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Este estudo tem como objetivo promover e conduzir a construção dos significados e dos raciocínios matemáticos dos alunos na aprendizagem da matemática, concretamente na aprendizagem dos parâmetros em funções em alunos do 11.º ano de escolaridade. Parte-se de um método de ensino construído para mediar os significados matemáticos dos alunos desde contextos matemáticos concretos até contextos matemáticos genéricos e estruturados. Pretendemos responder às seguintes questões: I) Como é que o método de ensino construído funciona como um mediador da atividade do professor? II) Como é que as tarefas matemáticas elaboradas e implementadas com base no método de ensino construído funcionam como artefactos de mediação semiótica de significados matemáticos dos alunos?
O método elaborado integra três níveis de ensino e quatro graus de significados. Os três níveis foram construídos sob um crescendo de independência e estrutura algébrica em concordância com os fundamentos teóricos do estudo. No 1.º nível o aluno atribui significado e representa em múltiplos registos de representação, num contexto aritmético em que as variáveis dependente e independente e o(s) parâmetro(s) são concretizados. No 2.º nível o aluno trabalha com significados e representações com que já trabalhou no 1.ºnível e constrói novos significados e novas representações em múltiplos registos de representação, num contexto algébrico em que as variáveis dependente e independente não são concretizadas e o(s) parâmetro(s) é(são) concretizado(s). No 3.º nível o aluno trabalha com significados e representações com que já trabalhou nos 1.º e 2.º níveis e constrói novos significados e novas representações em múltiplos registos de representação, num contexto algébrico em que as variáveis dependente e independente e o(s) parâmetro(s) não são concretizados. As tarefas da proposta pedagógica deste estudo foram construídas de acordo com a estrutura do método.
Para a construção dos graus de significados do método foi considerada a atividade sígnica dos alunos sob as etapas de primeiridade, secundidade e terceiridade definidas por Peirce (1978) (associadas aos signos ícone, índice e símbolo e ao raciocínio abdutivo, indutivo e dedutivo) e a mediação semiótica do professor, associada ao conceito de ciclo didático, definido por Bussi e Mariotti (2008). Em cada um dos três níveis (N1, N2, N3), o aluno constrói significados de grau um, dois e três (S1, S2, S3). Pode construir ainda significados de grau zero (S0), em resultado da construção de significados que não se enquadram no sistema de evidências e verdades definidas na própria matemática (erros, imprecisões/incorreções).
A metodologia de investigação usada é qualitativa. A investigadora é a professora do estudo. O método de ensino foi implementado em sala de aula com alunos do 11.ºano, no ano letivo 2010/2011. Os dados foram recolhidos aquando do estudo do tema Funções. Os alunos trabalharam em grupos. A recolha dos dados foi feita através de gravações áudio, de transcrições e de registos da investigadora. As gravações áudio e as transcrições foram usadas nas entrevistas realizadas. As entrevistas foram gravadas e posteriormente transcritas. Para dar resposta às questões de investigação, foram identificadas unidades de análise resultantes das intervenções escritas e orais que os alunos e a professora investigadora realizaram, durante as aulas em que o método foi implementado. Foram selecionadas sete unidades de análise: ‘A vedação do jardim’, ‘O passeio das amigas’, ‘Funções: composta e inversa’, ‘A caixa de volume máximo’, ‘O triângulo inscrito numa circunferência’, ‘Parâmetros em funções racionais’ e ‘Parâmetros, funções e geometria’. Sobre estas unidades de análise fizeram-se três tipos de análise: a de primeira ordem, a de segunda ordem e a de terceira ordem. A de primeira ordem é uma compilação do que se observou e registou. Para a análise de segunda ordem selecionaram-se os dados mais representativos de cada unidade de análise que foram analisados segundo as dimensões, categorias e subcategorias. A análise de terceira ordem baseia-se numa leitura transversal dos sete grandes blocos obtidos na análise de segunda ordem, permitindo colocar questões e relacioná-las, de onde resultaram as respostas às questões do estudo.
Como conclusões do estudo, salienta-se que a professora teve no método construído um referente diretamente associado àquilo que ela, em cada momento, quis ver trabalhado pelos alunos, conduzindo-os de forma sequencial e encadeada até que a sua intenção fosse alcançada. Neste método, essa intenção definiu-se por cada um dos três graus de significado em cada nível e tinha como meta atingir o significado de grau 3 no nível 3, que correspondeu ao que se pretendia na última questão das tarefas. Renovando-se o ciclo em cada novo episódio de mediação. O método resultou nas tarefas construídas sob três níveis, que conduziram os alunos desde contextos algébricos concretos até contextos algébricos genéricos. Em cada episódio de mediação, a professora identificou o grau de significado em que o aluno se encontrava e o significado de grau 3 que se intencionava que o aluno significasse. A mediação decorreu de três formas distintas, tendo havido, no mesmo nível ou entre os vários níveis: I) Construção de significados entre os alunos; II) Construção de significados de grau 3, a partir de significados de grau 0, de grau 1 e de grau 2; III) Construção de um novo significado de grau 3 que sintetizasse os significados de grau 3 já construídos. Foi a construção de um novo significado de grau 3 que sintetizou os significados de grau 3 já construídos que promoveu a criação de significados de grau 1 no episódio de mediação seguinte. A ocorrência de S0 gerou mediações da professora que conduziram os significados dos alunos a recapitulações e sínteses que promoveram a aprendizagem do conceito de parâmetro em funções.
Quanto às tarefas matemáticas apresentadas aos alunos, elas foram estruturadas com uma questão inicial onde o enunciado não tinha valores concretos e apelava à concretização dos alunos. As questões que constituíam cada tarefa obedeceram a uma determinada ordem: por três níveis algébricos, desde contextos concretos até contextos genéricos, o que promoveu nos alunos uma certa forma de raciocinar (desde o concreto até ao genérico). O que ocorreu na implementação do método de ensino foi que, no 1.º nível foram os alunos quem definiu o(s) valor(es) concreto(s) a experimentar. No 1.º nível, o reforço do estímulo à experimentação de diferentes valores concretos foi feito pelo professor. Contudo, após alguns raciocínios construtivos efetuados, os alunos iniciaram a generalização e construíram-na nos níveis seguintes (no 2.º e no 3.º nível). No 3.º nível ocorreram significados que estruturaram e sintetizaram os anteriores. Isto deu origem a significados que, na fase final das tarefas, facilitaram e tornaram naturais as sínteses que estruturaram os principais significados construídos no fim da última questão da tarefa.
As tarefas, com esta estruturação, complementadas com a mediação da professora, permitiram uma aprendizagem do conceito de parâmetro. Nos 1.º e 2.º níveis, os alunos fixaram o seu valor e, em consequência dessa concretização, compreenderam que a concretização do parâmetro condiciona os valores das variáveis. Essa compreensão estendeu-se a contextos em que o parâmetro é genérico, ou seja ao 3.º nível. Este raciocínio dedutivo mostrou a compreensão do conceito de parâmetro como algo que pode determinar o intervalo de variação de alguma variável. A interpretação que foi dada a uma letra (𝑥,𝑦,𝑝,…) no 1.º nível do método foi semelhante à que usualmente os alunos fazem para uma incógnita quando resolvem uma equação. Desta interpretação foi possível inferir que o significado de parâmetro que os alunos experimentaram foi: parâmetro é algo que pode ficar definido e que permite resolver os problemas matemáticos quando se lhe atribui um valor concreto. Esta sequência de significados promoveu a compreensão da noção de letra enquanto incógnita, enquanto variável de uma função e enquanto parâmetro, na transformação e conversão de representações em registos gráficos, algébricos, esquemáticos e em linguagem natural. No final do estudo, os alunos evidenciaram uma compreensão do conceito de parâmetro em funções e dos conceitos de variável, de constante e de incógnita. O estudo termina com algumas sugestões para trabalhos futuros.
This study aims to promote and lead the construction of meanings and mathematical reasoning of the students in learning mathematics, more specifically in learning of parameters in functions of 11th grade students. Starting from a teaching method constructed to mediate the mathematical meanings of students - from concrete mathematical contexts to generic and structured mathematical contexts. We intend to answer to the following questions: I) How does the constructed teaching method works as a mediator of teacher activity? II) How do the designed and implemented mathematical tasks based on the constructed teaching method work as artifacts of semiotic mediation of students’ mathematical meanings? The elaborated method includes three levels of teaching and four degrees of meanings. The three levels were constructed under a growing independence and algebraic structure in accordance with the theoretical foundations of the study. In the 1st level the student assigns meaning and represents in multiple registers of representation, in an arithmetic context in which the dependent and independent variables and the parameter/s is/are achieved. In the 2nd level the student works with the meanings and the representations with which s/he has worked in the 1st level and constructs new meanings and new representations in multiple registers of representation, in an algebraic context in which the dependent and independent variables are not achieved and the parameter/s is/are achieved. In the 3rd level the student works with meanings and representations with which s/he has already worked in the 1st and 2nd levels and constructs new meanings and new representations in multiple registers of representation, in an algebraic context in which the dependent and independent variables and the parameter/s are not achieved. The tasks of the pedagogical proposal of this study were constructed according to the structure of the method. In order to elaborate the construction of the degrees of meanings of the method it was considered the signic activity of students in steps of firstness, secondness and thirdness defined by Peirce (1978) (associated with the signs icon, index and symbol and abductive reasoning, inductive and deductive) and the semiotic teacher mediation, associated with the concept of educational cycle, defined by Bussi and Mariotti (2008). In each of the three levels (N1, N2, N3), the student constructs meanings of degree one, two and three (S1, S2, S3). S/he can still constructs meanings of degree zero (S0), as a result of the construction of meanings that does not fit in the system of evidences and truths defined in mathematics itself (errors, imprecisions/inaccuracies). The used research methodology is qualitative. The researcher is the teacher of the study. The teaching method was implemented in the classroom with 11th grade students, in academic year 2010/2011. The data were collected when the subject Functions was approached. The students worked in groups. Data was collected through audio recordings, transcripts and records gathered by the researcher. The audio recordings and the transcripts were used in the interviews. The interviews were recorded and transcribed later. To answer to the research questions, analysis units were identified as a result of the written and oral interventions that students and the researcher performed during the classes where the method was implemented. Seven units of analysis were selected: ‘The garden’s fence’, ‘The friends’ tour’, ‘Functions: composed and inverse’, ‘The box of maximum volume’, ‘The triangle inscribed in a circumference’, ‘Parameters in rational functions’ and ‘Parameters, functions and geometry’. These units of analysis were subjected to three types of analysis: of first, second and third orders. The analysis of first order is a compilation of what has been observed and recorded. For the analysis of second order it was selected the most representative data of each unit of analysis which were analyzed according to the dimensions, categories and subcategories created. The analysis of the third order is based on a cross-reading of the seven large blocks obtained in the second order analysis, allowing to make questions and relate them. From this, it was obtained the answers to the questions of the study. As conclusions of the study, it is noted that the teacher had on constructed method a referent directly associated to what she, at any time, wanted to see worked by students, leading them sequentially to this intention. In this method, this intention is defined by each of the three degrees of meaning in each level and had as goal the meaning of degree 3 at level 3, which corresponded to what was intended in the latest question of the tasks. Renewing the cycle at each new episode of mediation. The method resulted in the tasks constructed on the three levels that led the students from specific to generic algebraic contexts. In each episode of mediation, the teacher identified the degree of meaning in which the student was and the meaning of the degree 3 that is expected to be attributed by the student. Mediation occurred in three different ways, both in the same level and among the various levels: I) Construction of meanings among students; II) Construction of meanings of degree 3, starting from meanings of degree 0, degree 1 and degree 2; III) Construction of a new meaning of degree 3 that synthesized the meanings of degree 3 already constructed. It was the construction of a new meaning of degree 3 that synthesizes the meanings of degree 3 already constructed that promoted the creation of meanings of degree 1 in the next episode of mediation. The occurrence of S0 generated mediations of the teacher who led the students to recaps and synthesizes the meanings which promoted the learning of the concept of parameter in functions. Regarding the mathematical tasks presented to the students, they were produced in a way in which the initial question of the enunciation didn’t contain concrete values, prompting for student’s concretization. The questions composing each task obeyed to a specific order: with three algebraic levels, from concrete to generic contexts, which promoted in the student’s a specific reasoning (concrete to the generic). During the implementation of the teaching method, for the 1st level, the students were the ones who defined the concrete value/s to use. At this level, the teacher reinforced the stimulus to use different concrete values. However, after some constructive reasoning being made, the students begun the generalization and continued towards the next levels (2nd and 3rd levels). At 3rd level, some meanings structured and synthesized the previous ones. This allowed the arise of meanings that, at the final stage of the tasks, favored and made natural the synthesis supporting the main meanings constructed at the end of the last question of the task. The tasks, with the presented structure, complemented with the teacher mediation, allowed for an apprenticeship of the concept of parameter. At 1st and 2nd levels, the students have set their value, and consequently, they understood that the achievement of the parameter determines the values of variables. This understanding was extended to contexts in which the parameter is generic, i.e. the 3rd level. This deductive reasoning showed the understanding of the concept of parameter as something that can determine the variation interval of some variable. The interpretation that was given to a letter (𝑥,𝑦,𝑝,…) in the 1st level of the method was similar to the one that students usually use in an incognita when they solve an equation. From this interpretation it was possible to infer that the meaning of parameter that students experienced was: the parameter is something that can be defined and allows to solve mathematical problems when a specific value is assigned to it. This sequence of meanings promoted the understanding of the notion of letter while it is maintained as incognita, i.e. while variable of a function and parameter, in processing and conversion of graphic representations registers, algebraic, schematics and in natural language. At the end of the study, students showed an understanding of the concept of parameter in functions and of the concepts of variable, constant and incognita. The study concludes with some suggestions for future work.
This study aims to promote and lead the construction of meanings and mathematical reasoning of the students in learning mathematics, more specifically in learning of parameters in functions of 11th grade students. Starting from a teaching method constructed to mediate the mathematical meanings of students - from concrete mathematical contexts to generic and structured mathematical contexts. We intend to answer to the following questions: I) How does the constructed teaching method works as a mediator of teacher activity? II) How do the designed and implemented mathematical tasks based on the constructed teaching method work as artifacts of semiotic mediation of students’ mathematical meanings? The elaborated method includes three levels of teaching and four degrees of meanings. The three levels were constructed under a growing independence and algebraic structure in accordance with the theoretical foundations of the study. In the 1st level the student assigns meaning and represents in multiple registers of representation, in an arithmetic context in which the dependent and independent variables and the parameter/s is/are achieved. In the 2nd level the student works with the meanings and the representations with which s/he has worked in the 1st level and constructs new meanings and new representations in multiple registers of representation, in an algebraic context in which the dependent and independent variables are not achieved and the parameter/s is/are achieved. In the 3rd level the student works with meanings and representations with which s/he has already worked in the 1st and 2nd levels and constructs new meanings and new representations in multiple registers of representation, in an algebraic context in which the dependent and independent variables and the parameter/s are not achieved. The tasks of the pedagogical proposal of this study were constructed according to the structure of the method. In order to elaborate the construction of the degrees of meanings of the method it was considered the signic activity of students in steps of firstness, secondness and thirdness defined by Peirce (1978) (associated with the signs icon, index and symbol and abductive reasoning, inductive and deductive) and the semiotic teacher mediation, associated with the concept of educational cycle, defined by Bussi and Mariotti (2008). In each of the three levels (N1, N2, N3), the student constructs meanings of degree one, two and three (S1, S2, S3). S/he can still constructs meanings of degree zero (S0), as a result of the construction of meanings that does not fit in the system of evidences and truths defined in mathematics itself (errors, imprecisions/inaccuracies). The used research methodology is qualitative. The researcher is the teacher of the study. The teaching method was implemented in the classroom with 11th grade students, in academic year 2010/2011. The data were collected when the subject Functions was approached. The students worked in groups. Data was collected through audio recordings, transcripts and records gathered by the researcher. The audio recordings and the transcripts were used in the interviews. The interviews were recorded and transcribed later. To answer to the research questions, analysis units were identified as a result of the written and oral interventions that students and the researcher performed during the classes where the method was implemented. Seven units of analysis were selected: ‘The garden’s fence’, ‘The friends’ tour’, ‘Functions: composed and inverse’, ‘The box of maximum volume’, ‘The triangle inscribed in a circumference’, ‘Parameters in rational functions’ and ‘Parameters, functions and geometry’. These units of analysis were subjected to three types of analysis: of first, second and third orders. The analysis of first order is a compilation of what has been observed and recorded. For the analysis of second order it was selected the most representative data of each unit of analysis which were analyzed according to the dimensions, categories and subcategories created. The analysis of the third order is based on a cross-reading of the seven large blocks obtained in the second order analysis, allowing to make questions and relate them. From this, it was obtained the answers to the questions of the study. As conclusions of the study, it is noted that the teacher had on constructed method a referent directly associated to what she, at any time, wanted to see worked by students, leading them sequentially to this intention. In this method, this intention is defined by each of the three degrees of meaning in each level and had as goal the meaning of degree 3 at level 3, which corresponded to what was intended in the latest question of the tasks. Renewing the cycle at each new episode of mediation. The method resulted in the tasks constructed on the three levels that led the students from specific to generic algebraic contexts. In each episode of mediation, the teacher identified the degree of meaning in which the student was and the meaning of the degree 3 that is expected to be attributed by the student. Mediation occurred in three different ways, both in the same level and among the various levels: I) Construction of meanings among students; II) Construction of meanings of degree 3, starting from meanings of degree 0, degree 1 and degree 2; III) Construction of a new meaning of degree 3 that synthesized the meanings of degree 3 already constructed. It was the construction of a new meaning of degree 3 that synthesizes the meanings of degree 3 already constructed that promoted the creation of meanings of degree 1 in the next episode of mediation. The occurrence of S0 generated mediations of the teacher who led the students to recaps and synthesizes the meanings which promoted the learning of the concept of parameter in functions. Regarding the mathematical tasks presented to the students, they were produced in a way in which the initial question of the enunciation didn’t contain concrete values, prompting for student’s concretization. The questions composing each task obeyed to a specific order: with three algebraic levels, from concrete to generic contexts, which promoted in the student’s a specific reasoning (concrete to the generic). During the implementation of the teaching method, for the 1st level, the students were the ones who defined the concrete value/s to use. At this level, the teacher reinforced the stimulus to use different concrete values. However, after some constructive reasoning being made, the students begun the generalization and continued towards the next levels (2nd and 3rd levels). At 3rd level, some meanings structured and synthesized the previous ones. This allowed the arise of meanings that, at the final stage of the tasks, favored and made natural the synthesis supporting the main meanings constructed at the end of the last question of the task. The tasks, with the presented structure, complemented with the teacher mediation, allowed for an apprenticeship of the concept of parameter. At 1st and 2nd levels, the students have set their value, and consequently, they understood that the achievement of the parameter determines the values of variables. This understanding was extended to contexts in which the parameter is generic, i.e. the 3rd level. This deductive reasoning showed the understanding of the concept of parameter as something that can determine the variation interval of some variable. The interpretation that was given to a letter (𝑥,𝑦,𝑝,…) in the 1st level of the method was similar to the one that students usually use in an incognita when they solve an equation. From this interpretation it was possible to infer that the meaning of parameter that students experienced was: the parameter is something that can be defined and allows to solve mathematical problems when a specific value is assigned to it. This sequence of meanings promoted the understanding of the notion of letter while it is maintained as incognita, i.e. while variable of a function and parameter, in processing and conversion of graphic representations registers, algebraic, schematics and in natural language. At the end of the study, students showed an understanding of the concept of parameter in functions and of the concepts of variable, constant and incognita. The study concludes with some suggestions for future work.
Description
Keywords
Mediação pelo método de ensino Mediação pelas tarefas Significados matemáticos construídos pelos alunos Parâmetro em funções