Thursday 17 December 2015

MATHEMATICAL REASONING

MATHEMATICAL REASONING
Reasoning is fundamental to knowing and doing mathematics. Reasoning enables children to make use of all their other mathematical skills and so reasoning could be thought of as the 'glue' which helps mathematics makes sense. There are various terms used to refer to "reasoning": critical thinking, higher-order thinking, logical reasoning, or simply reasoning. Different subject areas tend to use different terms.
Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. Mathematical Reasoners are able to reflect on solutions to problems and determine whether or not they make sense. They appreciate the pervasive use and power of reasoning as a part of mathematics.

Inductive reasoning involves looking for patterns and making generalizations. For example, students use this type of reasoning when they look at many different parallelograms, and try to list the characteristics they have in common. The reasoning process is enhanced by also considering figures that are not parallelograms and discussing how they are different.  Students may use inductive reasoning to discover patterns in multiplying by ten or a hundred or in working with exponents. Learning mathematics should involve a constant search for patterns, with students making educated guesses, testing them, and then making generalizations.
Deductive reasoning involves making a logical argument, drawing conclusions, and applying generalizations to specific situations. For example, once students have developed an understanding of "parallelogram," they apply that generalization to new figures to decide whether or not each is a parallelogram. This kind of reasoning also may involve eliminating unreasonable possibilities and justifying answers. Although students as young as first graders can recognize valid conclusions, the ability to use deductive reasoning improves as students grow older. More complex reasoning skills, such as recognizing invalid arguments, are appropriate at the secondary level.

STRUCTURE OF MATHEMATICAL REASONING
Mathematical reasoning involves more than just deduction. Mathematical theories are systematized by axioms and definitions in a way exemplified by Euclid in his famous compilation of geometric knowledge in the Elements. Euclid's model of a how to structure a mathematical theory still dominates today. Euclid divided his theory into four parts, each of which he gave explicitly:
- Definitions
- Common Notions (Logic)
- Postulates (Axioms)
- Theorems
The Definitions are supposed to clarify the concepts used in terms of primitives that are completely clear and familiar. The Common Notions are to provide rules of logic, that is, rules for making inferences which preserve truth. The Postulates, or Axioms, are the substance of the theory. They provide the sum total of all that one need assume in order to derive the rest of the theory, which is separated into Theorems.
DEFINITIONS
        Deļ¬nition is a precise and unambiguous description of the meaning of a mathematical term . It characterizes the meaning of a word by giving all the properties and only those properties that must be true. In the book ‘Elements’ Euclid gave definitions for the phenomena observed with regards to solid objects and their parts. For example he defined ‘point’ as follows. Point is that which has no part. Line was defined as breadthless length. These definitions acted as the starting point for the logically bound structure.
AXIOM
The word "axiom" comes from the Greek word ‘axioma’, a verbal noun from the verb ‘axioein’ meaning "to deem worthy", but also "to require", which in turn comes from ‘axios’ meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.
Axiom is a premise so evident as to be accepted as true without controversy. In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
THEOREM
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.  
Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.



Wednesday 16 December 2015

EXPECTED QUESTIONS ON UNDERSTANDING DISCIPLINES AND SUBJECTS



EXPECTED QUESTIONS ON UNDERSTANDING DISCIPLINES AND SUBJECTS
1.   Define the term school subjects
2.  What do you mean by Academic Discipline ?
3.  Explain the term Discipline.
4.  ‘ Learning of school subjects bring a lot of benefits’- Illustrate.
5.  Briefly explain the contents of school subjects.
6.  Differentiate the term School subjects and Academic Discipline.
7.  Identify the similarities between Academic discipline and school subjects.
8.  Schooling clears the path of Every day life- justify your views.
9.  How schooling helps for university education ?
10.Define the term axioms.
11. Write an operational definition for the term ‘Theorems’
12.What is kutippallikoodam ?
13.Explain the terms Vedapatasala, Formal schools & Gurukulam.
14.Explain briefly the evolution of school subjects as a curricular area at school ?
15.Explain the evolution of school subjects before independence.
16.Give a brief summary on the evolution of subjects after independence of India.
17. Explain the historical development different school subjects.
18.Summarise the historical development and nature of Indian Languages.
19.Briefly explain the nature of subject science.
20.           Point out the origin of Science subject.
21.Differentiate the  historical developments of subjects Science and Social Science.
22.           Briefly explain the nature and history of Social science.
23.           ‘Science is the mother of all other subjects’- express your views.
24.           What do you mean by Theorems ?
25.           What is a definition ?
26.           Differentiate the terms Definition, Axioms and Theorems.
27.            Identify the similarities between the subjects Science and Mathematics.
28.           Explain briefly the history and nature of school subject – Mathematics.
29.           Explain briefly the subject arranging on different Curriculum Frame Works in India.
30.           Inclusion of work related subject areas in the curriculum brings all round development of Individuals. Do you agree with this ? Make this suggestions and views on inclusion of work related subjects.
31.Explain the concept of inter disciplinary approach.
32.           Briefly outline the benefits of Inter disciplinary approach in school education.
33.           Bring out the importance of including sex education at school level.
34.           Explain the term Horticulture.
35.           Develop a creative definition for the term Hospitality.
36.           Crete an operational definition for the term Horticulture.
37.            How learning of horticulture is useful for students.
38.           Define the term Horticulture.
39.           Define the term Hospitality.
40.           Define the term Life skills.
41.Write an explanatory definition for the term Life skills.
42.           How learning of hospitality and life skills are useful to pupils.
43.           Explain the concept Health Care.
44.           Bring out the importance of health education in developing awareness on health care.
45.           What is environmental protection ?
46.           Suggest the ways to develop environmental ethics in school childrens.
47.            Write the benefits of environmental education.
48.           Define the term sustainable development.
49.           Differentiate sustainable development and environmental protection.
50.           List out the different types of life skills.
51.Briefly explain the influence of socio – political factors in the curriculam change.

Monday 14 December 2015

SAMPLE QUESTION ON UNDERSTANDING DISCIPLINES AND SUBJECTS


EDU 04 UNDERSTANDING DISCIPLINES AND SUBJECTS
PART – A
Answer all the questions. Each carries one mark .
1.   Define the term school subjects.
2. What is Kutippallikoodam ?
3. List out the different subjects taught during Vedic period.  
4. Explain the Horticulture.
5. Write an operational definition for the term ‘Hospitality’
6. Write two objectives of  Sex Education .
PART – B
Answer all questions. Each carries two mark.
7.  Explain the necessity of learning School Subjects.
8. Differentiate the terms Academic Discipline and School subjects.
9. Why the learning of health care is necessary at school level ?
10.            Explain the concept of  Sustainable Development .
PART – C
Answer any four questions. Each carries four mark.
11.  Identify the need and benefits of Interdisciplinary Approach in todays curriculum.
12.            Explain the concept of life skills.
13.            Briefly explain the evolution of school subjects before and after Independence in India.
14.            Differentiate the terms Gurukulam , Vedapatasala and Formal Schools.
15.            Explain the structure of school subjects according to NCF 2005.
16.            “ School subjects are the gate way for the preparation of daily life” –          Express your views.
PART – D
Answer any one question . The question carries 10 mark.
17.             Explain the importance of bringing work related subjects and interdisciplinary approaches in school curriculum.

18.            Explain briefly the subject nature and subject history of  different school subjects.

Saturday 12 December 2015

HISTORY OF THE SOCIAL SCIENCES

     The history of the social sciences begins in the Age of Enlightenment after 1650, which saw a revolution within natural philosophy. Social sciences came forth from the moral philosophy of the time and was influenced by the Age of Revolutions, such as the Industrial Revolution and the French Revolution. The social sciences developed from the sciences (experimental and applied), or the systematic knowledge-bases or prescriptive practices, relating to the social improvement of a group of interacting entities.
     The beginnings of the social sciences in the 18th century are reflected in the grand encyclopedia of Diderot, with articles from Rousseau and other pioneers. The growth of the social sciences is also reflected in other specialized encyclopedias. The modern period saw "social science" first used as a distinct conceptual field. Social science was influenced by positivism, focusing on knowledge based on actual positive sense experience and avoiding the negative; metaphysical speculation was avoided. Auguste Comte used the term "science sociale" to describe the field, taken from the ideas of Charles Fourier; Comte also referred to the field as social physics.
     Around the start of the 20th century, Enlightenment philosophy was challenged in various quarters. After the use of classical theories since the end of the scientific revolution, various fields substituted mathematics studies for experimental studies and examining equations to build a theoretical structure. The development of social science subfields became very quantitative in methodology. The interdisciplinary and cross-disciplinary nature of scientific inquiry into human behavior, social and environmental factors affecting it, made many of the natural sciences interested in some aspects of social science methodology.
         In the contemporary period, Karl Popper and Talcott Parsons influenced the furtherance of the social sciences. Researchers continue to search for a unified consensus on what methodology might have the power and refinement to connect a proposed "grand theory" with the various midrange theories that, with considerable success, continue to provide usable frameworks for massive, growing data banks; for more, see consilience. The social sciences will for the foreseeable future be composed of different zones in the research of, and sometime distinct in approach toward, the field.
     The term "social science" may refer either to the specific sciences of society established by thinkers such as Comte, Durkheim, Marx, and Weber, or more generally to all disciplines outside of "noble science" and arts. By the late 19th century, the academic social sciences were constituted of five fields: jurisprudence and amendment of the law, educationhealtheconomy and trade, and art. The term "social science" first appeared in the 1824 book An Inquiry into the Principles of the Distribution of Wealth Most Conducive to Human Happiness; applied to the Newly Proposed System of Voluntary Equality of Wealth by William Thompson (1775–1833). Auguste Comte (1797–1857) argued that ideas pass through three rising stages,theologicalphilosophical and scientific. He defined the difference as the first being rooted in assumption, the second in critical thinking, and the third in positive observation.

     The social science disciplines are branches of knowledge taught and researched at the college or university level. Social science fields of study usually have several sub-disciplines or branches, and the distinguishing lines between these are often both arbitrary and ambiguous. The following are problem areas and discipline branches within the social sciences : Environmental Studies , Anthropology, Area studies , Business studies , Civics , Communication studies , Criminology , Demography , Development studies , Economics , Education , Geography , History , Industrial relations , Information science , Law ,Library science , Linguistics , Media studies , Political science , Psychology , Public administration , Sociology and Social work.
Nature Of Social Science
     Social Sciences are the advanced study of human society which are taught at different age levels. It is the theory part of human affairs. It posses the following characteristics ;
F Social science is the theory part of human relations.
F It is a combination of various disciplines such as Anthropology, Sociology, Linguistics, Law, History, Geography, Education, etc.
F It studies the development of man and society at different stages.
F It is a science of diversity of human relationships.
F It is one of the important Academic Discipline that tries preserve and transmit the culture of society.
F The motto of  teaching social science is to cultivate social values and making the pupils a social human being.
F Social science are those parts of cultural knowledge which have direct bearing on man’s activities in specific field.