FINAL REFLECTION

Hello everybody! 

During the last weeks I have written several posts on the geometry in different areas: didactic methods, techniques and materials for the learning of the geometry in Primary (the geoboard, the technological applications and the movie 'Donald Duck'), the geometry represented in different artistic styles (Picasso, urban art through Invaders …), the geometry in the sport (the basketball), the geometry in the cities (grid plans) and the history of some geometric figures in the flags. 

With these posts I have pretended the readers of this blog see different ways of learning geometry. Although several of the posts are not directly didactic, they are very valid to teach this mathematical area. Through real contexts learning is very enterteining and useful. For example, by means of Picasso's pictures children can identify flat figures, analyzing the national flags they can know something of the history of those countries, or as the lines of a basketball field can be used by the students to measure them and calculate the area and the perimeter.

This has been the aim of this blog, to teach geometry in a different way and to create an awareness of presence, influence and importance of the geometry in the real life.

I hope you have enjoyed it!

LEARNING GEOMETRY WITH DONALD DUCK

There is a cartoon film for children called 'Donald duck in the country of the mathematics' that is perfect to see geometric concepts, providing a context and explaining how from geometric figures certain objects and places are composed, in a entertaining way for children.

From the minute 7 up to 13 it talks about geometry. “Fue nuestro viejo amigo Pitágoras quien descubrió las matemáticas el que descubrió que el pentagrama está lleno de matemáticas” (It was our old friend Pythagoras, who discovered the Mathematics, the one who also discovered that a pentagram is full of Mathematics) In this manner the explanation of the geometric concepts begins. It centres on the golden proportion, detaling the law of proportionality and showing some representations of this proportion in the reality, like in the architecture or the painting. It also talks on the pentagons and its real examples, like flowers and starfishes.

In the minute number 16 it explains that the sports are practiced on geometric figures, and it shows some examples, like the baseball (rhomboid) or the basketball (rectangle). 

From the minute 22:30 it talks about the circles and triangles. It explains how with a circle and a triangle a sphere is formed, or how through the top part of a sphere a magnifying glass is done. Finally, it explains how from turning a triangle on an axis a cone is formed, and how this one, cutting it of diverse ways, can represent from the orbit of the planets to the glass of a telescope.

From my point of view, it is a great movie for the third cycle, although it would not be bad for 8 and 9-years-old children (the second cycle). They explain different key concepts to understand the geometry in a enjoyable way (the presence of Donald duck is a point of attraction) and promotes the curiosity in the pupils, since it shows real examples, so pupils think and reflect that the geometry is present in every space, that it is not anything that they are forced to learn in Primary although it does not make sense.

I would like to share the whole film with you that I honestly recommend.

NATIONAL GEOMETRY

In June the 14th the World Cup of football begins. From this date thousands and thousands of flags are going to be waved at the Russian stadiums (World Cup host) and for the streets and balconies of all the participant countries, flags composed of geometric figures. All these flags are rectangular (it is to say, four sides, right angles…) but inside them there are geometric elements that we are going to analyze then. 

In the majority of the flags there are interior rectangles delimited by horizontal (like the Spanish flag) or vertical lines (like the French) to separate the different colors. Others, like the Swedish or English flag, have a cross, which consists of two bars that intersect in right angle.

However, there are others that have more particular geometric figures. One of them is Brazil, whose flag has a blue circle -it reflects the position of the stars in the sky of Rio de Janeiro the proclamation of the Brazilian Republic Day in 1889- on a yellow rhombus that represents the House of Habsburg.

Finally, there are other flags whose symbol is a circle, like the Japanese or Korean flag. In case of the Korean one, this symbol is a ying-yang that means the heat and the light (red part) and the cold and the darkness (blue part). This flag possesses four blacks taeguks, which consist of a number of straight bars that according to each taeguk are three, four, five or six that represent the land, the water, the fire and the sky. 

In the end, with this post we can see how the nations have constructed their flags from geometric figures and forms that symbolize different events or beliefs. Children could investigate to know the symbology of the countries, so they would expand their cultural knowledge about the countries while they are working with different contents of geometry. It could be a transversal and entertaining activity.

PICASSO, EXPONENT OF THE GEOMETRY

Born in Málaga in 1881, Pablo Ruiz Picasso continues being considered by the majority of art historians as the most influential and representative artist of the modern art. For eighty five years he did not stop creating and rebranding oneself constantly and in this time the geometric shapes did not leave him.

Between his many contribu- tions to the art of the 20th century, we find the Cubism, an authentic revolution in all the areas of the Art: perspective, fragmentation of the objects, use of the color… After this artistic current, the contemporaneity started. This forefront began in 1907 when Picasso, who painted in the same year his Autorretrato (Self-portrait), dares to defy to the artistic world with Las señoritas de Aviñón (The young ladies of Avignon). The realism, finally, jumped over the airs. 

The geometry appeared in the world of the painting and the sculpture: parallel, perpendicular and oblique lines that delimit plans; bodies as cylinders, cones and spheres, three-dimensional in sculptural works reduces to the two-dimensionality in the Picasso's painting; basic geometric figures appear like the squares, rectangles, trapezes and trapezoids, triangles... that, combined with the colors, creates hieratic figures.

As a great example we can remember "El estudio" (The workshop, 1927-28), a work that presents the human anatomy and the objects with very simple plane figures: an isosceles triangle combined with another reversed rectangle, which shape the body of the painter; two parallel vertical lines, that represent the legs; other two horizontal ones to draw his arms; an isolated white circle that suggests the thumb that holds a non-existent palette; and an ellipse as head from which a trapezoidal face with three eyes arises. The painter is placed in front of the rectangular canvas on which he is going to paint a still life in which quadrilateral persist as decoration on the walls; on a table composed of trapezoids, rhomboids and triangles, a fruit bowl drawn by means of two triangles joined by two of its vertexes, and a circle inserted in the top triangle suggesting a piece of fruit, to end with an irregular hexagon representing a face. 

Picasso is one of the biggest artists in the history, and he used geometry to revolutionize the painting. Analyzing in a geometrical way Picasso's works would be an interesting activity to learn geometric figures and forms through a transversal topic, the art. Children could observe and point the geometric elements that they see and they could interprete the meaning of the figures.

DIDACTIC APPS TO LEARN GEOMETRY

The new technologies are revolutionizing all the areas of our lifes and have had an important impact on the learning methods of all the school matters. There are IT tools to learn geometry in a different way. These applications  can be downloaded in the mobile phone, so it is available for the majority of the population.

The first application that I have analyzed is 'Geometría'. This instrument calculates the area and the perimeter of all the plane figures and the volume of the geometric shapes too. In addition, he calculates diagonals, diameters… Only the user has to write the data and the program calculates it automatically, detailing every step up to verifying the final number. In my opinion it is a high quality application.

The second one is 'Pythagorea'. This application (in English) consists of twenty-six types of geometric contents (trapezoids, parallels, circles, rectangles…), and in each of them there are several minigames that the user must carry out. For example in the section ‘isosceles triangles’, one of the games is "constructing an isosceles triangle whose vertices are three of the given points”. On the screen there are several points and the player has to guess which points must be used to build the triangle. The minigames increases its difficulty when the easiest are completed. I have very good opinion about this application, I think it is entertaining and very rewarding to learn geometry.

Finally, I discovered 'Figuras geométricas', but it is the less elaborated application of the three I have been talking about. For using this one, the user has to guess types of triangles, the degrees of a polygon, elements of the circle… seeing the image on the screen. The application provides the letters and numbers of the response in a disperse way. Although it seems to be less valuable than the previous ones, also it is effective to know the variety of plane figures, to remember formulae...

Definitively, phones offer us new ways of learning geometry, to use them at any time and place.

GEOMETRIC INVADERS

The last year the urban French artist Invader chose Málaga to place 29 geometric works shaped like pixelated interventions in the style of the game 'Space Invaders'. 

His work consists of multiple mosaics that represent, principally, the spaceships of the arcade game who triumphed in the 80's. His creations can be in a wall of the promenade of Málaga or in diverse streets of the historical center. From this personal project, Invader has developed his own figures in all kinds of forms and colors, but always in the line of compositions based on small mosaics that simulate pixels.

His work tries, especially, to liberate the art "of the alienation that museums and institutions can suppose". But it also has as motive to dig up the characters of the game 'Space Invaders' of the screens to bring them to our physical world. 

Keeping to the geometry, the works are composed of small square units of diverse sizes which distribution forms different animations, as a "bailora", a crab, a beach or even Picasso. The contours also are squared, so there are not curves, but straight lines, whose angles are 90 degrees.

In its web page, Space-Invaders.com, appears as a location the capital of Malaga. Invader has an own application for mobiles, Flash Invaders, in which the user gains points whenever it takes a photo to one of his works, since the pixelated design of these ones allows the camera to detect about which it treats. In this way, the followers of this artist of French origin can accumulate points and compete between them. The 29 designs placed by the capital contribute to the users of the application 1.020 points. 

One year later, many of the works continue in the of Malaga streets. This one is one more sample of how the geometry is represented in art, and in this case a kind of art as curious as they are the pixelated squares to create geometric figures. 

THE GEOBOARD AND ITS DIDACTIC UTILITY

The geoboard is a very interesting didactic resource to work in the geometry, since it serves us to introduce the geometric concepts in a manipulative way. With it we also can discover the properties of the polygons or even to solve mathematical problems, to learn the areas, perimeters,… it is definitively a very useful resource to learn mathematics.

The geoboard was created by the Egyptian mathematician Caleb Gattegno in 1960, who was looking for a method to teach the geometry in a more didactic way. Although nowadays the majority of geoboards are made of plastic, the original one consisted in a square board of wood with nails forming a plot.

The method is very easy: we just need a rubber band and the children need to hook it to the nails to form geometric figures and to observe the area of polygons, for example.

For pupils between 3-7 years, this tool is very interesting to learn different concepts. In these ages the geometry should not be studied, but experimented. A good way to experiment is allowing them to promote their creativity. That is to say, allowing that the child freely explores his or her environment, without a specific goal. Perhaps they can discover the basic concepts and polygons, comparing different geometric forms, size and lenghts… In the following video we can observe how 6-7 yeared-ol children learn freely, form horizontal and vertical lines, etc.

                               
                       

Since they are 7 years old, the children can create complex plygons, classify them, recognize the basic elements of polygons (vertex, side and angle), observe the symmetry, calculate the perimeter and area of a polygon, solve the mathematical problems… In the next video we can see how it calculates the area of different polygons.

                                

THE BASKETBALL IS GEOMETRY

The geometry can be found in more areas of what we think, and one of these ones is the sport. Since the sport includes many disciplines, I have made concrete a bit, and I have chosen the basketball, sport that I know enough. There are different aspects of the basketball where the geometry appears. Let's have a look!

The first one is the basketball court. If we observe it, we can see that the court is a rectangle of 28 meters long and 15 meters wide, divided in two halves of 14 meters long. The zone is also a rectangle, of 5,8 meters long and 4,9 meters wide. Thirdly, I would highlight the central circle of the court, where the jump takes place between two players to begin the game. This circunference has a diameter of 3,6 meters. In the 2-points area there is a semi circunference that is 4,6 meters to the hoop, and whose diameter is identical to that of the central circle. Another semicircle that exists is in the zone, whose diameter is 1, 25 meters. There are other measures and plane figures as we can see in the image.

The second aspect encompasses other physical elements that integrate this sport. The basketball board is the first one, whose measures are 1,8 meters long and 1,05 meters wide. Inside it there is a small board, of 0,59 meters x 0,45 meters. The hoop has a diameter of 45,7 cm. Finally, the ball, whose diameter is 23 cm.

The last aspect is the game. During the practice of the basketball the players do movements that are studied in geometry. For example, to defend is advisable to place the legs in a 45-50º angle not to allow that the attacker can overcome the defender. Another example would be the shot. In this case, when a player throws the ball forms a parable in its distance, whereas the player has to place the arm in a 45º angle to throw correctly.

For children it might be an entertaining activity that they were finding the area of the rectangles and circumferences, the radius... in this sport, since they would relate these numbers to concrete things, not simply they would operate numbers of an abstract way. What is more, even a more dynamic and rewarding activity would be that the own pupils went to the court and they were measuring, that means, that they were extracting the information to be treated and analyzed then.

THE GRID PLANS

The geometry is present in the cities, and more concretely in the expansion projects of the big cities. The growth of the population was demanding the construction of new places, new neighborhoods in order that the people could have somewhere to live.

Many cities followed a grid plan. A grid plan is a type of urban development planning that organizes a city by means of the design of its streets in right angles, creating rectangular blocks. The appellative "grid" ("hipodámico" in Spanish) comes from the Greek architect Hippodamus of Miletus, considered one of the parents of the urbanism, whose organizations plans were characterized by a design of rectilinear streets that were crossing in right angle. An urban plane called orthogonal plane is used. The cities that present this type of urban planning have a perfectly distinguishable urban morphology in its tracing, as we can see in the image (tracing of Barcelona).

This type of planning has an advantage: its parceling is easier due to the regularity of the form of its blocks. In spite of this apparent simplicity, this type of plan presents some disadvantages, since it extends the length of the distances. To avoid it, diagonal streets are built. In order to increase the visibility in the crossings of the streets, bevels can be designed, which are an urban resource that consists in removing the corners of a building to have a better observation and to extend the crossings.

One of the cities in which it is possible to observe this planning is Barcelona, created by the engineer Ildefonso Cerdá. The "ensanche" extends along a big surface, with long streets and avenues, diagonals (one of them, the known one "Diagonal de Barcelona"), and 45º  bevels for a better visibility. 

To sum up, the city of Barcelona is a great example of how the geometry is a participant in the organization and distribution of the neighborhoods and places of a city. 

WHY STUDYING GEOMETRY IS THAT IMPORTANT?

I think that it doesn't make sense to write about geometry if I have never talked before about the importance of its study. Therefore, in the following lines, I give my opinion on why it is important.

From my point of view, the necessity of geometry teaching at school is for a simple reason: the role that geometry has in the daily life. A basic geometric knowledge is fundamental for all the people. We need to orient in the cities to move from one side to another, do estimates about the distance we have to travel… For these aims it is necessary a basic notions on geometry.

In addition, the geometry is present in several areas of our life. Some of these areas are religion, art, architecture, topography, distribution of the cities… In the following posts I will deepen more about some of these areas.

If we delimit the space to that in which a child grows there are lots of geometric elements: windows, tables, chairs, doors, shelf… That is to say, at home, at school, there are different geometric elements with which a child keeps a relationship.

For all of these reasons, I believe in the significance of geometry in the teaching-learning process.

INTRODUCTION

Hello and welcome to my blog! I am Andrés and I am a Primary Education degree student at the University of Málaga. As a future teacher (or I hope so) I am learning different geometry strategies and methodologies. 

Geo Didactic is a blog whose main objective is to make readers aware of the influence and the importance of the geometry in our lifes and to learn how to teach geometry in a different way, through entertaining methods and materials and by means of real contexts. The geometry is everywhere in our everyday life: buldings, panel of bees, the art, even in ourselves. That's the reason why it is important that children can acquire those knowledge. For that goal, I will be writing and sharing with you different methodologies and activities to teach geometry, representations of the geometry in real life, curiosities... 

I hope you like it! Let's go!