The Teaching Abstract Algebra for Understanding (TAAFU) project was centered on an innovative abstract algebra curriculum and was designed to accomplish three main objectives: to produce a set of multi-media support materials for instructors, to understand the challenges faced by mathematicians as they implemented this curriculum, and to study how this curriculum supports student learning of abstract algebra. Throughout the course of the project I took the lead investigating the teaching and learning in classrooms using the TAAFU curriculum. My dissertation is composed of three components of this research. First, I will report on a study that aimed to describe the experiences of mathematicians implementing the curriculum from their perspective. Second. I will describe a study that explores the mathematical work done by teachers as they respond to the mathematical activity of their students. Finally, I will discuss a theoretical paper in which I synthesize aspects of the instructional theory underlying the TAAFU curriculum in order to develop an analytic framework for analyzing student learning. This dissertation will serve as a foundation for my future research focused on the relationship between teachers' mathematical work and the learning of their students.

The purpose of this study was to investigate the effectiveness of the graphing calculator on the achievement, attitude toward mathematics, and confidence in learning mathematics of Algebra II students.

The purpose of this study was to survey Algebra II teachers in the State of Texas to determine the extent to which they use technology for the delivery of instruction. Additionally, the study attempted to determine reasons why teachers do or do not use technology when they have a choice.

This study investigated the use of computer-based instruction as a means of teaching Algebra I, compared to the teaching of the same topics using traditional methodologies. The achievement level of the two groups, and three aspects of attitude toward mathematics were considered. Achievement and attitude differences by gender were also analyzed.

The purpose of this study was to evaluate the influence of block scheduling and instructional strategies on student achievement in Algebra I. The study was conducted during the 1997-98 school year. This study was comprised of two components, a quantitative study and a qualitative study. The quantitative study focused on block and traditional scheduling and the influence identified through scores on the Texas End-of-Course exam for Algebra I. The sample for this study consisted of 59 school districts from five counties in the north Texas area. The qualitative portion of this study focused on 10 classrooms, 5 block and 5 traditional, taken from the sample of 59 districts. Data for the qualitative study included questionnaires, interviews, and observations. The End-of-Course scores were analyzed using an ANOVA at the .05 level of significance, no significant difference was identified in the achievement levels of the two groups. The qualitative data was organized by categories derived from the NCTM teaching standards. Data from this portion of the study indicated that teachers in both block and traditionally scheduled classes spend their class time in a similar manner, using similar materials, and using more traditional strategies. Additional analyses of data based upon usage of the graphing calculator and manipulatives also resulted in no significant difference. Although all comparisons between block and traditional scheduling and usage or non-usage of technology and/or manipulatives resulted in no significant difference, the block groups and those using technology and/or manipulatives had higher mean scores. This indicates that allowing teachers more time to use alternative instructional strategies would benefit the student, but this will not take place without the teacher receiving training and support.

The purpose of this study was to examine the effectiveness of a technology-enhanced unit on slope in algebra. The technology used in the study was the Topological Panorama Camera (Topocam). The research questions explored the learning and transfer of knowledge about slope and the engagement level of students during Topocam learning activities. The Topocam is a computer-controlled camera that moves on a modular track while it scans a scene through a vertical slit. Students can program the speed of the camera and frequency of pictures. They then witness the results of time and motion in the image created by the camera. Data for this study were collected from a pretest/posttest, as well as from observations of indicators of engaged learning. The research population consisted of 46 students from three classes of Algebra I students. Three classroom teachers each taught a unit on slope, while a fourth teacher conducted the activities with the Topocam for all the classes. The classroom activities focused on the concept of slope as a rate of change utilizing coordinate grids. The Topocam activities involved students in collaboratively making and testing predictions about slope. The findings of the study indicate that student learning did occur with this technology-enhanced unit on slope in algebra. Students showed statistically significant improvement in understanding slope and in transferring that concept to other situations. Since technology was only part of the unit presentation, the amount of learning attributed to the Topocam activities cannot be determined. However, students demonstrated a high degree of engagement in learning while working with the Topocam which suggests that the activities were a factor. A low correlation between students’ slope unit test scores and previous algebra performance may indicate that students who have not been successful in algebra were more successful in the technology-enhanced unit. Some variation was found between classes that could be attributed to other factors than the Topocam.

Complex systems abound on this planet, in the composition of the human body, in ecosystems, in social interaction, in political decision-making, and more. Analytical methods allowing us to better understand how these systems operate and, consequently, to have a chance to intervene and change the undesirable behavior of some of the more pernicious systems have developed and continue to be enhanced via quickly changing technology. Some of these analytical methods are accessible by pre-college students, but have not been widely used at that level of education. Jay Forrester, the founder of one of the methodologies, System Dynamics (SD), used to study complex system behavior involving feedback, laments the lack of understanding of complex systems evident in short-sited decisions made by legislators -- global climate change and fiscal policies being cases in point. In order to better prepare future decision makers with tools that could allow them to make more informed decisions about issues involving complex systems efforts have been underway to increase pre-college teacher understanding of the SD method. The research described in this dissertation introduces the mathematics education community to the value of System Dynamics modeling in pre-college algebra classes, indicates a path by which a traditional mathematics curriculum could be enhanced to include small SD models as a new representation for elementary functions studied in algebra classes, and provides an empirical study regarding conceptual understanding of functions by students. Chapter 2 indicates the numerous beneficial learning outcomes that empirical studies have shown accompany model-building activities. Chapter 3 indicates the need for students to become familiar with complex systems analysis, how SD modeling (one method of complex systems analysis) aligns with the Common Core State Standards in Mathematics, and the work that has transpired over the past two decades using SD in K-12. Chapter 4 focuses on the importance of the concept of function in high school mathematics, some limitations of exclusive reliance on the closed form equation representation for mathematizing problems and the SD stock/flow representations of some of the elementary functions that are studied in algebra classes. Chapter 5 looks at the issues affecting two traditional teachers and the challenges they faced when trying to reintroduce SD modeling into their algebra classes. Chapter 6 explains the student component of the classroom experiment that was conducted by the teachers who are highlighted in Chapter 5. The analysis of the results of student model-building activities in the two classroom studies that are part of the third paper did not indicate a statistical difference between the two experimental groups and the two control groups. Many environmental and scheduling issues conspired to adversely affect the experiment. However, positive outcomes were evident from the two pairs of students who were videotaped while they built the final multi-function drug model, the final student lesson in the experiment. Research focused on student outcomes is needed to further assess the strengths and weakness of the SD approach for student learning in mathematics.

This research report confirms the value of the Framework of Growth Points in a learner’s mathematical development in the area of functions in equation form. The study also shows that learners advance through the various growth points in a progressive, sequential fashion, which mirrors the results of Ronda’s study, on which a part of this study was based. The study was carried out in a high school in Johannesburg. Learners in Grades 9, 10 and 11 were required to do an assessment which tested for their achievement in different growth points. This study also explores the discourse of learners while they talked about the tasks in the assessment. A smaller sample of learners was interviewed so that the researcher could explore the nature of their discourse. This research report shows that there are patterns in the discourse of learners which can be related to the growth points that they achieve.

A research report submitted to the school of Education, Faculty of Science, University of the Witwatersrand in partial fulfilment for the degree of Master of Science Johannesburg 2017 / South African learners perform poorly in national and international mathematics assessments (Howie, 2004). A contributing factor to this poor performance is low mathematics knowledge of mathematics teachers in South Africa (Howie, 2003). One means of addressing this is professional development programs. The Wits Maths Connect Secondary Project runs such a program. A test is required by the project in order to assess whether learners are making learning gains after being taught by teachers who participated in this program. The focus of this study is the design of a test used to assess learners’ algebraic attainment. The aim is to design an informative and fair test using Rasch analysis. A sample of 235 learners’ responses to 47 questions was analysed using the Rasch model. In this study, the mean person measure was 2,87 (SD=1,38) logits, while the mean item measure was 0,41 (SD=2,25) logits, suggesting that overall, the test was too difficult. For the learners who wrote this test the person separation index is 1,78 and the person reliability 0,76. This implies that the test may not be not sensitive enough to distinguish between learners of high attainment from learners with low attainment. Various ways of improving the test are discussed. / MT 2018

iii ABSTRACT The study focused on two competent, qualified, experienced secondary Mathematics teachers working in contrasting South African school contexts (fee-paying and no fee schools). The study investigated: on how teachers chose and used examples and how they explained their choices and usage; and what considerations were in play when these teachers chose and used examples. These teachers were purposely selected because we can learn more from their experiences as Mathematics teachers especially when they teach quadratic functions. Quadratic functions were used as unit of analysis to illuminate their choice and use of examples.