Word problems. Applications. Practical problems. Story problems. Whatever you call them, it seems that no one likes them. Do your students say, “I don’t get what they want me to do!”
Despite their reputation, word problems can be an important tool for developing operation sense. You’ve heard of number sense, which is having an idea of how numbers are related to each other. But operation sense instead looks at how we use numbers to quantify what happens in our world. It’s knowing how we translate what we want to find out into mathematics.
For example, it’s number sense to know that 8 x 6 must be greater than 36 (6 x 6) yet less than 64 (8 x 8), but it’s another thing to know that 8 x 6 tells us how many sodas we get when we buy eight 6-packs for a party. Without operation sense we might not know how to use the 8 and 6 to efficiently figure out how many cans there are. Operation sense tells us that 8 groups of 6 can be figured out by finding 8 times 6.
The story problem below contains some complex language and vocabulary. Strategies like 3 Reads (Asturius, 2017 & n.d.) can help support students’ understanding of the words and context in the problem. That’s important, but it’s not operation sense.
|Today we are had a contest to see who can throw a crumpled ball of paper twice and get the greatest difference between the two throws. Alejandra’s first crumpled paper ball traveled 37 inches before it touched the floor. Her second crumpled ball of paper traveled 81 inches. How much farther did Alejandra’s second paper ball travel than her first?|
|Alejandra’s First Crumpled Ball||Alejandra’s Second Crumpled Ball|
Math lessons often jump right from understanding the problem situation to computation. Sometimes students try to use other clues to figure out the correct operation without even reflecting on the problem situation. They may assume that the operation matches the current chapter they are working on or they may rely on a key word. As a result, when students see word problems (or real-life situations) outside the textbook, they “number pluck” (SanGiovanni & Milou, 2018) and just guess what to do with the numbers in the problem. When students guess at what operation they aren’t using their operation sense.
What’s missing, in a word, is the mathematics.
We can focus instructional time on the mathematical actions and relationships that are represented by the operations. In other words, recognizing that learning to mathematize, or modeling reality with mathematical tools (Gravemeijer, 1999) is important. We call this working in the Mathematizing Sandbox.
Sandbox work happens after students understand the problem context, but before they start to calculate any numbers. In the sandbox we explore different representations, mimic the action in the problem, and make connections. Work in the sandbox is experimental and even playful. Students try out different ideas and consider all of the action and relationships in the problem before settling on a model that makes sense. Coming out of the sandbox the student has an understanding of what happens in the problem and what work the operations are doing. In the first book in the Mathematize It! Going Beyond Key Words to Make Sense of Word Problems series we take many trips to the mathematizing sandbox to explore the typical problem situations that emerge in a grade 3 – 5 curriculum. Not only does the book introduce the reader to the all of the common problem situations, each sample problem explores useful mathematical representations and tools that help students reveal what happens in a problem.
The number line below is a good example of a mathematical tool helping to make sense of a the crumpled paper ball problem situation.
How did you figure out the distance between Alejandra’s first crumpled paper ball and her second crumpled paper ball? Maybe you subtracted. Maybe you added. You probably didn’t multiply because you have enough operation sense to know that we don’t multiply distances.
How do you find the difference? You may have learned at some point that subtraction means “take away,” but in this case nothing is being taken away, yet subtraction can still find the answer to this problem. Subtraction has another meaning as well: it can describe how much distance there is between two numbers, as it does in the crumpled paper ball problem situation. Knowing that information about subtraction, and being able to find an operation that can solve this problem, is operation sense.
Teacher Tips for Engaging Operation Sense
- Use strategies like 3 Reads to help students with reading comprehension and understanding the context of the problem.
- Engage Operation Sense. Spend time in the sandbox using manipulatives, visual representations, and discourse to mathematize the problem and find different approaches to the problem.
- Calculate an answer, with confidence that you have the right operation.
To learn more about the mathematizing sandbox and about how addition can also be used to solve problems like the crumpled ball problem, visit corwin.com/math.
Asturius, H. (n.d.). The 3-read protocol. Retrieved August 15, 2018 from http://www.sfusdmath.org/3-read-protocol.html.
Asturius, H. (2017, December). CMC Leadership: Sixty Years of Taking Responsibility for What Matters! Lecture presented at the northern section of the California Mathematics Council (CMC-North) in Asilomar, CA.
Gravemeijer, K. (1999). How Emergent Models May Foster the Constitution of Formal Mathematics. Mathematical Thinking and Learning, 1(2), 155–177.
SanGiovanni, J. J., & Milou, E. (2018). Daily routines to jump-start math class, middle school: Engage students, improve number sense, and practice reasoning. Thousand Oaks, CA. Corwin.