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- Taking school outdoors: Mathematical trails as a teaching resourcePublication . Ambrósio, H.; Hall, A.; Pais, S.Everybody should have opportunities to experience mathematical learning, solve challenges, and become more knowledgeable. Indeed, governmental recommendations present in the introduction of the mathematics Portuguese curriculum state that “No human being should be deprived of the opportunity to learn and benefit from the unique scientific and cultural heritage that mathematics represents” [1, p. 2]. We have designed a mathematical trail, in the village of Vieira de Leiria, in which geometric /mathematical elements stand out. Our aim is, first of all, to promote the interest in the study of mathematics, through observation of the surrounding environment, in particular of outdoor elements with decorative art. We also intend to establish mathematical links that make mathematics a useful, important subject, with connections to the world around us. The trail has an educational component and explores specific topics of the mathematics curriculum of the 5th and 6th years: isometries and symmetry. This activity, in the form of a rally-paper with a sequence of challenges and problems along the trail, encourages students to explore the concepts in a different, more playful and interactive way. Furthermore, it involves students in the discovery, preservation and enhancement of Portuguese heritage. In order to assess the perceptions of the students regarding the pedagogical approach used in the experience, a questionnaire was applied. This paper describes this teaching experience and the main results of the applied questionnaire. We conclude that this teaching experience had a very positive impact on the students, contributing to a better understanding of the covered curriculum, the importance of mathematics and its relation to everyday life, as well as feeling more motivated to study mathematics.
- Interlacing mathematics and art: hands-on non-Euclidean geometryPublication . Hall, Andreia; Brás, Isabel; Pais, SóniaElliot Eisner (1933-2014), a pioneer in arts education, suggested that an artistic approach to education could improve its quality and lead to a new vision for teaching and learning. This is true for any subject, including mathematics. Geometry related topics make a perfect setting for a deeper contribution of art to education. One such topic is the study of non-Euclidean geometry. In this paper, we present some results of a professional development course for mathematics teachers where the participants studied non-Euclidean geometry concepts. The participants created ceramic pieces using the Poincaré disk and styrofoam balls covered with fabrics using spherical geometry.
- Learning and teaching symmetry by creating ceramic panels with Escher type tessellationsPublication . Hall, Andreia; Pais, SóniaElliot Eisner (1933-2014), a pioneer in arts education, suggested that an artistic approach to education could improve its quality and lead to a new vision for teaching and learning. This is true for any subject, including mathematics. Some topics of the mathematics school curriculum make a perfect setting for a deeper contribution of art to education and allow for a complete symbiosis between the teaching of mathematics and an artistic education. One such topic is the study of symmetry and isometry, present throughout the school mathematics curriculum from elementary to secondary levels. We believe that the learning and teaching of symmetry and isometry can be greatly facilitated by taking the role of an artist and creating works of art, eventually inspired by renowned artists. In this paper, we present some results of a professional development course for mathematics teachers where the participants created ceramic panels using the same techniques as the Dutch artist M.C. Escher did in his tessellation drawings.
- Using Kahoot! to enhance the motivation of undergraduate students of tourism in mathematics classes: A case studyPublication . Pais, Sónia; Hall, AndreiaNowadays we live in an ever-changing society. The educational context is no exception and requires a renewal of paradigms. Profound changes to the role and function of the professor and the students are particularly vital. Active, cooperative and participative methodologies of learning must be privileged, breaking with magisterial education and the mere transmission of knowledge. Mathematics is one of the school subjects where demotivation is a frequent concern. Relating mathematics with other areas usually draws students’ attention and increases their involvement in the classes. It is necessary to go beyond the traditional; the teacher must look for resources that make his classes flow, so that the students acquire or increase their interest in mathematics. In the last few decades, technology has advanced in multiple fields, including education. Some of its benefits include improving student performance and motivation, fostering active learning, and tracking student progress. Kahoot! is an example of a platform that can be used for reviewing content and motivating students for learning. The main reasons for choosing this tool are related to the fact that the application is accessible online free of charge, allows an intuitive use, both by the teacher/author and by the student/player and allows the participation of the whole class in an interactive, stimulating, and playful way. The teachers involved in this study began using Kahoot! in their classes as an alternative teaching methodology. The participants in the study are undergraduate students from a Portuguese higher education institution and encompassed the curricular unit of Statistics. The aim of the study is to investigate students’ perceptions of how Kahoot! can be used as a tool for reviewing class content or designing warm-up activities. A survey was conducted to gather information about students’ insights on the use of Kahoot!. Similar studies have shown that higher education students are usually receptive to the use of this tool, finding it useful to increase their motivation and consider technology can positively impact learning.
- Using an artistic approach to the teaching of non-Euclidean geometry in a professional development course for mathematics teachersPublication . Hall, Andreia; Brás, Isabel; Pais, SóniaElliot Eisner (1933-2014), a pioneer in arts education, suggested that an artistic approach to education could improve its quality and lead to a new vision for teaching and learning [1]. This is true for any subject, including mathematics. Geometry related topics make a perfect setting for a deeper contribution of art to education and allow for a complete symbiosis between the teaching of mathematics and an artistic education. One such topic is the study of non-Euclidean geometry which is now briefly addressed in the middle school mathematics curriculum in Portugal [2]. We believe that the learning and teaching of non-Euclidean geometry can be facilitated by taking the role of an artist and creating works of art, eventually inspired by renowned artists such as M.C. Escher. In this paper, we present some results of a professional development course for mathematics teachers where the participants studied basic non-Euclidean geometry concepts and created ceramic pieces using the Poincaré disk. The course took place in a Portuguese university, from January to March 2018, and involved 20 teachers of grades 1 to 12. The authors have developed a qualitative case study to evaluate how an artistic approach to the teaching of non-Euclidean geometry is perceived, by the mathematics teachers, as a contribution to the learning process. Overall, the activities developed have proved to be successful examples of interdisciplinary methodologies that bring into the teaching of mathematics usual procedures in the teaching of the arts. Moreover, the artistic approach followed during the course helped the teachers develop their geometric competences concerning non-Euclidean geometry in a more solid appropriation and application of the geometric concepts involved.
- Mathematical magic in undergraduate mathematical classes for pre-service teachersPublication . Hall, Andreia; Pais, SóniaNowadays we live in an ever-changing society. The educational context is no exception, and requires a renewal of paradigms. Profound changes to the role and function of the teacher and the students are particularly vital [1], [2], [3]. The current University students have different motivations, different attitudes towards the teaching and learning process, and different responses to specific classroom environments and instructional practices [4]. The challenge that teachers face today is to motivate students to learn, get them to commit to and have an active role in their learning [5]. Considering that it is imperative to make the teaching and learning process of mathematics more stimulating, taking into account modern society and student’s interests [6], the authors have developed several strategies to increase the interest and improve the success of students in mathematics. Thus, they decided to use mathematical magic tricks in their classes to raise the motivation of the students, captivating them and stimulating their interest in mathematics. In this work, a case study to evaluate how mathematical magic can contribute to increase students' motivation for learning mathematics is presented. The paper describes how it is conceived as well as the main results.
- Volume of geometric solids on the Desmos platform: A didactic experience in Cape VerdePublication . Machado, Daniel; Bastos, Nuno; Hall, Andreia; Pais, SóniaThis work intends to disseminate a didactic experience in mathematics, in times of pandemic, in an emergency remote teaching situation, at the Portuguese School of Cape Verde, using the Desmos digital platform. The topic addressed was the study of the volume of geometric solids. The main objective was to contribute to the learning of mathematical concepts, using digital tools that promote students’ autonomy while respecting their learning pace, thus improving teaching practices through more assertive methodologies and more innovative resources. In order to understand how the Desmos platform contributes to improve the learning of geometry an explanatory case study was conducted. The participants in the study are 9th grade students from a Cape Verde school. Preliminary analysis of the data collected through several techniques, using diverse data collecting mechanisms, indicates that the contribution is very positive with clear advantages in the construction of knowledge on the part of the students, centered in a logic of skill development.
- Using mathematical magic to improve student’s motivation in mathematics classesPublication . Pais, Sónia; Hall, AndreiaOne of the problems identified in the failure of mathematics teaching is the lack of motivation that students feel towards the discipline. Considering that it is imperative to make the teaching and learning processses of mathematics more stimulating, taking into account modern society and student’s interests, the authors have developed several strategies to increase the interest and improve the success of students in mathematics. Thus, they decided to use mathematical magic tricks in their classes to raise the motivation of the students, captivating them and stimulating their interest in mathematics. In this work, a case study to evaluate how mathematical magic can contribute to increase students’ motivation for learning mathematics is presented. In order to develop this experience, the techniques of inquiry, direct observation and analysis of documents were applied and the following instruments were used: questionnaires and respective analysis grids; production of a battery of tasks of a diversified nature and field notes. The analysis of the collected data allows us to conclude that the use of mathematical magic tricks in the classroom, with the purpose of motivating the students to learn mathematics, revealed to be effective.
- The Mathematical Circus ProjectPublication . Hall, Andreia; Pais, SóniaMagic and mathematics appear hand in hand when it comes to promoting the interest and motivation for learning mathematics. Portugal, like other countries, has a high failure rate in school mathematics, which of course causes a negative attitude towards this subject in society. Mathematics is generally thought of as a difficult subject with little practical importance. Motivated by the need to fight this stigma, the Mathematical Circus Project emerged in 2012. Linking mathematics with the performing arts (theater and circus) is not as common as other connections such as with the visual arts. The Mathematical Circus team performs mathematical magic shows with a circus atmosphere in schools, science centers and other public spaces. In a traditional magic show, one often realizes that what causes the fantastic effects is the magician’s ability to manipulate objects combined with the art of controlling the perception of the spectators. The tricks performed in the Mathematical Circus are exclusively based on mathematical principles. In this paper we describe the Mathematical Circus project: its context, goals, contents and achievements
- Mathematical magic in the classroomPublication . Hall, A.; Pais, S.Mathematics, considered one of the basic areas of various formations, has been the subject of concern for many authors and researchers due to its enormous academic and educational failure [1]. Its importance in day-to-day life and the formation of individuals is irrefutable [2]. One of the problems identified in the failure of mathematics teaching is the lack of motivation that students feel towards the discipline. According to [3], motivation is an essential factor in any learning since the quality of learning is not only related to the capacity to learn, but also to the level of motivation that we have to carry out this same learning. Considering that it is imperative to make the teaching and learning process of mathematics more stimulating, taking into account modern society and student’s interests [4], the authors have developed a qualitative case study to evaluate how "mathematical magic" can contribute to increase students’ motivation for learning mathematics. In order to develop this experience, the techniques of inquiry, direct observation and analysis of documents were applied and the following instruments were used: questionnaires and respective analysis grids; production of a battery of tasks of a diversified nature; field notes and interviews. Although this project is still ongoing and therefore not yet completed, a previous analysis of the collected data allows us to conclude that the use of mathematical magic tricks in the classroom, with the purpose of motivating the students to learn mathematics, was revealed effective. Students were curious about the new tricks and were positively surprised at the relative simplicity of their explanation, as if saying to themselves, "How can such mathematical concepts bring so much surprise?" The students showed that the topics gained more meaning after performing the tricks because they witnessed real applications of the concepts, with an extremely attractive purpose. They often stated that they were going to play the tricks on their friends / family outside the classroom context, which meant that they were mastering the concepts involved.