There’s no doubt that our students are seeing tougher word problems on state math accountability tests. In Texas, our students take the STAAR test, and the current version is much more rigorous than previous tests. Specifically, it includes more problemsolving and process skills. According to the state, 75% of the items on STAAR have process standards embedded. Instead of:
What is the perimeter of the figure shown below?
Now we see:
Figure 2 is a square. What is the difference between the perimeter of Figure 1 and the perimeter of Figure 2?
That’s a pretty big leap. And what about story (or word) problems? Yep, they are harder, too. More multistep problems.
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Kids need attack skills. They need to know how to tackle hard problems, and staring at a blank page just won’t do it. Yes, I know there are problemsolving strategies—draw a picture, work backward, make a table, etc.—but I think it really boils down to visualizing the math that’s going on in the problem. So, by far, my favorite problemsolving strategy—and the one I’ve had the most success with—is drawing a model. You and I know what math to do because we visualize the story in our heads. But that is a skill that is developed, and we need to help our kids learn how to draw models that show the math they need to do. The more they draw models, the more they will spontaneously develop the images in their heads. Yep, it’s that CRA stuff again. Drawing pictures to visualize the math is the representational part of the process. No, drawing a model will not work in every problem, but it will certainly work in problems that require kids to choose an operation—add, subtract, multiply, or divide.
So what does it look like? There are many ways to draw math pictures (or models). Probably the one most people are familiar with is Singapore model drawing. My process is a little less structured than that. Take this problem:
Susan is arranging flower vases for the Volunteer Appreciation Luncheon. She wants to put 4 roses in each of 12 vases. How many roses does she need to buy?
You probably read this and instantly saw multiplication in your head. Not so for a 3rd grader (who is just learning the difference between multiplication and addition) or a 4th or 5th grader who lacks comprehension skills.
I participated in a book study last summer on Laney Sammons’ new book, Building Mathematical Comprehension. One of the points she makes is that, for the most part, story problems follow a set structure. Typically, the first sentence is the “set up”. That is, it provides the setting of the story and often contains no information useful in solving the problem. Next comes the information—critical to solving the problem. The information might be in a table, chart, or graph, or it might be embedded in the story. This is the part of the problem that provides the numbers the students will need to solve the problem. Finally, comes the question. Sammons explains in her book that the question is really the main idea of the story problem. But by the time they get to it, kids have often made up their minds about what math they want to do. Haven’t you seen that? Kids answer a question, but it’s not the question that was asked.
So, in the problem above, the problem solving goes something like this.

Read the problem all the way through. Write an answer statement from the question.
How many roses does she need to buy? She needs to buy ______ roses. 
Now go back and carefully read the first sentence. Susan is arranging flower vases for the Volunteer Appreciation Luncheon. Okay, interesting enough, but this just tells me the setting of the story. No math here.

Read the next sentence. She wants to put 4 roses in each of 12 vases. Hmmm, what does that look like? Let me draw it. Well, there are 12 vases. Let me draw them (Yes, they look like rectangles. It’s not about creating art, it’s about seeing the math.).

Okay, that sentence also says that she wants to put 4 roses in each vase. How can I add that to my picture?

What kind of math does that look like (add, subtract, multiply, divide)? Why?

Now do the math you see. I see the number 4 twelve times. Hmm, 12 x 4 = 48.

Put your answer back into the answer statement to check for reasonableness.
She needs to buy 48 roses.
Trust me on this—teach kids this process and it will build their mathematical thinking and their problemsolving ability. It’s all about breaking the problems down into manageable parts.
Grab a little cheatsheet showing the steps here. It’s sized so students can glue it into their math journal.
Brilliant! I love the idea of starting with the end in mind (writing the answer statement first as opposed to after the problem has been solved). I know this will help my third graders stay more focused on what problems are asking them. 🙂
I do this, but I am having a really difficult time getting my kiddos to do this EVERy time. We use the UPS check problem and getting them to do it. Suggestions on that ??
It has to become a habit! I also find that putting it in terms of mathematical habits, rather than showing work, helps. Check out this post.
I just saw this blog referred to on twitter. Just wanted to add another idea. I think it is so powerful to help students use skills that are emphasized in reading…retelling, acting out, and then visualizing what is happening. To me, this entails engaging students in conversations with each other about what is happening in a word problem and then having them work to act it out, and see it in their minds. These skills can be taught and with practice and prompting become habit and lead to students being able to apply the skills independently. I just viewed an interesting clip about acting out word problems. It argues suggests that students begin with physical manipulation of objects in this acting out then move to visual manipulation. The video talks about a study where 3 days of training led to significant improvement in students work with word problems. http://www.watchknowlearn.org/Video.aspx?VideoID=53437&CategoryID=4032A“moved by reading”.
I love thinking about this stuff. Thanks Donna!
In a question like the one comparing the perimeters how important is it for the visuals to be accurate? The ones shown both have bases of 8 cm, but they are obviously different lengths. If they are labelled the same length shouldn’t they look the same length?
Excellent observation, Robert! Yes, the measurement on the square should not be 8 cm based on its relationship to the rectangle. Thanks for catching that.
Another great strategy that I have found to work really well for word problems is “act it out”. Get the kids up and moving and really turn that visual into something tangible (when possible)