Abstract Considering the development of reasoning as a central aspect in the process of learning mathematics, understanding ways to promote it and actions performed by the teacher in this process are important issues for investigations. In order to deepen these discussions, this article aims to understand how teacher’s actions can contribute to mathematical reasoning students’ development who study differential and integral calculus. Assuming a qualitative perspective of interpretative nature, we analyzed a class of this subject, in which it was promoted a discussion about an exploratory task involving a water and salt mixture concentration varying by the time, and, from it was introduced the concept of rational function and its long-run behavior. Six class excerpts are analyzed, based on a framework which classifies teacher’s actions from four categories (inviting; guiding/ supporting; informing/suggesting; and challenging). As results, we recognized a continuous and increasing movement in the teacher’s actions during the conduct of the discussion, highlighting the implications for the mathematical reasoning students’ development and the concept of rational function understanding.

Resumo Considerando o desenvolvimento do raciocínio como aspecto central no processo de aprendizagem matemática, compreender modos de promovê-lo e ações desempenhadas pelo professor neste processo são questões importantes para investigação. No intuito de aprofundar essas discussões, este artigo tem, por objetivo, compreender como as ações do professor podem contribuir para o desenvolvimento do raciocínio matemático de estudantes que cursam cálculo diferencial e integral. Assumindo uma perspectiva qualitativa de cunho interpretativo, analisamos uma aula dessa disciplina, na qual promoveu-se uma discussão a respeito de uma tarefa exploratória envolvendo a concentração de uma mistura de água e sal variando com o tempo, e, a partir dela, introduziu-se o conceito de função racional e seu comportamento a longo prazo. Com base em um quadro teórico que classifica as ações do professor a partir de quatro categorias – a constar: convidar; guiar/ apoiar; informar/sugerir; e desafiar, são analisados seis trechos da aula. Como resultados, reconhecemos um movimento contínuo e crescente nas ações do professor durante a condução da discussão, destacando as implicações para o desenvolvimento do raciocínio matemático dos estudantes e compreensão do conceito de função racional.

Resumo No intuito de desvelar como se constituem oportunidades de aprendizagem profissional do professor que ensina Matemática, propõe-se neste artigo compreender como interações discursivas entre professores participantes de um processo de formação continuada promovem oportunidades para eles aprenderem sobre o ensino de padrões e regularidades. Os dados desse estudo qualitativo-interpretativo incluem vídeo e áudio gravação dos encontros; arquivos das tarefas; e registros escritos da formação. A organização das tarefas e a dinâmica implementada nas discussões coletivas propiciaram momentos de comunicação dialógica, privilegiando um discurso pedagógico exploratório, atrelado a um discurso matemático exploratório, assim como favoreceram discussões matemáticas e didáticas, com argumentações que articulam os conhecimentos profissionais abordados. Sendo assim, refinar a capacidade de reconhecer formas de discurso e de motivá-las ou responder a elas contribui para o desenvolvimento profissional docente. Matemática propõese propõe regularidades qualitativointerpretativo qualitativo interpretativo encontros dialógica exploratório didáticas abordados motiválas motivá las docente

Abstract In order to comprehend how the professional learning opportunities of the teacher who teaches mathematics are constituted, the article proposed to understand how discursive interactions between teachers participating in a continuous formation process promote opportunities for them to learn about the teaching of patterns and regularities. The data of this qualitative-interpretative study include video and audio recordings of the meetings; archives of the tasks; and formation registers. The organization of tasks and the implemented dynamics in the collective discussions made possible the moments of dialogical communication, privileging the exploratory pedagogical speech, linked to an exploratory mathematical discourse, as well as favored the mathematical and didactic discussions, with arguments that articulate the professional knowledge approached. To refine the capacity to recognize forms of discourse and to motivate them or to answer them contributes to professional teaching development. constituted regularities qualitativeinterpretative qualitative interpretative meetings registers communication speech approached development

Resumo Assumindo a importância do desenvolvimento do raciocínio como competência básica na aprendizagem do Cálculo Diferencial e Integral, objetivamos compreender os processos do raciocínio matemático de estudantes dessa disciplina no trabalho com episódios de resolução de tarefas que envolvem a análise coordenada das mudanças que ocorrem em duas variáveis interdependentes. Adotando uma perspectiva qualitativa de cunho interpretativo, com princípios de uma Investigação Baseada em Design, utilizamos o protocolo escrito e gravações de áudios do trabalho de três grupos de estudantes ao resolverem tarefas envolvendo a criação, interpretação e reflexão a respeito de gráficos. Como resultados, destacamos um processo de construção de conjecturas apoiado em conhecimentos matemáticos que já possuíam, na percepção de relações presentes na tarefa ou, ainda, no senso comum. Além disso, ressaltamos a busca por motivos para validação ou refutação, momento no qual os estudantes resgataram conhecimentos que já possuíam ou construíram novos conhecimentos matemáticos, com a elaboração de novas conjecturas ou aprimoramento de uma já elaborada, novas investigações e tentativas de justificar. Por fim, destacamos o papel desempenhado pela discussão entre os estudantes, bem como o potencial e as limitações das tarefas.

Abstract Acknowledging the importance of reasoning development as a basic competence in learning Differential and Integral Calculus, this work is aimed at understanding the mathematical reasoning processes of students in this discipline, in the process involving the investigation of task resolution that contains the coordinated analysis of changes that happen in two interdependent variables. Adoptin a qualitative perspective of an interpretative nature, with the principles of Design-Based Research, we use the written protocol and audio recordings of the work of three groups of students when solving task involving the creation, interpretation, and reflection about graphics. As a result, we highlight a process construction of conjectures supported by mathematical knowledge that they already had, in the perception of relationships present in the task or, still, in common sense. In addition, we emphasize the search for reasons for validation or refutation, at which time students recovered knowledge they already had or built new mathematical knowledge, with the elaboration of new conjectures or improvement of an already elaborated on, new investigations and attempts to justify. Finally, we highlight the role played by the discussion among students, as well the potential and limitations of the tasks.

Abstract The population structure and growth of two populations (Arraial and Capivari rivers) of Aegla schmitti was evaluated. The following parameters were determined: size, age structure, sex ratio, recruitment period, and sampling method selectivity (traps and handnet). There was variation between males and females. The carapace length size class distribution was unimodal in the Arraial River, and polymodal in the Capivari River. The sampling method influenced the proportion of demographic categories. The recruitment occurred mostly between October and December. The growth curves for the Arraial and Capivari rivers, respectively, were Ct= 26,50[1- e -0,004(t+53,36)] and Ct= 34,79[1- e -0,0024(t+122,34)] for males, and Ct= 23,07[1- e -0.0027(t+23,18)] and Ct= 25.52[1- e -0,0033(t+19,51)] for females. The maximum CL estimated for males and females were 26.50 and 23.07mm for the Arraial River, and 34.79 and 25.52mm for the Capivari River. Longevity was estimated in 2 years for females of both populations, and 2.5 and 3 years for males of the Arraial and Capivari rivers, respectively. The results allow for the characterization of the growth patterns of A. schmitti in two hydrographic basins, providing information for the species’ preservation and also demonstrating the selectivity of the different sampling methods used for aeglids.

Resumo O presente estudo descreve a percepção de estudantes do Ensino Médio acerca da utilização de um instrumento de avaliação diferenciado em aulas de Matemática: uma prova realizada em várias fases. Buscou-se identificar indícios de como viram e julgaram o instrumento (suas percepções), bem como identificar possíveis implicações de sua utilização sobre as estratégias de estudo desses estudantes. Para tal, foram analisados dados coletados por meio de um questionário respondido ao final da última fase da prova, bem como apontamentos por eles realizados em seus diários de aulas. As reflexões propiciadas a partir dessa análise possibilitaram compreender os próprios estudantes como sujeitos do processo de avaliação e permitiram analisar criticamente tanto o instrumento quanto a prática avaliativa do professor.

Abstract This study describes the perception of high school students about the use of a different assessment tool in Mathematics classes: a test carried out in several phases. We tried to identify evidence of how they saw and judged this instrument (their perceptions) and identify possible implications of this use on the student's study strategies. The data were analyzed and collected through a questionnaire answered by the students at the end of the last stage of the test, as well as notes performed by them in their daily lessons. The reflections afforded from this analysis helped us understand the students themselves as subjects of the assessment process and allowed a critical analysis of the instrument as well as of the teacher's assessment practice.

Resumo Este artigo apresenta uma análise, à luz da Taxionomia revisada de Bloom, de itens de provas escritas de Matemática elaboradas por professores do Ensino Fundamental II e do Ensino Médio de escolas públicas. Esses professores participaram de oficinas que discutiram a temática "avaliação e aprendizagem", projeto de extensão que ocorreu no ano de 2013, e foram convidados a analisar provas escritas que eles mesmos elaboraram. Os resultados apontaram uma prevalência de itens classificados nos níveis mais baixos do domínio cognitivo da Taxionomia (lembrar, entender e aplicar) e uma quase ausência de itens nos níveis mais elevados (analisar, avaliar e criar). Tais resultados corroboram a necessidade de estudos e propostas de formação continuada que oportunizem aos professores repensar: (i) os instrumentos avaliativos que utilizaram (em geral, prioritariamente a prova escrita), (ii) a elaboração das tarefas que propõem (em geral, restritas aos modelos presentes em livros didáticos) e (iii) a sua própria prática avaliativa.

Abstract This article presents an analysis based on the Revised Bloom's Taxonomy of items of Mathematics written test questions prepared by teachers of Secondary School and High School public school. These teachers participated in workshops that discussed the theme "assessment and learning", a project in 2013 and they were asked to analyze the written test that they prepared. The results showed the prevalence of items classified in the lower levels of the cognitive domain of the Taxonomy (remember, understand and apply) and a near absence of items in the most developed levels (analyze, evaluate and create). These results support the need for studies and continuing education projects that encourages the teachers rethink: (i) the assessment instruments that they use (in general, mainly the written test); (ii) the development of the tasks proposed (generally restricted to "exercises-type" present in textbooks) and (iii) its own assessment practice.

ABSTRACT This study aimed to estimate the growth and size at the onset of sexual maturity in a population of Aegla georginae . Specimens were sampled from November 2007 to October 2008 in the Perau Creek (Ibicuí-Mirim sub-basin; Uruguay River basin). All captured specimens were sexed, and their cephalothoracic length (CL), minor and larger propodi length (MPL, LPL, respectively), height of major chela (HMQ), and abdomen width (AW) were measured. The lengthwise growth of males and females was evaluated using the Bertalanffy model, and the morphological sexual maturity was evaluated using the REGRANS program. The smallest ovigerous female presented a 10.45 mm CL. Males reached larger sizes than did females, but the growth rate was similar between the sexes. Longevity estimates for males and females were 670 and 736 days, respectively. The best relationship used to infer the size at morphological sexual maturity in males was CLxHMQ (12.78 mm) and in females was CLxAW (10.78 mm). Aegla georginae follows a common pattern for aeglids in terms of lifecycle, where sexual dimorphism is associated with the onset of morphological sexual maturity.

RESUMO Um estudo sobre o dimorfismo sexual no caranguejo Hepatus pudibundus (Herbst, 1785) foi realizado utilizando morfometria geométrica. A carapaça de 28 machos e 30 fêmeas e o própodo dos quelípodos direito e esquerdo de 22 machos e 26 fêmeas foram analisados. Foram estabelecidos 13 marcos anatômicos bidimensionais na carapaça e 10 no própodo para ambos os sexos. Uma Análise Generalizada de Procrustes baseada na configuração dos marcos anatômicos foi utilizada para separar os componentes de tamanho e de forma. Um Teste-t de Student foi utilizada para determinar a significância estatística do dimorfismo sexual da carapaça e do própodo dos quelípodos. A variação na forma das estruturas foi avaliada através de uma Análise de Função Discriminante. Houve dimorfismo sexual na forma e no tamanho das estruturas analisadas. Machos mostraram dimensões menores do que fêmeas. A carapaça das fêmeas foi mais larga na região posterior do que em machos, indicando uma ampliação da área para acomodação da massa de ovos. O dedo fixo do própodo dos quelípodos das fêmeas está mais voltado para a parte posterior e pode facilitar a limpeza dos ovos fixados na câmara abdominal. Nossos resultados trazem novas informações sobre o desenvolvimento de caracteres sexuais secundários e as suas consequências sobre a forma da carapaça e o própodo dos quelípodos em machos e fêmeas de H. pudibundus.

ABSTRACT A study on sexual dimorphism in Hepatus pudibundus (Herbst, 1785) was performed using geometric morphometrics. The carapace of 28 males and 30 females and the left and right cheliped propodus of 22 males and 26 females were analyzed. Thirteen landmarks were established in the carapace and 10 in the cheliped propodus. A Generalized Procrustes Analysis based on landmark configurations was used to separate the components of size and shape. A Student t-test was used to determine the statistically significant sexual dimorphism of the carapace and the cheliped propodus. The variation in the shape of the structures was evaluated with a discriminant analysis. There was sexual dimorphism in the analyzed structures. Males were smaller than females. The carapace of females was larger in the posterior region than in males, indicating an expansion of the area for accommodation of the egg mass. The fixed finger of the cheliped propodus of females is more posteriorly arched that can be used to facilitate the cleaning of eggs laid in the abdominal chamber. Our results provide new information about the development of sexual secondary characters and their effects on the shape of the carapace and the cheliped propodus in males and females of H. pudibundus.

Consciente que minhas (pré) concepções acerca da Matemática, de ensinar Matemática e, principalmente, de avaliar em Matemática provêm da própria história enquanto estudante e professor em formação, organizei o memorial que aqui apresento e que constituiu parte do capítulo inicial de minha tese de doutorado. Partindo de uma experiência com a utilização de uma prova em fases nas aulas de Matemática, numa tentativa de aproximá-la de sua perspectiva formativa, deparei-me com uma série de inquietações que me instigaram a interpretar quem sou, o que penso e por que assim penso. Este texto, que retrata parte do caminho trilhado, é resultado desse processo de reflexão, em que me tornei protagonista da minha própria pesquisa, na qual propus-me a investigar minha atuação em sala de aula e em especial minha prática avaliativa, a partir das relações com os vários contextos nos quais encontro-me inserido. Ao publicá-lo, vislumbro instigar outros professores a se tornarem também investigadores de suas práticas.

Aware that my (pre) conceptions of mathematics, mathematics teaching, and mainly assessment in mathematics come from my own history as a student and as a teacher in training, I organized this memorial which I present here and that constitutes the initial chapter of my doctoral thesis. Starting from an experience with the use of a stage test in Mathematics classes, in an attempt to bring it closer to its formative perspective, I came across a number of concerns that prompted me to interpret who I am, what I believe in, and why I think in a certain way. This text depicts part of the path and is the result of a reflective process, in which I became the protagonist of my own research, proposing to investigate my performance in the classroom and, in particular, my assessment practice, from relations with the various contexts in which I find myself inserted. By publishing it, I glimpse at “instigating” other teachers to become researchers of their own practices too.

OBJETIVO: O objetivo deste trabalho foi avaliar a densidade populacional, tamanho mediano de machos e fêmeas, distribuição etária, proporção sexual, período reprodutivo e de recrutamentoda espécie ameaçada Aegla manuinflata. MÉTODOS: Foram realizadas coletas mensais de maio de 2006 a abril de 2007 no Arroio Taquara, município de São Pedro do Sul e uma coleta adicional por estação do ano para determinação da densidade. Os animais foram coletados com armadilhas e puçá. Todos os exemplares capturados foram sexados e tiveram o comprimento cefalotorácico mensurado com um paquímetro digital com precisão de 0,01 mm e, em seguida, foram devolvidos ao riacho. RESULTADOS: Foram analisados 888 indivíduos sendo 17 jovens não sexados, 572 machos, 274 fêmeas e 25 fêmeas ovígeras. Foi estimada uma densidade populacional média de 1,52 indivíduos por metro quadrado. Os machos apresentaram tamanho mediano superior ao das fêmeas (p<0,01). A distribuição etária se mostrou bimodal tanto para machos quanto para fêmeas. Fêmeas predominaram nas classes de tamanho intermediárias enquanto machos foram mais numerosos nas classes de tamanho superiores, não havendo diferenças nas classes inferiores. A proporção sexual geral seguiu a esperada de 1:1 somente nos meses de maio, outubro e novembro (p>0,05). Machos e fêmeas quando coletados apenas com puça seguem a proporção esperada de 1:1 em todos os meses do ano (p>0,05), o que deixa claro que o método de coleta interfere significativamente nos resultados. Fêmeas ovígeras e juvenis foramregistrados em todas as estações do ano e não há diferença significativa na proporção destes animais entre uma estação e outra (p>0,05), caracterizando uma reprodução contínua. CONCLUSÕES: Os dados populacionais do presente estudo podem ser utilizados como base para a criação de medidas de conservação de A. manuinflata, uma vez que a espécie é considerada vulnerável.

AIM: We evaluate the population density, median size of males and females, age distribution, sex-ratio, reproductive period and recruitment period of the threatened species Aegla manuinflata. METHODS: Monthly samplings were made from May 2006 to April 2007 in the Passo Taquara stream, Brazil. An extra sampling was performed at each season of the year in order to determine population density. Animals were captured with traps and handnets. All the captured specimens were sexed and had their cephalotoraccic length measured with a digital caliper (0.01 mm), and were then released back in the stream. RESULTS: A total of 888 individuals were analyzed, including 17 unsexed juveniles, 572 males, 274 females and 25 ovigerous females. A mean population density of 1.52 ind/m² was estimated. Males showed median sizes larger than females (p < 0.01). Age distribution was bimodal for both males and females. Females predominated in intermediary size-classes, while males were more abundant in the larger classes, with no differences for the lower classes. The sex-ratio, in general, followed the expected 1:1 proportion only in the months of May, October and November (p > 0.05). Males and females caught only with handnet follow the expected 1:1 proportion in all months of the year (p > 0.05), demonstrating that the sampling method clearly affects the results. Ovigerous females and juveniles were registered in all seasons of the year and there is no significant different in the proportion of these animals among the seasons (p > 0.05), indicating continuous reproduction. CONCLUSIONS: Population data of this study can be used as a basis for the creation of conservation measures of A. manuinflata, since the species is considered vulnerable.

This work aims to estimate the average size at the onset of morphological and physiological sexual maturity and the reproductive period of Callinectes danae. Specimens were captured from a shrimp fishing boat equipped with a trawl net from March 2009 to February 2010. After sorting, crabs were sexed, and the following morphometric dimensions were measured: carapace width (CW), length and height of the major (MAP and HMAP), and the minor cheliped propodus (MIP and HMIP) for both sexes, and the abdominal width for females (AW). The onset of morphological maturity was estimated with the REGRANS program, in which the inflection point is calculated. The size at the onset of physiological maturity was determined by logistic regressions fitted to the relationships between the percentages of juvenile and adult individuals of each CW. A total of 893 individuals were sampled. Of these individuals, 389 were males, 472 non-ovigerous females, and 32 ovigerous females. The CW ranged from 19.81 to 117.17 mm for males, 19.91 to 113.11 mm for non-ovigerous females, and 70.55 to 88.60 mm for ovigerous females. Morphological sexual maturity was attained at 86.47 mm CW (males) and 67.87 mm CW (females). The size at the onset of physiological maturity was calculated to be 86.50 mm CW (males) and 67.00 mm CW (females). Due to the closeness of these values, both techniques were deemed suitable for determination of the onset of sexual maturity in C. danae. This species reproduces throughout the year, but reproduction is more intense during the winter. Mature males are present throughout the year. Data from this investigation will help resource managers to develop strategies that will ensure a sustainable harvest of this important species of swimming crab in southern Brazil.

A predação oportunista dos peixes Astyanax fasciatus e Crenicichla sp. emalhados em rede de espera pelos eglídeos Aegla grisella e Aegla singularis foi observada em rios do Sul do Brasil. Estes eventos ocorreram em dois rios na Floresta Nacional de Passo Fundo, município de Mato Castelhano, RS e no município de Tanguará, SC. Em ambos os locais os peixes apresentavam um padrão de mutilação semelhante, com olhos e nadadeiras sendo consumidos primeiramente. No entanto, a predação de todo o corpo também foi registrada, provavelmente em decorrência do elevado número de eglídeos se alimentando de um mesmo indivíduo. Esse registro visa contribuir para o entendimento da dieta natural dos eglídeos em ambientes de águas continentais.

The opportunistic predation on the fishes Astyanax fasciatus and Crenicichla sp. entangled in gill net by the aeglid crabs Aegla grisella and Aegla singularis was observed in rivers of southern Brazil. These events occurred in two rivers located at the National Forest of Passo Fundo, in Mato Castelhano municipality, Rio Grande do Sul State and Tangará municipality, Santa Catarina State. In both places the fish showed similar wound pattern, with eyes and fins being consumed first. However, predation of the whole body was also recorded, probably due to the high number of aeglid crabs feeding on the same individual. This record intended to contribute for understanding the natural diet of these crabs in freshwater environments.

A study on sexual dimorphism in Aegla marginata was conducted using geometric morphometric methods. The carapace of 47 females and 75 males and the left and right cheliped propodus of 29 females and 40 males were analyzed. Eighteen landmarks were established in the carapace and 10 in the cheliped propodus. A Generalized Procrustes Analysis based on landmark configurations was used to separate the components of size and shape. A Student t-test was used to determine whether statistically significant sexual dimorphism was shown by the carapace and the cheliped propodus. The variation in the shape of the structures was evaluated with a discriminant analysis. Our results show that there is no sexual dimorphism in the carapace of A. marginata. However, the size of the propodus differed statistically between the sexes. The carapace shape differed between the sexes: the females showed a wider posterior area and a narrower anterior area than the males. The shape of the cheliped propodus also differed between the sexes: overall, the females had a longer and narrower cheliped propodus than the males. The variations in the carapace shape found in this study confirm the results of other studies on aeglid morphology; however, the information presented by this study regarding variation in the shape of the cheliped propodus is new to the literature. The geometric morphometric approach applied in this study provided useful tools for achieving the proposed objectives.

O presente trabalho teve como objetivo avaliar o crescimento de Aegla manuinflata Bond-Buckup & Santos, 2009 em ambiente natural. Para isso foram realizadas coletas mensais de abril de 2006 a maio de 2007 no arroio Passo Taquara, no município de São Pedro do Sul, Rio Grande do Sul, Brasil. Os organismos foram coletados utilizando armadilhas plásticas e puçá. Os espécimes foram sexados e tiveram mensurado o comprimento do cefalotórax (CC), em seguida foram devolvidos ao córrego. O crescimento foi avaliado utilizando o modelo de von Bertalanffy. As curvas de crescimento do cefalotórax, estimadas para machos e fêmeas, são descritas pelas seguintes equações Ct = 28,00 [1 - e-0,0047(t+10,63)] e Ct = 25,16 [1 - e-0,0051(t+17,65)], respectivamente. Os machos atingem tamanhos maiores que as fêmeas. A longevidade foi estimada em três anos para os machos e dois anos e meio para as fêmeas.

The aim of this work was to evaluate the growth of Aegla manuinflata Bond-Buckup & Santos, 2009 in natural environment. Samplings were monthly performed, from April/2006 to May/2007, in a tributary of Ibicuí-Mirim River, which belongs to the Uruguay River basin, at São Pedro do Sul municipality, state of Rio Grande do Sul, Brazil. Traps and dip nets were used to collect the animals. The specimens were sexed and had their cephalothoracic length (CL) measured, using a digital caliper with 0.01 mm precision. The growth was evaluated using the von Bertalanffy model. The growth curves in length (mm), for the data obtained through the modal progression, are described by the equations: Ct = 28.00 [1 - e-0.0047(t+10.63)] for males and Ct = 25.16 [1 - e-0.0051(t+17.65)] for females. The maximum estimated age for males was three years and for females was two years and six months.

This study investigated the influence of land use and occupation on the density and distribution of aeglids in the hydrographic basin of the Jacutinga river (northern state of Rio Grande do Sul, Brazil). Sampling was carried out during the summer and winter of 2004, at 12 sites distributed in river spring, agriculture, grazing, and urban areas. Animals were collected using a Surber sampler. An analysis of variance (ANOVA) was conducted to evaluate differences in the density of species in each land use and sampling period. Biological data were classified using cluster analysis (Euclidian distance; UPGMA). A total of 226 aeglids were captured, including 167 (78.89%) specimens of Aegla platensis Schmitt, 1942 and 59 (26.11%) specimens of A. singularis Ringuelett, 1948. The aeglid distribution was associated with land use and occupation. In river spring and grazing areas, A. platensis was predominant, whereas in urban and agricultural areas A. singularis was the most prevalent species.