<p> The basis of this work is to lay the groundwork for relational thinking in mathematics by giving a general mathematical definition of relational thinking in mathematics that builds on the theory of relational thinking in arithmetic and then extends that theory to include all other mathematics subjects, especially algebra and geometry. The necessity to include all other mathematics subjects in relational thinking is predicated on the need for students at all levels to be able to think relationally. In an effort to further establish relational thinking in mathematics, this work attempts to merge mathematics and philosophy by examining Plato's <i>Meno</i> and Wittgenstein's <i>Philosophical Investigations</i> to show the importance of deductive reasoning, logic, and language in the use of relational thinking in mathematics. Further, this work also sets out to establish relations in a mathematical sense as a unifying concept in algebra and geometry. I therefore define relational thinking in mathematics as the skill and propensity to use deductive reasoning and logic in order to make connections between and among abstract mathematical concepts and specific instances thereof. This definition stems from mathematics being built on two pillars--that is, deductive reasoning and logic--and being of two different branches--that is, abstract mathematics and applied mathematics. </p>

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2007. / Source: Dissertation Abstracts International, Volume: 68-06, Section: A, page: 2377. Adviser: Klaus Witz. Includes bibliographical references (leaves 157-170) Available on microfilm from Pro Quest Information and Learning.

Through philosophical means, this thesis investigates the question: What mathematics can mean to students philosophically and psychologically? It is reasonable to assume that students may touch upon philosophical issues in trying to make sense of mathematics, since, in a sense, all individuals philosophise while searching for meaning in their own activities. Moreover, the existing literature indicates a substantial gap in our understanding of the meaning of mathematics and its philosophy in education. The thesis is based on a hermeneutical perspective. In this context, in-depth interviews were conducted with 17-year-old students in a Greek school. This method allowed me to obtain data which illuminated the objective meaning of students’ philosophical beliefs by way of the subjective, psychological meaning that they attributed to mathematics. The sample consisted of 28 students comprising both sexes and all levels of engagement with mathematics. The main issues that were examined were: whether mathematics exists; whether mathematical knowledge is certain, objective, true and immutable; whether mathematics consists of rules; and whether mathematical knowledge is based on logic or on experience. A thematic analysis helped me to move within the hermeneutical circle of understanding. As well as organising the objective meaning of students’ philosophical beliefs into themes and subthemes, analysis showed how for each student, there was an emergent a story which illustrating how they could combine such beliefs in order to find subjective meaning in mathematics. The most important finding of the study suggests that the students’ beliefs were influenced by common sense, and that students were able to find positive subjective meaning in mathematics when they were able to relate aspects of mathematical reasoning (e.g. certainty, subjectivity, rules, experience) to the operation of their everyday common sense. The study therefore shows that discussing philosophical issues, and in particular mathematical reasoning, could be of considerable benefit for students learning mathematics.

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2009. / Source: Dissertation Abstracts International, Volume: 70-06, Section: A, page: . Adviser: Walter Feinberg. Includes bibliographical references (leaves 201-210) Available on microfilm from Pro Quest Information and Learning.

Orientador: Antonio Miguel / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-08-09T09:57:52Z (GMT). No. of bitstreams: 1 Vilela_DeniseSilva_D.pdf: 1524315 bytes, checksum: 9bbabd38825517fb6c7294587eeedc4c (MD5) Previous issue date: 2007 / Resumo: Como o termo matemática vem sendo usado na literatura acadêmica da Educação Matemática? Esta é a questão inicial que orienta este estudo investigativo realizado com base em publicações e pesquisas acadêmicas recentes em Educação Matemática. Com base nesses documentos, verificou-se a ocorrência, em freqüência significativa, de diversas adjetivações do termo matemática tais como: matemática escolar, matemática da rua, matemática acadêmica, matemática popular, matemática do cotidiano, etc. A partir da análise de alguns desses textos, constatou-se que as adjetivações, que ocorrem geralmente aos pares, apontam especificidades das matemáticas, tais como, diferenças em resultados, processos, valores, significados, conceitos, etc. A partir de uma visão de conjunto das especificidades apontadas nos textos pesquisados, as diversas adjetivações são interpretadas como jogos de linguagem que não possuiriam uma essência, mas apresentariam semelhanças de famílias, no sentido dado por Wittgenstein a este conceito. Para formular a questão acima, inspiramo-nos nos conceitos desse filósofo, bem como em sua concepção de filosofia, que possui uma perspectiva de ampliação dos significados alcançada mediante as descrições dos usos de um conceito, a qual possibilita dissolver a noção essencialista e referencial de significado A partir disso, para alcançar um sentido sociológico dessas adjetivações à interpretação filosófica é ampliada com conceitos da sociologia de Bourdieu, notadamente com o conceito de campo científico. As adjetivações expressariam uma tensão no campo das matemáticas: o reconhecimento da produção de conhecimentos matemáticos em diversas práticas que não só a dos matemáticos profissionais, mas também as dos professores, as de grupos profissionais, etc., e também o questionamento do monopólio da definição e atribuições do campo por matemáticos profissionais. Ou seja, as adjetivações são entendidas como objetivações de novos termos da gramática do campo das matemáticas. Além disso, são indicados elementos para uma compreensão das matemáticas como práticas sociais, não simplesmente como determinadas por estratégias racionais intencionais, e sim como práticas condicionadas pela própria estrutura da linguagem, que implica em regularidades as quais limitam e regulam as possibilidades de inteligibilidade e de desenvolvimento das matemáticas nas práticas especificas, mas que não constituem regulamentos que impediriam novos usos / Abstract: How used the term Mathematic in Mathematics Education literature has been? This is the main question that guides this investigation, supported by recent academic researches and publications in the field of Mathematics Education. Based on these writings, we have noticed the existence of several ways of adjetivizing the term mathematics such as: school mathematics, street mathematics, academic mathematics, popular mathematics, everyday mathematics, and so on. After analyzing some of these works, it can be seen that these adjectives, that often show up in pairs, point to mathematics specificities, such as difference in results, processes, values, meanings, concepts etc. From a global view of the specificities pointed in all researched texts, adjectives are understood as language-games that do not have an essence, but would present family resemblances, in the sense given by Wittgenstein to these concepts. To answer the question stated above, we were inspired by these Wittgenstein¿s concepts, as well as by his conception of philosophy, which has a tendency to broaden meanings through the use descriptions of a concept. This allows dissolving na essentialist notion of meaning as a last and universal reference. In order to search for a sociological sense for these adjectivize, our goal is to historically rescue this tendency to adjective in the current context of cultural studies in which modernity values are questioned. Thus, a philosophical explanation is broadened by concepts of Bourdieu's sociology, specially the concept of scientific field. The adjectives would express a struggle within the mathematics field, and the recognition that there is mathematical knowledge production in many practices beyond the professional mathematicians¿, like teachers¿, professional groups¿ etc. That is, adjectives are understood as objectivations for new terms of a grammar of mathematics fields. Besides, this work indicates elements for a understanding of mathematics as social practices, not merely determined by rational and intended strategies, but also as practices conditioned by language structures, which implies regularities that limit and adjust the possibilities of understanding and developmenting of mathematics within specific practices, but not representing regulations that could hinder new uses / Doutorado / Educação Matematica / Doutor em Educação