mathematical education and psychological approaches
TRANSCRIPT
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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 learnedby 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 quantity
of 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: 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 students themselves. The student
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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.
1.3. - Confronting Differences
Behaviorism, Cognitivism and Humanism are the currents of psychology more widely
theorized about learning. However, only the behavioral and cognitive ideas figures in school practice.
Humanism which provides too, like Cognitivism, the student-centered learning, is not practical
when you want to apply it in the school systems as they are currently constituted or even when you
want to use it as the basis for individualized learning systems. Another fact to be mentioned is that in
Humanistic Psychology, the teacher, a "guiding non-directive" should create conditions for the student
to learn, and these conditions do not need to have connection or commitment to school curricula.
In Cognitivism Psychology applied in schools, teachers must create and provide learning
opportunities usually linked to school practice and the pedagogical project of that school.
Since the 60s, the Cognitivism has gained prestige as the current of psychology most important
being applied to human education, a fact that coincides with the reduction of the influence of
behaviorism.
It is justified the strong tendency to consider the Cognitivism as the one that best responds to
questions from psychologists and educators about the human cognition, learning styles and the student
models. Cognitivist Psychology embodies the explanation of all types of learning - from the simplest
to the most complex -, explaining the logical thinking, the conceptual learning, the understanding and
application of rules and principles, and especially the process of problems resolution.
While the behaviorist approach is presented according to the uniform principles of the sciences
on which it is based, the cognitive theory of learning is presented in the form of very creative and
diverse theories that seem to conflict, but which, in fact, can be grouped into a same category whose
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underlying ideas: people act according to what they believe or perceive, not only based on what they
know, as advocated the behaviorism.
Conflicting aspects of Behaviorism and Cognitivism not reach the conceptions of Humanistic
Psychology. The Humanistic Psychology virtually independent of the concepts they are behavioral or
cognitive.
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Chapter 2
Pedagogy and Didactics of Mathematics
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Chapter 3
Psychopedagogy and the Mathematics Education
Psychopedagogy is a combination of two main branches of study, Pedagogy and Psychology. The Psychology of
Education.
Field of Work
- In educational centers designing and implementing intervention strategies for people with learningdifficulties and with people with a disability.
- In educational centers favoring students development through mentoring actions.
- In Psychopedagogical and special education centers undertaking tasks related to assessment,intervention and psycho pedagogical assessment.
- In organizations fostering educational, working and social inclusion of people with specialeducational needs.
- In educational centers and organizations promoting the process of improvement througheducative innovation and paying attention to diversity, thus strengthening the quality of life.
- As an independent professional, working on his own Psychopedagogical consulting firm.