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Research on mathematical equivalence, and to some extent, on number structure in early grade mathematics, has shown that most students at primary and secondary levels understand these concepts instrumentally rather than relationally. The study reported in this paper uses a design research approach to explore both mathematical equivalence and number structure in a primary pre-service teacher education programme at a university in South Africa and reports on an intervention strategy intended to culminate in developing design principles comprising guidelines for the effective teaching of number structure and mathematics equivalence in primary pre-service teacher education programmes. The mixed method study comprises 70 first-year primary pre-service teachers (PSTs) and a teacher educator. The PSTs wrote pre- and post-tests, and were interviewed following the pre-test. Skemp's notion of relational and instrumental understanding guided the design of the intervention and the analysis of the data. The results from the study indicates that PSTs' understanding of the notion of equivalence is similar to what has been reported for students in schools, and that there were some learning gains in the post-test results. Based on the findings from the study in general, some emergent design principles are proposed. [ABSTRACT FROM AUTHOR]

Annual National Assessment (ANA) performance in Mathematics across the primary grades in South Africa indicates a decrease in mean performance across Grades 1-6. In this paper, we explore the apparently high performance in Grade 1 through a comparative investigation of learner responses on two assessments: the Grade 1 ANA taken in February 2011 by Grade 2 learners and a diagnostic oral interview test drawn from the work of Wright et al. (2006), administered at the same time. Our findings point to a predominant pattern of high performance on the ANA and low performance on Wright et al.'s tests. In-depth analysis of the responses of two learners in this group indicates that this discrepancy is due to acceptance in the ANA of correct answers produced through highly rudimentary counting strategies. The diagnostic test, in contrast, awards lower marks when correct answers are produced in inefficient ways. We conclude with concerns that acceptance of low-level counting strategies in the ANA may well work against persuading Grade 1 and 2 teachers to work towards more sophisticated strategies. (Contains 4 tables.)

Annual National Assessment (ANA) performance in Mathematics across the primary grades in South Africa indicates a decrease in mean performance across Grades 1--6. In this paper, we explore the apparently high performance in Grade 1 through a comparative investigation of learner responses on two assessments: the Grade 1 ANA taken in February 2011 by Grade 2 learners and a diagnostic oral interview test drawn from the work of Wright et al. (2006), administered at the same time. Our findings point to a predominant pattern of high performance on the ANA and low performance on Wright et al.'s tests. In-depth analysis of the responses of two learners in this group indicates that this discrepancy is due to acceptance in the ANA of correct answers produced through highly rudimentary counting strategies. The diagnostic test, in contrast, awards lower marks when correct answers are produced in inefficient ways. We conclude with concerns that acceptance of low-level counting strategies in the ANA may well work against persuading Grade 1 and 2 teachers to work towards more sophisticated strategies. [ABSTRACT FROM AUTHOR]