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Didactical design in mathematics education
Abstract
Paper available on Sense Publishers' website.
... A abordagem da Engenharia Didática proposta por Artigue (1996) é particularmente aplicável ao ensino de computação. Esta metodologia envolve a concepção, realização e análise de "situações de ensino" que são adaptadas aos conceitos complexos presentes nesses cursos. ...
... Esta pesquisa adotou uma abordagem qualitativa com base na Engenharia Didática proposta por Artigue (1996). O estudo foi conduzido em um curso introdutório de Ciência da ...
... Essas estratégias metacognitivas, caracterizadas por elementos reflexivos e regulatórios, têm um impacto direto e positivo na qualidade do processo de aprendizagem, corroborando as teorias de Artigue (1996) sobre a Engenharia Didática e sua aplicabilidade em contextos educacionais. ...
Este estudo investiga o impacto dos mapas conceituais digitais no desenvolvimento de estratégias metacognitivas em estudantes de Ciência da Computação e Engenharia da Computação. O objetivo geral é explorar a eficácia dos mapas conceituais digitais como ferramentas para melhorar a metacognição dos estudantes. Especificamente, o estudo visa, a) Analisar como os mapas conceituais digitais facilitam a compreensão e a integração do conhecimento em funções matemáticas, b) Avaliar a capacidade dos estudantes de mobilizar estratégias metacognitivas durante a criação e utilização dos mapas conceituais, c) Investigar o impacto das estratégias metacognitivas na qualidade do processo de aprendizagem e na autoregulação dos estudantes. Adotando uma metodologia qualitativa fundamentada na Engenharia Didática de Artigue, o estudo foi conduzido através de três fases principais: planejamento de atividades didáticas baseadas na Teoria das Situações Didáticas de Brousseau, implementação destas atividades em um ambiente de aprendizagem assistido por computador, e análise qualitativa dos dados coletados. Os resultados indicam que o uso de mapas conceituais digitais, em sinergia com abordagens teóricas sólidas, promove significativamente o desenvolvimento de estratégias metacognitivas, melhorando a compreensão conceitual e a autoregulação dos estudantes. Este estudo contribui para a integração de tecnologias digitais no ensino de disciplinas complexas, enriquecendo a experiência educacional com habilidades críticas de pensamento e autorreflexão
... As a methodological framework, we refer to Didactic Engineering (Artigue, 1988(Artigue, , 2008, which has an experimental scheme based on the conception, implementation, observation, and analysis of a didactic sequence. It consists of four phases. ...
... -To make primary school students formulate and validate effective programming strategies to construct symmetrical images with respect to an axis (RQ1) -To make primary school students able to identify key properties of axial symmetry (RQ2). For the design of the didactic sequence and milieu, we used the methodological framework of Didactic Engineering (Artigue, 1988(Artigue, , 2008. We carried out a preliminary analysis of the teaching content, that is, axial symmetry, and its usual teaching in primary school. ...
This paper outlines the design and application of a didactic sequence aimed at facilitating primary students’ understanding of axial symmetry, utilizing a combination of digital artefacts and paper tasks. We wondered to what extent the designed didactic sequence is able to make primary school students formulate and validate effective programming strategies to construct symmetrical images with respect to an axis and identify the key properties of axial symmetry. Data analysis from a study carried out with fifth-grade students shows a link between the evolution of students’ programming strategies and the construction of mathematical knowledge related to the definition of axial symmetry. The digital artefact and the paper tasks were effective in bringing out programming strategies and some of the key properties of axial symmetry. However, the designed didactic sequence was not enough to allow students to identify all properties related to axial symmetry, and a subsequent intervention by the teacher was necessary. The results of the experimentation led us to expand the paper tasks with additional questions for students.
... Brousseau (2006),Artigue (2009),Margolinas et al. (2011), Perrin Glorian (1994, Acosta, Blanco y Gómez (2010),Bessot (2009),Allard (2015) y otros. 4 Un instituteur qui aimait les mathématiques, qui connaissait l'épistémologie génétique et qui a été amené à utiliser puis à transformer ses dispositifs d'observation, pour enfin, sous bien d'autres influences, les assimiler dans une "théorie des situations didactiques" 5 El COREM, cuya finalidad había sido definida por Guy Brousseau a finales de los años 1960 y que pudo ser realizado con el apoyo de los poderes públicos a partir de 1972, va a permitirle realizar este estudio. ...
... La metodologíaLa TSD fue propuesta por Brousseau principalmente durante las décadas de 1970 y 1980, con una metodología de Ingeniería Didáctica. La Ingeniería Didáctica tiene un doble propósito, resolver problemas de enseñanza e investigar sobre los procesos de enseñanza(ARTIGUE, 2009).La Ingeniería Didáctica estuvo relacionada, en su origen, fundamentalmente con realizaciones didácticas en las clases, que toman la forma de secuencias de lecciones, pero esas realizaciones quieren ser la puesta a prueba de un trabajo teórico. Según Bessot(2009) laIngeniería Didáctica consiste en el diseño y realización de currículos, además, del estudio de las diversas posibilidades entre las que se hace la elección del diseño y realización de esos currículos.La Ingeniería Didáctica como metodología, segúnArtigue (1995) se caracteriza, en primer lugar, por ser un esquema experimental basado en las realizaciones didácticas en clase, es decir, sobre el diseño, realización, observación y análisis de secuencias de enseñanza. ...
Resumen La Teoría de las Situaciones Didácticas, desarrollada por Brousseau en la década del 1970, fue una propuesta innovadora especialmente por su convicción de que el campo naciente, la Didáctica de las Matemáticas, debía apoyarse en metodologías que otorgaran un papel esencial al diseño de situaciones capaces de hacer emerger el saber matemático, a partir de las interacciones de los estudiantes con un medio diseñado y controlado. Esta postura cambiaba, radicalmente, la concepción del aprendizaje en su propuesta: el aprendizaje directo y por imitación no tenía cabida en la teoría. Esta teoría ha impulsado diferentes trabajos teóricos y experimentales en los que se reflexiona sobre sus posturas teóricas. También, ha propiciado discusiones y controversias debido a que otras teorías difieren de sus concepciones sobre el diseño, el control de las variables o la concepción del sujeto en el sistema didáctico. El objetivo de este artículo es, luego de hacer un estudio del arte sobre la teoría, defender la tesis de que la Teoría de Situaciones Didácticas es un paradigma que no solo se mantiene vigente en el campo de la Didáctica de las Matemáticas sino, también, que evoluciona gracias a los aportes de diferentes investigadores del campo. Formulo que es necesario compartir con la comunidad de educadores matemáticos interesados o no en la teoría, las interpretaciones y nuevos desarrollos de los pensadores que siguen desarrollando su trabajo bajo estos principios teóricos.
... Estas habilidades permiten a los estudiantes reflexionar sobre sus propios procesos de pensamiento y ajustar sus estrategias según sea necesario. Por ejemplo, al resolver una ecuación cuadrática, los estudiantes deben decidir cuál de las diversas técnicas disponibles es la más adecuada para el problema en cuestión (Artigue, 2009). ...
Este artículo analiza la pertinencia del contenido de álgebra en el marco de prácticas situadas en el contexto social, destacando su relación con el desarrollo de habilidades y la generación de reglas y principios algebraicos. El objetivo es valorar cómo estas prácticas contribuyen al aprendizaje significativo, conectando los contenidos algebraicos con situaciones concretas y relevantes para los estudiantes. Se emplearon tres enfoques metodológicos complementarios: el método holístico para integrar perspectivas individuales y sociales; el método inductivo para construir reglas algebraicas a partir de experiencias específicas; y el método dialéctico para explorar las interacciones dinámicas entre los estudiantes, los contenidos y su contexto sociocultural. Los hallazgos indican que las prácticas situadas facilitan la comprensión de conceptos algebraicos mediante la resolución de problemas contextualizados y la interacción social. Los estudiantes logran identificar patrones y formular reglas algebraicas, fortaleciendo habilidades como el razonamiento abstracto, la reflexión metacognitiva y la comunicación matemática. Además, el uso de herramientas tecnológicas, como GeoGebra, potencia estas prácticas al ofrecer representaciones dinámicas que apoyan la construcción de conocimientos. El artículo concluye que las prácticas situadas en el contexto social son un enfoque pedagógico efectivo para enseñar álgebra, ya que promueven el aprendizaje significativo y la transferencia de conocimientos a contextos diversos. Estas prácticas implican la integración del contenido algebraico con experiencias relevantes, desarrollando habilidades que trascienden el ámbito académico. Las implicancias sugieren la necesidad de diseñar estrategias educativas que vinculen el aprendizaje algebraico con el entorno social y cultural de los estudiantes.
... Para Artigue (2009) es muy importante conocer cómo se diseña y se enseña la matemática, por lo que aborda el diseño curricular de esta asignatura en la educación superior, así como su impacto en la formación de profesionales; en cambio Kilpatrick, (2009) analiza la relación entre la matemática y la sociedad, y la necesidad de desarrollar habilidades matemáticas en la formación de profesionales. Mena (2013), aborda en su estudio, el impacto que ha tenido la enseñanza de la matemática en Colombia, en especial, en la formación de profesionales en diversas disciplinas. ...
The teaching of Mathematics in Higher Education is a topic addressed by numerous researchers, since the usefulness of its contents for the future professional performance of students is widely known; however, the difficulties that different careers face in this regard are highlighted, especially those in the area of technical sciences and administration, due, among other factors, to the diversity of knowledge, skills and motivations that students who begin their studies have. The
use of information and communication technologies (ICT) has been one of the most relevant trends in the teaching of Mathematics, however, difficulties for students in learning its contents still persist; in recent years, Artificial Intelligence (AI) has revolutionized this area, and in the case of the training of professionals, it offers new opportunities to personalize learning, improve the comprehensibility of concepts and encourage greater motivation in students. This article is part of this theme, analyzing the current applications of AI in the teaching of Mathematics, its
importance and contextualization in the training of future Business Administration professionals at the University of Guayaquil, Ecuador.
... They highlight that Polya's method remains a valuable tool for encouraging problem-solving skills in students, demonstrating its continuing effectiveness in modern instruction. Further supporting Polya's approach, Artigue [12] investigates the role of didactical design in mathematics education, noting that Polya's problem-solving strategies are essential to developing effective teaching practices. Artigue suggests that integrating these methods into the curriculum helps students become more skilled at undertaking unfamiliar problems, thereby developing their overall problemsolving competency. ...
The perpetual challenge of uncertain problem-solving skills among local education administrators is underscored by data from a public science and technology high school despite being a performing public science high school, students have been still struggling to enhance their critical thinking skills, posing an unceasing issue. The purpose of this study is to determine the impact of Polya's four-step method on the Grade 9 students' performance in mathematics. Utilizing a one-group pretest-posttest design, the researcher aimed to determine which aspects of Polya's method improved students' mathematics performance. The data analysis indicated a significant improvement in students' problem-solving performance using Polya’s four-step method from pre-test to post-test. This study reaffirms that Polya’s method helps students develop a structured way of thinking and solving problems, which is fundamental for their mathematical development.
... The construction of the DDMT in this manner allowed me to map the subdiscourses which were mentioned in a discussion, which connections were authored during the discussion and who authored them. This type of analysis aligns with didactical engineering methods (Artigue, 1994) which use a priori analysis and a posteriori analysis to identify crucial phenomena and then productively implement theoretical approaches regarding this phenomena (Artigue, 2009). Artigue posits that didactical engineering methods can establish effective connections between researchers and teachers to scale up developments and disseminate pedagogical suggestions. ...
Learner-centered teaching supports student engagement in meaningful mathematics, student collaboration for sense making and equitable instructional practices. Studies have described implementations of learner-centered teaching methods in tertiary settings. However, these were more focused on students outcomes, rather than on the processes of learning. Moreover, some researchers claimed there are drawbacks to inquiry-based and discussion-based teaching. These include learning without an expert, distracting social interactions and ineffectual communication in groups. Thus, this study had two goals. The first was to adapt instructional practices, shown to promote discourse-rich explorative participation, to a university linear algebra setting. The second goal was to examine learner-centered tertiary teaching and the processes involved to better understand what supports and what hinders student learning in this setting.
I pursued these goals through the use of the sociocultural commognitive framework, which has operational tools for describing and analyzing mathematical learning processes. The framework’s holistic treatment of content, as well as social and teaching-learning interactions, allowed the examination of whole-class discussions, and learner-learner interactions in workshops. The workshops were designed as extra-curricular enrichment for science and engineering students. Data analysis included, first, examination of the potential of tasks designed for these workshops to support explorative participation. Next, the extent that opportunities for explorative participation were taken up in the whole classroom discussion was studied. Finally, the learning processes in small groups of students were examined.
Based on a previously developed tool named the Realization Tree Assessment, I developed a tool called the Discourse Mapping Tree (DMT), for mapping the potential of the tasks through analyzing the subdiscourses that this task would invite. An extension of this tool, the Discussion Discourse Mapping Tree (DDMT), was used to map the actual implementation of the task in the whole classroom discussions. The learning processes in small groups were examined through an analysis of the mathematizing using the DDMT tool and commognitive tools. The communication channels in students’ peer-learning episodes were also examined.
The DMT and DDMT offered the opportunity to distinguish between object-level learning, where students develop new narratives about familiar objects, and meta-level learning, where students make new connections between realizations of objects treated as different. The analysis revealed that the designed tasks afforded opportunities for both object-level learning and meta-level learning and, in most cases, the whole-class discussions included numerous opportunities for meta-level learning. However, the small group discussions did not support meta-level learning. Object-level learning in peer discussions was supported only when the students’ communication patterns supported learning and the student’s objectification process was sufficiently advanced.
This study has practical, methodological and empirical implications. Practically, the tasks can be used by other instructors and utilizing the insights from this project, new tasks can be developed. Methodologically, the operational method of examining tasks and mathematical discussions can be used for other topics and other levels. Finally, this study showed that lesson-design in inquiry-based teaching should be attuned to the difference between object-level learning and meta-level learning, and the teaching methods chosen should be suited to the type of learning required.
... Desain pembelajaran atau bahan ajar yang disusun berdasarkan pada karakteristik hambatan belajar yang peserta didik alami kemudian disebut sebagai desain didaktis (Artigue, 2008). Adakah desain penelitian yang membahas mengenai desain didaktis sebagai fokus kajian? ...
... To find answers to the research question, an instructional sequence with which it could be possible to investigate how the suggested process could be fostered needed to be created. Consequently, a prominent role was covered by the process of task design (Watson & Ohtani, 2015), seen not only as limited to the selection and development of problems (Komatsu & Tsujiyama, 2013), but as a complex and multifaced process that involves several steps, such as the explication of a learning goal, the formulation of some hypotheses on students' learning, the design of specific learning activities that translate the previous hypotheses in concrete classrooms materials, in which the role of the task is central. Accordingly, hypotheses on students' learning, instructional activities and new teaching materials must be developed, making the design process an integrated part of the research. ...
This paper reports on a teaching experiment on mathematical modelling in an upper secondary school class that investigates how the constructs of model eliciting and emergent modelling may be brought together to inform teaching and learning. A prominent role is covered by the process of task design, seen as a complex process that involves several steps: the explication of a learning goal, the formulation of hypotheses on students’ learning, the design of specific learning activities. The study provides a case study of how model eliciting activities that start from rich context problems could play a central role to support emergent modelling. This result can be attributed to a combination of several factors: the choice of a rich context problem that stimulated students to elaborate formal mathematical concepts mathematising their informal solving strategies, and the use of a suitable artefact, that presented mathematics as a means of interpreting and understanding reality.
... Esta investigación es de corte cualitativo y siguió una metodología de estudio de casos (Thomas, 2015). La concepción del diseño didáctico y su análisis se basa en las cuatro fases de la ingeniería didáctica Artigue (2008), considerando la forma en que han sido adaptadas en la TAD (Barquero y Bosch, 2015) y que en esta investigación son las siguientes: 1) construcción del MER de las sucesiones reales infinitas, 2) conformación de la praxeología local "sucesiones reales infinitas" y análisis a priori, 3) implementación y análisis in vivo y 4) análisis a posteriori. ...
Se presenta un modelo teórico para el estudio del talento matemático, fundamentado en la Teoría Antropológica de lo Didáctico y la noción de creatividad. En dicho modelo se proponen dos componentes de la actividad matemática creativa: la Componente Matemática, que sustenta las técnicas matemáticas; y la Componente Creativa, definida por cuatro funciones: producir técnicas nuevas, optimizar técnicas, considerar tareas desde diversos ángulos y adaptar una técnica. Con base en los modelos Teórico y Epistemológico de Referencia sobre sucesiones infinitas, se genera un diseño didáctico conformado por seis situaciones problemáticas y se implementa en una institución creada para potenciar el talento matemático. El análisis de dos tareas realizadas por una pareja de niños constituye un estudio de caso, que permite ilustrar que enfrentar tareas retadoras de un mismo tipo, bajo condiciones institucionales propicias, posibilita el desarrollo del talento matemático.
... Meanwhile, according to Brousseau (2013), DE framework can be formulated as a series of steps in a didactical design aimed at improving the teaching process. On the other hand, Artigue (2009) and Godino & Batanero et al. (2013) explained that DE is used when research positions "didactical design" as an important product, so the research can be said to design a didactical ISBN : 978-602-50919-0-2 International Conference on Mathematics and Mathematics Education (ICM2E 2017) Universitas Negeri Padang, West Sumatera Indonesia Novotel Hotel, August 27 th -29 th 2017 399 engineering framework. Therefore, according to Suryadi (2013, p. 6), a didactical design basically consists of three stages: (1) analysis of didactic situational planning before instruction, (2) metapedadidactical analysis to support instruction, and (3) retrospective analysis of data generated between instructional planning and the results of metapedadidactical analysis. ...
This research is motivated by not yet optimal communication ability and problem solving of mathematical learners. Communication skills and mathematical problem solving are very important to develop because communication skills and problem-solving abilities is one of the goals of mathematics learning that must be mastered by learners. The ability of mathematical communication is the ability of learners in communicating ideas or ideas about mathematics material that learners learn, for example in the form of concepts, formulas, or strategies to solve a problem through oral, written, and demonstrate it and visualize it visually. Mathematical problem solving ability is the ability of learners in finding solutions of difficulties that require creativity, understanding, and thinking. Mathematical communication and problem solving skills of learners can be developed together in the learning process of mathematics through the model of learning PACE. The PACE learning model is a conceptual framework of learning based on constructivism that has stages/phases: Project, Activity, Cooperative Learning, and Exercise using the LKPD in the process Learning. In addition to applying PACE learning model, other factors that must be considered by educators are factors of learners themselves such as differences in learning styles and self-efficacy because the two factors have an influence in learning mathematics. PACE learning model is good to be applied in learning mathematics, because it can improve communication ability and solving mathematical problem of learners.
Keyword: Communication Skill, Problem Solving, PACE Model Learning, Learning Style, Self-Efficacy
... International Journal of Instruction, January 2022 • Vol.15, No.1 learning using Didactical Design Research (DDR) were developed in the preparation stage. DDR is a research methodology developed from tacit didactical and pedagogical knowledge (Artigue, 2009;Hudson, 2008;Prediger & Zwetzschler, 2013;Suryadi, 2013). ...
... The researcher uses a syntax consisting of three stages in the hybrid learning model: seeking information, acquiring information, and synthesizing knowledge. These stages have been modified and adjusted to the needs of prospective teachers who will get treatment [48]. ...
Abstract
AMT ability is one of the skills that is a priority to be developed in learning mathematics in universities. This research aims to test the effect of hybrid learning and enjoyment learning in improving Advance Mathematical Thinking of prospective teacher students moderated by learning style. This research is a quantitative research that aims to examine the effect of the independent variable on the dependent variable. The sample in this study is conducted by purposive sampling based on certain criteria. The research data is obtained by distributing questionnaires to the mathematics education program student at University X in Bandung City. The data collected is analyzed by path analysis assisted by Smart PLS application. Smart PLS analysis in this study is carried out in 2 stages which are the analysis of the outer model and the inner model. Analysis of the inner model is aimed at (1) testing the validity and reliability of the instrument and (2) performing the r-square test to determine the percentage of influence. The test results show that learning enjoyment and hybrid learning have a significant positive effect on advanced mathematics learning. The learning styles in this study have not been able to moderate the relationship between hybrid learning and enjoy learning on Advance Mathematical Thinking. Hybrid learning can be used as an alternative or learning solution in improving advanced mathematical learning.
Keywords: hybrid learning, Advance Mathematical Thinking, enjoyment learning, prospective teacher students, learning style
Abstrak
Kemampuan AMT merupakan salah satu kemampuan matematis yang menjadi prioritas untuk dikembangkan dalam pembelajaran matematika di perguruan tinggi. Penelitian Ini Bertujuan Untuk Menguji Pengaruh Hybrid Learning dan Enjoyment Learning Dalam Meningkatkan Berpikir Advance Mathematical Thinking of Prospective Teacher Students dimoderasi Gaya Belajar. Penelitian ini merupakan penelitian kuantitatif yang bertjuan untuk menguji pengaruh variabel independen terhadap variabel dependen. Sample dalam penelitian ini dilakukan dengan purposive sampling dengan berdasarkan beberapa kriteria tertentu. Data penelitian diperoleh dengan menyebar quesioner pada mahasiswa program studi Pendidikan Matematika di Universitas X di Kota Bandung. Data yang terkumpul dianalisa dengan analisis path berbantuan aplikasi Smart PLS. Analisis smart PLS pada penelitian ini dilakukan dengan 2 tahapan yaitu analisa outer model dan inner model. Analisa inner model untuk 1) menguji validitas dan reliabilitas instrumen serta 2) melakukan uji r square untuk mengetahui presentase pengaruh. Hasil pengujian hipotesis menunjukkan bahwa enjoyment learning dan Hybrid learning berpengaruh secara positif signifikan terhadap Advance mathematic learning. Adapun, gaya belajar dalam penelitianini belum mampu memoderasi hubungan hybrid learning dan enjoyment learning terhadap Advance Mathematic Thinking. Hybrid learning dapat dapat dijadikan alternatif atau solusi pembelajaran dalam meningkatkan Advance Mathematic Learning.
Kata kunci: pembelajaran hibrid, Advance Mathematical Thinking, kenikmatan belajar, mahasiswa calon guru, gaya belajar
... According to Artigue (2009), didactic design includes controlled intervention research into the processes of planning, delivering and evaluating mathematics teaching and learning, and further, it includes the problem of reproducibility of results from such interventions. Here, we examine the methodologies for instructional design of four frequently used theoretical approaches to mathematics teaching and learning: ATD, TDS, Realistic Mathematics Education (RME) and Lesson Study. ...
... Essa pesquisa é qualitativa e se estreita com a Engenharia Didática (ARTIGUE, 2008). Nesse caso, utilizar-se-á da avaliação a priori e a posteriori para se produzir uma compreensão em relação a influência do Excel, em quatro experimentos, realizados por onze alunos. ...
The changes brought about implementation of the Common National Curriculum Base (BRAZIL, 2017) which had affected the courses of formative physical education teachers, as well as recently published studies about training of physical education teachers, are the context in which discussions are presented in this paper. Supported by authors who critically analyze the issue of teacher’s formative process, as well as in those who discuss physical education in particular, the study aimed to analyze the issues of formative process to physical education teacher. It also aimed to verify the publications from 2010 to 2020 in the most popular virtual library of scientific articles, Scientific Electronic Library Online (SCIELO BR). The key words adopted were "teacher education" and "physical education". As a result of the crossing of unitemos, 16 papers were identified that dealt with themes adjacent to the teacher formation. However, about formation of the physical education teacher, just 7 papers are identified. The shortage of studies confirms lack of identity of the area of school physical education and the problems arising from the absence of critical discussions about the space of action of the physical education teacher. In the future, researches about this subject will need produce to offer support to teachers in formative process and their pedagogical practices.
Keywords: teacher training, physical education, education policies.
... Essa pesquisa é qualitativa e se estreita com a Engenharia Didática (ARTIGUE, 2008). Nesse caso, utilizar-se-á da avaliação a priori e a posteriori para se produzir uma compreensão em relação a influência do Excel, em quatro experimentos, realizados por onze alunos. ...
... Essa pesquisa é qualitativa e se estreita com a Engenharia Didática (ARTIGUE, 2008). Nesse caso, utilizar-se-á da avaliação a priori e a posteriori para se produzir uma compreensão em relação a influência do Excel, em quatro experimentos, realizados por onze alunos. ...
... It has been widely discussed, commented on, and expanded since its origin (Brousseau 1986;1997;2006;Perrin-Glorian 2008;Artigue et al. 2014). More particularly, the concept of milieu has been addressed in mathematics education (Margolinas et al. 2005;Artigue 2009;Kidron et al. 2014) but also outside mathematics education (Achiam et al. 2013); we refer to Margolinas and Bloch's refinements of the milieu structuring (discussed below) (Margolinas 2004;Bloch and Gibel 2011;Ainley and Margolinas 2015), which describe and organise the mutual roles of the student, the teacher and the milieu at different levels of interaction. The instrumental approach, born in the field of cognitive psychology (Rabardel 1995), has been adopted and adapted in mathematics education (Rabardel 1999;Artigue 2002); numerous researchers in their studies have developed this theoretical approach by highlighting and clarifying the concept of scheme, used in the psychologists' sense (Vergnaud 2009;Määttänen 2016;Roorda et al. 2016) or as technical and conceptual components (Ceratto Pargman et al. 2018;Buteau et al. 2019). ...
Formative assessment strategies have been studied for a long time. Drawing on data from the FaSMEd (Formative Assessment in Science and Mathematics Education) project, this paper has the aim of contributing to research about formative assessment and the use of technology, in the field of mathematics education, by claiming that digital technology does modify classroom assessment processes when mastered by teachers, especially regarding the implementation of formative assessment strategies, but also by discussing how and to what extent this occurs, taking into account the different perspectives of the actors involved. The methodology of this research is founded in the design-based research paradigm, and the work with teachers is detailed in order to show the contributions of the project both in providing research results and in examples of practical use in the mathematics classroom.
... Surgida oficialmente no início dos anos 80, durante a investigação da Didática Matemática (Artigue, 2008), a ED utiliza um esquema experimental criando situações didáticas de ensino de acordo com cada situação do estudante, tentando superar assim o obstáculo epistemológico identificado. No caso dessa pesquisa, o obstáculo encontra-se no ensino da Sequência de Padovan, abordando assuntos iniciais a respeito deste conteúdo matemático, buscando investigá-lo, evidenciando o seu processo histórico e evolutivo. ...
O artigo reporta uma experiência de ensino com recurso à metodologia da Engenharia Didática (ED) aplicada ao ensino e exploração da Sequência de Padovan. Esta experiência foi conduzida numa turma de licenciatura em Matemática no Ceará-Brasil e conta com a participação de cinco estudantes. Foi elaborado um questionário contendo três situações-problema, das quais foram analisadas as resoluções desses professores em formação. De modo específico, investigou-se, a partir de conceitos básicos referentes a esta sequência, a fórmula de Binet, função geradora e construção da fórmula de recorrência. Esse fato deu-se na fase de experimentação da ED, sendo então complementada, nesta fase, com uma metodologia de ensino (Teoria das Situações Didáticas-TSD). Nas demais fases os dados coletados são discutidos, e assim realizada uma validação interna desta pesquisa. Pode-se destacar que os estudantes apresentaram dificuldades em conjecturar seus argumentos formulados nas fases iniciais. O professor torna-se então responsável por intermediar essas discussões, mediando os estudantes para que haja demonstração e validação desses conceitos matemáticos.
Palavras-chave: Engenharia Didática; investigação; Sequência de Padovan; Teoria das Situações Didáticas.
... In previous research (e.g., Margolinas 2014;Pepin et al. 2013;Remillard 2005), mathematics teacher interaction with resources has been discussed. It has become clear that teachers interact with (curriculum) resources in different ways (e.g., adaptation, appropriation), and one of the forms of interaction has been labelled as "design" (e.g., Brown 2009;Pepin et al. 2017a). ...
... In previous research (e.g., Margolinas 2014;Pepin et al. 2013;Remillard 2005), mathematics teacher interaction with resources has been discussed. It has become clear that teachers interact with (curriculum) resources in different ways (e.g., adaptation, appropriation), and one of the forms of interaction has been labelled as "design" (e.g., Brown 2009;Pepin et al. 2017a). ...
In this chapter, we investigate the notion of “teachers as curriculum designers” from the literature and from six international perspectives. This is done in order to (1) develop a deeper understanding of the concept, and (2) provide an international perspective and illustrations of the different facets of teacher design. Based on this investigation, we could identify different modes of teacher design: from teacher design activities at micro level (e.g., lesson preparation alone or in small groups), over those at meso level (e.g., D/designing in collectives of colleagues for the purpose of use by others), to Design at macro level (e.g., involvement in the design of national frameworks by professional design teams for the use of many others). More generally, we claim that the often casually used term of “teacher design” has different meanings in different contexts and that teacher design activities may be for different purposes and for different expected end results. A major distinction is whether the design is more oriented towards the process, or the product. We argue that the most promising form of teacher design might lie at the crossroads between product and process orientation, with connections between the two. This has implications for teacher education and professionalism.
... Teaching includes the selection, modification, design, sequencing, installation, observation and evaluation of tasks. (Margolinas, 2013, p. 12, cited in Leung, 2017 Sinclair and Zazkis (2017) state that interesting and powerful tasks are those that make strong and consistent use of digital technologies and differ from tasks designed for non-digital environment. In this context, a challenge for teachers is to conceptualize the use of digital technologies as a set of tools for learner to think and involve in mathematical activities. ...
It has been widely recognized that problem-solving activities are crucial in developing and learning mathematics. Indeed, it is common to structure and frame both mathematical curriculum and learning environments through problem-solving activities. Currently, significant developments of digital technologies are shaping both students’ social interaction and ways of learning mathematics and solving problems. What types of strategies, representations, and resources emerge and are important in problem-solving approaches that rely on and foster the use of a Dynamic Geometry System affordances? The aim of this chapter is to analyze and discuss on how the use of a Dynamic Geometry System (GeoGebra) provides affordances to develop a geometric reasoning as a mean to work and solve mathematical problems. In this process, it becomes important to think of and represent problem statements and concepts geometrically, to construct dynamic models of problems, to trace and examine loci of particular objects, to analyze particular and general cases, and to communicate results.
... Our theoretical research includes an experimental facet, where we mainly rely on qualitative methodology, including participant and also ethnographic approaches (Alexander, 1982;Spradley, 1980) and simple and multiple case studies (Stake, 1995;Yin 2003) in the exploratory stage and didactical engineering in the sense of Artigue (2009) in the validating stage. Instead of adhering to the biomedical metaphor that sees control groups as a necessary tool for validation, we adhere instead to the enactivist methodology (Reid, 1996;Reid & Mgombelo, 2015), where the process of description of data is seen as an interrelationship, between researchers and a context that is not "out there" but is partially their own creation; moreover, the observers are modified by their observation. ...
We are interested in exploring and developing an enactivist approach to problem posing and problem solving. We use here the term “enactivist approach” to refer to Varela’s radically nonrepresentationalist and pioneering “enactive approach to cognition” (Varela et al., The embodied mind: Cognitive science and human experience. Cambridge, MA: The MIT Press, 1991), to avoid confusion with the enactive mode of representation of Bruner, which is still compatible with a representationalist view of cognition. In this approach, problems are not standing “out there” waiting to be solved, by a solver equipped with a suitable toolbox of strategies. They are instead co-constructed through the interaction of a cognitive agent and a milieu, in a circular process well described by the metaphor of the Ouroboros (the snake eating its own tail). Also, cognition as enaction is metaphorized by Varela as “lying down a path in walking.” In this vein, we present here some paradigmatic examples of enactivist, and metaphorical, approaches to problem solving and problem posing, involving geometry, algebra, and probability, drawn from our didactical experimenting with a broad spectrum of learners, which includes humanities-inclined university students as well as prospective and in-service maths teachers. Our examples may be metaphorized as cognitive random walks in the classroom, stemming and unfolding from a situational seed.
Mathematics instruction has traditionally emphasized procedural fluency, focusing on algorithms and step-by-step methods. However, recent educational research highlights the importance of fostering conceptual understanding alongside algorithmic skills to ensure long-term mathematical proficiency. This review examines various instructional strategies that aim to bridge this gap, evaluating their effectiveness in promoting both algorithmic competence and deep conceptual understanding. It contrasts traditional methods, such as direct instruction and rote memorization, with more contemporary approaches, including active learning, collaborative problem-solving, and technology-enhanced instruction. The paper also explores the integration of procedural and conceptual learning through scaffolding, real-world applications, and contextual learning, while addressing the challenges educators face, including teacher preparation, student misconceptions, and curriculum constraints. Finally, the review discusses emerging trends and future directions in math education, emphasizing the need for a balanced approach that prepares students for both practical and theoretical mathematical challenges. By adopting a comprehensive instructional framework, educators can better equip students to understand and apply mathematical concepts effectively, ensuring a more holistic and enduring math education.
En un mundo cada vez más tecnológico e interdisciplinario, resulta fundamental que la enseñanza de las ciencias exactas maximice el uso de herramientas digitales y metodologías innovadoras, con el propósito de preparar a estudiantes con los conocimientos y habilidades necesarias para enfrentar los desafíos del siglo XXI. El aprovechamiento de éstos recursos representa una oportunidad para enriquecer el aprendizaje, así como para fomentar una visión holística del conocimiento a través del enfoque STEAM (Ciencia, Tecnología, Ingeniería, Arte y Matemáticas, por sus siglas en inglés), promoviendo la creatividad, el pensamiento crítico y la capacidad de resolver problemas.
El presente estudio tuvo como objetivo desarrollar un prototipo didáctico basado en tecnologías LabVIEW y Arduino, que contribuye a la resignificación de conceptos matemáticos abstractos relacionados con el lenguaje variacional (funciones de una variable). La investigación adoptó un diseño metodológico mixto y se estructuró en tres fases: diseño y desarrollo del prototipo tecnológico, elaboración de materiales didácticos complementarios, así como la implementación del recurso en un grupo piloto de estudiantes.
Los resultados evidencian una contribución positiva en la resignificación de conceptos básicos asociados a las diversas representaciones semióticas de funciones matemáticas. Desde un enfoque cuantitativo, se observó que los retos propuestos para replicar gráficas de funciones matemáticas básicas fueron resueltos satisfactoriamente por poco más de dos tercios del total de los participantes. A nivel cualitativo, se identificaron mejoras en la interpretación conceptual, destacando la interacción y la aplicabilidad práctica del prototipo didáctico. Los participantes manifestaron opiniones positivas respecto a las características generales del recurso, calificándolo como útil, interesante y una mejora significativa en comparación con las metodologías tradicionales utilizadas en la enseñanza de las matemáticas.
Palabras clave: funciones matemáticas, tecnología educativa, STEAM, prototipo didáctico, LabVIEW, Arduino, educación media superior, progresiones de aprendizaje, Nueva Escuela Mexicana (NEM), pensamiento matemático.
La enseñanza de matemáticas en el Bachillerato General Unificado (BGU) enfrenta el desafío de promover el pensamiento crítico y la aplicación práctica del conocimiento. La resolución de problemas interdisciplinarios permite contextualizar los conceptos matemáticos en situaciones reales, fortaleciendo habilidades comunicativas, analíticas y colaborativas. Este estudio analiza la implementación de estrategias basadas en problemas interdisciplinarios y su impacto en las habilidades comunicativas de los estudiantes. Se utilizó un diseño cuasi-experimental con un grupo de control y otro experimental, aplicando instrumentos de evaluación cualitativos y cuantitativos. La muestra incluyó 150 estudiantes de 2° y 3° de BGU en tres instituciones educativas. Los resultados muestran un incremento del 18,5% en el rendimiento académico del grupo experimental, en contraste con el 5,2% del grupo control. Además, el 87% de los estudiantes reportó una mejor comprensión y aplicación de conceptos matemáticos en situaciones reales, mientras que el 78% indicó un aumento en la motivación hacia la asignatura. El análisis cualitativo evidencia mejoras en la argumentación, explicación y justificación de soluciones matemáticas en el grupo experimental. Estos hallazgos sugieren que la integración de problemas interdisciplinarios favorece un aprendizaje más significativo y el desarrollo de habilidades comunicativas esenciales. Se concluye que la enseñanza interdisciplinaria de las matemáticas debe considerarse en el currículo educativo del BGU, ya que no solo mejora el desempeño académico, sino que también potencia la comprensión y aplicación del conocimiento en diversos contextos.
The purpose of this research is to analyze the problem-solving abilities of Mathematics Education students in the Mathematical Modeling course. This study employs a descriptive method with a qualitative approach. The subjects consist of 37 students from the 2023 cohort of the Tadris Mathematics Study Program at UIN Syahada Padangsidimpuan. The primary instrument used is a problem-solving ability test. Data analysis involves three stages: data reduction, data presentation, and conclusion drawing. The results reveal several key findings: (1) 19% of the students are able to understand the problems effectively, 38% have limited understanding, and 43% are unable to comprehend the problems; (2) 14% of the students can formulate problem-solving plans, 27% have limited capability, and 59% cannot develop problem-solving plans; (3) 14% of the students are capable of implementing their problem-solving plans, 24% have limited ability, and 62% are unable to execute their plans; and (4) 11% of the students are able to review their solutions, 19% have limited ability, and 70% are unable to review their solutions. These findings indicate that the majority of students face challenges in every stage of the problem-solving process, from understanding the problems to evaluating their solutions. Consequently, this research recommends the adoption of more interactive and contextual teaching strategies to enhance students' problem-solving abilities in the Mathematical Modeling course.
Mathematical education requires innovative didactic strategies to enhance the understanding and application of mathematical concepts, as traditional teaching methods often lack relevance. This methodology aims to develop a problem-solving scientific approach called design thinking as a strategy for learning mathematics functions. The study was applied to a sample of 138 students of biochemical, biological, and industrial engineering careers attending the first academic cycle at the Faculty of Natural and Exact Sciences of the Particular Technical University of Loja-Ecuador. The methodology uses a quasi-experimental design with a convenience sampling method. All participants were divided into a control group (C, D, K) and an experimental group (P, Q, R). Knowledge, skills, perceptions, and engagement were measured through pretest, posttest, workshop, rubric, project, and survey instruments. The pretest results indicate that both groups had similar knowledge of mathematical functions (pretest mean experimental group: 1.42/2 and mean control group: 1.55/2). Moreover, after applying design thinking strategy to the experimental group, variables questionnaire, project, and workshop show statistical differences (p < 0.001) between groups related to the traditional learning strategy, increasing the experimental group’s score in the project (posttest mean experimental group: 1.62/2 points, and mean control group: 1.65/2). The survey opinion indicates that 53.5% of the experimental group highlighted the project’s development as positively impacting their academic training. In conclusion, problem-solving design thinking using scientific projects as a mathematical function learning strategy contributes to improving the comprehension of polynomial functions and developing mathematical competencies, abilities, and skills to generate tangible solutions for real problems.
This study explores the integration of Ethnomathematics within a Problem Based Learning (PBL) framework to enhance students' critical thinking skills in mathematics learning. Ethnomathematics contextualizes mathematical concepts within cultural practices, making learning more relevant and engaging. The PBL model encourages active problem-solving and critical thinking through real-world scenarios. This mixed-method research involved 120 students from various secondary schools in Indonesia, divided into experimental and control groups. The experimental group received Ethnomathematics-based PBL instruction, while the control group followed a traditional teaching approach. Pretests and posttests measured critical thinking skills, complemented by student interviews and classroom observations. Results showed a significant improvement in the critical thinking skills of the experimental group compared to the control group. Interviews and observations revealed increased engagement and a deeper understanding of mathematical concepts among students exposed to Ethnomathematics-based PBL. This study suggests that integrating cultural contexts into PBL can effectively enhance critical thinking in mathematics education, offering a more inclusive and culturally responsive teaching approach. Future research should explore the long-term impact of this integration on various student demographics and mathematical competencies.
Cognitive pragmatics is an interdisciplinary field that explores the mental processes underlying communication, specifically how individuals understand, interpret, and convey meaning in context. In mathematics education, communication plays a vital role as students exchange mathematical ideas, explain solutions, and collaborate to solve problems. Understanding how students mentally process mathematical concepts during communication is essential for fostering deeper learning. This systematic literature review (SLR) examines the intersection of cognitive pragmatics and mathematics education, focusing on the cognitive mechanisms that facilitate communication during mathematical problem-solving. The review synthesizes studies on cognitive pragmatics, highlighting its role in enhancing mathematical discourse, classroom interactions, and collaborative problem-solving. The main objectives of this review are to identify how cognitive pragmatics influences students' ability to communicate mathematical ideas effectively and to explore its impact on teachers' instructional strategies. Key themes emerging from the literature include the cognitive processes of inferencing, attention, reasoning, and memory in the context of mathematical communication. The review also discusses the application of relevance theory and theory of mind (ToM) in understanding students' mental processes during mathematical interactions. Findings suggest that cognitive pragmatics can improve students' comprehension of complex mathematical concepts by enabling them to decode symbols, interpret linguistic cues, and engage in effective mathematical discourse. Moreover, collaborative learning environments that emphasize communication can enhance students' ability to articulate mathematical reasoning and problem-solving approaches. Teachers who are aware of cognitive pragmatics are better equipped to facilitate discussions and guide students through the mental processes required for effective communication in mathematics. However, the review also highlights challenges in integrating cognitive pragmatics into mathematics education, such as the diversity of students' cognitive abilities and the complexity of mathematical language. Educators need to adapt their teaching strategies to meet students' varied cognitive needs and support their development of communication skills. In conclusion, cognitive pragmatics provides valuable insights into the mental processes that underlie communication in mathematics education. Further research is needed to explore how cognitive pragmatics can be systematically integrated into teaching practices, curricula, and assessments to enhance students' communication and problem-solving abilities in mathematics.
The study focuses on the development and implementation of open-ended discrete mathematics teaching materials to improve mathematical reasoning among students in the Mathematics Education Program at Syahada Padangsidimpuan. The research aims to address the challenge of enhancing students' mathematical reasoning skills, a critical component for their academic and professional growth. Open-ended problems, known for promoting higher-order thinking and deeper engagement, are utilized to create teaching materials that encourage exploration and problem-solving. The research employs a mixed-methods approach, combining qualitative and quantitative data to evaluate the effectiveness of these materials. The qualitative aspect involves the design and development of the teaching materials, while the quantitative aspect measures improvements in students' reasoning abilities before and after using the materials. The study's findings indicate a significant enhancement in students' mathematical reasoning skills, demonstrating the potential of open-ended materials in fostering critical thinking and problem-solving abilities. The results highlight the importance of integrating such innovative approaches into mathematics education to better prepare students for future challenges in their field.
GeoGebra, an interactive mathematics software, has gained significant traction in higher education due to its versatile applications in teaching and learning mathematics. This study aims to analyze the bibliometric data of research articles on GeoGebra in higher education indexed in the SCOPUS database. By employing bibliometric techniques, the study identifies key trends, influential authors, prominent journals, and collaborative networks within this research domain. Data were extracted from SCOPUS using the search terms "GeoGebra" and "higher education" covering publications from 2000 to 2023. The analysis revealed an upward trend in the number of publications, with a notable increase from 2010 onwards. The findings highlight key research areas, leading institutions, and geographical distribution of the studies. The results indicate that GeoGebra is increasingly being adopted as a tool to enhance mathematical understanding and pedagogy in higher education. This bibliometric analysis provides valuable insights for researchers and educators aiming to explore and expand the use of GeoGebra in their academic practices.
This study explores the integration of Ethnomathematics within a Problem Based Learning (PBL) framework to enhance students' critical thinking skills in mathematics learning. Ethnomathematics contextualizes mathematical concepts within cultural practices, making learning more relevant and engaging. The PBL model encourages active problem-solving and critical thinking through real-world scenarios. This mixed-method research involved 120 students from various secondary schools in Indonesia, divided into experimental and control groups. The experimental group received Ethnomathematics-based PBL instruction, while the control group followed a traditional teaching approach. Pretests and posttests measured critical thinking skills, complemented by student interviews and classroom observations. Results showed a significant improvement in the critical thinking skills of the experimental group compared to the control group. Interviews and observations revealed increased engagement and a deeper understanding of mathematical concepts among students exposed to Ethnomathematics-based PBL. This study suggests that integrating cultural contexts into PBL can effectively enhance critical thinking in mathematics education, offering a more inclusive and culturally responsive teaching approach. Future research should explore the long-term impact of this integration on various student demographics and mathematical competencies.
This study explores strategies for enhancing students' abstract thinking skills through the study of measurable functions in Advanced Real Analysis. Abstract thinking is essential for understanding complex mathematical concepts, and measurable functions offer a valuable context for developing these skills. Advanced Real Analysis often presents challenges due to its highly abstract nature, and the ability to navigate these challenges is crucial for students' success in higher-level mathematics. The research focuses on how students interact with and understand measurable functions, a fundamental component of measure theory. This study employs a mixed-methods approach, combining qualitative and quantitative data to assess the effectiveness of different teaching strategies. Data collection includes student surveys, in-depth interviews, and analysis of academic performance in assignments and exams related to measurable functions. The goal is to identify common difficulties students face and to determine which instructional methods are most effective in overcoming these challenges. Key findings indicate that students often struggle with the abstract nature of measurable functions, including the formal definitions and their practical applications. The study highlights that visual aids and interactive tools, such as mathematical software (e.g., GeoGebra and Wolfram Mathematica), play a significant role in bridging the gap between abstract theory and concrete understanding. Furthermore, integrating real-world applications and providing context for these mathematical concepts significantly enhances students' ability to grasp complex ideas. The study recommends several strategies to improve teaching practices. Incorporating visual and interactive elements into the curriculum can help demystify abstract concepts. Additionally, fostering a collaborative learning environment where students can discuss and solve problems together enhances their understanding. Professional development for educators, including training in the use of advanced educational tools and innovative teaching methods, is also crucial for improving instructional effectiveness. Ultimately, this study aims to contribute to the broader educational discourse by offering insights into effective methods for teaching abstract concepts in Advanced Real Analysis. By focusing on measurable functions, this research seeks to improve students' abstract thinking skills, thereby facilitating their overall success in mathematical studies.
This study introduces a novel framework designed to enrich mathematics teaching guides from a competency-based perspective. First, we narrow down the concept of a teaching guide in mathematics education, grounded in the documentational approach to didactics. This definition offers an updated perspective on the structure and function of math teaching guides in educational settings. Second, we provide a comprehensive definition of ‘richness’ in math activities, encompassing content, processes, cognitive demand, and classroom management. Lastly, we introduce an analytical tool developed for assessing and enhancing the richness of math teaching guides. This tool, formed through theoretical analysis and empirical testing, assists educators and curriculum developers in creating more balanced and integrative teaching guides. The results suggest that the tool holds potential for broader applications in curriculum design and teacher education. The findings contribute to the broader understanding of how teaching guides can effectively capture and communicate the richness of activities, thereby serving as a valuable tool for improving mathematical education resources.
The growing use of programming in mathematics classrooms presents a challenge linked to implementation in general and task design in particular. This article presents design ideas for mathematical problems incorporating programming in which the focus remains mainly on learning mathematics and less on learning programming. The article starts by reviewing the theoretical background for technology implementation and design, and then presents the methodology for the design, before exploring and discussing the design ideas with an in-depth example. Building on the idea of adidactical situations from the theory of didactical situations, the design illustrates a possible way of implementing programming in the mathematics classroom to facilitate mathematical learning.
Qualitative research is often identified as idiographic: a research focused on cases and avoided generalizations. Whereas quantitative research is identified as nomothetic by proposing the determination of general laws. Most of the time, quantitative and qualitative research should not be regarded as two paradigms, as two different worldviews are unable to communicate. In this paper, we first focus on the main characteristics of the two kinds of research analyses. Then, an approach based on Fuzzy Cognitive Maps is proposed as a mixed method analysis to be used in mathematics education to combine qualitative and quantitative research results. Finally, some examples of applications are discussed.
This paper introduces this special issue aiming to deepen and extend research on mathematics teachers’ work, from a resource perspective, by taking language and culture into account, and exploring two questions: How are teachers’ interactions with resources interpreted and modeled across contexts? And, What challenges and insights emerge through recent efforts to engage these models in cross-cultural (and linguistic) research? The fields of resources, language and culture in mathematics education are each extensive, and we do not attempt to survey comprehensively across them. We have chosen instead to propose three approaches on resources in mathematics teachers’ work that developed somewhat contemporaneously from three different countries with differing linguistic, curricular, and social contexts, corresponding to the work of the three guest editors. The models developed through these approaches are driven by the educational, and so cultural and material conditions of the time and the location of each author, and allow us to propose preliminary answers to our two guiding questions. We then move to pull the threads from these models together, and discuss the contributions to this Special Issue. This results in more robust and nuanced responses to our questions, and in identifying two themes that emerge from research that sit at the convergence of studies of teachers’ interactions with resources, languages, and cultures: an invisibility-visibility dialectic and a local-global tension. Finally, this study leads us to consider a new region of mathematics education research.
Several studies have shown the relationship between math skills and language skills. The main objective of this investigation is to identify preschool children’s spatial thinking skills, achieved through communication processes and also to identify the information transmission and reception skills acquired by children in relation to the development of spatial skills. The whole process of this research is based on the Didactic Engineering methodology, through which four didactic situations (DSs) were designed. The design is non-experimental with an established group. The sample consists of 73 preschool boys and girls aged between five and six years. The results show that once the ability to send and receive messages is acquired, students developed mathematical thinking about their location in space according to a coordinate system, as well as the ability to give instructions in which distance, direction, and orientation relationships appear. Keywords: language, communication, mathematics, spatial thinking, didactic engineering
ResumoO modelo matemático que, segundo os livros de História da Matemática, é atribuído, equivocadamente ao matemático Leonardo Pisano, preserva hodiernamente um vigor irrefreável de um progresso maior de evolução e de generalidade. Não obstante, um caráter de oblívio total pode ser observado no que concerne a uma maior divulgação e, também, melhor formação do professor de Matemática, concernentemente a determinados assuntos matemáticos específicos. Diante de tal entrave, o artigo atual apresenta a descrição das análises preliminares e análise a priori de uma Engenharia Didática, com o tema quaternions (generalizados) de Fibonacci. A relevância deste assunto se evidencia, na medida em que, o mesmo pode ser encontrado apenas em artigos científicos de Matemática que reproduzem propriedades formais cifradas. Assim, com arrimo da Teoria das Situações Didáticas – TSD, o trabalho apresenta, de modo pormenorizado, um planejamento para as fases dialéticas de ação, formulação, validação e institucionalização. Por fim, a atual proposta explora elementos que balizam sua eventual aplicação e experimentação em sala de aula, tendo como público alvo, professores de Matemática em formação inicial. Palavras-Chave: História da Matemática; Sequência de Fibonacci; Investigação histórica. AbstractThe mathematical model, according to the History of Mathematics books, is attributed wrongly to the mathematician Leonardo Pisano, in our times preserves an unstoppable force for further progress of evolution and generality. Nonetheless, a complete forgetfulness character can be seen in relation to greater disclosure and also improved training of the mathematics teacher, concernentemente to specific mathematical topics. Faced with this obstacle, the current article presents a description of the preliminary analysis and a priori analysis of a Didactic Engineering, with the theme quaternions (generalized) Fibonacci. The importance of this matter is evidenced, in so far as the same can be found only in scientific articles Mathematics reproducing encrypted formal properties. So with retaining the Theory of Didactic Situations - TSD, the work presents, in detail, a plan for the dialectics stages of action, formulation, validation and institutionalization. Finally, the current proposal explores elements that guide its possible application and experimentation in the classroom, with the target audience, mathematics teachers in initial training. Keywords: Didactical Engineering; Fibonacci´s sequence; Quaternion of Fibonacci; Historical investigation.
Alhamdulillahhirabbil’alamin, puji syukur kehadirat Allah Azza Wa Jalla yang telah memberikan petunjuk, hidayah, nikmat sehat dan sempat sehingga penulis dapat menyelesaikan buku referensi ini. Buku yang berjudul “RISET REKAYASA DIDAKTIS: Upaya Menyusun Pengetahuan Matematis Untuk Mengajar di Sekolah Dasar” merupakan buku referensi dari hasil penelitian dan dilakukan dengan tujuan mengkaji dan melakukan rekayasa didaktis yang kompatibel untuk membangun pengetahuan matematis dari calon guru sekolah dasar sebagai bekal untuk mengajar matematika di sekolah dasar. Buku referensi ini juga diharapkan menambah khazanah keilmuan penelitian dan pengembangan pendidikan matematika di sekolah dasar. Di samping itu, dapat digunakan sebagai landasan berpijak untuk dieksplorasi lebih lanjut dan juga contoh dari penerapan teori situasi didaktis matematis yang masih relatif baru dalam metodologi penelitian pendidikan matematika.
Buku ini menampilkan secara langsung proses pengetahuan matematis untuk mengajar yang dimulai dengan mengkaji learning obstacles dari pengetahuan matematis, kemudian bagaimana menyusun antipasi didaktis-pedagogis dari calon guru sekolah dasar, dan bagaimana merekayasa situasi didaktis yang digunakan untuk membangun pengetahuan matematis untuk mengajar. Keunikan buku ini adalah sebagai titik awal dalam penggunaan metodologi baru dalam pendidikan matematika. Di samping itu, kami menggambarkan secara utuh tentang rekonseptualisasi dan tipologi pengetahuan matematis untuk mengajar. Lebih dari itu, buku ini juga merujuk pada beberapa literatur terbaik sehingga harapannya dapat menjadi rujukan terbaik.
Buku ini membahas secara detail tentang teori situasi didaktis matematis berdasarkan rujukan dari the founding father dan ahli yang relevan dengan teori tersebut. Temuan penelitian ini juga menarik karena mengkaji tentang materi geometri yang dibahas secara holistik, menganalisis hambatan belajarnya, dan selanjutnya digunakan untuk antisipasi guna “memperbaiki” situasi didaktis yang didesain.
Buku ini terbagi menjadi 6 bab yang dimulai dari latar belakang pengetahuan matematis untuk mengajar, rekonseptualisasi dan tipologi pengetahuan matematis untuk mengajar, teori situasi didaktis, prosedur metode penelitian didaktis, temuan yang berisi learning obstacles dan antisipasi didakdis-pedagogis, dan pembahasan, implikasi, studi potensial di masa depan. Materi dalam buku ini disusun dan dikonstruksi secara sistematis dari teori ke aplikasinya dalam pembelajaran.
Buku ini mencoba mengajak Anda untuk mampu mengkaji secara mendalam tentang desain didaktis dan konteks terkait yang diperlukan untuk mendorong pengetahuan matematik yang digunakan sebagai pengajaran di kelas sekolah dasar. Selain itu, Anda diajak untuk dapat memahami apa itu pengetahuan matematis untuk mengajar? Bagaimana kritik terhadap kerangka kerja Ball dan Kolega? Bagaimana konsep teori situasi didaktis? Apa saja langkah prosedur metode penelitian didaktis? Apa saja temuan dan pembahasan terkait learning obstacles dan antisipasi didaktis-pedagogis? Anda juga dapat menemukan secara langsung hasil analisis dan hambatan belajar dalam materi geometri dalam konteks didaktis matematis yang merupakan hasil temuan dari peneliti. Dengan demikian, diharapkan buku ini dapat memperdalam wawasan, pengetahuan, dan keterampilan dalam kegiatan instruksional di kelas matematika sekolah dasar.
In this paper, we describe an experiment in using history to work on problem-solving and the relationship between arithmetic and algebra. The students involved attended the first year of the Italian upper secondary school (grade 9). The original sources we used are problems from Italian treatises on arithmetic and algebra that appeared in the Middle Ages and the Renaissance. At the beginning, we present these treatises, which belong to a widespread mathematical tradition. We then describe the classroom experiment: we administered 14 problems from ancient treatises accompanied by the request to solve them and write impressions and comments. Data were collected through written protocols and interviews. The analysis of the results focused on the strategies implemented in solving the problems, with particular reference to the use of arithmetic and algebra, and on the perception of the cultural aspect introduced by these problems in the students’ approach to mathematics. The findings show the difficulties of some students on topics, such as fractions and direct proportionality, which should have been acquired in earlier school years. Combining the use of history with aspects of research in mathematics education allowed us to outline some teaching implications linked to the use of history.
There is a consensus that numeracy is important for students to develop logical thinking and reasoning strategies in their everyday activities. As a result, teachers are encouraged to design numeracy rich tasks that incorporate real-life contexts across non-mathematics curriculum areas. However, it is not clear what types of knowledge teachers require to design such tasks. In this study, the authors used the stepwise generalization method and developed a framework for the knowledge required by teachers to design numeracy rich tasks across non-mathematics curriculum areas. The study begins with a brief introduction to numeracy and its definition in various contexts. The nature of knowledge required to design numeracy rich tasks (mathematics content knowledge, a non-mathematics curriculum area knowledge, activity design skills and knowledge of context)) are further described. In the later section of the study, each element of teacher knowledge and the framework to design numeracy rich tasks across non-mathematics curriculum areas are described. The knowledge framework developed in this study can be used to analyze teachers designed numeracy-rich tasks and identify their professional learning requirements.
O objetivo deste trabalho é apresentar uma sequência de orquestrações instrumentais visando o ensino dos conceitos de função integrável e integral de Riemann. Os sujeitos do estudo foram 21 discentes do curso de Licenciatura em Matemática da Universidade Federal do Amazonas. Para isso, realizou-se uma pesquisa qualitativa de caráter descritivo, tendo como metodologia norteadora a Engenharia Didática e numa perspectiva de complementaridade, adotamos um design de investigação, tendo em vista o uso da Teoria dos Registros de Representação Semiótica, com o escopo de descrever e estruturar situações de ensino, que buscou investigar se a sequência de orquestrações instrumentais proposta contribuiu para a compreensão dos objetos matemáticos aqui abordados. Na relação dos dados compilados e apresentados, como resultados, evidenciamos que a maioria dos discentes demonstrou compreender o conceito de função integrável, conseguiram encontrar a área sob uma curva e explicar seus procedimentos, inclusive em língua natural, definir a integral definida em palavras e símbolos, de interpretar e representar uma integral graficamente, de avaliar as integrais e de reconhecer o uso de integrais no mundo real.
This research develops resources to teach mathematics in French primary school by using the game of Go. A group of searchers, teachers and go players meet at university to produce teaching resources. These resources are implemented in the classroom. Then the group evaluate this implementation and improve the resources. The aim of this classroom research is to study the opportunities of the game of Go to learn mathematics and to propose a teacher training course to implement the game of Go in French primary schools in accordance with the French syllabus. Game of Go appears as a manipulative and semiotic tool to learn mathematics at primary school. Subject Classification: 97D50, 97U60
This chapter looks at an in-depth application of meaning equivalence reusable learning objects (MERLO) to mathematics education and teacher professional development. The study has been conducted during professional development courses for in-service teachers and is focused on mathematics teachers' praxeologies, namely their didactical techniques and theoretical aspects embraced to accomplish a task. Specifically, the task given to the teachers consists in designing MERLO items to be used in their classrooms, working in groups or individually, after having been trained by researchers in mathematical education. The chapter presents two case studies with data, one dealing with secondary school teachers in Italy and one concerning primary teachers in Australia. One of the main aims of the study is the analysis of the praxeologies of these teachers when they are engaged in designing MERLO items during professional development programs. The chapter demonstrates, with these examples, the generalizability potential of MERLO items and that they can be used in different cultural and institutional ecosystems.
In this article, we question the prevalent assumption that teaching and learning mathematics should always entail movement from the concrete to the abstract. Such a view leads to reported difficulties in students moving from manipulatives and models to more symbolic work, moves that many students never make, with all the implications this has for life chances. We propose working in ‘symbolically structured environments’ as an alternative way of characterising students’ direct engagement with the abstract and exemplify two such environments, both of which involve early number learning. We additionally propose some roles for the teacher working in a symbolically structured environment.
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