When people want to make a joke about how difficult, convoluted, or inaccessible word problems are, they often cite some version of the “two trains” problem. You can see an example of this problem here:
Maybe you want to try solving this problem yourself before reading on. (Maybe not!)
The “two trains” problem has become an emblem in popular culture. Saying the opening phrase, “Two trains leave different stations at the same time …,” invariably results in uncomfortable laughter. It surfaces memories of school problems that seemed convoluted and inaccessible and the inadequacy many people felt when faced with them.
The belief that what we used to call “word problems,” and now more often refer to as “story problems,” are more difficult than straightforward calculation also pervades teaching at the elementary grades. Teachers we work with often say, for example, that they teach calculation first, only later moving to what they think of as the more difficult challenge of story problems. Story problems are seen as a separate piece of instruction. What if, instead, we think of the idea of story as integral to learning about the operations from the beginning?
In recent years, I’ve been thinking, along with some of my colleagues, about the difference between the idea of story as a problem and story as a context. A story problem is a single instance, using specific numbers and requiring a specific solution. A story context might look exactly the same as a story problem, but its purpose is different: it is a representation used to ground calculation in meaning.
Consider the thinking of Fiona, a second grader. Fiona was working on this problem: There were 37 pigeons and 19 flew away. She recognized that this could be thought of as a subtraction problem, 37 – 19. To solve the problem, she decided on these two first steps:
Before you continue reading, pause and think through Fiona’s thinking. What parts of the problem has she solved? What does she need to do to finish the problem?
When students move from a story context to a numerical representation, they are decontextualizing, moving from the realm of context to the abstract. Sometimes students complete their solution within that abstract world of numbers and equations. They have lifted the numbers and operation off the context and are operating in that realm. Only when they have completed the problem do they contextualize again, returning to the original problem to see if their solution makes sense and to make sure they know what units their solution refers to.
At other times, students move back and forth during the problem-solving process, using the context as a tool for reasoning. In fact, at this point, Fiona knew that she hadn’t dealt with the 7 in 37, but she wasn’t sure how to proceed and seemed stuck. The teacher asked, “Did those 7 pigeons leave or stay?” Fiona said “stayed,” and then knew what to do with the 7. She completed the problem by writing 11 + 7 = 18.
With her question, Fiona’s teacher was modeling something that Fiona will gradually learn to do for herself, moving from the abstract back to the context—or, if the problem didn’t come with a context, inventing her own—in order to reason mathematically. Fiona had broken up the number of birds she started with, 37, into a group of 30 pigeons and a group of 7 pigeons, and also broken up the number of birds that flew away, 19, into 10 and 9. She removed both the 10 and the 9 from the group of 30, leaving 11 birds out of those 30 that stayed. By visualizing the context of birds that stayed and birds that flew away, Fiona realized that the group of 7 pigeons also still remained. Learning to contextualize and decontextualize is a part of the CCSS Math Practice 2, Reason abstractly and quantitatively. Part of the elementary elaboration of this Math Practice reads:
“Mathematically proficient students at the elementary grades … contextualize quantities and operations by using images or stories. They interpret symbols as having meaning, not just as directions to carry out a procedure. Even as they manipulate the symbols, they can pause as needed to access the meaning of the numbers, the units, and the operations that the symbols represent. … Mathematically proficient students can contextualize an abstract problem by placing it in a context they then use to make sense of the mathematical ideas.”
Along with pictures, drawings, diagrams, equations, and physical models, story contexts are a critical component of a student’s repertoire of representations. It is often useful to have three kinds of representations at play for a single problem: a numerical or symbolic representation, a picture or diagram, and a story. Moving among these to see how each relates to the others is powerful in understanding the mathematical relationships.
By the way, I had to solve the train problem for myself to see how I would approach it. It’s intended to be an algebra problem with two equations and two variables. But my first impulse was to draw a diagram, with which I was able to solve it with some simple addition. What worked for me about the representation is that I had to think out how all those numbers in the problem are related—the distance between the two cities, the two different speeds, and what the speed (in miles per hour) has to do with time (in hours). Now I’m working on how the algebraic representations relate to the story and representation.
In Part II of this blog, I’ll offer another example of students connecting story, visual images, and numbers and operations. In the meantime, if you have examples of how your students move back and forth between numbers or equations and story, we’d love to hear them.
This example is taken from a case written by a teacher, in Schifter, D., Bastable, V., & Russell, S.J. Building a System of Tens: Calculating with Whole Numbers and Decimals. Casebook, p. 48. Reston, VA: National Council of Teachers of Mathematics, 2016.
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