Pythagoras - latest Issue
Volume 37, Issue 1, 2016
The concepts of area and perimeter : insights and misconceptions of Grade 10 learners : original researchAuthor France M. MachabaSource: Pythagoras 37, pp 1 –11 (2016) http://dx.doi.org/http://dx.doi.org/10.4102/pythagoras.v37i1.304More Less
This 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 Olympiad-type test - the case of the selection validity in the South African Mathematics Olympiad : original researchSource: Pythagoras 37, pp 1 –14 (2016) http://dx.doi.org/http://dx.doi.org/10.4102/pythagoras.v37i1.333More Less
The 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.
Source: Pythagoras 37, pp 1 –11 (2016) http://dx.doi.org/http://dx.doi.org/10.4102/pythagoras.v37i1.331More Less
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.
Learners' errors in secondary algebra : insights from tracking a cohort from Grade 9 to Grade 11 on a diagnostic algebra test : original researchSource: Pythagoras 37, pp 1 –10 (2016) http://dx.doi.org/http://dx.doi.org/10.4102/pythagoras.v37i1.334More Less
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.