Utilization of teaching materials in first-grade mathematics

Utilization of teaching materials in first-grade mathematicsAuthor(s): DORA H. SHAWSource: The Arithmetic Teacher, Vol. 10, No. 1 (JANUARY 1963), pp. 37-41Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186690 .

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In the classroom Edwina Deans

Utilization of teaching materials in first-grade mathematics

DORA H. SHAW Durham County Schools, Durham, North Carolina

A variety of materials wisely used pro- vides an ideal situation for beginning in- struction in mathematics. Imaginative use of a few basic materials, however, is more important than an extensive collec- tion of materials.

The well-known number line, place- value boards, a hundreds board, a number board, thermometers for high and low temperature readings, and calendars are such basic materials. It is the purpose of this article to show a few of the ways in which these are used in a busy and produc- tive classroom.

The number line is one tool which is used from the first week of school (recognition of numbers) to the last week (division and multiplication for the mature children, more addition and subtraction for the others) . For this reason it is taped to the chalkboard at a comfortable level for the children so that it can be worked with easily. Its merits in aiding counting, visualization of the onward progression of numbers, and addition have long been pro- claimed, yet if used imaginatively it can help first graders to really understand their work as they subtract two-place numbers, multiply and divide by 2's, 3's,

4's, and 5's, and perform other number operations.

One of the early uses of the number line is in connection with recognition of numbers, their names and symbols, and is tied in with the reading lesson. At first the children are told the page number of their reading assignment and a child finds the numeral on the number line. Then all the children find the page in their books. In early trials the child will usually start at 1 and count until the right number is reached. Later he will learn to find both numerals on the number line and page numbers readily.

In the addition process one uses the number line from left to right, as in 5+3 = 8. (See Fig. 1.)

In multiplication the principle is the same and is really serial addition, as in 4X2 = 8. (See Fig. 2.) Here the child can easily see that you are working with four groups of 2's, and this concept of grouping is reinforced with objects.

In the subtraction and division however, you work from right to left. To subtract 2 from 7 you start at 7 and swing under two objects moving to the left. The action may be shown by an arrow picture start-

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Figure 1

January 1968 37



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Figure 2

ing with 7, and swinging under the numerals 7 and 6 to 5. The child can see readily that 5 remains. (See Fig. 3.)

, 1 23456789 100 ̂

Figure 3

For division of 6 by 2 (or better stated, 6 divided into groups of 2's), one begins at 6 and works to the left connecting groups of 2 numbers and enclosing two objects with each swing. (See Fig. 4.)

a I 23456780 100

t igure 4

A problem such as "I have 6 sticks of candy which I wish to divide equally among 3 children" is an example of parti- tion division. The action involved in this kind of division problem can be shown on the number line if the child draws three circles or boxes on the chalkboard under the area of the number line to represent the three children. The child then draws a line from a numeral to each circle, and re- peats until he has used up all the six numerals (sticks of candy). (See Fig. 5.)

S>i^</ ....юс (children)

Figure 5

After doing the work the child can see and understand that each child would get two pieces of candy.

Another use of the number line is found in weighing exercises. Each child weighs himself and puts an X under the numeral on the line which matches the numeral on the scale. From these marks the children can find out who weighs the most, the least, what children weigh the same as others, how much more or less one child

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weighs than another, etc. This activity is also adaptable to measurement of height. Any activity which involves the child di- rectly is, of course, much more meaningful to him.

The number board has four sets of the numerals from 1 through 9 painted on it in mixed order. Above each numeral there is a hole through which a colored peg can be pushed. (See Fig. 6.)

At the beginning of the year the board is used for additional work in number recognition. The teacher pushes a peg through the hole above the numeral 3, and the child holds up a number card with 3 written on it, which he has chosen from answer cards on his desk. Later the child responds by writing the numeral indicated, the numeral which comes before (2) and the numeral which follows (4).

For addition two colored pegs, neutral and red, are selected; the teacher pushes the two pegs through holes above the two numerals (2 and 3), and the children either write the combination (2+3 = 5) or show the answer with a number card (5). For subtraction a neutral and a green peg are used. The procedure is the same as for addition except the children are told which color signifies the total group or the sum and which color indicates the known part or the known addend.

Figure 6

The Arithmetic Teacher



Using a neutral peg as the multiplier and a yellow one as the multiplicand, the problem of 2 X3 is quickly posed. For division a neutral peg signifies the total to be regrouped and a black one the divisor. The more advanced children can give the answer to such problems by showing answer cards.

A good exercise in place value is derived by using a colored peg for tens and a neutral peg for ones. A colored peg through the hole over the numeral 6 and a neutral one over a 5 will signify 65.

Still another use is a game of ring toss played by tossing rubber jar rings at pegs over each numeral. Thus, you can add all numbers ringed, subtract misses, or add, subtract, multiply, or divide the peg ringed by a certain fixed number. Also, problem cards with answers on the back can be hung on the pegs for children who need work in a specific area.

The hundreds board (Fig. 7) is so named because it holds numerals from 1 to 100 and is arranged so there are ten numerals in a horizontal row. The numerals are painted on squares of wood

hundred's board

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Figure 7

which hang on pegs on the board. One activity used early in the year helps the children to learn the order of the numbers. The cards for the numerals 1 to 10 are removed from the board and mixed. A child then unscrambles the cards by placing them on the board in order. In a variation of this, ten children each take one of the numerals from 1 to 10, then go in mixed order to place them in their correct position on the board.

As the children master the order of all the numbers from 1 to 100, all the squares

January 1963

are removed from the board, mixed in order, then given to the children (three to a child) to be placed in their proper position on the board. To do this the child must be able to recognize the number, understand how many tens and ones it contains, and then must count down by tens to find the proper horizontal row and over by ones to find the proper position.

Each morning the children put the number of the day's date in the proper place on a calendar, prepare a place- value board showing the day's date in tens and ones, forecast the weather for the day on another calendar, and move the half red and half white ribbon tapes on large thermometers, putting the top of the red where the top of the column of mercury would be on a real thermometer for the low temperature for the night before and the high reading expected for that day. (See Fig. 8.)

After these tasks are completed, the morning milk order must be prepared and sent to the cafeteria. This requires the use of several of the materials already dis- cussed. There are thirty children in the room, so the children whose job it is to prepare our milk order put a line after the 30 on the number line. They then find out the number of children absent and sub-

Figure 8

39



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ABSENT

Figure 9

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NOT GETTING MILK ABSENT

Figure 10

tract these from the total children. The morning that the notes for this article were taken there were three children absent. (See Fig. 9.)

Next the children ask those not getting milk to stand and these are subtracted from the twenty-seven children present. Ten were not ordering milk. (See Fig. 10.) Here the children see that there should be seventeen children getting morning- milk if their work is correct. They check them- selves by counting the children who are ordering milk. As the year moves on the children begin doing more work with the milk order, tallying the number getting and not getting milk and finding the dif- ferent combinations that make up each number, for example, on the day of this record one child wrote combinations such as 8+9=17; another wrote such ones as 7+3 = 10. It was at this point that Debbie said, "I think that if I write down the number of children getting milk, the number not getting it, and the children absent I could add them all together at once and get 30."

17 10 3

30

Another activity follows this on some days. As each child puts his empty milk carton back on the tray, he goes to the

chalkboard and writes the amount of milk that his carton held, £ pt. One child converts the half pints to whole pints by hooking two half pints together. Then more children go to the board and write the number story for each pint found. § pt.+èpt. (See Fig. 11.)

After determining the number of pints of milk that have been enjoyed, children often like to figure out the total cost of their milk. This is done using the hundreds board. One half pint of milk costs each child 3 cents. Each child who has had milk is given a spool. The first child places his spool on the peg of the number which shows how much his milk cost (3). In turn, children add the cost of their milk onto the previous total, so that if one reads the numerals with spools above them he reads, 3, 6, 9, 12, 15, etc. The children soon learn how to count by 3's, and after they understand this well, by 2's, 5's, 10's, and 4's. Thus instead of starting with meaningless chants of 2, 4, 6, or 5, 10, 15, the children begin with a meaningful concept. The more mature children soon begin to write these series of numbers without looking at the hundreds board. With additional help from the number line, by the use of objects, and by continuous use of problems posed by teacher or children, they make the transition to actual multiplication and division. They learn that it is faster and easier to just know that three 7's are 21 than to add 7+7+7.

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Figure 11

40 The Arithmetic Teacher



An incident which occurred in the sec- ond week of May supports the theory that first graders can understand what multiplication is all about and actually perform these operations. Preparations were being completed for a birthday party celebrating the birthday of each child in the room at one time. Party whistles, three in a pack- age, had been tacked on the bulletin board still in their cellophane wrappings along with hats, cookies, and other party favors. Two little boys, Craig and Loyd, stood before the board, touching each thing with

only their happy, dancing eyes. Suddenly Craig turned to his friend and said, "Loyd, there are only 10 whistle bags up here. There are 30 children, so I wonder why only 10 of us are being given whistles? That isn't fair."

Loyd replied, "Oh, Craig, don't you see? There are 3 whistles in a bag and there are 10 bags. Ten 3's are 30 or three 10's are 30, so everybody will get a whistle."

"Oh," said Craig. "I didn't think of that."

For your information- Those in the region of Illinois State Nor- mal University may wish to plan to attend the Sixteenth Annual Mathematics Con- ference to be held on that campus on Saturday, March 30, 1963.

At the first general session Dr. Jack E. Forbes will address the elementary teachers and Dr. Robert B. Davis will address the secondary teachers. Following these sessions, discussion groups have been planned for primary, intermediate, junior

January 1963

high, senior high, and junior college teachers of mathematics and for school admin- istrators. For the luncheon meeting Dr. Davis has selected for the title of his address, "How Far Can Modern Mathe- matics Go?"

Further information may be obtained by writing to Kenneth A. Retzer, Depart- ment of Mathematics, Illinois State Nor- mal University, Normal, Illinois.

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Utilization of teaching materials in first-grade mathematics