On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin

De légendes pédagogiques à légendes psychologiques : analyse des critiques de N. Baillargeon et didactique des mathématiques

“…Par exemple, à travers une analyse historique de la spécificité du savoir (en particulier, les notions de nombre et de fonction), Sfard (1991) et Sfard et Linchevski (1994) montrent bien que les objets mathématiques apparaissent, premièrement, comme des processus (statut opérationnel) liés à des contextes spécifiques (par exemple le rapport entre deux nombres entiers émerge, d'abord, dans le contexte particulier des processus de mesure). Par la suite, ces objets prennent un statut structurel (en tant que « forme » réutilisable dans d'autres contextes) : le concept de nombre rationnel, pour continuer avec l'exemple récemment mentionné.…”

“…Cette première définition opérationnelle (en termes de processus) évoluera vers une autre définition structurelle, en termes de concepts de la théorie d'ensembles -définition présentée plus tard par Bourbaki (1977) -qui, en la réifiant, élimine la référence au processus computationnel impliqué. En ce qui concerne les processus d'apprentissages, les recherches développées par Sfard (1991) Par ces deux exemples, on voit que ce ne sont pas « les données empiriques » qui définissent par elles-mêmes l'interprétation et l'explication d'un phénomène déterminé : la théorie de référence, le cadre qui ancre les compréhensions données et les oriente, joue un rôle fondamental. La discussion de ce qui est un résultat « bien établi » ne peut rester au niveau des caractéristiques des données empiriques recueillies ni des questions strictement méthodologiques qui lui sont associées.…”

Examination of Prospective Teachers' Perceptions on Mathematical Concepts and Their Potential Teaching Strategies

“…Many educators have strived to explain various aspects of the learning and teaching process, though the terms they have used vary. Several frames suggested by research include procedural/conceptual knowledge (Hiebert&Lefevre, 1986), instrumental/relational understanding (Skemp, 1987), ritual/principled knowledge (Edwards & Mercer, 1987), and operational/structural conceptions (Sfard, 1991).The mathematics education community holds the general opinion that none of these distinctions are absolutely dichotomous.For example, the National Council of Teachers of Mathematics ' (2000) Principle and Standards for School Mathematics clearlydepicts the importance of "the alliance of factual knowledge, procedural proficiency, and conceptual understanding" (p. 20) as a powerful way of teaching and learning mathematics.However, it seems the public perception of these messages tells a slightly different story.For instance, in the survey conducted by Bossé& Bahr (2008), a group of mathematics teacher educators claimed that conceptual understanding and procedural knowledge are both necessary and important as general statements.However, what appeared in the features of each end identified by the participants was a strongly polarized view on the conceptual-procedural frame.The characteristics of conceptual understanding described by the respondents could be understood as unanimously positive, whereas procedural knowledge was described in an overwhelmingly negative tone and context.There is a line of research that reemphasizes the fact that various terminological distinctions ac-tually combine many dimensions into one. This demonstrates that multiple layers of characteristics and relationships can be learned and acquired in tandem, rather than independently, and there is a need to develop useful theory to explain how conceptual knowledge and procedural knowledge are related (e.g., Baroody, Feil, Johnson, 2007;Rittle-Johnson, Siegler, Alibali, 2001;Star, 2000Star, , 2005.The findings from prior research prompted this study to further investigate how future teachers generally understand the conceptual/procedural frame and what specific thoughts are ingrained in their perceptions.…”

Os "Mundos da Matemática" em Atividades de Modelagem Matemática

“…[...] pensamento é o "trabalho" efetuado pela reflexão do sujeito sobre um objeto, num movimento pelo qual a matéria-prima, que é a experiência, é transformada, de algo não-sabido, num saber produzido e compreendido. (JAPIASSU;MARCONDES, 2001, p. 209) No âmbito da Educação Matemática diversos autores (DREYFUS, 2002;TALL, 2002;SFARD, 1991SFARD, , 2007GRAY et al, 1999;DUBINSKY , 2002;DOMINGOS, 2003;COSTA, 2002, entre outros) têm se dedicado ao estudo do pensamento matemático. A nossa pesquisa fundamenta-se, essencialmente, nos trabalhos de Tall (1995;) e Dreyfus (2002 para tratar do pensamento matemático e dos processos cognitivos a ele relacionados.…”

“…A identificação ou caracterização de atividades com esse potencial e sua introdução nas aulas de Matemática, são aspectos importantes, considerando argumentações como a de Sfard (1991), por exemplo, de que a aparente dificuldade que muitos alunos revelam em relação à sua aprendizagem indica que deve existir algo diferente na maneira de pensar matematicamente: "[...] a Matemática parece ultrapassar outras disciplinas científicas, deve haver algo realmente especial e único no tipo de pensamento envolvido na construção de um universo matemático" (SFARD, 1991, p 2).…”