Η παρούσα διπλωματική εργασία εκπονήθηκε στα πλαίσια του Μεταπτυχιακού Προγράμματος Σπουδών του Παιδαγωγικού Τμήματος Δημοτικής Εκπαίδευσης του Πανεπιστημίου Πατρών και έθεσε ως κεντρικό ζήτημα τη διερεύνηση της ικανότητας που διαθέτουν οι υποψήφιοι μελλοντικοί εκπαιδευτικοί της πρωτοβάθμιας εκπαίδευσης, ως προς τη χρήση των αναπαραστάσεων και ειδικότερα τη χρήση των αναπαραστάσεων στην επίλυση προβλημάτων με κλάσματα. / This paper has been prepared within the Graduate Program of the Department of Education of the University of Patras and has os main issue to investigate the representational competence of pre-service teachers, and particularly the use of representations to solve problems with fractions.

A major purpose of this study was to investigate the relationship between fraction-identification rule abstraction and training exemplars. Fractions can be identified with at least two different rules. One of these, the denominator rule, is general in that it yields correct responses across a wide variety of fraction identification problems. The second, the one-element rule, is appropriate only when the number of elements in the set and the denominator-specified number of subsets are equivalent. Because these rules are not equally serviceable, a question of major importance is what factors determine which of the two fraction identification rules a child will learn during training. The main hypothesis within this study specified that the nature of the fraction identification rule abstracted by a learner would be influenced by the nature of the examples used in training. It was further hypothesized that mastery of the denominator rule would positively affect performance on one-element problems, and that denominator-rule problem errors consistent with the one-element rule would occur significantly more frequently than would be expected by chance. The study addressed two additional questions. These related to recent work in the area of information processing and concerned both changes in learner behavior across training/posttesting sessions and consistency between verbal reports of thinking processes and the fraction rules hypothesized to control fraction identification. A pretest was used to determine eligibility to participate in the study. Eighty-two children incapable of set/subset fraction identification participated in the study. Two additional children were involved in an exploratory phase. The children ranged in age from six to ten and in grade level from one to four. The participants were mainly from middle and lower-class Anglo and Mexican-American homes. Children were randomly assigned to one of two training groups. In both training groups learners were provided, through symbolic (verbal) modeling, the general denominator rule for fraction identification. Children in one training group were also provided examples of fraction identification requiring the denominator rule. In the second training group children were provided simple examples in keeping with the denominator rule stated as part of instruction and yet also in keeping with the unstated one-element rule. A modified path analysis procedure was used to assess the effects of training group assignment on fraction identification performance. Results of this analysis suggest a training group main effect. That is, children's performance on fraction identification posttest problems was in keeping with rules associated with the training examples they had been provided. The results suggest that the strongest effects of training were related to performance on denominator rule problems in that the odds of passing the denominator rule posttest were 11.52 times greater for children taught with denominator-rule exemplars than for children taught with the one-element exemplars. The findings also suggest that performance on either of the fraction identification tasks influenced performance on the other. A further finding was that training with the one-element exemplars was associated with performance congruent with inappropriate use of the one-element rule. Recall that the one-element rule was never stated and that the exemplars, while compatible with the one-element rule were equally compatible with the stated denominator rule. Protocols from the children in the exploratory portion of the study suggest that the child taught with the ambiguous exemplars did abstract the one-element rule while the child taught with the denominator rule exemplars abstracted the denominator rule. The protocols also suggest that the child taught with the denominator rule made changes in his thinking as training and posttesting progress progressed.

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Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Florida, 1980. / Description based on print version record. Typescript. Vita. Includes bibliographical references (leaves 268-275).

Thesis (Ph. D.)--Illinois State University, 1996. / Title from title page screen, viewed May 24, 2006. Dissertation Committee: Albert D. Otto (chair), Barbara S. Heyl, Cheryl A. Lubinski, Nancy K. Mack, Jane O. Swafford, Carol A. Thornton. Includes bibliographical references (leaves 188-198) and abstract. Also available in print.

Made available in DSpace on 2014-06-11T19:24:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2003Bitstream added on 2014-06-13T18:52:14Z : No. of bitstreams: 1 valera_ar_me_mar.pdf: 594283 bytes, checksum: 7fa747413b18f73739f058ca4ea1146e (MD5) / Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações...

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Continued fractions have been extensively studied in number theoretic ways. In this text we will consider continued fraction expansions with partial quotients that are in Z = f x : x 2 Zg and where = 2 cos( q ); q 3 and with 1 < < 2. These continued fractions are expressed as the composition of M obius maps in PSL(2;R), that act as isometries on H2, taken at 1. In particular the subgroups of PSL(2;R) that are studied are the Hecke groups G . The Modular group is the case for q = 3 and = 1. In the text we show that the Hecke groups are triangle groups and in this way derive their fundamental domains. From these fundamental domains we produce the v-cell (P0) that is an ideal q-gon and also tessellate H2 under G . This tessellation is called the -Farey tessellation. We investigate various known -continued fractions of a real number. In particular, we consider a geodesic in H2 cutting across the -Farey tessellation that produces a \cutting sequence" or path on a -Farey graph. These paths in turn give a rise to a derived -continued fraction expansion for the real endpoint of the geodesic. We explore the relationship between the derived -continued fraction expansion and the nearest - integer continued fraction expansion (reduced -continued fraction expansion given by Rosen, [25]). The geometric aspect of the derived -continued fraction expansion brings clarity and illuminates the algebraic process of the reduced -continued fraction expansion.