Este trabalho tem como objetivo principal contribuir para o aperfeiçoamento do professor de matemática seja ele em formação ou em atuação. Buscamos oferecer um material que possa servir de referência técnica, histórica e epistemológica para o estudo do Logaritmo Natural. Discutimos aqui o conceito de Conhecimento Especializado do Conteúdo, cunhado por pesquisadores da Universidade de Michigan e liderados por Deborah Ball. Em seu artigo Content Knowledge for Teaching: What Makes It Special? (2008), eles levantam a questão \"Qual matemática o professor deve conhecer para dar cabo do trabalho de ensinar?\", dado que o conhecimento matemático necessário para o docente difere do conhecimento matemático requerido em outras profissões. Fazemos aqui uma análise crítica da abordagem utilizada para o tema em alguns livros didáticos de Ensino Médio, descrevemos de modo detalhado a construção da Função Logarítmica como realmente ocorreu no século XVII, ou seja, por meio de áreas de regiões sob a curva xy = 1, e definimos a função exponencial como a inversa dela, enfoque esse com caráter fortemente geométrico e que deu origem à noção de integral definida. Mostramos também a estreita relação existente entre as Progressões Aritméticas, Geométricas, Trigonometria e o próprio tema principal. Obtemos ainda a formalização do número irracional e tanto pelo método tradicional usado em livros de Cálculo e Análise como a decorrente da teoria apresentada. Por fim, apresentamos algumas situações curiosas que envolvem direta ou indiretamente essa constante e que podem ser trabalhadas com alunos da Educação Básica. / The main objective of this work is to contribute to the improvement of the mathematics teacher, whether in training or acting. We seek to offer a material that can serve as a technical, historical and epistemological reference for the study of the Natural Logarithm. We discuss here the concept of Specialized Content Knowledge, coined by University of Michigan researchers and led by Deborah Ball. In your article Content Knowledge for Teaching: What Makes It Special? (2008), they raise the question \"What mathematics does the teacher need to know for teaching?\", since the mathematical knowledge required for the teacher differs from the mathematical knowledge required in other professions. Here we present a critical analysis of the approach used for the subject in some high school textbooks. We describe in detail the construction of the Logarithmic Function as actually occurred in the seventeenth century, that is, through areas of regions under the curve xy = 1, and we define the exponential function as the inverse of it, a focus with a strongly geometric character that gave rise to the notion of definite integral. We also show the close relationship between Arithmetic, Geometric, Trigonometry and the main theme itself. We also obtain the formalization of the irrational number e, both by the traditional method used in Calculus and Analysis books and by the theory presented. Finally, we present some curious situations that directly or indirectly involve this constant and that can be worked with Basic Education students.

Este trabalho tem como objetivo principal contribuir para o aperfeiçoamento do professor de matemática seja ele em formação ou em atuação. Buscamos oferecer um material que possa servir de referência técnica, histórica e epistemológica para o estudo do Logaritmo Natural. Discutimos aqui o conceito de Conhecimento Especializado do Conteúdo, cunhado por pesquisadores da Universidade de Michigan e liderados por Deborah Ball. Em seu artigo Content Knowledge for Teaching: What Makes It Special? (2008), eles levantam a questão \"Qual matemática o professor deve conhecer para dar cabo do trabalho de ensinar?\", dado que o conhecimento matemático necessário para o docente difere do conhecimento matemático requerido em outras profissões. Fazemos aqui uma análise crítica da abordagem utilizada para o tema em alguns livros didáticos de Ensino Médio, descrevemos de modo detalhado a construção da Função Logarítmica como realmente ocorreu no século XVII, ou seja, por meio de áreas de regiões sob a curva xy = 1, e definimos a função exponencial como a inversa dela, enfoque esse com caráter fortemente geométrico e que deu origem à noção de integral definida. Mostramos também a estreita relação existente entre as Progressões Aritméticas, Geométricas, Trigonometria e o próprio tema principal. Obtemos ainda a formalização do número irracional e tanto pelo método tradicional usado em livros de Cálculo e Análise como a decorrente da teoria apresentada. Por fim, apresentamos algumas situações curiosas que envolvem direta ou indiretamente essa constante e que podem ser trabalhadas com alunos da Educação Básica. / The main objective of this work is to contribute to the improvement of the mathematics teacher, whether in training or acting. We seek to offer a material that can serve as a technical, historical and epistemological reference for the study of the Natural Logarithm. We discuss here the concept of Specialized Content Knowledge, coined by University of Michigan researchers and led by Deborah Ball. In your article Content Knowledge for Teaching: What Makes It Special? (2008), they raise the question \"What mathematics does the teacher need to know for teaching?\", since the mathematical knowledge required for the teacher differs from the mathematical knowledge required in other professions. Here we present a critical analysis of the approach used for the subject in some high school textbooks. We describe in detail the construction of the Logarithmic Function as actually occurred in the seventeenth century, that is, through areas of regions under the curve xy = 1, and we define the exponential function as the inverse of it, a focus with a strongly geometric character that gave rise to the notion of definite integral. We also show the close relationship between Arithmetic, Geometric, Trigonometry and the main theme itself. We also obtain the formalization of the irrational number e, both by the traditional method used in Calculus and Analysis books and by the theory presented. Finally, we present some curious situations that directly or indirectly involve this constant and that can be worked with Basic Education students.

Aquesta tesi analitza l'aprenentatge de coneixement especialitzat a partir d'una proposta teoricometodològica, de base lingüística i conceptual, centrada en l'àmbit del dret de família català. L'anàlisi empírica se centra en l'estudi de les definicions d'estudiants universitaris sobre nou conceptes concrets al llarg de dos tests amb les mateixes definicions: un test a l'inici del període d'instrucció explícita i un altre al final. L'objectiu principal és determinar si es produeix increment i precisió de coneixement en els tres mesos d'instrucció explícita de l'assignatura. L'anàlisi de les dades confirma que es produeix una estabilització del coneixement més que no pas un increment. Aquesta estabilització s'ha de relacionar directament amb el període considerat i amb el fet que el coneixement previ (no especialitzat) dels conceptes sembla estar fortament arrelat en els estudiants i, per tant, en un període tan breu d'instrucció explícita és difícil d'assolir el canvi conceptual a la manera dels especialistes. / This Doctoral Dissertation deals with specialized knowledge acquisition having a theoretical and methodological orientation of a linguistic and conceptual nature. Empirical analysis is devoted to Catalan Family Law and, more precisely, to the definitions provided by university students regarding 9 concepts throughout two tests with the same definitions. The first test is given at the beginning of the period of explicit instruction and the second at the end.The main goal is to determine whether increase and precision of knowledge is achieved or not after the three months of explicit instruction. Data analysis tells that there is knowledge stabilization rather than knowledge increasing. This stabilization can be put down to the considered period, together with the fact that previous knowledge of concepts seem to be firmly rooted in students, which leads to state that conceptual change (i.e., that of specialists) is difficult to attain within such brief period of instruction.

Solicitud de embargo por publicación en revista indexada. / Los problemas aritméticos de enunciado verbal constituyen una parte fundamental del área de Matemáticas, ya que su enseñanza y resolución son una de las grandes dificultades que enfrentan los profesores y estudiantes. En este trabajo desde un enfoque cualitativo se realizará un análisis didáctico respecto al Conocimiento especializado del profesor de Matemáticas (MTSK) sobre los problemas aritméticos de enunciado verbal (PAEV). / Verbal arithmetic problems are established as one of the essential parts of the Area of Math, since their teaching and resolution are one of the great difficulties faced by teachers and students. In this work, from a qualitative perspective, a didactic analysis will be carried out with respect to the Mathematics Teacher Specialized Knowledge (MTSK) on the arithmetic problems of verbal statement (PAEV). / Trabajo de investigación

<p dir="ltr">This dissertation explores the intricate dynamics between the self-perceptions of undergraduate mathematics (UM) educators and their conceptions of excellent teaching practices conducive to student learning. Employing a sequential mixed methods approach, the study addresses two primary research questions. First, it investigates educators' self-perceptions within the realm of UM teaching, examining potential variances based on educators' Professional Status and Educational Institution (PSEI) affiliations and experience levels. Second, it delves into educators' perspectives on aspects of excellent UM teaching, scrutinizing potential disparities rooted in PSEI affiliations and experience levels, while also exploring the manifestations of Mathematics Teachers' Specialized Knowledge (MTSK) and teaching self-concept within these descriptors.</p><p dir="ltr">Drawing upon Shavelson's self-concept (1976) framework and Carrillo and colleagues' (2018) MTSK framework, data collection involved a Likert-style questionnaire augmented by open-ended inquiries, followed by qualitative case studies featuring eight participants from diverse Carnegie classifications. Findings demonstrate educators' overall confidence in their teaching abilities, with notable discrepancies observed among educators from associate's colleges and doctoral universities. Through thematic analysis, key dimensions of excellent teaching emerged, including active learning, student engagement, problem-solving, and positive learning environments.</p><p dir="ltr">This study yields implications for educational practice and institutional policy. Educators can leverage identified themes to inform professional development initiatives tailored to enhance UM teaching effectiveness. Furthermore, the validated instrument offers institutions a means to assess educators' confidence levels, facilitating targeted support within mathematics departments.</p><p dir="ltr">In conclusion, this dissertation contributes valuable insights into the multifaceted interplay between educators' self-perceptions, teaching practices, and student learning outcomes within the context of UM instruction.</p>