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.