BEGINNING ARITHMETIC

Press, Volume CIII, Issue 30560, 1 October 1964, Page 13

BEGINNING ARITHMETIC

Many parents, as they read in newspapers and magazines or hear from speakers that arithmetic must be modernised, feel uneasy about the unknown prospect. They want to know “What is this modem arithmetic? Will we have to go back to school to learn it? How can you change arithmetic anyway? Is this modem movement just a passing fad?”

Research into the teaching of arithmetic is continuing ■ll over the world and the work of psychologists and mathematicians such as Piaget, Cuisenaire, Gattegno, Dienes and others has given us new knowledge of how children learn and of the kinds of material that help them, explore, observe, think about and understand the relationships between quantities. It is now generally accepted that children should understand as fully as possible the arithmetic they are asked to learn and that this understanding comes best when they are helped to discover arithmetical facts and ideas for themselves. Active experience with material is the key to an understanding of the basic ideas of number that children develop if they are to enjoy arithmetic and use it successfully. The kinds of apparatus provided will determine the degree to which the child can experiment and discover the meaning of numbers and the processes that can be performed with them. This material is used to encourage the children to think and not just “get answers to sums.” This year our schools have been issued with a book called “Suggestions for teaching arithmetic in infant classes.” The purpose of this book is to describe ways in which teachers can help young children towards an understanding of the fundamental ideas of arithmetic. Systems of arithmetic and algebra have the same basic patterns of structure. Concepts appearing through the programme in arithmetic in infant classes develop and grow and understanding of the principles of arithmetic leads on naturally to an understanding of the same principles generalised still further in algebra.

Apparatus that is designed to make clear “basic laws” of number is referred to as structural apparatus. One such structural material in use in many schools is the “Cuisenaire rods” often referred to as “coloured rods.” These wooden rods are 1 sq. cm. in cross-section ■nd range from 1 cm. to 10 cm. in length. They represent a model of the set of the natural numbers.

This means that when they are handled they behave as numbers behave and reveal the principles that apply in arithmetic and algebra. The fact that they are coloured makes it possible for mathematical operations to be carried out without using number-names because they can be referred to by colournames which come naturally to children. In consequence, quite young children become proficient in the basic processes of arithmetic while still learning how numbers are named and written and when able to use numberlanguage they find that num-

bers act in accordance with principles already familiar to them through pattern making. It must be remembered that the “rods” are quantities in relative proportions and should not be given number-names until children understand the cardinal meaning of numbers. With children of all ages (and adults) a period of play is essential for it is during this time many vital discoveries are made. These will gain full significance later on but in the meantime the complete familiarity which the children are acquiring will enable them to select without waste of time the rods to perform the activities to be introduced.

Building on the above foundation organised games are introduced to achieve relative knowledge of the rods and absolute knowledge of each. The first means they identify the rods by comparison and the second that any rod is known by its own identity by feeling without resort to comparison. The children take the rods and compare them to find out which are equal and which are not. They place any two end-to-end and look for a single rod that is equal to the length formed. This is a very important fundamental concept. “Take a red rod and a crimson rod. Place them end-to-end on the table. Can you find a rod which is the same length as the two together?” We say red plus crimson equals dark-green, r+c = d. Here we are developing the idea that wholes may be added to wholes to form new wholes.

Children should talk about what they have done. They should now put the rods together in another way. Then crimson plus red is dark green. It is important for children to do this because even at this early stage children should experience the property of commutativity (commute to exchange).

"The order in which numbers are added does not affect the sum.” Patterns are made by forming as many equal rows as possible using two or more rods end-toend. The children compare the patterns they have made with those of neighbours. They learn to read the patterns using colour names only. Staircases are formed with one rod of each colour. Sizes are compared labelling the rods as smaller, bigger, smallest and biggest Lengths can be made of one colour. This is called making chains. These chains provide many important conceptions which can be evoked by questions. Absolute knowledge of the rods can be gained by having a selection of rods held behind the back. Children are asking to produce a certain colour. (Children love playing this game.)

When we are ready to go forward to a new notation—the familiar numerals of arithmetic—and use numbernames in relation to the rods it requires very careful thought and understanding on the part of the teacher. It is a serious mistake, and one easily made, to assume that specific whole number values are allotted to each of the rods. The number values 1 to 10 are measures of proportion only. Any of these rods may be taken as the unit and all the others will be related to it according to these proportions. Because the white rod is the smallest rod in our set it will represent one of these proportional values—the red two of them, the yellow five of them and so on.

It is particularly important that the children should understand that a number, which is an idea telling how many, does not necessarily refer to identical objects. Thus three rods or books, or coins need not be three of the same colour rods or the same book or the same coin. Within each group the objects may be different or two alike and one different The children should be asked to select three articles for themselves either all totally different or all identical or all in the same category but different from each other, like children and so on. The rods with their different sizes and colours provide a ready means of illustrating this.

As each number is met, the children make once again the patterns which they made during the pre-number stage. This time, however, they will be able to write down in terms of number, instead of saying the colour, all the patterns which they see. Thus the modern emphasis in arithmetic as part of mathematics is the understanding of ideas of patterns and relationships. The child can be helped to become aware of them and by encouraging this awareness early, teachers are giving a child time for these ideas to develop and grow.