Maths Education

In this paper, we examine Abel’s 1824 proof of the insolubility (in radicals) of the general polynomial of degree five. We give a precise description of what his proof establishes and what is does or does not imply, and we give a sense of the place that this work had in the overall development of algebra. The paper details the overall strategy, and some of the technical machinery, that were used in the proof. It also focuses on one key section of the proof so as to provide a basic sense of the ingenious methods that Abel brought to bear on this problem.

Tout le monde connaît la suite de Fibonacci et le problème de reproduction des lapins qui lui a donné naissance. Cependant Léonard de Pise est certainement un des plus grands mathématiciens de son temps, un de ceux, en particulier, qui ont largement participé à l'introduction des nombres indiens et de l'algèbre arabe en Occident. Son livre le plus connu, le Liber abaci ou Livre du calcul, présente et explique ces nouveautés venues des pays d'Islam. On trouvera dans ce petit classique Kangourou une introduction à la vie et à l'œuvre de Fibonacci, de courts passages sur l'écriture des nombres et des fractions et une vingtaine de problèmes assortis de leurs solutions et de commentaires détaillés.

The close and historical relationship between music and mathematics is well known. In fact, many musicians and scientists have highlighted existing connections between both disciplines. In contrast, there are very few contributions, experiences, and teaching
materials that link the two disciplines in schools, particularly in the Spanish context. This article sheds light on this situation. For this
purpose, we describe, on the one hand, the state of the art at an international level and, on the other hand, the limited academic
production available in Spanish. Taking this reality as a starting point, the project European Music Portfolio: Sounding Ways into Mathematics is presented. This European project seeks to cover this deficit using a pedagogical perspective that goes beyond the
narrowness of the disciplines, which still occupies a predominant position. Based on existing school experiences, this project aims to
outline good practices, create materials and activities, and design teacher training courses at a European continental level. Finally, the
article concludes by arguing in turn the interest and potential of the project given the current socio-political context and educational scenario in Spain.

Es bien conocida la estrecha e histórica relación entre música y matemática. De hecho, numerosos músicos y científicos han puesto
de manifiesto las conexiones existentes entre ambas disciplinas. En contraposición, existen muy pocas aportaciones, experiencias y
materiales didácticos que vinculen ambas disciplinas en el medio escolar, especialmente en el contexto español. En el presente artículo se ilustra esta situación exponiendo, por un lado, el estado de la cuestión a nivel internacional y, por otro, la escasa producción académica en castellano. Partiendo de esta realidad, se presenta un proyecto europeo, el European Music Portfolio: Sounding Ways
into Mathematics, que precisamente busca cubrir este déficit utilizando una perspectiva pedagógica que supere la estrechez disciplinar aún predominante. En base a las experiencias escolares existentes, dicho proyecto se propone perfilar buenas prácticas, crear materiales
y actividades, y diseñar cursos de formación del profesorado a nivel europeo. Finalmente, el artículo termina discutiendo en torno al interés y al potencial que contiene el proyecto dado el contexto sociopolítico y el escenario educativo españoles actuales.

This book provides an account of a large-scale, national STEM initiative in Australia, the Maths Inside Project, which is designed to increase secondary school students’ engagement and participation in mathematics. The project’s modules include videos illustrating how scientists use mathematics to find solutions to real-world problems, as well as themed activities linked to the school curriculum for mathematics.
Outlining the current debates concerning mathematics education in Australia and beyond, the book describes the development and implementation of the modules to guide their use by teachers in year 8-12 Australian mathematics classrooms. It concludes with a discussion of the research, showing how the project increased student engagement.
The book discusses the partners involved in the project, including scientists, a national mathematics teachers’ association and the authors’ university. It also offers insights into how to embark on pedagogical improvement through collaboration between individual institutional stakeholders. Providing details of the modules to enable teachers and teacher educators to help their students better understand and utilise the curriculum resources of Maths Inside, the book is a useful resource for educators around the globe wanting to make mathematics engaging, topical and relevant for secondary school students.

INTRODUCTION STEM is the combination of first letters of science, technology, engineering and mathematics. STEM is not a single subject discipline. The other way round it is a deep understanding of knowledge within four disciplines. It is believed that an integrated approach to STEM involves using multidisciplinary skills and will help learners in solving real world problems since solving a real world problem requires integrated knowledge and skills by its nature (Akgündüz, 2016). STEM education stemmed from a necessity of 21st century. It is a matter of concern for not only educators but also for business and industry organizations and even for policy makers of all developing countries. The world is changing rapidly and it is expected that the future will be different from what we are experiencing today. Future jobs are defined as multitasking and they require a combination of skills like questioning, reasoning, connecting, creative and critical thinking just like the way students learn while studying science, technology, engineering and maths (Akgündüz, 2016; Kelley & Knowles, 2016). STEM education differs from its content by being an integration of them which means that teaching maths or science singly is not STEM education unless it involves at least two of the subjects together (English, 2016). Over the past years, it is seen that the scope of STEM education is extended to embrace arts for innovative designs of performed STEM implementations and this inclusion transformed the acronym to STEAM. It is believed that STEM education is vital and necessary for raising new generations who are creative and capable of questioning, critical thinking, reasoning, analyzing, synthesizing, collaborating. Considering this, information about STEM is given and STEM education is elaborated by the related concepts and their associations with nature of a normally developing child. In this context, implementations of world wide STEM projects will be examined and profiles of participants including children, teachers and family will be depicted.Finally, future of STEM will be discussed along with conclusions and recommendations.

Resumen: En este artículo se presenta un decálogo que incluye diez ideas clave sobre la evaluación de la competencia matemática en Educación Primaria: 1) forma parte del proceso de enseñanza-aprendizaje de las matemáticas; 2) sólo tiene sentido si se trabaja en la línea de desarrollar la competencia matemática; 3) implica evaluar los procesos matemáticos, más que los contenidos; 4) requiere, a menudo, el uso de rúbricas o bases de orientación; e implica, además: 5) evaluar el grado de riqueza competencial de las actividades; 6) analizar la práctica docente del profesorado; 7) plantear claramente los aspectos que se quieren evaluar; 8) analizar si se han trabajado todas las competencias; 9) aportar evidencias; y, finalmente, 10) establecer niveles de adquisición. Se aportan también algunos recursos –principalmente en forma de rúbricas e indica-dores– para valorar el grado de riqueza competencial de las actividades, para analizar la práctica docente del profesorado y para valorar la competencia matemática de los alumnos, además de establecer el nivel óptimo de adquisición. Palabras clave: Evaluación, competencia matemática, procesos matemáticos, rúbrica, indicadores de evaluación. Mathematical competence assessment: key ideas and resources for the classroom Summary: This article presents a decalogue that includes ten key ideas on the evaluation of mathematical competence in Primary Education: 1) it is part of the teaching learning process of mathematics; 2) it only makes sense if you work in the line of developing mathematical competence; 3) involves evaluating the mathematical processes , rather than the contents; 4) it often requires the use of rubrics or guidance bases; and it also implies: 5) evaluating the degree of mathematical competence of the activities ; 6) analyze the teaching practice of teachers; 7) clearly state the aspects that are

Des pratiques des enseignants 1 pourraient être une des causes des échecs récurrents des élèves gabonais en géométrie. La recherche présentée dans cet article vise donc la conception et l'expérimentation d'un dispositif de formation initiale fondé sur la mise en oeuvre conjointe de deux cadres théoriques (double approche et prise en compte des postures épistémologiques des enseignants) afin d'installer des pratiques basées sur les conditions cognitives nécessaires au développement d'une pensée géométrique : articulation entre visualisation et différenciation des raisonnements. Il s'agit de promouvoir une meilleure adaptation des interventions des enseignants aux difficultés des élèves. Mots clés. Visualisation géométrique – Raisonnements – Preuves visuelles – Formation initiale des enseignants-Double approche – Postures épistémologiques des enseignants. I-Le contexte de la recherche : Des pratiques des enseignants en question L'école gabonaise est marquée par un paradoxe : quasiment tous les enfants en âge d'être scolarisés entrent à l'école (taux net de scolarisation d'environ 94%, taux de parité garçon-fille quasiment égalitaire de 0,97 2) mais ils se retrouvent trop souvent en situation d'échec (taux de redoublement de l'ordre de 40% au primaire et de 30 % au secondaire en 2012 3). L'enseignement des mathématiques n'échappe pas à cette morosité. En effet, les statistiques des examens nationaux du secondaire 4 montrent que les notes obtenues en mathématiques expliquent pour une très large part la faiblesse des résultats généraux (taux de réussite de l'ordre de 30% en moyenne depuis 10 ans). Certaines pratiques des enseignants pourraient être une des causes de cet échec. Quentin de Mongaryas (2013) met en évidence un faible engagement pédagogique des enseignants de mathématiques du secondaire se traduisant par une prise en compte insuffisante des difficultés des élèves. Il précise que ces difficultés se rencontrent avec le plus d'acuité en géométrie ce qui fait émerger deux questions préliminaires : 1) Qu'est-ce qui pourrait expliquer ces pratiques ? 2) Comment former les enseignants afin de favoriser un dépassement de ces pratiques et les inciter à une plus grande adaptation des interventions aux difficultés d'apprentissage en géométrie ? II – Le cadre théorique II – 1 La double approche didactique et ergonomique Robert (2010) propose de « partir des pratiques pour former les pratiques » (p. 96) afin d'établir une correspondance entre l'objet des formations initiales et les pratiques réelles telles qu'elles sont vécues en classe. Il s'agit de viser l'acquisition de connaissances nouvelles mais ayant suffisamment de proximité avec celles qui sont déjà là pour les 1 Dans cet article, le français est utilisé à titre d'épicène aussi son emploi ne saurait être considéré comme discriminatoire. 2

Resumen
Un objetivo de la Didáctica de las Matemáticas es la comprensión del proceso de
formación de definiciones de conceptos geométricos. Esta investigación lleva a cabo
una intervención en dos aulas de Educación Primaria para analizar el papel que realizan
los contraejemplos en este proceso. Un grupo ha recibido el temario de polígonos
trabajando con ejemplos y contraejemplos y el otro solo con ejemplos. Para evaluar el
aprendizaje se ha clasificado las respuestas de los estudiantes siguiendo los niveles de
razonamiento de Van Hiele antes y después de la intervención.
Palabras clave: ejemplos, contraejemplos, polígonos, definición inclusiva,
definición exclusiva y niveles de razonamiento de Van Hiele.



Abstract
One objective of the Didactic of Mathematics is to understand the creation
process of geometrics concepts definitions. This research carries out assistance in two
classes of primary education for analysing the role of examples and counterexamples in
this process. One group has learnt the subject matter of polygons using examples and
counterexamples and the other only with examples. For evaluating the learning, the
student’s answers have been classified according to the van Hiele levels before and after
the assistance.
Keywords: examples, counterexamples, polygons, inclusive definition, exclusive
definition and van Hiele levels.

Resumen: Calcular una raíz cuadrada a mano no resulta una gran dificultad conociendo el algoritmo estandar, pero si debemos calcular una al cubo, a la cuarta, a la quinta, sexta, etc. Las cosas cambian notablemente y se complica. Ya que cada raíz tiene su propio algoritmo o procedimiento. Entonces por lo general recurrimos a la calculadora. Ya que cuando más grande es el índice del radical más se complica.

The objectives of the study are to find out the effectiveness between the interactive e-Content module of teaching and chalk and talk method of teaching in learning mathematics among high school students. The investigator used the pre-and-post-test of two equivalent groups design in this study. The purposive sampling technique which was adopted by selecting a sample of 20 Ninth standard students of each group was assigned to the conventional and modern method of teaching. An e-content module and a teacher made achievement test was developed by the researcher under the guidance of research supervisor based on the Mathematics subject portion as set theory including drawing skills of Venn diagram. Fmax test, independent t-test, ANCOVA test, ηp2 test, R2 test, ω2 test, r2 test, Effect Size and Gain Ratio test are used for statistical analysis and interpretation for achievement scores (data) and the results are discussed in this paper along with the findings and recommendations.

The proof that pi is a transcendental number, first provided by Carl Louis Ferdinand von Lindemann in 1882, was and remains one of the most celebrated results of modern mathematics. The proof was of interest in its own right, and it also resolved a host of questions that had focused the attention of mathematicians since ancient times, including classical problems involving constructions with straightedge and compass. Here, von Lindemann’s result is conveyed in a form that will be accessible to the average undergraduate mathematics major.