# How is geometry used

## Educational construction of geometry and the use of concrete materials as a learning process

**ORIGINAL ITEM**

OLIVEIRA, José Luiz de Jesus Egues de ^{[1]}, SILVA, Edevaldo Maximo there ^{[2]}, SANTOS, Cleidison Da Silva ^{[3]}, PONTES, Dalila Fonseca ^{[4]}, SILVA, Jandaira Fernandes there ^{[5]}, QUEIROZ, Jose Pereira de ^{[6]}, SANTANA, Zenóbia Conceição Castro ^{[7]}

OLIVEIRA, José Luiz de Jesus Egues de. Et al.** Educational construction of geometry and the use of concrete materials as a learning process. **Revista Científica Multidisciplinar Núcleo do Conhecimento. Volume 05, Ed. 10, Vol. 10, pp. 46-61. October 2020. ISSN: 2448-0959, access link: https://www.nucleodoconhecimento.com.br/bildung-de/paedagogische-gestaltung, DOI: 10.32749nucleodoconhecimento.com.br/bildung-de/paedagogische-gestaltung

Contents

### SUMMARY

The research was characterized by a didactic reflection that meets the need for experimental constructions in the classroom, using concrete materials that support the process of teaching and learning geometry in elementary school. In this sense, ascribing an analysis to this problem, we have as an axis, mathematics education, as a way to stimulate students to develop workshops aimed at building didactic materials as a process of learning mathematics. The process of constructing the concept of geometry, which ascribes the skills and competencies developed in the students at this stage, allowed the same to develop a certain kind of thought in order to understand the geometric spaces in an organized way, to describe and represent in order to convey the importance that this content has in his social life as a citizen. By constructing geometric figures with manipulated materials, we examine all the concepts of the figure, through this tool, which stimulates the interest and motivation of the students and gives them geometric and mathematical reasoning. The definitions of geometry can be constructed successively, referring to the prior knowledge of the students or not, in which we emphasize that educators should create activities in which the students ask about situations related to geometrical dimensions and thus experiences of meaning and direction of themselves and a figure that is inserted into the room. For the start of this teaching and learning process, however, in addition to the presentation of theories, it was necessary for the students to show the geometric figures on the basis of their physical properties, i.e. in such a way that nature supplies us and not only through their properties and defined concepts. Finally, the research allowed us to analyze the intellectual capacity of each student selected for this purpose.

Keywords: mathematics education, educational construction, geometry.

### 1. INTRODUCTION

As professionals of education and various fields, we think of this research, driven by the desire for dialogue about the teaching of learning mathematics, more precisely observing the importance of geometry in the educational menu of the student and especially in elementary school. As researchers and connoisseurs of geometric spaces, we also highlight the argument of Malba Tahan (1967, p.31) in his text, who says: “What makes it difficult to teach mathematics is the unchangeable Latin habit of always dealing with the abstract start without going through concrete ”. Other authors are also very relevant to this part of mathematics. We will also look at the near abandonment of geometry in basic education, which is reported in some of the studies that we approach as the basis for this work. We also emphasize the specifics regarding the teaching of geometry content in the context of the physical world for the classroom, as well as the skills that this area of mathematics must offer for the inclusion of students in a de facto educational society.

However, the teaching-learning process in mathematics is the subject of much study and discussion, as new methods are constantly emerging to meet a demand that is constantly changing in the various teaching modalities.

Given the peculiarities of the learning process of mathematics in elementary school and the perspective of developing tools to stimulate the intellectual performance of students while appreciating the use of mathematics in everyday life, in which myriad problems can be solved, the discussion arises about the Importance of teaching content with geometry in the foreground because it is a very interesting way. It is extremely important to give the student a more organized view of the world in which they live. Indeed, it is very important to study this area of mathematics and it is defended by several educators in which it will be discussed here and discussed below, depending on the importance and specifics of this area.

With this research we try to contribute to the teaching of math education based on the development of skills and competencies in the students.

The development of workshops aimed at building didactic manipulable materials for the teaching and learning process and exploring the geometrical concepts that take into account the skills and competences to be developed by the students during the “educational building” phase will help Enable students to develop a particular type of thought in order to understand, describe, and represent the world in which they live in an organized manner. to let you know how important this content is. Therefore, one day, they can use this educational product in their social life or in daily life, apart from increasing the point of view of using basic geometric knowledge.

### 2. EDUCATIONAL DIDACTIC TRENDS IN GEOMETRY CLASSES

We believe that the pedagogical construction emitted the learning process in a fixed manner and thus saved the transmitted content that was carried out with manipulated concrete materials.

According to Duval (1995), geometry comprises three forms of the cognitive process: which fulfill specific epistemological functions;

Visualization for heuristic exploration of a complex situation; Construction of configurations that can be manipulated as models in which the actions performed are represented and the observed results are linked to the represented mathematical objects; Argumentation, this is the process that leads to evidence and explanation. (DUVAL, 1995).

According to the author, these three types of cognitive processes are intertwined in their synergy and are cognitively necessary for the ability of geometry. On the other hand, the heuristics of geometry problems refer to a spatial record giving way to forms of autonomous interpretations.

For Duval (1988), geometry problems represent great originality in relation to many mathematical problems that can be proposed to students. In addition, the author says that promoting the development of cognitive functions by organizing mathematically close geometry problems that require the same knowledge determines a cognitive categorization indispensable for learning.

According to Machado (2005), most problems in teaching and learning geometry are of didactic and linguistic origin. For Duval (1995), the coordination of the various representational records (algebraic writing, geometric figures, discourse in natural language) associated with the treatment of knowledge does not work spontaneously, not even in the course of a teaching that mobilizes this variety of records .

The proposed study, however, should allow the student to interpret the geometry around him differently because if he is in his memory through contact, i. H. “The concrete”, trained, he can acquire knowledge in this pedagogical context, and this will serve as a tool to be integrated into his daily life.

According to Fiorentini:

Several methodological changes are highlighted as teaching trends that seek to privilege student participation by viewing knowledge building as a form of learning. Mathematical knowledge arises from the physical world and is extracted by humans through the senses. (FIORENTINI, 1995, p.9).

The National Curricular Parameters (PCN, 1998) also recognize the lack of importance given to this area of mathematics and also emphasize that it is developing a fundamental role in the education of the student as a citizen.

For (MACHADO, 2005) most problems in teaching and learning geometry are of didactic and linguistic origin.

The authors named above cite these aspects in the learning processes, which are intertwined in their synergy and cognitively necessary for the ability of geometry. On the other hand, the heuristics of geometry problems refer to a spatial record giving way to forms of autonomous interpretations.

We hope that the methodology applied will be an educational trend in our daily life in this area, and that we can actually use it in our classes as a learning tool in geometry classes.

### 3. THE USE OF CONCRETE MATERIALS IN THE TEACHING OF GEOMETRY

The obstacles and difficulties that students encounter at the time of learning math themselves are innumerable. There are those who cannot understand, and even the teacher is rejected in a declaration, and those who, if approved, still do not apply the knowledge acquired. That is, they don't understand their real meaning.

Several studies show that the use of manipulated materials is outperforming the use of materials by students, at all ages and in all school years. This use is a methodological indication that is valued from the moment of acquisition to the time of the construction of concepts that can happen at all levels of education. So the question arises: Can specific material as educational use facilitate the teaching and learning of mathematics? According to Carraher and Schilemann (1988), they state in their research: “We don't need objects in the classroom, but goals. We need situations where solving a problem implies using mathematical principles of logic in order to be taught ”(p.179). Because the material “can be regarded as a series of‘ abstract ’objects, which are only found in school for the purpose of teaching, and has no connection to the world of the child” (p.180). For him, concrete is not necessary for children, because manipulative objects can be abstractly in the hands in different situations, which the child has to face socially. On the other hand, Fiorentini and Miorim (1990) confirm their words spoken above and explain that behind every material used there is hidden a vision of the mathematical education of man and the world. This means that there is an educational suggestion that justifies this use.

From this research we also found questions for the public of children with special needs. How would this use be done? What would the teaching and learning process look like for these children? So, the Italian pedagogue and also doctor Maria Montessori, answers us bringing results after research with extraordinary children, developed at the beginning of this century with various manipulative materials that relate to mathematics itself. It was believed that there is no learning without action: “Nothing should be given to the child in the field of mathematics without first presenting him with a concrete situation that makes him act, think, experiment, discover, and thus immerse himself in abstraction”. (AZEVEDO, 1979, p. 27)

Having considered this, we find that this specific material mentioned here and placed as an object of applicability for effective learning makes us realize that it is necessary to exercise the child's synthetic and analytical skills, as the author states below:

The child's synthetic and analytical skills should be exercised. Synthetic so the student can create a concept out of the concrete. Analytical, because in this process the child recognizes those elements in the object that make up globalization. For this, the object that can be subjected to a transformation must be mobile so that the child can identify the underlying operation (CASTELUNUOVO, 1970, p.82-91)

The manipulated materials can be fundamental to learning, and not always the most appropriate is what is already built or more beautiful. In promoting the development of children's education, we are thinking about making the learning process easier, and at this moment we realize that the concrete involvement of this process does indeed have an affirmation. So we are compared to the national basic education curriculum which states: Essential Competences (IDEB, 2001), related to the use of resources, states that:

Manipulated materials of various kinds are a privileged resource throughout school education as a starting point or support for many school tasks, especially those aimed at promoting research activities and mathematical communication between students ... [...] All students should learn not to use just the elementary calculator but also, as they progressed in basic education, students should have the opportunity to work with the table and various educational programs, including functional diagrams and dynamic geometry (2001, p.17).

### 4. METHODOLOGY

The methodology used is qualitative, bibliographical and experimental. The research was developed in pedagogical workshops organized by modules that at the same time made it possible to develop the activities related to “construction of didactic materials as a teaching and learning process”, which we refer to here as “educational construction”.

We conducted the workshops with meetings that were shared to bring the student into contact with the topic. We apply the concepts of the figures in connection with the selected texts from the mathematics book of the discipline of geometry. Further readings were selected according to the development of the work. We used the concepts of geometric figure planning in two-dimensional space and put together geometric figures of three-dimensional space. At this stage, the learning of geometry is realized through activities related to action, there must be a dominance of concrete over the symbolic, therefore the student manipulates and constructs objects of various shapes, in order then to analyze their physical and geometrical properties. Geometric activities should be undertaken in this perspective, which will allow the exploration of concepts related to space and shape and will help the course focus on the education of an inserted student because they need to understand the geometric world in which they live.

### 4.1 THE WORKSHOPS APPLIED

Two workshops took place in a 9th grade of the elementary school, which consisted of 27 students from the state school council Bento Muniz from the municipality of Tangará da Serra-MT. A diagnostic evaluation was carried out, which was also designed as a pre-test and with the aim of obtaining information about the previous knowledge of the students in the context of Euclidean geometry, since according to the LDB guideline law and the national basis (1961)

The guidelines and proposals of the MEC (Ministry of Education) underline and emphasize the need to recognize the prior knowledge of the students by using it as a learning tool and as a motivating factor, because in this way the student becomes active subjects in the school process and is no longer mere Marginal giftedness of knowledge.

The numbers include the submitted questionnaire as well as the responses from some students. The first three questions were aimed at testing the students' understanding of conceptualizing the definitions of polygons and polyhedra, as well as the designation of some elementary geometric figures. We found that most of the students had not conceived the definitions in question, which contradicts the statements made by Almouloud (2005), in which the author explains that several studies point to this discipline as one of the learning problems. And this becomes clearer when we look at the other questions included in the test.

Picture 01 - APPLIED PROPOSAL.

Figure 2: TEST CONTINUITY

Figure - 3. Constructions

Figure - 4. Constructions

Figure - 5. Constructions

In this research it was found that the vast majority of students do not recognize the polygonal and non-polygonal shapes and do not know how to relate them to the number of their sides and vertices, and yet show the geometry of their daily life too segment, although daily life is the promoter of the study of geometry, as the author explains below:

Mathematical knowledge arises from the physical world and is extracted by man through the senses ”and Miguel and Miorim (1986, p.66) who say:“ Geometry is so important to mankind that it makes it unimaginable, the need for them To study in school, to question because the world we live in is almost spontaneously geometric and its use in everyday life is almost a human necessity. (FIORENTINI, 1995 p. 9)

Perhaps this situation will be made worse by the way teachers present this content, which PCN says is just as important as it has been in the past.

The questions in connection with the forms and relationships between them, with the possibilities of occupying space, with the position and displacement of objects in space, seen from different angles, are just as necessary today as in the past. Everyday situations and the practice of various professions such as engineering, biochemistry, choreography, architecture, mechanics, etc. require the geometrically thinking ability of the individual (PCNs p. 122 - 1998).

In a second moment, constructions of flat numbers were proposed. During the constructions it was found that the students really understand the geometric shapes presented by the constructions with cardboard, that is, the concrete. Square, rectangular, triangular, circular, pentagonal, hexagonal, heptagonal, antitogonal, eneagonal, decagonal, dodecagonal, trideclagonal, tetradecagonal, pentadeagonal, all designed in cartolines to be cut and finally constructed, were assembled. We found that the construction by the students contributed to the creation of meanings by the students themselves, with an interest in more involvement than when these numbers were presented on the blackboard by the teacher. Which reinforces the sayings of Azevedo (2006, p.27), who believed that there is no learning without action: “Nothing should be given to the child in the field of mathematics without first presenting him with a concrete situation that enables him to do so brings to act, to think, to experiment, to discover and therefore to immerse oneself in the abstraction ”.

A room was also opened in which the students had the opportunity to question and argue about the activities carried out and the relationship of the forms in our coexistence. But after the discussion was opened for comment, some students only contributed when incited and asked to comment. The most common comment from students was that they had never attended a class this way and that they found it interesting and enjoyable to build the figures instead of just seeing the teacher introduce them. Machado (2005) states that most problems in teaching and learning are geometry of didactic and linguistic origin; and Pavanello (1993) explains that some factors related to such difficulties may be related to the lack of student interest generated by the lack of didactic resources, the lack of teachers interested in using these different teaching tools to find an attractive and To develop high quality teaching that can use new didactic concepts such as the use of concrete materials or educational games in the development of their didactic practices.

This activity made it possible to check whether the pedagogical potential offered by the constructions of flat digits is very relevant and attractive for pedagogical didactic purposes in order to establish a correlation between the content of the topic to be treated.

During this phase, the students were also asked to build the flat figures in groups where they should be painted with the crayons. This type of activity takes the student to a prior plan of the construction as it requires knowledge of the geometric shapes or polygons and motor skills to handle the materials.

The workshop for the construction of flat figures, besides dynamism, was free time, motivating and educationally attractive, because at different times the groups discussed which polygon would build, because secondly. Fiorentini and Miorim (1990) “says that behind every material used is hidden a vision of education, mathematics, people and the world; that is, there is a pedagogical suggestion that justifies it. "

It is noteworthy that it was possible to observe that the workshop, as a didactic process for the activities, through visual resources and experimentation, causes a change in the attitudes of the students, with a greater motivation and participation of the majority, the greater participation of the Students in relation to the content of geometry represents because, according to Castelunuovo:

Concrete must have a dual purpose: "to exercise the child's synthetic and analytical skills"; to enable the student to construct a concept out of concrete; analytical, because in this process the child succeeds in recognizing those elements in the object that make up globalization. To do this, the object must be mobile that can be transformed so that the child can identify the underlying process. (CASTELUNUOVO, 1970, p.82-91).

Geometric solids were also made, but before we present some concepts of geometric figures of three acrylic dimensions such as pyramids, prisms, cube, dodecahedron, icosahedron, tetrahedron, octahedron, sphere, cylinder to them to make a prior planning and only then to assemble the solids, but some students had already acquired the concepts of flat shapes in the previous workshop, because the assembly of solids requires knowledge of flat geometric spaces and rigor. , according to some students had never actually worked with these solids. Cardboard, grill stick, contact stick, contact glue, scissors, etc. were used to assemble the geometric solids.

The planning of the workshop was used in each constructed figure and the ratio of geometric solids to the flat figures was applied, the assemblies were manufactured one after the other and the acrylic model shapes were also analyzed for planning. This pedagogical potential interacts with the students in the constructions of the figures, opening the discussion about the assembly was an exercise of patience for the completion of these concretes.

We can also use interdisciplinarity in this educational process because, according to the National Curriculum Parameters (1998-MEC):

The teacher should be the facilitator who encourages the development of procedures that will help develop in students interest, seek new horizons and share knowledge acquired in different situations. Taking this orientation into account, there is a great need for derinterinteraction and contextualization of disciplines and especially mathematics

That is, to further facilitate the broad view of objects and their relationships. However, we emphasize that not all students actually have an overview of the three-dimensional figures, and we have noticed that some of them still had doubts about putting together one of the shapes of a solid. We also found that this type of action prepares the student to seek solutions while maintaining this dynamic process of pedagogical learning to explore the reasoning before the constructions of geometric shapes. We checked that this activity was carried out with great commitment and participation from the students, but at some point it was necessary to accompany some students in the construction of the geometric solids. We recorded some comments from them such as: B .:

- "We think that a geometry class learns a lot more through geometrical figure constructions and is not just fun."
- "In addition to theoretical learning, we can learn better in practice, the teacher not only passes the board on and it is more interesting to take practical courses."

It is obvious how important this activity evolves in the educational field because this didactic implementation shows the reality in the classroom and working with this collaborative activity we verify the importance of the common parts of this essential activity in the life of math students and educators.

This educational potential represents measures that can be implemented in the school environment. The experiment and approach of geometric theory in their life add to educational activities, which leads students to become more interested in learning geometry.

### 5. FINAL CONSIDERATIONS

Research shows that the inclusion of activities that encourage social interaction, collaboration, and experimentation in the classroom can make a difference in the school setting because the social context into which these students are placed is not isolated from the school, and one Its tasks are to promote an education that enables the social life of the pupils. Effective collaborative participation is another point that we emphasize with the workshops, considering that it has turned out to be a positive point, which Lorenzato (2006) reinforces when he says: teaching is different from teaching. The lessons give the students the conditions to build up their own knowledge.

We also emphasize that the use of concrete materials in the teaching of geometry in addition to promoting the interaction of the students with the aim of the study and from this interaction they can build up their knowledge, also promote discussions in the sense that in this way in daily life of students to identify the removal of mathematical concepts and their applicability alongside the more-interesting and meaningful one that serves as an alternative to teaching geometry that escapes the traditional.

It should be noted that the workshops developed have achieved their goals, which started from the development of skills that enable the abstraction of geometric content that can be used to understand the world in which we live in an organized manner describe and depict.

It is believed that research itself is feasible as a working methodology if it has been prepared beforehand and has potential for the simplicity presented in its development, thus adding to the learning process of what has been learned of geometry and mathematics.

Therefore we come to the conclusion that the elaborate “educational construction” had a worthwhile result, with a lot of interaction we have achieved our goals in the process of teaching and learning mathematics, we hope that with this work as a trend in the educational process of students and Educators are included.

### CREDENTIALS

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^{[1]} PhD student in education at UNR Argentina, professor of mathematics and criminal investigation at MT.

^{[2]} PhD in Education; Specialist in school management and human resource management; Degree in English Portuguese / English and Literature, Bachelor Degree and Bachelor Degree in Physical Education.

^{[3]} PhD student in Education at the National University Of Rosario - Unr, Argentina, holds a Degree in Letters - English from Castelo Branco College (2006). Law and philosophy. Specialization in English.

^{[4]} PhD student in Education at the National University of Rosario - UNR Argentina. Graduated in Vernacular Letters from UFBA - Federal University of Bahia. He has a postgraduate degree and Curriculum, Didactics and Evaluation from Bahia State University - UNEB, Postgraduate in Methodology of Portuguese Linguistics and Literature from the Regional Faculty of Philosophy, Science and Letters of Candeias-Bahia. Spanish language training certified by DUCLE - ARGENTINA. Spoken Spanish and Portuguese language teacher. Director of the State College Professor Maria Leal Lopes, Nova Ibia - Bahia.

^{[5]} PhD student in Education at the National University of Rosario - UNR Argentina, Bachelor es degree in Letters. University of Bahia State, UNEB and Education of the Regional Faculty of Philosophy, Sciences and Letters of Candeias-Bahia. Professional Master in Theology. College of Education, EST, Brazil. Specialization in linguistics and literary studies. Federal University of Bahia - UFBA. Specialization in planning and management of Sist. Educ. Distance. Bahia State University, UNEB. Specialization in university methodology. Integrated Faculties of Ducks - FIP, Specialization in Teaching Methodology for Vocational Education, UNEB-BAHIA. Brazil. Spanish language training certified by DUCLE - ARGENTINA.

^{[6]} Degree in Portuguese and English Letters from UEG, Post Graduate in University Teaching, from Ucan Rj; Postgraduate in Technology in Education from Puc Rio de Janeiro and Postgraduate in Elementary School Methodology from UFG. Participation in the doctorate in education at the UNR argentina.

^{[7]} Graduated from the Catholic University of Salvador in Geography. Year 1987. She graduated from the methodology of geography classes. Participation in the National University of Rosario -UNR Argentina.

Submitted: July 2020.

Approved: October 2020.

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