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 problem-solving 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 multi-step 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 problem-solving 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 problem-solving 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 problem-solving 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.