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Developing structure sense is an important part of learning algebra. We investigated learners’ structure sense of algebraic expressions involving brackets. This led us to propose the constructs surface structure sense and systemic structure sense. Using a random sample of 58 Grade 10 learners scoring above 40% in a test, we coded incorrect responses for surface and systemic structure errors. The initial analysis revealed that the presence of more brackets supports surface structure sense. However, learners overgeneralised the presence of brackets to represent multiplication in situations involving subtraction. The arrangement of brackets also led to errors in the order of operations. Further analysis suggested that regular application of procedures on familiar algebraic structures may give the illusion of systemic structure sense. We recommend that the teaching of algebraic expressions must emphasise what the arrangement of an expression means before focusing on how to operate on the expression. Contribution: The research contributes to mathematics teaching by suggesting teaching strategies to improve on learners’ understanding of the role of brackets in algebraic expressions, by considering the arrangement of the components in structures.

Background: Poor mathematics performance in South Africa is well known. The COVID-19 pandemic was expected to exacerbate the situation. Aim: To investigate Grade 7 learners’ mathematical knowledge at the end of primary school and to compare mathematical performance of Grade 7 and 8 learners in the context of the pandemic. Setting: Data were collected in term four of 2020 at 11 primary schools and five secondary schools. All schools drew learners from poor communities in Gauteng. Methods: A multiple-choice test covering mathematical content from Grades 4–7 was designed and piloted. Learner performance was measured through number of correct responses. Qualitative error analyses were conducted on learners’ choices of distractors. Results: The difference in performance of the two grade groups was not statistically significant. There were similar response patterns in learners’ choices of distractors with strong evidence of cue-based reasoning and evidence of additive reasoning in items requiring multiplicative reasoning. Conclusion: Grade 8 learners made very small gains, likely due to reduced learning time. Learner errors show many similarities with the international literature and show that Grade 7 learners are not yet ready for algebra. Contribution: The findings provide starting points for addressing the most common errors and highlight the need for: greater attention to whole and rational number concepts in Grade 8; strategies for teacher support in teaching primary maths content; and innovative teaching strategies to fast-track learning of this content.

Developing structure sense is an important part of learning algebra. We investigated learners' structure sense of algebraic expressions involving brackets. This led us to propose the constructs "surface structure" sense and "systemic structure" sense. Using a random sample of 58 Grade 10 learners scoring above 40% in a test, we coded incorrect responses for surface and systemic structure errors. The initial analysis revealed that the presence of more brackets supports surface structure sense. However, learners overgeneralised the presence of brackets to represent multiplication in situations involving subtraction. The arrangement of brackets also led to errors in the order of operations. Further analysis suggested that regular application of procedures on familiar algebraic structures may give the illusion of systemic structure sense. We recommend that the teaching of algebraic expressions must emphasise what the arrangement of an expression means before focusing on how to operate on the expression

It is well known that learners’ performance in mathematics in under-resourced secondary schools in South Africa is poor. However, little is known about the performance of learners in top-performing well-resourced secondary schools apart from their good results in Grade 12. In this study, the performance of Grade 9 learners in top-performing quintile 5 (i.e. well-resourced) schools was compared with that of learners in quintile 1–3 (i.e. poorly resourced) schools using a 45-item test. While the quintile 5 learners obtained higher test scores, the performance pattern across the test items was very similar for both quintile groupings. A detailed error analysis of 89 quintile 5 learners’ responses to equation items revealed difficulties in applying the standard equation-solving procedure, and in operating with negatives and subtraction, particularly on like terms. These problems may be related to a poorly conceived curriculum in the areas of integers and equations. It is recommended that Grade 8 and 9 teachers pay regular attention to all four arithmetic operations on integers. Furthermore, the teaching of equations should give greater attention to the cognitive shifts required in solving equations with letters on both sides of the equal sign. Significance: • Learner performance patterns on a test were similar for learners from top-performing quintile 5 schools and learners from lower quintile schools, although the quintile 5 learners obtained higher test scores. • Quintile 5 learners’ ability to solve linear equations correctly is substantially impacted by their difficulties in simplifying two algebraic terms to a single term, particularly when negatives and/or subtraction are involved. • Particular aspects of the curriculum may partly be responsible for the difficulties learners experience with integers and equations.

Background: Poor mathematics performance in South Africa is well known. The COVID-19 pandemic was expected to exacerbate the situation. Aim: To investigate Grade 7 learners' mathematical knowledge at the end of primary school and to compare mathematical performance of Grade 7 and 8 learners in the context of the pandemic. Setting: Data were collected in term four of 2020 at 11 primary schools and five secondary schools. All schools drew learners from poor communities in Gauteng. Methods: A multiple-choice test covering mathematical content from Grades 4-7 was designed and piloted. Learner performance was measured through number of correct responses. Qualitative error analyses were conducted on learners' choices of distractors. Results: The difference in performance of the two grade groups was not statistically significant. There were similar response patterns in learners' choices of distractors with strong evidence of cue-based reasoning and evidence of additive reasoning in items requiring multiplicative reasoning. Conclusion: Grade 8 learners made very small gains, likely due to reduced learning time. Learner errors show many similarities with the international literature and show that Grade 7 learners are not yet ready for algebra. Contribution: The findings provide starting points for addressing the most common errors and highlight the need for: greater attention to whole and rational number concepts in Grade 8; strategies for teacher support in teaching primary maths content; and innovative teaching strategies to fast-track learning of this content.

It is wellknown that the majority of South African learners achieve extremely poorly in Mathematics. Many claim that one of the causes of this poor attainment is teachers' weak knowledge of mathematics, and propose that improving teachers' mathematical knowledge would improve learner attainment. However, the evidence-base for this proposed solution is currently relatively weak. We report on a quasi-experimental study examining the learning gains of Grade 10 learners from five secondary schools in the Johannesburg area, whose teachers participated in a year-long professional development course aimed at improving the teachers' knowledge of mathematics for teaching. Statistical analyses of pre- and post-test results show that the intervention group of learners (N = 586) taught by teachers who had participated in the professional development (N = 14) outperformed a matched control group of learners (N = 217) taught by teachers in the same schools (N = 7). An effect size of d = 0.17 for the intervention group is equivalent to two months' additional progress. While the learning gains are small, they are statistically significant. These findings provide empirical support for claims that attending to teachers' mathematical knowledge can impact learners' attainment. Suggestions are made regarding the form and substance of such professional development.

It is well known that learner performance in mathematics in South Africa is poor. However, less is known about what learners actually do and the extent to which this changes as they move through secondary school mathematics. In this study a cohort of 250 learners was tracked from Grade 9 to Grade 11 to investigate changes in their performance on a diagnostic algebra test drawn from the well-known Concepts in Secondary Maths and Science (CSMS) tests. Although the CSMS tests were initially developed for Year 8 and Year 9 learners in the UK, a Rasch analysis on the Grade 11 results showed that the test performed adequately for older learners in SA. Error analysis revealed that learners make a wide variety of errors even on simple algebra items. Typical errors include conjoining, difficulties with negatives and brackets and a tendency to evaluate expressions rather than leaving them in the required open form. There is substantial evidence of curriculum impact in learners’ responses such as the inappropriate application of the addition law of exponents and the distributive law. Although such errors dissipate in the higher grades, this happens later than expected. While many learner responses do not appear to be sensible initially, interview data reveals that there is frequently an underlying logic related to mathematics that has been previously learned.

Shulman’s notations of subject matter knowledge (SMK) and pedagogical content knowledge (PCK) have been very influential in education research on teachers’ knowledge for teaching. However, there is little empirical evidence in support of these as separate analytical constructs. Furthermore, attempts to distinguish SMK and PCK highlight the complex and multidimensional nature of teachers’ knowledge and hence the difficulty of separating SMK and PCK. The author adopts the notion of mathematics-for-teaching (MfT) and argues that teachers’ knowledge for teaching annuities comprises knowledge of mathematical aspects, knowledge of pedagogical aspects and contextual knowledge of finance. Drawing from a larger study in which the author taught a financial mathematics course to pre-service secondary mathematics teachers, four examples of teachers’ knowledge for teaching annuities are identified, each of which illustrates how knowledge of mathematics, knowledge of pedagogy and contextual knowledge of finance are intertwined.

There is increasing acknowledgement that teachers’ knowledge for teaching mathematics is multifaceted and topic specific. Given the paucity of research on the teaching and learning of financial mathematics in general, little can be known about teachers’ knowledge for teaching compound interest. However, since financial mathematics is a component of the school curriculum in South Africa, and an important element of financial literacy more broadly, attention needs to be given to knowledge for teaching financial mathematics, and compound interest in particular. Drawing from a larger study in which the author taught a financial mathematics course to pre-service secondary mathematics teachers, a theoretical elaboration is provided of the underlying mathematics of compound interest, and connections with the world of banking. Based on findings from the study, two key student errors are identified: the over-generalisation of linear thinking in multiplicative scenarios, and the over-generalisation of reversible operations in percentage-change scenarios. Taken together, teachers’ knowledge of relevant mathematics, of the banking context and of learners’ conceptions will contribute to building a knowledge-base for teachers’ knowledge for teaching compound interest.