Onto-semiotic Approach Research Articles

Practices of the Random Variable Proposed in the Chilean Mathematics Curriculum of Secondary Education

Working with statistical data from real contexts has become fundamental in school-level statistics, because it enables the development of statistical reasoning. In this regard, the notion of the random variable is fundamental to statistical data analysis. Thus, the aim of this research was to characterise the meanings of the concept of the random variable that are promoted in the Chilean mathematics curriculum for secondary education. To achieve this, we examined the representativeness of the meanings of the random variable intended by the curriculum in relation to the meanings of reference for this concept, using theoretical and methodological notions from the onto-semiotic approach. The findings show that the meanings of reference were not fully represented in the textbooks and the national curriculum, and that the most promoted meaning was that of the random variable as a variable of interest. Regarding the types of representation, it was observed that the graphical and tabular representations were neglected. Lastly, it was noticed that definitions of the concept of variable, its classifications, and differences were omitted almost entirely from the textbooks.

Inferential Reasoning of Secondary School Mathematics Teachers on the Chi-Square Statistic

Statistics education has investigated how to promote formal inferential reasoning from informal inferential reasoning. Nevertheless, there is still a need for proposals that explore and progressively develop inferential reasoning of students and teachers. Concerning this, the objective of this article is to characterize the inferential reasoning that secondary school mathematics teachers show in the practices that they develop to solve problems regarding the Chi-square statistic. To achieve this, we use theoretical and methodological notions introduced by the onto-semiotic approach of mathematics knowledge and instruction. In particular, we have taken a theoretical proposal of levels of inferential reasoning for the Chi-square statistic. Based on the results, the main conclusion was that the proposal above effectively predicted the teachers’ practices, allowing us to distinguish characteristic elements of the levels of inferential reasoning.

Norms That Regulate the Theorem Construction Process in an Inquiry Classroom of 3D Geometry: Teacher’s Management to Promote Them

This paper aims to illustrate how a teacher instilled norms that regulate the theorem construction process in a three-dimensional geometry course. The course was part of a preservice mathematics teacher program, and it was characterized by promoting inquiry and argumentation. We analyze class excerpts in which students address tasks that require formulating conjectures, that emerge as a solution to a problem and proving such conjectures, and the teacher leads whole-class activities where students’ productions are exposed. For this, we used elements of the didactical analysis proposed by the onto-semiotic approach and Toulmin’s model for argumentation. The teacher’s professional actions that promoted reiterative actions in students’ mathematical practices were identified; we illustrate how these professional actions impelled students’ actions to become norms concerning issues about the legitimacy of different types of arguments (e.g., analogical and abductive) in the theorem construction process.

Mathematical Connections Activated in High School Students’ Practice Solving Tasks on the Exponential and Logarithmic Functions

The current paper aims to identify the mathematical connections activated by 10 Mexican high school students while solving mathematical tasks that involve the exponential and logarithmic function. We used the Expanded Mathematical Connections Model (EMCM) and the Onto-Semiotic Approach of Cognition and Mathematical Instruction (OSA) as theoretical frameworks. Task-based interviews were used to collect data that was analyzed using thematic and onto-semiotic analyses. It was found that the connection of reversibility is essential for achieving students’ full understanding of the existent relationship between the exponential and logarithmic function; however, this requires a network of connections.

Inferential Statistical Reasoning of Math Teachers: Experiences in Virtual Contexts Generated by the COVID-19 Pandemic

The COVID-19 pandemic generated a new scenario in education, where technological resources mediate teaching and learning processes. This paper presents the development of a virtual teacher training experience aimed at promoting inferential reasoning in practicing and prospective mathematics teachers using inference problems on the Chi-square statistic. The objective of this article is to assess the implemented or intended institutional meanings and the degree of availability and adequacy of the material and temporal resources necessary for the development of the training experience. For this purpose, we use theoretical and methodological notions introduced by the Ontosemiotic Approach to Mathematical Knowledge and Instruction (OSA), among which are the notions of practice and suitability criteria. The participants of this experience were divided into three groups; one of them was comprised of practicing teachers and the other two of prospective teachers. The intervention used different virtual modalities that enabled the development of the participants’ inferential reasoning in a similar way.

Students’ Obstacles in Learning Sequence and Series Using Onto-Semiotic Approach

Sequences and series is one of the mathematical topics that are related to everyday life. The topic is also taught at several levels of education in Indonesia. However, many students still experrienced difficulties in learning this topic. This study uses an interpretive paradigm that is part of the Didactical Design Research (DDR). This research aims to analyze students’ learning obstacles on the topic of sequence and series using the onto-semiotic approach. To do so, written test consists of five questions related to the conceptual understanding of an arithmetic sequences and series was administered to 23 students from one of the senior high schools in Kota Tangerang Selatan followed by interviews with 4 students. The results show that learning obstacles are classified into epistemological, ontogenic, and didactical obstacles. Based on the onto-semiotics approach, the students had difficulties in defining a mathematical idea on sequences and series topics. They could convert a problem into mathematical model but were confused to use a proper procedure. It can be concluded that students still experience obstacles in learning sequences and series topic. The results of this study can be used by teachers as considerations in designing learning situation on the topic of sequence and series.

Chi-square Reference Meanings: a Historical-epistemological Overview

The present article shows a historical-epistemological study on the Chi-square statistic. In which theoretical-methodological notions from the Onto-Semiotic Approach (OSA) of mathematical cognition and instruction were used to identify four problems that have been key to the evolution of the Chi-square statistic: the Goodness-of-fit-test, the test of independence, the test of homogeneity and distribution. Furthermore, various meanings of the Chi-square statistic were recognized in the mathematical-statistical practices that are used to solve each of those problems. These meanings could help to establish epistemic criteria that allow, on the one hand, to propose progressive levels of inferential reasoning for the statistic (from informal to formal); and on the other hand, to design tasks oriented to promote the understanding of the diverse meanings of the Chi-square.

Analysing theories of meaning in mathematics education from the onto-semiotic approach

The notions of meaning and sense, which are closely related to understanding, play an essential role in educational processes in general and, therefore, in the teaching and learning of mathematics. However, the terms ‘meaning’ and ‘sense’ are unevenly used in the theory of language, semiotics, philosophy and psychology. After describing the main characteristics of the referential and pragmatic theories of meaning, in this article, we detail various approaches to the notion of meaning, and present the perspective suggested by the onto-semiotic approach (OSA) of mathematical knowledge and instruction. This proposal partially incorporates the semiotic, ontological and epistemological assumptions of three general semiotic theories (by Hjelmslev, Peirce and Wittgenstein), with a holistic approach to the notion of object, sign and meaning, and takes into account some referential and operational theories, as well as other cognitive and sociocultural theories, about meaning. We also analyse how the meaning of mathematical objects is addressed in three models that have an impact on mathematics education: Frege's logical-realist position, Vergnaud's conceptual triplet and Steinbring's epistemological triangle. We finally suggest some implications for instructional design and teacher education of the OSA holistic approach to the meaning.

Onto-semiotic complexity of the Definite Integral

Teaching and learning Calculus concepts and procedures, particularly the definite integral concept, is a challenge to teachers and students in their academic careers. In order to develop an informed plan for improving instructional processes, it is necessary to pay attention to the nature and complexity of the mathematical features of the definite integral, that students are expected to understand and apply. In this research, we supplement the analysis made by different authors, applying the theoretical and methodological tools of the Onto-Semiotic Approach to mathematical knowledge and instruction. The goal is to understand the diverse meanings of the concept of the definite integral and potentials semiotic conflicts based on the given data. We focus attention on a first intuitive meaning, which involves mainly arithmetic knowledge, and the definite integral formal meaning as Riemann’s sums limit predominantly in the curricular guidelines. The recognition of the onto-semiotic complexity of mathematics objects is considered as a key factor in explaining the learning difficulties of concepts, procedures and its application for problem-solving, as well as to make grounded decisions on teaching. The methodology analysis of a mathematical text, which we exemplify in this work applying the tools of Onto-Semiotic Approach, provides a microscopic level of analysis that allows us to identify some semiotic-cognitive facts of didactic interest. This also allows for the identification of some epistemic strata, that is, institutional knowledge that should have been previously studied, which usually goes unnoticed in the teaching process.

Mathematical connections from a networking of theories between extended theory of mathematical connections and onto-semiotic approach

This article presents a networking of two theories, the Extended Theory of Mathematical Connections (ETC) and the Onto-semiotic Approach (OSA). In particular, concordances and complementarities are identified in the respective conceptions of mathematical connections, as a result of applying these theoretical frameworks to a student’s response protocol when solving a task on the derivative. First the mathematical connections were identified from the perspective of the ETC and, second, an onto-semiotic analysis was carried out where the mathematical practices, the primary mathematical objects activated in them and the semiotic functions established between these objects were analysed. The main conclusion is that both theories complement each other to make a more detailed analysis of the mathematical connections. In particular, the more detailed analysis carried out with the OSA tools, visualizes a mathematical connection like the tip of an iceberg of a conglomerate of practices, processes, primary objects activated in these practices and semiotic functions that relate them, which enables a thorough analysis that details the structure and function of the connection.

Epistemic Criteria for Designing Limit Tasks on a Real Variable Function

Abstract This article aims at presenting the results of a historical-epistemological study conducted to identify criteria for designing tasks that promote the understanding of the limit notion on a real variable function. As a theoretical framework, we used the Onto-Semiotic Approach (OSA) to mathematical knowledge and instruction, to identify the regulatory elements of mathematical practices developed throughout history, and that gave way to the emergence, evolution, and formalization of limit. As a result, we present a proposal of criteria that summarizes fundamental epistemic aspects, which could be considered when designing tasks that allow the promotion of each of the six meanings identified for the limit notion. The criteria presented allow us to highlight not only the mathematical complexity underlying the study of limit on a real variable function but also the richness of meanings that could be developed to help understand this notion.

Affective and interactive suitability indicators: an experience in physics from the onto-semiotic approach

The purpose of this research was to propose theoretical elements for the description of the students’ attitudes in the teaching of differential calculus and its applications to physics from the onto-semiotic approach in the affective and interactional suitability. The design of this research with a complementary approach of qualitative and quantitative methods, with predominance of the first one, under the interpretative paradigm. The field modality and a quasi-experimental design to evaluate the applicability of the concepts of differential calculus. The collection techniques were: first, the semi-structured interview, in order to guarantee the depth and density of the information so that it converges in an emerging system of meanings and categories; second, the application of a questionnaire to evaluate the cognitive capacity of students under the onto-semiotic and traditional approach; third, observation through surveys. The findings lead to an instructional process through which mathematical concepts are taught efficiently, the student’s role is valued as being assertive, autonomous, and responsible, while the teacher contributes to the development of emotional support by increasing the student’s motivation, self-confidence, and interest in learning mathematics.

Textbook: an Analysis in the Light of the Ontosemiotic Focus on Rational Numbers

Background: Many teachers consider the textbook the primary guide for the curriculum materialisation. This work analyses the content of the rational numbers in a textbook used in the 7th-grade of elementary school. For this analysis, the theoretical and methodological tools of the Ontosemiotic Approach to Knowledge and Mathematical Instruction (OSA) are used. Objective: Our goal was to understand the level of didactic suitability of the instruction process in the textbook. Design and setting: Thus, we analysed a section of the textbook about rational numbers using the categories described by OSA. Concepts, procedures, problem situations, definitions and arguments used by the authors were also analysed. Results: From the results obtained, we could infer that there are examples of the concepts worked in class, but without the proper definitions and arguments, hindering generalisation. The analysis also allowed us to highlight didactic-mathematical knowledge that can guide the teacher concerning the textbooks' possibilities and limitations, to achieve more didactic suitability in the process of teaching and learning rational numbers. Conclusions: the book under study should not be considered as planning for an instructional process to meet the current curriculum guidelines.

Prospective primary school teachers’ competence for analysing the difficulties in solving proportionality problem

In this paper, we describe the design, implementation and results of a formative intervention, meant to develop prospective primary school teachers’ competence in the analysis of the difficulties emerging in the resolution of proportionality tasks. Solving problems by different methods, identifying the knowledge put at stake in each problem and taking this information into account to recognize the difficulties that pupils may encounter in solving the problems using each strategy are essential aspects of the epistemic and cognitive facets of didactic-mathematical knowledge. The experience has been carried out with a sample of 88 prospective primary school teachers during the third year of their studies, by applying a didactic model that includes work in teams, institutionalization, and assessment of the individual learning achieved. To analyse the participants’ responses, we used some theoretical and methodological tools of the onto-semiotic approach in mathematics education. The identification of pupils’ potential difficulties in addressing problem-solving, based on the objects involved in the mathematical activity, was a challenging task for prospective teachers. The difficulties most frequently identified by the prospective teachers were those concerned with understanding the statement requirements, or the problem-situations context, and the mathematical procedures involved in the resolution of the task. They did not discriminate the difficulties according to the resolution strategies. Time constraints conditioned the degree of learning achieved. We conclude that it is necessary that teacher education programs consider this type of didactical analysis and should be articulated with situations focused on developing other complementary knowledge and competencies.

Complexities in university students’ understanding of parametric equations and curves

Despite the importance and many applications that parametric equations and curves have in mathematics and other sciences, there is a lack of research regarding this topic in mathematics education. The goal of this article is to explore university students’ understanding of parametric equations and curves based on two lenses: Action, Process, Object, Schema (APOS) and Onto-Semiotic Approach (OSA). Recent research showed that these two theories complement each other and the combination of them has good potential to analyse students’ performance of mathematical concepts. Three tasks of parametric equations and curves were proposed to characterize the schema development of twenty-seven university students in terms of triad (i.e. intra, inter and trans). Based on the results, most of the students participating in this study had difficulties regarding the practical tasks and developing the mental structures needed to solve parametric equation tasks, especially the mental structures that have to be made to draw parametric curves and identify their directions, and to compute the first and the second derivatives of parametric equations. Finally, we finish the article by making practical suggestions for instructors and researchers.

A geometria nos anos finais do Ensino Fundamental: uma análise de referenciais curriculares sob a perspectiva do enfoque ontossemiótico

Apresenta-se, neste artigo, uma análise em referenciais curriculares no que se refere aos conhecimentos geométricos a serem desenvolvidos nos anos finais do Ensino Fundamental. É parte integrante de uma pesquisa que está em andamento que tem como objetivo investigar possibilidades da constituição de um currículo para a Geometria nos anos finais do Ensino Fundamental, na região de abrangência da 36ª Coordenadoria Regional de Educação (CRE) do Estado do Rio Grande do Sul, tomando como referência os constructos do Enfoque Ontossemiótico do Conhecimento e da Instrução Matemática (EOS). A análise toma como suporte elementos da Análise Textual Discursiva e como referência a Idoneidade Epistêmica do EOS. Inicialmente se destaca a análise epistêmica produzida no documento que apresenta a Base Nacional Comum Curricular- BNCC, no que se refere aos conhecimentos geométricos e, em seguida, a análise realizada no documento que apresenta a proposta curricular de um município pertencente a 36ª CRE, tomando como referência a BNCC. Resultados apontam a presença, na BNCC, de modo equitativo, dos componentes epistêmicos situações-problemas, linguagem, regras, argumentos e relações. Na Proposta Curricular analisada foi possível perceber o encaminhamento de um trabalho com a Geometria focada em situações envolvendo grandezas e medidas, com pouca ênfase em situações envolvendo os objetos geométricos e suas propriedades. Ademais, a BNCC aponta para um trabalho com a geometria das transformações, além de aspectos formais de estudo dos objetos geométricos, que não estão presentes na Proposta Curricular. Destaca-se que o documento analisado se encontra em processo de reestruturação.

Helping Preservice Teachers to Reflect

Teacher reflection is a mechanism to evaluate and improve the educational practice. To achieve an efficient reflection, it is convenient to have guidelines that support this activity. In this paper it examines the incidence of studying basic notions of the onto-semiotic approach to develop the ability of future math teachers to reflect on teaching practices. The study was conducted during the second semester of 2018 with seven students, in three phases: first, participants reflected on the classroom practice observed in a video sequence, without intervention; second, training was given on some theoretical notions of the onto-semiotic approach; and third, a new reflective session was conducted on the continuation of a video sequence. Data analysis was focused on identifying and comparing elements referring to indicators of suitability present in both reflections. Results show that participants manage to express clearer ideas, and, mainly make justifications about their value judgments. It has been determined that, with the appropriate use of concepts from a theoretical framework of didactic analysis and guided reflection, there was a change in the nature of the reflection made by future teachers, moving from the description of events to the interpretation and analysis of situations.

Papel de las situaciones adidácticas en el aprendizaje matemático. Una mirada crítica desde el enfoque ontosemiótico

El postulado del aprendizaje por adaptación a un medio antagonista asumido por la teoría de situaciones didácticas en matemáticas se corresponde con el papel central que esta teoría atribuye a las situaciones adidácticas (momentos en los que tiene lugar la producción autónoma de conocimientos por parte de los estudiantes). Desde el punto de vista de las teorías socioculturales del aprendizaje se cuestiona la pertinencia de los planteamientos constructivistas cuando se trata del aprendizaje de conocimientos científicos. En este trabajo se justifica la importancia de un modelo didáctico dialógico-colaborativo para las situaciones de primer encuentro con los objetos de conocimiento matemáticos en el que el profesor y los estudiantes trabajan juntos en la resolución de las situaciones-problemas. La justificación de este modelo didáctico está basada en los supuestos epistemológicos, ontológicos, semióticos e instruccionales del enfoque ontosemiótico del conocimiento y la instrucción matemáticos.

Nuances of the philosophy of mathematics

Historically, the term philosophy of mathematical education is associated with the term philosophy of mathematics. However, mathematics as a mathematical education has its own foundations, schools, currents, and authors. Thus, it is possible to visualize from different research that one’s own research with foundations in the philosophy of mathematics is stronger than in the philosophy of mathematical education. In order to base the present research on the philosophy of mathematics education, we have made use of a mixed method, which involves a quantitative analysis of the references made in Scopus databases through their search algorithms, in addition to a qualitative design under the contributions of the onto-semiotic approach and the Kantian synthetic analysis. When trying to particularize the identity of the philosophy of mathematical education as an area of research in development, it is evident among the main approaches: (i) the identity process of the philosophy of mathematical education is still under construction; and, (ii) recognize the decoloniality of mathematical knowledge, imposed for centuries by Western mathematics, and backed by a single philosophy of mathematics.

Un análisis semiótico del objeto tabla estadística en libros de texto chilenos

En este artículo se realiza un análisis semiótico de los objetos matemáticos primarios en el trabajo con diferentes tipos de tablas estadísticas, utilizando elementos del enfoque ontosemiótico. Dicho análisis, se utiliza para extender los niveles de complejidad semiótica definidos por Arteaga para los gráficos estadísticos a las tablas estadísticas. También se aborda, a través de un análisis de contenido, la distribución de estos tipos de tabla en una muestra de doce libros escolares chilenos, dirigidos de 5º a 8º Básico (10 a 13 años). La representación tabular con mayor presencia en estos textos es la tabla de distribución, que se enmarcaría en un nivel 3 de complejidad semiótica, subdividido en tres niveles, y dentro de ellos el más frecuente es el primero, apareciendo el último sólo en 8º curso. La tabla de contingencia, asociada al máximo nivel de complejidad, aparece con poca frecuencia y paradójicamente disminuye su uso con el curso escolar.