What’s the Point of Expanded Form?

OK, I’m going back in time to tell you something totally ridiculous that I actually said to another teacher. It was because of my inexperience and my inability to understand math at a deep level during my second year of teaching.  I was frustrated that my students weren’t understanding what expanded form (that 145 is 100+40+5) was, and had no idea how to teach it for them to understand it conceptually. To the other teacher I said, “What is the point of teaching expanded form anyway? There is no point in learning this!”

Insert forehead smack emoji.

Now, 11 years later I am seeing the error of my ways. Expanded form is a way to deeply understand place value, but because back then I was a young and desperate, I thought anything that was hard to teach was worthless. Expanded form isn’t hard to teach with the right tools, and it will greatly enhance students understanding of place value. Place value simply is a way of understanding quantity in parts or chunks.

So what I do now is use my favorite approach, the concrete-representational-abstract string of learning. The concrete part is the tools, the actual quantity of what we are using.  The representational part is a picture form of the tools/number, and the abstract part is the actual number itself.

I STILL learned my lesson the hard way with assuming things of children. I knew they could make these numbers with tools, show them with ten frames and write them with much practice…but get this. They STILL didn’t understand what the numbers stood for when the number was in standard form. In the photo below they made the number 158 with bundles of sticks (in 10s and singles), cubes (in stacks of 10 and singles), place value blocks, and ten frames.  I asked, “What do you guys think that the number 5 stands for in this number?”

“Um…….5?”
“No! It’s for 50 tens!”
“What do you mean what does it stand for?”
“Tens…no…hundreds?”

So we did some practicing, I figured if they could make the number, they could identify parts of the number in what they made. So I asked them to put their hand on the hundred, what the 1 stands for:

Then we put our hands on the 50 (or what the 5 stands for):

We were getting somewhere! After practicing this a bit more, I was able to bring in the idea of expanded form with number cards, and then even write the equation:

Notice we kept as many tools in place as we possibly could. Making the links between the tools (place value blocks and bundles), the representations (ten frames), and the number cards finally seemed to solidify their understanding.  Eventually I began to remove the concrete tools, using only the ten frames and number cards…and then I could eventually remove the ten frames.

It was magical! Now I truly understand how wrong I was to teach it without the thinking behind it, just to get through it. I’m still learning, every day.