Our students “get” place value, right?

I mean it’s not rocket science; we teach them to know the names of the place values, the value of each place, the value of a digit sitting in a particular place. We are happy if our students can tell us that 345 = 300 + 40 + 5 or that the 2 in 325,908 really represents 20,000. We are satisfied if they can tell us how many thousands, hundreds, tens, and ones are in 6,794. And we are really happy if they understand that 415,112 is greater than 409,999 because there are more ten thousands. We test them, pat ourselves on the back when they all succeed, and move on.

So why all the fuss about place value?

The problem comes when students think that 300 – 196 = 296, or that 3 + 1.4 = 1.7; when students say that 2.319 is greater than 2.4 because 319 is greater than 4. Or when any student says that 5 tens and 12 ones is the same as 512.

Are these students careless? Did they just forget the rules? Or is our instruction lacking in a key concept that is causing these errors?

There are three fundamental components to our base ten place value system:

1. The base ten units: ones, tens, hundreds, thousands, tenths, hundredths, etc.

2. Positional notation, wherein each digit’s location specifies its base ten unit and the digit itself tells us how many copies of that unit are in the number.

3. Flexible bundling and unbundling, in which base ten units can be counted, broken down, and built up in different ways.

It is the third component that we often miss. Many kindergarten and first grade classes count the days of school every morning, bundling together groups of 10 popsicle sticks or straws until ten groups bring us to the hundredth day of school, which is then celebrated with activities surrounding the number 100. But how many classrooms beyond grade 1 make a regular habit of discussing different ways to express the values of numbers, or exploring what happens when we have more than 9 of a given place value?

If students really “get” place value, then they understand that 346 can be written as:

3 hundreds + 4 tens + 6 ones

OR

2 hundreds + 14 tens + 6 ones

OR

2 hundreds + 5 tens + 96 ones

OR

1 hundred + 23 tens + 16 ones

or many other ways.

Students who really “get” this flexible bundling and unbundling will then be able to apply this concept with understanding when subtraction calls for regrouping, when working with decimal numbers, when multiplying and dividing, when working with scientific notation. They will be much less likely to accept unreasonable answers because their sense about the magnitude of numbers has been sharpened.

So offer your students questions beyond the typical textbook fare; ask what number has the same value as 26 tens and 35 tenths. Ask for different ways to express the number 4207 using only hundreds, tens, and ones. Investigate how we could take 2641 items that have been packed into 3 crates that each hold 1,000 items. Discuss that no matter where along the place values you look, 1 larger unit decomposes to 10 of the next smallest unit, and 10 units compose into 1 of the next larger unit.

And for a simple app to practice and solidify this concept, download Place Value Cards.