One thing I’ve learned about teaching elementary mathematicians, is that you should assume NOTHING. We think because our young learners know how to count, that they know what the quantity of those numbers are. If you get a chance, watch Graham Fletcher’s progression video about Early Number and Counting. Actually, if you get a chance watch ALL of his progression videos. They are fascinating and full of information. I have watched each one at least half a dozen times. What hit home with me is that we must provide tools and visuals to all students. I repeat, we **MUST** provide tools and visuals!

I was also inspired to write this post because of this amazing visual of a multiplication table that helps students understand quantity when multiplying numbers:

So here was the situation. When working with some first and second graders, I noticed that they had just memorized the counting sequence. They were rattling off the numbers without thinking about their quantity. So when we asked how many tens or ones the number was made of, there was confusion about what that meant. Here’s an example of what happened last fall when I asked a student to get me 31 cubes. After she did that, I had to ask her what 13 cubes looked like. She was focused on tens and ones, because that’s what the instruction was focused on that week, instead of thinking about how many cubes she actually had.

So imagine trying to ask her what ten more or ten less of 31 is…

After a recent assessment uncovered nearly 20 students who were struggling with the concept of 10 more/10 less, I knew we had to do something with a visual. I introduced a visual 100 chart for the students that were struggling during our intervention block. (Except I only got to 60 before I ran out of paper which means technically it’s a 60 chart!)

I told the students that I wanted to put the tens and ones right on the chart (but all the pieces kept falling off!) so I drew them on instead.

After I explained how I made the chart, I asked them “What do you notice?”

Again this is why I say ASSUME nothing. The things they said sort of blew my mind. One student excitedly said, “The numbers are growing by one dot!”

THEY DID NOT KNOW THIS ALREADY?

I was simply amazed at all of the things that I assumed they knew! They started to build on one another, saying things like:

- The ones stay the same when you go down on the chart!
- When you go across, the tens change in the last box.
- There are no ones in the last column!
- If you don’t remember the name of a number, you can use this to count and figure out it.

These were all things that we assume students would know, but because they’ve only memorized the counting sequence they weren’t visualizing what the quantity of these numbers looked like. Instead of memorizing patterns with only the numbers showing, this gave them a visual model to anchor to.

The best part of it all was how easy it was to make that chart. It maybe took me 20 minutes, and can be used year after year. If you give it a try, I’d love to hear how it works for you.