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 Volume 37 Number 1, 2016
Pythagoras  Volume 37 Number 1, January 2016
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Volume 37 Number 1, January 2016

The concepts of area and perimeter : insights and misconceptions of Grade 10 learners : original research
Author France M. MachabaThis article focuses on learners' understanding and their descriptions of the concepts of area and perimeter, how learners solve problems involving area and perimeter and the relationship between them and misconceptions, and the causes of these misconceptions as revealed by learners when solving these problems. A written test was administered to 30 learners and clinical interviews were conducted with three of these learners, selected based on their responses in the test. This article shows that learners lack a conceptual understanding of area and they do not know what a perimeter is. Learners also hold misconceptions about the relationship between area and perimeter. It appears that inadequate prior knowledge of area and perimeter is the root cause of these misconceptions. This article provides suggestions on how to deal with the concepts of area and perimeter.

Investigating the treatment of missing data in an Olympiadtype test  the case of the selection validity in the South African Mathematics Olympiad : original research
Authors: Caroline Long, Vanessa Scherman, Johann Engelbrecht and Tim DunneThe purpose of the South African Mathematics Olympiad is to generate interest in mathematics and to identify the most talented mathematical minds. Our focus is on how the handling of missing data affects the selection of the 'best' contestants. Two approaches handling missing data, applying the Rasch model, are described. The issue of guessing is investigated through a tailored analysis. We present two microanalyses to illustate how missing data may impact selection; the first investigates groups of contestants that may miss selection under particular conditions; the second focuses on two contestants each of whom answer 14 items correctly. This comparison raises questions about the proportion of correct to incorrect answers. Recommendations are made for future scoring of the test, which include reconsideration of negative marking and weighting as well as considering the inclusion of 150 or 200 contestants as opposed to 100 contestants for participation in the final round.

Engaging with learners' errors when teaching mathematics : original research
Authors: Ingrid Sapire, Yael Shalem, Bronwen WilsonThompson and Ronel PaulsenTeachers 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 oneonone 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.

Learners' errors in secondary algebra : insights from tracking a cohort from Grade 9 to Grade 11 on a diagnostic algebra test : original research
Authors: Craig Pournara, Yvonne Sanders, Jill Adler and Jeremy HodgenIt 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 wellknown 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.

Where is the bigger picture in the teaching and learning of mathematics?
Authors: Satsope Maoto, Kwena Masha and Kgaladi MaphuthaThis article presents an interpretive analysis of three different mathematics teaching cases to establish where the bigger picture should lie in the teaching and learning of mathematics. We use preexisting data collected through preobservation and postobservation interviews and passive classroom observation undertaken by the third author in two different Grade 11 classes taught by two different teachers at one high school. Another set of data was collected through participant observation of the second author’s Year 2 University class. We analyse the presence or absence of the bigger picture, especially, in the teachers’ questioning strategies and their approach to content, guided by Tall’s framework of three worlds of mathematics, namely the ‘conceptualembodied’ world, the ‘proceptualsymbolic’ world and the ‘axiomaticformal’ world. Within this broad framework we acknowledge Pirie and Kieren’s notion of folding back towards the attainment of an axiomaticformal world. We argue that the teaching and learning of mathematics should remain anchored in the bigger picture and, in that way, mathematics is meaningful, accessible, expandable and transferable.

A square in drag as concrete universal; or, Hegel as a Sketchpad programmer
Author Zain DavisIn this article, by way of an analysis of a case of mathematics teacher training, I explore the general idea of pedagogic expectation of an alignment of pedagogic identities and specific realisations of mathematics in pedagogic contexts. The particular case analysed has a constructivist orientation, but the analytic resources brought to bear in the analysis can be used more generally for the description and analysis of pedagogic situations. The analysis is framed chiefly by the philosophical work of Georg Hegel alongside Basil Bernstein’s sociological discussion of evaluation in pedagogic contexts. The argument proceeds in three interrelated parts, the first of which produces an analytic description of the discursive production of the desired pedagogic subject in this case, a teacher/student of geometry in which I show how explication and abbreviation are used discursively in an attempt to construct the desired teacher/student that is, a particular pedagogic identity. The second part of the argument describes the discursive production of mathematics content in a manner intended to align content with the desired teacher/student and introduces the notion of a regulative orientation in order to grasp the differences in the mathematical work of students. The third part is a synthesis of parts one and two, showing how pedagogic identity and mathematics contents are brought together as correlative effects of each other.

Proportional reasoning ability of school leavers aspiring to higher education in South Africa
Authors: Frith Vera and Pam LloydThe ability to reason about numbers in relative terms is essential for quantitative literacy, which is necessary for studying academic disciplines and for critical citizenship. However, the ability to reason with proportions is known to be difficult to learn and to take a long time to develop. To determine how well higher education applicants can reason with proportions, questions requiring proportional reasoning were included in one version of the National Benchmark Test as unscored items. This version of the National Benchmark Test was taken in June 2014 by 5 444 learners countrywide who were intending to apply to higher education institutions. The multiple choice questions varied in terms of the structure of the problem, the context in which they were situated and complexity of the numbers, but all involved only positive whole numbers. The percentage of candidates who answered any particular question correctly varied from 25% to 82%. Problem context and structure affected the performance, as expected. In addition, problems in which the answer was presented as a mathematical expression, or as a sentence in which the reasoning about the relative sizes of fractions was explained, were generally found to be the most difficult. The performance on those questions in which the answer was a number or a category (chosen as a result of reasoning about the relative sizes of fractions) was better. These results indicate that in learning about ratio and proportion there should be a focus on reasoning in various contexts and not only on calculating answers algorithmically.