\input zb-basic \input zb-matheduc \iteman{ZMATH 2013e.00336} \itemau{Canavarro, Ana Paula; Pinto, Maria Elisa} \itemti{Mathematical reasoning by six year old students: characteristics and functions of representations of students. (O racioc\'\i nio matem\'atico aos seis anos: caracter\'\i sticas e fun\c c\~oes das representa\c c\~oes dos alunos.)} \itemso{Quadrante 21, No. 2, 51-79 (2012).} \itemab Summary: This article focuses on the use of representations by 6 year old students and analyses their contribute to the mathematical reasoning of the students. The main purpose of the article is to characterise the representations, specially the diagrams, that students create to solve the problems, and to analyse the functions of that representations. We analyse the mathematical productions of three students created to solve two different problems. The students created idiosyncratic iconic representations and also used conventional symbolic representations, but with very different functions. Diagrams were used by the students to interpret the problematic situation, to relate their elements, to unpack the mathematical structure, to obtain the solution, to review the problem solving process and to support the explanation and/or justification of the reasoning. Conventional symbolic representations were based on the iconic ones, and they were used by the students to display the solutions of the problems or the arithmetic expressions associated to the solution. \itemrv{~} \itemab Resumo: Este artigo foca-se no uso de representa\c c\~oes matem\'aticas por parte de alunos do $1.^{\text o}$ ano do $1.^{\text o}$ ciclo e analisa o seu contributo para o racioc\'\i nio matem\'atico desses alunos. O prop\'osito principal do artigo \'e caracterizar as representa\c c\~oes, em especial os diagramas, que os alunos criam para resolver problemas que lhes s\~ao propostos, bem como analisar as fun\c c\~oes com que as usam. Apresentamos os casos de tr\^es alunos, analisando as resolu\c c\~oes que produziram relativamente a dois problemas. Os alunos recorreram tanto a representa\c c\~oes ic\'onicas idiossincr\'aticas como a representa\c c\~oes simb\'olicas convencionais, mas com fun\c c\~oes muito distintas. Entre as representa\c c\~oes ic\'onicas criadas pelos alunos destaca-se o diagrama, que serviu de ferramenta fundamental ao racioc\'\i nio e comunica\c c\~ao matem\'atica. Mais concretamente, o diagrama foi usado pelos alunos para interpretar a situa\c c\~ao problem\'atica, relacionar os seus elementos, desocultar a estrutura matem\'atica presente, obter a solu\c c\~ao, rever o processo de resolu\c c\~ao adotado e apoiar a explica\c c\~ao e/ou justifica\c c\~ao do respetivo racioc\'\i nio para resolver os problemas. As representa\c c\~oes simb\'olicas convencionais decorreram das ic\'onicas e serviram essencialmente para apresentar as solu\c c\~oes dos problemas ou as express\~oes aritm\'eticas relativas \`a sua determina\c c\~ao. \itemrv{~} \itemcc{E52 D52} \itemut{mathematical reasoning; mathematical representations; diagrams} \itemli{} \end