Development of variational thinking for the teaching of preliminary notions of calculus. A class experience in basic education

Background: this work incorporates results that help to understand how to favour the development of variational thinking and language in students of middle school education. Objective: to relate variational thinking with some preliminary notions of calculus. Design: a sequence of activities is presented to analyse variational aspects of the function concept and the results of its implementation. Setting and participants: the experience takes place in a middle and high school education institution with a sample of 40 students attending the 9th grade. Data collection and analysis: a didactic sequence (ds) was elaborated from the results of a diagnostic test and applied to the students’ group. The information collected from the application of the ds was submitted to an inductive analysis. Results: we focused our attention on the way in which the study variation processes allowed students to build meaningful approaches to understanding and using functions as models of change situations. Conclusions: despite the institutional limitations and the difficulties of the students, the development and emergence of variational arguments is observed in young people, based on previous knowledge that they relate to experiences of daily life

Trigonometry vs. Trigonometric Integration

Background: Relating and applying trigonometric contents to solve problem situations involving trigonometric integration is not easy for university students. Objective: To recognise trigonometry as a mathematical tool with multiple applications in their professional context. Design: Exploratory, descriptive, analytical-interpretative. Setting and participants: University engineering students. Data collection and analysis: Two stages were programmed: preparation and formalisation. The classes were videotaped and transcribed into units of analysis in order to triangulate the information collected. Results: The importance of trigonometry as a mathematical tool that allows modelling different problem situations was highlighted. Conclusions: The development of mathematical competencies in young people was evidenced, manifested in the progress of advanced mathematical thinking skills, such as: abstracting, generalising, synthesising, defining, and demonstrating; manifested when facing different proposed situations where students understood that when using trigonometric integration, different trigonometric elements are also used, allowing for obtaining simpler expressions and facilitate integration

The OSA as an analysis tool for an instructional process

II Congreso Internacional Virtual sobre el Enfoque Ontosemiótico del Conocimiento y la Instrucción Matemáticos (CIVEOS-2), 23-26 de marzo de 2017. [http://enfoqueontosemiotico.ugr.es/]En esta comunicación se presenta un análisis realizado a la estructura y funcionamiento de una secuencia de clases de matemáticas, con estudiantes colombianos de segundo año de la licenciatura en matemáticas, desde el modelo de análisis propuesto por el Enfoque Ontosemiótico de la Cognición e Instrucción Matemática (EOS). Dicho análisis permitió encontrar una sesión de clases que podríamos considerar como una degeneración mecanicista de la clase formal donde fue característico encontrar aspectos propios del paradigma formal- mecanicista.This communication presents an analysis of the structure and functioning of a sequence of mathematics classes, with Colombian students of the second year of the degree in mathematics, from the analysis model proposed by the Onto-semiotic Approach to Cognition and Mathematical Instruction (OSA). This analysis allowed us to find a class session that could be considered as a mechanistic degeneration of the formal class where it was characteristic to find aspects specific to the formal-mechanistic paradigm

Instrumentalization vs Instrumentation of Microlearning in a Math Class

The study aims to innovate the teaching and learning process of mathematics with a group of elementary school students from a rural population of Colombia, where the use of information and communications technology resources, as well as internet access in limited. The teachers implement microlearning so that children learn to solve arithmetic problems. The experience is descriptive with a non-probabilistic convenience sampling, developed from the creation and application of a virtual learning object whose pedagogical strategy was the use of microlearning. The study shows that the use of information and communications technology resources assist the students to learn mathematics. It also develops the office content, skill to interpret, know and solve mathematical problems from everyday situation to student

Mathematical generalization from the articulation of advanced mathematical thinking and knot theory

Background: The professors of Topology and modern algebra expressed interest in the need to create a space that allows deepening the process of Mathematical Generalization from the articulation of some concepts of Theory of Knots with the development of Advanced Mathematical Thinking (PMA) skills. Objective: To offer students an additional space for disciplinary training that allows them to deepen the process of Mathematical Generalization. Design: The methodology used has a qualitative approach, as a strategy we take action research from the Whitehead (1991) proposal from three phases. Setting and participants: students of the Bachelor of Mathematics program who take the third to sixth semester. Data collection and analysis: we emphasized in the second phase (intervention), since it allowed us to articulate the holistic scheme of knot theory with the PMA as shown in Tables 2, 3 and 4 (results section). Results: The result was the creation of a syllabus and subject guide for an elective seminar, which is offered to undergraduate students. Conclusion: since 2019 this elective seminar is offered to students, which awards 3 credits

Development of mathematical thinking skill from the formulation and resolution of verbal arithmetic problems

Background: mathematics teachers interested in improving student performance in the face of the low academic results presented, we seek, Objective: articulate the skills of mathematical thinking with the formulation and resolution of verbal statement arithmetic problems (PAVE). Design: the methodology was focused on action research from the design and application of a didactic sequence developed from three categories of analysis: thinking skills, formulation, and solving of arithmetic problems. Setting and participants: basic education students starting high school. Data collection and analysis: we created and implemented a didactic sequence that includes two directions: one for the formulation and the other the resolution of PAVE. Each one was monitored from three activities: opening, development and closing. Results: difficulties in formulating and solving verbal statement arithmetic problems were evidenced in those students. Conclusions: after applying the intervention, changes were evidenced in the formulation and resolution of verbal statement arithmetic problems in the group of students. Some difficulties detected in the students are related to the length of the statement, the order of presentation of data, the situation of the question, the size of the numbers used, elements that affect the syntactic and mathematical structures of the PAVE

Significado global de la integral articulando su complejidad epistémica

Presentamos resultados de investigación sobre el significado global para el objeto matemático integral que se construye con estudiantes universitarios, que toman la asignatura calculo integral, articulando los significados que la conforman. Teóricamente se usó el Enfoque Ontosemiótico de la Cognición e instrucción matemática, desde un estudio de caso a tres profesores que enseñan cálculo integral a estos estudiantes. Se resalta la importancia de lograr una articulación de sentidos desde la reconstrucción y análisis de tres configuraciones epistémicas globales propuestas en Mateus-Nieves (2020b), para la integral y que conforman el significado global de referencia pretendido en el currículo de esta asignatura, contrastado con el trabajo declarado por los profesores

Modelling Improper Integrals, a Case Study

Background: University students show little clarity in applying content related to improper integrals due to a lack of meaning that does not allow them to connect them with the environment and everyday life problems. Objective: To recognise improper integrals as a mathematical tool with multiple applications inside and outside mathematics. Design: Qualitative action research. Setting and participants: University engineering students. Data collection and analysis: A proposal was designed and applied to articulate mathematical modelling by using software, where a sequence of ten selected situations involving this type of integrals was experimented and solved. Results: The importance of modelling as a didactic-dynamic resource is highlighted because it helps students to reach and understand real situations involving improper integrals in different contexts. Conclusions:  Despite the numerous errors detected in the students, this strategy made it possible to demonstrate that they developed mathematical competencies, which was manifested in the progress of advanced mathematical thinking skills

Development of Variational Thinking for the Teaching of Preliminary Notions of Calculus. A Class Experience in Basic Education

Background: This work incorporates results that help to understand how to favour the development of variational thinking and language in students of middle school education. Objective: to relate variational thinking with some preliminary notions of calculus. Design: a sequence of activities is presented to analyse variational aspects of the function concept and the results of its implementation. Setting and participants: the experience takes place in a middle and high school education institution with a sample of 40 students attending the 9th grade. Data collection and analysis: a didactic sequence (DS) was elaborated from the results of a diagnostic test and applied to the students’ group. The information collected from the application of the DS was submitted to an inductive analysis. Results: we focused our attention on the way in which the study variation processes allowed students to build meaningful approaches to understanding and using functions as models of change situations. Conclusions: despite the institutional limitations and the difficulties of the students, the development and emergence of variational arguments is observed in young people, based on previous knowledge that they relate to experiences of daily life

Development of Mathematical Thinking Skill from the Formulation and Resolution of Verbal Arithmetic Problems

Background: mathematics teachers interested in improving student performance in the face of the low academic results presented, we seek, Objective: articulate the skills of mathematical thinking with the formulation and resolution of verbal statement arithmetic problems (PAVE). Design: the methodology was focused on action research from the design and application of a didactic sequence developed from three categories of analysis: thinking skills, formulation, and solving of arithmetic problems. Setting and participants: basic education students starting high school. Data collection and analysis: we created and implemented a didactic sequence that includes two directions: one for the formulation and the other the resolution of PAVE. Each one was monitored from three activities: opening, development and closing. Results: difficulties in formulating and solving verbal statement arithmetic problems were evidenced in those students. Conclusions: after applying the intervention, changes were evidenced in the formulation and resolution of verbal statement arithmetic problems in the group of students. Some difficulties detected in the students are related to the length of the statement, the order of presentation of data, the situation of the question, the size of the numbers used, elements that affect the syntactic and mathematical structures of the PAV