Written resolution of a mathematical problem by 11th grade students

This work is supported by FCT through a PhD scholarship (SFRH/BD/147510/2019) and CIEd – UIDB/01661/2020 and UIDP/01661/2020, IE-UMinho, FCT/MCTES-PT

Strategies, difficulties, and written communication in solving a mathematical problem

In an age where we live surrounded by technology, it is increasingly important to develop capabilities that differentiate us from “machines”. The habit of solving problems can help us develop some of them, including the ability to solve problems, and stimulate critical thinking. It is, therefore, important to propose tasks of a diverse nature in the classroom, and to invest more in mathematical problem-solving by students. For students to solve those problems, it is essential that they know different strategies to use and it is necessary that the teacher can identify the difficulties experienced by students in solving mathematical problems, so the teacher can help students overcome them. This article aims to identify the strategies students use to solve a problem, acknowledge the difficulties students experience, and characterize students’ written communication in their answers. To achieve these objectives, the answers to a mathematical problem which was solved by students of three 12th grade classes were collected and analyzed. In the resolutions analyzed, the strategy students used the most was the construction of schemes/figures. Regarding the difficulties, they were felt more at the level of information selection, as the students tended to add data that were neither in the statement nor could be deduced from it. Finally, when communicating their answers in writing, over half of the students did it with a high level of clarity, and the most frequently used type of justification was the exclusive use of schemes. In addition, the type of representation most used by the students was iconic representation

A problem-solving experience: The teacher’s perspective

Problem solving is a skill that can be developed by students. For this to happen, the teacher must be prepared to teach classes in which they proposes that their students solve one or more problems. Teacher do not always feel ready and confident to have a class where the focus is on solving a problem. In this communication, the focus is on the planning and implementation of a class in which one intends to solve a mathematical problem. Thus, the experience of the first author is reported when planning a class, passing through the five practices that facilitate the discussion of mathematical tasks: anticipation, monitoring, selection, sequencing and connection.La résolution de problèmes est une compétence qui doit être développée par les étudiants. Pour ce faire, l’enseignant doit être prêt à donner des cours dans lesquels il/elle propose à ses élèves de résoudre un ou plusieurs problèmes. Les enseignants ne se sentient pas toujours prêts et confiants d’avoir une classe où l’accent est mis sur la résolution d’un problème. Dans cette communication, l'accent est mis sur la planification et la concrétisation d'une classe dans laquelle on entend résoudre un problème mathématique. Ainsi, l'expérience du premier auteur est rapportée lors de la planification d'un cours, en passant par les cinq pratiques qui facilitent la discussion des tâches mathématiques: anticipation, surveillance, sélection, séquence et connexion.This paper is a result of the project SmartEGOV: Harnessing EGOV for Smart Governance (Foundations, Methods, Tools) NORTE-01-0145-FEDER-000037, supported by Norte Portugal Regional Operational Programme (NORTE 2020),under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund (EFDR). Further support by CIEd –Research Centre on Education, projects UID/CED/1661/2013 and UID/CED/1661/2016, Institute of Education, University of Minho, through national funds of FCT/MCTES-PT

Written resolution of a mathematical problem by 11th grade students

[Excerpt] Problem solving and written communication are strongly connected, since the resolution of a problem presupposes the use of written communication to record the reasoning, either to communicate with another person or to review the resolution in the future. Bearing in mind this relation, and also considering the relevance of both in learning mathematics, our research question is: how students communicate their resolutions of a mathematical problem in writing? To answer this, we made a qualitative research with an interpretative paradigm. The participants were 29 students of 11th grade, divided into six working groups, who voluntarily signed up for a problem-solving project developed online and in an extracurricular format

Strategies used by 11th grade Portuguese students in solving mathematical problems

Problem-solving, organised as individual or group activity, should be highlighted in the practice of Mathematics. This is emphasized by the Portuguese Mathematics Working Group (GTM) in the report entitled "Recomendações para a melhoria das aprendizagens dos alunos em Matemática" [Recommendations for improving student learning in Mathematics] (2019). In addition to problemsolving, the authors of this document also mention the importance of communication and of resorting to different representations, among other mathematical skills. It was to highlight these capacities that the research presented in this paper was carried out, as part of a PhD project. The project's main objective is to understand the how problem-solving and written communication skill mutually reinforce each other. Within this broad context, one of the research questions enquires "What are the strategies used by students in problem-solving?". Problem-solving strategies is precisely the focus of this paper, where following strategies are considered: trial and error, search for a pattern, generalization, deduction, endto-start resolution, construction of diagrams or figures, construction of tables, construction of a model, resolution by parts, application of formulas, exhaustion, and particularization. Twenty-nine 11th grade students from two classes of the same school participated in this study. Throughout the academic year, students were invited to participate in 16 problem-solving sessions, lasting about 90 minutes each, conducted virtually over an online platform. These students were divided into six working groups which remained unchanged until the end of the project. In each session, a mathematical problem was proposed to be addressed and solved by the groups. At the end of each session, all groups delivered a single resolution to the proposed problem, which resulted in 92 resolutions at the end of the project. This data collection was carried out by the researcher and first author of this communication, who was present at all times of data collection, using a qualitative methodology within an interpretative paradigm. After collecting all the resolutions over the 16 sessions, the same author analysed the resolutions and identified which strategies the students used. After analysing the resolutions of all groups in all sessions, we concluded that all strategies listed were used by at least one group throughout the 16 sessions. Furthermore, it was found that all groups resorted to almost all the strategies listed, which means that they felt the need to address the problems with different processes. It was also possible to detect that the most used strategy by the students was the "construction of diagrams or figures" – 35 resolutions resort to it – while the least used were "search for a pattern", "generalization", "end-to-start resolution" and "particularization" – all with 4 resolutions only

Concordância entre dados obtidos em entrevistas repetidas com seis anos de intervalo

The objective of the study was to compare information collected through face-to-face interviews at first time and six years later in a city of Southeastern Brazil. In 1998, 32 mothers (N=32) of children aged 20 to 30 months answered a face-to-face interview with structured questions regarding their children's brushing habits. Six years later this same interview was repeated with the same mothers. Both interviews were compared for overall agreement, kappa and weighted kappa. Overall agreement between both interviews varied from 41 to 96%. Kappa values ranged from 0.00 to 0.65 (very bad to good) without any significant differences. The results showed lack of agreement when the same interview is conducted six years later, showing that the recall bias can be a methodological problem of interviews.O objetivo do estudo foi comparar a informação coletada em entrevista pessoal num primeiro momento e seis anos depois, em Minas Gerais. Em 1998, 32 mães (N=32) de crianças com idade entre 20 a 30 meses responderam à entrevista pessoal com questões estruturadas sobre os hábitos de escovação das crianças, sendo repetida seis anos depois. As duas entrevistas foram comparadas em concordância geral e em coeficientes kappa e kappa ponderado. A concordância geral entre as entrevistas variou de 41% a 96%. Os valores de kappa variaram de 0,00 a 0,65 (muito ruim a bom), sem diferença significativa. Os resultados mostraram que houve ausência de concordância quando a mesma entrevista foi conduzida seis anos depois, mostrando que o viés de memória pode ser um problema metodológico das entrevistas

Concordância entre dados obtidos em entrevistas repetidas com seis anos de intervalo

CAPES - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOThe objective of the study was to compare information collected through face-to-face interviews at first time and six years later in a city of Southeastern Brazil. In 1998, 32 mothers (N=32) of children aged 20 to 30 months answered a face-to-face interview with structured questions regarding their children's brushing habits. Six years later this same interview was repeated with the same mothers. Both interviews were compared for overall agreement, kappa and weighted kappa. Overall agreement between both interviews varied from 41 to 96%. Kappa values ranged from 0.00 to 0.65 (very bad to good) without any significant differences. The results showed lack of agreement when the same interview is conducted six years later, showing that the recall bias can be a methodological problem of interviews.The objective of the study was to compare information collected through face-to-face interviews at first time and six years later in a city of Southeastern Brazil. In 1998, 32 mothers (N=32) of children aged 20 to 30 months answered a face-to-face interview with structured questions regarding their children's brushing habits. Six years later this same interview was repeated with the same mothers. Both interviews were compared for overall agreement, kappa and weighted kappa. Overall agreement between both interviews varied from 41 to 96%. Kappa values ranged from 0.00 to 0.65 (very bad to good) without any significant differences. The results showed lack of agreement when the same interview is conducted six years later, showing that the recall bias can be a methodological problem of interviews422346349CAPES - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOCAPES - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOsem informaçãoO objetivo do estudo foi comparar a informação coletada em entrevista pessoal num primeiro momento e seis anos depois, em Minas Gerais. Em 1998, 32 mães (N=32) de crianças com idade entre 20 a 30 meses responderam à entrevista pessoal com questões estruturadas sobre os hábitos de escovação das crianças, sendo repetida seis anos depois. As duas entrevistas foram comparadas em concordância geral e em coeficientes kappa e kappa ponderado. A concordância geral entre as entrevistas variou de 41% a 96%. Os valores de kappa variaram de 0,00 a 0,65 (muito ruim a bom), sem diferença significativa. Os resultados mostraram que houve ausência de concordância quando a mesma entrevista foi conduzida seis anos depois, mostrando que o viés de memória pode ser um problema metodológico das entrevista

Direitos educacionais em relação a alunos com Transtorno do Espectro Autista

This paper aims to present the definition of Autism Spectrum Disorder (ASD) and the laws regarding inclusive education for students diagnosed with ASD. We did an extensive bibliographic research using the Brazilian Law of Inclusion (LDI), articles on the subject, and publications from international organizations. We came to the conclusion that Brazil is  investing in new ways to ensure that the person with disabilities can integrate into society. .O presente trabalho pretende apresentar a definição de Transtorno do Espectro Autista (TEA) e as leis a respeito da educação inclusiva para alunos diagnosticados com o TEA. Fizemos uma extensa pesquisa bibliográfica utilizando a Lei Brasileira de Inclusão (LDI), artigos sobre o tema e publicações de órgãos mundiais. Chegamos a conclusão de que o Brasil está  investindo em novas formas de garantir que a pessoa com deficiência possa integrar a sociedade

Types of tasks in Mathematics textbooks: a study with Portuguese’s textbooks of 10th and 11th grades

A diversidade de tarefas é essencial para a aprendizagem, considerando as diferentes funções que desempenham. Assim, é importante fornecer aos alunos a oportunidade de ter contacto com diferentes tipos de tarefas em Matemática. Neste estudo, fez-se uma análise a todos os manuais portugueses de 10.º e 11.º ano, autorizados pela Direção-Geral de Educação para o ano letivo 2020/2021, da disciplina de Matemática A, para se perceber a diversidade de tarefas que cada um propunha. Estes são dois dos três anos finais da escolaridade obrigatória em Portugal, inseridos no chamado “Ensino Secundário”, e os alunos normalmente têm entre 15 e 17 anos de idade. Considerando que as tarefas podem ser divididas em quatro categorias principais, de acordo com a sua estrutura e o seu nível de dificuldade, concluiu-se que cerca de 88% têm uma estrutura fechada e nível de desafio reduzido, aproximadamente 11% têm também estrutura fechada, mas grau de desafio elevado, e as restantes têm estrutura aberta e desafio reduzido, com uma percentagem inferior a 1%. Quanto a tarefas de estrutura aberta e desafio elevado, foi apenas encontrada uma.The diversity of tasks is essential for learning considering the different functions they perform. Thus, it is important to provide students with the opportunity to have contact with different types of tasks in Mathematics. In this study, an analysis was made of all Portuguese manuals of 10th and 11th grade, which are authorized by the General Direction of Education for the academic year 2020/2021, of the subject of Mathematics A, in order to understand the diversity of tasks that manual proposed. These are two of the final three years of compulsory education in Portugal, inserted in the “Secondary Education”, and students are usually between 15 and 17 years old. Considering that tasks can be divided into four main categories, according to its structure and its level of difficulty, it was concluded that about 88% have a closed structure and reduced level of challenge, approximately 11% also have a closed structure but a high level of challenge, and the remaining tasks have an open structure and reduced level of challenge, with a percentage below than 1%. As for open structure and high level of challenge tasks, only one was found.FCT – Fundação para a Ciência e a Tecnologia através de uma bolsa de doutoramento (SFRH/BD/147510/2019). CIEd – Centro de Investigação em Educação, Instituto de Educação, Universidade do Minho, projetos UIDB/01661/2020 e UIDP/01661/2020 e UID/CED/1661/2016, através de fundos nacionais da FCT/ MCTES-PT