Place value means that each position in a number has a different value associated with it, that is, 362 is three 100s, six 10s, and two 1s. Often, we teach children about place value by asking them to think about, for example, the difference between the 2 in 12 and the 2 in 24. However, children learn more about place value by solving problems that involve collections of ten. When children learn about place value, they come to understand that groups of 10 can be created and counted just like units of one. The multiplication and division problems that you have worked on with your child are good places for children to learn about the advantages of grouping by ten.

Here is a multiplication problem that can help us understand how a child is thinking about ten:

*The class has 4 boxes of doughnuts with 10 doughnuts in each box, and Carl
brought in a bag from home with 17 doughnuts.
How many doughnuts do they have
altogether? *

This problem can be solved in many ways. Each way tells us about how a child is thinking about tens.

Rich: |
Puts out 40 counters and then puts out 17 counters. He counts all the counters to get 57. |

Bob: |
Puts out 40 counters and 17 counters. He groups the first 40 into tens, counting, 10, 20, 30, 40, and then counts on from 40 by 1's until he reaches 57. |

Misha: |
Counts by 10 up to 40. As she counts to 40 by ten, she puts up 4 fingers. When she reaches 40, she puts her fingers down so she can use them to keep track of the 17. As she counts from 40 to 50, she puts up a finger with each count. At 50 she puts them down again so that she can use them to keep track as she counts 7 more from 50. |

Robin: |
"Well, that's 40 and 10 more is 50, then 7 more is 57." |

These four children demonstrate a range of understanding about groups of ten. Rich views ten just like any other number and does not group by ten or anything else-he counts all the counters. Bob and Misha use the structure of the problem to form 4 groups of 10, but they are not comfortable enough yet to recognize that 17 could also be thought of as 10 and 7. Robin is more flexible. She recognizes that different groupings of 10 are possible: 57 can be thought of as five 10's and seven 1's, as four 10's and seventeen 1's, or even as two 10's and thirty-seven 1's.

Here's a division problem that can help us understand how a child is thinking about 10:

* Mary has 64 stickers. She pastes them in her sticker book
so that there are 10 stickers on each page.
How many pages can she fill
with 10 stickers? *

**Martin:** Counts out 64 counters and puts them into groups with 10 in each group.
Counts the number of groups and says 6.

Children invent their own ways to solve problems involving larger numbers where place value is important like those given above. However, people often try to help children be more efficient about place value by teaching them algorithms-tried and true ways to find answers using ideas like carrying and borrowing and so on. These algorithms were invented before the era of machine calculation so they were designed to be efficient and quick. However, they were not designed to help people understand why they work-in fact, algorithms often make understanding harder to achieve. Consider, for example, this problem:

*There were 302 animals at the zoo.
104 of the animals were monkeys.
How many animals were not monkeys? *

Zoa: |
302 -104 = 202 |

Ndidi: |
"Well if I take away the 100 that's 202. 4 less than that is like 2 less than 200; that's 198." |

Zoa shows one of the errors children commonly make when they try to apply an algorithm - she knows that smaller numbers are subtracted from larger numbers, so she decides that the reasonable thing to do is take 2 from 4. In contrast, Ndidi uses her understanding of place value to reason about the answer.

Here's another problem:

* There were 35 geese & 47 ducks in the marsh.
How many birds were in the marsh ? *

Matt: |
35 + 47 = 712 |

Jeff:
| 30 + 40 is 70, and 5 + 7 is 10 + 2, and 70 + 10 is 80 and 2 more is 82 |

People often think that children who use the traditional method-the algorithm where one puts one number above the other and adds ones with ones, tens with tens, etc., and carries or borrows-must understand place value very well. This is simply not so. Algorithms are often memorized without understanding. The child may get the right answer but have no idea what s/he is really doing. Even worse, calculations and results like those above are commonplace.

When children invent their own ways of working with place value, they often do the following:

(1) They add and subtract from left to right.

125 x 4 | ---> 100 + 100 + 100 + 100 = 400 |

---> 25 + 25 + 25 + 25 = 100 | |

---> 400 + 100 = 500 |

* Gary had 73 dollars. He spent 51 dollars on a pet snake.
How many dollars did Gary have left? *

"The 70 take away the 50, that's 20. And 3 take away 1 is 2, that's 22."

Children's understanding of place value is strengthened by inventing their own ways to handle large numbers. They also learn more about numbers in general. Encourage your child to invent her or his own ways of solving problems. There's plenty of time to learn traditional algorithms later or to simply use calculators that are even more efficient than algorithms.

** NOTE: **Many parents have asked how they can make the homework problems harder
for their children. One way is to use larger numbers!!