A curricular approach to develop autonomies from the mathematics and scientific education

The purpose of this work is to highlight how the current Spanish education system supports the development of both scientific and mathematic competences. We understand these competences as the set of skills and capabilities needed to use mathematics or science when the situation requires. For that purpose, we have carried out a curricular didactic analysis of the current regulations in Andalusia. This analysis intends to reveal how the new educational systems emphasize the development of the formative character of teaching-learning processes, which has been overshadowed by the functional and the instrumental characters in Science and Mathematics Education. The findings of the research show that the objectives set out in the law are aimed at developing in students a number of attributes such as personal autonomy, creativity, tolerance, empathy, critical spirit, etc. From our point of view, all of these attributes contribute to developing the students' formative character. Besides, the analysis shows that course contents are not presented as an end in itself, but as a means for students to acquire these attributes and be whole human beings. Finally, it can be inferred from the analysis that both the methodological guidance and the evaluation criteria highlight the need to educate citizens to be capable of adapting to the environment and positively transforming it. Ultimately, promoting teaching-learning processes where scientific and mathematic competences are developed, educating students to be intellectually, socially and morally autonomous.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.“Criterios e instrumentos de evaluación de unidades de enseñanza y aprendizaje” (PPIT.UMA.B1.2017/16) financiado por la Universidad de Málaga en la convocatoria de 2017-2018

Infinite limit of sequences and its phenomenology

In this document, we search for and define an infinite limit of sequences that is correct and accepted by the mathematical experts, the final purpose of which is to analyze its phenomenology, in Freudenthal’s sense. To make the choice, experts were consulted on two issues. The first one was not decisive because of the effect that the divergence term causes, and for this reason, we did a second expert consultation where this term was removed and we selected the definition we have analyzed in this document. Once the definition was chosen, two approaches were considered for analysis: the intuitive approach and the formal approach. Based on these two approaches, we specify certain phenomena organized by the definition: unlimited intuitive growth and unlimited intuitive decrease (intuitive approach) and one way and return infinite limit of sequences (formal approach), and show examples of such phenomena by graphical, verbal and tabular representation systems. All this aim to be a help to overcome the difficulties that pre-university students have with the concept of limit

The numeric-vectorial component for analysing the infinite limit of a sequence in preservice teachers

This article studies how preservice teachers consider the concept of the infinite limit of a sequence in relation to their future teaching practice through the numeric-vectorial component. During the research, 27 prospective secondary education and Baccalaureate (secondary school)-level teachers were organised into groups and were asked to debate different fragments setting out the above-mentioned limit that had been extracted from textbooks from different publishers. In the analysis, phenomenology was considered in the sense defined by Freudenthal, from two possible approaches—intuitive and formal, and by applying four systems of representation: verbal, tabular, graphic, and symbolic; in addition, Elementary and Advanced Mathematical Thinking levels were considered in order to classify the phenomena chosen. Based on this analysis, we were able to determine five individual and three group level phenomenological profiles. We have used these data to offer some insights into teaching and learning of the infinite limit of a sequence from the perspective of prospective teachers

The Flipper classroom as an alternative methodology for mathematics education in university teaching

Este trabajo de investigación se ha realizado con los objetivos de conocer los resultados obtenidos tras haber implementado propuestas didácticas basadas en la metodología de Clase Invertida (Flipped Classroom) en distintas asignaturas de matemáticas en diferentes cursos y titulaciones de la Universidad de Málaga; analizar la viabilidad didáctica de esta innovación; y su continuidad. Todo ello orientado a buscar nuevos modelos educativos que fomenten el trabajo colaborativo, la adaptación a distintos ritmos de aprendizaje, clases más prácticas y experimentales, que impulse el trabajo autónomo, y la autorregulación en el aprendizaje. Los resultados obtenidos, una vez realizado el estudio del grado de satisfacción del alumnado universitario con la metodología flipped classroom, concluyen que existe una amplia satisfacción con la introducción de una metodología que le proporciona mayor participación en el aula, que le facilita el acceso a contenidos didácticos en un formato más cercano (que puede consultar cuándo y cómo quiera), y que se adapta a sus necesidades y ritmos de aprendizaje. Consideran que el tiempo en el aula es más eficaz, estiman que realizan un mejor aprovechamiento del tiempo y que además mejora su aprendizaje y la comprensión de contenidos. Estas conclusiones se han podido contrastar en los distintos grupos y en diferentes etapas en las que se ha llevado a cabo la experiencia.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Tratamiento del límite finito en libros de texto españoles de secundaria: 1933–2005

En este escrito describimos un estudio de libros de texto de matemáticas españoles (nivel de secundaria). Se trata de un estudio en el que se usó una muestra intencional, con un total de 40 libros de textos, para buscar por separado el desarrollo realizado por los respectivos autores, en lo que respecta a la noción de límite finito de una sucesión y el límite finito de una función en un punto. Describimos, de nuestro marco teórico, la fenomenología del límite, los fenómenos que hemos buscado en los libros de texto. Esto se ha hecho siguiendo un método sistemático para analizar los libros, plan que también presentamos. Resumimos los resultados obtenidos integrándolos en rangos temporales relevantes para la bibliografía en español, e indicamos algunas vías de futuro de nuestras investigaciones

Maatalousyrittäjien tapaturmat Kiuruvedellä

Vuonna 2006 Kiuruvedellä sattui tapaturmia 13 %:lle viljelijöistä, kun koko Suomessa vastaava luku oli vain 7 %. Näin ollen Kiuruveden maatalousyrittäjille sattuu enemmän työtapaturmia kuin muualla Suomessa. Yleisimmät tapaturmatyypit ovat nyrjähdykset ja venähdykset, haavat sekä luunmurtumat. Tapaturmien riskitekijät luokitellaan fyysisiin ja teknisiin, biologisiin ja kemiallisiin riskitekijöihin. Miehille tapaturmia sattuu eniten konetöissä ja naisille karjanhoitotöissä. Maatalousyrittäjien työturvallisuutta tukevat monet eri toimet, kuten työterveyshuolto ja Mela. Tutkimus oli määrillinen ja strukturoitu kyselylomake lähetettiin kaikille Kiuruveden karjatiloille. Tutkimukseen vastasi 41,4 % maatalousyrittäjistä. Tutkimuksen tulokset käsiteltiin SPSS-tilastoohjelmalla ja esitettiin suorina jakaumina, ristiintaulukointina ja prosenttilukuina. Tutkimustuloksista ilmeni, että tapaturmia oli sattunut yhdestä kahteen kertaan 68,5 % vastaajista. Tutkimuksesta selvisi, että Kiuruvedellä tilakoot ovat keskimääräistä suuremmat, lypsykarjatiloja on paljon ja karjan koot ovat suuremmat. Nämä kaikki tekijät altistavat tapaturmille. Tapaturmatyypit tukivat aiempia tutkimustuloksia. Myös työvaiheet, missä tapaturmia sattui, olivat jakautuneet samoin kuin aiemmissa tutkimuksissa. Tutkimuksesta ilmeni että maatalousyrittäjät kokivat kiireen, väsymyksen ja poikkeavat tilanteet suurimmiksi tapaturmia aiheuttaviksi tekijöiksi. Opinnäytetyön tuloksien avulla Kiuruveden työterveyshuolto saa tietoa siitä, millaista tietoa maatalousyrittäjät tarvitsevat ja haluavat. Tulosten pohjalta työterveyshuolto pystyy myös paremmin ohjaamaan maatalousyrittäjiä tapaturmien ennaltaehkäisyyn.In year 2006 in Kiuruvesi occupational accidents happened to 13% of farmers, when in whole Finland corresponding figure was only 7 %. Consequently farmers in Kiuruvesi have more accidents than elsewhere in Finland. The most common types of accident are sprains, strains, wounds and fractures. Risk factors for accidents are classified as physical, technical, biological and chemical risk factors. With men the injuries occur mostly with machines at work and women in animal husbandry at work. Agricultural Entrepreneurs’ safety at work is supported by many different activities, such as Occupational Health Care and Mela, the Farmer’s Social Insurance Institution. The study was quantitative and a structured questionnaire was sent to all livestock farms in Kiuruvesi. The study was answered by 41.4 % of agricultural entrepreneurs. The results of the study were dealt with SPSS statistical software, and provided a breakdown of direct, cross tabulations and percentage figures. The results of the study showed that accidents had occurred between one and two times in 68.5 % of the respondents. The study showed that in Kiuruvesi farm sizes are larger than average, there are a lot of dairy farms with bigger livestock.. All of these factors endanger accidents. Types of accidents supported the earlier findings. Even phases where the accidents occurred were divided as at previous trials. The investigation revealed that the agricultural entrepreneurs considered the rush, fatigue and abnormal situations as major factors for accidents. With the help of the results of the Thesis the Occupational Health Care in Kiuruvesi gets information about the type of information farmers need and want. Based on the results the occupational health care is also better able to guide the agricultural entrepreneurs for prevention of accidents

Sucesión convergente y sucesión de Cauchy : equivalencia matemática y equivalencia fenomenológica

Esta investigación está enmarcada por la Fenomenología, el Pensamiento Matemático Avanzado y los Sistemas de Representación; en ella estudiamos una equivalencia fenomenológica, junto a la conocida equivalencia matemática, entre sucesión convergente y sucesión de Cauchy. Enunciamos dos fenómenos organizados por definiciones de límite finito de una sucesión y de sucesión de Cauchy; con el apoyo de libros de texto de secundaria, reconocemos en los libros de texto españoles dos fenómenos organizados por la primera definición mientras que, en los libros británicos, solamente observamos uno de los dos fenómenos. (Los libros de texto españoles se eligieron al azar y no constituyen muestras representativas. Los libros de texto británicos se eligieron "ad hoc" por lo que, evidentemente, tampoco constituyen muestras representativas.) Comparamos los fenómenos organizados por cada definición; establecemos analogías y diferencias entre ellos; introducimos un criterio de equivalencia entre fenómenos y un criterio de 'equivalencia fenomenológica' entre definiciones matemáticamente equivalentes; concluimos afirmativamente acerca de la equivalencia fenomenológica entre ambas definiciones. Ésta es algo más compleja que la equivalencia matemática, ya que involucra dos pares de fenómenos: un par se observa bajo un enfoque intuitivo mientras que el otro se observa bajo un enfoque formal. Este artículo prolonga resultados presentados en Claros (2010).This research is framed by Phenomenology, Mathematical Advanced Thinking and Representation Systems; we study a phenomenological equivalence to be added to the well known mathematical equivalence between a convergent sequence and a Cauchy sequence. We introduce two phenomena associated to each definition (the finite limit of a sequence, and a Cauchy sequence); by using several high school mathematics textbooks, we recognize, in Spanish high schools textbooks, two phenomena organized by the first definition and, in British high schools textbooks only one of these phenomena. (Spanish textbooks were randomly selected, but they are not a representative sample; British textbooks were selected by "ad hoc" criteria and do not should be considered as representing the corresponding population.) We compare phenomena organized by each definition; we establish analogies and differences among them; we introduce a criterion of equivalence between phenomena and a criterion of phenomenological equivalence between mathematically equivalent definitions; finally, we answer affirmatively about the phenomenological equivalence among both definitions. Phenomenological equivalence appears to be more complex than mathematical equivalence, since it involves two couples of phenomena: the first one is observed within an intuitive context while the other couple is observed within a formal context. This paper extends results presented in Claros (2010)

Service-learnig and games as a strategy for the transfer of mathematical knowledge

Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech4 researcher-professors and 168 students 2nd year of Primary Education Degree at the University of Malaga, subject Didactics of Arithmetic, carried out a Service-Learning experience. The objective was to design and develop games with manipulative materials, which were taken to two Public Primary Education centers. We wanted to show how Service-Learning promoted greater practical training and formation of values in future teachers who will have a key role in society (Rodríguez-Gallego, 2014), and to assess the groups’ opinion. According to Alsina (2011), the game is a symbolic content activity, through which students carry out a process of adaptation to reality and solve problems that they could not do with real content in the real world. Games and the handling of materials promote mathematical skills development. The university students organized themselves into 32 groups of 4 to 6 people. They went to the schools with mathematical games adapted to the different primary levels. They took some traditional materials, such as Cuisenaire rods, vertical abacuses, tables of 100 number and Multibase Arithmetic Blocks. Furthermore, the university students made some of the materials by hand, and some of them were also customized with different themes. In addition, they adapted traditional games such as Monopoly (called "Mathpoly"), Bingo, dominoes, roulette and cards with numerical operations, among many others. These university students showed the pupils from primary schools how to play and played with them.Universidad de Málag