In my last post, I shared some abominable strip diagrams. Last night, my friend messaged me again about some different models. Also pretty terrible.
“Sorry to hit you up for math help but I can’t find any like this on the internet.”
There are two reasons for this, the second of which I’ll get to later in this post. The first is because this model is too bloated and trying to show competing ideas.
Here’s a cleaned up version of the model.
Any (good) area model should simultaneously represent multiplication and division. They’re inverses of each other. If you understand the components of the model, you should be able to write equations related to the model using both operations.
If I look at this model in terms of multiplication, I know I can multiply the length (7) times the width (13) to find the area (91). This area model represents 7 × 13 using the partial products of 7 × 10 and 7 × 3.
If I look at this model in terms of division, I know I can divide the area (91) by the width (7) to find the length (13). This area model represents 91 ÷ 7 using the partial quotients of 70 ÷ 7 and 21 ÷ 7.
All that from this one model. I don’t need all the “noise” included in the original model. For example, what is the purpose of writing the dimensions along the top as “10|70” and “3|21”? Knowing how an area model works, the only place 70 and 21 appropriately appear are inside the rectangle to show they represent area. Putting them along the top edge creates confusion about their meaning. Our students don’t need more confusion in their lives.
The repeated subtraction underneath isn’t terrible, but it’s unnecessary if you just want to know what multiplication or division sentence this model represents. Now, if a student were building the area model while using the partial quotients strategy, then the subtraction might be a useful recording strategy, but that’s not the same as being part of the model itself. I think it’s important to distinguish between those two things: features of the model itself and recording strategies a person might use as they build the model.
So the first problem my friend shared wasn’t great, but of course there was a second problem.
And it’s worse.
Holy cow! Bring on the tears.
I get that a student solving 46 ÷ 2 might think about and possibly even jot down potential options for partial quotients, but there is no reason this needs to be shown to children on their homework. And there’s still the problem of there being two numbers side-by-side along the length. Does someone think interpreting bad models is a sign of rigorous math instruction? I don’t.
Here’s the cleaned up version.
While the original model was terrible, the question wasn’t bad at all. I’d probably revise it slightly though. I might say, “Gina found partial quotients to solve 46 ÷ 2. She recorded her work in the area model shown. Circle the number(s) in Gina’s model that shows the quotient of 46 ÷ 2. Convince me you circled the right numbers in the model.”
So earlier in the post I mentioned there are two reasons my friend couldn’t find anything like this on the internet. The first is because these were bad drawings. I tried looking for videos of someone solving a division problem using partial quotients and an area model which led me to the second problem. So many videos out there of varying quality. And by varying, I mean it’s easy to find videos that aren’t all that great. Many demonstrate either a limited view of partial quotients or a limited understanding of the area model.
One of the great things about using partial quotients to divide is the flexibility in how you can choose to decompose the dividend. In the first problem in this post, for example, the dividend (91) was decomposed into 70 and 21, which are both easy to divide by 7. It could just as easily been decomposed in to
- 90 and 1
- 35, 35, and 21
- 63 and 28
While looking for videos to share with my friend, I found these (Video 1 | Video 2). What I noticed is that the partial quotients method is carried out in a rigid way that maps closely to the long division algorithm. In one of the videos, the presenter even connects the area model to long division notation.
The emphasis on place value is appreciated, but students deserve to know that they do have choice in how they decompose the dividend. Place value isn’t the only way.
These were the good videos. They might have missed out on sharing the power of this strategy, but at least the math is good. (I still didn’t share them with my friend.)
Sadly, there were also the bad videos. My major beef with these is that if you aren’t familiar or comfortable with partial quotients, you could just as easily watch a bad video and think you’re getting good information. These videos are so bad because, intentionally or not, they demonstrate big misunderstandings about the area model.
Example 1
In this example, the students are writing the numbers in the wrong place on the model. The partial areas (800, 370, and 23) should all be inside the rectangle while the lengths (100, 70, and 4) should all be outside along the top. I’m not blaming the kids. From what I can tell, they invented this strategy in their class (Cool!) but their teacher helped them make this video to share their strategy far and wide on the internet (Not cool!). Rather, as a teacher, I would have noted the students’ misunderstandings, helped them develop a better understanding of the area model, and then helped them create a video to show off their strategy.
Example 2
This one doesn’t even try to represent the values of the numbers. For whatever reason, the long division algorithm is carried out in boxes. Which, by the way, I don’t care if your video calls this the “box method” or “rectangle method.” It does not excuse you from misrepresenting the area model, because that’s what you’re doing. So many people believe math is confusing enough. Don’t add fuel to the fire.
Considering the time and effort that goes into building an understanding of area as a model for multiplication and division, we shouldn’t be making or showing these bad models to our students. We shouldn’t be showing them to our parents either. Seriously, if you share YouTube videos with your parents, please preview them and make sure the mathematics is good. Make sure they model the kinds of thinking, reasoning, and representing we want our own students to be developing.
Remember, the only people who should be making bad drawings are our students because they’re still figuring all of this out. Our job is to help them so that over time they get better.
Teaching to the Beat of a Different Drummer 
I do not see where the area comes into the area model.
“side multiplied by top left” is an explanation, but nowhere is it spelled out, and particularly not in terms of area.
It seems to me that “the volume model” is no less appropriate in the case of the examples.
I agree that in these particular models area is assumed to be understood. The idea is that *if* you’ve already developed an understanding of *how* area and multiplication are related, then you can leverage that understanding when multiplying or dividing, especially with larger numbers. These sorts of models are a shorthand of what is assumed you already understand. The problem is that it is very easy to teach these as a “trick” short circuiting the relationship students should have developed. Hence the reason it is often called the “box method” or “rectangle method” rather than an area model.
I’ll go for the rectangle method or the box method.
It’s a strange idea that any particular one of the “methods” to help in the calculations has become a calculation in its own right, and ends up as “a method” itself. Maybe the common core implementations have got hold of the wrong end of the stick.
If we have larger numbers then the common sense approach is to break up the multiplier into its bits, as in 234 = 200 + 30 + 4, but I’m old fashioned.
I’ve been working w/ students w/ division for the past couple of weeks — my adults in pre-algebra — possibly the victims of this kind of ‘instruction,’ so they’re in college and trying to figure it out…
It’s exciting that they’re getting exposed to it then! I did not develop a robust understanding of multiplication until I was in my mid-20s. Prior to that, I knew how to memorize facts and carry out the standard US algorithms for multiplication and division. I didn’t understand either operation much beyond that. I’m so grateful for those experiences I was able to have as an adult to help me see that I could *understand* what I was doing.
I’ve found it helps a *little* with longer division (things like they’d have to do to change an improper fraction to mixed number) to talk about how many 13’s I can take *out of* 73 instead of how many times it “goes in,” because then I can follow up with “okay, now we’ll take them out!” when it comes to the subtracting part, which basically completely befuddles them without lots of concept building.
You raise a great point about not really understanding multiplication until later in life. We learned the procedural steps of math but didn’t always understand the “why.” One of the great things about Common Core is the emphasis on conceptual understanding and not simply knowing the steps. It’s important to expose students to multiple ways of looking at math, so THEY can determine which one makes sense to them. Thanks for the post!
Thank you, Mike! I completely agree that students need multiple ways of looking at math. They need to develop robust understandings, not limited and narrow ones.
Have you seen Jo Boaler’s latest video about area models? https://t.co/AnnXG7pGhk Quite good. I think the teacher that has been using the poor models needs a little refresher on the mathematics. I think that visual math is a good idea as it stimulates both sides of the brain, but a bad drawing is like a bad driver – you just end up in the wrong place.
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I love math!