+Theory

B.   Who are the people that have worked in this intelligence?

-Aristotle, Plato, René Descartes, Isaac Newton all this thinker mans were talking about logical mathematical intelligence through the philosophy.

-Lev Vygotsky (1896-1934): Vygotsky was a very important Russian physiologist and he pertains to the phase of the constructivism. His theory said that apart from the genetic code, children were conditioned to the society and the culture they were born into at the moment they were growing up and learning. He used to say that it was not the same to be born in an occidental culture or in an oriental culture because the environment was the main factor to develop the intelligence.

Another factor that he said was very important was the language and the people around the child, like his parents or classmates. And he made emphasis on a guided learn.

-David Paul Ausubel (1918-2008): Ausubel was an American physiologist and pedagogue. He’s also a important person inside the phase of constructivism. David  made the theory the significant learning and it was based on Piaget’s theory. This theory said that the significant learning is when the new concept the child learns is well understood, linking up this new information on previous concepts. To make this possible, he made three conditions: the concept must have a logical and psychological meaning and the attitude of the child must be correct and positive

-Joseph Donald Novak: Novak was born on 1932 in America and he’s a teacher and a research scientist. His theory is based on the David Ausubel’ theory and he is famous for inventing the concept mapping which proves the quantity of concepts children have learned and can understand as well.

-Jean Piaget (1896-1980)

B.1)  Who is Piaget and why is he the most famous?

Jean Piaget (1896-1980) was born on Switzerland and he was a biologist, philosopher and psychologist who developed some theories, like the “assimilation and accommodation theory” and he made studies about childhood. He also published some studies about the child’s psychology and he made his most famous theory about the cognitive development. This theory talks about the development of the knowledge on human beings.

For him, this development is totally connected with the development of some cognitive structures which are previously formed and they're more simple.

With these studies, Piaget helps us understanding better the human development and knowing how to help children in all parts of their development.

-What is his input?

He wants to answer his question”How the brain passes from a stage with less knowledge to a stage with more knowledge?” 

This is why he starts working with children, making different tests on them and analysing all their answers and having a lot of interest on their mistakes, because he thought those mistakes were very constructive.

Piaget saw that children with the same age were making the same mistakes, so with all the patterns of mistakes of all ages, he made the conclusion of the existence of one evolutional sequence on intellectual growth.

He finally made 4 periods of cognitive development: SENSORIOMOTOR, PREOPERATIONAL, SPECIFIC OPERATIONS and FORMAL OPERATIONS.

B.1.1) What are the Piaget’s tests? How many tests are there?

 The tests of Piaget are a formal description of the evolution of our intelligence, taking into account different stages of the development. They are a sequence of tasks which evaluate a group of thoughts that make the functions of the reasoning. Those tests are not intellectual standard tests; they only contrast universal patterns of children reasoning.

B.1.2) General characteristics of the tests:

·         Category: Intelligence tests.

·         Objective: Evolution of the logical reasoning. They examines why, the process or the structure through the child can reason.

·         Application: Individual.

·         Time: Variable.

·         Age: 0-12

·         Material: It depends on the test + protocol of answers.

·         Ambit: Educational-Investigation.

-Other characteristics:

·         They can be used as a way to stimulate cognitive abilities.

·         With those tests we can identify clearly psychological changes.

·         Those tests must be done when the child and the investigator have a little relationship between them.

·         We have to fix our attention on 3 characteristics of the answer of the child:

o   They have to recognize the materials

o   The vocabulary used

o   Intention to answer the test

B.1.3) Types of tests:

There are 3 general types of Piaget’s tests, but there are some different tests to prove each one:

·      Series

·      Conservation

·      Classification

1) Definitions:

A.     What’s Logical-Mathematics intelligence?

To start the definition of Logical Mathematics Intelligence, first, we have to know some previous information to understand better all we are going to work on. This is why we are going to ask our self some questions.

•What is intelligence?

Intelligence is the ability to understand, reasoning, assimilate, plan, think abstractly, learn quickly and solve problems. The Dictionary: Spanish Royal Academy, defines intelligence, among other meanings, as the "ability to understand or comprehend" and as "capacity of solving problems". But intelligence seems to be linked to other mental functions as perception or ability to receive information or memorize.

•How many types of intelligences exist?

There are 9 different intelligences known:

            1. Linguistic Intelligence

            2. Musical Intelligence

            3. Spatial Intelligence

            4. Bodily-kinesthetic Intelligence

            5. Interpersonal Intelligence

            6. Intrapersonal Intelligence

            7. Emotional Intelligence

            8. Naturalistic Intelligence

            9. Logical-Mathematics Intelligence

•What factors are needed to have these intelligences?

The development of those intelligences depends on different factors on children’s life:

            a) Hereditary factors.

            b) Environmental factors such as education, motivation or healthy habits.

            c) Other factors like a bigger amount of knowledge neurons.

•On how many parts is divided the brain? All parts have the same intellectual abilities?

The brain is divided in two different parts or hemispheres, and those of these parts have different functions and have different abilities:                                                                    -Left hemisphere: it works with the logic, with analysis and sequential thoughts.                -Right hemisphere: it’s very creative.  It works with music, paintings, art... It has a global view.

Taking all this as a base, we are going to focus on the last one intelligence.

First of all, we are going to define logic and then mathematics for trying to make a simple definition of what we call Logical-Mathematics Intelligence:

-Logic is a formal science which studies the valid reasoning and demonstration.

It examines general forms that arguments may take; which forms are valid, and which are invalid.

-Mathematics studies the valid inferences within some formal language and doing an individual division of the contents.

*Reasoning is the capacity that human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions, and beliefs.

*One inference is the deduction of a thing, a conclusion or a demonstration.

So ... Logical Mathematical Intelligence is:

-The ability to use logic, reason and mathematics to solve problems.

-The ability to apply principles of cause-and-effect and predict possible answers.

-The appreciation of patterns as well as different mathematical contents.

2) Piaget's Stages of Cognitive and L.M. Development:

Stage	Age	Characteristics of Stage
Sensorimotor	0–2	The child learns by doing: looking, touching and sucking. The child also has a primitive understanding of cause-and-effect relationships; this is called practical intelligence. Object permanence appears around 9 months. They only think on immediate present.
Preoperational	2–7	The child uses language and symbols, including letters and numbers. Egocentrism is also evident.
Concrete Operations	7–11	The child demonstrates conservation, reversibility, serial ordering, and a mature understanding of cause-and-effect relationships. Thinking at this stage is still concrete.
Formal Operations	+12	The individual demonstrates abstract thinking, including logic, deductive reasoning, comparison, and classification.

On Sensorimotor and Preoperational periods children are able to:

-Discover and play with objects

-Explore and establish relations between objects and people.

-Making classifications and groups of things taking into account their property.

-Deducting, taking conclusions and making general views, through chance.

·All this actions start at a very young age and from here, children can form new ways of thinking which are called: operative logic relations. 

3) Piaget’s Research is divided in 3 different knowledge developments:

 1.   Physical

 2.   Logical-Mathematical; also is divided in:

a.   Classification

b.   Series

c.   Numbers

 3.  Social

4) Human's Brain:

Chart of ages about useful information of the development in different cases

	AGE: 0-3 years old [NURSERY education]
How is the logical-mathematical development?	0-1 years old: The brain is not prepared for this development. 1-2 years old: symbolic thought development and appearance with symbolic and language function on the brain. +2 years old: decomposition of the thoughts in: a) images b) symbols and concepts.  -Children start using imitation, memory and their own thoughts. Also, they start to know that objects which are hidden  are still existing.
How this development must be worked on educative centres?	-He must work this development taking into account the perspective and constructive conception of the materials or objects we are working on. -Dynamic learning by doing games, songs, images, stories... based on their around and adapting them to all that information. -The development must be constant, individual and sometimes in groups; it’s very important not to work it only at school, it has to be in all cases of the live of the child (at home, with friends...) -It’s important to prevent wrong ways of thinking and wrong aptitudes.
What “Generalitat” says about how to apply this development?	-Numbering and calculation: -Space and forms: -Measurement: -Statistics and random:
Do the schools apply what scientists say, what “Generalitat” says or what they think is the best?
	AGE: 4-6 years old [PRIMARY education]
How is the logical-mathematical development?	-They start to have intuitive thinking, and it helps the child to strengthen the logical operations. -Children are able to think in a symbolical way and make simple operations.
How this development must be worked on educative centres?	-In this stage the development must be work by handling objects and linking their together. An atmosphere with lot of stimulation can help in a very positive way to make observations, manipulations and descriptions. It also can help to distinguish particular things between abstract thoughts.
What “Generalitat” says about how to apply this development?	Maths is an obligatory subject at school because all the contents are essential for the personal, social and academical live of the child. But, all this contents aren’t gained if teachers don't show children how to use them in an everyday life; because this is the main object of this subject. -Numbering and calculation: Knowing how to calculate simple addition and subtraction operations, knowing how to use  numbers in a graphical, symbolic and verbal way up to 1 000, Knowing how to use the Euro and developing the mental calculations. -Relations and changes: Making a selection, a classification and a rank with objects, taking into account different standards. -Space and forms: Knowing figures in 2D and 3D on the reality and their properties. -Measurement: Recognizing the magnitudes: length, weight and time. -Statistics and random: Starting to learn how to make own questions and picking, organizing and present information to solution them.
Do the schools apply what scientists say, what “Generalitat” says or what they think is the best?
	AGE: 7-9 years old [PRIMARY education]
How is the logical-mathematical development?	-From 7 years old: At this stage the child is able to make his thought reversible as well as solving problems if the object is present. This stage also develops the ability to serialize, classify and mentally organize. The development is taking place in the socialization process and relationships become more complex. -This stage is suitable for solving concrete problems in a logical way. -Children start to understand conservation rules and they are able to classify and order things.
How this development must be worked on educative centres?
What “Generalitat” says about how to apply this development?	Maths is an obligatory subject at school because all the contents are essential for the personal, social and academical live of the child. But, all this contents aren’t gained if teachers don't show children how to use them in an everyday life; because this is the main object of this subject. -Numbering and calculation: Knowing at all the decimal system, knowing how to make multiplications and divisions, developing mental operations and dominate the Euro. -Relations and changes: Learning how to create numerical and geometrical series and apply it in different ways. -Space and forms: Representing and investigating about the relations between different geometrical figures, describing, understanding, make maps and knowing how to use technological objects. -Measurement: Knowing how to use magnitudes of length, height, volume, area, time and degrees and making activities with them. -Statistics and random: Making questions based on usual facts and knowing how to answer them with researched information previously well analysed. Also, starting to describe facts in their degree of truth and predicting the results of some simple experiments.
Do the schools apply what scientists say, what “Generalitat” says or what they think is the best?
	AGE: 10-12 years old [PRIMARY education]
How is the logical-mathematical development?	-They start  to stablish serial relations. +12) - Children have the capacity to solve abstract problems in a logical way. -From 11 to 15 years old: In this period the teenager can do operations of second-degree. At this stage the qualitative development reaches the highest points and they develop idealist feelings.
How this development must be worked on educative centres?
What “Generalitat” says about how to apply this development?	Maths is an obligatory subject at school because all the contents are essential for the personal, social and academical live of the child. But, all this contents aren’t gained if teachers don't show children how to use them in an everyday life; because this is the main object of this subject. -Numbering and calculation: Recognizing and using fractions, decimal numbers and percentages and manipulating them in different cases. -Relations and changes: Analysing the proprieties of numbers and knowing how to make numerical and geometrical series. -Space and forms: Knowing perfectly all 2D and 3D figures. Representing figures on coordinate axes, etc. -Measurement: Using magnitudes of length, height, volume, area, time and degrees and using their international system. -Statistics and random: Solving difficult problems with other’s information and searching for useful data, knowing how to use the calculator and other technological objects and using statistics, random and logic for having good decisions in real life.
Do the schools apply what scientists say, what “Generalitat” says or what they think is the best?
	AGE: 13-15 years old [SECUNDARY education]
How is the logical-mathematical development?	-Reflections are understood in a scientific way. -From 11 to 15 years old: In this period the teenager can do operations of second-degree. At this stage the qualitative development reaches the highest points and they develop idealist feelings.
How this development must be worked on educative centres?
What “Generalitat” says about how to apply this development?	-Numbering and calculation: -Space and forms: -Measurement: -Statistics and random:
Do the schools apply what scientists say, what “Generalitat” says or what they think is the best?
	AGE: 16-18 years old [HIGHER education]
How is the logical-mathematical development?	-Adolescents are capable of highly abstract forms of thinking and reasoning.
How this development must be worked on educative centres?
What “Generalitat” says about how to apply this development?	-Numbering and calculation: -Space and forms: -Measurement: -Statistics and random:
Do the schools apply what scientists say, what “Generalitat” says or what they think is the best?

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