studies. As Aarifah Gardee and Karin Brodie ask in their conclusions, what is the play between socialisation through the authority of the teacher and learners’ agency in determining their own identities? What will be the long-term outcome of the tension between affiliation and marginalisation on each learner’s ongoing journey through the education system, and each of their successes at the assessment gateways ahead of them? This study was carried out at a school that is described as “well resourced”; what would be the advantages of focussing on errors to teach mathematics at a poorly resourced Quintile 1 school in South Africa?
Developing a “mathematical identity” as a marginalised learner in Grade 9 and 10 of High School will have a profound effect on a person’s lifetime prospects. In South Africa, they are unlikely to take Mathematics as a subject in their National Senior Certificate examinations at the end of Grade 12. In turn, further study at college or university will either be closed to them, or their qualification options will be limited. With South Africa’s persistently high levels of unemployment, there is a high probability that a marginalised mathematical identity, however it formed, will evolve into economic marginalisation.
This is why theoretically informed and painstakingly detailed studies such as these, which have the potential to improve teaching and learning, are so important.
Gardie, A. (2019). “Social Relationships between Teachers and Learners, Learners’ Mathematical Identities and Equity.” African Journal of Research in Mathematics, Science and Technology Education 23(2): 233-243.
Gardie, A. and K. Brodie (2022). “Relationships Between Teachers’ Interactions with Learner Errors and Learners’ Mathematical Identities.” International Journal of Science and Mathematics Education 20(1): 193-214.
Metcalfe, J. (2017). “Learning from errors.” Annual Review of Psychology 68: 465-489.
Mullis, I. V. S., M. O. Martin, P. Foy, D. L. Kelly and B. Fishbein (2020). TIMSS 2019 International Results in Mathematics and Science. Boston, TIMMS & PIRLS International Study Center, Lynch School of Education, Boston College: 606.