Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 2002. / In this thesis I consider how mathematics students in a traditional firstyear Calculus course at a South African university appropriate mathematical objects which are new to them but which are already part of the official mathematics discourse. Although several researchers have explained mathematical object appropriation in process-object terms (for example, Sfard, 1994; Dubinsky, 1991, 1997; Tall, 1991, 1995, 1999), my focus is largely on what happens prior to the object-process stage. In line with Vygotsky (1986), I posit that the appropriation of a new mathematical object by a student takes place in phases and that an examination of these phases gives a language of description for understanding this process. This theory, which I call “appropriation theory”, is an elaboration and application of Vygotsky’s (1986) theory of concept formation to the mathematical domain. I also use Vygotsky’s (1986) notion of the functional use of a word to postulate that the mechanism for moving through these phases, that is, for appropriating the mathematical object, is a functional use of the mathematical sign. Specifically, I argue that the student uses new mathematical signs both as objects with which to communicate (like words are used) and as objects on which to focus and to organise his mathematical ideas (again as words are used) even before he fully comprehends the meaning of these signs. Through this sign usage the mathematical concept evolves for that student so that it eventually has personal meaning (like the meaning of a new word does for a child); furthermore, because the usage is socially regulated, the concept evolves so that its usage is concomitant with its usage in the mathematical community. I further explicate appropriation theory by elaborating a link between the theoretical concept variables and their empirical indicators, illustrating these links with data obtained from seven clinical interviews. In these interviews, seven purposefully chosen students engage in a set of speciallydesigned tasks around the definition of an improper integral. I utilise the empirical indicators to analyse two of these interviews in great detail. These analyses further inform the development of appropriation theory and also demonstrate how the theory illuminates the process of mathematical object appropriation by a particular student.

NAVORSINGSVERSLAG voorgele ter gedeeltelike vervuiling van die vereiste vir die graad MAGISTER IN NATUURWETENSKAPPE in die FAKULTEIT NATUURWETENSKAPPE aan die UNIVERSITEIT VAN DIE WITWATERSRAND. / There is widespread concern over some of the problems encountered in the teaching of Euclidean Geometry in secondary schools and also over the fairly negative attitudes experienced by pupils towards Geometry. This piece of research was designed to improve attitudes of Std.9 pupils towards Euclidean Geometry by making use of guided discovery and the integration of the history of Mathematics into the teaching method used. The latter was done in order to humanise the subject and to make it more interesting to pupils who otherwise experience it as very rigid and abstract. Active participation of pupils in developing the Geometry was central to the method employed. The outcome of the research was positive. It showed that attitudes towards Geometry can be improved if a deliberate attempt to do so is made. The results of this research suggest guidelines by which the teaching of Euclidean Geometry in secondary schools could be improved. / AC 2018

A research report submitted to the Faculty of Science, University of Witwatersrand, in partial fulfilment for the degree of Masters of Mathematics Education by coursework and research report. Johannesburg, March 2017. / The current mathematics curriculum in South Africa require that learners are provided with opportunities to develop abilities to be methodical, to generalise, to make conjectures and try to justify and prove their conjectures. These objectives call for the use of teaching strategies and tasks that support learners’ participation in the development of mathematical thinking and reasoning. This means that teachers have to be cautious when selecting tasks and deciding on teaching strategies for their classes. Tasks differ in their cognitive and difficulty levels and opportunities they afford for learner to learn mathematics competently. The levels of tasks selected by the teachers; the kinds of questions asked by the teachers during the implementation of the selected tasks and how the questions asked by the teachers and the teachers’ actions at implementations affected the levels of the tasks were the focus of this research report. The study was carried out in one high poverty high school in South Africa. Two teachers were observed teaching and each teacher taught their allocated grades. One teacher was observed teaching Grade 9s while the other taught Grade 11s. Both teacher taught number patterns at the time their lessons were observed. The research was qualitative. Methods of data collection and instruments included lesson observations; collection of tasks used in the observed classes, audio-taping and field notes. Pictures of the teachers’ work and copies of learners’ workbooks also provided some data. The analysis of data shows that the teachers not only selected and used lower-level cognitive demand and ‘easy’ tasks, that did not support mathematical thinking, but also did not lift up the levels and/or maintain the ‘difficulty levels’ of the task at implementation. Teachers were unable to initiate class discussions. Their teaching focused on ‘drill and practice’ learning and teaching practices. / LG2017

For several reasons, Guilford's psychological theory, "The Structure-of-Intellect" (SI), seems a good candidate for relating to the learning of mathematics. The general purposes of this study were to identify SI factors which would be significantly related to achievement in a junior-college mathematics course for non-science, non-mathematics majors and to determine whether semantic factors would be better predictors than symbolic for students classified as having high verbal ability. The two topics in the mathematics course which were selected for study were (1) numeration in other bases and (2) finite systems. / Typescript. / "August, 1975." / "Submitted to the Area of Instructional Design and Personnel Development, Program of Mathematics Education, in partial fulfillment of the requirements of Doctor of Philosophy." / Advisor: Eugene D. Nichols, Professor Directing Dissertation. / Vita. / Includes bibliographical references (leaves 151-153).

"This study investigated whether there is a relationship between first grade children's ability to tell a story about a dynamic picture or a sequence of three dynamic pictures and their ability to describe the picture(s) by a number sequence. The artistic variables characterizing the pictures were controlled so as to provide information concerning which types of illustrations best facilitated interpretation of the pictures and perception of mathematical relationships. An 8 x 2 design allowed analysis of the effects of the form of the drawing, the number of pictures, and the response condition. Ninety-six first grade children were individually tested using an instrument designed by the investigator. Statistical analysis revealed that neither drawing style nor the number of pictures had a significant effect on either the level of assimilation within the stories, the perception of motion, or the number sentence responses. Analysis of the response condition revealed a significant difference favoring the force condition on number sentence responses. Also, initially viewing and interpreting sequences provided a learning experience to significantly effect the interpretation of single pictures"--Abstract. / Typescript. / "August, 1976." / "Submitted to the Area of Instructional Design and Personnel Development, Program of Mathematics Education, in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: Eugene D. Nichols, Professor Directing Dissertation. / Vita. / Includes bibliographical references (leaves 162-172).

"The present investigation was designed to study the effect of two instructional treatments on the achievement of students of different abilities--Verbal and Spatial. This was achieved by studying the interaction between the two treatments and each of the verbal and the spatial abilities. The instructional treatments were Figural and Verbal programmed units designed to teach concepts related to modulus seven arithmetic. Subjects for the study were 90 students enrolled in the first year mathematics course at Elmansoura College of Education in Egypt for the academic year 1978-1979"--Abstract. / Typescript. / "December, 1979." / "Submitted to the Department of Curriculum and Instruction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: Eugene D. Nichols, Professor Directing Dissertation. / Includes bibliographical references (leaves 115-117).

"The purpose of this research is the development and testing of an instrument to be used in prediction of success in college freshman mathematics courses"--Introduction. / Typescript. / "April, 1967." / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Doctor of Education." / Advisor: R. Heimer, Professor Directing Dissertation. / Vita. / Includes bibliographical references (leaves 118-120).

"The purpose of the study was to determine whether there is a difference in the ability of two equally capable groups of subjects to generalize the winning strategy of a simple game when one group learns the perfect strategy for one form of the game by the discovery method and the other group learns it by reading an explanation of the strategy"--Introduction. / Typescript. / "August, 1970." / "Submitted to the Department of Mathematics Education of Florida State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: E. D. Nichols, Professor Directing Dissertation. / Includes bibliographical references.

This experimental study was intended to examine whether game-based learning (GBL) that encompasses four particular game characteristics (challenges, a storyline, rewards, and the integration of game-play with learning content) in the OpenSimulator-supported virtual reality (VR) learning environment can improve mathematical achievement and motivation for elementary school students toward math learning. In this pre- and post-test experimental comparison study, data were collected from 132 fourth graders through an achievement test, and a Short Instructional Materials Motivational Survey (SIMMS). The same tasks were provided to the experimental and control groups. Tasks for the experimental group involved the following four game characteristics: (1) challenges, (2) a storyline, (3) rewards, and (4) the integration of game-play with learning content. The control group was given the same tasks and learning environment setting (OpenSimulator-supported VR) that was used for the experimental group. The exception was that the control group tasks did not include the game characteristics: (1) challenges, (2) a storyline, (3) rewards, and (4) the integration of game-play with learning content. Analysis of covariance (ANCOVA) using a treatment (treatment vs. control) on the achievement indicated a significant effect of GBL in the VR environment on math knowledge test performance. For motivation, the results indicated that there was no significant difference on the post-test scores for the perceived motivational quality of the learning activity (MQLA) between the experimental group and the control group. / A Dissertation submitted to the Department of Educational Psychology and Learning Systems in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2015. / September 15, 2015. / fractions, Game-Based Learning, math performance, OpenSim, Real-life application skills, Virtual Reality / Includes bibliographical references. / Fengfeng Ke, Professor Directing Dissertation; Young-Suk Kim, University Representative; Allan Jeong, Committee Member; Insu Paek, Committee Member.

This study looks at the use of integrated mathematics and music lessons at the high school level. Four lessons were taught by the researcher in both a research and a control class to determine how mathematically motivated music instruction affects students understanding of operations of functions, composition of functions, inverse functions and domain and range. A pretest-posttest was used to determine the effect of these lessons and a questionnaire was used to identify differences between groups and to help determine the effect of musical applications of mathematics on students’ mathematical perceptions, self-efficacy and grit. The pretest-posttest included both a standard mathematics section and a section involving non-musical applications. A gain score approach using independent sample t tests was used to determine the impact of the integrated instruction. The research group demonstrated significantly greater gains both overall and on the applications portion of the exam. Additional qualitative analysis was done to determine how the posttests differed between groups. Three major differences were identified: the research group used function notation more frequently than the control group, the control group demonstrated confusion between composition of functions and inverse functions while the research group did not and the research group showed more mathematical work for the applications portion of the exam than the control group. Qualitative analysis was also done to identify trends in the questionnaire data. Among the major differences between groups was the increased willingness to work with mathematical applications in the future by the research group compared to the control group. The integrated instruction led to comparable and in some cases significantly better mathematics outcomes than the control group and led students to an increased willingness to work with mathematical applications both on the posttest and moving forward.