Abstract The objective of this study was to explore the feasibility of using the One-Variable Linear Equation Situational Test to investigate the development of the concept of one-variable linear equation for middle school students. The conduct of the first stage of this study was as follows: first, a thorough literature review was made; which was followed by interviews with middle and high school math teachers; finally a survey of the eighth-grade students was made using the ¡§One-Variable Linear Equation Concept Unstructured Questionnaire¡¨. The second stage of this study was to devise the One-Variable Linear Equation Situational Test. First, a detailed concept map of the one-variable linear equation was made based on the results obtained in the first stage of this study. Then, a situational test of one-variable linear equation was constructed according to the map. This test was to be used later in the one-to-one interviews with twelve seventh-grade students from a middle school in Kaohsiung, who did not learn the one-variable linear equation before. These twelve subjects were randomly devided into two groups: with guidance and without guidance. The data of the math achievement tests were also collected for these subjects. The results of the test interviews were analyzed and the feasibility of using the One-Variable Linear Equation Situational Test to investigate the development of the concept of one-variable linear equation was discussed. The analysis results of individual questions of the situational test of one-variable linear equation indicated that the concepts of one-variable linear equation for middle school students were detectable. This suggested that it was feasible to use the One-Variable Linear Equation Situational Test to investigate the development of the concept of one-variable linear equation for middle school students. All subjects who participated in this study already had some preliminary ideas of the one-variable linear equation. The factor of providing guidance would enhance the development of the concept of one-variable linear equation and it would reduce the differences in numbers of various concepts of one-variable linear equation developed among the high, medium and low achievement students. Therefore, the variable of with- or without-guidance would have some effects on the detectable concepts of one-variable linear equation by the One-Variable Linear Equation Situational Test.

The main purpose of this study is to investigate eighth-grade students¡¦ understanding of the equal sign and analyzed error types of quadratic equation in one variable. To achieve this purpose, the investigator did a survey and development instruments. Participants were 215 eighth-grade students who formed a convenient sample. There are three results. First, participants with a relational definition of the equal sign added to about 80% of the sample. Second, the performance of students with relational definitions is higher than the performance of students with operational definitions. Third, students¡¦ understanding of the equal sign was related to their respective problem-solving performance on quadratic equation in one variable. In this study, participants with an operational definition of the equal sign tended to guess randomly or leave a blank. Problem-solving performance of participants with a relational definition of the equal sign involved multiple strategies. The researcher suggested that teachers should strengthen students¡¦ understanding of equal sign and related students¡¦ prior algebraic knowledge.

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In the experiment, with forty-eight students as subjects, a serie s of nonsense syllables (DAX, MEF, TOV, VIC, YOP, ZIP, and ZIL) were to be associated with four geometric figures. The task was so arranged that Zip applied to all figures, Dax and Vic to subsets of two figures each, and the remaining were individual labels. In each of three experiments there was an experimental group that received pre-response cueing by means of an analogy which involved hierarchic concepts in the same general form, i. e ., animal, wild, tame, and individual names. The results suggest that the order and timing of the presentation of the cues were varied in the three separate experiments. Groups that received prior analogy versus groups not given the analogy were more successful in ordering the random stimuli. Further, the order of presentation of the cues had no significant effect on the ability of the subjects to order the random stimuli.

This study reanalyzed a part of the national data of the responses of 288 students in grades 6 to 8 on the ¡§One-Variable Linear Equation Virtual Situational Test¡¨ collected by Professor Pao-Kuei Wu from August 1, 2001 through July 31, 2003. The analyses were based on the ¡§One-Variable Linear Equation Conceptual Tables¡¨. The results of the analyses are the following. I. The use of variables A. Compared to 7th and 8th graders, 6th graders would first solve the numerical arithmetic and solve the unknown parts next. But if the students could not handle the unknown parts, the 6th graders tended to ignore or even not list the unknown variable in the equations. B. When encountering the unknown situations, most 6th graders are not accustomed to using symbols to represent unknown variables. Instead, they would observe the numerical components first to try to deduce what the unknown variable would be, and proceed from there. Some students would even set up some constants to represent those unknown variables. These results indicate that the 6th graders¡¦ ability to use symbolic representation is still in the beginning stages. C. In the unknown virtual situations, the majority of 7th graders were able to use symbolic representations. However, most of them would use pictorial representations such as ¡¼, instead of alphabetical representations such as x, y and z. Moreover, many students use the same symbols to represent different variables; this shows that although the 7th graders know to use symbols to represent unknown variables, they still are not able to fully comprehend unknown variables. Hence, the 7th graders¡¦ ability to use symbolic representation is in the transitional stage. D. When encountering unknown virtual situations, the majority of the 8th graders would able to use the numerical symbols such as x, y and z to represent the unknown variables. The frequency of using pictorial representations such as ¡¼ becomes less and less, and the tendency to use the same symbols to represent different variables is decreasing. All these indicate that the 8th graders¡¦ development of the concept of unknown variables is maturing. II. The concept of problem solving A. The 6th graders¡¦ ability to use symbolic representation is still in the beginning stages: 1. They only deal with the simple part; for the more complicated part, they chose to ignore. 2. Due to their immature development of symbol representation, when encountering the two variable linear equation problems, they even do not have the ability to write the ¡¥complete¡¦ equation, not to mention to solve the equations. B. The 7th graders¡¦ ability to use symbolic representation is in the transitional stage: 1. Compared to the 6th graders, the 7th graders are more able to draw relationships among the different components of the problem. 2. The fact that the substantially decreasing proportion of 7th graders conceiving the unknown variable as a certain numeric compared with 6th graders means that the 7th graders have deeper recognition of unknown variables. 3. When encountering ¡¥simple¡¦ two-variable linear equation virtual situations, some 7th graders can translate at least one condition into an equation. This result shows that the 7th graders have developed some ability to translate the conditions embedded in the virtual situation into some equations. But when the situation gets more complicated, due to conception immaturity of solving two equations simultaneously, the 7th graders either solve each equation independently, or mess up and tangle the clues of all the conditions together. Moreover, they would use the same symbol to stand for different variables. C. The 8th graders¡¦ development of the concept of unknown variables is maturing: 1. Most of the 8th graders can use the clues of all the conditions in the virtual situation in a sufficient way. 2. Only a few 8th graders would use the same symbol to stand for different variables during their problem-solving procedure. This result indicates that the ability to use the symbolic way to represent unknown variables is more mature among the 8th grade students. 3. When encountering two-variable linear equation virtual situations, the 8th graders can formulate two independent equations and solve them simultaneously. This result shows that the 8th grade students possess more profound skills to solve two-variable linear equations. III. Proportion of answering questions correctly: In general, for simpler virtual problems, there does not exist many differences among grades. Whereas, for the more difficult virtual problems, the 8th graders outperform the 7th graders, and the 7th graders, in turn, outdo the 6th grade students.

This study used the multiple methods of classroom observation, interview with teachers and students, and paper-and-pencil test to investigate the ability development of seventh-grade students in learning two-dimensional linear systems of equations and the corresponding error analysis. Hopefully, the results of this study can be as a reference for the middle school math teachers to plan the suitable teaching strategies when they teach two-dimensional linear systems of equations to their students. At the beginning, the researcher entered two seventh-grade classrooms of one middle school in Kaohsiung to make the preliminary observations and let students (also the teachers) to get used to the appearance of the researcher in the classroom during the period that one-variable linear equations were taught. Subsequently, the formal observations were carried out for 40 class periods that two-dimensional linear systems of equations were taught. All the observations made about how teachers taught and how students learned were recorded and content analyzed. Two paper-and-pencil tests were administered during the period of preliminary observations. And three paper-and pencil tests were given during the period of the formal observations. All the test results were collected and analyzed in numerous ways. Based on the literature survey and the interviews with six middle school math teachers, all relevant abilities of mastering two-dimensional linear systems of equations were classified into three categories: Character Symbols (10 sub-abilities), Operational Principals (five sub-abilities), and Other Abilities (16 sub-abilities). Based on the results of the content analyses of classroom observations and the error analyses of five paper-and-pencils tests for each sub-abilities of mastering the subject, it was observed that during the period of developing the abilities on solving two-dimensional linear systems of equations, most students showed some signs of obstacles and puzzles. Even by the end of the course on two-dimensional linear systems of equations, most students still did not master the subject well. Based on the results of this study, it is proposed that the length of teaching period needs to be increased and more efficient learning strategies need to be introduced to the students when two-dimensional linear systems of equations are taught.

Since the time of Cauchy, integration theory has in the main been an attempt to regain the Eden of Newton. In that idyllic time [. . . ] derivatives and integrals were [. . . ] different aspects of the same thing. -Peter Bullen, as quoted in [24] The theory of integration has gone through many changes in the past centuries and, in particular, there has been a tension between the Riemann and the Lebesgue approach to integration. Riemann's definition is often the first integral to be introduced in undergraduate studies, while Lebesgue's integral is more powerful but also more complicated and its methods are often postponed until graduate or advanced undergraduate studies. The integral presented in this paper is due to the work of Ralph Henstock and Jaroslav Kurzweil. By a simple exchange of the criterion for integrability in Riemann's definition a powerful integral with many properties of the Lebesgue integral was found. Further, the generalized Riemann integral expands the class of integrable functions with respect to Lebesgue integrals, while there is a characterization of the Lebesgue integral in terms of absolute integrability. As this definition expands the class of functions beyond absolutely integrable functions, some theorems become more cumbersome to prove in contrast to elegant results in Lebesgue's theory and some important properties in composition are lost. Further, it is not as easily abstracted as the Lebesgue integral. Therefore, the generalized Riemann integral should be thought of as a complement to Lebesgue's definition and not as a replacement. / Ända sedan Cauchys tid har integrationsteori i huvudsak varit ett försök att åter finna Newtons Eden. Under den idylliska perioden [. . . ] var derivator och integraler [. . . ] olika sidor av samma mynt.-Peter Bullen, citerad i [24] Under de senaste århundradena har integrationsteori genomgått många förändringar och framförallt har det funnits en spänning mellan Riemanns och Lebesgues respektive angreppssätt till integration. Riemanns definition är ofta den första integral som möter en student pa grundutbildningen, medan Lebesgues integral är kraftfullare. Eftersom Lebesgues definition är mer komplicerad introduceras den först i forskarutbildnings- eller avancerade grundutbildningskurser. Integralen som framställs i det här examensarbetet utvecklades av Ralph Henstock och Jaroslav Kurzweil. Genom att på ett enkelt sätt ändra kriteriet for integrerbarhet i Riemanns definition finner vi en kraftfull integral med många av Lebesgueintegralens egenskaper. Vidare utvidgar den generaliserade Riemannintegralen klassen av integrerbara funktioner i jämförelse med Lebesgueintegralen, medan vi samtidigt erhåller en karaktärisering av Lebesgueintegralen i termer av absolutintegrerbarhet. Eftersom klassen av generaliserat Riemannintegrerbara funktioner är större än de absolutintegrerbara funktionerna blir vissa satser mer omständiga att bevisa i jämforelse med eleganta resultat i Lebesgues teori. Därtill förloras vissa viktiga egenskaper vid sammansättning av funktioner och även möjligheten till abstraktion försvåras. Integralen ska alltså ses som ett komplement till Lebesgues definition och inte en ersättning.

Made available in DSpace on 2016-04-27T16:58:33Z (GMT). No. of bitstreams: 1 Shirley Ferreira Signorelli.pdf: 4346228 bytes, checksum: a37d23996da101385ce1df722242946c (MD5) Previous issue date: 2007-10-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The purpose of this investigation was the elaboration and implementation of a virtual environment for a part time distance course for students of the Computer Science and Systems of Information. The students failed in a former discipline that approaches topics of elementary Mathematics. It happened in a particular university in the city of São Paulo. Our focus was on analyzing this environment, the tools and interactions that happened on distance mode, about the content of real function of one variable. The methodology used, design-based research, favored proposing and analyzing the activities about this topic, as well as their reorganization and, the re design of the environment. The Blackboard platform and the used tools were analyzed based on Chaves (2000) criteria, and it seemed efficient as a virtual environment for the learning in our course. However, we leaved some critical and suggestions for future works, mainly about the role of the tools for communication within part time distance courses. The analysis of the students and teacher speeches' was based on the Model of Argumentative Strategy (CASTRO et al, 2004) and allowed to raise some aspects on the understanding of Real Functions of one variable, as they were privileged in the different interactive spaces such as forum, chat, email and daily log. Aspects like the meaning production in Mathematics can be produced due to the authority of a teacher or of another student who is considered good by the classroom peers, or based on everyday language usage or on cultural characteristics. Moreover, we found that besides the students lack of prerequisite elementary mathematics, there is a lack of a culture for on-line courses / Nesta pesquisa objetivamos a elaboração e implementação de um ambiente virtual para um curso semipresencial, para estudantes dos cursos de Bacharelado em Ciência da Computação e Sistemas de Informação de uma instituição particular na cidade de São Paulo, dependentes na disciplina que aborda tópicos de pré-calculo. Nosso foco recaiu na análise do ambiente, da viabilidade das ferramentas e das interações que ocorreram a distância, no que tange o conteúdo de Funções de uma Variável Real. A metodologia utilizada, design research, permitiu propor e analisar as atividades sobre este tópico, incluindo a reestruturação e complementação deste ambiente. A plataforma Blackboard e as ferramentas foram analisadas segundo critérios definidos por Chaves (2000) e se mostraram eficazes como ambiente virtual de aprendizagem atendendo, para nosso curso, os critérios necessários. Entretanto, deixamos algumas críticas e sugestões para trabalhos futuros, principalmente quanto o papel do uso de ferramentas de comunicação em cursos semipresenciais. A análise dos discursos dos alunos e docente baseados no Modelo de Estratégia Argumentativa (CASTRO et al, 2004) permitiu levantar alguns aspectos sobre a compreensão de Funções de uma Variável Real que foram privilegiados nos diferentes espaços interativos como fórum, chat, e-mail e diário de rotina, tais como o fato de que a produção de significados em Matemática pode estar apoiada na autoridade do professor ou alunos bem vistos pela classe, na linguagem cotidiana e no aspecto cultural. Observamos ainda que além da falta de pré-requisitos de matemática básica, ainda há falta de cultura de trabalhos on-line