Taken from the CA 2nd grade math framework. |

Now I have a handful of students decomposing numbers in ways that aren't really "helpful." For example, last week a student was working on the problem 9+7. He broke up the 9 into 5+4, then he added the 5+4 to make 9, so now he had 9+7 again, which he said is equal to 16.

I have watched a few students do this. I observe that they understand that they can break a number up into parts and those parts are equal to the number it was decomposed from, but maybe they don't see decomposing as a STRATEGY to use. I watched a handful of students do this last week over and over again. I'm wondering... Did I say or do something to encourage this strategy which created an "answer getting" type of feel or are they trying to copy someone in class without really understanding the purpose? I'm wondering, what do I do here? I would worry that a talk about efficiency during a Number Talk would be ill timed and work against the positive disposition toward math that I am trying to cultivate. Part of me thinks to leave it alone and let students work it out in their own time. The other part of me wonders if I could tackle this outside of Number Talks during a math lesson or during our procedural fluency/math intervention time. I would love to hear your thoughts. Thanks in advance!

In Elham Kazemi and Allison Hintz's book Intentional Talk, there's a chapter on comparing strategies - "What’s Best and Why?" - where the teacher picks a couple of the strategies being used - with really sensitive exemplification - and gets students to compare them for... which works best. I wonder if a session or two along those lines might go down well?

ReplyDeleteThanks Simon! What I'm now wondering is if this session should take place inside or outside of Number Talks time? Thinking maybe... create a scenario in a math lesson and look for them to do this and use it to as a centerpiece for a session like the one you shared. Thank you so much.

DeleteI meant to say - you've already achieved so much, with that list of what they've discovered and worked on!

DeleteThanks, Simon!

DeleteGreat minds think alike! That exact chapter came to mind while I was reading this post. They key thing is that you position both students as competent, but you are still using the conversation as an opportunity to look for efficiency in strategy use.

ReplyDeleteAnother possibility might be that perhaps these students already know the answer so the idea of a strategy isn't feeling like a tool because they're retrofitting it into a solution they already have? (I don't know your students, so you'll know better than me, but I could see students not getting the point of a strategy if they have the number combination automatic already.)

Just some thoughts. Thank you for sharing this! Not that I want things to go badly for others, but it's such a rich source of thinking and discussion to hear about times when things aren't working out perfectly. Thank you for sharing!

Apparently I really need to buy that book! This is the second time this week you've brought it up to me! :)

DeleteI think you're right about the students retrofitting the solution to a strategy they don't need to solve it. I realize the numbers were a little low, but at the time I was looking for them to have an opportunity to "discover" the relationship between addition and subtraction - aka fact families. Maybe sometime this week I can use the same mathematical goal, but beef it up now that they've already noticed the relationship with smaller numbers??? hmm Thank you for sharing your expertise.

I'm going to chime in here, too, and agree that you DEFINITELY need to get your hands on Intentional Talk! We actually refer to the book as "Number Talks 2.0" and use it as a follow-up training for teachers who have been implementing number talks and want to "hone their practice." It really opens up new ideas for how your number talks could inspire more targeted math discussion. I'm going to try to catch your session at NCTM next week!

DeleteA couple of thoughts. First, this seems like a great problem to have, especially since it's happened several times! This opens up a great possibility for discussion and thinking about WHY we compose/decompose differently when we have different number combinations. Now you know your students, at least some of them, are really in a place where this could be meaningful for them. "Ohhhh! THAT'S why we've been doing this!" At first they may have just been thinking that there are different strategies to learn and try. Now they get to see that there's another layer where they *choose* strategies. I third the suggest for the chapter in Kazemi and Hintz's book. I think this could be a great time for a math journal entry, if you do that. Ask the students what they think/notice about someone doing something like 9+6=(4+5)+6=9+6=15 (you could use an example that happened in your class or mock one up if you feel like that would be better with your students). It might be interesting to read their thoughts before having a whole class discussion on the idea.

ReplyDeleteP.S. What is your twitter handle?

DeleteDang! I really need that book! :) I really like you're suggestion about the math journal! Why didn't that cross my mind?! Thinking I could show one problem with two different strategies under it and ask students to compare them and their efficiency. Thank you for planning my math journal center for me for tomorrow!

DeleteI can completely relate Jamie, even in a 5th grade classroom. I agree with all of the recommendations for Intentional Talk, super helpful. I found in my classroom that students started decomposing in inefficient ways because they were just trying to find "another way" to get at the answer. It became more about the fun of playing around with the numbers but it definitely started to distract from the goals I had in mind in terms of standards. I am wondering if, since they are really great at making 10s, if you have tried some true/false statements to have them start noticing and move to conjectures they can start to generalize?

ReplyDeleteThank you, Kristin! I know the next book I need in my Amazon cart! I wish I could have a week of time to just sit and read and map out ideas! I always feel so behind! I'm also wondering if the numbers were too easy for the majority of the class like Brian suggested and they just retrofitted a strategy after the fact. I'm thinking more students would have used making ten as a strategy if they were forced to because they would need it to figure it out. This particular problem was part of a string where I was hoping they would notice the relationship between addition and subtraction (aka fact families), and I think when I chose the numbers I wasn't as focused on the strategies as I should have been as I was on them possibly noticing the numbers.

DeleteThe string was:

7+9

9+7

16-7

16-9

My students are pretty great at making tens. We have worked with true false statements, but not in the way you asked. That's a good idea. Thanks for sharing!

We need to place students in a place where they need to be efficient. In Number Talks we discuss efficiency but it's not out of necessity. A lot of the time we don't provide students an opportunity to practice their efficiency. For example, if students are working on making a 10 strategy in Number Talks, are they playing games that allow them to become fluent in making tens? A lot of time students will find shortcuts within their own thinking. Maybe use more games that mirror the Number Talks strategies your using that require students to become efficient. Just a thought:-)

ReplyDeleteThank you for sharing, Graham! I totally know what you mean about not always allowing students to practice their efficiency. I was totally guilty of that last year. I was so focused on creating good tasks, that my students could have used more time with things to really get into a groove with things. This year, I do have a block of time every day for procedural fluency and math interventions. I use games a lot in that chunk of the day. Most of the games have been heavy on making tens, like make 10 go fish, make ten castles and five fives - some partner and some individual games.

DeleteI haven't thought about making/finding games that would mirror all of the strategies that come out in Number Talks though. I love that idea! Thank you for sharing! Looking forward to seeing you in a couple months!

Dr. Ruth Parker emailed and said,

ReplyDelete"It’s hard to know what to suggest without knowing more about what Number Talks are like in your classroom. They play out so differently with different teachers from what I’ve seen.

One of the things I’ve watched a lot teachers struggle with during Number Talks, is taking care to NOT slip into a “teaching” mode. It’s so easy and natural to do so; and yet it is so important to keep this time all about student thinking and sense making, not about us.

It sounds like you’ve already considered your options. There are several ways to go here, and no best way as far as I can see. You could try to find a time to sit with the students you’ve seen doing this inefficiently in order to get more information about what they’re thinking. And you could have them think about it more in a small group. I’m sure, if asked, they could figure out and tell you what makes the strategy efficient (easy) and what doesn’t.

How old are these kids? If 2nd or 3rd graders you could ask the class to take a closer look at the strategy of “breaking apart the subtrahend.” Have them think about and investigate when the strategy is useful and when it really doesn’t help. And why.

It’s okay to let kids know that efficiency is a goal, and that we want them to use strategies that they find most helpful. I’ve rarely seen all kids go to the same strategy for any given problem if the focus is on their making sense in their own ways.

You could also just leave it alone and watch what happens. Unless kids are just trying to find more strategies (not the goal), they will naturally leave behind strategies or ideas that are not helpful and gravitate to more efficient ways.

During Number Talks, you could sometimes ask students why they broke the subtrahend up like they did. Most would probably tell you that they did it to get to a friendly number. It might help the others to hear the student-to-student interactions around why they used the strategy.

So many choices! Let me know how it goes. Best wishes!"

I am reading this several months after your post. So please bear with me...

ReplyDeleteSome thoughts I had were possibly using a series of true/false number sentences such as:

9 + 6 = (5 + 4) + 6

9 + 6 = (6 + 3) + 6

9 + 6 = 9 + (1 + 5)

Discussions could center around why these equations are true. (supports your students strategies of decompostion)

The next day possible provide this number string:

5 + 4 + 6

6 + 3 + 6

9 + 1 + 5

What strategies do they use today? Are some ways more efficient than others? Did they make any connections to yesterday's equations?

I've found that sometimes kids decompose numbers in ways that simply make more sense to them than to me. On the other hand, decomposing 9 into 5 and 4 and then recomposing it back into 9 really wasn't worth decomposing to begin with. Just knowing 9 + 6 is a good thing too!