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A research project submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Masters in Science Education. 2015 / The Curriculum Assessment Policy Statements (CAPS) re-introduced some mathematics topics such as probability. An immediate effect of this re-introduction is that most teachers and learners were not well equipped to deal with this topic. To at least begin addressing this problem, this research explored the errors and misconceptions that learners have when solving probability problems using different representations. The study draws from Nesher’s (1987) theory of errors and misconceptions as well as Sfard’s (2007) theory of commognition in explaining representations and prevalence of errors in learning mathematics. Twenty two Grade 10 learners wrote probability tasks after which their scripts were analysed for errors. Six of those learners were interviewed on the errors they made in solving probability problems with different representations. The findings reveal five main categories of errors and misconceptions. These are: (1) difficulty with construction of visual representations; (2) improper distinction between simple and compound events; (3) application of inappropriate routines; (4) errors associated with familiarity; and, (5) misinterpreted language. The findings also showed that inappropriate choice of representations was caused by misinterpretation of probability terminology. Concurring with Zahner and Corter (2010) the researcher found that learners made a multitude of errors if they constructed and used their own probability representations. Further, learners committed fewer errors where the task provided representations. Results also show that learners were most confident in using tree diagram representations even though they struggled to construct them from scratch. Most learners avoided Venn diagrams, outcome listings and matrix representations even though they would be the most useful in answering the questions. As a result many errors and misconceptions resulted when learners tried to use these representations. The study recommends that teachers take time to discuss probability terminology and the use of different representations with their learners. This promotes both the conceptual and procedural knowledge of probability. Also, to reduce learners’ errors and misconceptions on the topic, teachers need to scaffold the construction of representations by providing partially constructed representations and gradually encourage learners to construct their own probability representations.

The Revised National Curriculum requires that mathematics teachers make shifts in their pedagogical content knowledge about teaching non-routine problems. One of the shifts is to move from a traditional approach of teaching routine tasks to an approach which includes problem solving. Teaching non-routine problems develops flexible forms of knowledge, which allow learners to construct their own strategies and not merely follow steps shown to them by the teacher. In this study I explore what pedagogical content knowledge grade 6 teachers have about problem solving in everyday and mathematical contexts. A qualitative study was used to explore Grade 6 teachers’ pedagogical content knowledge about problem solving in these contexts. Data was collected by means of interviews and lesson observations. The study found variations in the teachers’ content knowledge but very similar pedagogical content knowledge among the teachers. Teachers have different understandings of teaching non-routine problems and they are faced with the challenge of how much to make knowledge accessible to their learners in order to lay a conceptual foundation to solve non-routine problems while not lowering the demands of the task. Teachers taught everyday context tasks differently to mathematical context tasks by focusing on the linguistic aspects of the tasks.

"The purpose of this study was to present descriptively and subjectively analyze the learning problems experienced by a graduate social work student with a ninety-seven per cent loss of vision. This student's second-year block field work placement of four months took place at the Institute of Human Development, Florida State University, Tallahassee, Florida. The focus of this study was to show how this student resolved certain professional learning problems, thus leading to positive learning experiences"--Introduction. / Typescript. / "August, 1960." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Social Work." / Advisor: Dorothy D. Hayes, Professor Directing Study. / Includes bibliographical references (leaves 80-81).

Typescript (photocopy). / Digitized by Kansas Correctional Industries

The effects of order of presentation and amount of experience on errors and solution time were investigated. An interactive narrative puzzle was presented on a computer screen to 60 undergraduate students. Solution of the problem involved the integration of two path segments. Subjects in the Forward Condition were presented the path segments in the order in which they had to be traversed for solution. Subjects in the Backward Condition were exposed to the opposite order of presentation. Amount of experience was varied by permitting one, three, or five readings of the narrative.

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