The History of Algebra in Mathematics Education

In this chapter, we analyse key issues in algebra history from which some lessons can be extracted for the future of the teaching and learning of algebra. A comparative analysis of two types of pre-Vietan languages (before 16th century), and of the corresponding methods to solve problems, leads to conjecture the presence of didactic obstacles of an epistemological origin in the transition from arithmetic to algebraic thinking. This illustrates the value of historic and critical analysis for basic research design in mathematics education. Analysing the interrelationship between different evolution stages of the sign system of symbolic algebra and vernacular language supports the inference th…

<p>Sentido y elaboración del componente de competencia de los modelos teóricos locales en la investigación de la enseñanza y aprendizaje de contenidos matemáticos específicos</p>

El marco teórico y metodológico para la investigación en Matemática Educativa que Filloy denominó hace años “de los Modelos Teóricos Locales” tiene como uno de sus componentes la descripción de la competencia en el dominio cuya enseñanza y aprendizaje va a ser investigado. En este trabajo exponemos en qué sentido se usa el término competencia, con ejemplos de la elaboración de modelos de competencia en algunas investigaciones. Esto nos conduce al examen de la relación entre la competencia en un dominio matemático y el análisis fenomenológico de dicho dominio. Incidentalmente examinamos el uso del término competencia en varios documentos del estudio PISA. Meaning and Elaboration of the Compe…

Teaching and Learning of Algebra

Topic Study Group 9 aimed to bring together researchers, developers and teachers who investigate and develop theoretical accounts of the teaching and learning of algebra. The group sought both empirically grounded contributions focussing on the learning and teaching of algebra in diverse classrooms settings, the evolution of algebraic reasoning from elementary through university schooling as well as theoretical contributions throwing light on the complexities involved in teaching and learning of algebra. Prospective contributors were requested to address one or more of the following themes: early algebra, use of ICT in algebra classrooms, proof and proving in algebra, problem solving, semio…

Researching (Algebraic) Problem Solving from the Perspective of Local Theoretical Models

Concepts such as Local Theoretical Model, Mathematical Sign System, logico-semiotic outline, analytical reading, and formal competence model, elaborated within long term research programs developed in Mexico and Spain, are presented. Empirical data from a study carried out in Spain with students of a teacher training school are analyzed using this theoretical framework. The cases analyzed show to which extent the logico-semiotic outline determines the arithmetic or algebraic nature of the entire problem solving process, even in the case of pupils which had been instructed in general problem solving heuristics, management and control, including the explicit training in changing the plan of s…

Actuaciones de alumnos instruidos en la resolución algebraica de problemas en la hoja de cálculo y su relación con la competencia en el método cartesiano

Presentamos resultados de una investigación en la que, entre otros objetivos, se pretendía determinar cómo influía la enseñanza de la resolución algebraica de problemas en la hoja de cálculo en la competencia en el método cartesiano. La comparación de los cuestionarios administrados antes y después de la enseñanza puso de manifiesto un aumento del uso polisémico de la equis cuando se resolvía con lápiz y papel, y una disminución del uso del lenguaje del álgebra en los problemas de edades. Mostramos que estos resultados pueden atribuirse a la aparición de estrategias de resolución en la hoja de cálculo en las que las situaciones descritas en el enunciado se modelizaban mediante relaciones fu…

Solving word problems algebraically in a spreadsheet environment in a primary school

This paper presents some results from an investigation into the teaching of the algebraic solving of word problems in a spreadsheet environment in the sixth grade of primary school in Spain (11-12 year old pupils). The main aim of the study was to investigate whether the spreadsheet could be a mediator to the teaching of algebraic problem solving. Through the analysis of excerpts from a case study, the core of the paper is focused on two different types of the difficulties that students showed when solving problems algebraically in a spreadsheet environment.

Modelos emergentes en un primer curso de economía y administración

[EN] In this paper we present an outline of an ongoing research in which we intend to investigate on the teaching of the modelling process and families of functions in the context of a course in Economics and Management. Specifically, we introduce some of the theoretical foundations that support the research and we present an example of one of the activities that we are designing for teaching.

Topic Study Group No. 54: Semiotics in Mathematics Education

Fundamentals of the design and the operation of an intelligent tutoring system for the learning of the arithmetical and algebraic way of solving word problems

Designers of interactive learning environments with a focus on word problem solving usually have to compromise between the amount of resolution paths that a user is allowed to follow and the quality of the feedback provided. We have built an intelligent tutoring system (ITS) that is able to both track the user's actions and provide adequate supervision during the resolution. This is done without imposing any restriction on the resolution paths that are allowed. Instead, the system attempts to enforce metacognitive learning by requiring an appropriate definition of quantities before they are used. The program (a) supports both the arithmetical and algebraic way of solving problems; (b) allow…

Using Modelling and Tablets in the Classroom to Learn Quadratic Functions

In this chapter, we present teaching material to work with a quadratic function and the meaning of its parameters through the mathematical modelling of a real-life phenomenon: the relation between the time and the height of a ball during a complete vertical rebound and fall. The teaching material uses electronic tablets to collect and process real data in the classroom. After analysing a year 11 implementation, we note that a qualitative analysis of the phenomenon and the families of functions and the students’ prior knowledge about these functions are key elements to manage and control the modelling process, especially, to choose the model and to interpret the results in terms of the pheno…

Learning the concept of family of functions through the modelling process using tablets

International audience; In this study, we present a teaching situation to work on the modelling process using real data taken in classroom with tablets. The situation allowed us to work on the concept of the quadratic function with a group of 11th grade students and take into account the hypothesis that the qualitative analysis of the phenomenon, the knowledge of the qualitative properties of the families of functions and the meaning of the parameters are crucial elements for the management and control of the modelling process. We observed they are key elements to choose the function used as a model and to check its adequacy throughout the modelling process. Some student's conceptions on he…

Tres textos en diálogo con la teoría del análisis didáctico de Luis Rico

En este monográfico, presentamos tres textos que, con motivo de la jubilación del profesor Luis Rico, dialogan con su concepto de “análisis didáctico”. En las teorías, los términos propios de ellas suelen estar definidos para perfilar su significado, en especial cuando se está usando un término que ya tiene un campo semántico más o menos grande en su uso civil o en otras teorías, como es el caso de “análisis” y de “didáctico”. En el caso de los términos propios de las teorías puede pues relajarse la consideración de que el significado está en el uso y quedarse uno con lo que diga la definición que da quien elabora la teoría –en todo caso, la definición explícita es uno de los lugares en que…

The Influence of Technology on the Mathematical Modelling of Physical Phenomena

A study is presented in which students are asked to model two physical phenomena using applications on electronic tablets : a bounce of a ball and the extension of a spring. The analysis focusses on (a) the influence of characteristics of the applications on the tablets on the decisions that groups of 16-year-old students made during the modelling phases in which reality and mathematics are related, (b) mathematisation of the phenomena and (c) interpretation of the models. The phenomena were recorded using an app that requests users establish a set of references during the mathematisation process, which makes students focus on the way the references have been set to interpret the model prop…

Análisis de las actuaciones de los estudiantes de secundaria cuando resuelven problemas verbales en el entorno de la hoja de cálculo

En este trabajo presentamos una división en pasos ideales del Método de Resolución de la Hoja Electrónica de Cálculo y analizamos las dificultades propias del método que su uso plantea al resolutor. También mostramos los resultados de un estudio exploratorio realizado con alumnos de primer curso de secundaria en el que pretendíamos observar cómo resolvían problemas en el entorno de la hoja de cálculo y cómo se enfrentaban a las dificultades propias del método.