When we talk about learning mathematics, we are considering the process through which the children establish a relation more frequent and systematic with the mathematics knowledge, and the children are encouraged by teacher to participate of this process in way that the new notions are learned, accepted as natural extension of knowledge they already have been developing.

Learning mathematics in school should represent, a continuous process of knowledge construction logic mathematics that the children have been developing through their observation and experiences realized with physical objects of the world, of this own space, of her own cultural universe.

This research concerns about material and its need in the classroom, noticing the relation between the educational games and mathematical apprenticeship.

The simple consisted of 78 students in second grade of elementary level in one public school in Ehime, Japan. The class was heterogeneous. The children were around 7 or 8 years old, all of them Japanese, 50% boys, and were assigned at this grade level after completed the first grade.

The students were observed in the context where naturally they were environment, I means in the classroom during the third part of school period and the task was applied by the research student and their teachers. They had 25 minute to fill in the printed paper. It was enough time. At the end we took the material for analysis.

In this present study 78 students were requested to fill in some printed shapes in a paper with colors pencil triangle (three side shape) and quadrilateral (four side shape) were among others geometrical figures.  The game goal is to find or recognize some animal designers after painting all triangles and quadrilaterals.

Were given by the teachers to students one paper for each of them which was printed the shapes and the game rule. The material used by students was colors pencils to paint the triangle with one color and the quadrilateral with a different color such identify the two different shapes. All students who were involved in this activity already knew that it was a part of our research aim and they accepted to collaborate with us. They absolutely were not pressing to do the activity. We asked them before if they want to make part of the research.

It was verified that 95% of pupils made “mistakes”, they didn´t have ability yet to recognized triangles either quadrilaterals; but always has relation with concepts, still if it was poor or uncompleted. This activity or this game involves many concepts like comparison, difference, resemblance, similarity, rotation, inclusion, union, notion of set group, geometry idea also arithmetic. Remember that geometry topic was already experienced along the first grade, but students still made “mistakes” and were in “doubt” about these concepts.

The responses to the task were analyzed and classified according of these geometrical contents were 13 triangles and 26 quadrilaterals. Altogether, there were 39 shapes to be found. Note that only one triangle shape was not recognized by the children. In fact, the identification of quadrilaterals was much more difficult than triangles, therefore 15 or 57,6% of quadrilateral shapes were not discovered for all students.

The identification of appropriate forms among others is conditioned on concepts that the students had acquired before and previously experienced. But not only geometrical concepts the current task also involves arithmetic, set group, etc. 82% did not find all triangles either quadrilaterals.

Analysis of not recognized geometrical shape per each animal designer: the butterfly was made by 2 triangles and 2 quadrilaterals. 50% of these shapes ere not found for all children; the fish was composed by 6 quadrilaterals. 60% of these shapes were not found for all children; the bird was made by 3 triangles and four quadrilaterals. 42,8% of these shapes were not found for all children; The fox was composed by 8 triangles.12% of these shapes were not recognized for all children; the elephant was made by 14 quadrilaterals. 35% of these shapes were not recognized by all children.

In this game the children had the opportunity to (re) create geometrical concepts; the mistakes that happened was necessary step to knowledge construction, the child had an activity way of reasoning, which involve with stages, until intellectual maturity. These mistakes only show the particular way of child´s thinking. We are considering Piaget humnan´s development stages. As Piaget says, in the cognitive aspect the error is only transitory, sooner or later takes up to the contradiction.

The two important findings of this study are those mathematical concepts thinking develops slowly, and the various reason for young subject´s inability or limits. First to find the correct shape depends on multiples mathematical ideas. The game involves inclusion; triangle is different of quadrilateral, but to see the animal designer the subject must use union concept to join to objects of the same class in a big set. This characterizes the union of sets. For instance look at the bird or the butterfly shape.

Let´s consider now the addition aspect that occurs in the task. The problem situation which was given to the subject was to find some animal designers made by triangles and quadrilaterals; so to do the task the subject needs to observe and join the same type of object (only triangle or only quadrilateral). This is the addition idea, join collection of resembling elements.

The significant fact was that the mathematics was troublesome task for students. Consequently, it was hard for them to learn. Most of student´s difficulties identified as belonging geometric concepts. Still for Upper Level students some basic geometric concepts are obscure or unknown although they have been experimented it along Lower Level.

However the current research has shed more light on the causes for the student´s difficulties through various studies about the knowledge construction which exercise influence in Education now a days. Thus we noted a valid tenor that considers the knowledge construction through game connected with child´s real life. Those studies suggest the game as a important and fundamental instrument to acquire mathematical concepts. Moreover, the game develops intellectual, social also affective aspects. Whereas offering a general development for students.

In fact, the teacher can evaluate the student´s performance in a good classroom atmosphere; because the game task is not a formal test, so the student is not tensioned. Hence, the student can do his mathematical activity spontaneously. Thus, the teacher can make a better valuation of student´s performance.

The game was an excellent chance to children constructs knowledge in a pleasure and satisfied way, without boring or tedious from school exercises as Freinet wanted. It´s very impressive how dedicate they were with their activity without dispersion, since they liked it. It´s clear, for us, that children needed games as fundamental school activity.