They call it problem-solving for a reason. When students routinely engage in productive struggle, they become problem solvers!
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There is an amazing book called Adding It Up: Helping Children Learn Mathematics. It is not a new book (Copyright 2001), nor is it an easy book to read. I waded through it once a few years back. The book takes a hard look at the state of mathematics instruction in the United States through the lens of the Third International Mathematics and Science Study (TIMSS) conducted in the mid-1990s. Throughout the book, comparisons are drawn between mathematics instruction in the United States, Japan, and Germany.
One finding of the TIMSS that I found particularly striking, and that has really stuck with me, is the use of problem-solving in United States classrooms. They studied the types of problems used in classrooms in the three countries and, actually, the problems were not significantly different. What was different was the way in which the teachers in each country allowed students to work the problems. Not surprisingly, Japanese teachers typically allow students to struggle the most with problems. Teachers in Germany stepped in to help a little quicker than the teachers in Japan. In the United States, however, teachers typically offered assistance at the first sign of struggle or frustration. In other words, we often deny students the opportunity to engage in productive struggle.
Wait, isn’t it called problem-solving?
So here’s what I want you to think about. How quick are you to jump in and help your students solve their problems? How much do you allow them to struggle? Do you often catch yourself saying, “But they can solve the problems when they work with me!”? I challenge you to let kids grapple with problems. I know that, as teachers, it’s our nature to nurture. But kids can’t learn to solve problems if we always give them the answers. You might be surprised at the solutions they come up with!
Take a look at this Sharing Cookies problem.
If you had to guess, what grade level would you say this problem was designed for? Second grade? Third?
I have actually used the Sharing Cookies problem in both Kinder and 1st-grade classrooms. Let’s take a peek at what it sounded and looked like.
You can grab a copy of the Sharing Cookies problem as well as a couple of others using the link at the end of this post.
Solving the Problem
We start with a choral reading, one sentence at a time. Next, we modeled it using students volunteers. Here’s how it sounds:
Teacher: It says Suzy gave her 3 friends 5 cookies each. Hmmm. I need a friend to come up and help me (chooses a student to come up). This is my first friend. Friend, use your fingers to show me how many cookies you have (student holds up 5 fingers). Right, you got 5 cookies. Leave that hand up! Now I need another friend (chooses a second student to come up). This is my second friend. Show me how many cookies you got (student holds up 5 fingers). Great. Let’s see, how many cookies has Suzy given out so far? (Teacher holds her hands over each of the two students while the class counts 5, 10) So I’m done, right??
Students: NO!!! You need another friend.
Teacher: Oh, right (chooses another student to come up). Show me how many cookies you have (student holds up 5 fingers–repeat counting by fives). Hmmm, let’s see. The problem asks how many cookies Suzy has left (the teacher stands beside the 3 friends). Okay, I am Suzy. Turn and talk to your partner and see if you can figure out how many cookies I have left.
Lots of great math conversations! I see them looking at the fingers the friends are holding up (their arms are getting tired!) and looking back at the numbers in the problem. When I asked for their ideas, I got lots of different (wrong) answers, and that’s okay! A couple of students said 20, and I reminded them that Suzy only had 18 cookies to start with. And, yes, one pair of students came up with 3! They were so proud when they explained how they got their answer, and the other kids gave them their complete attention.
After we modeled with bodies, we modeled the same problem with two-color counters. Even though we had just done it with bodies, some of the kiddos still had a little bit of difficulty. But you know what, that’s why they call it problem-solving!!
Finally, we put our work on paper. I told them that mathematicians usually like to show their work at least two different ways. I suggested we draw a diagram first. You can see in the picture below how it turned out. I will tell you that they did most of the work on this poster. I asked how we could show the 18 cookies, and they suggested 18 circles. I had them turn and talk to a partner about how to show the cookies each friend got, and they suggested I circle 5 cookies. I circled the first group of 5, but I had helpers do the other two. Notice that we made a key to show that each circle represented one cookie. Finally, we did a tally mark chart.
Now I’m wondering, did you catch that Kinder and 1st-grade babies solved a division problem where they had to interpret the remainder? To me, that’s the amazing thing about great, rich problems and utilizing problem-solving strategies–kids can actually solve problems way beyond their actual skill set. A great diagram goes a long way!
Just a reminder here about CRA. Notice that we solved this problem using concrete and representational methods. No abstract here, because that wouldn’t be appropriate for these kiddos. The abstract would mean writing a division number sentence, and that would be meaningless to them. Now, if it had been an addition or subtraction problem, we would have written a number sentence and taken it through all stages of CRA.
Encouraging Multiple Representations
In order to be able to engage in productive struggle, students need tools. Otherwise, the struggle is not likely to be productive at all—just frustrating.
Prior to working on the Sharing Cookies problem, we discussed all the different ways mathematicians show their work, using the anchor chart shown below. The picture chart doesn’t really show it, but for the model, I actually glued counters on the poster. We talked about that when we use models, it’s things we can touch. Models also include acting out problems. You’ll notice on the anchor chart, all the different types of representations show the same thing—three friends with 2 cookies each. The students totally got that, and it was really cool.
Students should be able to use the representations that make sense to them. Of course, we teach lessons to introduce them to each of the representations, but then they become tools in their problem-solving toolboxes.
So there you have it. Go forth and let those students struggle!! Click here to grab your copy of the Sharing Cookies problem as well as a couple of others.