By Corey Drake, posted May 8, 2017 —

In the previous two posts on this blog, authors have written about the importance of the context of word problems—focusing specifically on the use of food as a context and representations of gender in word problems. In this post, I shift the discussion to a focus on the mathematics of word problems and, specifically, the numbers used in word problems.

Too often, we as teachers take the numbers in a word problem for granted and think that one number choice will work just as well as another. But researchers and practitioners alike (e.g, Carpenter et al. 2014; Land 2017; Land et al. 2014) have found that is not the case—number choice matters for both the accessibility and productivity of problems. Additionally, number choice can be used to meet specific learning goals, differentiate instruction, and build relational thinking (Land 2017) in each and every learner.

Below are four considerations for you to keep in mind when you are choosing numbers for a word problem.

• Consider the numbers in relation to your students. Often, we start with a problem from a textbook or other resource. Instead, we need to start with our students and their mathematical understandings. Then, look at the numbers in the problem: Are they the right ones to support “productive struggle” and engagement with the mathematical ideas for your students? Or are they too small, too large or complex, or the wrong kinds of numbers (e.g., decade numbers [10, 20, 30, . . . ] vs. non-decade numbers [17, 24, 32 . . . ]) to meet your students’ needs? Can all students access the mathematics of the problem through the given numbers?
• Consider the numbers in relation to your learning goals and standards. Different number choices provide opportunities for students to work on different mathematics, even within the same word problem context and problem type. Think about the numbers that will support—even prompt—students to work on the strategy and concept with which you want them to engage. Land and her colleagues (2014) provide many examples of number choice combinations and progressions that work well for different Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010).
• Consider the numbers in relation to each other. Many word problem types involve two given numbers; students are required to find the third using various operations and properties. Consider adapting one number or the other (or both) so that the relationship between the numbers supports students in learning about patterns in our number system and ways in which patterns and properties of numbers can be used to solve problems. For example, if you want students to try using a compensation strategy for addition, using 39 and 42 is likely to yield better results than 36 and 43. Questions to consider here include whether both numbers are close to decade numbers, whether they have some of the same factors, whether they are close to each other or far apart, and whether they are each close to the same landmark number. Consider also if and how the numbers are progressing within the day or across a week or a unit (Land et al. 2014).
• Consider the numbers in relation to the context. Finally, as this blog is part of a series focused on both the content and context of word problems to promote problem solving, consider the numbers of a given problem in relation to the problem’s context. Are the numbers reasonable in the given context? Are they the kinds of numbers that might be experienced in this context outside the mathematics classroom (i.e., in the “real world”)? Or do the numbers require students to suspend their out-of-school funds of knowledge because they are not numbers that would typically be found in the given context? Although matching contexts and numbers in ways that match students’ experiences may not always be possible, if the contexts are intended to connect to students’ lives, then connecting the numbers is important as well.

As I often tell the teacher candidates with whom I work, there is no single right or perfect number choice for a given problem. And, as Land and her colleagues note, the best way to learn about how number choices work is to “try different numbers and see what happens,” (2014, p. 8). Many resources are available to help you do that, including those cited here. The more purposeful you can be in your choices of numbers and contexts, the more you will be able to facilitate productive engagement with mathematics for all your students.

References

Carpenter, Thomas P., Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan B. Empson. 2014. Children’s Mathematics: Cognitively Guided Instruction. 2nd Ed. Portsmouth, NH: Heinemann.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

Land, Tonia J. 2017. “Teacher Attention to Number Choice in Problem Posing.” Journal of Mathematical Behavior 45 (March): 35–46. doi:10.1016/j.jmathb.2016.12.001

Land, Tonia J., Corey Drake, Molly Sweeney, Natalie Franke, and Jennifer M. Johnson. 2014. Transforming the Task with Number Choice, Grades K–3. Reston, VA: National Council of Teachers of Mathematics.

Corey Drake is an associate professor of teacher education and Director of Teacher Preparation at Michigan State University. She teaches elementary school mathematics methods courses, and her research interests include teachers learning from and about curriculum materials as well as the roles of policy, curriculum, and teacher preparation in supporting teachers’ capacity to teach diverse groups of students.