Mathematical problem solving: an evolving research and practice domain

Abstract

Research programs in mathematical problem solving have evolved with the development and availability of computational tools. I review and discuss research programs that have influenced and shaped the development of mathematical education in Mexico and elsewhere. An overarching principle that distinguishes the problem solving approach to develop and learn mathematics is to conceptualize the discipline as a set of dilemmas or problems that need to be explored and solved in terms of mathematical resources and strategies. In this context, relevant questions that help structure and organize this paper include: What does it mean to learn mathematics in terms of problem solving? To what extent do research programs in problem solving orient curricular proposals? What types of instructional scenarios promote the students' development of mathematical thinking based on problem solving? What type of reasoning do students develop as a result of using distinct computational tools in mathematical problem solving?Mathematics instruction should help students develop mathematical power, including the use of specific mathematical modes of thought that are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols (Schoenfeld, 1992, p. 345).

A Utilização do Geogebra na Demonstração Matemática em Sala de Aula: o estudo da reta de Euler

“…A construção do triângulo e dos pontos notáveis permitiu aos alunos chegarem à reta de Euler, mas a visualização da figura e a sua manipulação contribuiu decisivamente para que os alunos formulassem conjeturas, nomeadamente em relação às distâncias entre os pontos e que estas se mantinham inalteradas quando arrastavam elementos do triângulo. Tal como refere Santos-Trigo (2007), o recurso ao Geogebra permitiu desenvolver conhecimentos que apenas o trabalho com papel e lápis não tornaria facilmente acessível nem compreensível pelos alunos; é este o verdadeiro sentido da utilização da tecnologia na aula de Matemática. Os alunos, a partir do trabalho feito no Geogebra, compreenderam qual era o objetivo que pretendiam alcançar e foram capazes de traduzir os seus raciocínios dedutivos com papel e lápis.…”

“…Os ambientes de geometria dinâmica atualmente existentes possibilitam a construção e a manipulação das figuras, com rigor e rapidez, envolvendo os alunos mais ativamente na realização das tarefas que têm por base a análise de propriedades e relações geométricas. Um das grandes vantagens proporcionadas pelas tecnologias e, em particular, pelo Geogebra, é permitir que os alunos adquiram novos conhecimentos que de outra forma dificilmente estariam ao seu alcance pelo elevado grau de abstração que exigiriam na ausência de meios de teste e experimentação (SANTOS-TRIGO, 2007). Vários estudos mostram que o recurso ao Geogebra pode trazer importantes benefícios, ao permitir, de uma forma mais ou menos intuitiva, construir e explorar figuras, formular conjeturas e relacionar propriedades que se evidenciam durante o processo de manipulação (NCTM, 2000).…”

Preferences of Teaching Methods and Techniques in Mathematics with Reasons

“…Teachers are also expected to both use teaching techniques effectively and to have modern management skills in classroom environments [15] in order to establish learning that can be defined as permanent changes in behavior. Those factors which most impact students' learning and performance are not only teachers' attitudes, choice of methodology, and the content of curriculum, but also students' socioeconomic background, behavior, and personal characteristics [32,38,22,33,14,28,27,12]. Effective teaching, therefore, must place equal emphasis on teacher, student, environment, curriculum and other factors [42].…”

“…The study began with a self-evaluation of the teachers' strengths and weaknesses regarding their teaching preferences. As teachers develop their teaching skills, they may help students integrate their mathematical knowledge with other activities, and find out what works best for their personalities and curriculum [33].…”

La résolution de problèmes en mathématiques au Québec : évolution des rôles assignés par les programmes et des conseils donnés aux enseignants

“…Sur le plan international, plusieurs études se sont intéressées au développement de la résolution de problèmes dans les curricula (voir notamment Coppé et Houdement, 2009, pour la France; Artigue et Houdement, 2007, pour la France aussi;Santos-Trigo, 2007, pour le Mexique; Cai et Nie, 2007, pour la Chine;D'Ambrosio, 2007, pour le Brésil;Schoenfeld, 2007, pour les États-Unis; Hino, 2007, pour le Japon; ou encore Burkhardt et Bell, 2007, pour la Grande-Bretagne). Ces études ont contribué notamment à éclairer différentes composantes de la notion de problème et à dégager les multiples fonctions que joue la résolution de problèmes dans l'apprentissage et l'enseignement des mathématiques : appliquer des connaissances, permettre la construction de connaissances nouvelles, solliciter une activité de recherche mathématique, permettre l'utilisation partagée de plusieurs connaissances dans des situations complexes, travailler la modélisation, développer des heuristiques générales, etc.…”