A good math story problem is messy. Students have to parse the language carefully to understand the context and identify the operation that is required. Word problems can also be misleading. Superfluous information and nuanced keywords like “take away,” “total,” and “in all” tempt students to make rash calculations. Story problems challenge a young mathematician’s understanding—and sometimes reveal its fragility—in a way that rote number calculations do not. “At Fay, we focus on the conceptual as well as the procedural,” says Head of Lower School and fifth grade math teacher Lainie Schuster. “A balanced mathematician needs both. The engineers of the world are not given a list of calculations—they are faced with a problem, and they use calculations to solve it.”
In third grade, students are encouraged to write about and discuss their mathematical thinking. For example, after learning a new multiplication strategy, they write one thing that they learned about the strategy on a sticky note and place it on the anchor chart on the wall. Often, students will share their sticky notes with a classmate and then update their reflection based on the feedback they receive. The teacher brainstorms keywords with students ahead of time to stress the importance of being specific and using rich mathematical vocabulary. Students start writing their own word problems to demonstrate their understanding of a particular procedure, and often their daily math work contains a question that asks them to explain how they got a specific answer, why their answer is correct, or whether a particular method of solving a problem is efficient.
Fourth graders write about their mathematical thinking in math journals, where they respond to writing prompts and are required to explain their thinking efficiently. “The students have a limited amount of space, which requires them to use precise language and a set structure,” says fourth grade math teacher Maura Oare. Blending reading comprehension and math comprehension skills, students state their opinion, provide support, and then add a visual representation such as a drawing or graph to bolster their proof. Students use graphic organizers to decode different story problem structures, sorting important from unimportant information, defining what the problem is asking them to do, and identifying the operation that is required to find the solution. Lower School math teachers often require students to solve a problem in multiple ways. “Not every solution fits every problem,” explains Maura. “As mathematics becomes more complex, some strategies expire.”
Throughout Lower School, students are encouraged to write their own word problems, which stretches their mathematical thinking even further. “When kids write story problems, they have to manipulate so much mentally to create the context, determine how the context relates to the numbers, and ensure their calculations make sense,” says Lainie. In fifth grade, students hone their math storytelling skills in a collaborative project between their math and digital literacy classes. Each student has to demonstrate their mastery of five different types of multiplication and division story problem structures by creating and solving their own set of themed word problems. Students choose themes that reflect their interests and passions, from hockey to chickens! Each problem set has to include a question with a rectangular array, an equal grouping, a rate problem, a cartesian product, and a multiplicative comparison problem. Students display their problems, complete with graphics, on posters along the Lower School hallways.
Sixth graders culminate their study of fractions by demonstrating how to scale a family recipe up or down by multiplying by 2, by 1⁄2, by 2⁄3, and by 1 1⁄2. Students must use the three multiplication strategies they have learned during the year: repeated addition, the waffle method, and the butterfly method. Students then create a video of themselves making one of the new versions of their family recipe, and they explain how they calculated each ingredient amount. Math teacher Emily Samperi encourages her students to see story problems and numbers and symbols as two ways to tell the same story. “Sometimes, I’ll write random numbers on the board and ask someone to tell me a story that goes with them. The exercise helps me realize who doesn’t understand and where their misconceptions are, and it helps the students to connect math to real-world situations.”
The emphasis on reading, writing, and mathematical language in Lower School helps develop the pre-algebraic thinking that students will need to unravel complex problems in Upper School math’s problem-based learning units. “Algebra is all about the graph, the rule, and the word problem all telling the same story,” says Lainie. “When kids learn the calculation in isolation from the story problem, they don’t see the relationships, and math is all about building upon relationships.”