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.

Title: Football match of AC Sparta Praha as a product for marketing Objectives: The main goal of the thesis is to elaborate suggestions and recommendations for improvement of marketing activities on the match-day. Methods: The improvements are based on electronic interviewing AC Sparta Prague fans and personal interviews with marketing experts in the field of football. Direct observation without use of technique was used as well. Results: Recommendations and suggestions are focused on making more effective use of the marketing elements of AC Sparta Prague football matches. The main reasons for fans dissatisfaction are lack of toilets and its uncleanness, low capacity of food stall and slow operation. Safety and match-day program can be improved as well due to fans' opinion. Key words: Sport marketing, fans' satisfaction, marketing research

Orientador: Marcos David Ferreira / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Agrícola / Made available in DSpace on 2018-08-18T02:09:22Z (GMT). No. of bitstreams: 1 Spricigo_PolianaCristina_M.pdf: 1670459 bytes, checksum: a93fd04b87a2e895813ed5e1bb330237 (MD5) Previous issue date: 2011 / Resumo: Em flores de corte, como crisântemos, a manutenção da turgescência é um dos principais fatores a serem controlados na pós-colheita. A perda de água, e consequentemente da qualidade, acarretam prejuízos tanto a produtores e consumidores e pode ocorrer em diversas etapas desde a colheita até a comercialização. O transporte ou armazenamento a seco, além de não permitir reidratação imediata, compromete a resposta à absorção posterior da água. Hastes que permanecem fora da água após o corte apresentam bloqueio vascular, levando ao entupimento dos vasos xilemáticos. Além desses fatores, a qualidade da água utilizada também influência na conservação pós-colheita. A perda de água está ligada à aceleração de processos que levam a senescência da haste, como aumento na atividade respiratória e na transpiração. Para a avaliação da turgescência em tecidos florais, é utilizado com frequência o teor relativo de água. No teor relativo de água, dentre outras características, é necessário que se faça a destruição da amostra para chegar a um valor estimado em porcentagem de água nos tecidos. Para suprir a necessidade de avaliar a turgescência em tecidos florais sem a necessidade de destruí-la, foi criado o equipamento Wiltmeter®. Por meio da pressão de turgescência ele oferece valores instantâneos da condição hídrica dos tecidos, sem que para isso haja necessidade de destruir a amostra. O objetivo deste trabalho foi avaliar a turgescência de hastes de crisântemos ao longo do manejo pós-colheita, pelos métodos do teor relativo de água (%) e pressão de turgescência (kPa) correlacioná-los e avaliar a mudança de parâmetros de qualidade em decorrência da perda de água. Foram realizadas análises físicas e químicas como: variação da massa fresca, teor relativo de água, pressão de turgescência, taxa de absorção, taxa de transpiração, coloração, carboidratos solúveis e número de botões, flores entreabertas e abertas. Para avaliar as hastes, estas foram submetidas a tratamentos com água destilada e potável, e neste experimento a água potável obteve melhor desempenho na manutenção da turgescência das hastes. Diferentes períodos de armazenamento a seco e posterior reidratação foram avaliados, onde o tratamento testemunha que não foi submetido a seca manteve melhor a hidratação de flores e folhas. Diferentes alturas de corte da base da haste foram testadas, onde o maior corte inicial obteve melhor resultado. Avaliou-se a turgescência em flores de corte ao longo do manejo pós-colheita, evidenciando a eficiência e sensibilidade do Wiltmeter®, sendo possível correlacionar os resultados obtidos com os métodos de teor relativo de água, e também verificar alteração da qualidade das hastes em decorrência da perda de água / Abstract: In cut flowers such as chrysanthemums, maintenance of turgor is one of the main factors to be controlled in postharvest. The loss of water, and consequently the quality, is detrimental to producers and consumers and can occur at various stages from harvesting to marketing. The transportation or dry storage did not allow immediate rehydration, compromising the subsequent response to absorption of water. Stems that remain outside the water after cutting show vascular blockage, leading to clogging of the xylem. Besides these factors, the quality of water used also influence the postharvest conservation. Water loss is linked to the acceleration of processes leading to senescence of the stem, such as increased respiratory activity and transpiration. For the evaluation of turgidity in floral tissues, is often used the relative water content. In relative water content, among other characteristics, it is necessary to make the destruction of the sample to reach an estimated percentage of water in tissues. To meet the need to evaluate the turgidity in floral tissues without the need to destroy it, was created the equipment Wiltmeter ®. By means of turgor pressure values it offers the water status of tissues, without this being necessary to destroy the sample. The aim of this study was to evaluate the turgidity of stems of chrysanthemums during the postharvest management, by the methods of relative water content (%) and turgor pressure (kPa) and correlate them to evaluate the change in quality parameters due water loss. Were performed physical and chemical analysis: variation of fresh mass, relative water content, turgor pressure, absorption rate, transpiration rate, color, soluble carbohydrates and number of buds, flowers and open ajar. To evaluate the stems, they were subjected to treatment with distilled water and tap water, and tap water in this experiment showed the best performance in maintaining the turgidity of the stems. Was also executed various periods of dry storage and rehydration, where the control treatment that was not subjected to drought maintained better hydration of petals and leaves. Also tested were cut at different heights from the base of the stem, where the largest initial cut had the best results. Evaluation of turgidity in cut flowers throughout the postharvest management, demonstrated the efficiency and sensitivity of Wiltmeter®, was possible correlate their results with the methods of relative water content, and also check the deterioration of the stems due water loss / Mestrado / Tecnologia Pós-Colheita / Mestre em Engenharia Agrícola

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.

I denna uppsats undersöker vi några gymnasieelevers förståelse av begreppet derivata. Vi gör detta genom att presentera teoretiska verktyg för att kunna mäta begreppsförståelse. Vi försöker bedöma elevernas kunskaper genom att eleverna får göra ett skriftligt prov och några av eleverna en efterföljande intervju. Undersökningen gjordes på en skola och omfattar 8 elever i en naturvetare klass i årskurs 2. De resultat som vi presenterar visar på att eleverna inte uppnår någon högre konceptuell förståelse. Det verkar inte heller finnas någon större skillnad i begreppsförståelse mellan elever som har höga betyg, MVG, och elever med lägre betyg.

Syftet med studien har varit att öka kunskapen om hur lärare använder konkret material i sin undervisning för att elever därigenom ska nå ett abstrakt matematiskt tänkande. Studien har genomförts med en kvalitativ metod i form av intervjuer med lärare i årskurs F-3. Insamlade data har sorterats, analyserats och presenterats i relation till APOS-teorins fyra faser av matematikinlärning. Enligt teorin är eleven till en början i behov av att använda konkret material, för att sedan börja göra kopplingar mellan det konkreta och det abstrakta. Därefter når de ett helt abstrakt tänkande och kan slutligen skapa sig matematiska strukturer som de kan använda i andra sammanhang. Resultatet av studien visar att lärare upplever att elevers väg från konkret till abstrakt matematik är högst individuell och att det inte är säkert att alla elever når ett abstrakt tänkande. Slutsatsen är att detta skulle kunna bero på att syftet med det konkreta materialet eller kopplingen mellan det konkreta och det abstrakta inte tydliggörs tillräckligt. En ytterligare anledning skulle kunna vara att matematikundervisningen i alltför hög grad kopplas till situationer och material som eleverna redan har en vardaglig relation till.

The intent of this thesis is to investigate student approaches to linearity within a linear algebra context, focusing on definitional, computational, and theoretical skills. Linear algebra’s abstract nature constitutes a major challenge for a significant sector of STEM students, with the course often serving as undergraduates’ first encounter with mathematical proofs and extrapolations. The current student struggle is reflected through the prominent gap in knowledge derived from a lack of a concrete understanding of rudimentary concepts (like linearity), pivotal to student success. As such, this investigation aimed to bridge this gap by considering students’ modes of thinking regarding the elementary notion of linearity to improve the current course delivery and curriculum. Students were given three assessment questions targeting different skills integral to the mastery of linearity. Their responses were categorized using Action, Process, Object, Schema (APOS) and analyzed through Sierpinska’s (2000) proposed modes of thinking. About 26% of the participants responded correctly to question 1, 77% to question 2, and 59% to question 3. The analytic mode proved pivotal, specifically when considering definition application and computational abilities. The synthetic-geometric mode, however, was integral to the practical application of the concept. Further discussion and suggestions regarding the results and their implications on the current structure of linear algebra instruction are provided.

Made available in DSpace on 2016-04-27T16:59:05Z (GMT). No. of bitstreams: 1 Eneias de Almeida Prado.pdf: 1754113 bytes, checksum: 4f347ae679598e17485474c1d80187f6 (MD5) Previous issue date: 2010-05-07 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / The purpose of this study was to indentify the basis conception of the IR vetorial space finitely generated by students who concluded an extension course in Linear Algebra. The relevance of the research is in the importance attributed to this discipline in the professional education of Exact Sciences and others, and in the need to investigate its teaching and its learning, according to the opinion of many researchers, as Dubinsky (1991;2001); Dorier et al (1997); Machado e Bianchini (2009). For such, the theoretical ground used was APOS, developed by Dubinsky and collaborators that allowed the refinement of a genetical decomposition for the notion of basis which approached three points of view from this notion: maximal group of vectorials linearly independent; minimal group of generating vectors and justaposition between the two latters. The data survey was held by semistructured interviews to 10 subjects graduating from the same extension course, characterizing it as a qualitative case study. The analysis held indicates that five students built an object conception and incorporated the notion of dimension to their scheme, using indistincvely the dimension to one of three notions of basis. A student was able to build a process conception and another, an action conception. After two courses of Linear Algebra, the studens conceived basis, mainly, with being the independent linearly generating group, and only two of the interviewed perceived the equivalence among the points of view approached / Este estudo teve o objetivo de identificar a concepção de base de um IR-espaço vetorial finitamente gerado de alunos que concluíram um curso de extensão em Álgebra Linear. A relevância da pesquisa reside na importância atribuída a essa disciplina na formação de profissionais das Ciências Exatas e afins, e na necessidade de investigar seu ensino e sua aprendizagem, conforme opinião de vários pesquisadores, como Dubinsky (1991; 2001); Dorier et al. (1997); Machado e Bianchini (2009). Para tanto, utilizou-se o aporte da teoria APOS, desenvolvida por Dubinsky e colaboradores que permitiu o refinamento de uma decomposição genética para a noção de base que abordou os três pontos de vista dessa noção: conjunto maximal de vetores linearmente independentes; conjunto minimal de vetores gerador e a justaposição entre as duas anteriores. A coleta de dados foi realizada por meio de entrevistas semiestruturadas a 10 sujeitos concluintes de um mesmo curso de extensão, caracterizando-se como um estudo qualitativo de caso. A análise realizada indica que cinco estudantes construíram uma concepção objeto e incorporaram a noção de dimensão a seu esquema, utilizando indistintamente a dimensão a uma das três noções de base. Um estudante mostrou ter construído concepção processo e outro, concepção ação. Após dois cursos de Álgebra Linear, os estudantes concebem base, sobretudo, como sendo um conjunto gerador linearmente independente, e só dois dos entrevistados perceberam a equivalência entre os pontos de vista abordados

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)