Dr. Cotter on Place Value

Place value seems to be a difficult concept for many children and even adults. Western children traditionally are taught to add by counting. Counting ignores place value. For children, thinking in terms of tens and ones becomes an unnecessary burden when counting. 

Yet, place value is the very foundation of arithmetic. Packing numbers into groups of tens and tens of tens and so on makes calculating with large numbers similar to adding single-digit numbers. It was considered so important in the fifteenth century that the first printed arithmetic text, Treviso Arithmetic of 1478, states that there are five fundamental operations: numeration (now called place value), addition, subtraction, multiplication, and division. The term place value didn’t make it into the dictionary until 1911.

Asian children do not experience difficulty learning place value because Asian languages are transparent in number naming. For example, eleven is ten-1, twelve is ten-2, and sixteen is ten-6. This contrasts with English teen numbers where the ones and the tens are reversed and the word for ten has morphed into teen, as in sixteen. Also, twenty-one is 2-ten 1, and twenty-three is 2-ten 3. Here the word for ten is morphed into -ty.

Children who learn to name numbers with transparent words tend to use strategies based on tens. They make tens whenever possible. To add 9 + 5, they take 1 from the 5 to make the 9 into a 10, making the sum 10 + 4 = 14. Unfortunately, many English-speaking first graders do not know what 10 and 4 is. 

Is there anything we can do about this? Yes! We can teach children to refer temporarily to the numbers from 11 to 99 by their transparent number names. My research shows that children using these names gain the same benefits as the native Asian speakers. With transparent names, the sum of 10 and 4 is obviously ten-4.

Next, the children are given two groups of tens and are told it is called 2-ten. This is repeated for three groups of tens, 3-ten, up to nine groups of tens, 9-ten. They are also given place-value cards printed with: 20, 30, . . . 90. Point to the 2 while saying two and point to the 0 while saying ten. The zero converts 2 into 2-ten.

Continuing on with ten groups of ten, we show the children the place-value card of 100. Ask them to point out how it says 10-ten. The first two digits show ten and the last zero says it’s ten. Next, we tell them it has another name, one hundred. Point to the 1 while saying one, then point to the first zero while saying hun, and point to the last zero while saying dred.

When combining numbers with different place values, the corresponding cards are stacked. The cards are stacked by length with the longest card on the bottom and edges aligned on the right. Thus, cards for 100, 50, and 9 when stacked show 159. And cards for 200 and 6 stack to 206. Notice any unneeded zeros are hidden, but necessary zeros are visible.

Place-value cards encourage reading numbers beginning at the left. The child first states the digit, notes how many digits follow, and then states the correct place value word. The old method suggests starting at the right and naming column headings to decide the place value of the leftmost digit. Starting from the left matches the direction we read.

Transition to the traditional names by saying 4-ten has another name, forty, with the ty meaning ten. Repeat for 60, 70, 80, and 90. Additionally, the numbers 30 and 50 have a variation in pronouncing the three and five parts of the word, also found in thirteen and fifteen. The number twenty enunciates the w as does twelve, twin, and twice, although it is silent in two. The teen names from 13 to 19 are said in reverse order with teen meaning ten.

This leaves 11 and 12 as mysteries. Centuries ago people decided the call the number 11, a one left, because it was one left over ten. However, they reversed the one left and called it a left one, which turned into eleven. Twelve evolved from two left, with the w pronounced. Mystery solved.

Sometimes people think children can understand place value better if the ones, tens, and hundreds are color-coded. Unfortunately, 8 percent of boys are color deficient, so color coding is not helpful. Also, children relying on color schemes may view numbers having “color value” rather than place value.

We need to regard place value as a wondrous gift. Instead of giving our children a seemingly endless string of number words, place value allows us to neatly package our numbers and work with them efficiently.

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