About the Developing of Mathematical Thinking in Children

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ABOUT THE DEVELOPING OF

MATHEMATICAL THINKING IN CHILDREN:

A WORKSHOP EXPERIENCE WITH

MEXICAN STUDENTS

L. Sbitneva1, N. Moreno Martínez2, M. Cervantes1 1 Morelos State University (MEXICO)

2 CINVESTAV-Matemática Educativa (MEXICO)

[email protected], [email protected],[email protected]



INTRODUCTION.

• In the frames of the National Plan of Development 2001-2006the Council of Basic Education in Mexico has assumed acompromise to attend the special educational necessity ofchildren with extraordinary aptitudes in primary education [1].

• Thus at the state level, Morelos State University took part of this

program within the Pilot Project• Which offered a program of logic Mathematics Thinking for

160 children selected from different public schools on thebase of special tests.

• There have been formed four groups for each category,starting from 3-4 grade of primary school up to 3d grade of

secondary school.• Compared to the traditional methods used to teach

mathematics. In our sessions there have been applieddifferent techniques.



OVERVIEW OF THE WORKSHOP

WITH CHILDREN WITH

EXTRAORDINARY APTITUDES



OVERVIEW OF THE WORKSHOP WITH

CHILDREN WITH EXTRAORDINARY

APTITUDES

• During the sessions of workshops dedicated to childrenwith extraordinary aptitudes, detected by their schoolteachers, we have observed that every one of them

demonstrates a higher level of sensibility correspondingto the grades 3 and 4 of primary education comparedwith general case.

•

The gifted children or those who are capable todemonstrate significant efforts and goals comparing toother members of their social group in the activities ofdiverse fields such as scientific, technological, socialhumanity, artistic and/or kinematic activities areconsidered to be “Children with extraordinary aptitudes”





APTITUDES

To participate in our Project ofMathematics and Logic ThinkingDevelopment only two types ofextraordinary aptitudes aresupposed to be treated, namelyintellectual and creative one, sothat the recommended childrenhave been selected by their school teachers according to thecriteria corresponding to theseaptitudes. Some children havebeen presenting only one or bothextraordinary aptitudes butalways with interest towardmathematics activities.





APTITUDES

For some children it was more easy to grasp an ideapresenting a graphic interpretation of a givenproblem, others was making some sort of estimations

or some metaphoric o even fantastic comparisons,also there were children who needed tangibleobjects.





APTITUDES

The objectives of theproject was focused onthe development of themore adequate

perception of classactivities and on theformation of abilities to use(generate) mathematicallogic thinking in order toproduce a structured planin order to obtainadequate results in solvingproblems





APTITUDES

It has been noted in some occasions that a child hasbeen recognizing his mistakes (errors) at the verymoment of presentation of his/her solution, and

realizing immediately appropriate corrections to theinitial solution.





APTITUDES

For some students the favorable factors that facilitated theprocess of assimilation to the conditions of the workshops

happened to be such as the opportunity to have permanentcontacts with other children with similar points of view andhabits. These classmates have been forming small groups ofdiscussions of the problems with similar solutions giving a varietyof equivalent arguments, which has been producing greatbenefits because they never have had such sort of opportunityof mutual intellectual activities as well feedback





APTITUDES

For another group of children itseemed to be enough to listen toargumentations and explanation

of variety of points of view, so thatlater on they could be able togenerate their own point of view or solutions. These children apparentlydid not like conversations with their

classmates neither presentationson the blackboard. So that we hadto revise their solutions in writtenform.



MATHEMATICS: PRACTICE AND

DIDACTIC. DESCRIPTION OF SOME

PRACTICES REALIZED IN THE PROCESS OF

SOLVING SOME PROBLEM SITUATIONS



DESCRIPTION OF SOME PRACTICES REALIZED IN

THE PROCESS OF SOLVING SOME PROBLEM

SITUATIONS

We have proposed some problem situations for children to initiate their activities which aim togenerate a process of personalization so that they

could construct a mathematical object and/or thelogic procedure which leads to a solution.





SITUATIONS

At the first step every childreceived a labyrinthcorresponding to a level lower than that it is supposed to be

for their age, so that couldfamiliarize with the problemand to gain believe andassurance to such sort of tasks

Labyrinths

Children have been asked to find the exit of a series oflabyrinths.





SITUATIONS

The second step

required a little moreeffort and initiativebut still inaccordance to their school level, 3 and 4

primary grade





SITUATIONS

Finally, the thirdlabyrinth, which wasdesigned supposing for the higher level of

primary school, for theage corresponding to 5and 6 grade. Thepurpose was to seewhether children can

follow the same type ofstrategy although thelevel of complexitygrew up considerably





SITUATIONS

Children accepted this Challenge with enthusiasmand to our surprise accomplished with their task almost within the same time range (approximately 5minutes for each labyrinth).

During another session of Workshops, children weregiven the so called “impossible” labyrinths, so nameddue to rather irregular shapes, with contours far awayfrom the rectangular one as in the previous task. Thereaction was to choose firs the labyrinth with a formwhich resembled the familiar one or which seemedto be more attractive to some respect. Every childhas been found identified with one particular type ofthe representations.





SITUATIONS

Define abbreviations and acronyms the first time they

are used in the text, even after they have beendefined in the abstract. Do not use abbreviations inthe title or heads unless they are unavoidable





SITUATIONS

Tantrix

Tantrix is a game from New Zealand, which has quicklybecome more popular worldwide thanks to the propertiesthat provides recreational and educational. In recognitionof this, has been awarded many times in differentcountries [2].





SITUATIONS

Children formed two teams and each was given puzzlepieces unconventional: Tantrix. The challenge was to geta ring of a certain colour, using all hexagonal pieces ofthe puzzle. Considering that wherever stay connected

hexagons, the colours should match. Moreover, they mustnot be holes in the solution.





SITUATIONS

The activity took place outside the classroom. The pieces were50 cm. The implementation of this dynamic and interdisciplinaryinterconnection was first performed with very good results in thegroup. They worked in collaboration with students ofoutstanding student group of 5th and 6th grade. Both, boys and

girls teams, included the 4 degrees





SITUATIONS

The purpose of thesegame activities wasdifferent due to another perspective comparedto problems with

labyrinths. It wasexpected theperforming of anadequate distribution ofthe plastic figures in thespace so that the

correct edges ofhexagons emergedwould coincidegenerating closedsimple curves of theindicated colour.





SITUATIONS

The solution should be selected as the result of a collectivedecision. The colours and big size of figures played amotivational role to initiate the activities





SITUATIONS

It was observed that ateam formed by malepupils as a major partarrived to a solution by

means of previousexplorations of the figuresplaced on the yard.Meanwhile as the teamwith the majority of

female pupils arrived totheir solution by trail anderrors approaches to smallpatrons emerged as thesolution of early attempt.





SITUATIONS

Despite of the distinct approaches of perception of the task andexploration of the solution both teams obtained (gained) theequivalent results. Finally it was announced 50/50

Children often demonstrated better acceptance of the tasksproposed in Workshop sessions when the presentation of the

problems possessed colors, pictures, figures, etc.





SITUATIONS

Among the problems that have been preferred andresolved with notable interest there are problems of theOlympiad type suggested for the national competitionKangaroo.

Problem K3

• The figures of the same shape represent the sameLumber. Which values could correspond to a triangle?





SITUATIONS

Problem K1

• The Square on the Figure has four circles en everyone of its Edge. The sum of the numbers which arefound inside the four circles of each edge is thesame. Find the value of the sum (A+B+C).





SITUATIONS

Problem K2

• The rectangle on the Figure is formed by thesquares. The edge of shaded square has longitude2 cm. Find the longitude of the mayor edge of therectangle.





SITUATIONS

Problem K3

• The figures of the same shape represent the sameLumber. Which values could correspond to atriangle?



CHILDREN'S OPINIONS ABOUT

ACTIVITIES OF WORKSHOP



CHILDREN'S OPINIONS ABOUT ACTIVITIES OF

WORKSHOP

At the end of the semester there has been appliedan interview in order to have an idea what opinionshave children with respect to the mathematics andthe activities which have been undertaken within the

project:Among the 11 questions there was one as follows:

What is the mathematics for you?

Do you believe that to do mathematics means onlyto treats numbers?

How would you like to be taught the mathematics?




WORKSHOP

What is the mathematics for you?

25% of children responded that it may be

considered as a tool instrument for achieving atriumph.

35% consider the mathematics activities as a sort ofmental game.

And 40% have written that it is series of activitieswhere the arithmetic operations are involved




WORKSHOP

Do you believe that to do mathematics means only totreats numbers?

All children responded negatively, pointing out that

there are mental problems also.




WORKSHOP

How would you like to be taught the mathematics?

Giving the following answers:

“Playing games”— ----35%,

“Reading”------------ — 10%

“Solving problems”— 15%

And 40% of responds included these tree answers.



REFERENCES

[1] Secretaría de Educación Pública (2006).Propuesta de intervención: Atención educativa a

alumnos y alumnas con aptitudes sobresalientes.

[2] http://www.tantrix.com.es/

About the Developing of Mathematical Thinking in Children

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