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Welcome fellow Recovering Traditionalists to Episode 151: What Do Students Need to Know About Division?
Before I start this episode I’ve decided to add a positive comments area to the podcast. With all the negativity in the world it’s nice to hear nice things and I’m no different. Plus it’s a way for me to acknowledge & share what great things you all are out there doing to help Build Math Minds. When I get emails and comments on the YouTube channel that are THANK YOUs it always brightens my day. If anything I’ve done has helped you and your students out, please email in and let us know. It’s firstname.lastname@example.org. This week’s positivity comes from Molly. Molly posted this in the chat at the end of our recent Math Strategy Session about Compensation: “Wow, I learned SO much today, thank you!! I’m excited to attend the upcoming sessions.” Rosalba Serrano and I had a great time doing the session. If you missed it, the recording is now inside the Build Math Minds PD site for our members (you can join at buildmathminds.com/bmm) but we also still have more sessions where we are investigating other math strategies. During each session we play around with a specific math strategy that kids use for Addition/Subtraction/Multiplication/Division. You learn how the strategy works and what types of math situations help kids see the usefulness of the strategy. We also provide a resource to help you get started using word problems, number routines, and games for that specific strategy. You can register at https://buildmathminds.com/strategy-sessions.
In this week’s episode we are taking a look at the Essential Understandings kids need for division. If you aren’t familiar with the Essential Understandings series by the National Council of Teachers of Mathematics, those books should definitely be something you look into getting. There’s actually 2 different series of the Essential Understandings. The original series is Developing Essential Understandings and those books detail out what the essential understandings are for ALLLLLL the math concepts in PreK-12th grade. I believe there are 30 of those books. After those books were written the next series came out, Putting Essential Understandings into Practice, which details how to help students build the understandings in the classroom. You are given tasks to use and investigate student work samples from those tasks to better understand how children develop their understandings. I’ll add links to both series in the show notes page buildmathminds.com/151
This week I was looking back through the book Putting Essential Understanding of Multiplication & Division into Practice 3-5, trying to find things to share with you in this podcast. I actually have quite a few sticky notes throughout that book now, but the reason I picked the one for today was because of something that kept coming up in the Math Strategy Session Rosalba and I did last weekend.
In the first Math Strategy Session, Rosalba focused on the Compensation Strategy. That’s the broad term and can look different with different types of problems, but generally it’s when kids change one (or sometimes both) numbers to make the problem a bit nicer to deal with and then they have to compensate for what they did. A few examples include when solving 18 + 26 and they decide to change the 18 to a 20 and so to compensate for that they take two from the 26 and the problem is now 20+24. On that same problem, another kid might change the 18 to a 20, but add the 20 to the original 26 for an answer of 46 and do the compensating, or subtracting out the extra 2, at the end. Those strategies look different, but the mathematics they are doing is exactly the same just in a different order. Those are both compensation. For multiplication, a kid solving 19 x 8 might decide 20 x 8 would be easier but then they need to compensate for having too many groups of 8.
Rosalba spent time looking at this strategy with addition, subtraction, and multiplication with all different sets of numbers, whole numbers, decimals, & fractions. But one thing kept coming up, what does this strategy look like with division.
Now typically kids don’t use this strategy to actually solve a division problem, but they do use it to help them get started. Which is what we talked about in the live strategy session. You could think through this with any division problem but for the sake of us being all on the same page, let’s use 2753 ÷ 29. Kids aren’t going to find the exact answer using compensation, but they often use their understanding of compensation to think something like: that problem is kind of close to 3000 ÷ 30, so my answer is going to be close to 100 but it’s going to be less than 100 because I have way more in the total than I had in the original problem.
I thought that was a good explanation but we still had people throughout the session keep asking about division….and it wasn’t until I was reading back through this Essential Understandings book about Multiplication & Division that a line in it made me think maybe this was why.
On page 55, they list out a couple of Essential Understandings that relate to division and Essential Understanding 1e states: Division is defined by its inverse relationship with multiplication.
That’s the part I think not everyone sees. Division isn’t supposed to be seen as a whole separate operation. It is defined by its inverse relationship with multiplication.
Students need to understand the meanings of division and multiplication and how they relate to each other; they should not be taught separately.
Yet our textbooks do teach it separately and so as teachers, we feel like it needs to be. I believe that’s why so many teachers on the live session kept asking for more info about how to do this strategy with division. But the answer is: you don’t. You help kids develop it for multiplication AND help them see how multiplication & division are connected and then they will naturally use the idea of compensation to help them make sense of division problems.
On pages 55-56 they go on to write:
“Mulligan and Mitchelmore (1997) explain that the term multiplicative describes situations that lead to either multiplication or division and that every multiplication situation can lead to various division problems. Therefore, students can develop an understanding of the meaning of multiplication and division at the same time. Students can make connections between multiplication and division as they ‘think multiplication’ when working with division. Students should not simply identify some problems as ‘multiplication problems’ and others as ‘division problems.’ However, this view is at odds with the typical approach to multiplication and division in U.S. textbooks, which commonly introduce multiplication and division in separate chapters. When students focus on a single problem type or operation, they often limit their thinking to the ‘strategy of the day’ while failing to make important connections with other ideas.”
So whether or not you were in that strategy session, I want this episode to remind you that the teaching of mathematics should be about helping your students build connections. Mathematical concepts should not be taught in isolation. Even the strategies we are discussing in the strategy sessions should not be done in isolation. Different kids use different strategies on the same problems, but you can help students see how those strategies are connected.
One of the biggest things we need to help students to understand about division is its connection to multiplication. If your students are struggling with division, make sure your students are solid with multiplication and start working on building a connection between multiplication and division.
Alright, that’s all for this week, so until next week my Fellow Recovering Traditionalists, keep Building Math Minds.