Teaching of mathematics

INTRODUCTION

As mathematics is a compulsory subject upto the secondary level, access to quality mathematics education is the right of every child. Developing children’s abilities for mathematisation is the main goal of mathematics education. In the words of David Wheeler, it is “more useful to know how to mathematise than to know a lot of mathematics”. The higher aim is to develop the child’s resources to think and reason mathematically, to pursue assumptions to their logical conclusion and to handle abstraction. It includes a way of doing things, and the ability and the attitude to formulate and solve problems (NCF, 2005).

In 1937, when Gandhi ji propounded the idea of basic education, the Zakir Husain committee was appointed to elaborate on this idea. It recommended:

“Knowledge of mathematics is an essential part of any curriculum.”

The National Policy on Education 1986 went further:

“Mathematics should be visualized as the vehicle to train a child to think, reason, analyze and to articulate logically. Apart from being a specific subject, it should be treated as a concomitant to any subject involving analysis and reasoning.”

 As per NCF 2005, Vision for School Mathematics

• Children learn to enjoy mathematics rather than fear it.

• Children learn important mathematics: Mathematics is more than formulas and mechanical procedures.

• Children see mathematics as something to talk about, to communicate through, to discuss among them, to work together on. Children pose and solve meaningful problems.

• Children use abstractions to perceive relation-ships, to see structures, to reason out

things, to argue the truth or falsity of statements.

• Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry and trigonometry, the basic content areas of school Mathematics, all offer a methodology for abstraction, structuration and generalization.

• Teachers engage every child in class with the conviction that everyone can learn mathematics.

 As per NCF 2005, Some problems in school Mathematics education

1. A majority of children have a sense of fear and failure regarding Mathematics. Hence, they give up early on, and dropout of serious mathematical learning.

2. The curriculum is disappointing not only to this non-participating majority, but also to the talented minority by offering them no challenges.

3. Problems, exercises and methods of evaluation are mechanical and repetitive, with too much emphasis on computation. Areas of Mathematics such as spatial thinking are not developed enough in the curriculum.

4. Teachers lack confidence, preparation and support.

ALGORITHM- MEANING

In the most basic sense, an algorithm is a process- a set of detailed instruction that must be carried out in a particular order and follows logic to attain a given result. An algorithm is a well-defined procedure or set of rules guaranteed to achieve a certain objective.

You use an algorithm every time you follow the directions to put together a new toy, use a recipe to make cookies, or defrost something in the microwave (personal algorithms).

When the term algorithm used in math, it typically refers to a set of steps or procedures used to solve a mathematical computation. In mathematics, an algorithm is a specific series of steps that will give you the correct answer every time. For example, in grade school, you and your classmates probably learned and memorized a certain finite steps or procedures for addition, subtraction and multiplying etc. (standard algorithms).

Algorithm are of two type: informal (personal) and formal (standard) algorithm.

An informal algorithm is a procedure that the student him/herself figured out while a formal algorithm is a process and procedure that has been taught to them. It may or may not be similar to a conventional algorithms (formal algorithms).

Examples on Informal Algorithms Examples on Formal Algorithms

1. ADD the given numbers

109 + 207

Jimmy did like this. David did like this.

=100+9+200+7 109-9=100

=300+16 207-7=200

=316 200+100=300

9+7=16

300+16=316

2. Multiply the given number

25×13

25(20+5)

x13(10+3)

————–

15

60

50

+ 200

—————

325

1. ADD the given numbers

109+207

1

109

+ 207

——–

316

——–

2. Multiply the given number

25×13

1

25

x13

————

75

+25x

————

325

MERITS OF INFORMAL ALGORITHMS

They learn to think and use their common sense, as well as new skills and knowledge. Students who invent their own procedures:

 Learn that their intuitive methods are valid and that mathematics makes sense.

 Become more proficient with mental arithmetic.

 Are motivated because they understand their own methods, as opposed to learning

by rote.

 Become skilled at representing ideas with objects, words, pictures, and symbols.

 Develop persistence and confidence in dealing with challenging problems.

ADVANTAGES OF THE FORMAL ALGORITHMS

Plunkett (1979), Thompson (1997), Usiskin (1998) and other writers offered several reasons for this. These included :

• Standard algorithms are powerful in solving classes of problems, particularly where the computation involves many numbers, where memory may be overloaded.

• Standard algorithms contracted, summarizing several lines of equation involving distributivity and associativity.

• Standard algorithms are automatic, being able to be taught to, and carried out by, someone without having to analyze the underlying basis of the algorithm.

• Standard algorithms are fast, with a direct route to the answer.

• Standard algorithms provide the written record of computation, enabling teachers and students to locate any errors in the algorithm.

• Standard algorithms can be instructive.

• For teachers these are easy to manage and assess.

DISADVANTAGES OF THE FORMAL ALGORITHMS

Kamii and Dominick (1998), McIntosh (1998), and Northcote & McIntosh (1999) have potential dangers that can be summarized as follows :

• They do not correspond to the ways in which people tend to think about numbers.

• They encourage children to give up their own thinking and creativity, leading to loss of ‘ownership of ideas’.

• The traditionally-taught (standard) algorithm may no longer be the most efficient and easily learned.

• They tend to lead to blind acceptance of results and over-zealous applications. Given the focus on procedures that require little thinking, children often use an standard algorithms when it is not at all necessary.

• There is a high probability that the students will lose conceptual knowledge in the process of gaining procedural knowledge.

There is also the use of relevance. Students use standard algorithms for only a small proportion of their calculation.

NEED OF THE STUDY

There has been a rapid expansion of knowledge in recent years. Realization of the relevance of education as reflected in human thought, style, social values and culture have made it imperative to upgrade the curriculum and learning approaches in order to improve the quality of life.

To make a student (learner) a scientifically literate citizen as envisaged in the National Policy on Education (NPE) 1986 and NCF-2005, there is an imperative need for the learner to:

o Understand and apply the basic concepts of mathematics;

o Develop desirable attitudes and value appreciation for truth and objectivity.

o Learn scientific method to apply it in solving problems and making decisions to improve everyday living.

It has been seen that emphasis are given on formal ways of solving mathematical problems in the school setting even though we know this practice overburden and increase the cognitive load over the students. To make mathematics meaningful and enjoyable, alternatives ways of solving problems should also be embraced and encouraged so that mathematics fear and phobia reduce and interest and attitude accelerate.

Position paper on teaching of mathematics, NCERT,(2006), also talked about the use of personal algorithms as mental algorithms or folk algorithms under the head of mathematics that people use. Appreciating the richness of these methods can enrich the child’s perception of mathematics.

The use of the constructivism in teaching of mathematics has been highly recommended in NCF-2005 for enjoyable, meaningful and effective learning.

For achieving these objectives, it is necessary to shift emphasis from rote memory based, content-oriented and teacher-centered method of teaching to child-centered methods based on constructivism approach.

Source: Essay UK - http://www.essay.uk.com/essays/education/teaching-of-mathematics/


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