By Corey Drake,
posted May 8, 2017 —
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.
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.
are four considerations for you to keep in mind when you are choosing numbers for
a word problem.
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?
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).
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).
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.
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.
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
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
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.
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.