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The purpose of this paper is to explore how master mathematics teachers use the concrete-representational-abstract (CRA) sequence of instruction in mathematics classrooms. Data was collected from a convenience sample of six master teachers by observations, video recordings of their teaching, and semi-structured interviews. Data collection also included focus-group interviews with learners. In South Africa, master teachers are considered expert teachers in their discipline. The master teachers in this study were selected by the Department of Education based on their many years of teaching experience; in addition, these selected master teachers taught at six different Dinaledi schools in KwaZulu-Natal (South Africa). A key finding of this research demonstrated that the use of the CRA instructional sequence was paramount for the effective teaching of mathematics. This instructional sequence was found to be predetermined as well as intuitive. The CRA instruction may be used in classrooms where learners are not streamed into ability levels, as is the case with the majority of schools in South Africa. These findings are important for shaping both teacher and curriculum development.
Keywords: Visualisation, representations, activity theory, CRA sequence
Introduction
Success in mathematics is becoming increasingly important for learners since it is essential for a variety of employment opportunities (Witzel, Riccomini & Schneider, 2008). Our many years of experience in the field of mathematics education have convinced us that the teaching of mathematics can be highly frustrating. For example, algebra often requires the manipulation of equations that have little to do with the purpose of the equation; the arbitrary symbols may be manipulated with little regard to the numerals in the original equation (Witzel, 2005). Researchers have shown that the use of the CRA sequence of instruction has been very effective and beneficial to learners who struggle with understanding mathematical concepts and procedures (Flores, 2009; Witzel et al., 2008). Anecdotal evidence and research suggest that learners often claim that mathematics is difficult and abstract. Some teachers claim that learners are no longer prepared to work hard. Furthermore, these teachers maintain that either they do not have the proper resources to teach mathematics adequately or they find learners uncooperative and unprepared. Despite these negative feelings, there are teachers who are able to assist their learners in making meaning of what may sometimes be...