Engaging with learners’ errors when teaching mathematics

Ingrid Sapire,Yael Shalem,Ronél Paulsen,Bronwen Wilson-Thompson

doi:10.4102/pythagoras.v37i1.331

Ingrid Sapire, Yael Shalem + Show 2 more

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https://doi.org/10.4102/pythagoras.v37i1.331

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Abstract

Teachers come across errors not only in tests but also in their mathematics classrooms virtually every day. When they respond to learners’ errors in their classrooms, during or after teaching, teachers are actively carrying out formative assessment. In South Africa the Annual National Assessment, a written test under the auspices of the Department of Basic Education, requires that teachers use learner data diagnostically. This places a new and complex cognitive demand on teachers’ pedagogical content knowledge. We argue that teachers’ involvement in, and application of, error analysis is an integral aspect of teacher knowledge. The Data Informed Practice Improvement Project was one of the first attempts in South Africa to include teachers in a systematic process of interpretation of learners’ performance data. In this article we analyse video data of teachers’ engagement with errors during interactions with learners in their classrooms and in one-on-one interviews with learners (17 lessons and 13 interviews). The schema of teachers’ knowledge of error analysis and the complexity of its application are discussed in relation to Ball’s domains of knowledge and Hugo’s explanation of the relation between cognitive and pedagogical loads. The analysis suggests that diagnostic assessment requires teachers to focus their attention on the germane load of the task and this in turn requires awareness of error and the use of specific probing questions in relation to learners’ diagnostic reasoning. Quantitative and qualitative data findings show the difficulty of this activity. For the 62 teachers who took part in this project, the demands made by diagnostic assessment exceeded their capacity, resulting in many instances (mainly in the classroom) where teachers ignored learners’ errors or dealt with them partially.

Highlights

Teachers come across errors in the mathematics classroom virtually every day
In answering the research questions, the evidence presented below suggests, firstly, that the teachers shied away from engaging with learners’ errors during teaching and, secondly, that when they did engage with learners’ errors, both during the teaching and interviews, they were not always successful in coming to grips with the nature of the error nor did they enable learners to clarify their own thinking and develop a deeper understanding of the mathematical concepts underpinning the error. In other words they did not increase the germane load of the learners. We argue that this is evidence of teachers not coping with the cognitive load they face when errors arise in discussions, which impacted on the way they managed their pedagogical load
The above findings point to the cognitive difficulty involved in applying error analysis, when teachers engage with learners, more so in teaching than in interviews

Summary

Introduction

Teachers come across errors in the mathematics classroom virtually every day When they respond to learners’ errors in their classrooms, during or after teaching, teachers are actively carrying out formative assessment (Black & Wiliam, 2006). Responding to learners’ errors is a specialised activity of formative assessment, which relies on teachers’ deep knowledge of content, and requires teacher’s professional judgement on how to respond to learners’ needs when teaching that content. 2) and develop better lessons on the basis of these interpretations This requirement implies that teachers are expected to use learner data diagnostically, which places a new and complex cognitive demand on teachers’ pedagogical content knowledge (for example, deciding on what to focus, and how, and what to leave out or postpone)

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