Mathematical Education and Educational Psychology - A Radical Approache.pdf

7/30/2019 Mathematical Education and Educational Psychology - A Radical Approache.pdf

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Mathematical Education and Educational Psychology:

A Radical and Definitive Approach

Aury de Sa Leite,

BOOK


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Preface

There is no a specific psychology of mathematics, but exist a psychological theory that can give a

strong support for the Mathematics Education: the Cognitivist Psychology generally based on the

ideas about learning of Piaget, Vygotsky and Papert, only to mention three examples.

No teacher can teach math for the new generations using old methods, because they are habitual

users of innovations produced by computers. Particularly in the case of mathematics, one cannot

talk about 'teaching', but only in 'learning', and this learning should occur through contact of

students with meaningful learning opportunities, i.e., interested participation in performing

significant tasks to them.

Meaningful opportunities for learning, according to the age of the students can be presentedby

the teacher through:

(1) games for thought,

(2) the use of concrete materials or experimental devices,

(3) addressed readings in selected schoolbooks of mathematics either history of

mathematics,

(4) projects involving mathematical modeling,

(5) the solving model problems well chosen in didactical books, taking during resolution,

the teacher as the most competent peer,

(6) the indication of challenge exercises to be discussed in groups in the classroom,

(7) to display of mathematical ideas in electronic devices,

and amazingly,

(8) even through lectures with a very active participation of students.


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Chapter 1

Psychological Approaches to Mathematics Education

1.0.- Mathematics Education

In contemporary education, mathematics education is the practice of teaching and learning mathematics,

along with the associated scholarly research.

Researchers in mathematics education are primarily concerned with the tools, methods and approaches that

facilitate practice or the study of practice. However, mathematics education research, known on the continent of

Europe as the didactics or pedagogy of mathematics, has developed into an extensive field of study, with its

own concepts, theories, methods, national and international organizations, conferences and literature.

My concept of Mathematics Education (in Brazil: Educao Matemtica; in Europe: Didactics

of Mathematics and/or Psychology of Mathematics) is the following:

Mathematics teachers must have a deep knowledge of the subject and must know to

transform this knowledge into meaningful learning opportunities for students.

Nobody teaches mathematics to others, they must learn for themselves through the

learning opportunities created by educators.

Students only learn mathematics effectively when are faced with real learning

opportunities.

All the learned ideas (concepts included) through these learning opportunities should be

perfectly understood, discussed and assimilated significantly by him and his colleagues

in a study group.

From that, all the new learned ideas should apply to solve new problems, preferably

connected to the real world or linked to everyday life, when possible.

Teachers should present the Mathematics through a cognitive point of view, that is, by

applying concepts of cognitive psychology in everyday classroom full time.

I believe and I experienced this through more than 35 years working with

Mathematical Education in K-12 and Universities and talking about theories of

Didactics of Mathematics, Mathematical Education and its Learning in schools

mathematics should learned by students through a cognitive approach, that is, by

applying concepts of Cognitive Psychology in all mathematics lessons all the time.


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In this book, we are suggesting and working with Cognitivism, but it is well to remember that

there are three basic types and completely different approaches to mathematics education through

psychology: Behaviorism, Cognitivism and Humanism. Moreover, we will consider the following: we

will include the Gestalt Theory inside the Cognitive Theory, fact that is quite natural for many scholars

of this field.

When we talk about to present (by teachers or educators) and to learn (by students)

mathematical ideas according to cognitivism, would be a great heresy to talk about "TO TEACH"

mathematics, in this case we will have to talk about Mathematics Education or the Learning

Mathematics or the Psychology of Mathematics, but never "teach mathematics". A teacher do not

teaches Mathematics because it is not possible to teach Mathematics, for all students (and for you too)

only is possible to discover and self learn the mathematical ideas.

1.2.- Behaviorism, Cognitivism and Humanism

Here we will discuss, rather quickly, the main lines of psychology interested in human learning.

1.2.1.- The Behaviorist Theory - a summary

The initial ideas of behaviorism (or behaviourism), described by John B. Watson, were based

on experimental studies of the classical conditioning carried out by Pavlov.

Pavlov's experiments with dogs are a much known example of the classical conditioning. A dog

was presented to a stimulus such as a light or a sound and after this stimuli, after this, an small quantityof food is supplied to the animal. After a few repetitions of this procedure, the light or sound by itself

caused the dog to salivate.

The behaviorism conceives that: events only objective and publicly observables may constitute

the true basis of all scientific psychology, do not involving the mental constructs or events considered

happen in the private conscience. A theorist of this school was only interested in the relationship

between the "stimuli" and "responses to these stimuli" (RS) ignoring aspects or processes that connect

the stimulus and the occurrence of measurable response or observable behavior.

In short, behaviorism encompasses the basic assumptions of the construction of scientific

knowledge: facts strictly observed and controlled they can be reproduced again in these same

conditions. Behaviorism is the science of observable behavior without reference to consciousness, i.e.,

does not consider the private mental events of person.


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1.2.2.- The Cognitive Theory - a summary

Cognitive Psychology, Cognitivist Psychology or simply, Cognitivism, is a specific area of

psychological research characterized by the study of how the human brain grasps, stores, processes,

retrieves, and uses information.

In other words, the cognitive scientist observes how humans assimilate information from the

environment and how it transforms this information through internal processes and uses these

information already processed and properly adopted to act, handle objects, interact with others, make

decisions, solve problems, modify the environment, to invent, to fable, to create, discover and finally,

to live. This is the cognitive model of information processing performed by humans.

By extension, the Cognitive Psychology decides completely leave the exclusive concern with

external components and publicly observable behaviors, starting to worry about the organization of

knowledge, information processing, thinking styles and behaviors related to decision-making.

Jean Piaget and Vygotsky are the two most prominent psychologists of Cognitivism. Many

educators and teachers apply many educational ideas of Piaget and Vygotsky in the schools with

large success.

Many of the theoretical concepts of these two psychologists (and the ideas of many others

cognitive scientists) will serves as the basis for many of the original ideas introduced along this book.

1.2.2.1.- The Gestalt Theory

The Gestalt psychological theory formulated in the late nineteenth century in Germany and

Austria, was a protest against the tendency of to consider human experiences in isolated parts, that is,

by dividing the whole into pieces or atoms. The Gestalt proposes that an analysis of the parts could

never provide an understanding of the whole, since the whole defined by the interaction and

interdependence of these pieces. The many parts of a gestalt not maintain their identity when they are

separated.

The German word "gestalt" means "organized totality", that is, an ordered set of elements

(objects, facts, ideas, occurrences, etc). The Gestalt which was initially conceived as a psychological

theory, modernly is considered as a philosophy, when then is called "Theory of Form", whose interests

are the "good shapes", or by simplification: a "gestalt".

Some authors do not consider the Gestalt psychology as an independent area in Psychology, but

as a relevant part of the Cognitivist Theory. The most notable contribution of Gestalt theory to


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understanding cognition concerns the study of insight (sudden enlightenment or revelation). In Gestalt

theory, exactly this kind of sudden illuminations receives the name: insights.

The human learning, which most often occurs through problem solving, assumes that this

phenomenon (insight/enlightenment/illumination) is not entirely clear to the individual conscience.

The phenomenons that accompany the processes of problem solving often occur abruptly in regions

not accessible by consciousness. Humans think deliberately and consciously, but the solutions of some

problems occur suddenly and inexplicably as coming from somewhere undetectable of their minds.

Another issue involved in these processes of problem resolution by humans is that who solves

a problem usually does not bother to describe the mental process used to solve it. Some persons are

incapable to report on each step considered valid by them involving the resolution and the right

response to the problem. Many others will be incapable of describe or report the steps attempted, but

abandoned; some of these were discharged because are paths that leads to to a dead end or to a wrong

answer.

1.2.3.- The Humanistic Psychology - a summary

The focus of Humanistic Theory takes place directly on the person, from the analysis of their

development and their particular manner of acquiring knowledge, i.e., is a person-centered

psychological theory. In it, the human will being viewed from various existential angles, not just

through the evaluation of your intellectual behavior; persons are essentially left free to make their

choices and to live your lifes.

Humanism considers the human being potentially able to expand, of grow and develop,

making you more independent of the environment as much as possible using their skills creatively,

reaching ever higher levels of efficiency and performance. This innate tendency of human self-

realization and this quest for self-actualization is that underlie the humanist theory.

Let's find out the proposed ideas for educators by the Third Force Psychology, as it is also

called the Humanistic Psychology: (1) the learning student-centered and (2) the non-directive guidance

in the classrooms:

(1)The student-centered learning: The student-centered learning: in this approach, we

defend the idea that the educator must not transmit content or must direct the educational

process. Rather, the teacher should guide and assist, regarded in the educational process as

a facilitator of learning, a creator of meaningful learning opportunities for students. All

contents of the learning will be established based on the previous experiences of the


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students themselves. The student knowledge-base must be rebuilt and expanded from

stimulations and guidance of the teacher; the teacher will does not directs the learning

process, but only organizes lessons, printed papers, and the necessary 'concrete objects' to

realize all the works in the classroom, as well as the involved logical procedures or the

crucial experimental actions.

(2)The non-directive guidance: to guide in a non-directive intervention is to stimulate the

processes of intellectual construction taking into account the personality of the individuals,

their proposals, their aspirations and their desires and anxieties. The non-directive guidance

must integrate the students and the teacher in order to accomplish actions as people

committed to the process of growth of the group, without loss of their personalities and

their interpersonal discoveries.

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Mathematical Education and Educational Psychology - A Radical Approache.pdf