Made available in DSpace on 2016-04-27T16:57:16Z (GMT). No. of bitstreams: 1 Aspectos do Pens Matematico na RP-Segurity printing.pdf: 2060132 bytes, checksum: c0711be8b2de82245dd639a3d704a6be (MD5) Previous issue date: 2005-11-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This doctorate thesis aims to identify the characteristics and the dimensions of mathematical thinking in experimental and theoretical terms which may be useful to teachers with respect to teaching processes, development of mathematical ideas and the delineation of learning contexts. Our study began with a detailed analysis of the work of Krutetskii (1968). This book is very rich in theoretical examples and reflections. It is, however, a completely psychological work and provided few indications of the more general mathematic knowledge and thinking. For this reason we added detailed information about the work of other authors such as Gowers, Poincaré, Boutroux, Otte and Kurz, and this added other dimensions which assisted our understanding of the nature of mathematics. These authors were concerned with the problems of cognitive styles, cultural and historical differences, differences that are the results of mathematics itself and distinctive ways of representing mathematics. The experimental dimensions consisted of analysis of data obtained from qualitative research with students whereby one was taken from the literature (Krutetskii) and the other an exploratory survey which we carried out for the purposes of this thesis. Krutetskii carried out an experimental investigation involving 201 Russian students with different mathematic abilities, attending elementary school. These students were presented with a number of different series of mathematic problems and their mathematic abilities were observed during the problem solving process. In our survey we carried out case studies exploring mathematic problem solving involving 13 students from the Federal University of Mato Grosso with 9 students from the Mathematics/Education course and 4 students from the Computer Sciences Course. The exploratory survey was organized into 3 phases. The first was the completion of a questionnaire with subjective questions about Mathematics and preferred ways of thinking and dealing with this subject. The second phase was reserved for the solution of 13 varied mathematical problems. The final phase was the completion of another questionnaire with subjective questions which sought to obtain information about the experiences of the students when solving the problems set. With our exploratory survey we were able to document and verify several parameters and characteristics of mathematical thinking which were described in the theoretical chapters as well as being able identify the problems themselves and the experience of solving them also influenced mathematical thinking. As a general result we concluded that mathematical thinking must be considered in the light of different parameters since this can help to characterize more complete mathematical thinking / A presente Tese de Doutorado pretende indicar características e dimensões do pensamento matemático, em termos teóricos e experimentais, que podem ser úteis aos professores no que se refere aos processos de ensino, ao desenvolvimento de idéias matemáticas e ao delineamento de contextos de aprendizagem. Nosso estudo começou com uma análise detalhada do trabalho de Krutetskii (1968). Esse livro é muito rico em exemplos e reflexões teóricas. No entanto, é um trabalho completamente psicológico e forneceu poucas indicações a respeito dos aspectos mais gerais do conhecimento matemático e do pensamento matemático. Por esse motivo, adicionamos informações detalhadas sobre o trabalho de outros autores como Gowers, Poincaré, Boutroux, Otte e Kurz que acrescentaram outras dimensões que auxiliaram a nossa compreensão da natureza da Matemática. Esses autores se preocuparam com problemas de estilos cognitivos, de diferenças culturais e históricas, de diferenças que são resultados das várias áreas da própria Matemática e distintas formas de representação na Matemática. As dimensões experimentais consistiram na análise de dados obtidos em pesquisas qualitativas com estudantes, sendo uma da literatura (Krutetskii) e outra uma pesquisa exploratória realizada por nós para a presente Tese. Krutetskii realizou uma investigação experimental envolvendo 201 estudantes russos do Ensino Fundamental, com diferentes habilidades matemáticas. A esses estudantes foram propostas diversas séries de problemas matemáticos, em que foram observadas suas habilidades matemáticas durante o processo de resolução. Na nossa pesquisa, realizamos estudos de caso exploratório na resolução de problemas matemáticos envolvendo 13 estudantes da Universidade Federal de Mato Grosso, sendo 09 do Curso de Licenciatura Plena em Matemática e 04 do Curso de Ciências da Computação. A pesquisa exploratória foi organizada em três momentos. O primeiro foi destinado a responder um questionário com perguntas subjetivas acerca da Matemática e de preferências na forma de pensar e de lidar com a mesma. O segundo momento foi reservado para a resolução de 13 problemas matemáticos variados. E o último momento foi destinado para responder a outro questionário com perguntas subjetivas que procurava obter informações sobre a experiência dos estudantes na atividade de resolução dos problemas propostos. Com a nossa pesquisa exploratória pudemos documentar e verificar vários parâmetros e características do pensamento matemático que foram descritos nos capítulos teóricos, bem como identificar que os próprios problemas e as experiências com a resolução dos mesmos também influenciam o pensamento matemático. Como resultado geral, concluímos que o pensamento matemático deve ser considerado sob diferentes parâmetros, pois eles podem auxiliar na caracterização mais completa do pensamento matemático

Each year thousands of students enrol in introductory statistics courses at universities throughout Australia, bringing with them formal and informal statistical knowledge and reasoning, as well as a wide range of basic numeracy skills, mathematical inclinations and attitudes towards statistics, which have the potential to impact on their ability to develop statistically. This research develops and investigates measures of each of these components for students at the interface of secondary and tertiary education, and investigates the relationships that exist between them, and a range of background variables. The focus of the research is on measuring and analysing levels and abilities in statistical reasoning for a range of students at the tertiary interface, with particular interest also in investigating their basic numeracy skills and how these may or may not link with statistical reasoning allowing for other variables and factors. Information from three cohorts in an introductory data analysis course, whose focus is real data investigations, provides basis for the research. This course is compulsory for all students in degree programs associated with all sciences or mathematics. The research discusses and reports on the development of questionnaires to measure numeracy and statistical reasoning and the students' attitudes and reflections on their prior school experiences with statistics. Students' attitudes are found to be generally positive, particularly with regard to their self-efficacy. They are also in no doubt as to the links that exist between mathematics and statistics. The Numeracy Questionnaire, developed to measure pre-calculus skills relevant to an introductory data analysis course which emphasises real data investigations, demonstrates that many students who have completed a basic algebra and calculus senior school subject struggle with skills which are in the pre-senior curricula. Direct examination of the responses helps to understand where and why difficulties tend to occur. Rasch analysis is used to validate the questionnaire and assist in the description of levels of skill. General linear models demonstrate that a student's numeracy score depends on the result obtained in senior mathematics, whether or not the student is a mathematics student, gender, whether or not higher level mathematics has been studied, self-efficacy and year. The research indicates that either the pre-senior curricula need strengthening or that exposure to mathematics beyond the core senior course is required to establish confidence with basic skills particularly when applied to new contexts and multi- step situations. The Statistical Reasoning Questionnaire (SRQ) is developed for use in the Australian context at the secondary/tertiary interface. As with the Numeracy Questionnaire, detailed examination of the responses provides much insight into the range and features of statistical reasoning at this level. Rasch analyses, both dichotomous and polychotomous, are used to establish the appropriateness of this instrument as a measuring tool at this level. The polychotomous, Rasch partial credit model is also used to define a new approach to scoring a statistical reasoning instrument and enables development and application of a hierarchical model and measures levels of statistical reasoning appropriate at the school/tertiary interface. General linear models indicate that numeracy is a highly significant predictor of statistical reasoning allowing for all other variables including tertiary entrance score and students' backgrounds and self-efficacy. Further investigation demonstrates that this relationship is not limited to more difficult or overtly mathematical items on the SRQ. Performance on the end of semester component of assessment in the course is shown to depend on statistical reasoning at the beginning of semester as measured by the partial credit model, allowing for all other variables. Because of the dominance of the relationship between statistical reasoning (as measured by the SRQ) and numeracy on entry, some further analysis of the end of semester assessment is carried out. This includes noting the higher attrition rates for students with less mathematical backgrounds and lower numeracy.

The study constituted an investigation for concept images and mathematical reasoning of Grade 11 learners on the concepts of reflection, translation and stretch of functions. The aim was to gain awareness of any conceptions that learners have about these transformations. The researcher’s experience in high school and university mathematics teaching had laid a basis to establish the research problem. The subjects of the study were 96 Grade 11 mathematics learners from three conveniently sampled South African high schools. The non-return of consent forms by some learners and absenteeism during the days of writing by other learners, resulted in the subsequent reduction of the amount of respondents below the anticipated 100. The preliminary investigation, which had 30 learners, was successful in validating instruments and projecting how the main results would be like. A mixed method exploratory design was employed for the study, for it was to give in-depth results after combining two data collection methods; a written diagnostic test and recorded follow-up interviews. All the 96 participants wrote the test and 14 of them were interviewed. It was found that learners’ reasoning was more based on their concept images than on formal definitions. The most interesting were verbal concept images, some of which were very accurate, others incomplete and yet others exhibited misconceptions. There were a lot of inconsistencies in the students’ constructed definitions and incompetency in using graphical and symbolical representations of reflection, translation and stretch of functions. For example, some learners were misled by negative sign on a horizontal translation to the right to think that it was a horizontal translation to the left. Others mistook stretch for enlargement both verbally and contextually. The research recommends that teachers should use more than one method when teaching transformations of functions, e.g., practically-oriented and process-oriented instructions, with practical examples, to improve the images of the concepts that learners develop. Within their methodologies, teachers should make concerted effort to be aware of the diversity of ways in which their learners think of the actions and processes of reflecting, translating and stretching, the terms they use to describe them, and how they compare the original objects to images after transformations. They should build upon incomplete definitions, misconceptions and other inconsistencies to facilitate development of accurate conceptions more schematically connected to the empirical world. There is also a need for accurate assessments of successes and shortcomings that learners display in the quest to define and master mathematical concepts but taking cognisance of their limitations of language proficiency in English, which is not their first language. Teachers need to draw a clear line between the properties of stretch and enlargement, and emphasize the need to include the invariant line in the definition of stretch. To remove confusion around the effect of “–” sign, more practice and spiral testing of this knowledge could be done to constantly remind learners of that property. Lastly, teachers should find out how to use smartphones, i-phones, i-pods, tablets and other technological devices for teaching and learning, and utilize them fully to their own and the learners’ advantage in learning these and other concepts and skills / Mathematics Education / D.Phil. (Mathematics, Science and Technology Education)

Made available in DSpace on 2014-12-17T14:36:01Z (GMT). No. of bitstreams: 1 JoaoCBQ.pdf: 1363815 bytes, checksum: 1e07e6a070ddb0ed8acc6e7cea8e04c5 (MD5) Previous issue date: 2009-02-19 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / This work aims to analyze the historical and epistemological development of the Group concept related to the theory on advanced mathematical thinking proposed by Dreyfus (1991). Thus it presents pedagogical resources that enable learning and teaching of algebraic structures as well as propose greater meaning of this concept in mathematical graduation programs. This study also proposes an answer to the following question: in what way a teaching approach that is centered in the Theory of Numbers and Theory of Equations is a model for the teaching of the concept of Group? To answer this question a historical reconstruction of the development of this concept is done on relating Lagrange to Cayley. This is done considering Foucault s (2007) knowledge archeology proposal theoretically reinforced by Dreyfus (1991). An exploratory research was performed in Mathematic graduation courses in Universidade Federal do Par? (UFPA) and Universidade Federal do Rio Grande do Norte (UFRN). The research aimed to evaluate the formation of concept images of the students in two algebra courses based on a traditional teaching model. Another experience was realized in algebra at UFPA and it involved historical components (MENDES, 2001a; 2001b; 2006b), the development of multiple representations (DREYFUS, 1991) as well as the formation of concept images (VINNER, 1991). The efficiency of this approach related to the extent of learning was evaluated, aiming to acknowledge the conceptual image established in student s minds. At the end, a classification based on Dreyfus (1991) was done relating the historical periods of the historical and epistemological development of group concepts in the process of representation, generalization, synthesis, and abstraction, proposed here for the teaching of algebra in Mathematics graduation course / O presente estudo analisa o desenvolvimento hist?rico-epistemol?gico do conceito de Grupo a luz da teoria do pensamento matem?tico avan?ado, proposto por Dreyfus (1991) e apresenta subs?dios did?ticos que contribuam para o ensino-aprendizagem das estruturas alg?bricas, visando dar maior significado ao referido conceito abordado na gradua??o em Matem?tica. Nesse sentido, o estudo responde a seguinte pergunta: de que maneira uma abordagem de ensino, inicialmente, centrada na Teoria dos N?meros e na Teoria das Equa??es se constituiria em um modelo de efetiva??o do ensino do conceito de Grupo? Para responder a quest?o fizemos uma reconstru??o hist?rica do desenvolvimento desse conceito, de Lagrange a Cayley, em uma reescrita orientada na arqueologia do saber proposta e discutida por Foucault (2007) e com o apoio te?rico em Dreyfus (1991) analisamos o material hist?rico elaborado. Em seguida, fizemos uma pesquisa explorat?ria com turmas da gradua??o em Matem?tica da Universidade Federal do Par? (UFPA) e da Universidade Federal do Rio Grande do Norte (UFRN), para avaliar a forma??o de imagens conceituais nos alunos participantes de dois cursos de ?lgebra baseado em um modelo tradicional de ensino. Al?m disso, realizamos outra experi?ncia, na UFPA, com o ensino de ?lgebra envolvendo, conjuntamente, a inclus?o da componente hist?rica (MENDES, 2001a; 2001b; 2006b), o desenvolvimento de m?ltiplas representa??es (DREYFUS, 1991) e a forma??o das imagens conceituais (VINNER, 1991). Avaliamos a efic?cia da abordagem em termos da profundidade no alcance do aprendizado, ou seja, a imagem conceitual estabelecida na mente dos alunos. Ao final, apresentamos uma classifica??o, baseada em Dreyfus (1991), que relaciona per?odos hist?ricos do desenvolvimento hist?rico-epistemol?gico do conceito de grupo aos processos de representa??o, generaliza??o, s?ntese e abstra??o, e uma proposta para um curso de ?lgebra na gradua??o em Matem?tica

<p>У докторској дисертацији презентовано је педагошко истраживање које се односи на увођење<br />математичког моделирања као савременог методичког приступа учењу математике, његову примену у<br />наставној пракси, као и на могућност да се кроз математичко моделирање проблема, појава и процеса који<br />се обрађују у неком другом наставном предмету оствари модернији, интердисциплинарни приступ<br />средњошколској настави. Све то у циљу афирмације иновативног, креативног и напредног математичког<br />мишљења, квалитетног, структуираног и функционалногзнања, које се у овом истраживању односи на<br />одређене наставне садржаје математичке анализе. У наставни процес интегрисан је осмишљен и израђен<br />пројекат чија је реализација омогућила успостављањемеђупредметне корелације, изучавање функција и<br />њихових примена у научним проблемима кроз процесе математичког моделирања. Нарочита пажња<br />посвећена је реализацији когнитивних активности ученика у свакој етапи процеса математичког<br />моделирања и остваривању њиховог вертикално-кумулативног поретка. Као посебна активност усмерена на<br />потребе и циљеве одабраних процеса математичког моделирања и усклађена са наставним садржајима<br />који су изучавани, уведен је и примењен нови методички приступ у обради функција датих помоћу<br />одређеног интеграла, са посебним акцентом на анализу логаритамске функције дате помоћу одређеног<br />интеграла, а затим и анализу експоненцијалне функције, као инверзне логаритамској. Један део<br />истраживања фокусиран је на увођење и реализацију назначене нове методичке концепције у обради<br />функција датих помоћу одређеног интеграла и на високошколском нивоу учења математичке анализе.<br />Испитивање ефикасности примењених нових, савременихметодичких приступа у раду са ученицима,<br />односно студентима, уз планску и систематску употребу рачунара са одговарајућом софтверском<br />подршком, обрађено је компаративном анализом резултата педагошких експеримената. На основу<br />резултата истраживања утврђен је позитиван утицај предложених методичких приступа на квалитет<br />математичког знања ученика, односно студената, и оствареност оптималних резултата у учењу и изучавању<br />наставних садржаја из области функције.</p> / <p>U doktorskoj disertaciji prezentovano je pedagoško istraživanje koje se odnosi na uvođenje<br />matematičkog modeliranja kao savremenog metodičkog pristupa učenju matematike, njegovu primenu u<br />nastavnoj praksi, kao i na mogućnost da se kroz matematičko modeliranje problema, pojava i procesa koji<br />se obrađuju u nekom drugom nastavnom predmetu ostvari moderniji, interdisciplinarni pristup<br />srednjoškolskoj nastavi. Sve to u cilju afirmacije inovativnog, kreativnog i naprednog matematičkog<br />mišljenja, kvalitetnog, struktuiranog i funkcionalnogznanja, koje se u ovom istraživanju odnosi na<br />određene nastavne sadržaje matematičke analize. U nastavni proces integrisan je osmišljen i izrađen<br />projekat čija je realizacija omogućila uspostavljanjemeđupredmetne korelacije, izučavanje funkcija i<br />njihovih primena u naučnim problemima kroz procese matematičkog modeliranja. Naročita pažnja<br />posvećena je realizaciji kognitivnih aktivnosti učenika u svakoj etapi procesa matematičkog<br />modeliranja i ostvarivanju njihovog vertikalno-kumulativnog poretka. Kao posebna aktivnost usmerena na<br />potrebe i ciljeve odabranih procesa matematičkog modeliranja i usklađena sa nastavnim sadržajima<br />koji su izučavani, uveden je i primenjen novi metodički pristup u obradi funkcija datih pomoću<br />određenog integrala, sa posebnim akcentom na analizu logaritamske funkcije date pomoću određenog<br />integrala, a zatim i analizu eksponencijalne funkcije, kao inverzne logaritamskoj. Jedan deo<br />istraživanja fokusiran je na uvođenje i realizaciju naznačene nove metodičke koncepcije u obradi<br />funkcija datih pomoću određenog integrala i na visokoškolskom nivou učenja matematičke analize.<br />Ispitivanje efikasnosti primenjenih novih, savremenihmetodičkih pristupa u radu sa učenicima,<br />odnosno studentima, uz plansku i sistematsku upotrebu računara sa odgovarajućom softverskom<br />podrškom, obrađeno je komparativnom analizom rezultata pedagoških eksperimenata. Na osnovu<br />rezultata istraživanja utvrđen je pozitivan uticaj predloženih metodičkih pristupa na kvalitet<br />matematičkog znanja učenika, odnosno studenata, i ostvarenost optimalnih rezultata u učenju i izučavanju<br />nastavnih sadržaja iz oblasti funkcije.</p> / <p>In this PhD dissertation pedagogical research related to introduction of mathematical modeling as a<br />modern methodical approach to learning mathematics is presented, its application in teaching practice,as well<br />as possibilities to achieve modern, interdisciplinary approach to high school teaching, using mathematical<br />modeling of problems, phenomenons and processes which are being looked at in another ciruculum subject. The<br />intention is to affirm innovative, creative and advanced mathematical thinking, high quality, structured and<br />functional knowledge, which in this study refers tospecific areas of mathematical analysis. A well thought<br />project was designed and constructed in such a way&nbsp; to integrate within the teaching process, and its<br />implementation has enabled the establishment of interdisciplinary correlation and the study of functions and<br />their use in scientific problems through the process of mathematical modelling. Special attention is paid to the<br />realization of cognitive activities of pupils in each stage of the process of mathematical modelling, and acheiving<br />their vertically-cumulative order. As a separate activity focused on the goals and needs of selected processes of<br />mathematical modeling, harmonized with teaching content being studied, a new methodical approach was<br />introduced and applied to treat functions given by&nbsp; a definite integral, with special emphasis on treatment of<br />logarithmic function given by a definite integral,&nbsp; and followed by treatment of the exponential function as the<br />inverse logarithmic. A part of the research is focused on the introduction and implementation of new methodical<br />concept in treatment of functions given by a definite integral and on university level of studying mathematical<br />analysis. Efficiency of these new and modern methodical approaches was tested on pupils and students,&nbsp; with<br />planned and systematic usage of computers and appropriate software, and processed by comparative analysis of<br />results obtained by pedagogical experiments. Results have shown positive impact of proposed methodical<br />approaches to the quality of pupils&rsquo; and students&rsquo;&nbsp; mathematical knowledge, and the achievement of optimal<br />results in learning and studying of teaching content on mathematical functions.</p>

In our experience of teaching Calculus to engineering undergraduates we have had to grapple with many different problems. A major hurdle has been students’ inability to appreciate the importance of the theory. In their view the theoretical part of mathematics is separate from the computing part. In general, students also believe that they can pass their exams even though they do not have a real understanding of the theory behind the problems they are required to solve. In an effort to surmount these difficulties we tried to find ways to make students better understand the theoretical part of Calculus. This paper describes our experience of teaching Calculus. It reports on the continuation of our previous research.

Today’s research, as well as current Swedish governance documents, stress the high importance of developing students’ abstract, individual and critical thinking in mathematics education. The needs for such quality thinking however stand in stark contrast to the ‘traditional mathematics education’, which cannot be expected to develop such thinking. This study suggests an evolutionary instead of revolutionary approach in leaving the old paradigm of ‘traditional education’ behind, since project support for such a transformation project is claimed to be lacking on central level while the feasibility of such a paradigm shift on individual (teacher or even school) level is claimed to be very low in today’s time-pressured reality. By investigating teacher actions which generate occurrences of activated, exposed and elicited student thinking, this study purports to suggest alternative ways forward. A series of semi-structured classroom observations, complemented with questionnaires and interviews, has therefore been carried out of 18 mathematics teachers in upper secondary level in Sweden. The study is based on empowerment theory (as well as an adaptation from the industry of an applied empowerment model), a triangulation mixed-method design was adopted and a thematic analysis, underpinned by a latent theoretical approach from an epistemologically constructionist perspective as described by Braun and Clarke (2006), was used. The study identified several occurrences of the aforementioned student thinking, as well as their corresponding teacher actions and proposes a way for organizing the actions as well as their outcome into an overarching model. / Dagens forskning och samtliga svenska styrdokument betonar den stora vikten av att inom matematikundervisningen utveckla det abstrakta, individuella och kritiska tänkandet hoseleverna. Behoven av sådant ”kvalitetstänkande” står dock i kraftig kontrast till ”traditionell matematikundervisning”, som inte kan förväntas utveckla sådant tänkande. Denna studie föreslår ett evolutionärt, istället för revolutionärt, angreppssätt för att lämna det gamla paradigmet, eftersom projektstöd för ett sådant transformationsprojekt saknas på central nivå och genomförbarheten för ett paradigmskifte på individuell (lärar- eller t.o.m. skol-) nivå hävdas vara låg i dagens tidspressade verklighet. Genom att studera lärarhandlingar som genererar förekomster av aktiverat, exponerat och stimulerat elevtänkande, menar denna studie att hitta alternativa vägar framåt. En serie semistrukturerade klassrumsobservationer, kompletterade med enkäter och intervjuer, har därför genomförts hos 18 matematiklärare på gymnasiet i Sverige. Studien är baserad på empowerment-teori (tillsammans med en industriell tillämpning av en empowerment-modell), en blandad metoddesign (triangulation mixed-method) samt en tematisk analys kännetecknad av ett latent-teoretiskt angreppssätt och av ett epistemologiskt konstruktionistiskt perspektiv enligt Braun och Clarke (2006). Studien identifierade flera förekomster av ovannämnda elevtänkande, likväl som motsvarande lärarhandlingar, och föreslår ett sätt att strukturera både handlingar och deras resultat i en övergripande modell.

The present study investigated imagery and visualisation characteristics of undergraduate students’ thinking processes in learning selected concepts of mathematical analysis. The aim was to discover the nature of images evoked by these undergraduate students and the role of imagery and visualisations when students were solving some selected problems related to mathematical analysis. The study was guided by the theory of registers of semiotic representations. Psychological notions on imagery were also fused to cater for a cognitive approach to the study. A sample of 50 undergraduate mathematics students participated in the study. The researcher employed both quantitative and qualitative methods. Before the main study, a pilot study was conducted to account for the reliability and validity of the research instruments. The data were collected through use of a cognitive test that was composed of 12 tasks with items selected from mathematical analysis. These tasks were specially designed to capture the variables of imagery and visualisations. A structured interview was also conducted as a follow-up to the results of the cognitive test. The study found that visual images were noticeable in the thinking processes of undergraduate students in solving problems related to mathematical analysis. The nature of the visual images evoked by the students varied from person to person. The nature of these images was also determined by the nature of the task. The most common types of imagery were diagrams, prototypes and symbols. On rare occasions the students also evoked metaphoric images. It was also observed that these images were used for illustrative purposes and to spark the idea for a proof. It was also interesting to note that some images were used to discover the limit of a converging series. The results confirmed the need to use visualisation with caution, especially when treating concepts which involve infinity. The study recommends that instructors of mathematics should encourage visual thinking in the learning and teaching of mathematical analysis. Knowledge of the students’ concept images helped the researcher to understand the nature of the learning difficulties of the students. Further research should focus on the strengths and weaknesses of visual-mediated learning and also on the relationship between creativity and visual thinking. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education (Mathematics Education))

CARVALHO, Liliane Maria Teixeira Lima de. O papel dos artefatos na construção de significados matemáticos por estudantes do ensino fundamental II. 2008. 239 f. Tese (Doutorado em Educação) – Universidade Federal do Ceará. Faculdade de Educação, Programa de Pós-Graduação em Educação Brasileira, Fortaleza-CE, 2008. / Submitted by Raul Oliveira (raulcmo@hotmail.com) on 2012-07-05T14:31:58Z No. of bitstreams: 1 2008_Tese_LMTLCarvalho.pdf: 4899664 bytes, checksum: df61d0611ea9db1257467f902e9384de (MD5) / Approved for entry into archive by Maria Josineide Góis(josineide@ufc.br) on 2012-07-05T14:36:14Z (GMT) No. of bitstreams: 1 2008_Tese_LMTLCarvalho.pdf: 4899664 bytes, checksum: df61d0611ea9db1257467f902e9384de (MD5) / Made available in DSpace on 2012-07-05T14:36:14Z (GMT). No. of bitstreams: 1 2008_Tese_LMTLCarvalho.pdf: 4899664 bytes, checksum: df61d0611ea9db1257467f902e9384de (MD5) Previous issue date: 2008 / A pesquisa investiga se diferentes formas de conceber o papel dos artefatos e apresentação da informação influenciam a construção de significados matemáticos por estudantes de 11 a 14 anos. A cognição humana é concebida como processo mediado pela tradição cultural e histórica das representações enquanto artefatos, inserindo-se essa análise no âmbito do raciocínio matemático. Utilizou-se o método experimental aliado a uma pesquisa-ação envolvendo o design intencional de tarefas. Explorou-se o papel mediacional das tarefas, desde a sua confecção e introdução na sala de aula de matemática, até o seu uso pelos estudantes. Essa abordagem se concretizou por meio de seis experimentos, dos quais participaram 922 estudantes: 598 oriundos do key Stage Three (corresponde em idade ao 7º, 8º e 9º anos do Ensino Fundamental II no Brasil) de quatro escolas inglesas, e 324 oriundos do 7º, 8º e 9º anos de duas escolas brasileiras. O Experimento 1 investiga se gráficos, tabelas ou casos isolados influenciam o raciocínio dos estudantes sobre variáveis discretas. O Experimento 2 verifica se diferentes informações sobre variáveis contínuas influenciam a interpretação gráfica dos estudantes. O Experimento 3 analisa se interações de aspectos visuais e conceituais da informação sobre variáveis contínuas influenciam a interpretação gráfica dos estudantes. O Experimento 4 investiga se gráficos, tabelas ou a combinação de ambas as representações influencia interações de aspectos visuais e conceituais da informação. Esses quatro experimentos foram realizados nas escolas inglesas. As tarefas usadas no primeiro e quarto experimentos foram aplicadas nas escolas brasileiras, sendo designados Experimentos 5 e 6, respectivamente. As tarefas foram potencialmente facilitadoras ao uso de conteúdos matemáticos. Os Experimentos 1 e 5 oferecem evidências de que estudantes já familiarizados com representações em tabelas e gráficos para representar variáveis discretas não se beneficiam em atividades em que eles precisam organizar os dados por eles mesmos. Estudantes ingleses tiram proveito igualmente de tabelas e gráficos. Estudantes brasileiros não se beneficiam do uso de tabelas. Os Experimentos 2 e 3 confirmam resultados de estudos prévios de que informações gráficas sobre variáveis contínuas possuem diferentes níveis de complexidade. Ler pontos é significativamente mais fácil do que interpretar problemas globais. Os Experimentos 2 e 3 também confirmam a hipótese de que os problemas de inferência inversa explicam as dificuldades com informações globais. Essa dificuldade é acentuada em gráficos com inclinação negativa. O Experimento 4 mostra que a forma de apresentação da informação não afeta o desempenho dos estudantes na resolução de problemas sobre variáveis contínuas. O raciocínio dos estudantes sobre variáveis contínuas, no entanto, é influenciado pela forma de apresentação da informação. A pesquisa sugere a necessidade de uma discriminação da informação não apenas quanto ao tipo de variável, discreta ou contínua, ou tipo de relação proporcional, direta ou inversa, mas também quanto ao tipo de inferências requeridas dos estudantes.

This study explores the predominant mathematical thinking style that students with academic talent used in solving mathematical problems. Thinking styles are preferences by subjects in the way of expressing mathematical skills against a task, in this case, visual, formal and integrated. We assessed 99 students from an academic support talent program, in a retrospective ex post facto study with only one group. We administered the questionnaire mathematical thinking styles of Borromeo-Ferri and determined that these students exhibited mostly an integrated style of thinking, which involves the use of symbols and verbal representations with visual expressions in solving mathematical exercises. They also show a strong orientation to address the problems of combined mode, which involves considering them as a whole at a time. / El presente estudio establece el estilo de pensamiento matemático predominante que utilizan los estudiantes con talento académico en la resolución de problemas matemáticos. Los estilos de pensamiento son preferencias por parte de los sujetos en la forma de expresar las habilidades frente a una tarea matemática, en este caso, visual, formal e integrado. En el marco de un estudio ex post facto retrospectivo de grupo único, se evaluó a un total de 99 estudiantes pertenecientes a un programa académico de apoyo al talento con el cuestionario Estilos de Pensamiento Matemático de Borromeo-Ferri. Los resultados indican que los estudiantes declararon orientarse hacia el estilo de pensamiento integrado, que supone el uso de simbología y representaciones verbales junto con expresiones visuales en la resolución de los ejercicios matemáticos, así como una significativa orientación a abordar los problemas de modo combinado, que supone considerar los problemas como un todo. / La présente étude établit le style de pensée mathématique prédominant utilisé par les étudiants ayant un talent académique dans la résolution de problèmes mathématiques. Les styles de pensée sont des préférences de la part des sujets sous la forme d’exprimer les capacités face à une tâche mathématique, dans ce cas, visuelle, formelle et intégrée. Dans une étude rétrospective sur un seul groupe ex post facto, un total de 99 étudiants appartenant à un programme de soutien aux talents universitaires ont été évalués, à qui le questionnaire Styles de Pensée mathématique de Borromeo-Ferri a été appliqué et déterminé que ce type de sujets déclare principalement un style de pensée intégré, ce qui implique l’utilisation de la symbologie et des représentations verbales ainsi que des expressions visuelles dans la résolution des exercices mathématiques. En outre, ils montrent une forte orientation pour aborder les problèmes de manière combinée, ce qui implique de les considérer dans leur ensemble dans le même temps. / Este estudo estabelece o estilo predominante do pensamento matemático usado por os alunos com talento acadêmico na resolução de problemas matemáticos. Os estilos de pensamento são as preferências dos indivíduos sobre a forma para expressar as capacidades em uma tarefa matemática, neste caso, visual, formal e integrada. Como parte de um estudo ex post facto retrospectivo de grupo único, foram avaliados um total de 99 estudantes de um programa de talento acadêmico. Foram aplicados nos alunos o questionário “Estilos de Pensamento Matemático de Borromeo-Ferri” e determinou-se que a maioria dos participantes declararam um estilo de pensamento integrado, que envolve o uso de símbolos e representações verbais com resolução de expressões visuais de exercícios matemáticos. Eles mostram também uma forte orientação para resolver os problemas de modo combinado, o qual envolve a considerá-los como um todo de uma vez.