DescriptionLearning to solve mathematical word problems is a complex endeavor, particularly in the primary grades when young students are simultaneously developing language proficiency to make sense of what the problems are asking to be solved. Word problems present the cognitive challenge of figuring out who is doing what, which numbers refer to which actions, actors or objects, and how to use the known number(s) to perform which mathematical operation to solve for the unknown. Sometimes manipulative objects are made available to help students, yet students still need to decide how to use those objects to represent the mathematical situation and perform the appropriate operation to solve the problem. In other instances of problem solving, students need to construct their own representations without the use of manipulative objects. Either way, students need to make sense of problems and attend to precision while persevering to solve them. These acts are two of the Standards for Mathematical Practices articulated in the Common Core State Standards for Mathematics (NGACPB & CCSSO, 2010).
In this multimedia narrative, we follow young Stephanie through the primary grades as she works with her peers in small groups to solve mathematical word problems. As a participant in the Rutgers-Kenilworth longitudinal study (Maher, 2010), Stephanie has been given opportunities to revisit the same word problem. Sometimes this occurs during a single classroom session, as we see in events 1 and 2. During this scenario from first grade, Researcher Alice Alston scaffolds Stephanie’s group in attending to precision by asking Stephanie to read the problem aloud multiple times. With each reading, potentially more can be noticed and cognitively processed by the children. It is argued that both revisiting this problem after completing the other assigned problems and thinking more carefully about how the language maps onto an appropriate mathematical operation that can be carried out precisely with use of manipulative objects helped the students find the correct answer the second time around.
Other times revisiting the same word problem occurs after several months have elapsed, as we see in events 3 and 4. Stephanie and Dana first worked together on the Shirts and Pants problem in a small group with Jaime when they were in second grade. At that time, Stephanie used the letters Y, B and W to represent, respectively, the colors Yellow, Blue and White. She used the numerals 1-5 to keep track of different possible outfits that could be made by combining various colored shirts and pants. However, her representation of outfits using this notation was not as precise as it needed to be for her to find the correct solution. When Stephanie and Dana were again partnered to work on this same problem as third graders, Stephanie’s notation had evolved to include drawings of shirts and pants with color letters written inside the garment, and connecting lines drawn between them to represent outfits. Although not perfect (one of the lines was not drawn as directly as it might have been), the increased attention to precision in representing the problem and performing appropriate operations led Stephanie to finding the correct solution the second time around. Taken all together, the four events provide evidence to support the claim that having an opportunity to revisit the same problem enables students to develop their skills at making sense of problems and attending to precision when learning mathematics in the primary grades.
References
Maher, C. A. (2010). The longitudinal study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 3-8). Springer: New York, NY.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards Initiative: About the standards. Retrieved from http://www.corestandards.org/about-the-standards