Counting groups or “grouping” counts?

Recently, I read a post by Kristin Gray titled What is it About these Questions?  It got me thinking about our 4th grade students know about area and multiplication.  We have four small elementary schools in our district.  About every 6 weeks, we hold monthly elementary grade level meetings.  During that time, we do learning lab in one of the classrooms per grade level. The purpose of the learning labs is to collaboratively and plan a lesson that allows us to learn from each other and our students.  We usually pose a guiding question to help us identify what it is we want to learn about.

After I read Kristin’s post, I thought it would be interesting and informative to use  explore these questions in our upcoming fourth grade learning lab.  I shared her post with the fourth grade teacher with whom I will be planning the learning lab.

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As you can see, Deb thought this was a great idea! Deb and I decided that we would give her students the same exact questions that Kristin used, analyze them together, and decided what we still wondered. Then, we would think about how we could structure the learning lab to explore what we still wondered.




Since Deb and I will not have a lot of time to plan when we meet, I thought it would be helpful if we could process what we notice virtually – through Twitter and  blogging.

I am going to get the ball rolling by sharing my thoughts about what I noticed. The first thing I did was sort the student work into some groups. The first group represents students that got all three problems correct:







The first thing I notice is that even though all of these students got all three answers correct, I think they demonstrate very different understandings of multiplication and area. Student A decomposed all three problems to find partial products.  Student B used partial products to solve 19×7, but his explanations for the other two problems makes me wonder whether he added or multiplied to find his answers. Student C’s answers are really interesting to me because this student decomposed the rectangle but used addition to arrive at a solution. I wonder what student C and student A would say if we asked them to compare and contrast their strategies.  Student D got me thinking about my own understanding of doubling and halfing. He found 9 groups of 14 plus 7 more.  Do you see it?

What does student F know about multiplication and area? He didn’t show any work for the first problem, used partial products to answer the second one, and appears to have counted individual squares to answer the third problem.

I want to ask all of these students if they can use the array to show me where their decomposed problems are.  Students A, B, C, and E seem to be referencing the array in their strategies. Can student D see the 9 x7?  Does he understand what that partial product represents in terms of the array?

Even though these students got these answers correct, I still have a lot of math I would like to explore with them and I am not sure it would suit them best to put them in a group together just because they all got the answers correct.

The students below all got two of the answers correct.  I would like to ask students H and K why they both got different answers for 7×9.  I would like to show the whole class student I’s work and ask them what they think he is doing. Where is his work for 7 x 19 in the array?  I don’t even know if I would show the answer he arrived at. I think I would just show them some of his beautiful number arrays.  I think he might have a natural affinity for the associative property






The following students did not get any of the problems correct and  I don’t think I would put them in a group together. I need to ask student L more about how he arrived at his answers. I would like to get L and M to look to the array to support them. I wonder about putting N, M, and J in a group and asking them to compare and contrast all the different ways they decomposed 19 x 7.  Can they use different colors to show where the partial products are?  Would this help them revise some of their thinking?




I think it would be really interesting to ask all of the students, how could we adapt the 7 x 19 array so it shows 19 x 4?  What about 21 x 8?  Maybe we could start the class with this problem.  What do you think?


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