Dissertations / Theses on the topic 'Calculus, Study and teaching (Higher) - Lesotho'

The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective.

Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for f(x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate f(0) as the limit value. This method is appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually.

This study investigates the effects of implementing mathematical problem solving instruction in a regular calculus course taught at the college level. Principles associated with this research are: i) mathematics is developed as a response to finding solutions to mathematical problems, ii) attention to the processes involved in solving mathematical problems helps students understand and develop mathematics, and iii) mathematics is learned in an active environment which involves the use of guesses, conjectures, examples, counterexamples, and cognitive and metacognitive strategies. Classroom activities included use of nonroutine problems, small group discussions, and cognitive and metacognitive strategies during instruction. Prior to the main study, in an extensive pilot study the means for gathering data were developed, including a student questionnaire, several assignments, two written tests, student task-based interviews, an interview with the instructor, and class observations. The analysis in the study utilized ideas from Schoenfeld (1985) in which categories, such as mathematical resources, cognitive and metacognitive strategies, and belief systems, are considered useful in analyzing the students' processes for solving problems. A model proposed by Perkins and Simmons (1988) involving four frames of knowledge (content, problem solving, epistemic, and inquiry) is used to analyze students' difficulties in learning mathematics. Results show that the students recognized the importance of reflecting on the processes involved while solving mathematical problems. There are indications suggesting that the students showed a disposition to participate in discussions that involve nonroutine mathematical problems. The students' work in the assignments reflected increasing awareness of the use of problem solving strategies as the course developed. Analysis of the students' task-based interviews suggests that the students' first attempts to solve a problem involved identifying familiar terms in the problem and making some calculations often without having a clear understanding of the problem. The lack of success led the students to reexamine the statement of the problem more carefully and seek more organized approaches. The students often spent much time exploring only one strategy and experienced difficulties in using alternatives. However, hints from the interviewer (including metacognitive questions) helped the students to consider other possibilities. Although the students recognized that it was important to check the solution of a problem, they mainly focused on whether there was an error in their calculations rather than reflecting on the sense of the solution. These results lead to the conclusion that it takes time for students to conceptualize problem solving strategies and use them on their own when asked to solve mathematical problems. The instructor planned to implement various learning activities in which the content could be introduced via problem solving. These activities required the students to participate and to spend significant time working on problems. Some students were initially reluctant to spend extra time reflecting on the problems and were more interested in receiving rules that they could use in examinations. Furthermore, student expectations, evaluation policies, and curriculum rigidity limited the implementation. Therefore, it is necessary to overcome some of the students' conceptualizations of what learning mathematics entails and to propose alternatives for the evaluation of their work that are more consistent with problem solving instruction. It is recommended that problem solving instruction include the participation or coordinated involvement of all course instructors, as the selection of problems for class discussions and for assignments is a task requiring time and discussion with colleagues. Periodic discussions of course directions are necessary to make and evaluate decisions that best fit the development of the course.
Education, Faculty of
Curriculum and Pedagogy (EDCP), Department of
Graduate

The purpose of this study was to identify the nature of students' conceptual understanding of two concepts of calculus namely, derivative and function. As a way of collecting data two methods were employed: (a) modification of Piagetean clinical interview; and, (b) tutorial sessions. Whenever the students seemed to be confused about the issues being discussed, the researcher provided instructions through the tutorial sessions. The analysis of data was done by developing individual profiles and by response categories. It was found that the interview methodology was effective in revealing some aspects of students' concept images. The students were found to have little meaningful understanding of derivative. A number of students held proper concept images of function which should lead to the development of an appropriate concept definition. It was also evident from the study that students had adequate skill in using algorithm to solve problems. The results of the study would be useful to the instructors of calculus. It was suggested that introducing a concept by its formal definition would contribute to students' confusions and difficulties. Yet if a concept is presented by means of meaningful examples, students had better opportunity to develop their concept images. Thus leading them to form concept definitions. The researcher strongly recommended that more challenging exercises be posed to the students in problem-solving situations.
Education, Faculty of
Curriculum and Pedagogy (EDCP), Department of
Graduate

Making the transition from calculus to advanced calculus/real analysis can be challenging for undergraduate students. Part of this challenge lies in the shift in the focus of student activity, from a focus on algorithms and computational techniques to activities focused around definitions, theorems, and proofs. The goal of Realistic Mathematics Education (RME) is to support students in making this transition by building on and formalizing their informal knowledge. There are a growing number of projects in this vein at the undergraduate level, in the areas of abstract algebra (TAAFU: Larsen, 2013; Larsen & Lockwood, 2013), differential equations (IO-DE: Rasmussen & Kwon, 2007), geometry (Zandieh & Rasmussen, 2010), and linear algebra (IOLA: Wawro, et al., 2012). This project represents the first steps in a similar RME-based, inquiry-oriented instructional design project aimed at advanced calculus. The results of this project are presented as three journal articles. In the first article I describe the development of a local instructional theory (LIT) for supporting the reinvention of formal conceptions of sequence convergence, the completeness property of the real numbers, and continuity of real functions. This LIT was inspired by Cauchy's proof of the Intermediate Value Theorem, and has been developed and refined using the instructional design heuristics of RME through the course of two teaching experiments. I found that a proof of the Intermediate Value Theorem was a powerful context for supporting the reinvention of a number of the core concepts of advanced calculus. The second article reports on two students' reinventions of formal conceptions of sequence convergence and the completeness property of the real numbers in the context of developing a proof of the Intermediate Value Theorem (IVT). Over the course of ten, hour-long sessions I worked with two students in a clinical setting, as these students collaborated on a sequence of tasks designed to support them in producing a proof of the IVT. Along the way, these students conjectured and developed a proof of the Monotone Convergence Theorem. Through this development I found that student conceptions of completeness were based on the geometric representation of the real numbers as a number line, and that the development of formal conceptions of sequence convergence and completeness were inextricably intertwined and supported one another in powerful ways. The third and final article takes the findings from the two aforementioned papers and translates them for use in an advanced calculus classroom. Specifically, Cauchy's proof of the Intermediate Value Theorem is used as an inspiration and touchstone for developing some of the core concepts of advanced calculus/real analysis: namely, sequence convergence, the completeness property of the real numbers, and continuous functions. These are presented as a succession of student investigations, within the context of students developing their own formal proof of the Intermediate Value Theorem.

Thesis (MCur (Interdisciplinary Health Sciences. Nursing Science))--University of Stellenbosch, 2010.
ENGLISH ABSTRACT: HIV/AIDS is considered as a global problem with the number of people living with HIV infection continuing to increase. At the end of 2007 HIV/AIDS had already claimed 25 million lives. Of all new HIV infections 71% were diagnosed in the Sub-Saharan region in 2008, remaining the worst affected region globally. UNAIDS (2008:43) indicated that heterosexual intercourse remained the main origin for HIV infection in the Sub-Saharan region. Therefore the researcher is of the opinion that prevention strategies should focus mainly on sexual transmission of the disease. HIV/AIDS affects mainly people between the ages 15-24 years, notably the age group of most of the learners in Higher Education Institutions (HEIs). Lesotho, a country in the Sub- Saharan region, presents with the third highest HIV adult prevalence (23.2%) in the world and in the region. In an attempt to address the prevailing situation, Lesotho has a number of programmes geared towards addressing HIV/AIDS in the country. However, all these attempts exclude the learners in HEIs, yet the majority of learners are found within the most affected age group. It is also to be noted that Higher Education provides the bedrock for socio-economic and political development in Africa. Some studies have identified insufficient knowledge as being at the root of the increasing HIV infections among youth. However, other studies have shown that there is adequate knowledge among the young people, but still a challenge remains and that is to facilitate changes in behavioural patterns as a component to be linked to the knowledge. Studies conducted in other African countries have shown that there are anti-AIDS programmes and clubs for learners in HEIs where learners are involved in the fight against HIV/AIDS. No publication indicating the same for Lesotho’s HEIs could be found, except for the National University of Lesotho (NUL) that only launched its HIV/AIDS policy for learners in 2009. The researcher is of the opinion that HEIs in Lesotho are not doing enough to combat HIV/AIDS and hence intends to focus on HEIs in Lesotho. This study had two objectives namely:  To determine the knowledge of learners in a specific HEI in Lesotho regarding HIV/AIDS prevention and care.  To explore the needs of learners in a specific HEI in Lesotho regarding HIV/AIDS prevention and care. This mixed method study was conducted, comprising of both quantitative and qualitative designs. Quantitative phase used a questionnaire for determining the knowledge of learners. The questionnaire was adopted from a study that was performed to determine knowledge of South African educators in public schools with some modifications. The qualitative phase was used to explore the needs of the learners through the focus group discussions with the leaders of the learners. Sample was drawn from the entire population using stratified random sampling for the quantitative phase. The qualitative phase used the purposive sampling to obtain in-depth information concerning learners’ needs. Quantitative data was analysed through the use of statistical package for social sciences (SPSS) and qualitative data was analysed using the thematic analysis and open-coding. All ethical principles were adhered to especially the principle of respect for persons. The findings from the quantitative phase of the study showed that learners had adequate knowledge regarding HIV/AIDS prevention and care and the findings from the qualitative phase showed the various needs of the learners with regards to prevention and care of HIV/AIDS in a specific HEI in Lesotho. Recommendations have been proposed based on the findings from the two phases of the study. Limitations observed by the researcher have also been identified. In conclusion the objectives of the study were met and the research questions had been answered.
AFRIKAANSE OPSOMMING: MIV/Vigs word as ‘n internasionale probleem erken, siende dat daar ‘n verhoging in die toename van MIVgeïnfekteerde indiwidue tans is . Einde 2007 het MIV/Vigs het reeds 25 miljoen lewens ge-eis . In 2008 is 71% van al die nuwe MIV-infeksies in die Sub-Sahara streek gediagnoseer, wat aandui dat die streek die mees geaffekteerde streek tans is. UNAIDS (2008:43) het aangedui dat heteroseksuele omgang die hoofoorsaak van MIV-oordrag in die Sub-Sahara-streek is. Laasgenoemde het daartoe gelei dat die navorser van mening is dat voorkomende strategieë meestal op seksuele oordrag van die siekte moet fokus. MIV/Vigs affekteer meestal mense in die ouderdomsgroep 15-24, opmerklik is dit die ouderdomsgroep waarby meesste leerders in Hoëronderwysinstellings (HOI) is. Lesotho, ‘n land in die Sub-Sahara-streek, het tans die derde-hoogste MIV-voorkoms (23.2%) in die wêreld en in die streek. Lesotho het verskeie programme ontlont om MIV/Vigs te bekamp in ‘n poging om die huidige situasie te beredder . Nieteenstaande sluit al die programme leerders in HOI uit, alhoewel die leerders in die ouderdomsgroep van die mees-geaffekteerde groep val. Dit is ook duidelik dat Hoëronderwys die fondasie vir sosio-ekonomiese- en politieke ontwikkeling in Afrika verskaf. Sommige studies het onvoldoende kennis as die wortel van die verhoging van MIV-infeksies onder die jeug geïdentifiseer. Ander studies, daarenteen, wys dat kennis voldoende is onder jeug, alhoewel veranderinge in gedragspatrone om by die kennis aan te sluit ‘n uitdaging bly. Studies uit ander Afrikalande dui daarop dat daar anti-Vigs programme en klubs is waarby HO leerders betrokke is om teen die verspreiding van MIV/Vigs te veg. Geen publikasies in hierdie verband word in Lesotho aangetref nie, behalwe ‘n MIV/Vigs-beleid wat in 2009 deur “National University of Lesotho’ (NUL) gepubliseer is. Dus is die navorser van mening dat HOI nie genoeg doen om MIV/Vigs te beveg nie, daarom fokus sy op HOI in Lesotho. Hierdie studie het twee doelstellings ten doel gehad, naamlik om die leerders in ‘n sekere HOI in Lesotho se kennis aangaande MIV/Vigs voorkoming en sorg te bepaal en die behoeftes van die leerders aangaande MIV/Vigs voorkoming en sorg te verken. ‘n Studie met beide kwantitatiewe- en kwalitatiewe metodes is gebruik om die doelstellings te verwesenlik. In die kwantitatiewe fase is ‘n vraelys gebruik om leerders se kennis te bepaal. Die vraelys is verkry uit ‘n vorige studie wat in RSA gedoen is, maar aangepas om in die Lesotho-konteks te gebruik. Gedurende die kwalitatiewe fase is fokusgroep besprekings met die leiers van die leerders gehou om die behoeftes indiepte te verken. Die steekproef was uit die totale populasie getrek deur van gestratifiseerde streekproefneming gebruik te maak in die kwantitatiewe fase en ‘n doelgerigte steekproefneming is in die kwalitatiewe fase te gebruik. Die navorser het ‘n kwantitatiewe data-analise sagteware (SPSS)gebruik om kwantitatiewe data te ontleed en tematiese- oopkodering is gedurende die kwalitatiewe fase gebruik. Etiese kode is ten volle gerespekteer, veral die respek vir mense gedurende navorsing. Bevindinge van die kwantitatiewe fase het bewys dat leerders voldoende kennis aangaande die voorkoming en sorg van MIV/Vigs besit en die kwalitatiewe bevindinge het die behoeftes van leerders met betrekking tot die voorkoming en sorg van MIV/Vigs in ‘n spesifieke HOI in Lesotho geopenbaar. Die aanbevelings is gemaak, gebaseer op die bevindinge uit die twee fases. Beperkinge in die studie is uitgelig. Ter afsluiting is die doelstellings in die studie bereik en die navorsingsvrae beantwoord.

The purpose of this dissertation was to investigate the relationship between the Advanced Placement Calculus AB exam and student achievement in college level Math 1710-Calculus I. The review of literature shows that this possible relationship is based on Alexander Astin's longitudinal input-environment-outcome (I-E-O) model. The I-E-O model was used to analyze the relationship between the input and outcome of the two variables. In addition, this quantitative study determined the relationship between a score of 3 or lower on the Advanced Placement Calculus AB exam and student achievement in college level Math 1710-Calculus I. The sample population of this study contained 91 students from various high schools in Texas. Spearman's rank correlation revealed there was a statistically significant relationship between Advanced Placement Calculus AB exam scores and final grades in Math 1710-Calculus I.

The purposes of this study were (a) to develop, implement, and evaluate a computer-oriented instructional program for introductory calculus students, and (b) to explore the association between a computer-oriented calculus instructional program, a non-computer-oriented calculus instructional program, student achievement on three selected calculus topics, and student attitude toward mathematics. An experimental study was conducted with two groups of introductory calculus students during the Spring Semester, 1989. The computer-oriented group consisted of 32 students who were taught using microcomputer calculus software for in-class presentations and homework assignments. The noncomputer-oriented group consisted of 40 students who were taught in a traditional setting with no microcomputer intervention. Each of three experimenter-developed achievement examinations was administered in a pretest/posttest format with the pretest scores being used both as a covariate and in determining the two levels of student prior knowledge of the topic. For attitude toward mathematics, the Aiken-Dreger Revised Math Attitude Scale was administered in a pretest/ posttest format with the pretest scores being used as a covariate. Students were also administered the MAA Calculus Readiness Test to determine two levels of calculus prerequisite skill mastery. An ANCOVA for achievement and attitude toward mathematics was performed by treatment, level, and interaction of treatment and level. Using a .05 level of significance, there was no significant difference in treatments, levels of prior knowledge of topic, nor interaction when achievement was measured by each of the three achievement examination posttests. Furthermore, there was no significant difference between treatments, levels of student prerequisite skill mastery, and interaction when attitude toward mathematics was measured, at the .05 level of significance. It was concluded that the use of the microcomputer in introductory calculus instruction does not significantly effect either student achievement in calculus or student attitude toward mathematics.

The purposes of the study were: to determine the relationship between class size and academic achievement among university students in first-semester calculus classes, and to compare opinions about the instructor, course, and classroom learning environment of university students in small first-semester calculus classes with those in large classes. The sample consisted of 225 university students distributed among two large and two small sections of first-semester calculus classes taught at the University of Texas at Arlington during the fall of 1987. Each of two tenured faculty members taught a large and small section of approximately 85 and 27 students, respectively. During the first week of the semester, scores from the Calculus Readiness Test (CR) were obtained from the sample and used as the covariate in each analysis of covariance of four periodic tests, a comprehensive final examination, and final grade average. The CR scores were also used in a logistic regression analysis of attrition rates between each pair of large and small sections of first-semester calculus. Three semantic differentials were used to test the hypotheses relating to student opinion of the instructor, course, and classroom learning environment. It was found that for both pairs of large and small first-semester calculus classes there was no significant difference in the adjusted means for each of the four periodic tests, the final examination scores, the final grade averages, and the attrition rates. It was also found that the means of the student evaluation of the course by students in small and large classes were not significantly different, and the results of the student evaluations of the instructor and classroom learning environment by students in small and large first—semester calculus classes were mixed.

Acompanha: Manual didático para aplicação de testes estatísticos na análise do desempenho de alunos em disciplinas da graduação
Esta pesquisa teve como objetivo analisar variáveis a fim de entender se elas são significativas para a reprovação dos alunos ingressantes nos cursos de Engenharia na disciplina de Cálculo Diferencial e Integral I. Para tanto, adotou-se como hipótese básica que o comprometimento acadêmico é um dos fatores que interfere de forma expressiva neste contexto. O referencial teórico faz um breve apanhado sobre a origem e evolução dos cursos de Engenharia, sobre a importância do Cálculo, bem como sobre as reprovações e possíveis agravantes. Além disso, aborda as principais variáveis associadas à reprovação em Cálculo I apontadas na literatura existente. Trata-se de uma pesquisa com abordagem mista, sendo que as hipóteses secundárias buscavam confirmar ou descartar a influência de seis variáveis - nota obtida pelos estudantes na prova de Matemática do Exame Nacional do Ensino Médio (ENEM), pesos atribuídos às provas de Matemática do ENEM, período de ingresso no curso, carga horária semanal de aulas, conhecimento matemático prévio e metodologia de avaliação diferenciada - no desempenho obtido pelos calouros na disciplina em questão. Para tanto, estudou-se o desempenho de 3.010 alunos da UTFPR, pertencentes aos campi Pato Branco e Ponta Grossa, que ingressaram na instituição de 2010 a 2014. Os dados referentes às variáveis quantitativas foram coletados por meio de consultas ao sistema acadêmico institucional e aplicações de testes aos calouros. Em seguida, estes dados foram analisados com auxílio de ferramentas estatísticas. A coleta de dados referentes à variável qualitativa (comprometimento acadêmico) ocorreu por meio de entrevistas semiestruturadas realizadas junto a dezessete alunos, sendo que a análise se amparou na metodologia de Análise do Conteúdo, proposta por Bardin (1977). Os resultados sugerem a dependência entre cinco variáveis quantitativas analisadas e o desempenho obtido na disciplina de Cálculo I. Além disso, apontam que as posturas discentes adotadas frente a disciplina de Cálculo Diferencial e Integral I foram determinantes para o bom ou mau desempenho na disciplina. Como produto final foi confeccionado um aplicativo web que permitirá a reaplicação da metodologia de análise dos dados quantitativos nos outros câmpus da UTFPR e em outras instituições de ensino superior.
This research aims to analyse factors in order to understand their significance to the failure of Engineering freshmen students in Differential and Integral Calculus I. To this purpose, the basic hypothesis adopted is that academic commitment is a variable that expressively affects this setting. The theoretical framework summarizes the origin and evolution of Engineering courses, the relevance of the subject and respective failures, as well as potential aggravating circumstances. In addition, it approaches key factors related to failure in Calculus discussed in current literature. This is a mixed approach research and secondary hypotheses intended to either confirm or disregard the impact of certain variables, namely: grade achieved by students in Mathematics exam conducted in Brazilian High School National Exam (Exame Nacional do Ensino Médio, ENEM); weights assigned to ENEM Mathematics test; term of course admission (fall or spring); quantity of courses per week; previous knowledge on Mathematics; and distinct evaluation methodology. The research studies the performance of 3,010 students of UTFPR of both Pato Branco and Ponta Grossa campuses enrolled in the institution from 2010 to 2014. Data related to quantitative variables were collected through searches in the institution’s academic system and conduction of tests to first-year students. Subsequently, this data was analysed using statistics tools. The data accrual related to the qualitative variable (academic commitment) occurred through semi-structured interviews conducted along with some students and analysis was supported by Content Analysis methodology proposed by Bardin (1977). Results suggest the dependency among the five quantitative variables analysed and the performance achieved in the subject Calculus I. Furthermore, they indicate that students’ behaviour regarding the subject Differential and Integral Calculus I was definitive for either good or poor performance in the subject. The final product was the construction of a web applicative which allows the reutilization of quantitative data analysis methodology in other UTFPR campuses and college institutions.

It is common for students to make mistakes while solving mathematical problems. Some of these mistakes might be caused by the false ideas, or misconceptions, that students developed during their learning or from their practice. Calculus courses at the undergraduate level are mandatory for several majors. The introductory course of calculus—Calculus I—requires fundamental skills. Such skills can prepare a student for higher-level calculus courses, additional higher-division mathematics courses, and/or related disciplines that require comprehensive understanding of calculus concepts. Nevertheless, conceptual misunderstandings of undergraduate students exist universally in learning calculus. Understanding the nature of and reasons for how and why students developed their conceptual misunderstandings—misconceptions—can assist a calculus educator in implementing effective strategies to help students recognize or correct their misconceptions. For this purpose, the current study was designed to examine students’ misconceptions in order to explore the nature of and reasons for how and why they developed their misconceptions through their thought process. The study instrument—Calculus Problem-Solving Tasks (CPSTs)—was originally created for understanding the issues that students had in learning calculus concepts; it features a set of 17 open-ended, non-routine calculus problem-solving tasks that check students’ conceptual understanding. The content focus of these tasks was pertinent to the issues undergraduate students encounter in learning the function concept and the concepts of limit, tangent, and differentiation that scholars have subsequently addressed. Semi-structured interviews with 13 mathematics college faculty were conducted to verify content validity of CPSTs and to identify misconceptions a student might exhibit when solving these tasks. The interview results were analyzed using a standard qualitative coding methodology. The instrument was finalized and developed based on faculty’s perspectives about misconceptions for each problem presented in the CPSTs. The researcher used a qualitative methodology to design the research and a purposive sampling technique to select participants for the study. The qualitative means were helpful in collecting three sets of data: one from the semi-structured college faculty interviews; one from students’ explanations to their solutions; and the other one from semi-structured student interviews. In addition, the researcher administered two surveys (Faculty Demographic Survey for college faculty participants and Student Demographic Survey for student participants) to learn about participants’ background information and used that as evidence of the qualitative data’s reliability. The semantic analysis techniques allowed the researcher to analyze descriptions of faculty’s and students’ explanations for their solutions. Bar graphs and frequency distribution tables were presented to identify students who incorrectly solved each problem in the CPSTs. Seventeen undergraduate students from one northeastern university who had taken the first course of calculus at the undergraduate level solved the CPSTs. Students’ solutions were labeled according to three categories: CA (correct answer), ICA (incorrect answer), and NA (no answer); the researcher organized these categories using bar graphs and frequency distribution tables. The explanations students provided in their solutions were analyzed to isolate misconceptions from mistakes; then the analysis results were used to develop student interview questions and to justify selection of students for interviews. All participants exhibited some misconceptions and substantial mistakes other than misconceptions in their solutions and were invited to be interviewed. Five out of the 17 participants who majored in mathematics participated in individual semi-structured interviews. The analysis of the interview data served to confirm their misconceptions and identify their thought process in problem solving. Coding analysis was used to develop theories associated with the results from both college faculty and student interviews as well as the explanations students gave in solving problems. The coding was done in three stages: the first, or initial coding, identified the mistakes; the second, or focused coding, separated misconceptions from mistakes; and the third elucidated students’ thought processes to trace their cognitive obstacles in problem solving. Regarding analysis of student interviews, common patterns from students’ cognitive conflicts in problem solving were derived semantically from their thought process to explain how and why students developed the misconceptions that underlay their mistakes. The nature of how students solved problems and the reasons for their misconceptions were self-directed and controlled by their memories of concept images and algorithmic procedures. Students seemed to lack conceptual understanding of the calculus concepts discussed in the current study in that they solved conceptual problems as they would solve procedural problems by relying on fallacious memorization and familiarity. Meanwhile, students have not mastered the basic capacity to generalize and abstract; a majority of them failed to translate the semantics and transliterate mathematical notations within the problem context and were unable to synthesize the information appropriately to solve problems.

The purpose of this study was to evaluate whether supplemental instruction and online homework can improve student performance and understanding in a first-semester calculus course at a large urban four-year college. The study examined the metacognitive and study skills and posttest scores of students. The study also focused on students’ and instructor’s perception and experiences of supplemental instruction and online homework using WebAssign. The study used a modified version of the Motivated Strategies for Learning Questionnaire (MSLQ) to reveal any significant differences in metacognitive and study strategies between students in a class with supplemental instruction/online homework and students in a traditional class. Students’ scores on their final examination were analyzed to reveal the effect of mathematical achievement between the control and experimental groups. Surveys and interviews were utilized to provide anecdotal evidence as to the overall effectiveness of the online homework management system and supplemental instruction. Results of the study showed no substantial difference between the control group and the experimental group in seven out of eight sub-scales of metacognitive and study strategies: metacognitive self-regulation, time and study environment, effort regulation, help seeking, rehearsal, organization, and critical thinking. But, students with supplemental instruction/online homework showed a higher level of elaboration learning strategies. The interaction of pretest and type of class (traditional or treatment) did not have a significant effect on students’ posttest score. There was no substantial effect of pretest on posttest, but the treatment influenced students’ posttest score. Students’ gender, race, class level, or the number of courses they registered for were insignificant predictors of their posttest scores. The instructor and students agreed that time spent in supplemental instruction sessions and on WebAssign were worthwhile and beneficial. They believed supplemental instruction and online homework using WebAssign may have influenced students’ understanding and performance in the course.

The study examined classroom instructional practices and teacher's professed conceptions about teaching and learning college calculus in relationship to the implementation of scientific-programmable-graphics (SPG) calculators. The study occurred at a university not affiliated with any reform project. The participants were not the catalysts seeking to implement calculus reform, but expressed a willingness to teach the first quarter calculus course with the SPG calculator. The research design was based on qualitative methods using comparative case studies of five teachers. Primary data were collected through pre-school interviews and weekly classroom observations with subsequent interviews. Teachers' profiles were established describing general conceptions of teaching calculus, instructional practices, congruence between conceptions and practice, conceptions about teaching using SPG calculators, instructional practice with SPG calculators, and the relationship of conceptions and practice with SPG calculators. Initially, all the teachers without prior experience using SPG calculators indicated concern and skepticism about the usefulness of the technology in teaching calculus and were uncertain how to utilize the calculator in teaching the calculus concepts. During the study the teachers became less skeptical about the calculator's usefulness and found it effective for illustrating graphs. Some of the teachers' exams included more conceptual and graphically-oriented questions, but were not significantly different from traditional exams. Findings indicated the college teachers' conceptions of teaching calculus were generally consistent with their instructional practice when not constrained by time. The teachers did not perceive a dramatic change in their instructional practices. Rather, the new graphing approach curriculum and technology were assimilated into the teachers' normal teaching practices. No major shifts in the role of the teachers were detected. Two teachers demonstrated slight differences in their roles when the SPG calculators were used in class. One was a consultant to the students as they used the SPG calculators; the other became a fellow learner as the students presented different features on the calculator. Use of the calculator was influenced by several factors: inexperience with the calculator, time constraints, setting up the classroom display calculator, preferred teaching styles and emphasis, and a willingness to risk experimenting with established teaching practices and habits.
Graduation date: 1996

The results of this thesis can be used to help identify students at risk of an unsuccessful entry level undergraduate mathematics (ELUM) outcome early in their course by using a student's score on a diagnostic test, using a student's high school performance or tracking graded homework submissions. The quantitative results suggest that optional additional remediation is not addressing students' need for remediation and that struggling students do not frequently engage with departmental supports offered. They also suggest how a suite of pedagogical changes to an ELUM course can be associated with increased student success rates if managed carefully. It is commonly known that prior mathematical knowledge and success in current math courses are strongly linked. Through observational studies at UVic it is found that a significant proportion of students beginning one of several ELUM courses do not demonstrate the high levels of preparation required to succeed and that optional additional remediation is not addressing this issue. In Calculus I, we can infer that more than half of the population of graduating high school students in British Columbia meet or exceed the minimum prerequisite of a B in Principles of Math 12 at UVic. However, entering Calculus I students scored a mean of only 51% on a diagnostic test of important topics from Grade 7 to Grade 12 level mathematics. Almost 90% of the Calculus I students that were identified as at risk by results of the diagnostic test had an unsuccessful outcome. In addition, although nearly half of Calculus I students who entered with a B in Principles of Math 12 failed the Calculus I final exam, passing the remedial course Precalculus is only required for students with a C+ or less in Principles of Math 12. Thus an insignificant proportion of students are adequately prepared for Calculus I by passing the remedial course while many struggling students do not get the remediation required for them to succeed. The quantitative results indicate that the strength of the link between levels of prior mathematical knowledge and ELUM success varies by course. According to Astin (1984) increasing student engagement positively correlates with higher satisfaction levels. Thesis results show that a student who misses more than one graded homework (used as a measure of engagement) will very likely fail the final exam. They show that a student who does not express satisfaction with his/her individual performance is also very unlikely to engage frequently with departmental supports offered such as instructor office hours or the UVic Math Assistance Centre. The results of these observational studies influenced pedagogical changes to the course Logic and Foundations that were designed to increase student engagement. These changes included a more accessible textbook, giving in-class quizzes on assigned readings, fostering a positive course experience, intervening with students at-risk and assigning less weight to the final exam. Analysis of course outcomes shows that the failure rate significantly decreased during this term. Student outcomes were not initially improved after similar modifications to Calculus II, but many of these changes to Calculus II have been maintained through subsequent terms in which improved student outcomes have been observed.
Graduate

This thesis investigates the mathematical cognitive errors made in elementary calculus concepts by first-year University of Technology students. A sample of 34 first year students, the experimental group, from the Durban University of Technology Faculty of Engineering were invited to participate in project in elementary calculus using computer technology (CT). A second group, the control group, also consisted of 34 first year engineering students from the same University were given a conventional test in elementary calculus concepts. The experimental group was then given the same conventional test as the control group on completion of the project in elementary calculus using computer technology (CT). The purpose of the analysis was to study the effect of technology on the understanding of key concepts in elementary calculus. The major finding was that technology helps students to make connections, analyse ideas and develop conceptual frameworks for thinking and problem solving. The implications include: • Improvement of curriculum in mathematics at tertiary level; • New strategies for lecturers of elementary calculus; • An improved understanding by students taking the course in elementary calculus. • Redesign of software to improve understanding in elementary calculus.
Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2007.

It has been held that heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. Many educational researchers have proposed problem-based curricula to improve students' views of mathematical thinking. The present study reports findings regarding effects of a problem-based calculus course, using historical problems, to foster Taiwanese college students' views of mathematical thinking. The present study consisted of three stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by a six-item, open-ended questionnaire and nine randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week problem-based calculus course in which mathematical concepts were problematized in order to challenge their personally expressed empirical beliefs in doing mathematics. Several tasks and instructional approaches served to reach the goal. Near the end of the semester, all participants answered the same questionnaire and the same students were interviewed to pinpoint their shift in views on mathematical thinking. It was found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students leaned toward a conservative attitude in the certainty of mathematical knowledge. Participants focus seemingly shifted from mathematics as a product to mathematics as a process.
Graduation date: 2003

The quality of mathematics knowledge attained by students entering university in Science, Technology, Engineering and Mathematics (STEM) fields has been decreasing. There is a need to enhance students’ mathematical knowledge in order to maintain the standards of STEM curriculum at university. The rationale of this study was to investigate the influence of Pre-Calculus Mathematics Refreshment module taught using Meta-cognitive skills and Co-operative Learning (MCL), or Co-operative Learning (CL) only, or Traditional lecture (T) intervention method to First Year pre-engineering Students on their Applied Calculus 1 in an Ethiopian university. The study further investigated the influence of Pre-Calculus Mathematics Refreshment module for MCL, or CL, or T intervention method on male and female students’ achievement. The refreshment module and Applied Calculus 1 scores were measured through posttest and normal class room score of Applied Calculus 1 result. The dependent variables were student achievement in pre-calculus refreshment Module and Applied Calculus 1. Out of 29 universities in Ethiopia only four were selected to participate in this study. Population of this study was all pre-engineering first year students in those universities in 2016/2017. The sample consisted of 200 pre-engineering university students who studied in four of Ethiopian universities and one class was randomly selected by lottery method from existing pre-engineering classes in each university. Two experimental groups which were taught MCL and the other CL intervention method and two of them were control groups upon whom the control novice with traditional lecture method and control without intervention was applied. In each group 50 students of 25 males and 25 females were purposely selected from sampled class. A pre-calculus mathematics Pre-test was administered first, where the average scores of all students Pre-test result was below 33%. Then, first MCL and CL intervention methods were discussed and exercised for one week before implementing the study. For the study, selected pre-calculus mathematics topics was taught in all classrooms for 32 periods i.e. 50min x32= 26.7hrs at the beginning of the first semester parallel with Applied Calculus 1 for the academic year 2016 / 2017. The statistical tools used under this procedure include descriptive statistics percentage, mean and standard deviation and inferential statistics, T-test, and one-way analysis of variance (one-way ANOVA). The results show statistically significant differences (Sig 0.00) at the significance level (0.05) between students that learnt pre-calculus refreshment module and control group which did not. Among the students those learned pre-calculus refreshment module through MCL, CL and T method students in the MCL and CL groups’ posttest scores significantly different from T group in pre-calculus results both with Sig of 0.00. But there was no significant difference between MCL & CL groups were Sig is 0.97. Additionally, the female students in the MCL group was not significant different from CL and T group, on an impact of refreshment module, in Applied Calculus 1 mathematics where Sig is 0.994 and 0.237 respectively, and CL female group scores significantly different from T group in Applied Calculus 1 results with Sig 0.042. The male students in the MCL and CL groups were significantly different from T group in Applied Calculus 1with Sig of 0.07 and 0.012 respectively. Also, there was a positive correlation between Pre-Calculus refreshment module and Applied Calculus 1 with correlation coefficient of 0.835. Lastly, the result of pre-calculus mathematics posttest scores with the female students in MCL relatively increased than male students, than in CL and T groups, which indicated that MCL benefit more female students than male students. The differences were more in favor of pre-calculus mathematics refreshment with MCL intervention method. To improve success in engineering participation of all students, recommended that a pre-calculus module should be offered by all universities for first year engineering students, structured co-operative learning with purpose has significant gains for effective instruction, and to increase the success rate of female students this study has proven that they are trainable and therefore, meta-cognition skills have to be nurtured for female students.
Mathematics Education
D. Phil (Mathematics Education in Science and Technology)

This study was prompted by serious problems regarding specific teaching and learning problems in calculus at the technikon. The general aims were to identify and analyze particular teaching and learning problems relating to 2nd level engineering courses in calculus and to recommend improvements which could increase student performance in engineering calculus courses. An extensive study revealed world wide concern in calculus reform. The empirical research instruments consisted of structured questionnaires given to staff and students from nine technikons plus interviews. Five serious problem areas were identified: student ability in mathematics, content difficulty, background difficulties, timetable pressures and lecturer's presentation. The impact of training technology on calculus was investigated. Recommendations were that routine exercises can be done on computer with extra tutorial time for computer laboratory projects. Background recommendations suggested that schools give more time to trigonometry and coordinate geometry and that bridging courses at technikons for weaker students be developed.
Curriculum and Instructional Studies
M. Ed. (Didactics)

In this investigation an attempt was made to determine the misconceptions that engineering students have of the idea of a limit. A comprehensive literature study showed that there are a number of common misconceptions that students normally form. The empirical investigation was done in two phases. A questionnaire on the idea of a limit was given to the students during the first phase. During the second phase six interviews were conducted. The findings were grouped according to the nature of a limit and students' views on the relationship between the continuity of a function at a point and the limit at that point. An analysis of these findings led to the identification of the misconceptions that these students have of the idea of a limit.
In hierdie ondersoek is gepoog om die wanbegrippe wat ingenieursstudente van die limietbegrip vorm, bloot te stel. 'n Omvattende literatuurstudie het 'n aantal algemene wanbegrippe aan die lig gebring. Die empiriese ondersoek het in twee fases plaasgevind. Tydens die eerste fase is 'n vraelys aan die studente gegee in 'n poging om meer te wete te kom van hulle begrip van 'n limiet. Die vraelys is opgevolg deur ses onderhoude. Die responsies is gegroepeer in terme van die aard van 'n limiet en studente se sienings van die kontinuiteit van 'n funksie by 'n punt en die limiet by daardie punt. Die analisering van hierdie responsies het die identifisering van 'n aantal wanbegrippe by hierdie groep studente moontlik gemaak.
Educational Studies
M.Ed. (with specialisation in Mathematics Education)

Mathematics achievement has been of great concern to researchers involved in mathematics education. This concern has resulted in research seeking to determine for example, the factors that positively or negatively contribute to student performance in mathematics. Many of the reported studies in the literature have investigated the factors within the context of mathematics teaching and learning in general. Very few studies have investigated the factors contributing to student achievement in mathematics when learning takes place in a computer aided environment. With the pervasiveness of computers in education in general, studies in this direction become imperative. The present study fills this gap in the literature by examining the extent to which selected variables (mathematics attitude, mathematics aptitude, computer attitude, computer prior experience, computer ownership, proficiency in language of instruction, and learning style) contribute to students' achievements in pre-calculus algebra classes that are supplemented with a computer lab program. The participants in the study were 120 students sampled from the population of students enrolled in the second pre-calculus algebra course at the preparatory year program of King Fahd University of Petroleum & Minerals during the 2003/2004 academic session. The instruments used to measure the study constructs were the mathematics attitude scale (Aiken, 1979), the computer attitudes scale (Loyd & Gressard, 1984a), and the learning styles questionnaire (Honey & Mumford, 1992). New instruments to measure computer prior experience and computer ownership were developed for the present study. Hypotheses formulated for the study were tested using multiple regression and other statistical techniques. The results show that mathematics aptitudes and English language proficiency are the most significant contributors to students' mathematics achievement. No other variables show statistically significant effects on students' achievement. Together, the selected variables explain more than 41 percent of the total variance of students' achievement. Theoretical and policy-making implications of the results are outlined and discussed.
Mathematical Sciences
D. Phil. (Mathematics, Science and Technology Education)

Students’ performances in mathematics in an Open Distant Learning setting have not always been impressive. An exploratory study into the problem solving skills of the University of South Africa students in the Calculus module MAT112 is being conducted using past examinations scripts between 2006 and 2009. The study re-assesses the work done in the end-of-year Calculus examinations, by both looking at the distribution of marks awarded and assigning new scores based on an assessment rubric adapted for the problem at hand. Further assessment of qualitative dimensions that is important for problem solving in Calculus is developed from the data obtained from the assessment rubric. Using factor analysis, a hesitation factor, transfer-of-knowledge factor as well as ingenuity factor, are identified in successful Calculus problem solving. The study proposes two conceptual models; the first is to guide students in solving Calculus problems while the second one is meant to assist lecturers in the assessment of students of Calculus.
Science and Technology Education
M. Ed. (Technology Education)

Intermediate calculus bridges secondary school and advanced university mathematics courses. Most mathematics education research literatures indicated that the conceptual knowledge in intermediate calculus has challenged first year undergraduate mathematics and science learners to a great extent through the lecture method. The content knowledge attained by them has been tremendously decreasing. Negative attitude exhibited by students toward calculus was highly influenced by the lecture method used. Generally, students have not looked at the learning of all mathematics courses offered in universities as normal as other courses. Due to this lack of background conceptual knowledge in learners, they have been highly frustrated by the learning of advanced mathematics courses. Taking the understanding of teaching and learning challenge of conceptual knowledge of calculus into consideration, Ethiopian public universities have been encouraging instructors to devise and implement active learning methods through any professional development training opportunity. The training was aimed to enhance learners’ content knowledge and attitude towards calculus. This is one of the main reasons for the motivation of this study that experimental group learners were allowed to be nurtured by the lecture method in their mainstream class, and then also the active learning intervention method integrated with GeoGebra in the mathematics laboratory class. Only conventional lecture method was used to teach the comparison group in both the mainstream and mathematics laboratory class. The purpose of the study was to explore the Gambari and Yusuf (2016) stimulus of the jigsaw co-operative learning method combined with GeoGebra (JCLGS) on statistics and chemistry learners’ content knowledge improvement and change of their attitude towards calculus. The post-positivism mixed methods tactic was used in a non-equivalent pre- and post-test comparison group quasi-experimental design. The population of the study was the whole freshman mathematics and science degree program learners of two public universities in Ethiopia in 2017. Samples of the size 150 in both the experimental and comparison groups were drawn utilizing two-stage random sampling technique. A questionnaire using a Likert-scale on attitudes and an achievement test were sources used for data collection. Data analysis employed descriptive statistics conducting an independent samples t-test and a Two Way ANOVA for repeated measures using SPSS23. Each of the findings on content knowledge, conceptual knowledge, and procedural knowledge development produced through the TWO-Way ANOVA, respectively as F(1,148)=80.917; 𝜂2=.353; p<.01, F(1,148)=106.913; 𝜂2=.419; p<.01, and F(1,148)=7.328; 𝜂2=.047; p<.01, revealed a statistically significant difference between the treatment and comparison groups from pre-test to post-test. These findings show that the experimental group participants were highly beneficial in developing their content knowledge and conceptual knowledge through the active learning approach and technology-based learning strategy using Vygotsky’s socio-cultural learning theory. The JCLGS learning environment representing Vygotsky’s socio-cultural learning theory modestly influenced the procedural knowledge learning of the experimental group learners’. Although the lecture method affected the comparison group students’ knowledge development in calculus during the academic semester, the impact was not comparable to that of the active learning approach and technology-based learning strategy. The major reason for this was the attention and care given to the active learning intervention integrated with GeoGebra by the researcher, data collectors, and research participants. Overall findings showed that the active learning intervention allowed the experimental group students to considerably enhance their conceptual knowledge and content knowledge in calculus. Learners also positively changed their opinion towards calculus and GeoGebra. The intervention was a group interactive environment that allowed students’ to be reflective, share prior experience and knowledge, and independent learners. As a matter of fact, educators are advised to model such a combination of active learning approach and technology-based learning strategy in their classroom instructional setting and practices. Consequently, their learners will adequately benefit to understand the subject matter and positively change their opinion towards university mathematics.
Mathematics Education
Ph. D. (Mathematics, Science and Technology Education)