Understanding Early Mathematical Modelling: First Steps in the Process of Translation Between Real-world Contexts and Mathematics

The aim of this study is to provide data to better understand the processes of early mathematical modelling. According to this, an early mathematical modelling activity carried out by 21 Spanish schoolchildren aged 5–6 years is analysed, using the validated tool “Rubric for the Evaluation of Mathematical Modelling Processes” (REMMP). The results show that children link the content of the problem with their prior knowledge (understanding); identify the important data of the problem and simplify it (structuring); show some difficulties in substituting the elements of the real context for mathematical objects (mathematizing); use progressively mathematical objects and strategies in order to propose solutions for the problem (working mathematically); compare the solution with the initial problem (interpretation); justify the proposed model via valid arguments (validation); and also communicate the decisions taken throughout the modelling process and the concrete model obtained applied to the real context (presenting). We conclude that the description of this type of activities and the tools for their analysis could be used for grading and teaching tool in order to promote mathematic modelling in early childhood education.

Modelagem Matemática e Programação de Computadores: uma Possibilidade para a Construção de Conhecimento na Educação BásicaMathematical Modeling and Computer Programming: a Possibility for the Construction of Knowledge in Basic Education

ResumoA modelagem matemática é uma das tendências em educação matemática que valoriza, entre outras coisas, o percurso do aluno na resolução de atividades de cunho investigativo. Essas investigações, por vezes, são permeadas e mediadas pelas tecnologias digitais que tanto estão imbricadas em nosso cotidiano, ressignificando a dinâmica ocorrida durante as aulas, a relação aluno-professor e a interação com os colegas. Neste artigo, apresentamos os resultados de uma pesquisa de mestrado ocorrida em uma escola pública do interior de Minas Gerais, na qual propusemos a um grupo de estudantes do ensino médio uma atividade de modelagem matemática que deveria ser explorada e resolvida em um ambiente de programação de computadores. Essa atividade de modelagem foi proposta no intuito de auxiliar-nos a compreender a interrogação: O que se revela de uma tarefa de modelagem matemática, no ambiente de programação de computadores, desenvolvida por estudantes da educação básica? A partir de uma análise qualitativa dos vídeos das aulas, auxiliados pelo software Atlas.ti e assumindo uma postura fenomenológica, pudemos observar, dentre outras coisas, que a tarefa de modelagem matemática norteou o desenvolvimento do pensamento computacional em um ambiente dialógico, no qual o aprender-com-outro foi valorizado a partir da sinergia entre modelagem matemática e programação de computadores.Palavras-chave: Modelagem matemática, Programação de computadores, Pensamento computacional.AbstractMathematical modelling is one of the trends in mathematical education that values, among other things, the student's path in solving investigative activities. These investigations are sometimes permeated and mediated by digital technologies that are so interwoven in our daily lives, giving new meaning to the dynamics that occurred during classes, the student-teacher relationship, and the interaction with colleagues. In this article, we present the results of a master's research that took place in a public school in the interior of Minas Gerais, in which we proposed to a group of high school students a mathematical modelling activity that should be explored and solved in a computer programming environment. This modelling activity was proposed to help us understand the question: What is revealed by a mathematical modelling task, in the computer programming environment, developed by students of basic education? From a qualitative analysis of the videos of the classes, aided by the Atlas.ti software and assuming a phenomenological stance, we could observe, among other things, that the task of mathematical modelling guided the development of computational thinking in a dialogical environment, in which the learning-from-others was valued based on the synergy between mathematical modelling and computer programming.Keywords: Mathematical modelling, Computer programming, Computational thinking.ResumenLa modelación matemática es una de las tendencias en la educación matemática que valora, entre otras cosas, el camino del estudiante en la resolución de actividades de investigación. En ocasiones, estas investigaciones están permeadas y mediadas por tecnologías digitales que están tan entretejidas en nuestra vida cotidiana, dando un nuevo significado a las dinámicas que se dieron durante las clases, la relación alumno-profesor y la interacción con los compañeros. En este artículo presentamos los resultados de una investigación de maestría que se llevó a cabo en una escuela pública del interior de Minas Gerais, en la que propusimos a un grupo de estudiantes de secundaria una actividad de modelación matemática a ser explorada y resuelta en un ambiente de programación de computadoras. Esta actividad de modelación se propuso para ayudarnos a comprender la pregunta: ¿Qué revela una tarea de modelación matemática, en el entorno de programación de computadoras, desarrollada por estudiantes de educación básica? A partir de un análisis cualitativo de los videos de las clases, ayudado por el software Atlas.ti y asumiendo una postura fenomenológica, pudimos observar, entre otras cosas, que la tarea de modelación matemática guió el desarrollo del pensamiento computacional en un ambiente dialógico, en en el que se valoró el aprender-con-otros con base en la sinergia entre la modelación matemática y la programación informática.Palabras clave: Modelación matemática, Programación de computadoras, Pensamiento computacional.

Modelling Activity and Durability of Electrocatalysts from a Statistical Perspective

<p>We develop a theory to investigate how energetic nonhomogeneity of active sites determines the overall activity of an electrocatalyst and how the evolution of the nonhomogeneity determines the overall durability. The simple theory is amenable to exact analytical solutions and thus fosters an in-depth transparent analysis. It is revealed that nonhomogeneity does not necessarily diminish the electrocatalytic activity; instead, the highest overall activity is obtained with a suitable level of nonhomogeneity that is commensurate with the mean property. The evolution kinetics of nonhomogeneity is described by using the Fokker-Planck theory. Exponential decay of the activity is predicted theoretically and confirmed experimentally. The present work represents a first step toward closing the gap between model and practical electrocatalysts using statistical considerations.</p>

Modelling Activity and Durability of Electrocatalysts from a Statistical Perspective

<p>We develop a theory to investigate how energetic nonhomogeneity of active sites determines the overall activity of an electrocatalyst and how the evolution of the nonhomogeneity determines the overall durability. The simple theory is amenable to exact analytical solutions and thus fosters an in-depth transparent analysis. It is revealed that nonhomogeneity does not necessarily diminish the electrocatalytic activity; instead, the highest overall activity is obtained with a suitable level of nonhomogeneity that is commensurate with the mean property. The evolution kinetics of nonhomogeneity is described by using the Fokker-Planck theory. Exponential decay of the activity is predicted theoretically and confirmed experimentally. The present work represents a first step toward closing the gap between model and practical electrocatalysts using statistical considerations.</p>

The Mathematization Process as Object-oriented Actions of a Modelling Activity System

In this paper, we describe and characterize the mathematization process that happened in an open mathematical modelling project, in which all decisions, from the choice of a real-world problem to its solution, were taken collectively by a group of students and researchers. The study was developed according to a qualitative methodological approach. Based on modelling in mathematics education studies and on the Cultural-Historical Activity Theory (CHAT), we could interpret the mathematization process as a set of actions oriented to the activity system object, and the modelling project objective as the activity object itself. We concluded that the mathematization process, permeated by tensions, took place concurrently with the delimitation of the modelling project objective, which caused a series of advances and setbacks in the subjects' actions.

Diseño de una actividad didáctica con base en un diálogo entre TAD y APOEDesign of a didactic activity based on a dialogue between ATD and APOE

We present the design of a modelling activity based on the dialogue between anthropological theory of the didactic (ATD) and the Action-Process-Object-Schema (APOS) theory. We focus in illustrating the potentiality of this dialogue for the design and analysis of research experiences.Palabras-clave: TAD, APOE, Modelación, Álgebra lineal; Transformación inversa.ResumenSe presenta el diseño de una actividad de modelación a partir del diálogo entre la teoría antropológica de lo didáctico (TAD) y la teoría Acción-Proceso-Objeto-Esquema (APOE). Nos centramos en ilustrar la potencialidad de dicho diálogo para el diseño y análisis de experiencias de investigación.Keywords: TAD, APOS, Modeling, Linear algebra; Reverse transformation.