Sunday 26 June 2016

CURRICULUM AND RESOURCE MATERIALS FOR TEACHING MATHEMATICS

CURRICULUM AND RESOURCE MATERIALS FOR   TEACHING MATHEMATICS
Curriculum construction and organization in mathematics
Meaning of Curriculum According to Cunningham “The curriculum is the tool in the hands of the artist (the teacher) to mould his material (the pupil) according to his ideals (objectives) in his studio (the school)”.
CURRICULUM
1)   Curriculum is “ A general overall plan of the content  or specific materials of instructions that the school should offer the students by way of qualifying him for graduation or certification or for entrance into a professional or vocational field”.(Good,1959).
2)  “All experiences a learner has under the guidance of the school” (Foshay, 1959).
3)   “Curriculum consists of all the situations that the school may select and consciously organize for the purpose of developing the personality of its pupils and for making behavioural changes in them” (Payne).
PRINCIPLES OF CURRICULUM CONSTRUCTION
1)    Principle of Disciplinary value: The topics and contents of mathematics which help in the task of disciplining the mind.
2)   Useful for Higher Education: The child aims to go higher and higher on the Education ladder. Therefore education at one stage must aim to prepare the child for the education at the higher stages. Therefore, curriculum of mathematics at any stage must cater to the needs of the higher classes.
3)    Principle of utility: According to this principle all that which is useful should be minded in the curriculum.
Mathematics curriculum should include all those topics which are,
i)    Helpful in day to day life.
ii)    Helpful in learning other subjects.
iii)   Helpful in realization of aesthetic and artistic value of the subject mathematics.
iv)    Helpful in understanding the scientific and technological progress and rendering help for the further research work in the field of mathematics and science.
4)   Child centeredness: In curriculum construction we must give proper weightage to the needs and requirements of the students for whom we are going to prepare a curriculum. Therefore   in any scheme of curriculum construction, the needs-ability, interest and other developmental characteristics of the children of particular age, interest and society should be kept in view.
5)    Integrity of theory with practice: Mathematics curriculum requires the topics, contents, experiences and activities in such a way that we may have enough opportunities of integrating theory with practice.
6)    Principle of flexibility: Curriculum by all means should have a flexible nature, so that it can be modified and reshaped according to the circumstances and demands of the resources in hand. 
7)  Principle of community centeredness: Curriculum be constructed and shaped for the welfare of the local community.
Principle of consulting experts: Curriculum construction needs the help and guidance of those persons whose interests are to be served by a particular curriculum
Principles of curriculum organization:
The following methods in general are adopted while framing the curriculum of a subject.
1)    1)  Topical and Spiral method:  Topical approach suggested by its name advocates to cover a topic as a whole in a particular grade. Here a few topics of the subject mathematics may there by marked for being included as in the curriculum of one grade or the other then it is expected to cover all the content or learning experiences related to that very topic only in that very class and not to repeat it in any way in the junior or senior grades. Thus a topic marked for a particular grade should have its beginning and end it that every grade without having its need to be taught in the earlier and later grades. We have different sets of topic for their inclusion in the curriculum of different grades of secondary stages of school education.
      It has been described under the head criterion of difficulty that easier topics should be dealt with earlier than the difficult ones. Certain portions of a topic are always easier than the other portions of the same topic. Thus Square Measure is easier than cubical measure. Again V=LXBXH is easier than the area of a circle or a triangle. Similarly in profit and loss, certain problems are easier than other. Inverse problems are generally more difficult than the direct ones.
     Spiral system is based on the principle that a subject cannot be given an exhaustive treatment at the first stage. To begin with, a simple presentation of the subject-matter is given, gaps are filled in the following year and more gaps a year or two later, in accordance with the amount of knowledge which pupils are capable of assimilating. Spiral method demands the division of the topic or the subject into number oif smaller independent units to be dealt with in order of difficulty suiting the mental capacities of the pupils, while the topical method demands that a topic once taken should be finished in its entirety. Spiral method is more natural and less tiring to the pupil. The child loses nothing in accuracy and gains considerably in the power of intelligent application of rules to problems.       
     Spiral approach may run contrary to the topical approach there we do not include topic as a whole and finish it entirely in a particular grade as practiced in topical approach but try to spread it over to different grades by covering easier portion   in the lower grades and the difficult ones in the higher and higher grades. In this way while expecting concentric (Spiral) approach in any single topic may find its place in the curriculum of different grades on that school education in accordance of the difficulty level of the subject matter/learning experience suiting to the mental level of the students.
     The chief defect in the topical plan is that it introduces in the curriculum a large mass of irrelevant material for which the pupil finds no time and no immediate need or the use of which cannot be appreciated by the pupil at that stage. They are introduced with a view to making the teaching of the topic complete and thorough.
Ex. –Multiplication and division with 6 or 7 digits in the first or second standard.
2)      2. .Logical and Psychological Arrangement:  Logical arrangement leads to the rigorous treatment of the subject-matter which is based on logical reasoning whereas psychological arrangement is from the point of view of the students. It seems that both the approaches are different but these can be easily merged. The organization can both be psychological and logical. All thinking is psychological. Psychology throws light on the power of understanding of students at a particular stage. We can be logical in various ways. Psychology should decide which logical approach will suit for a particular topic. Logic will help in maintaining proper sequence of topics, so we should organize the topics in such a way that we may follow psychology and logic at the same time. The happy combination of two is always desirable.
Psychology should decide what kind of logic is appropriate for the pupil of a certain age and what type of topics will be most suitable for the development of such logical thinking. Logic will help in maintaining the link and sequence of topics found useful and meaningful for the child.
3)      3Principle of correlation: While organizing the content in mathematics the principle of correlation should always be given due weightage. Correlation may be of different varieties. The following types of correlation must be kept in mind while organizing curriculum in Mathematics:
                                                                                I.            Correlation of Mathematics with the problems of everyday life.
                                                                              II.            Correlation of Mathematics with other subjects.
                                                                            III.            Correlation between different branches of Mathematics.
                                                                            IV.            Correlation between different topics of a particular branch of Mathematics.
                                                                              V.            Correlation with craft or work experience.
           For correlating subject-matter we must know the following:
Ø  Day to day life activities of the students.
Ø  The nature of topics included in other subjects at the same stage.
Ø  The topics included in different branches of the subject e.g., Arithmetic, Algebra, Geometry etc.
Ø  The sequence of topics of the same branch of the subject.
Ø  The nature of work experience or projects undertaken by the students.
4)     4.  Concentric approach:  
          This is a system of organizing a course rather than a method of teaching. It is therefore better to call it concentric system or approach. It implies widening of knowledge just as concentric circles go on extending and widening. It is a system of arrangement of subject-matter. In this method the study of the topic is spread over a number of years. It is based on the principle that subject cannot be given an exhaustive treatment at the first stage. To begin with, a simple presentation of the subject is given and further knowledge is imparted in following years. Thus beginning from a nucleus the circles of knowledge go on widening year after year and hence the name concentric approach. 
 Procedure:
     A topic is divided into a number of portions which are then allotted to different classes. The criterion for allotment of a particular portion of the course to a particular class is the difficulty of portion and power of comprehension of students in that age group. Thus it is mainly concerned with year to year teaching but it its influence can also be exercised in day-to-day teaching. Knowledge given yesterday and should lead to teaching on following day.
Merits of Concentric approach:
Ø  This method of organization of subject-matter is decidedly superior to that in which one topic is taken up in particular class and an effort is made to deal with all aspects of the topic in that particular class.
Ø  It provides a frame work from science course which is of real value to students.
Ø  The system is most successful when the teaching is in hands of one teacher because then he can preserve continuity in the teaching and keeps the expanding circle concentric.
Ø  It provides opportunity for revision of work already covered in a previous class and carrying out new work.
Ø  Since the same topic is learnt over many years so its impressions are more lasting.
Ø  It does not allow teaching to become dull because every year a new interest can be given to the topic. Every year there are new problems to solve and new difficulties to overcome.
Drawbacks:
·         For the success of this approach we require really capable teacher. If a teacher becomes over4 ambitions and exhausts all the possible interesting illustrations in the introductory year then the subject loses its power of freshness and appeal and nothing is left to create interest in the topic in subsequent years.

·         In case the topic is too short or too long then also the method is not found to be useful. A too long portion makes the topic dull and a too short portion fails to leave any permanent and lasting impression on the mind of the pupil.

Perspectives of Mathematical knowledge and objectives of teaching mathematics

Perspectives of Mathematical knowledge and objectives of teaching mathematics
INTRODUCTION:
   Mathematics plays a vital role in the day to day life. It is a very important subject. Therefore before imparting and transmitting its knowledge it is necessary to understand that ‘what is mathematics?’ and its nature etc. There are various definitions of mathematics has been interpreted and explained in various ways. Mathematics deals with the quantitative facts and relationships as well as with problems involving space and form.
     Though mathematics has been with us for more than 5000 years, the subject has never been made as lively as it is today. The pace of mathematical discovery and invention has accelerated amazingly during the last few decades. It has been said that mathematics is the only branch of learning in which theories of two thousand years old are still valid.
MEANING AND DEFINITIONS OF MATHEMATICS:
     The dictionary meaning of mathematics is that ‘it is either the science of number and space or the science of measurement, quantity and magnitude’.
    According to Webster’s dictionary “Mathematics is the science of number and there operations inter relations, combinations, generalizations and abstraction and of space configurations and generalizations.”
     “Mathematics may be defined as the subject in which we never know what we are talking about not whether what we are saying true.”------Bertrand Russel.
     “Mathematics is the gate and key of the sciences”---Roger Bacon. Neglect of mathematics work injury to all knowledge since who is ignorant of it cannot know the other science world. And what is worse, men who are thus Ignorant are unable to perceive their own ignorance and so do not seek a remedy.”
     “Mathematics is the language in which god has written the universe. ----Galileo.
“Our entire civilization depending on the intellectual penetration and utilization of nature has it s real foundation in the mathematical science.”-----Prof.Voss.
According to Locke-“Mathematics is a way to settle in the mind of children a habit of reasoning.
     On the basis of above definitions we can say or conclude that,
·         Mathematics is the science of Space and Number.
·         Mathematics is the science of calculation.
·         Mathematics is the science of measurement, Quantity and magnitude.
·         Mathematics is a systematized, Organized and exact branch of science.
·         It deals with quantitative facts and relationship.
·         It is the abstract form of science.
·         It is the science of logical reasoning.
·         It is an inductive and experimental science.
·         Mathematics is the science which draws necessary calculations.
NATURE OF MATHEMATICS:
 Mathematics is the gate way of all science. In school those subject which are included in the curriculum must have certain aims and objectives on the basis of which its nature is decided. Now we are in position to conclude the nature of mathematics. The nature of Mathematics are enlisted in the following points,
·         Mathematics is an exact science. Mathematical knowledge is always clear, logical and systematic and that may be understood easily.
·         It is the science of space, numbers, magnitude and measurement.
·         Mathematics involves conversion of abstract concepts into concrete form.
·         It is the science of logical reasoning.
·         It helps the man to give exact interpretation to his ideas and conclusion.
·         Mathematics is that science which is by product of out empirical knowledge.
·         Mathematical propositions are based on postulates and axioms from our observations.
·         It may exhibit abstract phenomenon into concrete. Thus abstract concepts may be explained and understood with the help of mathematics.
·         It is related with each aspect of human life.
·         Mathematical knowledge is developed by our sense organs therefore it is exact and reliable.
·         The knowledge of Mathematics remains same in the whole universe, everywhere and every time. It is not changeable.
·         The knowledge of mathematics has no doubt. It provides clear and exact response like yes or no, right or wrong.
·         It involves inductive and deductive reasoning and can generalize any proposition universally.
·         It helps the self evaluation.


CHARACTERISTICS OF MATHEMATICS:
 Mathematics has certain unique features which one could hardly find in other disciplines. The following are the important characteristics of mathematics.
1) LOGICAL SEQUENCE:  The study of mathematics begins with a few well –known uncomplicated definitions and postulates, and proceeds, step by step, to quite elaborate steps. It would be difficult to find a subject, in which a better gradation is possible, in which work can be adapted to the needs of the pupil at each stage, than in mathematics. Mathematics learning always proceeds from simple to complex and from concrete to abstract.
2) Structure in mathematics:  In English language structure denotes ‘the formation, arrangement, and articulation of parts in anything built up by nature or art’  It  seems reasonable to assume then that a mathematical structure should be some sort or arrangement, formation, or result of putting together of parts.
     For example, we take as the fundamental building units of a structure the members a,b,c,…. Of a non empty set ‘S’. We hold together these building units by using one or more operations.
The familiar operations of addition denoted by +, and multiplication denoted by X, of natural numbers are operations on set N of natural numbers. Subtraction is not an operation on the set of natural numbers since the difference of two natural numbers may not be a natural number(Example:3€N, 3-6=-3₵N) But subtraction is an operation on the set  ‘I’ of all integers.)
3) PRECISION:  Mathematics is known as an ‘exact’ science because of its precision. It is perhaps the only subject which can claim certainty of results. In mathematics the results are either right or wrong. Mathematics can decide whether or not its conclusions are right. Mathematicians can verify the validity of the results and convince others or its validity with consistency and objectivity. This holds for all not only the experts in mathematics.
     Even when there is a new emphasis on approximation, mathematical results can have any degree of accuracy required. Although precision and accuracy are distinctly different as criteria for the measures of approximation, they can be most effectively discussed when contrasted with each other. The most effective measures of both precision and accuracy are in terms of the errors (positive or negative) involved. The precision of a measure or a computation is evaluated in terms of the apparent error. The accuracy of a measure or a computation is evaluated in terms of the relative error or percent of error made.
4) ABSTRACTNESS:  Mathematics is abstract in the sense that mathematics does not deal with actual objects in much the same way as physics. But, in fact, mathematical questions, as a rule cannot be settled by direct appeal to experiment. For example, Euclid’s lines are supposed to have no width and his points no size. No such objects can be found in the physical world. Euclid’s geometry describes an imaginary world which resembles the actual world sufficiently for it is a useful study for surveyors, carpenters and engineers.
    Infinity is something that we can never experience and yet it is a central concept of mathematics. Our whole thinking is based on the assumption that there are infinitely many numbers, so that counting need never stop; that there are infinitely many fractions between
0 and 1, that there are infinitely many points on the circumference of a circle etc.
     Again someone whose thinking was essentially physical might refuse to believe in negative numbers on the ground that you cannot have a quantity less than nothing. Still more, such a person would refuse to believe in the square root of minus one.
5) SYMBOLISM:  The language for communication of mathematical ideas is largely in terms of symbols and words which everybody cannot understand. There is no popular terminology for talking about mathematics. For example, the distinction between a number and a numeral could head the list. A number is a property of a set; that property tells how many elements are there in a set. A numeral is a name or a symbol used to represent a number.
     A teacher ought to be very careful to use correct terms, since this helps children to learn and think better. It is important that a student understands the distinction between a number and a numeral so that he may realize the differences between actually operating with numbers and merely manipulating symbols representing those numbers. This manipulating symbols representing those numbers.
     Without language, we cannot talk about anything. Mathematical talk consists of making use of mathematics symbolism. Understanding mathematics is realizing what symbolism corresponds to the structure that has been abstracted.   The process of speaking of the mathematical language runs as follows: an abstraction process, followed by a symbolization process, followed again by the learning of the use of the symbols.
     The use of symbols makes the mathematical language more elegant and precise than any other language. For example, the commutative law of addition and multiplication inb real number system can be stated in the verbal form as ‘ the addition and multiplication of two real numbers in independent of the order in which they are combined’.
      This can be stated in a concise form as: a +b =b +a and a X b = b X a, where a and b are elements of R. Almost all mathematical statements, relations and operations are expressed using mathematical symbols such as +, -, X, %,<, >, ≤ , √, ∑,  € and so on.                       

PROCESSES IN MATHEMATICS:   
·         Mathematical reasoning.
·         Pattern recognition.
·         Algebraic thinking.
·         Geometric thinking ( Van Hiele model of geometric thought.
·         Problem solving in mathematics.
·         Creative thinking in mathematics.
Mathematics is also called the science of reasoning. According to Locke, “Mathematics is a way to settle in mind a habit of reasoning. “ Reasoning in mathematics is of two types: i) inductive reasoning and ii) deductive reason.
Inductive reasoning:   When statements containing mathematical truths are based on general observations and experience, reasoning is called inductive reasoning.
Deductive reasoning: This type of reasoning is based on certain postulates or axioms and in this statements are products of mind. We try to compare and contrast various statements and then draw out conclusions from such a comparison. Essentials of a deductive reasoning are as follows:
      * Undefined terms-----point, surface etc.
      * Definitions
      * Postulates and Axioms.
Reasoning:
·         Justify why an answer or approach to a problem is reasonable;
·         Make and test generalization or observation;
·         Make prediction or draw conclusions from available information;
·         Analyze statements and provide examples which support or refute them;
·         Judge the validity of arguments by applying inductive and deductive thinking;
 Problem Solving:
·         Use information to identify and define the question/s within a problem;
·         Make a plan and decide with information and steps are needed to solve the problem;
·         Choose the appropriate operation/s for a given problem situation;
·         Select and apply appropriate problem-solving strategies  to solve a problem from visual(draw picture, create a graph), numerical (guess and check, look for a pattern), and symbolic(write an equation) perspectives;
·         Organize, interpret, and use relevant interpretation;
·          Select and use appropriate tools and technology;
·         Show that no solution or multiple solutions may exist;
·         Identify alternate ways to find a solution;

·         Apply what was learned to a new problem.