One of the cool things about small group instruction is that your lessons often take you down roads you never expected to travel. This week, my third graders were working on adding 3-digit numbers. We were using base-ten blocks to provide the concrete support they need to understand the regrouping process associated with the standard algorithm. So, for example, when adding 347 + 286, the students built both numbers with base-10 blocks, combined the ones and traded for a ten, combined the tens and traded for a hundred. Normal stuff, right? But it also meant that they were counting a lot of little unit cubes, and I noticed they were counting them by ones. Now I don’t know when or how I learned about counting objects by twos, pushing aside each group of two as I count it, I just know it’s what I do. So I suggested to one student that he might count the unit cubes by twos. He pushed one unit cube aside and said “two”. Hmmm. I reminded him that if we are counting by twos, we need to move two unit cubes. He nodded his head and started again, this time pushing two cubes at a time. There were a total of 13 unit cubes, and he counted 12 of them by twos. When he pushed the last single cube aside, he said 14. I asked, “Because 12 and one more is 14?” He shook his head, counted again, and this time said 13 when he counted the last single cube.

You might think this was a lone case–that’s certainly what I thought. So I decided to ask my other third grader to count the cubes by twos. Pretty much the same story. And then my three fourth graders. I think you can guess the result.

It’s just so interesting because we teach skip-counting by twos and we teach the concept of one more than a number, and those are the skills needed to count objects by twos. But somehow those skills don’t transfer to this application. Skip counting is also connected to multiplicative thinking, yet it’s often taught as simply a rote memorization process–saying the numbers without understanding what they mean.

I’ll wrap this up with a pitch for small group instruction. It’s really difficult to make this type of discovery using whole-group instruction. Working with small groups of students is like mining for treasure every day!

I think you pointed out a big part of the problem- counting by twos (and other numbers) “is often taught as simply a rote memorization process–saying the numbers without understanding what they mean.” How can we teach counting by twos to develop deeper understanding. Do students have opportunities to discuss what happens when they count by twos starting at different even numbers, starting at odd numbers, counting odd amounts of objects. Do they create visuals in their heads that help them understand patterns that occur. #elemmathchat

I agree, but I do not think skip counting is the same as understanding skip counting. I think skip counting can be learned by rote as early as pre/k. That once learned, it can be built upon to develop the number sense concepts at various intervals later on.. I appreciated my students having a background in skip counting when it came time to introduce multiplication. They could count an array or set more quickly, etc…..

I started my small group this week. With a class of 28 my small groups are 2 of 14. I know that’s not ideal, but it’s better than whole class. We just got 14 chrome books so I have the independent workers on that. Can you recommend any “free” math sites? I have them using FrontRow and I like that. The reports you get are good ( and a bit scary when I see 12 of my 3rd graders working in the 1st grade range!) I wonder how I could have so many in that range???

I was just talking to a teacher about this the other day before her class started a counting collections activity. I was with a little guy who had 120 chart and this day she wanted them to count by 10s. He said “I get it.” and counted by 10s using the hundred chart. He then progressed to counting his single pom poms by ten. It was a great teaching moment. I called her over because I had just mentioned to her why that rote counting might translate to this… I was like “Come see this!” It was great for the teacher to see too.

I’m going to put circles on the floor in pairs, counting to 20. Then the kids will “walk it out” by putting left foot on the odd number, right foot on the even number. Then progress to counting by 2’s. Thoughts?

I love it!