Since the 3rd grade classes are about to begin their multiplication unit, the teachers and I wanted to hear how they talk about equal groups to get a sense of where they are in their thinking. What better way to do that than a dot image? I chose the first image because of the 3’s and “look” of 5’s and the second image because of the 2’s,3’s, and 6’s, all of which students can count by easily.

Image 1 went relatively the same in both classrooms and much like I anticipated. There were two things that stood out to me as a bit different between the class responses:

- There were more incorrect answers shared in the 1st class than the 2nd class.
- In the second class, multiplication came out during the discussion. The “4 groups of 7 and 4 x7 = 28” in the 1st class came out after both images were finished and one student said she knew some multiplication already. She asked to go back to the first image and gave me that.

After the first image, I anticipated Image 2 would go much the same, however it was quite different.

After the 1st image, I was really surprised at the difference in responses and I have to say it even felt really different. My assumption at this point is that in the 2nd Class one of the early responses was multiplication.

I am left wondering:

- Does that early multiplication response shut down other students who don’t know anything about multiplication yet? While I asked her to explain what she meant when she said 4 x 12, I wonder if that intimidated others?
- How could I have handled that response differently so others felt OK using skip counting or addition to count the dots?
- Can we anticipate that type of reaction from other students when someone starts the discussion with something that may be beyond where the majority of the class is in their thinking?
- Was this even the issue at all? Did the 2nd class just see so many more dots and wanted to avoid adding and skip counting?

The 2nd Class ended with a journal entry after a student remarked, “If we know it is 8 groups of 6, then it is also 6 groups of 8.”

I asked if 8 groups of 6 is the same as 6 groups of 8 and the class was split on their response, so they set off to their journals.

The yes’s went with multiplication expressions representing the same product and commutative property:

I loved this no because the picture changes:

I am not sure about this argument but I would love to talk to the student a bit more about the bottom part!

After that talk, I am excited to see what these guys do when they actually start their multiplication unit!

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deniseI love this idea of “is it the same?”. When we ask students to identify sameness, to some it has to look the same, to others the total quantity is the same. I like the student who justified his

“No” with the picture of how they aren’t the same. 8 baskets of 6 oranges is not the same as 6 baskets of 8 oranges. The total number of oranges is however the same. That is why we need to be careful with our Direct Modelers when given problems in context and expect them to see it as the same. Those that use a counting strategy will also use different counting strategies. 8 groups of 6 would be counted by 6’s. 6 groups of 8 would be counted by 8’s. Until they truly are able to use relational thinking, this idea of “sameness” is complicated.

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