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Effective learning planning requires information on the obstacles to learning a given concept that students experience to create a learning flow that matches the students’ characters, including in the learning of the concepts of relations and functions. The research aims to determine the learning obstacles (LOs) that students experience in learning the concepts of relations and functions as per the criteria of the theory of didactic situations (TDS). Employing the case study method, this qualitative research was conducted with subjects that included 19 prospective teacher students who had taken a diagnostic test and participated in in-depth interviews. An analysis was conducted based on three TDS criteria, namely action, formulation, and validation. The results revealed two types of LOs that students experienced in learning the concepts of relations and functions: 1) ontogenic obstacles and 2) epistemological obstacles. The first type of obstacles included 1a) the inability to properly understand the definition of the intersection of two relations, referred to as conceptual ontogenic obstacle, and 1b) the inability to relate the concept of relations in drawing a Hasse diagram, referred to as instrumental ontogenic obstacle. Meanwhile, the second type included 2a) the difficulty in expressing relations in the forms of mapping diagrams and graphs, 2b) the difficulty in determining the range of ​​a function, 2c) the difficulty in identifying relations which are also functions, 2d) the inability to represent relations in the forms of mapping diagrams, and 2e) the inability to write a function as a set of ordered pairs. It is recommended that these LO findings be taken into consideration in designing learning tools.

The golden legend of modern astronomy often reduces its history to two defining moments : the ‘Copernican revolution’ and Galileo’s challenge to the presumed obscurantism of the Church. Inevitably, its real history is richer and more subtle, in constant interaction with the scientific imagination of its time. We propose to reread it in the light of the history of science fiction, to examine how the latter may have contributed to destabilizing the astronomical paradigms of the past ; we will also attempt to identify some of the didactic issues associated with these conceptual changes. SF authors offer those who know how to spot them situations that make it easier for students to identify choices linked to paradigmatic simplifications that have never been questioned ; it’s then up to the teacher to build a system that enables them to better understand these choices, so as to overcome arguments of authority and better accept the proposed paradigm shifts. We’ll begin by identifying a number of classic as well as less-known SF works whose potential for destabilizing astronomical paradigms can be exploited in the classroom. We’ll then show how similar processes are at work in science fiction images, such as Manchu’s « Framed Earth », and illustrate how these works can be used in the case of astronomical paradigms, using a few examples of investigative sequences.

Epistemological obstacles are obstacles that cannot be avoided by students. There are many epistemological obstacles experienced by students in understanding scientific knowledge. In this case, students experience cognitive constraints in understanding a learning topic, such as Relations and Functions. The topic of Relations and Functions is one of the important topics for learning further topics. However, there are still students who cannot understand this topic perfectly due to various epistemological obstacles they experience. Therefore, this study aims to describe the epistemological obstacles on the topic of relation and function. This study was qualitative research with 22 junior high school students who have studied the topic of relation and function. The results of this study are in the form of epistemological obstacles experienced by students, namely obstacles to understanding the concepts of relations and functions, obstacles to understanding the concepts of relations and functions in different contexts, obstacles to determining the domain, codomain, and range, and obstacles to presenting relations and functions to various representations. This research is expected to be used as a reference by teachers to minimize the occurrence of epistemological obstacles in the learning process. In addition, one of the efforts that can be made by teachers and researchers to minimize epistemological obstacle is by making Didactical Design Research (DDR) that supports the development of student knowledge epistemically. Hambatan epistemologis merupakan suatu hambatan yang tidak dapat dihindari oleh siswa. Terdapat banyak kendala epistemologis yang dialami oleh siswa dalam memahami pengetahuan ilmiah. Dalam hal ini, siswa mengalami kendala kognitif dalam memahami suatu materi pembelajaran, seperti Relasi dan Fungsi. Topic Relasi dan Fungsi merupakan salah satu topic penting untuk mempelajari topic-topic selanjutnya. Namun, masih terdapat siswa yang tidak dapat memahami topic ini dengan sempurna dikarenakan berbagai hambatan epistemologis yang dialaminya. Oleh karena itu, penelitian ini bertujuan untuk mendeskripsikan hambatan epistemologis pada topic relasi dan fungsi. Penelitian ini merupakan penelitian kualitatif dengan subjek berupa 22 siswa sekolah menengah pertama yang telah mempelajari topic relasi dan fungsi. Hasil penelitian ini berupa hambatan-hambatan epistemologis yang dialami oleh siswa yaitu hambatan memahami konsep relasi dan fungsi, hambatan memahami konsep relasi dan fungsi pada konteks yang berbeda, hambatan menentukan domain, kodomain, dan range, serta hambatan menyajikan relasi dan fungsi ke berbagai representasi. Penelitian ini diharapkan dapat digunakan sebagai acuan bagi guru untuk meminimalkan terjadinya hambatan epistemologis dalam proses pembelajaran. Selain itu, salah satu upaya yang dapat dilakukan oleh guru maupun peneliti untuk meminimalkan hambatan epistemologis yaitu dengan membuat Didactical Design Research (DDR) yang mendukung perkembangan pengetahuan siswa secara epistemic.

The invention by Western science (natural sciences, anthropology, eugenics) of the concept of race made possible three mental operations—exclude, enslave, purify—which were applied to the management of human populations, paving the way for crimes and genocides in colonial empires and fascism. By examining the social uses of these mental operations in the modern world, this article seeks to grasp the epistemological obstacles and the mythological and religious backgrounds that allowed such a concept to be invented.

This research aims to determine and obtain actual data on the types of epistemological learning obstacles in solving probability story problems. This study was motivated by the low learning outcomes of students in solving probability story problems. The research uses a qualitative descriptive method with data collection techniques, including written tests, interviews, and documentation. Based on the analysis of the written test, interviews, and documentation data, it was found that out of 20 students who took the learning obstacle test, 11 scored below 60. The percentage of students who scored below the minimum completion criteria was 55%. These obstacles include errors in understanding the problem concept, errors in selecting and using solution procedures, errors in writing answers, and errors in operational techniques. It indicates that half of the students who took the learning obstacle test on probability story problems still experience learning obstacles, particularly epistemological obstacles, in solving such issues. It is reinforced by analysis and in-depth interviews with three vocational students that these students still participate in epistemological obstacles ranging from conceptual, procedural, and technical operational obstacles.

3D geometry is an essential concept in mathematics because it relates to real-world experiences and numerous mathematical topics. However, some students still have difficulty understanding this geometry. Therefore, this study aims to identify and describe students' epistemological obstacles to the dimensions of representation, spatial structure, and measurement of 3D geometry in junior high school. This study was carried out using an exploratory case study design with the purposive sampling method used to obtain data from26 students from three junior high schools in Indramayu Regency, Indonesia, through tests and interviews. The data collected were analyzed, synthesized, and described based on the framework that had been prepared. The result showed that students experience epistemological obstacles in performing a series of 3D geometric thinking tasks. They experience difficulties translating the representative model from 2D to 3D and determining the number of unit cubes of the 3D shape. Meanwhile, in the measurement dimensions, students experience difficulties in calculating surface area and comparing volumes of 3D geometric shapes. Therefore, to minimize the epistemological obstacle experienced by students, this study suggests teachers or prospective teachers pay attention to the characteristics of the material to be taught to students. [ABSTRACT FROM AUTHOR]

This study presents a qualitative and theoretical analysis, based on bibliographical research to enlighten whether fake news can be characterized as an epistemological obstacle to the learning process of scientific knowledge, regarding Science and Biology subjects, during Covid-19 pandemic context and fake news dissemination. Thus, to develop this proposal, it aims at identifying the notion of epistemological obstacle in Gaston Bachelard's work by briefly reconstructing the author's narrative, in an attempt to indicate possibilities to scientific knowledge and implications of fake news along this process. It is known that an overflow of information, the infodemic, has been a barrier due to the difficulty of criteria to select relevant and safe information as allies in the educational process. There is an intense anti-scientific movement, which is not new, but it is possible to observe its worldwide increase in recent years. It will be important for educators to assume an investigative and reflective attitude so that the feeling of failure does not go unnoticed when instinctive behavior does not appear in their pedagogical practice, as they hold knowledge. Thus, any advice regarding pedagogical mistakes will be absolutely useless. At last, fake news was configured as a phenomenon of weak thought or an epistemological obstacle to the first experience, which means let you be carried away by impulse, by the natural rapture caused by information, without critically consider it, since criticalness helps on scientific background, reforms and rectifies knowledge, when we do not offer resistance to new knowledge. [ABSTRACT FROM AUTHOR]

Konsep peluang merupakan konsep dasar yang perlu dipahami oleh siswa karena pemahaman konsep tersebut akan terus digunakan pada jenjang selanjutnya. Namun, masih banyak siswa yang mengalami hambatan dalam mempelajari topik tersebut. Penelitian ini bertujuan untuk mengidentifikasi dan menganalisis hambatan belajar (learning obstacle) siswa pada materi konsep peluang. Metode yang digunakan adalah metode penelitian kualitatif dengan pendekatan fenomenologi yang melibatkan 22 siswa SMP kelas IX dan 24 siswa SMA kelas X di dua sekolah di Kabupaten Wonosobo, Jawa Tengah, Indonesia. Pengumpulan data menggunakan triangulasi dengan pemberian tes diagnostik, wawancara guru dan siswa, serta analisis buku ajar yang digunakan (analisis dokumen). Hasil penelitian menunjukkan bahwa siswa mengalami hambatan belajar dimana kurangnya ketertarikan siswa terhadap materi konsep peluang, ketidaksiapan siswa berkaitan dengan hal teknis yang bersifat kunci dari proses pembelajaran seperti ruang sampel dan titik sampel, dan siswa masih kesulitan dalam menguasai konsep prasyarat himpunan, yaitu kesulitan memahami dalam mendaftarkan anggota suatu himpunan (ontogenic obstacle). Hambatan lain yang dialami siswa yaitu, banyaknya miskonsepsi siswa pada penyelesaian permasalahan terkait konsep peluang (epistemological obstacle) serta alur pembelajaran yang belum sesuai dengan kebutuhan dan karakteristik siswa (didactical obstacle). Hambatan-hambatan tersebut harus dipertimbangkan dalam pengembangan desain didaktik yang lebih optimal untuk mencapai tujuan pembelajaran yang diharapkan. The concept of probability is a basic concept that students need to understand because this concept will continue to be used at the next level. However, many students still have obstacles in learning this topic. This study aims to identify and analyze student learning obstacles in the concept of probability. The method used is qualitative research method with phenomenological approach involving twenty two 9th grade students and twenty four 10th grade students in two school in Wonosobo Regency, Central Java, Indonesia. Data collection uses triangulation by giving diagnostic tests, teacher and student interviews, and analysis of the textbooks used (document analysis). The results showed that students have learning obstacle which there was a lack of student interest in the concept of probability, students' unpreparedness related to key technical matters of the learning process such as sample space and sample points, and students still have difficulty mastering the concept of sets, they have difficulty writing members of a sets (ontogenic obstacle). Another obstacle is the number of misconceptions in solving problems of probability (epistemological obstacle) and the learning trajectory that is not in accordance with the needs and characteristics of students (didactical obstacle). These learning obstacles must be considered in the development of a more optimal didactical design to achieve the expected learning objectives.

The article queries the theoretical foundations of cosmopolitan attitudes from the perspective of anthropology and educational philosophy. In a first step, a cosmopolitan attitude is defined as an expression of inner structures of the self towards the ›world‹. In a second step, ›world‹ is differentiated using sociology and anthropology into ›the world of everyday life‹ and ›the world as a totality‹. To bridge the gap between these two conceptions, ›everyday cosmopolitanisms‹ are introduced. The potentials and limits of everyday cosmopolitanisms are discussed in a final step on the example of the Hiroshima Peace Memorial. [ABSTRACT FROM AUTHOR]

Multiplication of fractions is one of the operations on fractions that tends to be easy, but it becomes a problem for students. Only a little previous research has examined the multiplication of fractions and the factors that cause this problem at the tertiary level. This study aims to describe the factors that cause problems in terms of the types of learning barriers that students experience. The research design is hermeneutics phenomenology. The participants in this study were 15 students aged 18-25 years, 7 boys and 8 girls, 9 students majoring in natural sciences during high school, and 6 students from social studies majors. The researcher is the main instrument, with a fraction multiplication test consisting of 2 questions developed by NCTM and a semi-structured interview guide as an additional instrument. Data is processed using NVivo-12-assisted thematic analysis to simplify the coding process. The study revealed that students experienced more learning difficulties when working on non-routine fraction multiplication problems. For routine problems, students experience learning difficulties with the type of ontogenic obstacle that is psychological (not careful). For non-routine problems, students experience learning barriers with epistemological types because they rarely work on word problems, as well as ontogenic obstacles that are conceptual and psychological.

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Copyright of Cumhuriyet International Journal of Education is the property of Cumhuriyet University, Faculty of Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract The inverse function in secondary schools is still a difficult concept to learn. Meanwhile, in national exams or competitions this problem often arises. Didactic design is an attempt to improve the learning process. The purpose of this study is to identify the epistemological obstacle, to develop the learning designs, and to describe the students’ responses regarding the implementation of inverse function’s concept learning designs in the class. This research was held in one of Senior High School in South Tangerang of 38 people from class X. The research method is Didactical Design Research (DDR). This method is conducted from three stages, prospective analysis, metapedia didactic analysis, and retrospective analysis. The result of the study was indicated that the student’s obstacles are according to predictions and didactical design still generates some new epistemological obstacle. The revised didactical design was obtained by updating the initial didactical design to resolve the obstacle. The revised didactical design includes choosing vocabulary, adding new instructions, and expanding predictions and anticipating student responses. Abstrak Fungsi invers di sekolah menengah masih merupakan konsep yang sulit dipelajari. Sedangkan dalam ujian atau kompetisi nasional masalah ini sering muncul. Desain didaktik merupakan upaya untuk meningkatkan proses pembelajaran. Tujuan penelitian ini adalah untuk mengidentifikasi hambatan epistemologis, mengembangkan desain pembelajaran, dan mendeskripsikan tanggapan siswa terkait penerapan desain pembelajaran konsep fungsi invers di kelas. Penelitian ini dilaksanakan di salah satu Sekolah Menengah Atas di Tangerang Selatan yang berjumlah 38 orang dari kelas X. Metode penelitian yang digunakan adalah Penelitian Desain Didaktik. Metode ini dilakukan dari tiga tahap yaitu analisis prospektif, analisis metapedadidaktik, dan analisis retrospektif. Hasil penelitian menunjukkan bahwa kesulitan yang dialami siswa sesuai dengan prediksi dan desain didaktis masih menimbulkan hambatan epistemologis baru. Desain didaktis yang direvisi diperoleh dengan memperbarui desain didaktis awal untuk menyelesaikan kesulitan. Desain didaktis yang direvisi terdiri atas: memilih kosakata, menambahkan instruksi baru, dan memperluas prediksi dan mengantisipasi respon siswa. How to Cite: Muin, A., Pratiwi, A. A., Satriawati, G. (2020). Didactical Design for Overcoming Students' Learning Obstacles on the Inverse Function Concept. TARBIYA: Journal of Education in Muslim Society, 7(2), 183-191. doi:10.15408/tjems.v7i2.13041.

Abstract Background Teleology is one of the critical aspects of students’ intuitive concepts about living beings and, specifically, their evolution. This cognitive bias imposes a substantial restriction on the process of learning such content. In this work, we rely on epistemological, psychological and pedagogical analyses to substantiate an educational proposal centered on the concepts of epistemological obstacles and metacognitive vigilance. Results Based on Michael Ruse’s epistemological analysis, according to which teleology in biology persists because the scientific explanation of adaptation necessarily involves appeal to the metaphor of design, and on research in cognitive psychology, especially in relation to metacognition and self-regulated learning, we argue that the primary educational aim must be to encourage students to develop metacognitive skills to regulate the use of teleological reasoning. We develop our instructional proposal based on the didactic concepts of epistemological obstacles and metacognitive vigilance (consistent with epistemological and psychological analyses). Conclusion We briefly discuss the instructional implications of our analysis and some possible relationships between our proposal and other lines of research in psychology and science education.

In modern mathematics curricula in primary and secondary education, number line is an important supervisory tool for understanding many concepts, such as different types of numbers, equations, and more. The use of the number line is supported by a large number of researchers, but there are also studies showing that students find it difficult to use. Although the concept of number line is important for teaching and there is a great deal of debate about its use, as far as we know, there are very few systematic studies that examine the epistemological development of some components regarding the concept of number line throughout history and correlate this development by learning this concept from the students. However, there are no studies that examine the concept of historical development of the number line as a whole or relate it to student behavior. In this paper, therefore, the first attempt has been made to examine the overall development of the concept of number line in the history of mathematics. We have therefore studied the historical evolution of the concept of number line and divided it into periods, according to the characteristics of this evolution. It seems that based on the slow mathematical integration of the concept of number line at the end of the 19th century, but also on some other critical points in the four historical periods that we have analyzed, some of the difficulties that students encounter when using it are likely to be epistemological obstacles.

This article focuses on a crucial topic of epistemology in French philosopher Gaston Bachelard: epistemological obstacle. Through bachelardian psychanalysis of scientific mind and its neuroses the author puts a question, if a term of "the guardian of threshold", used by phenomenological psychotherapy based by Robert Desoille within a method of daydreaming, is or is not narrative reference to analytical thinking about the meaning of epistemological obstacle, which stops scientific knowledge in its progress. The author develops the issue of Bachelard's rationalism through his own genuine phenomenology, or anthropology of studying man as a man of "twenty-four hours", but also through Bachelard's analyses of epistemological obstacle described in his The Formation of the Scientific Mind (originally from year 1938). Between philosophy of reached erudition (by Bachelard own words, a senile state of science) and philosophy, which permanently visits a school, learns and reorganises its knowledge (youth of science) is a dialectical motion of historical epistemology. The science opened to consciousness of its own errors in ways, by which it reaches its knowledge, is for Bachelard materialized in the imagination of overcoming an obstacle, for example such obstacle, which is described by image of the guardian of threshold. [ABSTRACT FROM AUTHOR]

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Background: Teleology is one of the critical aspects of students' intuitive concepts about living beings and, specifically, their evolution. This cognitive bias imposes a substantial restriction on the process of learning such content. In this work, we rely on epistemological, psychological and pedagogical analyses to substantiate an educational proposal centered on the concepts of epistemological obstacles and metacognitive vigilance. Results: Based on Michael Ruse's epistemological analysis, according to which teleology in biology persists because the scientific explanation of adaptation necessarily involves appeal to the metaphor of design, and on research in cognitive psychology, especially in relation to metacognition and self-regulated learning, we argue that the primary educational aim must be to encourage students to develop metacognitive skills to regulate the use of teleological reasoning. We develop our instructional proposal based on the didactic concepts of epistemological obstacles and metacognitive vigilance (consistent with epistemological and psychological analyses). Conclusion: We briefly discuss the instructional implications of our analysis and some possible relationships between our proposal and other lines of research in psychology and science education. [ABSTRACT FROM AUTHOR]

Copyright of Paradigma is the property of Universidad Pedagogica Experimental Libertador and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article polemizes with the interpretative key that affirms that after the passage to the underground in 1974 there was a process of increasing militarization in the organization Montoneros. To this end, it systematizes and analyzes the political and military orientations and initiatives of Montoneros between 1974 and 1980, based on a hypothesis that affirms that, despite clandestinization, during this entire period efforts were devoted both to military and political actions. [ABSTRACT FROM AUTHOR]

In this study we first attempt to present relevant research and analyze the types of number lines, then present two studies examining components of number line, namely the direction and negative numbers based on historical evolution. Then a historical analysis of the evolution of number line is presented. After that, we present the difficulties encountered by students in using the number line as a representation tool, and through the historical evolution of the number line we try to contrast these difficulties with critical points in its historical evolution. 1st period. Mathematics up to Euclid: Separation of numbers and lines. In Greek mathematics there was a clear separation between number and magnitude. Numbers (natural numbers) were simply collections of discrete units that measured a multitude. Magnitude Size, on the other hand, was usually described as a continuous quantity divided into parts and is infinitely divisible. This distinction between number and magnitude also resulted in the distinction between arithmetic and geometry. Arithmetic dealt with a discrete or non-extended quantity, while geometry dealt with a continuous or extended quantity. It also imposed a different way of solving and handling many problems. 2nd period. Until the 16th Century: Foundations of Integers - Rational Numbers - Empirical Geometry From the 12th to the 16th century an orientation in empirical Geometry is observed, as well as its relation to computational methods and the use of measuring tools. Michael Stifel (1487-1567) was the first to define negative numbers as numbers less than zero and positive numbers as greater than zero. He was the first to describe zero, fractional and irrational numbers as numbers. In the 16th century a transformation began in the classical conception of number and magnitude. Francois Viete (1540-1603) introduced a new form of symbolism to denote unknown magnitudes and numbers, stating that numbers and magnitudes can be interchanged. This relationship between numbers and magnitudes encouraged the idea that numbers could also be treated as if they were continuous, in the Aristotelian sense of continuity. However, even during this period there is no concept of the number line and the identification of each point with a number. Nevertheless, the elements that make it possible to display it are beginning to emerge. 3rd period. From the 17th century to the beginning of the 19th century: The first connection of numbers and geometric lines - Algebraization of geometry Mapping lines of numbers was not a common idea among mathematicians until the end of the 16th century. However, the concept of the number line begins to emerge in the 17th century by some pioneering mathematicians. John Wallis (1685) was the first to use a number line in his book, Treatise of Algebra, in order to interpret addition and subtraction with negative numbers. The development of algebraic symbolism and the connection of curves to their equations led Descartes to the algebraization of geometry with the help of the coordinate system. Descartes does not use the terms abscissa, ordinate or axis in his work. Descartes did not introduce the number line through the discovery of the coordinate system in 1637 in his work La Géometrie, as he never mentions the concept of axis, and none of his illustrations depicts an axis or a numerical system of coordinates, coordinates, even when values for specific magnitudes are specified. The first recording of number lines appears in the first half of the nineteenth century in the work of Ernst Gottfried Fischer (1754-1831). Fischer works with negative and positive quantities without limitation. He explicitly associates each point on the axis of the abscissa with the values of x and corresponds to one point on the axis of the ordinate. 4th period. From the beginning of the 19th century to the present: The foundation of the number line in its present form. The first attempt to develop a theory of real numbers was made in the early 1830s by Bolzano, who saw real numbers as limits of progressions of rational numbers. Around the same time, Rowan Hamilton (1805-1865) made an attempt to define real numbers, but could not escape the logic of ordinary geometric tradition. Karl Weierstrass (1855-1897), wanting to base infinite calculus solely on the concept of number, believed that he had to define irrational numbers irrespective of the concept of limit. He therefore considered the convergent sequence itself as the number or the limit. Thus, irrational numbers are defined as sets of rational numbers, rather than ordered sequences of rational numbers. In 1871 Georg Cantor (1845-1918), launches a new numbering program, similar to the Meray and Weierstrass programs. At the same time Heine (1821-1881) proposed some simplifications that led to the so-called Cantor- Heine development, which resembles that of Meray, in which convergent sequences that do not converge to rational numbers are considered to define irrational numbers. Dedekind tried to give a clear definition of continuity, first for the points of a straight line and then for a set of numbers starting from the set of rational numbers, after observing that the ordering properties of rational numbers apply just as the relations between the points of a straight line. Dedekind considered that the set of rational numbers could be extended to a continuous set of real numbers if the Cantor - Dedekind principle accepted that the points of a straight line could be mapped one by one with the real numbers. Therefore, we have the foundation of the number line in its current form. First of all, we should note that the mathematical integration and constitution of the notion of the number line, as we know it today, took place very slowly in the history of mathematics. As we have seen, it was only with the foundations of Dedekind and Cantor in the late 19th and the early 20th century that we can now consider that there is a one-to-one correspondence between the points of a straight line and the numbers of the set of real numbers. This in itself shows that while the concept of the number line appears to be simple, its composition to its present form has taken a long time. As critical points in this mathematical constitution of the notion of the number line over the course of mathematical history, we can in principle consider the separation between the numbers and the magnitude or the separation between the numbers and the straight line. We can see this separation, as we indicated above, in the mistakes of students who often manage numbers separately from the measures on the straight lines of numbers. The second critical point that appears in the historical evolution, but also as a difficulty for students, is the negative numbers and the orientation on the number line in the positive or negative direction. The third critical point is the density of rational numbers and the extra unit intervals needed to place them on the line of numbers. This manifests as a problem for students, when they have to place fractions, and generally rational numbers, on the number line where it is necessary to determine the extra unit space such as, for example, specifying the unit of ¼ to place ¾ on the number line. The fourth critical point focuses on the density of the irrational numbers, the separation of the rational from the irrational numbers and the representation of the irrational numbers on the number line. In this work we examined the difficulties of students and teachers of mathematics when dealing with the density and separation of rational and irrational numbers and the representation of irrational numbers on the number line. As it was pointed out earlier and drawing on historical information, the identification of real numbers and their correspondence with points on the number line reaches its conclusion very late, actually between the late 19th and the early 20th centuries. [ABSTRACT FROM AUTHOR]

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During the learning of the natural numbers in Primary Education, the student acquires the conception of number as quantity of magnitude with support in the physical and concrete world. Later, the generalization of this conception to other numerical fields supposes an epistemological obstacle that hinders its learning. This study aims to determine these epistemological obstacles in the learning of the integer numbers, verify their presence in the errors and in the justifications made by the students and present pedagogical suggestions that facilitate the break with the obstacles to improve the teaching of the integer numbers. [ABSTRACT FROM AUTHOR]

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