The purpose of this article is to present a bibliographic survey of the current trends in **Colombia** regarding teacher training **and** professional development located for the **teaching** of Language **and** **Mathematics** at primary **and** secondary academic education levels. The aim is to focus on three categories of analysis: learning communities, situated professional development **and** pedagogical practice, demonstrating in a concrete way how the quality of education in the country has been impacted, from the official sector, by analyzing the results obtained in the research carried out in this educational field. These categories develop some dimensions that allow to solve the question of what are the changes that occur in the pedagogical practice of the teachers after applying a training in the methodology of the **study** of classes. The analysis of this system of categories **and** dimensions has been carried out under the process of **teaching** Language **and** **Mathematics**, **and** how it contributes significantly in improving the processes of reading comprehension **and** solving mathematical problems. The studies

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Este artículo tiene como propósito presentar un rastreo bibliográfico de las tendencias actuales en **Colombia** acerca de la formación docente y desarrollo profesional situado para la enseñanza del Lenguaje y Matemáticas en los niveles de educación básica primaria y media académica. Se pretende focalizar en tres categorías de análisis: comunidades de aprendizaje, desarrollo profesional situado y práctica pedagógica, demostrando de manera concreta cómo se ha impactado en la calidad de la educación del país, desde el sector oficial, mediante el análisis de los resultados obte- nidos en las investigaciones realizadas en este campo educativo. Estas categorías desarrollan algunas dimensiones que permiten resolver el interrogante de cuáles son los cambios que se dan en la práctica pedagógica de los docentes luego de aplicar un entrenamiento en la meto- dología del estudio de clases. El análisis de este sistema de categorías y dimensiones se ha realizado bajo el proceso de la enseñanza del Lenguaje y Matemáticas, y de cómo este contribuye significativamente en el mejora- miento de los procesos de comprensión de lectura y resolución de problemas matemá- ticos. Los estudios abordados demuestran un efecto positivo sobre la enseñanza e instruc- ción orientada por los docentes, especialmente en sus metodologías, estrategias didácticas, desarrollo profesional situado y las prácticas evaluativas, así como en el aprendizaje de los estudiantes. Esto se materializa en la conso- lidación de comunidades de aprendizaje conformadas por los docentes de lenguaje y matemáticas del país –teniendo en cuenta áreas de conocimiento y perfil de cada uno de ellos-, donde se realizan constantemente procesos de planeación, ejecución, observación y retroalimentación de las clases, por medio de equipos de estudio

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For this empirical exercise, RMCAB data for 2011 was gathered so it could be merged as our air quality variable for estimations of the Hedonic Pricing Model. Using PM10 concentration as a measure of air quality data offer two advantages: first, it allowed us to work with the city’s most relevant pollutant, **and** second, it is one of the most visible, helping to mitigate possible biases due to households´ subjective perception of pollution. For obtaining a measure of air quality for each of the households in our sample of the EMB we use Inverse Distance Weighted interpolation from the monitoring readings, a standard interpolation technique. Figure 4 shows the results of the interpolation showing the spatial variability suggested in Figure 1. Even though, our **study** area encompasses a single urban area, we identified some spatial variation of pollution levels: important PM10 concentrations are observed in the south **and** southwest areas of the city, while the north **and** center areas presented relatively low measures. Additional measures of air quality were gathered so that we could test for robust results. Particularly, the Voronoi Neighborhood Averaging methodology is used for replicating the interpolation of monitoring data method used by EPA’s Environmental Benefits Mapping **and** Analysis Program (RTI International, 2015). For both interpolation methods we used the mean annual stock of concentrations as

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In the design of our **teaching** proposal, we consider the three worlds of mathematical thinking: embodied, symbolic **and** formal, given by Tall (2004, 2008). The embodied world is where we make use of visual **and** physical attributes of concepts, combined with enactive sensual experiences to build mental conceptions. The symbolic world is where the symbolic representations of concepts are manipulated **and** where it is possible to “switch effortlessly from processes to do **mathematics**, to concepts to think about” (Tall, 2004, p. 30). The formal world is where properties of objects are formalized as axioms, **and** logical deduction is used to build **and** prove theorems. This theory suggests the existence of different ways of thinking in **mathematics** which are not isolated but interact offering advantages.

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Abstract: The modern world faces the paradigms of traditional **teaching**-learning in contrast to new technologies, which, in addition to the attention of young people **and** adults, covers almost all branches of science **and** academia. In this paper we propose the analysis of computer tools for the development of didactic strategies for the **teaching** of **mathematics**, an area of **study** that has historically generated great difficulties in the student sector, From a constructivist perspective **and** applying the qualitative-quantitative method, the information collected is managed through interviews, surveys **and** observation forms; those applied to 52 first-level PUCESE students, two teachers in the area **and** one manager; lead to the analysis **and** generation of results that show that in the educational environment in question, a traditional **teaching**-learning style is maintained, which is not necessarily mediated with educational technological tools, but also reflects the interest of students to integrate educational software **and** simulations, such as support resources, for math classes. Based on the results, free educational software Exe-Learning is proposed **and** it is demonstrated, through statistical analysis, that this resource strengthens the **teaching**-learning process of **mathematics**.

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In recent years, through different types of **teaching** innovation experiments **and** participation in the above European projects we have built a stable, open **teaching** team to adapt the different math subjects in our University Schools of Engineering to the new **teaching** scenario. We encapsulate the evolution of our thinking in a few references. References [10] y [11] address the possibilities of the use of CAS in math subjects, specifically Calculus in Several Variables **and** Differential Equations. Also, the use of CAS may involve an innovation in **teaching**, as seen from references [12] **and** [13]. We also encourage the students for the construction of toolboxes [14]. The next step is the adaptation of the Bologna Accord specifications in **teaching** with CAS [15], [16] **and** [17]. For a globalized process of **teaching** it is crucial to choose a suitable CAS. One important argument in this is the ease of access **and** in general, it is preferable to use open-code, freely distributed software. Experience with different types of CAS **and** the comparative **study** described in [18] support this priority. Now may be the time to consider the new devices than can be called hand-held technology. Drijvers **and** Trouche [19] suggest that we should be aware that hand-held technology is no longer an isolated artifact but integrated in **and** articulated within a network of resources, particularly on-line resources.

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Journal for Educators, Teachers **and** Trainers JETT, Vol. 10 (1); ISSN: 1989-9572 25 In the **study**, first a problem-based education program for **teaching** middle school **mathematics** in a training course was designed **and** implemented for middle school **mathematics** teachers based on the theoretical frameworks of Polya (1978) **and** Schoenfeld (1985). After the problem- based trainings in four levels, as it was outlined in the procedure section of the **study**, a test **and** a questionnaire were used, where the standard questionnaire based on the four levels of Kirkpatrick were distributed among the teachers. After examining the pre-tests **and** post-tests of the problem-solving performance, a significant difference was found between the problem- solving performance in two groups of control **and** experimental after the training, with regard to the teachers' scores before the training, then we designed a new mode to compare the relationship between Kirkpatrick's levels with the post-test of **mathematics** teacher's problem- solving performance. Therefore, by structural equations in R software, we showed that each level of Kirkpatrick model has a positive effect on the **mathematics** teachers’ post-test scores. Moreover, with regard to the standard load factor values, since the third level of the Kirkpatrick model has a larger standardized coefficient, it can be said that the third level of the Kirkpatrick's model has the greatest impact on the post-test score of the **mathematics** teachers. Also, the second most important level is the fourth level, the third most important level is the second level, **and** finally, it can be said that the first level of Kirkpatrick model has the least effect on the post- test scores of the **mathematics** teachers. When the assessment is done in a coherent way **and** in a pre-determined manner, such as in Kirkpatrick's model, educational approaches **and**

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The purpose of this article is to identify if the pedagogical work of teachers in the General direction of army schools, is part of the pedagogical model that characterizes to the Lyceum. The methodology, studied the literature referred to the concepts of pedagogical practice, pedagogical model, educational management. Through documentary analysis, we studied the institutional education project (PEI) **and** the pedagogical model. They were five (5) focus groups involving teachers in the institution, to investigate concepts taken as references of this **study** were conducted. The results identified that the pedagogical model is not consistent with the pedagogical practices of teachers; the educational management of the institution should promote open **and** reflective professionalism of the actors of the educational community; in addition, it is necessary to define clear objectives that will facilitate **and** enhance the pedagogical practices of teachers.

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Polyhedra are very interesting objects for architecture **and** construction, mainly because of their possibilities to be understood as a set of planes **and** lines. Given that simplicity, from the geometrical point of view, such objects have a lower formal complexity **and** therefore are less interesting than curves **and** surfaces, which become main characters of the experience. The great contribution of mathematical knowledge applied to this matter, is that it allows the student to determine the variables that define the forms, it is what characterizes them radically since their inception. The mathematical formulation let us identify the precise parameters of a curve or surface, **and** then we aproach through the numerical abstraction. Mathematical geometry **and** programming eases rigorous approach to the **study** of the object (Coloma, Mesa 2012).

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In any long journey, there is likely to be some terrain that is very difficult to cross, such as getting over a mountain range moving up through narrow mountain passes. In learning **mathematics** there are many such ‘mountain ranges’, where students have to work hard to build new concepts **and** develop new skills, **and** might lose track of the direction in which they are heading. **Teaching** through problem solving can be very effective here if there is a very carefully designed sequence of tasks. This approach has been developed to a high degree in Japan, **and** forms the basis of some Japanese school textbooks. The example below is from a presentation by Akihiko Takahashi of DePaul University Chicago drawing on material from the Lesson **Study** Alliance (http://www.LSAlliance.org) following the Japanese curriculum (Takahashi, 2018). The overarching goals are that students should learn the ideas deeply **and** have experience of using them flexibly.

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However, rationality in planning **and** organizing the society was definitely a characteristic of Sweden. The narrative about the centralization of school governance, mentioned above, concerns this. **And** the new **mathematics** syllabus of 1969, which was based on New Math, was prepared in this spirit. The preparations began in 1960, when the Nordic Committee for the Modernization of **Mathematics** **Teaching** (Nordiska kommittén för matematikundervisningens modernisering) was formed. Sweden, Norway, Denmark **and** Finland had decided to cooperate in the reformation of school **mathematics**. Apart from producing a new syllabus, a key element in the project was textbook development. About 30 authors were contracted to produce new textbooks that fitted a radically new syllabus. The textbooks were also tried in **teaching**, in some cases for up to three years. In total, the trials involved 1,310 school classes. By the end, the project received an even more scientific character as comparisons between experimental classes **and** control classes were made, but with much fewer classes (Prytz & Karlberg, 2016, p. 73). It is important to note that the results from the trials indicated that new types of textbooks were possible to use in **teaching**. For further details about the trials **and** their results, see Prytz & Karlberg (2016).

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In this article we present one of the proposed games in the laboratory: The Mate-Trivial, a board game to play individually, although competitions can also be established through a ranking. In it, students have the same capacity to move across **and** about the board with the object of reaching its centre, at which point the game is considered to be over. In order to achieve this end they will have to face up to a series of mathematical questions, as a test, which will allow them to achieve their ultimate objective. Why have these games been chosen? The game has the nature of a pastime **and** entertainment which, although at first sight is opposed to the idea of **study**, can also be used in its benefit. In fact, the idea of entertainment **and** the possibility of going into action quickly which underlies all games, can be used to initiate students into **mathematics**. The game can be an attractive **and** motivating element which turns the task into a much more pleasant **and** stimulating one.

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recognizing the work of teachers, for example the Global Teacher Prize, also referred to as the Nobel Prize for **teaching** recognises the work of teachers all over the world. Their objective which is to support the **teaching** profession **and** quality of education by awarding teachers who have impact on the children, schools **and** community. Their website reveals that a best teacher is one that 1) employs effective instructional strategies that influence **and** improve the quality of education. 2) uses innovative strategies to address challenges of a school, community **and** country 3) achieves the learning outcomes of his class **and** improve the **teaching** process by raising its bar 4) impacts his community beyond the classroom **and** 5) help students become global citizens of the world who are informed socially **and** economically **and** equip them for the world where they would live, work **and** socialise with people from different cultures **and** religions. Maarit Rossi **and** Colin Hegarty are **mathematics** teachers who were recognized as among the 10 best teachers of the world, Colin Hegarty from the United Kingdom teaches students ages 11-18 **and** he believes that no child is bad at **mathematics** as they just need the right support. While Maarit Rossi from Finland who also teaches **mathematics** by incorporating fun activities **and** enabling students see the relevance of it in their lives, these teachers are innovative who employ effective instructional strategies that influence **and** improve the quality of education globally. There is a substantial amount of research regarding the calibre of a good teacher. Students are the most significant people who are close to the teachers because they spent most of the times with them. Therefore, it is important that views **and** opinions of students representing different backgrounds **and** culturally diverse are considered. Although the context of this research **study** is **mathematics** teachers **and** **teaching**, it also contributes to a broader discussion of teachers in different subjects **and** disciplines. Defining a good **mathematics** teacher can sometimes be difficult. At hand, many researchers have made attempts to describe good **mathematics** teachers **and** **teaching**. According to Ball et al (2008) a

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This **study** investigates K-1 teachers´ perceptions, practices, **and** equity issues in the integration of contextualized problem solving into the curriculum as a way to adapt instruction to meet the needs of Latino students who were English language learners. We draw from three bodies of literature: Cognitively Guided Instruction, language **and** culture as pedagogical resources, **and** equity issues in **mathematics**. This qualitative **study** takes place in the context of a situated professional development between bilingual teachers **and** researchers in a southwestern elementary school in the United States. Data sources included classroom observations, videotaped lessons, **and** interviews with the teachers. Through classroom episodes, this **study** illustrates how both teachers: (1) valued the integration of contextualized problem solving into their curriculum **and** used language **and** culture to scaffold understanding of complex **mathematics** concepts, (2) prioritized the use of the native language in their **teaching** **and** the development of different ways to communicate mathematically their ideas **and** strategies, **and** (3) prioritized equitable **teaching** by valuing **and** promoting a student centered approach to **teaching** as well as students´ identity as **mathematics** learners.

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El trabajo de G. Cirade (2006) muestra la enorme problematicidad que encierran las matemáticas que se enseñan en Secundaria y cómo los recursos matemáticos que permiten abordarla están, todavía, muy alejados de la cultura matemática de la comunidad docente dado que, en muchos casos, requieren elaboraciones matemáticas originales que evolucionarán para convertirse en herramientas matemáticas de uso didáctico. Por otra parte, estas elaboraciones necesarias distan mucho de ser triviales desde el punto de vista de los recursos matemáticos que solicitan. Un trabajo como The **mathematics** of physical quantities, de Hassler Whitney (1968) 3 , puede ser un buen ejemplo de este tipo de recursos, elaborado para aportar una respuesta funcional a la necesidad de justificar matemáticamente expresiones del tipo:

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Although the surface of the curricular ocean may sometimes appear to have been swept by a tsunami, therefore, down in the depths curricular life goes on unper- turbed. For example, one can argue that every wave of curriculum reform in U.S. school **mathematics** (“unified **mathematics**,” “new math,” **and** “standards”) has left the curriculum unreformed. Changed, perhaps, but not reformed. **And** the changes that these reform efforts did bring about were not the changes the re- formers intended. The strong rhetoric of reform has actually been a mask for dis- unity, contradiction, misinterpretation, **and** indifference. A reasonable estimate might be that for the reform efforts in the U.S., fewer than 10% of **mathematics** teachers have been professionally involved in curriculum change. Nonetheless, the public is very much aware of reactions to reform efforts even if they are not necessarily well informed about the substance of the reforms.

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The reflective-hermeneutic model describes **teaching** as a complex activity located in an unstable ecosystem, over-determined by the content, temporary **and** sociopolitical space loaded with conflicts of value that require ethical **and** political options. In this model, the teacher must engage in a commitment that goes beyond presenting content or **teaching** processes. The teacher must, rely on his or her own experience in the classroom, reflect in order to be able to make changes to the class **and** create new spaces in which he or she can become a methodological **and** theoretical reference. Moreover, the teacher must be committed to solid values **and** skills.

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The current phenomenon of globalization, understood as the liberalization of the markets, linked with the concept of economic development as a primary objective of the countries, entails the constant change of their internal normativity to increase competitiveness **and** achieve benefits **and** opportunities for the individuals. It is needed then, an interdisciplinary work, especially the interaction between economic development **and** law, to create or modify, both normative **and** administrative aspects, that leads to the effective economic performance of the countries in a globalized world. However, the existence of the differentiating gap between countries, makes the benefits, as an effect of globalization, not homogeneous, **and** instead, allows that some countries receive more benefits than the other ones, which in despite their efforts of adaptation of their systems to the global scenario, represent not only an economic delay, but also social. All of this occurs due to the diversity of characteristics **and** needs of the countries, which must be taken into account for the creation of policies **and** the adoption of economic models that leads them to economic development. **Colombia** as a case of **study**, is a clear example of the effort to adapt its system through the relationship between economic development **and** law in contexts of globalization, efforts that even though, in some cases has helped the country, in others, has represented negative effects for itself.

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integrally consider the territory (sánchez, 2005) involves, besides exploring it as a “so- cial **and** cultural” continent of productions, understand it as a complex network, where multiple frames are constructed, based on the simultaneous existence of internal **and** external elements that overlap or merge, causing changes in the population life structure, as reflected in their physical, political, economic **and** social organization, by the way territory is configured, pos- sibilities for integration **and** sociocultural articulation or danger of breakdown of the social bond.

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- how do the changing needs of the workplace (travel, meetings, changing workloads) fit with the inflexibility of formal education **and** language institutes? In addition, how will these two aspects fit in with other normal **and** frequently non-schedulable requirements of adult family life?

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