Yesterday I shared the following thread on Twitter about an exchange between me and my daughter:
The tl;dr version is that my daughter was introduced to something called “turtle head multiplication” – something I am well aware of and that makes me cringe – and I explained to her why I want her to focus her time and energy thinking about *math* while she’s doing math rather than getting distracted by stories of turtles, butterflies, or cowboys on horses eating donuts.
So what is turtle head multiplication?
Essentially, it’s the standard US algorithm for multiplying multi-digit numbers. I have nothing against the standard algorithm. What I dislike is dressing it up in a story that distracts from learning actual mathematics, such as why the 0 (not an egg) is needed. It’s not an egg, nor is it a “magic 0” (another name I’ve heard it called). Rather, it’s because when you multiply 94 × 3, you’re really multiplying 94 × *30*. That product is going to be 10 times greater than the product of 94 × 3, so all of the digits in 242 shift one place to the left:
2 hundreds become 2 thousands (200 × 10 = 2,000)
8 tens become 8 hundreds (80 × 10 = 80)
2 ones become 2 tens (2 × 10 = 20)
The digit in the ones place was shifted into the tens place, so now there are 0 ones left, which is why the product 2,820 ends in a 0. This is also referred to as a placeholder 0 because it helps us accurately recognize the values of the other digits. We use placeholder 0s in all sorts of numbers, such as:
Why do some teachers teach turtle head multiplication?
I believe teachers introduce it with the best of intentions because teaching the standard US algorithm for multiplication is hard.* They have likely tried teaching the standard US algorithm in the past, noticed common mistakes students make, and eventually looked to the internet or a fellow teacher for advice about making it easier for students to remember and follow the steps of the algorithm.
* If it’s taught in isolation from other ways of thinking or working with numbers. If it’s taught as a series of steps without connection to other concepts such as place value or other strategies such as partial products.
I appreciate these teachers being reflective and looking for ways to support their students. That I love. However, the small change I want to suggest is bringing these questions along whenever a teacher is looking for tips or advice about how to support students:
Is this idea I’m hearing about, and potentially going to implement, focusing on mathematics, or is it distracting from mathematics with a cute picture/rhyme/story?
If my students are struggling with remembering steps, why might that be? What mathematical ideas might they not understand that is making these steps hard to remember?
In the case of turtle head multiplication, I would say it distracts from the mathematics. If students are telling a story about a turtle head (not even a body!) wearing a collar and laying an egg, then we’re not focusing our limited and precious time with our students on actual mathematics. We’re not stopping and identifying the important mathematical ideas our students need to go deeper with that will help them make more sense of the algorithm and remember the steps they need to take (and why they’re taking them!).
If not turtle head multiplication, then what?
I’m going to pause here, but I plan to write a follow up post shortly providing one answer to this question. I’ll give you a hint that it has to do with having a clear progression of understandings and experiences that build students’ knowledge of multiplying with multi-digit numbers.
[Update – I’ve added a four more Quizlet study sets to my Multiplication Facts Practice folder. The three “Practice Doubling” study sets are designed to provide students practice doubling a number, a necessary skill to be able to efficiently use the Doubling Multiplication Fact Strategy The “Practice Halving” study set is designed to provide students practice halving a multiple of ten, a necessary skill to be able to efficiently use the Use-Ten Multiplication Fact Strategy.]
As a member of NCSM, I get a weekly email called the Marshall Memo that shares summaries of a variety of education-themed articles. What I like about the Marshall Memo is that I get exposed to articles I may never have encountered on my own. Even better, while many articles are on topics that aren’t math-specific, I’m still often able to able to make connections to my own work.
It also connected to something I’ve been thinking a lot about lately, which is the strong research evidence that retrieval practice promotes learning:
“Retrieval practice” is a learning strategy where we focus on getting information out. Through the act of retrieval, or calling information to mind, our memory for that information is strengthened and forgetting is less likely to occur.”
“…quizzes or tests that require students to actively recall specific information (e.g., questions that use fill-in-the-blank or short-answer formats, as opposed to multiple-choice items) directly promote learning and help students remember information longer.”
IES Practice Guide, Organizing Instruction and Study to Improve Student Learning, page 21
This also brings to mind “Rachel,” a thought-provoking blog post from Michael Pershan that has had me thinking about the interrelationships between deriving and recalling facts.
Suppose a student has just derived 9 x 4. If they’re confident and successful, they might have an opportunity to share that solution with the class — I might ask them to share their solution, and they might have a moment where they ask themselves, “wait, what was 9 x 4 again?” This is recall practice. Or, maybe, they are working on a larger problem in which 9 x 4 is merely a step, and their later work calls on them to remember the product of 9 x 4. They derive it, and then turn back to the problem and ask themselves, “what was 9 x 4?” Or perhaps, while working on a large set of multiplication problems, a student derives 9 x 4 and is then asked to derive 90 x 4. They ask themselves: what is 9 x 4?
All of this thinking got me inspired to give Quizlet a try for creating study sets that provide students practice both deriving and recalling multiplication facts. I organized my study sets around the thinking strategies shared in The Book of Facts: Multiplication, published by ORIGO Education.
“Research show that the most effective way for students to learn the basic facts is to arrange the facts into clusters. Each cluster is based on a thinking strategy that students use to help them learn all of the facts in that cluster.”
The Book of Facts: Multiplication, ORIGO Education
If you’re unfamiliar with these thinking strategies, ORIGO has kindly created a one-minute overview video of each one:
For each strategy I created three levels of study sets in Quizlet. Level 1 focuses on reinforcing the thinking strategy. As students practice the flashcards, they are presented a pictorial representation of the multiplication fact that reinforces the thinking strategy. For example, if students are solving 8 × 5, the reverse side of the flashcard shows the product as well as a visual that reinforces the idea that each fives fact is half of the related tens fact. In this case, the array model shows that 8 × 5 is half of 8 × 10.
Level 2 focuses on a verbal reminder of the related thinking strategy. The front of the card remains the same, but the back of the card includes a reminder of what students can think about to help them derive the fact. Here’s the back of the 8 × 5 card in Level 2:
Finally, in Level 3, the focus is on recalling the multiplication facts. The back of the card does not include any reminders; it just shows the product. If students get stuck, the teacher can ask the student to recall the thinking strategy they’ve learned, otherwise students should focus on recalling the facts.
In addition to the strategy-focuses study sets, I’ve also included three study sets that practice a variety of multiplication facts when students are ready to focus on recalling across all of the facts. Version 1 focuses on the x0, x1, x2, x3, x4, and x5 facts. Version 2 includes a wide variety of all facts. Version 3 focuses on the x6, x7, x8, and x9 facts.
You can access all 21 study sets on Quizlet. If you’re not familiar with Quizlet, there is a free version and a paid version. I’d recommend starting with a free account. If you’re a teacher, be sure to indicate it when creating your account because teachers get extra features.
Some words of advice, Quizlet offers a wide variety of modes for practicing study sets.
I’ve noticed that many of these activities show the product and students are supposed to answer with the multiplication expression. If you want to start by presenting the multiplication fact to the students, all you have to do is click the Options button and then change “Answer with” to “Definition” instead of “Term.” I recommend doing this because generally we want students to recall the product not the multiplication expression.
In the Flashcards activity, I recommend turning on Shuffle. If students are at a point of focusing on recall rather than deriving each fact, then I also recommend turning on Play. This will make the flashcard automatically turn over after a few seconds. This prevents students from falling back on counting strategies.
In the Learn activity, I recommend going into the options and deselecting “Multiple choice questions.” For retrieval practice, research does not recommend multiple choice questions. Rather, the “Flashcards” and “Written questions” are preferable Question Types for this activity.
In the Test activity, I recommend only the “Written” and “True/False” question types. Again, in all of these activities, don’t forget to change the “Answer With” option from Term to Definition.
And finally, if your students are not familiar with the thinking strategies in these study sets, then they may be very confusing and unhelpful to students. In The Book of Facts series, ORIGO recommends four teaching stages:
Introduce the strategy – Hands-on materials, stories, discussion, and familiar visual aids to introduce the strategy or sub-strategy
Reinforce the strategy – This stage make links between concrete and symbolic representations of the facts being examined. Students also reflect on how the strategy or sub-strategy works and the numbers to which it applies.
Practice the strategy – This stage aims to develop accuracy and increase ‘speed’ of recall. In this stage, a range of different types of written and oral activities is used.
Extend the strategy (to greater numbers) – Students are encouraged to apply the strategy to numbers beyond the range of the basic number facts. The activities in this stage are designed to further strengthen students’ number sense, or “feel” for numbers.
The Quizlet study sets I created fall within the Practice stage. If you’d like to teach these strategies to your students, I do recommend checking out The Book of Facts: Multiplication because it provides several activities at each of the four stages for each strategy.
If you try out these study sets with your students, let me know how it goes! I’m excited to be able to share this resource for retrieval practice to the teachers in my district. If I hear feedback from them, I’ll be sure to let you all know how it goes.
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.
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.
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.