Today, I gave the 4th graders four questions to get a glimpse into how they think about multiplication and division before starting their multiplication and division unit. Michael Pershan had given the array question to his 4th graders last week and shared the work with me. As we chatted about next steps with his students, I became curious if the students think about multiplication differently depending on the type or setup of the problem.

Here were the questions:

After sorting 35 student responses I found the following:

- 17 students got the area question wrong but the two multiplication problems on the back correct. Not only correct, but with great strategies based on place value.
- 8 students got all of the problems correct, however the area was found in many ways, some not so efficient with lots of addition.
- 10 got more than two of them incorrect. Some were small calculation errors on the back.

So, what makes almost half of the students not get the area?

Here is the perfect example of what I saw on the majority of those 17 papers:

Then I did a Number String with them to hear how they shared their mental strategies. I wanted to get more insight into some of their thinking because a few students had used the algorithm on the back two problems.

They did great. They used the 10 and 20 to help them solve the problems and talked about adding and removing groups of one of the factors. I was surprised on the final problem of 7 x 18 that no one used the 7 x 20 but instead broke the 18 apart to find partial products.

This makes me think there is something about that rectangle that makes them not use the 10s to help them decompose for partial products. I would love others thoughts and ideas!

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After reading the comments about area and perimeter, I wanted to throw another typical example of what I saw to see what others think of this (when I asked her she could easily explain partial products on the second and third problem)

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howardat58My simple observation is that the 17 students were “good” at breaking the numbers apart and putting them together (call it multiplication) but they haven’t got a clue about the process of multiplying. Not enough time with bricks.

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howardat58Maybe they were happier with rows of dots.

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mathmindsblogPost authorI am unsure about the intent of this comment. Is this suggesting the dots are more helpful?

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howardat58Idly consulting “area model”, today, in the CCSSM I found this in the Table 2, at the end.

“5 Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations.”

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mathmindsblogPost authorI dont agree they dont have a clue of the process. I believe they have a structural understanding of multiplication and can flexibly talk about groups of things and relate the problems to one another. But the thinking behind area of a rectangle is different.

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howardat58The word “area” was not given, so a picture of 7 by 19 dots or separated little squares should have given them a clue. Maybe the bunched up picture has made them think of area and then got them confused.

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Nili PearlmutterI love seeing this work – so interesting! I am wondering if the students conflated area and perimeter. It seems to me that the student who multiplied 19 x 19 and 7 x 7 might have been thinking about the way that when you calculate the perimeter, you add all the side lengths. This reflects a misunderstanding of the relationship between area and multiplication. Perhaps some students primarily think of multiplication as addition of equal groups, but do not visualize area as made up of equal sized “groups” of rows/columns.

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mathmindsblogPost authorHi Nili! Thanks so much for your comment. I completely agree, I think they confused area and perimeter. The funny thing about that is we purposely did not use the word area in the problem for that reason. We thought asking how many little squares would avoid that confusion. I think there is a jump from arrays of things (in groups) to groups of rows and columns. So interesting. I put another example at the bottom of the post of something I was seeing often as well!

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FranAs a fourth grade teacher, has been my experience when dealing with this particular group of standards, area and multiplication, it seems that not enough time is spent working on the two standards that build the understanding of the concept. I have tried to give this type of problem, with the drawing of boxes, and find that Ss will count the boxes individually and not make the connection to multiplying the sides. The don’t seem to understand the connection. It is very frustrating.

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mathmindsblogPost authorI agree it can be so frustrating but I think it really makes me wonder more about the connection-making experiences we need to spend more time on with students. I think the array to area model is a perfect example of that! I also don’t think it helps when we start talking area and perimeter that becomes one more thing in the mix! This teaching thing is tough, right?

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ProfSmudgeThis is fascinating stuff.

I’ve observed an obverse situation with 12 year old students in the UK. In two group-interviews about the number of dots in a 12×25 array, students have happily agreed you can find the total number of dots by multiplying (the dimensions), but when I’ve asked them to explain why, they have struggled to do so or even to understand the question. Their responses have been about the process of multiplying (how to do it) or about it being quicker or less error-prone than counting or using (repeated) addition, or they have simply declared that it’s the way you find area. None has spoken about ’12 rows of 25 dots’ or the like.

I’m beginning to think that the very ‘elegance’ of the array is one reason why students struggle with it. I might construe an array as ’12×25′ while a student sees it as ’25×12′. Or we might both go for 12×25 but while I see this as ’12 lots of 25′ the student might be seeing ’12, 25 times’. Also, if we describe an array in terms of rows, say, then we have to think both of the number of elements in a row and the number of rows. Of course the same numbers crop up, but with a different sense, if we’re thinking in columns.

And while I’m seeing a row of 12 dots, the student might be seeing two adjacent rows of 6 dots so that the array becomes a collection of 50 6s.

Ultimately, all these features make the array a rich and powerful model. But other models, such as those ‘quick images’ made of dot patterns seen on dice, might be easier to grasp, initially.

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mathmindsblogPost authorSo much insight in the comment, thank you so much! I think you are exactly right with your thoughts about the array. I think we miss the elegance and complexity when making the jump between groups of things and groups of rows and columns! So interesting. I put another example at the bottom of the post of something I was seeing often as well!

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mathontheedgeThis is fascinating. Thanks so much for sharing. I am really intrigued by this. I am a math coach. We have a fourth grade learning lab coming up on October 26th. I think this would be a great thing to explore with our fourth grade teachers and students. I am going to ask the host teacher if she is interested in trying these same problems with her students next week. Then, she and I can analyze and wonder. Last year, the fourth grade teacher group started to investigate which properties of operations were more prevalent in our student strategies. We found that they predominantly used the distributive property. We also found that our curriculum and instruction placed heavy emphasis on the distributive property. We wanted to find ways to highlight factoring and it’s connection to the associative property. Simultaneously, I was working with a teacher who was helping a student who was struggling to “let go” of repeated addition. I wrote about that on my blog ( Called Simmer Down – https://mathontheedge.wordpress.com/2016/05/). If you don’t mind, I would like to use the same problems that you did. I am wondering what the students would say if you asked them if they could find the numbers from 21 x 8 and 19 x 4 inside the array OR what if you presented it as a Which One Doesn’t Belong?? I am really interested to see where our fourth grade teachers and students go with this. Thanks for the inspiration.

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howardat58And just for fun……….

https://www.learner.org/courses/learningmath/number/session4/part_b/division_manip.html

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