Wednesday, 25 July 2012

Unit-6 Psychological Theories and factors influencing the Learning of Mathematics



Psychology of Learning Mathematics
Teaching and learning mathematics have many myths associated with them. Two of them are : mostly men can learn and enjoy mathematics, and math is a combination of rote computational activities and the logic of word problems- nothing more, nothing less.
As is the case, understanding, and using mathematics involves many more strategic operations than implied by memory and logic.
Gagnes Type Of Learning
Gagne suggested that there are five categories of learning. Such categories can be formed because each leads to a different class of human performance and each requiresa different set of instructional conditions for effective learning (Gagne.19147)
The categories are :
1. Intellectual skills: These skills are the capabilities that make the human individual competent. They enable him to respond to conceptualizations of his environment.
2. Cognitive strategies : These skills are the ones that govern the individual capability to learn, think and remember.
3. Verbal information : Stored in our memory, to recall when needed, such as names of months, days of week, letters, numeral etc.
4. Motor skills : The capability to learn : to ride a bike, drive a car, write, draw a straight line.
5. Attitudes : All of us possess attitudes of many sorts towards different things, persons and situations. These attitudes may affect our position toward those things.

Gagne’s Learning out Comes
Verbal Information
Stating previous knowledge : facts, concepts, principles, procedures
Draw attention to distinctive features in print or speech.
Present information so that it can be made into chunks.

Intellectual Skills
Discriminations-Distinguishing objects, features, symbols Concrete concepts-Identifying classes of concrete objects, features or events.
Defined Concepts-Cassifying New Examples of Events or Ideas by Their Definition
Rules-Applying a single relationship to solve a class of problems.
Higher Order Rules-Applying a New Combination Rules to Solve a Complex Problem
Call attention to distinctive features.
Stay within the limits of working memory.

Cognitive Strategies
Using personal ways to guide learning, thinking, acting and feeling.
Describe or demonstrate the Strategy.
Provide a variety of occasions for practice using the strategy.
Attitudes
Choosing personal actions based on internal states of understanding and feeling
Establish an expectancy of success associated with the desired attitude.
Assure student indentification with an admired human mode.

Motor Skills
Executing performances involving the use of muscles :
Present verbal or other guidance to cue the excutive subroutine.
Arrange repeated practice.

Jean Piaget
Piaget considers the most critical factor in a child’s cognitive development to be interaction with peers. Interaction lends opportunities for the child to have cognitive conflict, which results in arguing or debating with peers. This type of interaction requires children to decenter, or consider another person’s point of view. Piaget observes that children are most challenged in their thinking when they are with peers, because they all are on an equal footing and are more free to confront ideas than when interacting with adults.
Jerome Bruner
Jerome Bruner was born in New York in 1915. At the age of 2 he underwent operations to correct vision impaired due to cataracts. His father died when Jerome was 12, after which the family moved frequently and Jerome had an education interrupted by frequent changes of school. Despite this, Bruner’s grades were good enough to enter Duke University in Durham, NC where he obtained a B.A. in 1937 followed by a ph.D. from Harvard In 1941.
Bruner’s cognitive approach to his work in childhood learning and perception has made him a key figure in educational reform in the United States and Britain.


Psychological Aspect of Mathematics
Recently, attempts to create a unified theory of learning have been made by psychologists, and one may try to approach practical problems via such a unified theory. We have adopted three principles from this theory
Ø Progress in small steps.
Ø Active response.
Ø Immediate confirmation and reinforcement.

Interest in Psychology
The psychologist’s challenging job not only involves studying individuals and groups but also requires the researcher to take into account the environmental, physiological, and cultural factors that influence human behaviour. Students of psychology employ a variety of clinical and scientific methods to discover how and why the human mind behaves as it does.
Health psychology studies how to prevent and treat illness using biological, psychological, and social perspectives. Physiological psychology examines the biological bases of behaviour, especially those involving the nervous system. Experimental psychology focuses on the basic processes of sensastion, perception, learning, memory, thinking, language, emotion, and motivation.
Attention and Concentration
Concentration and attention, concentrating and focusing, have been described as the power which if one possessed would make all other known power insignificant to one. Focusing attention, concentration, is what intelligence is, is about. Here is how to acquire focus-concentration, attention.
Intelligence is knowing how to avoid distractions and concentrate on issues-it is concentration, focusing attention; concentrating, knowing how to concentrate and focus attention success in life situations by effective concentration, effective focus of attention.
Ø Formation of mathematics concepts
Ø Abstracting and classifying
Ø Abstracting
Invariant properties of examples are abstracted: these memories persist longer in the memory than particular instances an activity by which we become aware of similarities among our experiences product of abstraction concept.
Classifying
similarities of an instance with a class are noted: Collecting together our experience on the basis of these similarities
classification
Naming an object classifies it.
We then know how to behave in relation to it.
Once classified, less open to other classifications.
Concept
Requires for its formation a number of experiences which have something in common
Everyday concepts come from everyday experiences

Processes involved
. Generalization
. Discrimination
Learning of mathematical concepts
1. The particular problem of mathematical lies in the great abstractness and generality, achieved by successive generations of particular
2. But only indirectly from other mathematicians in conjunction with one’s own reflective intelligence.
Principles of learning mathematical concepts
Concepts of a higher order than those which people already have cannot be communicated them by a definition but only by arranging for them to encounter a suitable collection examples.
Since in mathematics these examples are almost invariably other concepts, it must first be ensured that these are already formed in the mind of the learner.


Learning and teaching
Although we have to create all the concepts in our own minds, we are only able to do this by using the concepts arrived at by past mathematicians.
Very dependent on good teaching
To know mathematics
To communicate it to those at a lower conceptual level
Factors influencing the learning of mathematics
The conference the future of mathematics education and learning mathematics will consider the mathematical. Learning process from the most important perspectives and discuss the relationship to mathematics teaching. Two leading world-experts working in each of these perspectives will be invited to present a review of state-of-the-art research in their field, and they will discuss connections to the other approaches.



Motivation for learning of mathematics
Mathematics is a universal subject, so much a part of life that anyone who is participating, member of society must know basic mathematics.
The mathematics study panel also emphasized the important of mathematics. Furthermore, the panel declared that students develop math proficiency.
Motivation math incorporates research-based strategies and pedagogically sound principles for teaching and learning. This product is designed to support and enhance the best practices for teaching the TEKS. Motivation math is founded on the modeling of ―active teaching, which is teacher-directed instruction that proceeds in small steps.
Ø Introduction to skills
Ø Guided practice
Ø Independent practice

Assessment:
Ø Critical thinking activities
Ø Homework with parent activities

Perception in mathematics
Teacher perceptions of students.
While much has been written about teacher perceptions, surprisingly little concerns teacher perceptions of their students and the implications of these of their students and the implications of these judgment, and even less deals with teacher perceptions of mathematics learning and aboriginal students.
Some studies to deal with teacher perception generally examining the various different perceptions that teacher have of their students. A study by uhlenberg and brown (2002), for example, looked at teacher perceptions as the reason for the black-white achievement gap in the united states.
In this study, a group of 15 white teachers were asked list the major teachers were drawn from classrooms where the aboriginal school.

Attitude evaluation
Attitude affects everything that you attempt.
They affect your relations with other people and your openness to new experiences. If your attitude towards a task is positive you will most likely enjoy the process of doing it and look for opportunities to do it. If your attitude is negative, you will most likely avoid for delay the occasion of doing it and , if you most do the task, you will probably not enjoy it or do well at it.
Your attitude toward mathematics affects how well or how often you do to, and how much enjoyment you derive from it. many students taking mathematics courses in college have a negative attitude toward mathematics that can be described as ―math anxiety or ―math avoidance. Math Anxiety is a state of such tension or fear that the learning process in mathematics is blocked or interrupted.

THINKING SKILLS
"The whole of Science is nothing more than a refinement of everyday thinking"
- Albert Einstein
Introduction
Thinking is the highest mental activity present in man. All human achievements and progress are simply the products of thought. The evolution of culture, art, literature, science and technology are all the results of thinking.
Thought and action are inseparable - they are actually the two sides of the same coin. All our deliberate action starts from our deliberate thinking. For a man to do something, he should first see it in his mind's eye -- he should imagine it, think about it first, before he can do it. All creations-- whether artistic, literal or scientific --first occur in the creator's mind before it is actually given life in the real world.

The Purpose of Thinking
The purpose of thinking, paradoxically, is to arrive at a state where thinking is no more necessary at all. In other words, thinking starts with a problem and ends in a solution. Thus, thinking is a tool for adapting ourselves to the physical and social environment in which we are in.
Can We Improve Our Thinking Ability?
Dr. Edward de Bono says that thinking CAN be improved just like any skill because thinking according to him is a skill. He has developed many useful techniques for training thinking skills.
Why Should We Improve Our Thinking Skills
The benefits of developing thinking ability are manifold. By developing one's thinking skills one can make achievements; can become successful; can shine in social life; can attain emotional, social and economic maturity and so on. By developing one's thinking abilities it is possible to transform one's aggressive tendencies, bad temper and other negative tendencies creatively and constructively. It has been found by Dr.Edward de Bono that when school students were taught to think effectively, their ill-temper and aggressive tendencies reduced significantly. Clinical Psychologists have also found that those who have neuroses are poor thinkers compared to normals. Neurotics scored significantly lower scores in decision making, problem solving and creative thinking. Interestingly, when neurotics were taught to think effectively, they showed a remarkable reduction in their neurosis.

Misconceptions Regarding Thinking
There are a lot of misconceptions about thinking even in highly educated people. These notions have been deeply rooted in the society and thus have become very detrimental to the development of thinking skills. It is time and enough that we removed them. The most important of such misconceptions are:
Misconception No.1:
The present education system develops and enhances thinking and so the more educated you are the better thinker you are.
Fact:
Actually, education suppresses free thinking. Creative thinking has almost no place in current education. Moreover, education even destroys creative thinking abilities by its over emphasis on logical thnking and critical thinking which are relatively lower types of human thinking.
Since reasoning, argument, problem solving are given over importance a need to become correct and successful all the time is developed in the student. Again, our present education system is so information oriented that it gives ready-made answers. This kills the student's natural tendency to explore, experiment and to experience. Thus, the highly educated person ends up having lesser ability to think creatively although he/she may have a lot of information at his/her disposal and also have admirable abilities in logical and critical thinking.
Misconception No.2:
Less Educated or Uneducated can never become good thinkers.
Fact:
Actually, less educated display higher abilities in creative thinking. This is because they do not have an inflated ego that demands cent percent correct answers or success in all endeavors. Again, they do not have ready-made-answers (i.e., book-answers) and so are forced to explore, experiment and experience things themselves. This empowers them to go through less explored answers and even find original answers.
Misconception No.3:
IQ and thinking ability are the same. The more IQ one has, the better thinking ability one has. On the contrary, those who have lower IQ have only low thinking abilities.
Fact:
It is true that those who have greater thinking ability, as a rule, have high IQ. But this does not mean that all those who have high IQ are good thinkers. Usually high IQ people use their thinking skills for logical thinking, arguments, critical thinking. They very rarely use creative thinking. Thus high IQ is actually a blockage to creative thinking. It has also been found that those who have average IQ can become better thinkers.
Misconception No.4:
Thinking ability, decision making ability and problem solving ability are inherent and there is very little we can do to develop these.
Fact:
This is the most terrible misconception regarding thinking. In fact, Dr.Edward de Bono (and also many others) have proved that thinking is a skill that can be enhanced by training and practice. Thus decision making, problem solving and creative thinking can be developed and improved.
 Errors in Thinking
It is necessary that we should be aware of the errors in thinking. There are five such errors in thinking: 1. Partialism 2. Adversary Thinking 3. Time Scale Error 4. Initial Judgement and 5.Arrogance and Conceit.
 Partialism
This error occurs when the thinker observes the problem through one perspective only. That is, the thinker examines only one or two factors of the problem and arrives at a premature solution.
1.     Adversary Thinking
This is a "you are wrong. So, I should be right." type of reasoning. Politicians are the masters in this type of thinking and they use it to their advantage.
2.     Time Scale Error
This is a kind of partialism in thinking in which the thinker sees the problem from a limited time-frame. It can be likened to short-sightedness.
3.     Initial Judgement
Here, the thinker becomes very subjective. Instead of considering the issue or problem objectively, the thinker approaches it with prejudice or bias.
4.     Arrogance and Conceit
This error is sometimes called the "Village Venus Effect" because like the villagers who think that the most beautiful girl in the world is the most beautiful girl in their village, the thinker believes that there is no better solution other than that he has already found. This blocks creativity. Not only individuals but societies and even the whole mankind sometimes fall prey to this error. For example, before Einstein, the whole scientific community (and thus the whole mankind) believed that time was absolute.

Thinking Formulae
Dr. de Bono has devised some thinking formulae to help us practice thinking as a skilll. Most important of them are: 1. AGO 2. CAF 3. PMI 4.OPV 5.APC
   
1.     AGO
AGO stands for Aims Goals Objectives. Since deliberate thinking is actually the manifestation of deliberate use of Will Power, it is important that the thinker should be well aware of the aims, goals and objectives. In other words, a sense of direction is required if one is to use his thinking effectively.
2.     CAF
CAF stands for Consider All Factors. It reminds us that all factors or parameters of a problem should be considered to analyze it. By doing so, one can avoid the error of partialism in thinking.
3.     PMI
PMI is the abbreviation for Plus Minus Interesting. When making decisions, this technique is very useful. First write down all the plus (i.e., positive) suggestions or aspects of a solution. Then write down all the minus (i.e., negative) aspects. Lastly, write down the interesting ideas or suggestions or aspects of the same. Now it is easy to arrive at the best suitable solution of the issue or problem at hand. 
4.     OPV
OPV means Other People's Views. In this technique, the thinker thinks from the perspectibve of the different people involved in or affected by the decision or solution. For example, a change in syllabus mainly affects the students, teachers, management and parents either directly or indirectly. So, before implementing a new syllabus, we should think from the perspective of all these people. This is especially important when the decision is enacted upon and through other people.
5.     APC
APC is the short form for Alternatives Possibilities Choices. In this technique, the thinker generates as much alternatives, possibilities and choices for the solution of the problem. The best suitable one can then be selected by applying PMI or OPV.

Lateral Thinking
Dr. de Bono speaks about two types of thinking -- Vertical thinking and Lateral thinking. Vertical thinking is high probability thinking, whereas lateral thinking is low probability thinking. In the former type, the thinker selects the most logical solution possible. This will be the one that is the most used and the most tested one. For example, suppose you want to cut a cake or hard boiled egg into two. Using a knife or a sharp blade to cut it is the most probable and most logical solution. But there are other less probable ways: for example, you can use a twine. The former is an example of a solution arrived at Vertical thinking while the later is an example of a solution arrived at using Lateral thinking.
Dr. de Bono argues that creativity comes from Lateral thinking. He also says that it is possible to improve lateral thinking ( and thus creativity) by deliberate application and practice of the techniques he has devised. He gives two main techniques for improving our problem solving by lateral thinking:
1.     Divide the Problem into Small Sub-Problems
This allows the thinker to perceive the problem in detail because he is able to approach each sub-problem individually and thus he is able to consider more parameters.
2.     Move Attention from One Part of the Problem to Another Part
When we move our attention from one part of the problem to another, it is possible to view the problem from a new perspective. This would in turn help you solve the problem easily.
Creative Thinking in Mathematics
Mental maths is the simple calculations you can perform in your brain. This does not need tables or calculators to get answers. There are many benefits of engaging in such mathematics like: you get your answers fast. For instance, in an exam situation you might be able to calculate some of the problem easily. This helps you to save time to concentrate on the problems that need extra time. This can result to good grades in the subject.
Other than the classroom, one can use mental maths in different aspects in your daily life. When you are out shopping for a few items, you can use this to help you come with the total amount of money that you will spend. This allows you to stay within the budget that you had as you only use the amount of money you have. It also saves you the embarrassment of having to return items when you are at the counter because you do not have enough money. You can also calculate discounts in your head when you are out eating or shopping. This is necessary to avoid confrontations with people because of the price.
Performing mental maths helps one to be more attentive and sharpens your critical skills. This should be taught to children especially at a tender age. When children learn the subject, it will help them be good in mathematics, a subject that is disliked and performed poorly by a majority of students. This is because it improves their confidence in the subject and makes it more enjoyable. This also helps to improve a sense of achievement within the children once they are able to come up with problem answers without using too much effort.
One of the greatest benefits of the subject is it enhances intuitive thinking and creativity. This helps one to think out of the box. You can come up with creative ideas that can help you be successful in life. These also help one to gain mental endurance and improved concentration. It also increases the interest of many students in mathematics. This is because they learn other techniques other than the normal routine. Some of the alternatives emerge from the Asian continent where teachers use verdict and Abacus maths to help improve mental maths skills. This is where your child can be able to come up with the solution of a complicated sum in seconds.
Creative thinking in philosophy is possible because philosophical questions are open questions: an investigation into the natural of the problem is the start of the enterprise and the way followed is often much more interesting than the final results.
Mathematical problems are well-defined they have strict solutions, and –at for school mathematics- the way to success is clear and unilinear for the initiated.

Divergent thinking in mathematics
Based on this evidence, some creativity researchers have argued that creativity is domain specific knowledge and skills are more important than domain general creative thinking abilities for creative performance in a given domain.
The purpose of the current study was to investigate and compare the effects of domain knowledge divergent thinking and different types of interest on creative performance in art and math, as well as the hypothesized moderating and mediating effects of different types of interest the relationships among domain knowledge divergent thinking and creative performance.


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