The Difference Between Knowing and Understanding Place Value - Math Coach's Corner

Understanding Place Value
My 2nd graders have been exploring place value, so yesterday I wrote the number 43 on the table in front of one sweet friend and asked what he could tell me about the number 43. He proudly told me it was 4 tens and 3 ones. This is very common, right? We ask students to name the place value positions or tell us what digit is in the tens place, etc.

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I handed him a student ten-frame kit and asked him to build the number 43 using ten-frames. He immediately reached for a 4 tile. Uh oh. I asked what the 4 represents in the number 43, and again he told me 4 tens. He had the tile for 4 in his hand, so I asked if that was a ten. I got a quizzical look in return. I changed directions and asked him to show me 10 using the ten-frames. I also made headings for tens and ones on the table in front of him (see the photo). He put the 4 back and took out a 10. I asked him how many tens he had, and he told me 1. I then asked how many dots were on the ten-frame, and he told me 10. I asked him where he should place the 10, and he told me under the tens. Next, I asked him to show me another 10, so he took another ten out of the kit. I repeated my questions–how many tens? how many dots (ones)?–and he told me 1 ten and 10 dots and put the 10 next to the other one under the tens heading.  Then I asked him how many tens he had under the tens heading, and he told me 20. Hmmm, I think I see 20 dots, but how many groups of 10? Two. Great! So there are 2 tens and how many dots on those two tens? Twenty. His responses were still nowhere near automatic. That is, he really had to think about his answers. I repeated this process until we had 4 tens. Finally, I asked him what he would need to add to make the number 43. He considered the question, looked at the ten-frames in his kit, pulled out a 3, and placed it in the ones column. To tie it all together we wrote that 43 is 4 tens and 3 ones, and we also wrote it in expanded form, 40 + 3. We repeated this same process for several more 2-digit numbers.

Truly understanding place value requires students to be able to unitize. What this means is that they must be able to understand that a group of ten ones can be counted as a single unit, called a ten. That’s a huge mathematical step, because the idea is so abstract. As with any abstract concept, students need lots of varied hands-on experiences building tens and ones and talking about the meaning to develop a true understanding. That means ten-frames and counters, linking cubes, beans on a stick, linking chains, etc.–as many different representations as you can think of.

The ten frame kits can be a bit pricey, and have limited availability, but you can create your own student kits using this file. I also uploaded a B&W version, in case you want your kiddos to color their own. If you need ten-frames to use with counters, download those here.

I’d love hearing your experiences with early place value understandings! Please share what has worked for you.

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