Place value is the foundation of arithmetic. Unfortunately, many children and even some adults find it a difficult concept to understand. In Western cultures, children are usually taught to add two numbers together by counting. However, this counting strategy does not acknowledge place value. When children think of arithmetic in terms of tens and ones, counting becomes a burden.
When numbers are grouped into tens and tens of tens, and so forth, adding large numbers becomes similar to adding single-digit numbers. In the fifteenth century, the first printed arithmetic text, Treviso Arithmetic of 1478, considered place value so significant that it was listed as one of the five fundamental operations: numeration (place value), addition, subtraction, multiplication, and division. It wasn’t until 1911 that place value was included in the dictionary. Before that, the concept was referred to as numeration.
Transparent Number Naming
As previously stated, people in Western cultures struggle with place value. However, Asian cultures do not experience the same struggle. Why is that? Because most Asian languages use transparency in their number naming. For example, the number eleven is spoken as ten-1, twelve is ten-2, and fifteen is ten-five. In English, the teen numbers are reversed. The word for the ones place is named first, and the ten (or teen) is named second. For example, four is in the ones place in fourteen, yet it is said first. In addition, the tens place is spoken as teen, effectively masking place value. Another example is twenty-one. The Asian languages say this number as 2-ten 1, but in English, the ten is spoken as -ty, again veiling place value.
When children learn the transparent words for numbers, they can more easily use tens-based strategies when calculating quantities. For example, when adding 9 + 5, they know they can take 1 away from the 5, making the 9 a 10. Now, they have modified the problem to be 10 + 4, which is easier to solve. Unfortunately, many English-speaking children in the first grade do not readily know what 10 + 4 is.
What can we do about this? We can temporarily use transparent number naming for quantities 11 to 99 when we teach our students. In my experience and research, I have seen that children using transparent number naming gain the same understanding as Asian students. When using transparent number names, the sum of 10 and 4 is simply ten-4. Easy as that!
Place Value Cards
Students should also use place value cards while saying transparent number names and counting by tens. We can help the student learn place value through syllables. When saying the number 2-ten, we can show the place value card of 20 and point to the 2 when saying two and point to the 0 when saying ten.
As the student continues to count by 10 using place value cards, we will show them the 100 place value card. This number can be called 10-ten, with the first two digits showing a ten and the last zero showing a ten. This number has another name, one hundred. We can point to the 1 while saying one, then point to the first zero as we say hun, and finally, point to the last zero while we say dred. Students can see and hear place value when syllables are added to this activity.
As students grow in their understanding of numbers, we can begin to combine numbers with different place values and stack the corresponding place value cards. Start with the longest place value card on the bottom and align the edges of the other cards on the right side. For example, for the number 159, the 100 card would be on the bottom, and the 50 card would be next and aligned on the right edge. Finally, the 9 card would be on top, again lined up on the right edge, showing 159. Place value cards are especially beneficial for numbers such as 206. The student only hears the hundreds place and the ones place. However, when they stack the cards, they see that the tens place has a 0, notating that there are no tens. While it is not heard, it still needs to be seen.
Another benefit of place value cards is that they encourage the student to read numbers from left to right. The student first says the name of the number, observes how many digits follow it, and then says the correct place value word. Older strategies had students start on the right, name the column heading, and determine the place value of the digit furthest on the left. However, teaching our students to read numbers from left to right is most effective because we read words from left to right.
Transition to Traditional Number Names
Once your student understands place value well, you can easily transition them to the traditional number names. You can start by telling them that ten can also be said as ty. So, 4-ten becomes forty, 6-ten becomes sixty, and 9-ten becomes ninety.
The numbers 13, 15, 30, and 50 use the ordinal prefixes of thir and fif. So, help your students learn these by starting the instruction by having them say first, second, third, fourth, and fifth. Encourage them to see the relationship between the ordinal numbers and the traditional number names.
Because the w in two-ten is silent, when you introduce the word twenty, emphasize the w in the same way as it is pronounced in twelve, twin, and twice.
When you teach the teen numbers 13 through 19, tell your students that ten is also pronounced teen. The teen numbers are also said in reverse order. You can introduce this to your students by having them play a game. Give them a compound word and have them say the reverse. For example, you say fireplace and the student says place fire. You say, box-mail and the student replies, mailbox. Then they do the same with the teen numbers: ten-four becomes teen-four, and the reverse is fourteen.
History of Eleven and Twelve
Where do the words eleven and twelve come from? They do not fit the same pattern as the rest of the numbers. In the Middle Ages, people mostly worked with quantities up to ten. When they had one more than ten, they called it a one left because there was one more than ten. Over time, it was reversed to a left one, which morphed into eleven.
Twelve has a similar story. When the people had two more than ten, they said they had two left. During that time, the w was not silent like now. Instead, the tw made the sound as in twin and twist. Over time, two left morphed into twelve.
Place value is a powerful gift. Instead of memorizing an endless string of number words, place value puts quantities into a neat package to help us work with them effectively. Let’s unburden our students by assisting them to understand place value and become solid and efficient math students.
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