Philosophiae Doctor - PhD / The purpose of this study was to explore errors that are displayed by students when learning derivatives of trigonometric functions in an extended curriculum programme. The first aim was to identify errors that are displayed by students in their solutions through the lens of the APOS theory. The second aim was to address students' errors by using the two principles of Vygotsky's socio-cultural theory of learning, namely the zone of proximal development and more knowledgeable others. The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The participants in this study comprised a group of students who registered for mathematics in the ECP at Cape Peninsula University of Technology, Cape Town, South Africa. The study was piloted in 2008 with a group of twenty students who registered for mathematics in the ECP for Chemical Engineering. In 2009 thirty students from the ECP registered for mathematics in Chemical Engineering were selected to participate in the main study. This study was conducted over a period of four and a half years. Data collection was done through students' written tasks; classroom audio and video recordings and indepth interviews. Data were analysed through categorising errors from students' written work, and finding common themes and patterns in audio and video recordings and from the in-depth interviews. The findings of this study revealed that students committed interpretation, arbitrary, procedural, linear extrapolation and conceptual errors. Interpretation errors arise when students fail to interpret the nature of the problem correctly owing to over-generalisation of certain mathematical rules. Arbitrary errors arise when students behave arbitrarily and fail to take account of the constraints laid down in what is given. Procedural errors occur when students fail to carry out manipulations or algorithms although they understand concepts in problem. Linear extrapolation errors happen through an overgeneralisation of the property f (a + b) = f (a) + f (b) , which applies only when f is a linear function Conceptual errors occur owing to failure to grasp the concepts involved in the problem or failure to appreciate the relationships involved in the problem. The findings were consistent with literature indicated that errors are based on students’ prior knowledge, as they over-generalise certain mathematical procedures, algorithms and rules of differentiation in their solutions. The use of learning activities in the form of written tasks; as well as classroom audio and video recordings assisted the lecturer to identify and address errors that were displayed by students when they learned derivatives of trigonometric functions. The students claimed in their interviews that they benefited from class discussions as they obtained immediate feedback from their fellow students and the lecturer. They also claimed that their performances improved as they continued to practice with the assistance of more knowledgeable students, as well as the lecturer. This study supports the view from the literature that identification of errors has immense potential to address students’ poor understanding of derivatives of trigonometric functions. This thesis recommends further research on errors in various sections of Differential Calculus, which is studied in an extended curriculum programme at Universities of Technology in South Africa.

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Doctor of Philosophy / Department of Mathematics / Andrew Bennett / Engineering majors and Mathematics Education majors are two groups that take the basic, core Mathematics classes. Whereas Engineering majors go on to apply this mathematics to real world situations, Mathematics Education majors apply this mathematics to deeper, abstract mathematics. Senior students from each group were interviewed about “function” and “accumulation” to examine any differences in learning between the two groups that may be tied to the use of mathematics in these different contexts. Variation between individuals was found to be greater than variation between the two groups; however, several differences between the two groups were evident. Among these were higher levels of conceptual understanding in Engineering majors as well as higher levels of confidence and willingness to try problems even when they did not necessarily know how to work them.

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Doctor of Philosophy / Electrical and Computer Engineering / Steven Warren / Linear Systems (a.k.a., Signals and Systems) is an important class in an Electrical Engineering curriculum. A clear understanding of the topics in this course relies on a well-developed notion of lower-level mathematical constructs and procedures, including the roles these procedures play in system analysis. Students with an inadequate math foundation regularly struggle in this class, as they are typically able to perform sequences of the underlying calculations but cannot piece together the higher-level, conceptual relationships that drive these procedures. This dissertation describes an investigation to assess and improve students’ higher-level understanding of Linear Systems concepts. The focus is on the topics of (a) time-domain, linear time-invariant (LTI) system response visualization and (b) Fourier series conceptual understanding, including trigonometric Fourier series (TFS), compact trigonometric Fourier series (CTFS), and exponential Fourier series (EFS). Support data, including exam and online homework data, were collected since 2004 from students enrolled in ECE 512 - Linear Systems at Kansas State University. To assist with LTI response visualization, two online homework modules, Zero Input Response and Unit Impulse Response, were updated with enhanced plots of signal responses and placed in use starting with the Fall 2009 semester. To identify students’ conceptual weaknesses related to Fourier series and to help them achieve a better understanding of Fourier series concepts, teaching-learning interviews were applied between Spring 2010 and Fall 2012. A new concept-based online homework module was also introduced in Spring 2011. Selected final exam problems from 2007 to 2012 were analyzed, and these data were supplemented with detailed mid-term and final exam data from 77 students enrolled in the Spring 2010 and Spring 2011 semesters. In order to address these conceptual learning issues, two frameworks were applied: Bloom’s Taxonomy and APOS theory. The teaching-learning interviews and online module updates appeared to be effective treatments in terms of increasing students’ higher-level understanding. Scores on both conceptual exam questions and more traditional Fourier series exam questions were improved relative to scores received by students that did not receive those treatments.

The main purposes of this study is to investigate first year calculus students&rsquo / understanding of formal limit concept and change in their understanding after limit instruction designed by the researcher based on APOS theory. The case study method was utilized to explore the research questions. The participants of the study were 25 students attending first year calculus course in Middle East Technical University in Turkey. Students attended five weeks instruction depending on APOS theory in the fall semester of 2007-2008. Limit questionnaire including open-ended questions was administered to students as a pretest and posttest to probe change in students&rsquo / understanding of limit concept. At the end of the instruction a semi-structured interview protocol developed by the researcher was administered to all of the students to explore students&rsquo / understanding of limit concept in depth. The interview results were analyzed by using APOS framework. The results of the study showed that constructed genetic decomposition was found to be compatible with student data. Moreover, limit instruction was found to play a positive role in facilitating students&rsquo / understanding of limit concept.

This dissertation reports on an investigation of transition-to-proof students' understanding of proof by contradiction. A plethora of research on the construction aspect of proof by contradiction is available and suggests that the method is one of the most difficult for students to construct and comprehend. However, there is little research on the students' comprehension of proofs and, in particular, proofs by contradiction. This study aims to fill this gap in the literature. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, this study proposes a preliminary genetic decomposition for how a student might construct the concept `proof by contradiction' and a series of five teaching interventions based on this preliminary genetic decomposition. Data was analyzed in two ways: (1) group analysis of the first two teaching interventions to consider students' initial conceptions of the proof method and (2) case study analysis of two individuals to consider how students' understanding developed over time. The genetic decomposition and teaching interventions were then revised based on the results of the data analysis. This study concludes with implications for teaching the concept of proof by contradiction and suggestions for further research on the topic.

Philosophiae Doctor - PhD / 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./{x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate ./{O) as the limit value. This method is Within the context of a sequence everyday language acted as an epistemological obstacle in interiorising actions into processes. For example, in finding lim (_1)n ,the majority of x~oo n the subjects obtained the correct answer O. It was however revealed that such an answer was obtained by using an inappropriate method. The subjects substituted one big value for n in the formula. The result obtained was the number close to O. Then 0 was taken as the limit value because the subjects interpreted the word 'approaches' as meaning 'nearer to'. Other subjects rounded off the result. In everyday life when one object approaches another, we might say that they are nearer to each other. It seems that in this case the 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. Within the context of a sequence everyday language acted as an epistemological obstacle in interiorising actions into processes. For example, in finding lim (_1)n ,the majority of x~oo n the subjects obtained the correct answer O. It was however revealed that such an answer was obtained by using an inappropriate method. The subjects substituted one big value for n in the formula. The result obtained was the number close to O. Then 0 was taken as the limit value because the subjects interpreted the word 'approaches' as meaning 'nearer to'. Other subjects rounded off the result. In everyday life when one object approaches another, we might say that they are nearer to each other. It seems that in this case the subjects used this meaning to get 0 as the limit value. We also round off numbers to the nearest unit, tenth, etc. The limit value is however a unique value that is found by using the limiting process of 'tending to' or 'approaching' which requires infinite values to be considered. Some are computed and others are contemplated. In constructing the coordinated process schema, f(x) ~ L as x ~ a, over-generalisation and everyday language were still epistemological obstacles. Subjects still perceived the limit value to exist where the function is defined. The limit was also taken as a bound, lower or upper bound. In a case where the function was represented in a tabular form, the first and the seemingly last functional value that appeared in the table of values were chosen as the limit values. Limit values were also approximated. In constructing the coordinated process an ~ L as n ~ 00, representation, generalisation and everyday language also acted as epistemological obstacles. An alternating sequence was perceived as not one but two sequences. Since the subjects will have met situations where convergence means meeting at a point, as in the case of rays of light, a sequence was said to converge to a number that did not change in the given decimal digits. For example, the limit of the sequence {3.1, 3.14, 3.141, 3.1415, ... } was taken to be 3 or 3.1 as these are the digits that are the same in all the terms. In encapsulating processes into objects, everyday language also acted as an epistemological obstacle. When subjects were asked what they understood the limit to be, they said that the limit is a boundary, an endpoint, an interval, or a restriction. Though these interpretations are correct they are however, inappropriate if used in the technical context such as the mathematical context. While some subjects referred to the limit as a noun to show that they refer to it as an object, other subjects described the limit in terms of the processes that give rise to it. That is, it was described in terms of either the domain process or the range process. This is an indication that full encapsulation of processes into objects was not achieved by the subjects. The role of language and symbolism has been identified in making different connections in building the concept of limit as: representation of mathematical objects, translation between modes of representation, communication of mathematical ideas, manipulation of surface or syntactic structures and the overcoming of epistemological obstacles. In representation some subjects were aware of what idea some symbolism signified while other subjects were not. For example, in the context of limit of a sequence, most subjects took the symbolism that represented an alternating sequence, an = (-lr, to represent two sequences. The first sequence was seen as {I, I, 1, 1,... } and the second as {-I, -1, - 1, -1, ... }.This occurred in all modes of representation. In translating from one mode of representation to another, the obscurity of the symbol lim/ex) = L was problematic to the students. This symbol could not be related to its X~a equivalent form lex) ~ L as x ~ a. The equal sign, '=', joining the part lex) and L does not reflect the process ofj{x) tending to L, rather it appears as if it is the functional value that is equal to L. Hence, instead of looking for the value that is approached the subjects chose one of the given functional values. The part of the symbol lim was a x~a source of difficulty in translating the algebraic form to the verbal or descriptive. The subjects saw this part to mean "the limit of x tends to a" rather than seeing the whole symbolism as the limit of j{x) as x tends to a. Some subjects actually wrote some formulae in the place of L because of this structure, e.g., lim/ex) = 2x. These subjects x~a seemed to have concentrated on the part lex) = .... This is probably because they are used to situations where this symbolism is used in representing functions algebraically. In communicating mathematical ideas the same word carried different meanings for the researcher and for the subjects in some cases. For example, when the subjects were asked what it means to say a sequence diverges, one of the interpretations given was that divergence means tending to infinity. So, over-generalisation here acted as an epistemological obstacle. Though a sequence that tends to infinity diverges, this is not the only case of divergence that exists and therefore cannot be generalised in that way. The manipulation of the surface structures was done instrumentally by some subjects. For 1 . ti di 1·.J x 2 examp e, m in mg im +29 - 3 ,urdmugri the mam.pu 1ati.on some subjiects 0 bttaaime d x.... o x 2 part of the expressions such as ~ by rationalising or .:;- by using L'Hospital's rule 2x x which needed to be simplified. Instead of simplifying the expressions further at this stage, the substitution of 0 was done. So, .o2. = 0 was obtained as the answer. This shows that neither the reasons for performing the manipulations, nor the process of rationalising for example was understood. The result was still an indeterminate form of limit. The numerator was also not yet in a rational form. In using language to overcome epistemological obstacles, subjects were exposed to a piece of knowledge that falsified the knowledge they had so that they could rethink replacing the old with the new. In some cases, this was successful but in others, the subjects did not surrender these old pieces of knowledge. For example, when asked what they understood the 'rate of change' to mean, the majority of the subjects associated the rate of change with time only. However, when referred to a situation that required them to find the rate of change of an area with respect to radius, some subjects changed their minds but others did not. Those who did not change their minds probably did not make any connections between ideas under discussion. The implications for practice of the findings include: In teaching one should discuss explicitly how answers to tasks concerning limits are obtained. The idea of the limit value as a unique value can only be recognized if the process by which it is obtained is discussed. It should not be taken for granted that students who respond correctly understand the answers. It is evident from the study that even when correct answers are given, improper methods may have been used. Hence, in investigating epistemological obstacles attention should also be paid to correct answers. Also beyond this, students should be exposed to different kinds of representation of the limit concept using simple functions and using a variety of examples of sequences. Words with dual or multiple meaning should also be discussed in mathematics classrooms so that students may be aware of the meanings they carry in the mathematical context. Different forms of indeterminate states of limit should be given attention. Relations should also be made between the surface structures and the deeper structures.

Made available in DSpace on 2016-04-27T16:57:20Z (GMT). No. of bitstreams: 1 Fabio Rodrigues de Siqueira.pdf: 2570104 bytes, checksum: 932e47e8f7aba3ad1adce0aba39a8a34 (MD5) Previous issue date: 2012-10-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This research has as objective to check if the proposal of an algorithm to be converted into a computer program can contribute to the learning of an object of mathematical study. For that, we have formulated the following research question: how can the elaboration of a converted algorithm into a computer program help high school students in the learning of the zeros of the 2nd degree polynomial function? The research was conducted in two stages. The first stage, with two initial activities, consisted of two parts: the first one with a 1st year high school student in order to verify if the activities were appropriate to the research and in the second part, after the application of the two activities, we have selected four participants for the second stage that had presented different results. After the analysis of the first stage development, the activities were improved to the second one, composed of three activities, among which the software Visualg 2.0 was used for the algorithm edition and its conversion to a computer program. The APOS theory by Ed Dubinsky, theoretical support of the research, presents the action levels, process, object and scheme, that allow the verification of the individual`s capacity to develop actions over an object and think about its properties. The research participants had improvements in their learning, because besides developing a computer program to determine the zeros of 2nd degree polynomial function, they have started to elaborate other functions previewing their possible solutions, presenting all the levels of the APOS theory. As research methodology we have adopted the Design Experiments. We have justified its use, for adjustments could be done during the work development. Analyzing the activities which were done we have concluded that the students have achieved a satisfactory learning level over the object of study / Esta pesquisa tem como objetivo verificar se a proposta de um algoritmo a ser convertido em um programa de computador pode contribuir na aprendizagem de um objeto de estudo matemático. Para tanto, formulamos a seguinte questão de pesquisa: como a elaboração de um algoritmo convertido em um programa pode auxiliar alunos do ensino médio na aprendizagem dos zeros da função polinomial do 2º grau? A pesquisa foi realizada em duas etapas. A primeira etapa, com duas atividades iniciais, foi composta por dois momentos: o primeiro com um aluno da 1ª série do ensino médio a fim de verificar se as atividades estavam adequadas à pesquisa e no segundo momento, após a aplicação das duas atividades, selecionamos quatro participantes para a segunda etapa que apresentaram resultados diferenciados. Após análise do desenvolvimento da primeira etapa, as atividades foram aprimoradas para a segunda, composta de três atividades, entre as quais foi utilizado o software Visualg 2.0 para edição do algoritmo e sua conversão para programa. A teoria APOS de Ed Dubinsky, aporte teórico da pesquisa, apresenta os níveis ação, processo, objeto e esquema, que permitem a verificação da capacidade do indivíduo em desenvolver ações sobre um objeto e raciocinar sobre suas propriedades. Os participantes da pesquisa tiveram melhorias em seu aprendizado, pois além de desenvolver um programa de computador para se determinar os zeros de funções polinomiais do 2º grau, passaram a elaborar outras funções já prevendo as possíveis soluções, apresentando todos os níveis da teoria APOS. A metodologia adotada nessa pesquisa foi o Design Experiments. Justificamos seu uso, pois adequações puderam ser realizadas durante o desenvolvimento do trabalho. Ao analisarmos as atividades realizadas concluímos que os alunos atingiram um nível de aprendizagem satisfatório acerca do objeto de estudo

This study investigated the effect of integration of GeoGebra into the teaching of circle geometry on Grade 11 students’ achievement. The study used a quasiexperimental, non-equivalent control group design to compare achievement, Van Hiele levels, and motivation of students receiving instruction using GeoGebra and those instructed with the traditional ‘talk-and-chalk’ method. Two samples of sizes n = 22 (experimental) and n = 25 (control) drawn from two secondary schools in one circuit of the Vhembe district, Limpopo Province in South Africa were used. A pilot study sample of size n = 15, was carried out at different schools in the same circuit, in order to check the reliability and validity of the research instruments, and statistical viability. The results of the pilot study were shown to be reliable, valid and statistically viable. The study was informed by the action, process, object, schema (APOS) and Van Hiele theories, as the joint theoretical framework, and the literature search concentrated on technology integration, especially GeoGebra, in the teaching and learning of mathematics. The literature was also reviewed on the integration of computer technology (ICT) into mathematics teaching and learning, ICT and mathematical achievement, and ICT and motivation. The study sought to answer three research questions which were hypothetically tested for significance. The findings of this study revealed that there was a significant difference in the achievement of students instructed with GeoGebra compared to those instructed with the traditional teaching method (teacher ‘talk-andchalk’). The average achievement of the experimental group was higher than that of the control group. Significant differences were also established on the Van Hiele levels of students instructed with GeoGebra and those instructed without this software at Levels 1 and 2, while there were no significant differences at Levels 3, 4 and 5. The experimental group achieved a higher group average at the visualisation and analysis Van Hiele levels. It was also statistically inferred from questionnaires through chi-square testing, that students instructed with GeoGebra were more motivated to learn circle geometry than those instructed without the software / Mathematics Education / M. Sc. (Mathematics Education)