Dissertations / Theses on the topic 'Visual algebra'

Master of Science<br>Mathematics<br>Andrew G. Bennett<br>This study utilizes eye-tracking technology as a tool to measure college algebra students’ mathematical noticing as defined by Lobato and colleagues (2012). Research in many disciplines has used eye-tracking technology to investigate the differences in visual attention under the assumption that eye movements reflect a person’s moment-to-moment cognitive processes. Motivated by the work done by Madsen and colleagues (2012) who found visual differences between those who correctly and incorrectly solve introductory college physics problems, we used eye-tracking to observe the visual attention difference between correct and incorrect solvers of college algebra problems. More specifically, we consider students’ visual attention when presented tabular representations of linear functions. We found that in several of the problems analyzed, those who answered the problem correctly spend more time looking at relevant table values of the problem while those who answered the problem incorrectly spend more time looking at irrelevant table labels x, y, y = f(x) of the problem in comparison to the correct solvers. More significantly, we found a noteworthy group of students, who did not move beyond table labels, using these labels solely to solve the problem. Future analyses need to be done to expand on the differences between eye patterns rather than just focusing on dwell time in the relevant and irrelevant areas of a table.

The purpose of this study was to examine the effectiveness of a technology-enhanced unit on slope in algebra. The technology used in the study was the Topological Panorama Camera (Topocam). The research questions explored the learning and transfer of knowledge about slope and the engagement level of students during Topocam learning activities. The Topocam is a computer-controlled camera that moves on a modular track while it scans a scene through a vertical slit. Students can program the speed of the camera and frequency of pictures. They then witness the results of time and motion in the image created by the camera. Data for this study were collected from a pretest/posttest, as well as from observations of indicators of engaged learning. The research population consisted of 46 students from three classes of Algebra I students. Three classroom teachers each taught a unit on slope, while a fourth teacher conducted the activities with the Topocam for all the classes. The classroom activities focused on the concept of slope as a rate of change utilizing coordinate grids. The Topocam activities involved students in collaboratively making and testing predictions about slope. The findings of the study indicate that student learning did occur with this technology-enhanced unit on slope in algebra. Students showed statistically significant improvement in understanding slope and in transferring that concept to other situations. Since technology was only part of the unit presentation, the amount of learning attributed to the Topocam activities cannot be determined. However, students demonstrated a high degree of engagement in learning while working with the Topocam which suggests that the activities were a factor. A low correlation between students’ slope unit test scores and previous algebra performance may indicate that students who have not been successful in algebra were more successful in the technology-enhanced unit. Some variation was found between classes that could be attributed to other factors than the Topocam.

Apesar de muito trabalho ter sido dispendido sobre linguagens de consulta a Sistemas de Gerenciamento de Bancos de Dados Relacionais, existem somente dois paradigmas básicos para essas linguagens, que são representados pela Structured Query Language – SQL e pela Query by Example – QBE. Apesar dessas linguagens de consultas serem computacionalmente completas, elas tem a desvantagem de não permitir ao usuário nenhuma interação gráfica com a informação contida na base de dados. Um dos principais desenvolvimentos na área de base de dados diz respeito às ferramentas que proveêm aos usuários um entendimento simples da base de dados e uma extração amigável da informação. A linguagem descrita neste trabalho possibilita que usuários criem consultas graficamente por meio de diagramas de fluxo de dados. Além da linguagem de consulta gráfica, este trabalho mostra também a ferramenta de apoio Data Flow Query Language - DFQL, que é um editor/executor de consultas construído para suportar essa linguagem, através de um conjunto de operadores representados graficamente, e a execução desses diagramas, analisando a rede e gerando os comandos correspondentes em SQL para realização da consulta. Esses comandos são submetidos ao sistema de gerenciamento de banco de dados e o resultado é mostrado/gravado conforme a consulta feita.<br>In spite of many works done on query languages, all existing languages are direct extensions of Structured Query Language – SQL and query-By-Example – QBE. These two languages were developed in the beginning of the Relational Database Management Systems – RDBMS development. Althoug these languages are computationally complete, they take the disadvantage of not supporting graphical interaction with data. One of the the main developments in the database area concerns tools to provide users a simple understand of database content, and friendly extraction of the information. The language described in this work enables users to create graphical queries using data flow diagrams. Besides the graphical query language, this work also shows the Data Flow Query Language - DFQL tool. This tool is a query editor/executer that supports this language, using a set of operators represented graphicaly, and the diagram execution is done by analising the network and producing the respective commands in SQL to realize the query. This commands are sent to the DBMS and the result is shown/recorded according to the query.

The process of coset enumeration has become a significant factor in group theoretical investigations since the advent of modern computing power, but in some respects the process is still not well understood. This thesis investigates some features of coset enumeration, working mainly with the group F(2,7). Chapter 1 describes the characteristics of coset enumeration and algorithms used for it. A worked example of the method is provided. Chapter 2 discusses some features which would be desirable in computer programs for use in investigating the coset enumeration process itself, and reviews the Havas/Alford program which to date best meets the requirements. Chapter 3 deals with the use of coset ammeration in proofs, either in its own right or as a basis for other workings. An example of one attempt to obtain a proof by coset enumeration is given. Chapter 4 reviews techniques designed to reduce the length of coset enumerations and proposes the 'equality list' technique as a way to reduce enumeration length for some groups. Extra insights obtainable using the equality list method are also discussed. Chapter 5 summarises the factors by which the success of different coset enumerations can be compared and proposes an algorithm for making systematic comparisons among enumerations. Chapter 6 reports five coset enumerations, obtained manually by three main methods on the group F(2,7). All these enumerations were shorter than is so far obtainable by machine and one is shorter than other known hand enumerations. The enumerations were compared by applying the process developed in Chapter 5. Chapter 7 presents a shorter proof of the cyclicity of the group F(2,7) than was hitherto available. The proof derives from the workings for one of the coset enumerations described in Chapter 6. There are eight appendices and an annotated bibliography. The appendices contain, inter alia, edited correspondence between well-known coset-enumerators, a guide to the Havas/Alford program, further details on the equality list method and listings of various enumerations.

One purpose of this study was to determine whether any differences in immediate achievement or retention existed between students using manipulatives and students not using manipulatives. Also addressed in this study is whether or not the use of manipulatives is more beneficial for girls than boys and whether the use of manipulatives is more beneficial for low-ability students than for high-ability students. Students selected for this study were from a large suburban school district in Texas. The students were from eight intact classes, four of which were designated as the experimental group and the other four as the control group. The sample consisted of one hundred eighty-seven students. All students were tested with a test developed by the researcher. This same test was administered as a pretest, posttest, and retention test. The following supplemental data were also gathered on the students: mathematics scores from the California Test of Basic Skills and scores from the mathematics section of the Texas Educational Assessment of Minimum Skills test. Analysis of the data revealed no statistical difference in the mean scores of students instructed with or without manipulatives when the test was administered immediately after instruction. Nor was there any statistical difference in the mean scores when the test was administered two months after instruction. There was no statistical difference in the mean gain scores from the pretest to the posttest between boys and girls or between high- and low-achieving students. Nor was there any statistical difference between the mean gain scores from the pretest to the retention test between boys and girls or between high- and low-achieving students. It is recommended that further studies be conducted to investigate achievement and retention of students using manipulatives at the secondary level. It is also recommended that variables other than achievement be studied to determine the effects of manipulatives on secondary students.

Orientadora: Profa. Dra. Ana Carolina Boero<br>Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016.<br>O Teorema Fundamental da Álgebra é um tópico de grande relevância para a Matemática, com o qual o aluno toma contato na 3a série do Ensino Médio. Talvez porque todas as demonstrações conhecidas desse resultado utilizem argumentos que não podem ser apresentados de modo preciso nessa etapa de ensino, sua abordagem em diversos livros didáticos resume-se, basicamente, a destacar algumas de suas consequências e aplicações. O propósito deste trabalho é fornecer um material que possa ser utilizado por professores da Educação Básica no intuito de explorar esse fascinante resultado. Para atingirmos esse objetivo, apresentamos uma breve contextualizaçãohistória do Teorema Fundamental da Álgebra, que serve tanto para apontar sua utilidade em outros ramos da Matemática como também para observar a evolução de certos conceitos matemáticos. Em seguida, apresentamos uma prova rigorosa desse resultado, com o menor nível de complexidade possível, além de duas abordagens alternativas com apelo visual que podem ser utilizadas para apresentar uma justificativa de sua validade aos alunos do Ensino Médio.<br>The Fundamental Theorem of Algebra is a topic of great relevance to Mathematics, with which the student makes contact in the 3rd grade of High School. Perhaps because all known demonstrations of this result use arguments that can not be accurately presented at this stage of teaching, its approach in several textbooks basically boils down to highlighting some of its consequences and applications. The purpose of this work is to provide a material that can be used by teachers of Basic Education in order to explore this fascinating result. To reach this goal, we present a brief history of the Fundamental Theorem of Algebra, which serves both to point out its usefulness in other branches of mathematics and also to observe the evolution of certain mathematical concepts. Next, we present a rigorous proof of this result, with the lowest level of complexity possible, as well as two alternative approaches with visual appeal that can be used to present a justification of its validity to high school students.

In this multiple-case study, I developed cases describing three students' (Mary, Ryan and David) solution methods for algebra word problems and investigated the effect of technology on their solution methods by making inferences about their preferences for visual or nonvisual solutions. Furthermore, I examined the students' solution methods when presented with virtual physical representations of the situations described in the problems and attempted to explain the effect of those representations on students' thinking preferences. In this study, the use of technology referred to the use of the dynamic software program Geogebra. Suwarsono's (1982) Mathematical Processing Instrument (MPI) was administered to determine their preferences for visual and nonvisual thinking. During the interviews, students were presented with paper-and-pencil-based tasks (PBTs), Geogebra-based tasks (GBTs) and Geogebra-based tasks with virtual physical representations (GBT-VPRs). Each category included 10 algebra word problems, with similar problems across categories. (i.e., PBT 9, GBT 9 and GBT-VPR 9 were similar). By investigating students' methods of solution and their use of representations in solving those tasks, I compared and contrasted their preferences for visual and nonvisual methods when solving problems with and without technology. The comparison between their solutions of PBTs and GBTs revealed how dynamic software influenced their method of solution. Regardless of students' preferences for visual and nonvisual solutions, with the use of dynamic software students employed more visual methods when presented with GBTs. When visual methods were as accessible and easy to use as nonvisual methods, students preferred to use them, thus demonstrating that they possessed a more complete knowledge of problem-solving with dynamic software than their work on the PBTs.; Nowadays, we can construct virtual physical representations of the problems in technology environments that will help students explore the relationships and look for patterns that can be used to solve the problem. Unlike GBTs, GBT-VPRs did not influence students' preferences for visual or nonvisual methods. Students continued to rely on methods that they preferred since their preferences for visual or nonvisual solutions regarding GBT-PRs were similar to their solution preferences for the problems on MPI that was administered to them to determine their preferences for visual or nonvisual methods. Mary, whose MPI score suggested that she preferred to solve mathematics problems using nonvisual methods, solved GBT-VPRs with nonvisual methods. Ryan, whose MPI score suggested that he preferred to solve mathematics problems using visual methods, solved GBT-VPRs with visual methods. David, whose MPI score suggested that he preferred to solve mathematics problems using both visual and nonvisual methods, solved GBT-VPRs with both visual and nonvisual methods.<br>ID: 030422900; System requirements: World Wide Web browser and PDF reader.; Mode of access: World Wide Web.; Thesis (Ph.D.)--University of Central Florida, 2011.; Includes bibliographical references (p. 293-303).<br>Ph.D.<br>Doctorate<br>Education

<p>This is a literature study of learning mathematics in general and algebra in particular. The goal is to explore and investigate the learning procedure and the difficulties pupils have with their understanding. Furthermore, to understand what qualities is requested of pupils in their effort to be successful in mathematics and algebra. I have also explored different pupil thinking styles in mathematics and what consequences that has on their learning and understanding. Moreover, I have investigated some different learning styles and how they can be addressed in teaching of mathematics in general and algebra in particular. Finally, the goal has been to suggest how you can improve the teaching in order to enhance the understanding of algebra. As a result I have presented a variety of methods to literally illustrate the basic concepts of algebra. The conclusions of this work are that it is possible to make algebra more visual or kinaesthetic by illustrating the function of the equal sign as a similarity and not an operational command by scales for instance, in order to enhance the understanding of algebra. Additionally, it is for example possible to make the multiplication of binomial visual or kinaesthetic by treating the product of the binomials as an area of a rectangle with the binomials as sides. Finally, the importance of team work should not be underestimated as it lets the pupils learn from each other, toss ideas and develop a more flexible thinking style.</p>

Swedish education needs to change in order to reverse the negative trend in PISA and TIMSS. At the same time digital artifacts are increasingly present in the classroom and commercial applications more available. It is difficult to determine which technical tool that provide real results and it is not yet clear what role technology should have in the classroom. This study examines how visualisation and technology can be used to promote deeper understanding of algebra by addressing the misconception that frequently occurs among students. Observation and interviews were held with students from two schools, one in Norrköping and one in Östersund. Theory regarding algebraic misconceptions have been studied and compiled into two categories. In addition, three prototypes were developed with and later evaluated by a focus group. This work has resulted in suggestions on what technical and interactive activities could be used in algebraic education. The study shows that more research is needed to determine how technology best can be implemented and used effectively in teaching. The work also shows that such research and development needs to be implemented with a user-centered approach with the involvement of relevant target groups.<br>Svensk undervisning är i behov att förändras för att vända den negativa trenden i PISA och TIMSS. Samtidigt är digitala artefakter alltmer närvarande i klassrummet och kommersiella applikationer alltmer tillgängliga. Det är svårt att avgöra vilka tekniska verktyg som ger faktiska resultat och det är ännu inte klarlagt vilken roll tekniken bör spela i en klassrumssituation. I detta arbete undersöks hur visualiseringar och teknik kan användas för att främja djupare förståelse inom algebra genom att adressera missuppfattningar som ofta förekommer hos elever. Observation och intervjuer genomfördes med elever på två skolor, en i Norrköping och en i Östersund. Teori om algebraiska missuppfattningar har studerats och sammanställts i två kategorier. Dessutom har tre prototyper utvecklats tillsammans med, och utvärderats av, en fokusgrupp. Arbetet har resulterat i förslag på vad tekniska och interaktiva aktiviteter skulle kunna innebära i algebraisk matematikundervisning. Det visar att mer forskning behövs för att avgöra hur teknik på bästa sätt kan implementeras och utnyttjas effektivt inom undervisning. Arbetet visar också att sådan forskning och utveckling behöver genomföras med en användarcentrerad ansats med delaktighet från berörda målgrupper.

Resumen En esta tesis proponemos GEA (Global Epipolar Adjustment), un método de corrección de movimiento para aplicaciones de Visual SLAM y SfM altamente eficiente basado en restricciones algebraicas epipolares, que no hace uso de la información sobre la estructura. Este método obtiene resultados muy cercanos, y en muchos casos equivalentes a los obtenidos mediante bundle adjustment (BA), a pesar de su naturaleza algebraica, requiriendo por contra un coste computacional mucho menor. En este trabajo describimos cómo usar esa técnica para estimar las poses de cámara de forma incremental en aplicaciones SfM, sin necesidad de estimar en el proceso información sobre la estructura tal como correspondencias multivista de características de imagen. Estas poses de cámara pueden ser usadas a posteriori para obtener una estimación de alta precisión para la estructura. Demostramos que las técnicas descritas son robustas a los errores típicos de correspondencia obtenidos por los métodos habituales usados en SfM.<br>In this work we propose GEA (Global Epipolar Adjustment), a high-performance structureless BA correction method based on algebraic epipolar constraints. The GEA cost can be optimized very efficiently, in most cases using a small fraction of the time required by BA to obtain the optimal configuration. Despite of the algebraic nature of the cost, under general circumstances the accuracy of the obtained camera poses is very close to that obtained with classical BA methods. We also describe how to obtain accurate initializations for the camera poses incrementally using GEA and pairwise image feature correspondences, without requiring the composition of feature trackings or triangulation of scene landmarks. The resulting camera poses can be used afterwards to obtain highly accurate sparse or dense estimations of the scene structure. The techniques described are robust against outliers found by common pairwise image matching techniques.

Made available in DSpace on 2016-04-27T16:57:49Z (GMT). No. of bitstreams: 1 dissertacao karina andrezzo.pdf: 1135963 bytes, checksum: 3dd93f8487f3f29db82548fd1e1745ac (MD5) Previous issue date: 2005-04-08<br>ABSTRACT This study aimed to identify factors that contribute to the understandings of algebraic expressions constructed by blind students. To this end, it involved the design of situations that would enable the participation of blind high-school students (Brazilian Ensino Médio) in generalizing activities. The research was inspired by Vygotsky s visions of the social integration and intellectual development of learners with special needs. The use of an approach to algebra emphasizing generalizations based on the patterns underlying sequences presented as spatial arrangements has been supported by various researchers in Mathematics Education, this study was informed particularly by the work of Küchemann (1981), Booth (1988) and Mason (1996). In the first phase of the research, materials based on the use of magnets to represent dynamic elements of sequences were designed and tested. The second phase involved five blind students in completing a preliminary activity, elaborated to obtain data about their performance on traditional algebra exercises followed by a series of interviews, during which each students worked through a set of seven generalizing activities. The analysis of the students responses to the preliminary activities indicated that the students had difficulties in dealing with three of six categories for interpreting letters in algebraic expressions described by Küchemann (1981), letter as specific unknown , letter as generalized number and letter as variable ; and identified errors committed during the simplification of expressions. Analysis of the interview data was based on the strategies the students used to construct mathematical generalizations. These analyses highlighted a process by which the external marks (the magnets and their organizations) were gradually internalized through the manipulations and articulations by which the learners strived to get a sense of algebraic expressions.<br>Nesta pesquisa, buscamos identificar fatores que contribuíram na apreensão de expressões algébricas por alunos sem acuidade visual. Para isto elaboramos atividades que facilitassem sua participação em atividades de generalização. O estudo se desenvolveu com alunos do Ensino Médio e tem como ponto de partida a posição de Vygotsky referente à integração social do aluno portador de alguma deficiência e seu potencial para um desenvolvimento normal. A sugestão de abordagem algébrica através de generalizações de padrões figurativos teve suporte de vários pesquisadores em educação Matemática, em particular, os estudos e considerações de Küchemann (1981), Booth (1988) e Mason (1996). A primeira fase da pesquisa envolveu a elaboração das atividades e dos materiais manipulativos, através dos quais os alunos s.a.v. poderiam ter acesso às seqüências de padrões figurativos. A segunda fase constituiu-se de uma Atividade de Sondagem composta por itens que envolviam noções algébricas e cinco entrevistas onde foram trabalhadas sete seqüências cujos padrões eram representados por ímãs de formas geométricas. A análise da Atividade de Sondagem possibilitou que identificássemos as dificuldades dos alunos em lidar com três das seis categorias descritas por Küchemann (1981), letra como uma incógnita específica , letra como número generalizado e letra como variável ; e alguns erros cometidos pelos alunos durante as simplificações. Quanto à análise das tarefas, esta, procedeu das estratégias usadas na evolução das generalizações matemáticas. Nossas análises indicaram um processo gradual de internalização das marcas externas (os imãs e suas organizações) através das manipulações , articulações e na tentativa de atribuir significado algébrico às letras em expressões algébricas ( constituindo um sentido para ).

Questa tesi presenta una descrizione dei primi aspetti della Teoria dei Reticoli e della Teoria delle Categorie e una loro applicazione alla descrizione di un sistema di corrispondenze fra geometrie proiettive di incidenza in senso ampio e reticoli modulari nel caso finito dimensionale. Nell'ordine: dualità fra insiemi e reticoli booleani; dualità fra poset e reticoli distributivi (Birkhoff, 1937); equivalenza fra geometrie proiettive e reticoli modulari complementati (Birkhoff, 1935), equivalenza fra geometrie proiettive su poset e reticoli modulari (Faigle ed Herrmann, 1981).

La réduction des informations redondantes et/ou non-pertinentes dans la description de données est une étape importante dans plusieurs domaines scientifiques comme les statistiques, la vision par ordinateur, la fouille de données ou l’apprentissage automatique. Dans ce manuscrit, nous abordons la réduction de la taille des signatures des images par une méthode issue de l’Analyse Formelle de Concepts (AFC), qui repose sur la structure du treillis des concepts et la théorie des treillis. Les modèles de sac de mots visuels consistent à décrire une image sous forme d’un ensemble de mots visuels obtenus par clustering. La réduction de la taille des signatures des images consiste donc à sélectionner certains de ces mots visuels. Dans cette thèse, nous proposons deux algorithmes de sélection d’attributs (mots visuels) qui sont utilisables pour l’apprentissage supervisé ou non. Le premier algorithme, RedAttSansPerte, ne retient que les attributs qui correspondent aux irréductibles du treillis. En effet, le théorème fondamental de la théorie des treillis garantit que la structure du treillis des concepts est maintenue en ne conservant que les irréductibles. Notre algorithme utilise un graphe d’attributs, le graphe de précédence, où deux attributs sont en relation lorsque les ensembles d’objets à qui ils appartiennent sont inclus l’un dans l’autre. Nous montrons par des expérimentations que la réduction par l’algorithme RedAttsSansPerte permet de diminuer le nombre d’attributs tout en conservant de bonnes performances de classification. Le deuxième algorithme, RedAttsFloue, est une extension de l’algorithme RedAttsSansPerte. Il repose sur une version approximative du graphe de précédence. Il s’agit de supprimer les attributs selon le même principe que l’algorithme précédent, mais en utilisant ce graphe flou. Un seuil de flexibilité élevé du graphe flou entraîne mécaniquement une perte d’information et de ce fait une baisse de performance de la classification. Nous montrons par des expérimentations que la réduction par l’algorithme RedAttsFloue permet de diminuer davantage l’ensemble des attributs sans diminuer de manière significative les performances de classification<br>In several scientific fields such as statistics, computer vision and machine learning, redundant and/or irrelevant information reduction in the data description (dimension reduction) is an important step. This process contains two different categories : feature extraction and feature selection, of which feature selection in unsupervised learning is hitherto an open question. In this manuscript, we discussed about feature selection on image datasets using the Formal Concept Analysis (FCA), with focus on lattice structure and lattice theory. The images in a dataset were described as a set of visual words by the bag of visual words model. Two algorithms were proposed in this thesis to select relevant features and they can be used in both unsupervised learning and supervised learning. The first algorithm was the RedAttSansPerte, which based on lattice structure and lattice theory, to ensure its ability to remove redundant features using the precedence graph. The formal definition of precedence graph was given in this thesis. We also demonstrated their properties and the relationship between this graph and the AC-poset. Results from experiments indicated that the RedAttsSansPerte algorithm reduced the size of feature set while maintaining their performance against the evaluation by classification. Secondly, the RedAttsFloue algorithm, an extension of the RedAttsSansPerte algorithm, was also proposed. This extension used the fuzzy precedence graph. The formal definition and the properties of this graph were demonstrated in this manuscript. The RedAttsFloue algorithm removed redundant and irrelevant features while retaining relevant information according to the flexibility threshold of the fuzzy precedence graph. The quality of relevant information was evaluated by the classification. The RedAttsFloue algorithm is suggested to be more robust than the RedAttsSansPerte algorithm in terms of reduction

碩士<br>國立交通大學<br>理學院科技與數位學習學程<br>99<br>In this study, we use different visual guiding materials to the worked-example teaching. The purpose is to discuss how the visual guiding materials can have effect both on students’ performance and their cognitive load. Our study designed the unit of systems of first-order algebraic equations in variables of mathematics for the seventh grade, and with quasi-experimental design, the experiment targets the classes of normalized class assignment. Meanwhile, we try to develop the rating scales of the cognitive load as a pilot study and explore the expertise reversal effect. 　　The results are as follows. 　　Firstly, the worked-example teaching with different visual guiding materials remarkably encourage learners of low prior knowledge. The material with the algebra material design principles is really helpful to the learners in their performance. 　　Secondly, the worked-example teaching with different visual guiding materials remarkably encourage learners of low achievement in their performance. The material with the algebra material design principles is also helpful to the learners. 　　Thirdly, the worked-example teaching with different visual guiding materials cause significant effect on the cognitive load of the difficulty from the learners of high prior knowledge. Visual guiding materials, especially those by algebra material design principles, can reduce the cognitive load of the difficulty more prominently. 　　Finally, the worked-example teaching with different visual guiding materials principles lead to expertise reversal effect in view of the effect size and learners’ involvement. However, there is no negative impact on the learners of high prior knowledge and of high achievement in their performance. 　　Nevertheless, there are still problems for the pilot study of the rating scales of cognitive load, and thus it remains to be improved with future studies.

Thesis (Ph. D.)--Florida State University, 2006.<br>Advisor: Leslie Aspinwall, Florida State University, College of Education, Dept of Middle and Secondary Education. Title and description from dissertation home page (viewed Sept. 15, 2006). Document formatted into pages; contains vii, 173 pages. Includes bibliographical references.

碩士<br>中原大學<br>工業與系統工程研究所<br>106<br>In recent years, A lot of universities and educational institutions have been developing distance learning courses. In the past, traditional distance education provided web-based platform rather than considering whether the learning material was suitable for students. This study aims to learning materials design and students&apos;&apos; trait. The study aimed at students&apos;&apos; learning styles and different visual learning materials. The purpose of this study is: (1) The influence of learning material design on students&apos;&apos; learning performance. (2) Analyzing learning style for students&apos;&apos; learning performance. (3) Cross-analyzing the influence of learning styles and learning materials on students&apos;&apos; learning outcomes. In this study, we use Silverman and Felder learning style as the classification of the individual student’s learning trait. According to the visual and auditory dimension, the students were divided into two groups. The results show that learning material is presented through handwritten have better learning performance than color prompt. The visual learning style students who are in the handwriting reminder group have better learning performance than the color-prompt group. In addition, the distance course can provide students with a good review effect. The results show the students who have reviewed the distance courses have better learning performance than who did not review. The number of times the students playing back the distance course slides positive correlation with the students&apos;&apos; attendance.

We investigate the differences --- conceptually and algorithmically --- between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. We then use the affine invariant to derive new algebraic connections between perspective views. It is shown that three perspective views of an object are connected by certain algebraic functions of image coordinates alone (no structure or camera geometry needs to be involved).

Η παρούσα διδακτορική διατριβή επικεντρώνεται σε δύο ερευνητικά προβλήματα που αποτελούν τα αντικείμενα δύο ερευνητικών φάσεων. Στην Α΄ ερευνητική φάση, διάρκειας ενός έτους, ασχολείται με τη διερεύνηση μαθηματικών πρακτικών σε τρεις ομάδες τεχνικών του Οργανισμού Τηλεπικοινωνιών Ελλάδας αναζητώντας παράλληλα την ύπαρξη αμετάβλητων στοιχείων της μαθηματικής επιστήμης τα οποία διαπερνούν την ακαδημαϊκή και την παρούσα εργασιακή κοινότητα. Στη Β΄ ερευνητική φάση, διάρκειας οκτώ μηνών, εξετάζει κάτω και υπό ποιες προϋποθέσεις πέντε σπουδαστές ενός Τεχνολογικού Εκπαιδευτικού Ιδρύματος που πραγματοποιούν την πρακτική τους άσκηση στον ίδιο Οργανισμό είναι σε θέση να αναγνωρίσουν τα αμετάβλητα αυτά στοιχεία. Στην Α΄ ερευνητική φάση η Θεωρία Δραστηριότητας των Vygotsky, Leont’ ev και των συνεχιστών του έργου τους, Engeström & Cole, αποτελεί τη θεωρητική βάση της εργασίας. Τα ερευνητικά δεδομένα προκύπτουν από εθνογραφικής φύσης παρατηρήσεις αλλά και συζητήσεις με τους συμμετέχοντες. Η μαθηματική δραστηριότητα που αναγνωρίσαμε στο χώρο εργασίας ήταν πολύπλοκη και πλούσια αλλά πλήρως ενταγμένη στο πλαίσιο αναφοράς της. Ειδικότερα, αναγνωρίσαμε και ταξινομήσαμε τα μαθηματικά εργαλεία τα οποία διαμεσολαβούσαν στις κεντρικές καθημερινές εργασιακές δραστηριότητες των τεχνικών και αναδείξαμε τους τρόπους με τους οποίους αυτά εμπλέκονταν με τα τεχνικής φύσης εργαλεία τους. Ταυτόχρονα αναγνωρίσαμε αμετάβλητα μαθηματικά στοιχεία στις μαθηματικές έννοιες, στο τρόπο κατανόησής τους από τους τεχνικούς και σε μαθηματικές διαδικασίες που οι ίδιοι χρησιμοποιούσαν για την επίτευξη των εργασιακών τους στόχων. Στην Β΄ ερευνητική φάση τα ερευνητικά δεδομένα προέρχονται από διερευνητικής και παρεμβατικής φύσης συνεντεύξεις με τους σπουδαστές και εθνογραφικές παρατηρήσεις. Μέσα από τις διερευνητικής φύσης συνεντεύξεις καταγράψαμε τις στάσεις των σπουδαστών ως μέλη της σπουδαστικής και της συγκεκριμένης εργασιακής κοινότητας και αναζητήσαμε μαθηματικές πρακτικές που ανέπτυξαν ως μαθητευόμενοι στην παρούσα εργασιακή τους κοινότητα. Οι μαθηματικές πρακτικές που ανέπτυξαν οι σπουδαστές, έστω και ασυνείδητα, είχαν άμεση εξάρτηση από τα εργαλεία και τους εργασιακούς στόχους της κάθε κοινότητας και αφορούσαν την ικανότητα οπτικοποίησης και την ανάγνωση και ερμηνεία σύνθετων οπτικών αναπαραστάσεων. Τέλος, μέσα από μια σειρά παρεμβατικής φύσης συνεντεύξεων αναλύσαμε με εργαλεία σημειωτικής τη δραστηριότητα που ανέπτυξαν οι ίδιοι σπουδαστές στην προσπάθεια ερμηνείας αυθεντικών αναπαραστάσεων με σκοπό τη σύνδεση κοινών μαθηματικών εννοιών που συναντώνται στην ακαδημαϊκή και στην παρούσα εργασιακή κοινότητα. Οι έννοιες αυτές αφορούσαν το θεσιακό σύστημα αρίθμησης και τη συναρτησιακή σχέση αντίστασης, μήκους, διαμέτρου χάλκινων καλωδίων. Καταλήγουμε, καταγράφοντας τα χαρακτηριστικά που προάγουν και αναστέλλουν, τη μεταφορά της γνώσης στο νέο κοινωνικό-πολιτισμικό πλαίσιο. Στο τέλος της διατριβής καταγράφονται και αναλύονται οι εκπαιδευτικές προεκτάσεις της έρευνας.<br>This dissertation thesis focuses on two different research problems carried out in two research phases. In the first research phase, lasting one year, it focuses on the exploration –identification of mathematical practices of three different groups of technicians of the Greek Telecommunication Organization. In parallel, it investigates the existence of invaried mathematical elements that are crossing the academic and the current workplace community. In the second research face, lasting eight months, it investigates how and whether five students of a Technological Educational Institute who were doing their practicum in this setting could recognize these invariant mathematical elements. In the first research phase, the theoretical framework is guided by Vygotsky and Leont’ev work on Activity theory and their followers, Engeström & Cole. Our data are coming from ethnographic observations and discussions with the participants. The mathematical activity we identified was complex and rich but completely contextual. Especially, we recognized and categorized the mediated mathematical tools in technicians’ central workplace activities and we were showing off how these are interrelated with their physical mediated tools. At the same time we recognized invariant mathematical elements in the category of mathematical concepts, the meanings the technicians attributed to these concepts and in the category of mathematical processes they were using in order to achieve their workplace goals. In the second research phase, our data are coming from eexploratory and intervention interviews with the students and ethnographic observations. In the exploratory interviews we recorded their experiences and their attitudes as members of the academic and the workplace community and we identified mathematical practices they developed as apprentice members of this community. Τhe main mathematical practices the students developed, mainly unconsciously, were attached to the tools and the goals of the workplace community and referring to visualization and reading and interpreting complex visual representations. Finally, through the intervention interviews, we analyzed with the help of semiotic tools the activity the same students developed in order to interpret mathematical objects that are common to the academic and workplace community. The mathematical objects were referring to the place value concept and the functional relation between the resistance, the length and the diameter of the copper wires. In the conclusion, we recorded the characteristics that support and block students’ transfer of knowledge in their new socio-cultural context. In the end of the thesis we discuss and analyze the educational implications of our findings.