Students’ Dichotomous Experiences of the Illuminating and Illusionary Nature of Pattern Recognition in Mathematics

Published ArticleThe concept of pattern recognition lies at the heart of numerous deliberations concerned with new mathematics curricula, because it is strongly linked to improved generalised thinking. However none of these discussions has made the deceptive nature of patterns an object of exploration and understanding. Yet there is evidence showing that pattern recognition has both positive and negative effects on learners’ development of concepts. This study investigated how pattern recognition was both illuminating and illusionary for Grade 11 learners as they factorised quadratic trinomials. Psillos’s fourconditions model was used to judge the reasonableness of learners’ generalisations in six selected examples. The results show that pattern recognition was illuminating in the first three examples where learners made use of localised pattern recognition. In one example, pattern recognition was coincidental but not beneficial in terms of conceptual understanding. In the last two examples localised pattern recognition was at the centre of learner confusion as they failed to extend its application beyond the domain of the examples that generated the pattern. Learners’ confusion with pattern recognition could be attributed to teachers’ failure to meet four important conditions for good generalisations. Results from this study confirm earlier studies showing that abduced generalisations developed out of a few localised instances might be illuminating at first but might not provide the best explanation when extended beyond the localised domain. Further studies are needed that assist in developing patternaware teachers

Computer assisted assessment and the role it plays in educational decision-making and educational justice: a case study of one teacher training college in Zimbabwe

A ZJER research on computer - assisted assessments and educational decision-making in the Zimbabwe education system.Although the use of computers in data-driven decision making in education was initially focused on education's core business i.e. computer aided learning (CAL), educational leaders are now using this approach to transform other aspects of their operations e.g. computer-assisted assessment (CAA). The full potential of CAA has yet to be realized and its implementation within higher education can be fraught with difficulties. This paper draws on a research that was carried out in one teachers' college in Zimbabwe. The main aim was to engage with the final grading system used on the teaching practice phase ofa group of600 newly qualified teachers with a view of identifying how the computer was being used to allow humans to benefit from machine decision-making without losing the opportunity for rational thought. This was driven by a sincere conviction that better data-driven decisions in education benefit everyone, including the learners, teachers, administrators, patrons, taxpayers and the state. The researcher employed an approach commonly used in IT, which is called Data Mining. The findings seem to point to a grading system which is using a computer more as a data capture and calculation instrument without questioning the moral argument for letting the computer decide. Such a grading system has potential for loss of human autonomy and for being unfair to the subjects

Lessons for South Africa from Singapore’s gifted education – A comparative study

Since 1999 South African learners have participated in various international studies but sadly the learners have continued to perform dismally, which brings to question the quality of their education. Meanwhile, Singaporean students have been among the top achievers in all these competitions. Many comparative studies have been done between different nations and Singapore, but in few, if any, of these studies the focus has been on comparisons regarding gifted education. Singaporean policies and practices on gifted education generally prioritise a commitment to engaging learners from all ability levels with appropriately challenging curricula and instruction. In this article we report on a comparative study between the Singaporean and South African education systems. Three frames, (a) political context (b) curriculum structure and (c) loose coupling shaped the analysis. Results show that both countries had similar  challenges at the point of independence from colonial rule and yet, they responded differently to those challenges. Singapore  implemented inclusive education driven by excellence while South Africa’s inclusive education is driven by equity without excellence. South Africa has a one-size-fits-all curriculum, whereas Singapore has alternatives that create multiple pathways for learners to reach their full potential. Although gifted education is being proposed in current South African pronouncements, there is no evidence of coherence in terms of its implementation. Meanwhile, Singapore has a coherent system that ensures their policies move from theory  into practice. All these are lessons that South Africa can learn. Keywords: comparison; coupling; frames; gifted education; inclusive education; Singapore; South Afric

Investigating learners’ meta-representational competencies when constructing bar graphs

Published ArticleCurrent views in the teaching and learning of data handling suggest that learners should create graphs of data they collect themselves and not just use textbook data. It is presumed real-world data creates an ideal environment for learners to tap from their pool of stored knowledge and demonstrate their meta-representational competences. Although prior knowledge is acknowledged as a critical resource out of which expertise is constructed, empirical evidence shows that new levels of mathematical thinking do not always build logically and consistently on previous experience. This suggests that researchers should analyse this resource in more detail in order to understand where prior knowledge could be supportive and where it could be problematic in the process of learning. This article analyses Grade 11 learners’meta-representational competences when constructing bar graphs. The basic premise was that by examining the process of graph construction and how learners respond to a variety of stages thereof, it was possible to create a description of a graphical frame or a knowledge representation structure that was stored in the learner’s memory. Errors could then be described and explained in terms of the inadequacies of the frame, that is: ‘Is the learner making good use of the stored prior knowledge?’ A total of 43 learners were observed over a week in a classroom environment whilst they attempted to draw graphs for data they had collected for a mathematics project. Four units of analysis are used to focus on how learners created a frequency table, axes, bars and the overall representativeness of the graph vis-à-vis the data. Results show that learners had an inadequate graphical frame as they drew a graph that had elements of a value bar graph, distribution bar graph and a histogram all representing the same data set. This inability to distinguish between these graphs and the types of data they represent implies that learners were likely to face difficulties with measures of centre and variability which are interpreted differently across these three graphs but are foundational in all statistical thinkin

Theory-practice dichotomy in Mathematics Teacher Education: An analysis of Practicum Supervision Practices at one Teachers’ Training College in Zimbabwe

Published ArticleWhile a symbiotic companionship is expected to exist between the theory courses and practicum experiences, traditional practices in primary teacher education continue to create dichotomous gaps in this relationship. Such practices needed to be investigated as they negatively affect teacher-self efficacy which in turn compromises both teacher quality and learner performance in mathematics. This paper draws from a case study which investigated the extent to which mathematics practicum experiences at one teachers’ college in Zimbabwe were coordinated with methods courses. Structured reflective journals were used as a data collection tool where 47 trainee teachers documented their experiences of the practicum. A total of 226 reports were then analysed using an analytical tool that classified trainees’ supervision experiences as either embedded or separated depending on whether the supervisor was a specialist or not. Results show that in 87.6% of the cases, practicum experiences were separated from the methods courses because they were supervised by non-specialist lecturers. Analysis of verbatim entries shows that teacher-self efficacy was generally negative. A number of recommendations are made to bridge this theory-practice gap

Performance appraisal systems — equity perceptions of mathematics teachers: an exploratory study

Published ArticleThis study investigated equity perceptions of a school based performance appraisal system in a subject discipline. Participants were 110 mathematics teachers (females = 10%) and 12 school principals (females = 25%). They completed a questionnaire on distributive, procedural and interactional justice regarding the performance evaluation process. The School principals and eight of the teachers were qualitatively interviewed to elaborate on their performance appraisal equity perceptions. The quantitative data were descriptively analysed to characterize negative and positive perceptions and the qualitative data were thematically analysed. The findings show that respondents perceived inequity in the distributive, procedural and interactional aspects of the performance based scheme. Equity was marginally more endorsed regarding just one category of interactional justice. The study underscores the importance of perceptions of equity in performance appraisal systems in work organisations.85 815 bytes, 1 fil

Is Rote Learning of Number Concepts ‘Inherently Rotten’ or Is It Just a Blame and Shame Game that Vitiates Principles of Natural Progression?

Published ArticleThis paper questions the validity of the foundational premise of the ‘problem-solving movement’ following a perennial dichotomization of the ‘old’ and ‘new’ math curricula globally. Essentially, the substance of the controversies that places the ‘problem solving movement’ and the ‘back-to-basics movement’ on the opposite ends of the ‘battle-fields’ is the view within the former that rote learning is unimportant or counterproductive as it does not enhance the development of application skills of critical thinking, logical analysis, and creative problem-solving. The paper then uses Tall’s framework to show how counting is foundational to number concept formation and how rote learning is a necessary first step in this counting process. I then argue that the suggestion that critical thinking could be developed without the lower level concepts is possibly founded on a flawed premise. Globally, there is a current wave of back-to-basics across mathematics curricula. However it would appear in the South African curricula for Foundation Phase Grade R – 3, rote learning is still not valued despite going back-to-basics. This suggests a continued dichotomization of the old and new approaches which does not seem to be beneficial to learners who tend to end up with neither the problem solving skills nor the basic foundational knowledge. The paper recommends that we develop teaching methodology for mathematics and other subjects that incorporates rote learning in an effective way so that knowledge is better conveyed and represented in the minds of students

Regular classroom teachers’ recognition and support of the creative potential of mildly gifted mathematics learners

Published ArticlePost independent reforms in South Africa moved from separate education for the gifted learners to inclusive education in regular classrooms. A specific concern that has been totally ignored since then is whether or not the regular classroom would expand or limit the gifted child’s creativity. This study aimed at investigating the extent to which South African mathematics teachers recognised and supported the development of gifted students’ creative potential. Four teachers were each observed teaching over a week and the analysis focused on their representational fluency and how they responded to gifted students’ creative ideas. The results show that in 70% of the episodes teachers’ representations were either mathematically faulty or correct but with no further justification or explanation. In 63% of the micromoments students’ creative ideas were considered disruptive and were therefore not recognized. These results suggest that currently regular classrooms in South Africa might not be conducive to the development of the gifted students’ creative potential

From coherence in theory to coherence in practice : a stock-take of the written, tested and taught National Curriculum Statement for Mathematics (NCSM) at Further Education and Training (FET) level in South Africa.

Initiatives in many countries to improve learner performances in mathematics in poor communities have been described as largely unsuccessful mainly due to their cursory treatment of curriculum alignment. Empirical evidence has shown that in high achieving countries the notion of coherence was strongly anchored in cognitively demanding mathematics programs. The view that underpins this study is that a cognitively demanding and coherent mathematics curriculum has potential to level the playing field for the poor and less privileged learners. In South Africa beyond 1994, little has been done to understand the potential of such coherent curriculum in the context of the NCSM. This study examined the levels of cognitive demand and alignment between the written, tested and taught NCSM. The study adopted Critical Theory as its underlying paradigm and used a multiple case study approach. Wilson and Bertenthal’s (2005) dimensions of curriculum coherence provided the theoretical framework while Webb’s (2002) categorical coherence criterion together with Porter’s (2004) Cognitive Demand tools were used to analyse curriculum and assessment documents. Classroom observations of lesson sequences were analysed following Businskas’ (2008) model of forms of mathematical connections since connections of different types form the bases for high cognitive demand (Porter, 2002). The results indicated that higher order cognitive skills and processes are emphasized consistently in the new curriculum documents. However, in the 2008 examination papers the first examinations of the new FET curriculum, lower order cognitive skills and processes appeared to be emphasized, a finding supported by Umalusi (2009) and Edwards (2010). Classroom observations pointed to teachers focusing more on rote learning of both concepts and procedures and less on procedural and conceptual understanding. Given the widespread evidence of the tested curriculum impacting on the taught curriculum, this study suggests that this lack of alignment between the advocated curriculum on one hand, the tested and the taught curricula on the other, needs to be investigated further for it endangers the teaching and learning of higher order cognitive skills and processes in the FET mathematics classrooms for the poor and less privileged. Broader evidence suggests that this would work against efforts towards supporting the upward mobility of poor children in the labour market

Towards empowering learners in a democratic mathematics classroom: to what extent are teachers' listening orientations conducive to and respectful of learners' thinking?

In an effort to make education accessible, to 'heal the divisions of the past and establish a society based on democratic values', the South African Department of Education claims that a series of mathematics reforms that has so far been introduced is underpinned by the principles of 'social justice, fundamental human rights and inclusivity'. Critics however argue that the system has remained 'undemocratic' in that those groups of learners who were supposed to be 'healed' continue to underperform and hence be disempowered. In this study, we conceptualised a democratic and mathematically empowering classroom as one that is consistent with the principle of inclusivity and in which a hermeneutic listening orientation towards teaching promotes such a democratic and mathematically empowering learning environment. We then worked with three different orientations teachers might have towards listening in the mathematics classroom: evaluative, interpretive and hermeneutic. We then used these orientations to analyse 20 video-recorded lessons with a specific focus on learners' unexpected contributions and how teachers listened and responded to such contributions. The results were consistent with the literature, which shows that teachers tend to dismiss learners' ways of thinking by imposing their own formalised constructions