Errors as Learning Opportunities: Cases from Mathematics Teaching Learning

Abstract

The hierarchical and abstract nature of mathematics makes it a perceivably difficult subject to understand. There is also a prevailing focus on the computational aspect of mathematics, learning of procedures and algorithms in mathematics classrooms, which has promoted ‘instrumental learning’ as coined by Skemp in 2006. Errors in mathematics have no significant value and position in the ‘instrumental’ way of learning. They are seen as inconsequential pieces and are treated as extraneous with reference to learners’ mathematical learning. What do teachers do with these errors? They simply turn a blind eye towards them or, at best, tell learners to correct them. These errors are treated as cessation in itself and hence find no place in teachers’ pedagogy and mathematics classrooms. With the longstanding concern of reforming teachers’ pedagogy to involve learners, as active meaning-makers, errors have the potential to uncover a lot of information about the mathematical understanding of learners. The study of these errors can help teachers to understand the mathematical knowledge of learners. An in-depth analysis of these errors can help teachers adopt a dynamic pedagogy which is ‘responsive’ and attuned to the mathematical learning of the learners. This chapter discusses in detail, the attempts of pre-service teachers to use ‘error analysis’ to get insights into the mathematical understanding of learners. It also presents the attempts of pre-service teachers to focus on the process of understanding and learning mathematics. The position of the learners as the co-constructors of mathematical knowledge, along with the pre-service teachers, is also discussed in this chapter. The Newman Error Analysis (NEA) framework forms the theoretical background of this study and is discussed in detail in this chapter. The sequential stages of NEA are explained with examples. This framework was used by the pre-service teachers for identification and categorization of the errors. Vignettes are presented to give first-hand accounts of the whole process of interaction with the learners by these pre-service teachers. They also worked on the errors made by the learners and planned follow-up tasks and activities to help them resolve these errors. This chapter also reports the experiences of these pre-service teachers in using the NEA and making it a subliminal part of their pedagogy. This chapter concludes with the implications of these pre-service teachers using error analysis, with specific reference to NEA, and shows how errors of learners can become learning opportunities.