Browsing by Author "Wessels, Dirk"
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- ItemDevelopment of affective modelling competencies in primary school learners(AOSIS Publishing, 2011) Biccard, Piera; Wessels, DirkLearner affect and beliefs about mathematics are complex and multifaceted aspects of mathematical learning. Traditional teaching and learning approaches in mathematics education often result in problematic beliefs about mathematics. Since beliefs influence what learners learn and how they deal with learning mathematics, it is essential that the roles of beliefs and affect in mathematics classrooms are carefully examined. In solving modelling problems, learners and teachers take on new roles in the classroom: learners are placed in an active, self-directing situation in which they solve real-world problems. When learners engage in modelling tasks, they display and integrate cognitive, meta-cognitive and affective competencies. A modelling approach therefore allows one to detect learner beliefs in an authentic learning environment. Will this environment lead to students having more positive and productive dispositions towards mathematics? This article presents partial results of a study documenting the development of modelling competencies in learners working in groups over a period of 12 weeks. Through a design research approach, 12 learners working in groups solved three modelling problems, and transcriptions of learner interactions, questionnaires and informal interviews revealed that learner beliefs improved over this short period when exposed to modelling tasks. The results are encouraging, and may provide mathematics education with an avenue to develop more positive learner beliefs in mathematics.
- ItemAn instrumental approach to modelling the derivative in Sketchpad(AOSIS Publishing, 2011-11) Ndlovu, Mdutshekelwa; Wessels, Dirk; De Villiers, MichaelEncouragement to integrate information and communication technologies into mathematics education curricula is an increasingly universal phenomenon. As a contribution to the discourse, this article discusses the potential use in the classroom of The Geometer’s Sketchpad® (Key Curriculum Press, Emeryville, CA, United States) mathematics software in modelling the derivative and related concepts in introductory calculus. In an empirical study involving first-year non-mathematics major undergraduate science students, a hypothetical learning trajectory (HLT) was conjectured and implemented for students to experience the visualisation and multiple representations of calculus concepts on the Cartesian plane with a computer graphic interface. The utilisation scheme is interpreted through the lens of the instrumental1 approach proposed by Trouche. The HLT was partly informed by the historical development of the derivative as synthesised from the literature on the history of calculus and partly by the affordances, enablements, constraints and potentialities of Sketchpad itself. The findings of the study suggest that when exposed to the capabilities of this software, learners can experience Geometer’s Sketchpad® as an effective visualisation tool or instrument for the representation and learning of the derivative and related concepts in introductory calculus. However, the effectiveness of this tool is not a given or a foregone conclusion − it is a product of the teacher’s instrumental orchestration, gradual learner mastery of the software syntax and careful resolution of theoretical-computational conflicts that can arise during early use of the instrument.
- ItemDie moontlikhede van 'n modelleringsperspektief vir skoolwiskunde(AOSIS OpenJournal, 2009) Wessels, DirkENGLISH ABSTRACT: The ﬁndings of the international TIMSS investigations of a few years ago into the position and application of problem solving strategies in school mathematics in about 50 countries caused serious concern globally. During each survey South Africa was found to be among the poorest performers of the participating countries. The main problem was that the majority of school learners in South Africa do not have the ability to solve mathematical problems; in fact, it would appear that they lack the total spectrum of mathematical problem solving competencies. The present school system does not develop their mathematical abilities or competencies. While Outcomes-based education, which became very popular in the Western world, has the ability to improve participants’ affective values of mathematics, it proved to be inadequate in improving the quality of their mathematical performances. Mathematics teachers are unsuccessful in teaching in a manner that will make a difference with respect to the way learners do, learn or perform in mathematics. The pedagogical and mathematics content knowledge of the teachers are lacking in conceptual depth, clarity and connectedness (integration). The language proﬁciency of the learners is poor, which means that they do not understand what they should do with a problem and how to interpret, present and verify their ﬁndings. Learners still do not know how to handle mathematics and how to utilise mathematics in order to solve problems. They seriously lack the ability to approach problems in a meaningful and constructive way. Real-life and open-ended problems are being perceived as huge obstacles to most learners. Teachers are not trained and educated to assist their learners in bridging this gap. The teaching methodology that will make a difference in the classroom falls in the broad category of problem solving. The day-to-day teaching method should be the problem-centred teaching and learning approach. This rather complex teaching methodology requires in-depth thinking about the role of the teacher, the role of the learner, the nature of the classroom culture, the nature of the negotiation of meaning between the teacher and individuals or groups, the nature of selected problems and material, as well as the kind of integrative assessment used in the mathematics classroom. Modelling is closely related to the problem-centred teaching approach, but it also smoothly relates to bigger and longer mathematical tasks. This article gives a theoretical exposition of the scope and depth of mathematical modelling. It is possible to introduce modelling at every school phase in our educational sytem. Modelling in school mathematics seems to make the learning of mathematics more effective. The mastering of problem solving and modelling strategies has deﬁnitely changed the orientation, the competencies and performances of learners at each school level. It would appear from research that learners like the application side of mathematics and that they want to see it in action. Genuine real life problems should be selected, which is why a modelling perspective is so important for the teaching and mastering of mathematics. Modelling should be integrated into the present curriculum because learners will then get full access to involvement in the classroom, to mathematisation, to doing problems, to criticising arguments, to ﬁnding proofs, to recognising concepts and to obtaining the ability to abstract these from the realistic situation. Modelling should be given a full opportunity in mathematics teacher education so that our learners can get the full beneﬁt of it. This will put the mathematical performances of learners in our country on a more solid base, which will make our learners more competitive at all levels in the future.
- ItemThe potential of teacher development with Geometer’s Sketchpad(AOSIS Publishing, 2008-12) Stols, Gerrit; Mji, Andile; Wessels, DirkIn this paper we document the advantages of utilising technology to enhance teachers’ instructional activities. In particular we showcase the potential and impact that the use of Geometer’s Sketchpad may have on the teaching and learning of geometry at school. A series of five, two-hour teacher development workshops in which Geometer’s Sketchpad was used were attended by 12 Grade 11 and 12 teachers. The findings revealed that teachers had a better understanding of the same geometry that they initially disliked. This finding was supported by a quantitative analysis which showed a positive change in the understanding of and beliefs about geometry from when the teachers started to the end of the workshops.
- ItemThe role of visualisation in data handling in grade 9 within a problem‐centred context(AOSIS Publishing, 2009-07) Makina, Antonia; Wessels, DirkIn the recent past, data handling has been neglected at secondary school level, perhaps partially due to the strong emphasis on developing arithmetic, algebra and geometry. For the first time, the South African curriculum includes substantial amounts of data handling at all grade levels. The introduction of more data handling in the secondary school curriculum in South Africa and the prevalence of many problems in the teaching of probability and statistics argues for a serious reconsideration of the way it is taught to the pupils. Currently this concern has been the focus of a call for reform in mathematics education by a body like the National Council of Teachers of Mathematics (NCTM) at all levels of schooling (NCTM, 1989; 2000). The importance of visualisation in mathematics, at all levels of mathematical problem solving is well documented in the literature (Bishop, 1989; Maher & Alston, 1989; Moses, 1982; Wheatley, 1991) but almost nothing was done to appreciate visualisation in the learning of data handling. The paper therefore provides a qualitative examination from a Masters dissertation (Makina, 2005) of the role of visualisation in the learning of data handling. This is done through examining the thought processes involved by Grade 9 learners during visualisation while solving data handling tasks. Several roles of visualisation were identified and most were found to improve the critical and creative thinking of pupils during their learning of data handling. The results show that learners are likely to improve their performance in data handling if the awareness of the need to use visualisation creatively as a tool for understanding are highlighted.
- ItemStudent mathematical activity as a springboard to developing teacher didactisation practices(AOSIS Publishing, 2015-12-07) Biccard, Piera; Wessels, DirkThis article is part of a larger study on teacher development. The main study investigated teacher development within primary school Mathematics teachers’ classrooms to determine if teaching practices could be enhanced through a didactisation-based programme. It sought to develop teachers within their own environments and classrooms. Design research (both designing the conditions for change and studying the results of those conditions) enabled the researchers to design a programme that was congruent with teachers’ own needs and experiences. The programme ran for a period of a year with regular contact between the teachers and the researcher conducting the programme (the first author). The programme set out nine didactisation practices: active students, differentiation, mathematisation, vertically aligned lessons, accessing student thinking and ideas, probing student thinking and ideas, connecting student ideas, assessing students and reflecting on practice. One practice, student activity, is the focus of this article. It was found that by initiating discussion and cognitive conflict in teachers by using modelling problems, and further allowing teachers to observe pupils working in groups with modelling problems, teachers were starting to incorporate the didactisation practices within their own classrooms. This article documents specifically the fundamental role of student mathematical activity and the importance of improving student mathematical experiences, both for teacher development and for student mathematical learning. The study may be valuable in structuring and planning further effective teacher development programmes.