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From the Mathematical Didactics’ viewpoint about teaching Calculus, this paper aims to characterize the knowledge and mathematical work that emerges when high school students solve tasks about the rate of change mediated by GeoGebra. Based on the theory of Mathematical Working Spaces, we analyze the mathematical production of Chilean students (16- 17 years old) and conclude that making sense of the change rate starts from a work based on algorithmic processes and an interpretation/experimentation approach to the idea of variation and change emphasizing the role of the artefact, being particular to establish proof and demonstration processes. [ABSTRACT FROM AUTHOR]

This paper analyzes the treatment of the random variable and its applications on probability distributions in Chilean high school textbooks, from the Ontosemiotic Approach to Mathematical Knowledge and Instruction. Thus, a model is used for the analysis of mathematical objects in the Chilean textbooks and the Chilean school curriculum by means of content analysis. The results show a diversity of languages, concepts, propositions, procedures and arguments linked to the topics, although some of them are identified in the curriculum and excluded in books or inversely, showing a lack of coherence between the documents. [ABSTRACT FROM AUTHOR]

In this paper we present the results of an investigation carried out in Higher Education mathematics classes, in which we evidenced students' actions when develop experimentation in mathematical modeling activities. Our analyzes are supported in the development of two activities by a group of five students in a discipline of Differential and Integral Calculus of the Degree in Chemistry at a Brazilian university. The qualitative analysis inspired by Research Design of the speeches, gestures and written reports of the students allowed us to highlight differences in actions when the theme is suggested by the teacher and when it is chosen by the students, as a way of "dealing" with phenomenon and concepts mathematicians in experimentation. [ABSTRACT FROM AUTHOR]

This paper establishes semantic conceptions for the terms mathematical thinking and mathematical culture, which are present in the Theory Mathematics in the Context of Sciences. We used the method called comparative constant. This is fundamental to avoid misconceptions or carry out contradictory analyses in regard to the precepts of the referential adopted. The acknowledgement of the elements that characterize these conceptions can enable the design and implementation of adequate programs of Mathematics to develop mathematical thinking, as well as mathematical culture in students of Higher Education. [ABSTRACT FROM AUTHOR]

This paper shows the mathematical connections that university students make when solving tasks involving first-order Ordinary Differential Equations. We consider mathematical connections as a process by which a person relates one or more ideas, concepts, representations, theorems, or meanings to each other, to those in other disciplines or to real-life situations. For data collection, task-based interviews were used, which were analyzed with the Thematic Analysis method. The results indicate the use of seven types of mathematical connections: different representations, procedural, meaning, characteristic, part-whole, implication and reversibility. However, it was observed that a high academic performance is not an indicator that the connections emerge adequately in the resolution of tasks. [ABSTRACT FROM AUTHOR]

The knowledge of Mathematics teachers has been a very prominent focus of attention in the last decades. However, it leaves aside one of the dimensions involved in the development of this type of knowledge, specifically the knowledge of Mathematics teacher educators. In this paper, we discuss a mathematics teacher educator's knowledge in the context of classes on Euclid's division algorithm theorem in a Number Theory course for prospective secondary teachers. Some indicators of this specialized knowledge of mathematics teacher educators are presented and discussed. El conocimiento de los profesores de matemáticas ha sido un foco de atención muy activo en las últimas décadas. Sin embargo, tales focos dejan de lado una de las dimensiones involucradas en el desarrollo de dicho conocimiento, específicamente el conocimiento del formador de profesores de matemáticas. En este artículo, discutimos el conocimiento de un formador en el contexto de un curso de Teoría de números, para futuros profesores de secundaria, al probar el teorema del algoritmo de división de Euclides. Se presentan y se discuten algunos indicadores de este conocimiento del formador. [ABSTRACT FROM AUTHOR]

In this paper, we carry out the research and the analysis of phenomena organized by the infinite limit of a sequence in 35 Spanish textbooks of Mathematics published from 1936 to 2019. From a construction of an instrument based on the phenomena characterized by an infinite limit of a sequence and the analysis of the information collected by it, we establish the identification of these phenomena in a different way of representation and formats. The historical evolution of these phenomena and the comparison between some of legislative periods shows these phenomena are presented in an isolated way without using different systems of representation and formats, making difficult with it their teaching-learning process. [ABSTRACT FROM AUTHOR]

Competencies training implies active learning that requires motivation, the ability to formulate critical judgments, and the ability to know how to learn. This paper shows the impact of an Adaptive Tutoring System in the development of three mathematical competencies, and its effect on Mexican undergraduates' attitudes towards mathematics and attitudes towards mathematics learned with a computer. Results showed a greater development of the mathematical competencies in the experimental group than in the control one, as well as a significant difference between the groups in the attitude towards the mathematics learned with the computer. No differences reported in the attitude towards mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the arguments given by teachers in training by justifying the truthfulness of statements referring to the derivability of a function at one point. Using an adaptation to Toulmin's model, we focus on the guarantee or justification given and whether support is presented. We identify the mathematical structure on which the justification is based, how it is used and how the argument is represented. Findings reveal that, to argue, mathematical results or rules used are often used without backing. Likewise, we detect invalid arguments, mainly because the necessary conditions for derivability are not considered. [ABSTRACT FROM AUTHOR]

The first part of this paper provides a brief analysis of the historicalepistemological component of the measure and, additionally, different strategies and specific resources to promote situations that facilitate interaction, negotiation, and dialogue in the classroom are described. The second part presents a single case study showing a specific example of the teaching of mass in a context of experimentation and communication. The results show that the formulation of good questions produces a significant effect on the construction of mathematical knowledge. In view of this, it is essential that teachers progressively acquire skills to formulate good questions. [ABSTRACT FROM AUTHOR]

This paper deals with the specialized knowledge of pre-service secondary Mathematics teachers from Peru, about the conceptualization of quadrilaterals, using the MTSK model. We focus on understanding what their knowledge is like when they define quadrilaterals in an openresponse questionnaire, plan a lesson and execute it. The research is an instrumental case study belonging to a broader interpretative investigation. They propose descriptive and partitional definitions, without questioning the necessity and sufficiency of the properties that they include. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we study the interactions between two mathematics teachers of high school and resources using the theoretical framework of documentational approach. According to this approach, these interactions produce documents associating resources and schemes of use. These schemes comprise professional knowledge described in terms of operational invariants. Teachers work together to prepare the teaching on tolerance intervals. These two teachers constitute a theory of practice. We show that, in spite of the collective work, the documents developed by each teacher always comprise different elements, because their initial operational invariants do not evolve enough to lead to common documents. [ABSTRACT FROM AUTHOR]

In this paper we categorise students' errors when solving linear equations and we identify the causes for those errors. The research has involved 266 13-16 years old high school students. Data from a questionnaire on linear equations solving and cognitive interviews are reported and analysed. Results complete and reformulate other error categorizations found in the literature and deepen in the identification of the origin of these errors. [ABSTRACT FROM AUTHOR]

This paper describes how to establish relationships between the subdomains of the Mathematics Teacher's Specialised Knowledge model and the conceptions of the mathematics teaching, according to the Conceptions of Teaching-Learning Mathematics model. We present a methodology for identifying and analyzing these relationships, which has allowed to understand the practice in her classroom of one fifth grade primary teacher. As a result, the description of the indicator "conception of school mathematics" and the established relationships are provided. Finally, the potential use of the proposed instrument is discussed. [ABSTRACT FROM AUTHOR]

In this paper we examine some of the most common terms used in the theory of knowledge objectification to carry out a philosophical/epistemological analysis based on several meanings suggested by different authors. The objective is to contribute to a clarification that does not want to be only terminological, but also meaningful of the objectification, knowledge and ontology, knowledge and gnoseology, work, semantics and communication. [ABSTRACT FROM AUTHOR]

In this paper we analyze the results and processes used by third-year secondary students and prospective elementary teachers in estimation tasks of continuous quantities of length, surface, capacity and mass. The results show no differences between both groups of students. Both are good estimators of length and surface but underestimate capacity and mass. No big differences are detected in the processes and strategies used, except a higher and more effective use of formulas in the case of prospective teachers. [ABSTRACT FROM AUTHOR]

In this paper, we study the answers that 285 prospective elementary teachers provide in Chile about the constructs self-confidence, motivation and anxiety, which are considered personal-dimension descriptors of students' attitudes towards mathematics. We used a questionnaire that considers the psycho-social component of the attitudes and its multidimensional nature, considering the personal, social, professional, and institutional dimensions. We note that motivation has a higher positive score, while the scores of selfconfidence and lack of anxiety are positive and closer to zero. [ABSTRACT FROM AUTHOR]

In this paper we reflect on two components of number sense: The use of benchmarks and the use of graphical representation to estimate amount or the result of a calculation (with whole numbers, decimals and fractions). The results of a written test answered by 47 high school students are presented, in whose analysis the success and the strategies were examined. Results indicate difficulties for students when making use of numerical benchmarks and for building their own graphical representations, preferring rule based methods. Better results were obtained in numerical estimate from graphs already given. The lowest test results correspond to tasks in which fractions are involved. [ABSTRACT FROM AUTHOR]

Promoting the elaboration of students' mathematical knowledge and understanding requires teachers' having a knowledge allowing connecting students' prior and future knowledge. In this paper we focus on aspects of the mathematical horizon as one of teachers' knowledge dimensions, presenting an interpretation of such knowledge, with a practice-based approach, grounded in the analysis of three levels of teachers' practices. Using examples from tasks in geometry and measurement in elementary school practices, some indicators are presented and discussed with the aim of describing and analyzing how teachers' horizon knowledge allows enriching teaching practices. [ABSTRACT FROM AUTHOR]

Mathematics Teacher's Specialised Knowledge Detected in the Circle Chord Problem Solution In this paper we show an analysis of the knowledge that evidences a secondary teacher to solve the circle chord problem (if there are n points in a circumference, is it possible to determine the number of all the possible cords?) using the analytical model of professional knowledge: mathematics teacher's specialised knowledge (MTSK). The results show the potential of the model as an analytical tool to deep in the comprehension and characterization of the mathematics teacher's knowledge, especially the knowledge of topics. [ABSTRACT FROM AUTHOR]

Empirical evidence is provided in the paper to support the idea that the conviction experienced by the teacher with respect to the mathematics contents that arise in class --in interaction with his level of comprehension- has an impact on the mathematics practices of sustentation that the teacher promotes. The empirical data, that come from an exploratory case study, undertaken in a natural classroom scenario are analyzed using an instrument to identify mathematics convictions. This instrument was designed within the framework of the research presented here. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we present some reflections on the application of the technique of content analysis in the study of historical texts in mathematics. This work is based on a study conducted in 2009 on the treatment of the metric system in mathematics texts in Spain in which a selection of texts published in the second half of the nineteen century was analyzed. The study provided the identification of three stages that characterize the texts used for the dissemination of new Spanish weights and measures in the selected period. [ABSTRACT FROM AUTHOR]

Spain produces a large amount of research in mathematics education and its results are presented in the symposia of the Spanish Society for Research in Mathematics Education (SEIEM). This paper presents a study of the proceedings of these symposia. Its aim is to identify some bibliometric indicators and networks of institutional collaboration and co-authorship. The analysis was performed with 268 SEIEM documents. We found a low level of scientific collaboration between researchers in the field. Similar results are obtained for international cooperation. [ABSTRACT FROM AUTHOR]

We present the results of an exploratory study whose purpose was to explore the use by 12-13 years old students of Cabri-Géomètre II and a mathematical strategic virtual game in a Mexican state school. For this paper, the mathematical focus was on symmetry. The data analysis allowed us to observe the functionality of this learning environment to transform the students' conceptions and to recognize that a different type of mathematical thinking can be developed with these computational learning environments. [ABSTRACT FROM AUTHOR]

This paper presents the results of a research project about the communication between families and schools. The objective is to identify the mathematical contents taught in current schools as well as to build dialogue bridges between schools and families, in order to improve children's performance in mathematics. First we introduce the context of the study. Then, we present the study and the methodology. After that, we discuss part of the results obtained, which highlight the interest of the connection between families and schools, particularly at High School. We end with some contributions to teacher training in mathematics. [ABSTRACT FROM AUTHOR]

In this paper we present a developmental model of ordinal competence which consists of 6 empirically validated states of knowledge. An exploratory empirical study was carried out with individual clinical interviews. The aim was to analyze children's knowledge about the logicalordinal relationship of the immediately next term. The results obtained served to determine the developmental stages of the model. The model was validated through a test comprising 6 tasks, each of which presented the mathematical and logical characteristics of a given state of the model. [ABSTRACT FROM AUTHOR]

In this paper we propose specific meanings for the notions of capacity and competence in the context of a Secondary mathematics teacher training program. We briefly describe this program, and present a procedure with which prospective teachers can reflect on students' learning in the classroom and use these notions for planning didactical units. [ABSTRACT FROM AUTHOR]

In this paper we explore the patterns and the generalization developed by 359 students in years 9 and 10 in the resolution of the tiles problem. We pay special attention to the kinds of patterns identified, to the written ways in which students express generalization and, using inductive strategies, we present some characteristics of the generalization relating to the elements and the representations used. [ABSTRACT FROM AUTHOR]

One of the components of the methodological and theoretical framework that Filloy named "Local Theoretical Models" is the description of the competence in the domain of which teaching and learning are going to be researched. In this paper we show in which sense the term competence is used in this framework and provide examples of the construction of competence models in some investigations. This leads us to examine the relationship between the competence in a mathematical domain and the phenomenological analysis of this domain. Incidentally, we study the use of the term competence in several PISA documents. [ABSTRACT FROM AUTHOR]

This paper shows some partial results of an empirical study on the development of multiplicative thinking in the first school levels. In particular, we posed manipulative situations to children of third course of second cycle of Kindergarten (five years). These situations are constituted by several divisions problems that could not be solved by sharing, as well as some questions on (proportional) relational thinking. [ABSTRACT FROM AUTHOR]

In this paper we characterize the term reflexive mathematics teacher in order to elucidate teacher training programs and analyze teachers' development and its relation to their professional knowledge. We describe some of the research variables that are used to characterize and train reflective mathematics teachers. With this purpose, we introduce some theoretical considerations and present some of the research projects we are currently working on concerning the reflection processes of preservice teachers in a methods course of the mathematics degree. [ABSTRACT FROM AUTHOR]

In this paper we show the presence of the phenomena of approximation organized by a definition of limit in the cases of the successions of real numbers and the real functions of one real variable. The exposition includes the characterization of such phenomena, a description of the comparative analysis, developed using them, between two formal definitions of limit of succession and function, and a synthesis of the study carried out on an intentional sample of mathematics textbooks. [ABSTRACT FROM AUTHOR]

This paper is related to three main issues: (a) preservice teacher training and mathematics teacher competences, (b) preservice teachers' knowledge of mathematics for teaching and learning, and (c) knowledge established by Mathematics Education and didactical analysis. I explain briefly these three points and then I describe the particular way in which we address the transformation of formal mathematics into school mathematics in a methods course in the University of Almería. [ABSTRACT FROM AUTHOR]