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.

published_or_final_version / Education / Master / Master of Education

This was a qualitative study carried out with one grade 8 multicultural, multiethnic, mathematics class. This research study began with the idea of finding out whether the learners home language (especially Zulu Xhosa) could be linked to algebraic misconceptions. The 40 learners (participants) in my study had just been introduced to algebra. I chose the school and participants through “convenience sampling”. This made sense since I am an educator at this particular school. I had explained the meaning of the word "variable" in depth. The concepts "like terms" and "unlike terms" had been explained. The index laws for multiplication and division of the same bases had been discussed. It was within this context that the algebra worksheet was given to the learners, in the first cycle. I examined the algebra errors made by the grade 8 learners after marking the worksheets. I linked the errors to past literature on algebraic misconceptions as well as to Bernard's (2002b) error classification list. The conclusion was that the learners were making common errors which were not affected by their home language. I spent time on reflection since the outcome was not exactly what I had anticipated (that is, I had harboured strong suspicions that English second language learners would commit more algebraic errors than the English home language learners). I then considered a possible link between culture and algebraic misconceptions. Videotaped lessons were used for this purpose. However, observations of these videotaped lessons did not produce much data. I honestly could not reach a conclusion. This formed the second cycle of my action research. Prompted by the obvious lack of interaction in the video recordings from my teaching, I changed my focus to what I, the teacher, did during the lessons, and how these actions may or may not have supported some of the algebraic misconceptions. I reflected on my teaching method and recognized the need to change to a more interactive teaching style. I needed to give the learners the space to think for themselves. I would merely facilitate where necessary. In the third cycle, I drew up a set of problems which matched the new teaching style (interactive teaching).The lessons during which the new set of problems were discussed and solved, were videotaped. These videotaped lessons were analyzed and a completely different picture emerged. The learners were absolutely responsive and showed a side of them that I had not seen before! This study came to be an action research study because I went through three cycles of reflecting, planning, acting and observing and then reflecting, re-planning, further implementation, observing and acting etc. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2010.

The purpose of this exploratory study was to look at the importance of Indiana's Algebra I academic standards in relationship with essential, entry-level work skills as perceived by employers and guidance counselors in northern and northeastern Indiana. Counselors and employers were sent electronic (e-mail and web-based) or hard (paper) copies of a survey instrument. Section II of the survey asked for demographic data, (age, race, gender) as well as information related to the size of the business or school, type of business, and years in current position. Respondents were also asked to rate their knowledge of the Indiana academic standards, the Americans with disabilities Act and Section 504 of the Vocational Rehabilitation Act. In Section III, respondents were presented with the nine Indiana Algebra I standards, along with examples to illustrate and further explain each standard. Employers rated each standard, and the standards combined, on its importance to skills required from people seeking jobs immediately after leaving high school using a Likert scale. Guidance counselors rated the importance of the standards based on what they knew of the labor market within school district and surrounding areas using the same Likert scale. Comments were solicited at the end of the survey instruments. Employers disagreed with the importance of any of the Algebra I standards, or the standards as whole, as entry-level work skills. Their highest mean rating was 2.4. Counselors also disagreed, although on two standards their mean rating was above the Neutral rating. Sample size was deemed too low for generalization, but study does supply a framework for future research. / Department of Special Education

<p>This thesis investigates grade seven learners perceived experiences in learning algebra.Things that learners do and say during algebra lessons and about algebra were investigated. The study was done at one of the previously disadvantaged schools in Cape Town, South Africa.The data were collected through observations, a questionnaire and interviews. Observations were made from the day the topic was started in two grade seven classes. Two different teachers taught the two classes. Focus group interviews were conducted, two group of learners, ten learners from each of the two classes were interviewed. Learners devised a number of strategies for solving problems related to sums and differences. The principal learning difficulties experienced by learners in algebra related to the transition from arithmetic conventions to those of algebra, the meaning of literal symbols and the recoginition of structures. It became obvious then that developing algebraic thinking is not necessarily dependent upon algebraic notation and that the presence of algebraic notation says little about the level of problem solving.</p>

No description available.

This thesis presents findings made during an examination and comparison of the individualized unit and traditional group methods used to teach ninth grade algebra in Hillsboro, Texas.