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This paper aims to identify and analyze the theme of Financial Education (EF) in academic research in the latest editions of ENEM, CIBEM and SIPEM. The methodological approach is qualitative of the bibliographical. We believe that the SBEM electronic database enabled a bibliographic survey of academic production. The results pointed to the need to promote discussions about EF in the school environment, recognizing its contribution to the integral and critical formation of the subject, its articulation with social, political, economic, environmental and cultural problems, making education a progressively active social force. They also highlighted the urgent need to implement continuing education programs that enable the planning and implementation of more assertive teaching practices in this area.

Since the publication of the Base Nacional Comum Curricular (National Common Curricular Base), Financial Education has become a topic to be developed in the classroom, mainly related to the mathematics curriculum. In this sense, teachers are in charge of addressing it. In view of this, in this paper we aim to discuss understandings of Financial Education aimed at Basic Education and produced by mathematics undergraduates, mobilized in two training meetings. Based on Critical Mathematics Education and the qualitative research paradigm, we identified five topics using content analysis procedures. The results of the analysis of these topics indicate that the future teachers have knowledge of School Financial Education, although they relate the subject to economic aspects and Financial Mathematics.

The undoubted benefits and intrinsic virtues of mathematics should not blind us to potential collateral damage as the huge juggernaut of mathematics rolls over education and society towards a reshaped future. Because of its great power and influence across the whole of society we need to undertake an ethical audit. This paper conducts an ethical critique that considers four interrelated aspects of mathematics in education and society and their negative impacts. 1. The overvaluation of mathematics and the effects this power has in maintaining privileges. 2. The powerful negative impacts of mathematical studies have on many individual learners. 3. Visible but problematic applications of mathematics within society that are shielded from criticism because mathematics continues to be viewed as neutral. 4. The profound and performative effects of the hidden applications of mathematics that reformat society and change our everyday lives but remain unchecked in every sense. Most of the paper is devoted to revealing these damaging effects. The solutions proposed are for ethical awareness to be fostered in mathematical study at all levels, as well as challenging the widespread stereotype of mathematics as ethically neutral.

The purpose of the present paper is to show how some current researches – published in papers and books – approach the difficulties in Mathematics learning. We intend to reflect on teaching and learning Mathematics by analysing some aspects that have been pointed out, especially about learning difficulties some students have in Mathematics area. We used the research qualitative approach with procedures of document analysis. At first, we described the meaning of the word ‘difficulty’, passing through learning difficulties and learning difficulties in Mathematics. The latest researches point the urgent need of a didactic overhaul in Mathematics teaching and maturation of all school communitiy, regardless the teaching level, concerning learning difficulties.

This paper describes an emerging approach to the design of task sequences and the theory that undergirds it. The approach aims at promoting particular mathematical concepts, understood as the result of reflective abstraction. Central to this approach is the identification of available student activities from which students can abstract the intended ideas. The approach differs from approaches in which learning to solve the problem posed is the intended learning. The paper illustrates the approach through data from a teaching experiment on division of fractions.

This paper is part of a wider study focused on divisibility. Participants were prospective elementary teachers. One of the aims of the research is to characterize the meanings of multiple shown by prospective teachers. In this paper, we present the results concerning this aim. We analyse the productions of 37 prospective elementary teachers collected in a practice session, designed and developed in the context of a teaching experiment. We characterize the meanings through the following elements of the didactic analysis: conceptual structure, representation systems and phenomenology. Prospective teachers showed mostly three meanings of multiple: product, relationship and dividend in an exact division.

En este artículo se exponen los resultados parciales de un estudio centrado en el análisis del papel que juega el convencimiento en la construcción del conocimiento matemático que se da en el aula. Se describen los resultados de una investigación empírica, centrada en el análisis de un estudio de caso longitudinal, en el que se examinan las prácticas de justificación y promoción de convencimiento a las que sistemáticamente recurre una profesora de sexto grado de primaria. Además de describir los patrones de racionalidad identificados en las clases observadas, se muestra que en el aula pueden converger, en un mismo recorrido discursivo, argumentos por razones y argumentaciones por motivos y que estas justificaciones son acumulativas, suelen ser implícitas, tienen límites borrosos y carecen de una estructura lineal. The paper contains the partial outcomes of a study focused on analyzing the role played by convincingness in building mathematical knowledge in the classroom setting. The paper describes the findings of an empirical study that is centered on the analysis of a longitudinal case study, in which we analyze the justification practices and convincingness promotion systematically resorted to by a sixth-grade elementary school teacher. In addition to describing the patterns of rationality identified in the classes observed, this paper serves to show that in the classroom setting, reasons-based arguments and motive-based lines of argument can converge within one discursive path. Consequently the justifications are cumulative, apt to be implicit, with blurred outer borders and lack a linear structure.

In this paper, I argue that international Mathematics Education Research (MER) is dictated by the interests and agenda of the imperialist countries, which dehistoricizes the material conditions of imperialized and impoverished nations while deliberately obfuscating the economic reasons underlying the “failure” of mathematics learning outcomes of their students. Instead, in the guise of improving mathematics teaching/learning, such researches usually blame the very people of these imperialized countries for the ‘low learning outcomes’ of their school students, attributing it to “cultural” and “local” beliefs, mindsets, practices, etc., thereby producing academic justifications for the neoliberal destruction of their public education systems. I begin my argument by drawing from my PhD research study with visually challenged mathematics learners from socio-economically marginalized communities in India. I then follow it up with a brief overview of some mathematics education research works in the context of Sub-Saharan African countries to illustrate my point.

In this paper, I argue that international Mathematics Education Research (MER) is dictated by the interests and agenda of the imperialist countries, which dehistoricizes the material conditions of imperialized and impoverished nations while deliberately obfuscating the economic reasons underlying the “failure” of mathematics learning outcomes of their students. Instead, in the guise of improving mathematics teaching/learning, such researches usually blame the very people of these imperialized countries for the ‘low learning outcomes’ of their school students, attributing it to “cultural” and “local” beliefs, mindsets, practices, etc., thereby producing academic justifications for the neoliberal destruction of their public education systems. I begin my argument by drawing from my PhD research study with visually challenged mathematics learners from socio-economically marginalized communities in India. I then follow it up with a brief overview of some mathematics education research works in the context of Sub-Saharan African countries to illustrate my point.

This article aims to analyze the evaluations made by the teachers of the participation of elementary school teachers in the research project entitled “Professional Development of Teachers who Teach Mathematics” (Project D-Estat), in the year of 2018. The Project aims to investigate the statistical learning experiences of teachers who teach mathematics in elementary school, within the scope of a collaborative group, aiming at professional development during the training process. The research uses qualitative methodology in an action research perspective. The paper presents theoretical reflections on professional development and teacher education, considerations on Teaching Sequence and Investigative Cycle. As a methodological resource, the narrative interview was used, in the format of a conversation circle. The analysis of the results gives evidence of a path taken by the teachers towards professional development and a better understanding of how to work the statistical thinking of their students. The teachers' oral and written narratives demonstrate the importance of participation in training processes for professional development.

In this article, our objective was to identify and analyze what and how the texts of the Mathematical Modeling discourse can be developed in the pedagogical practice and its implications in the formation of teachers. To this end, we selected scientific papers from the last three editions of two events in the country, the National Conference on Modeling in Mathematics Education and the International Research Seminar in Mathematics Education, which specifically bring together the entire community that researches on Mathematical Modeling. The research has a qualitative approach, inspired by Basil Bernstein's model of description language. From the analyzed texts, we observed that there are different possibilities to develop modeling activities and these different possibilities have different approaches: focus on content, focus on technology, focus on ethnomodelling, which allows the teacher to create different opportunities for learning students.

Considering the need for differentiated pedagogical practices enabling the association of student’s reality and other areas of knowledge, so that scientific/school concepts/content have greater meaning, this paper presents the approximation between methodologies Mathematical Modeling and STEAM. Such discussion is indispensable, because when researching the Catalog of Theses and Dissertations of the Coordination for the Improvement of Higher Education Personnel (CAPES), no studies involving both methodologies were identified. With this, research presents elements associating similarities in the stages/phases indicated by each methodology, which allow the student, mediated by the teacher, to develop mathematical and physical models to solve current problem situations. Based on the theoretical studies carried out, it is possible to consider that the two methodologies have similarities and can be inserted into the process of Mathemathics teaching and learning, allowing the student to be the protagonist in the construction of his knowledge, not only of mathematical concepts, but also in an interdisciplinary approach.

This paper sought to discuss the extent to which cultural issues allied with Mathematical Modeling can contribute and provide students with Meaningful Learning. Students from elementary schools in the countryside of Rio Grande do Sul participated in the research. Interventions were carried out in the school environment and classroom observations with the purpose of verifying the subsumers of the students, making a data survey and analyzing the school context. This survey made it possible to list three themes of interest to the students: Livestock, Fisheries and Agriculture. In this work we present an activity that addresses the topic of Fisheries as a basis for studying statistical concepts and analyze whether the learning of these concepts was significant. From the analysis of the results it was possible to infer that learning becomes significant when we insert cultural issues as a central theme, because these can underpin the learning of new concepts; strengthen and mean concepts already known.

This paper aims to analyze the evidence that constitutes the professional identity of future mathematics teachers when participating in a study group within the scope of the Supervised Curricular Internship. The qualitative research was carried out in 2019 at a public school in Formiga (MG), and included the participation of two students from a Mathematics Degree course and their Internship supervisors. The data discussed in this text were produced through written narratives of the interns, audio and video recordings of the meetings, evaluation and self-evaluation made by the undergraduate students. The formative experiences, experienced by the graduates in the internships and in the study group, contributed to the manifestation of aspects of the teacher identity, such as: emotion, self-knowledge, autonomy and political commitment. The results show the importance of formative contexts for the development of professional identity.

This paper presents the partial results of a Master's-degree-level research project on the contribution of employing the teaching-learning-assessment methodology through problem solving using GeoGebra, in connection with the different registers of representations, in teaching and learning the maxima and minima of functions. The educational product derived from this research was a didactic booklet, developed using the online GeoGebraBook tool, composed of several applications that seek to assist in the resolution of function optimization problems. Adopting a qualitative and interpretative approach, data were collected on the solutions of an activity tested in two pilot classes, one in high school and another in an undergraduate program. In the results, we will point out the potential regarding the use of technology that favored the resolution of the problem due to the interactive approach and the possibility of transition between different representations, as well as some challenges in certain stages of the adopted methodology.

This paper aims to reflect on the practice of Mathematical Modeling in the sociocritical perspective articulated to the model of investigative cooperation that deals with dialogical communication. It is part of a master's thesis whose theme for the development of modeling is water scarcity, a problem that emerged from the students' reality. The research is qualitative whose instruments used in the production of data were the researcher's logbook, students' textual production and recordings. Data were collected at a school located in Castelo, Espírito Santo, with students from the 8th grade of Elementary School. The results reveal that the Mathematical Modeling in the sociocratic perspective is par excellence, an environment that favors dialogical acts; in addition to bringing mathematical content closer to social problems, it promotes the collective investigation of perspectives through dialogue, favoring the teaching and learning process and the citizens' education of students.

This paper aims to recognize the research carried out on the theme Mathematical Modeling and Continuing Education and to understand from them the professional development (DP) of the participating teachers. For this, we used as a methodological approach qualitative research, of bibliographic type, based on the search for data in the Brazilian Digital Library of Theses and Dissertations. We found fourteen surveys, between 2010 and 2019 using the advanced search field and the title option, among which we selected eleven for analysis. The analysis procedures are based on thematic content analysis. From the unit of analysis professional development we present two categories in which we discuss that: 1) the research deals with continuing education with modeling from the perspective of the teachers DP and the spaces/time constituted for the training favor the DP and 2) the transition from continuing education to the classroom is a DP movement situated in the teaching profession.

This paper aims to analyze the use of Mathematics grammar in teaching and learning situations. Such analysis is made on the basis of Wittgenstein's second language philosophy and some of his commentators. To do so, our theoretical text, we discuss the universality of mathematical language; the need to translate mathematical language into natural language; and the use of mathematical grammar and some of its developments. The mathematical grammar is the set of rules that determine the correct use of symbols and words of the mathematical vocabulary and it is in the application of these rules that the apprentice understands the meaning of the mathematical concepts.

The importance of good mastery of specialised mathematics knowledge for mathematics teachers and prospective teachers is now recognised internationally. This paper presents an exploratory study aimed at investigating how mathematics education courses affect the Mathematics Teachers’ Specialised Knowledge (MTSK) of prospective primary teachers. A questionnaire was administered to a sample of 118 future teachers, and an analysis of their answers was carried out with the MTSK model. The research highlights positive feedback on the effectiveness of the mathematics education laboratory that was staged, and provides an outline of the beliefs and the knowledge of the prospective teachers.

In this paper we present a research whose objective was to verify the effectiveness of the active methodology of Learning Based on Problems in the teaching of Probability in High School, with a view to developing the competences and abilities foreseen in the Common National Curricular Base. For that, we proceeded to a qualitative research through a case study, applying a didactic sequence to a class of 2nd year High School students from a public school in Maceió (AL). In order to verify the effectiveness of the didactic sequence and the development of competences and skills provided for in the National Common Curricular Base, we created analysis tools to identify the types of errors and skills and competences developed. The results demonstrated the need for teachers to know the PBL to use it in classes, in addition to understanding that its application differs in several aspects from application in Higher Education, with the need to adapt it Basic Education.

The Differential and Integral Calculus (DIC) course is taught in six of the nine undergraduate courses of the Federal Institute of Education, Science and Technology of the North of Minas Gerais (IFNMG), campus Januária and has a high failure rate compared to other subjects. This paper shows the results of a research which sought to find answers to the main difficulties faced by students and teachers of the DIC course, in relation to the teaching and learning process. In order to obtain the data, questionnaires were applied to the students of the course and to the teachers who teach or have already taught this course on the Campus, as well as monitoring for a group of students. The results show the reality of DIC teaching and learning at Campus Januária and opens the door for discussions about this issue that covers not only the IFNMG, but also other higher education institutions in Brazil.

This paper presents an analysis of a chapter of Luiz Roberto Dante's high school textbook “Mathematics: Context and Applications” on the content of functions, based on some assumptions of the Semiotic Representation Records Theory. The work chosen for the analysis is volume 1 of a collection approved by the National Textbook Program (PNLD 2018). For this, a qualitative study was carried out from Raymond Duval's Theory of Semiotic Representation Records, which aimed to identify and analyze the representation records that the book brings in the explanatory text of the chosen chapter and how it occurs the coordination between them. In the end, we conclude that the way the articulation between the representation records displayed in the book occurs favors the understanding of the mathematical object functions, since the various representation records are used simultaneously when approaching each part of the explanation.

This paper aims to analyze academic research that addresses the inclusion of deaf students and focuses on problem solving. For this, mapping in educational research was used as a methodological procedure. Searches were conducted in Qualis A1 to B2 journals, CAPES thesis and dissertations portal, Academic Google and Scielo, using the keywords 'inclusion of deaf people', inclusion and problem solving. 'and' problem situations'. After filtering the results were found six articles that were part of the corpus of analysis. The results showed that inclusion is not occurring effectively, because the teacher is not yet prepared to attend deaf students and, consequently, teaching Mathematics for these students. It is also worth noting the limited number of researches that deal with the inclusion of deaf people, focusing on problem solving.

This paper presents some results of a professional development course for teachers where the participants studied basic solid geometry and developed applied projects in an interdisciplinary context. The course took place in a Portuguese university, from February to May 2020, and involved 19 teachers of preschool and primary levels (grades 1 to 4). The authors have developed a qualitative case study to evaluate how an interdisciplinary approach to the teaching of solid geometry is perceived, by the mathematics teachers, as a contribution to the teaching/learning process of geometry. Overall, the activities developed have proved to be successful examples of interdisciplinary methodologies. Moreover, the approach followed during the course helped the teachers develop their geometric competences concerning solid geometry in a more consistent appropriation and application of the geometric concepts involved.

This paper describes a study of mathematical argumentation in primary school. The principal aim is to explore the nature of complex argumentation at a structural level. The context of the study was a teaching experiment involving nine tasks that promoted argumentation among fifth graders. We use the framework and method of reconstructing complex argumentation in the classroom proposed by Knipping (2008). The findings show that complex argumentation at a structural level in the context of refuting conclusions is characterized by a source-like structure with the addition of a new refutation argument element.

This paper investigates ten mathematics textbooks distributed between the years of 2013 to 2018 in countryside schools in Brazil within the scope of the Textbook National Program in its version for the Countryside, the PNLD Campo. The analyzes take the Animal Studies as a motto to discuss the subjectivities produced in this context. The Foucaultian theorizations are adopted to examine the discursive networks described and inquired from the use of two qualitative and quantitative analysis software, Atlas TI and Gephi. The articulation of the theorization employed with the software is supported by the analysis of the Connected Concept Analysis. The results show that the animals are positioned in these materials sometimes take them as a pedagogical device, in order to capture students' attention and engagement in their teaching and learning processes in school Mathematics, while at other times they are engaged in a process of exploration of their lives to sustain a rationality that distributes, organizes, and maintains policies of life and death.

In this paper, the effect of holding a math training workshop using GeoGebra software has been studied on the changes on teachers' knowledge and beliefs . The selected sample is 40 male and female teachers in Iran. Before and after the intervention were administered a pre and post questionnaire with two components: TPACK knowledge and teachers’ beliefs. Fuzzy logic and Fuzzy TOPSIS methods were used to analyze the data. The results of this method showed a significant difference between the results before and after the workshop.

In this paper we aim to analyze how a figure proposed as part of the statement of a problem situation about flat Geometry can influence the resolution procedures presented by students entering a Mathematics degree course. This article is a clipping from an ongoing research that deals with teaching and learning of flat Geometry in the initial training of math teachers. In it, we use as theoretical contribution Raymond Duval's Semiotic Representation Registration Theory. We developed a research with qualitative approach of the case study type. As possible results, we observed that the students used mereological, positional and reconfiguration modifications in their resolution procedures. We also observe that in problem situations that contain pictures as part of the utterance, attention must be paid to the purpose of the situation.

In this paper we focus on the cardinal aspect of the natural number, where the number is associated with a set to indicate the amount of elements it has. It is about studying how cardinal thinking is in children in pre-school education. Presenting tasks with the logical-mathematical schemes of the cardinal number, and taking into account that each task corresponds to the child's maturational age, we analyze the strategies followed, as well as the mistakes made by those students who have not overcome this task, achieving all this diagnose the numerical thinking in its cardinal aspect.

The paper situates mathematics teaching-learning processes within a multimodal perspective and discusses a semiotic approach apt to seize this dimension, namely the Semiotic Bundle lens. This analytical tool considers the great variety of semiotic resources through which mathematical meanings emerge and evolve in the classroom, ranging from embodied ones such as gestures, to symbolic systems. In particular, the analysis considers them in a systemic and dynamic way. The theoretical account is illustrated by means of an example on children spatial conceptualization, carried out in kindergarten. The data analysis will constitute a background against which the connections with the Theory of Objectification will be highlighted, showing a constructive resonance between the two theories.

This paper aims to discuss relativism in Mathematics Education and its consequences for learning, especially when the objectivity of Mathematics is put in suspicion, considering the production of knowledge relative to each culture, society and individual. To do so, from a theoretical essay, we analyze the relation between relativism, science and Mathematical Education. Reflecting on the formation of subjects capable of understanding reality in an increasingly elaborate way, we point the school Mathematics as a socializing instance of systematized knowledge. Our analysis seeks to show that objectivity has always been present in the construction of science and in this sense we highlight the objectivity of mathematics in the philosophy of language, clarifying how it is intertwined with language, allowing an objective knowledge of the world, contributing, in our view, for the debate on the valorization of school knowledge, in particular the mathematical one, so important for the development of any society.

This paper is part of the research line Elements and Mathematics Teaching Methodology and was aimed to analyze the articles on the teaching of mathematics and emphasize aspects related evasion in Engineering courses presented at the Brazilian Engineering Education Congress (COBENGE) in the period between years 2000 to 2014. The choice to analyze the work from the year 2000 is due mainly to the fact that at that time, the enrollment of higher education increased by over 150,0%. The methodological approach that supported the study was quantitative through content analysis, however, the immersion in COBENGE the work was given by a study on the state of the art. Research conducted showed that there is strong relationship between evasion and failure in the disciplines of basic cycle, especially the discipline of Differential and Integral Calculus.

The present paper has as purpose to analyze the potential of an extension course for the professional development of teachers and future mathematics teachers in an inclusive perspective. This study, of qualitative nature, is a cutting of a master research´s, and it is inserted in the field of Teacher Education for Inclusive Mathematics Education and involved: observation, questionnaire, field journal, tape and video records from the meetings and produced by the participants involved like as sources of the datas production. The results show that, the extension course, as well as raising awareness in the participants with relation to the inclusive mathematics, contributed to mobilization of knowledge related to the teaching of mathematics of the students with visual impairments both in relation to the mathematical content and its didactics and related to the organization of the class and production of materials.

This paper presents a study of networking of theories between the theory of registers of semiotic representation (TRSR) and the onto-semiotic approach of mathematical cognition and instruction (OSA). The results obtained show complementarities between these two theoretical perspectives, which might allow more detailed analysis of the students’ performance.

O presente artigo tem por objetivo discutir numa perspectiva contemporânea os conteúdos de Lógica, Matemática, Filosofia da Matemática e História da Matemática presentes no livro A Lógica na Matemática, escrito por Malba Tahan. Para isso, mediante o uso da historiografia, foram selecionados temas concernentes com os assuntos da pesquisa. Foram tratados os seguintes temas: a base lógica da Matemática, a definição de conceito, os princípios para se definir um objeto, as definições e a natureza dos axiomas em Matemática, o método axiomático e as diversas axiomáticas para a geometria euclidiana, a estrutura lógica de um sistema dedutivo, os métodos de demonstração em Matemática, a indução, analogia e dedução em Matemática. Palavras-chave: Lógica Matemática; História da Matemática; Filosofia da Matemática. A TOUR BY THE LABYRINTH OF MATHEMATICAL LOGIC IN THE COMPANY OF MALBA TAHAN Abstract In this paper we discuss the Mathematics, the Logic of Mathematics, the Philosophy and History of Mathematics that presents in the book A Lógica na Matemática of the Malba Tahan, in a contemporary approach. For that, we use the historiography to select matters in adherence with the research. Are treated this topics: the basis of the Logic of Mathematics; the concept definition; principles to define an object; definitions and nature of the axioms in Mathematics; the axiomatic method and the diverse axiomatic to the Euclidean Geometry; the logical structure of a deductive system; demonstration methods in mathematics; the induction, analogy and deduction in mathematics.

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.

This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove.

In this paper, we focus on topological approaches to space and we argue that experiences with topology allow middle school students to develop a more robust understanding of orientation and dimension. We frame our argument in terms of the phenomenological literature on perception and corporeal space. We discuss findings from a quasi-experimental study engaging 9 grades 5-8 students (10-13 years old) in a 6-week series of school-based workshops focused on knot theory. We discuss video data that shows how students engage with the intrinsic disorientation of mathematical knots through the use of gesture and movement.

In this paper we focus on variation of the design and the implementation of a specific task during three mathematics lessons in the 8th grade in a learning study (Marton & Tsui, 2004; Runesson, 2008). The theme of the lesson was division, with a denominator between 0 and 1. The teachers wanted their students to understand that when dividing with a denominator less than 1, the quotient is larger than the numerator. Four teachers collaboratively planned, analyzed and revised three lessons in a cyclic process. The study shows that the implementation of the task changed between the lessons. Although the same task was used in the lessons, the way it was enacted provided different possibilities to learn. RESUMEN En este artículo nos centramos en la variación del diseño y la implementación de una tarea específica durante tres sesiones de clase de matemáticas en octavo grado en un estudio de aprendizaje (Marton y Tsui, 2004; Runesson, 2008). El tema de las clases fue la división con un divisor entre 0 y 1. Los profesores querían que sus estudiantes entendieran que, cuando se divide por un divisor menor que 1, el cociente es mayor que el numerador. Cuatro profesores colaboraron, en un proceso cíclico, en la planificación, análisis y revisión de las tres sesiones de clase. El estudio muestra que la implementación de la tarea cambió entre las sesiones. A pesar de utilizarse la misma tarea en las sesiones, la manera en que se implementó proporcionó diferentes posibilidades para aprender.

A recent discussion involves the elaboration on possible design principles for sequences of tasks. This paper builds on three principles, as described by Bokhove and Drijvers (2012a). A model with ingredients of crises, feedback and fading of sequences with near-similar tasks can be used to address both procedural fluency and conceptual understanding in an online environment. Apart from theoretical underpinnings, this is demonstrated by analyzing a case example from a study conducted in nine schools in the Netherlands. Together with quantitative results of the underlying study, it is showed that the model described could be a fruitful addition to the task design repertoire. RESUMEN Una discusión reciente implica la elaboración de posibles principios para el diseño de secuencias de tareas. Este documento se basa en tres principios, descritos en Bokhove y Drijvers (2012a). Un modelo que comprende las componentes de crisis, realimentación y desvanecimiento de secuencias con tareas muy similares puede ser utilizado para abordar tanto la fluidez procedimental como la comprensión conceptual en un entorno en línea. Además de estar fundamentado teóricamente, esto se demuestra mediante el análisis de un ejemplo de caso de un estudio realizado en nueve centros educativos de los Países Bajos. Junto con los resultados cuantitativos del estudio subyacente, se muestra que el modelo descrito podría ser una incorporación útil en el repertorio del diseño de tareas.

En este trabajo se exponen los resultados parciales de una investigación sobre el conocimiento profesional de los profesores evaluadores de olimpiadas matemáticas puesto en acción al analizar errores relativos al álgebra. A través de dos pruebas de interpretación de errores, y a partir de un análisis inspirado en la teoría emergente de los datos, obtuvimos dimensiones para la caracterización de saberes en la comprensión de los errores. En los resultados describimos los saberes de un profesor estrechamente relacionados con la práctica y con el conocimiento sobre el aprendizaje de los alumnos. ABSTRACT This paper presents the partial results of a research on the professional knowledge of mathematics olympiad evaluators teachers put into action when analyzing errors relating to algebra. By means of two tests on errors interpretation and by using analysis inspired by the grounded theory, we obtained dimensions for the characterization of knowledge in understanding about errors. In the results section, we describe a teacher's knowledge related to practice and knowledge about students’ learning.

Even though statistical thinking and critical thinking appear to have strong links from a theoretical point of view, empirical research into the intersections and potential interrelatedness of these aspects of competence is scarce. Our research suggests that thinking skills in both areas may be interdependent. Given this interconnection, it should be possible to stimulate both forms of thinking through the one task. This paper explores the implications of an exploratory qualitative study into processes when working on tasks encompassing both these areas for the design of tasks that simultaneously stimulate critical thinking and domain-specific thinking. Resumen Aunque desde un punto de vista teórico, el pensamiento estadístico y el pensamiento crítico parecen tener fuertes conexiones, la investigación empírica sobre la intersección y la potencial interrelación entre estos aspectos de competencia es escasa. Nuestra investigación sugiere que las habilidades de pensamiento en ambas áreas pueden ser interdependientes. Dada esta interconexión, debería ser posible estimular ambas formas de pensamiento mediante una tarea. Este artículo explora las implicaciones de un estudio cualitativo exploratorio sobre los procesos de pensamiento en tareas que abordan ambas áreas para el diseño de tareas que simultáneamente estimulan el pensamiento crítico y pensamiento específico de un dominio.

The aim of this paper is to show how engaging students in real-life mathematical situations can stimulate their mathematical creative thinking. We analyzed the mathematical modeling of two girls, aged 10 and 13 years, as they worked on an authentic task involving the selection of a track team. The girls displayed several modeling cycles that revealed their thinking processes, as well as cognitive and affective features that may serve as the foundation for a methodology that uses model-eliciting activities to promote the mathematical creative process. RESUMEN El objetivo de este artículo es mostrar cómo involucrar a los estudiantes en situaciones matemáticas de la vida real puede estimular su pensamiento matemático creativo. Analizamos la modelización matemática de dos chicas, de 10 y 13 años, cuando trabajaban en una tarea auténtica que involucraba la selección de un equipo de atletismo. Las chicas mostraron varios ciclos de modelización que revelaron sus procesos de pensamiento, así como las características cognitivas y afectivas que pueden servir como fundamento para una metodología que usa actividades que suscitan modelos para promover los procesos matemáticos creativos.

Este trabajo trata sobre el concepto de función, básico en el Análisis Matemático, y, en particular, su representación gráfica. Nos centramos en aspectos relacionados con la forma; es decir, el trazado de dicha representación. Analizamos las representaciones gráficas de funciones existentes en los cuadernos de matemáticas de estudiantes de varias aulas de 1º de Bachillerato. Encontramos deficiencias en el trazado de gráficas que se repiten en un alto número de estudiantes, relacionadas con los conceptos de función y asíntota, con el uso de las escalas en los ejes del diagrama cartesiano y con las características de algunas funciones. Además, discutimos sobre las limitaciones técnicas y las dificultades didácticas y cognitivas que pueden dar lugar a su aparición y hacemos algunas recomendaciones didácticas al respecto. ABSTRACT This paper deals with the concept of function, basic in mathematical analysis, and, in particular, with its graphical representation. We focus our attention on plotting graphs of functions. We analyzed the graphical representations of functions found in mathematical notebooks of high school students. We encountered several deficiencies related to the concepts of function and asymptote, the use of scales in diagram axes and the characteristics of some functions. Besides, we discuss the technical limitations and the didactic and cognitive difficulties that may promote their emergence, and, make some didactic recommendations for teachers.

Este trabajo describe tendencias detectadas en las actitudes de estudiantes de ingeniería de nuevo ingreso hacia el uso de la tecnología en la enseñanza y aprendizaje de las matemáticas, mediante opiniones emitidas en un cuestionario sobre actitudes diseñado para tal efecto. Se describe el proceso seguido para establecer categorías para las actitudes hacia el uso de tecnología puestas de manifiesto por los estudiantes en sus respuestas. Los resultados indican una tendencia de actitudes positivas.This paper describes trends detected in first year engineering students’ attitudes towards the use of technology in teaching and learning mathematics through opinions gathered in a survey about attitudes designed for this purpose. We describe the process followed to establish categories for the attitudes towards the use of technology evidenced by the students in their answers. The results indicate a trend of positive attitudes.

In this paper, we report some findings from an investigation of a topic related to affect and mathematics which is not well-represented in the literature. For some mathematicians, mathematics itself is a source of security in an uncertain world, and we investigated this feeling and experience in the case of 19 adult mathematicians working in universities and schools in Greece. The focus reported here is on ways that a relationship with mathematics offers a sense of permanence and stability on the one hand, and an assurance of novelty and progress on the other. En este trabajo presentamos algunos resultados de una investigación sobre un tema relacionado con el afecto y las matemáticas que no está bien representado en la literatura. Para algunos matemáticos, las matemáticas en sí mismas son una fuente de seguridad en un mundo incierto; nosotros investigamos este sentimiento y experiencia en el caso de 19 matemáticos adultos que trabajan en universidades y escuelas de Grecia. El foco de este artículo es en las maneras en que una relación con las matemáticas ofrece un sentido de permanencia y estabilidad por una parte, y garantía de novedad y progreso, por otra.

In this paper, we present an emerging set of learning conjectures and design principles to be used in the development of professional learning tasks that support elementary teachers’ learning of mathematics learning trajectories. We outline our theoretical perspective on teacher knowledge of learning trajectories, review the literature concerning mathematics professional learning tasks, offer a set of initial conjectures about teacher learning of learning trajectories, and articulate a set of principles to guide the design of tasks. We conclude with an example of one learning trajectory professional learning task taken from our current research project. En este artículo, presentamos un conjunto emergente de conjeturas de aprendizaje y de principios de diseño para ser empleados en el desarrollo de tareas de aprendizaje profesional que apoyan el aprendizaje de trayectorias de aprendizaje de matemáticas de maestros de primaria. Describimos brevemente nuestra perspectiva teórica sobre el conocimiento del profesor acerca de trayectorias de aprendizaje; revisamos la literatura sobre tareas de aprendizaje profesional, presentamos un conjunto de conjeturas iniciales acerca del aprendizaje del profesor sobre trayectorias de aprendizaje; y articulamos un conjunto de principios para guiar el diseño de tareas. Concluimos con un ejemplo de una tarea de aprendizaje profesional que ha sido tomada de nuestro proyecto de investigación actual.

This paper addresses contributions that dynamic geometry systems (DGSs) may give in reasoning by contradiction in geometry. We present analyses of three excerpts of students’ work and use the notion of pseudo object, elaborated from previous research, to show some specificities of DGS in constructing proof by contradiction. In particular, we support the claim that a DGS can offer “guidance” in the solver’s development of an indirect argument thanks to the potential it offers of both constructing certain properties robustly, and of helping the solver perceive pseudo objects.Este artículo aborda las contribuciones que los sistemas de geometría dinámica (DGSs) pueden dar al razonamiento por contradicción en geometría. Presentamos un análisis de tres extractos del trabajo de estudiantes y el uso de la noción de pseudo-objeto, elaborado a partir de investigaciones anteriores, para mostrar algunas especificidades del DGS en la construcción de pruebas por contradicción. En particular, afirmamos que un DGS puede orientar en el desarrollo de un argumento indirecto gracias a las posibilidades que ofrece tanto para construir sólidamente algunas propiedades como para ayudar a percibir los pseudo-objetos.

The Early Years Generalising Project (EYGP) involves Australian Years 1-4 (age 5-9) students and investigates how they grasp and express generalisations. This paper focuses on data collected from six Year 1 students in an exploratory study within a clinical interview setting that required students to identify function rules. Preliminary findings suggest that the use of gestures (both by students and interviewers), self-talk (by students), and concrete acting out, assisted students to reach generalisations and to begin to express these generalities. It also appears that as students become aware of the structure, their use of gestures and self- talk tended to decrease.El Early Years Generalizing Project (EYGP) implica a estudiantes de primer a cuarto curso de la educación primaria australiana (de 5 a 9 años) e investiga cómo comprenden y expresan las generalizaciones. Este artículo se centra en los datos recogidos de 6 estudiantes de primer curso en un estudio exploratorio con entrevista clínica que requería que los estudiantes identificaran patrones funcionales. Los resultados preliminares sugieren que el uso de los gestos (de estudiantes y entrevistadores), las conversaciones con ellos mismos (de estudiantes), y las actuaciones concretas, ayudaron a los estudiantes a buscar generalizaciones y a comenzar a expresar estas generalidades. También parece que cuando los estudiantes tomaron conciencia de la estructura, el uso de gestos y de las conversaciones con ellos mismos tendió a disminuir.

This paper extends existing research regarding content knowledge for teaching (CKT) and role it plays in advancing student learning. Two teachers (with high and low measured CKT) are observed on the same day teaching similar content. Many studies have recently been published linking student achievement to teacher’s CKT and many U.S. schools have begun including CKT measures in teacher hiring and retention decisions. Teaching observed for this study illustrates that content can be taught effectively by teachers across the spectrum of CKT levels, but observable and significant differences in teaching leads to important questions for inservice and preservice teacher educators.Este artículo amplía la investigación existente sobre el conocimiento del contenido para la enseñanza (CKT) y su papel en el progreso del aprendizaje del alumno. Dos profesores, con medidas altas y bajas de CKT respectivamente, fueron observados el mismo día enseñando contenidos similares. Numerosos estudios publicados recientemente relacionan el logro de los estudiantes con el CKT del profesor y muchos colegios estadounidenses han empezado a considerar medidas de CKT en la toma de decisiones para la contratación y permanencia de profesores. La enseñanza observada en este estudio muestra que el contenido puede ser enseñado efectivamente por profesores con niveles distintos de CKT. No obstante, se observan diferencias significativas en la enseñanza que conducen a cuestiones importantes para formadores de profesores en formación y en ejercicio.