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Dr. Jennifer Krawec is the author of Solve It! Teaching Mathematical Problem Solving in Inclusive Classrooms (Grades 5–6). One of Dr. Krawec’s research areas involves children’s understanding of the concepts of mathematical operations. We asked her to share some of her latest observations on how students’ understanding of concepts of operations affects their ability to solve math word problems.

Exinn: Solve it! teaches students how to solve math word problems. What are concepts of operation and why would students’ performance be affected by these operations?

Dr. Krawec: Concepts of operations include addition, multiplication, subtraction, and division. Students should understand the relationships among operations. They also should know the definitions for each of the operations, be able to apply that knowledge by identifying word problems that reflect the specific operations, modify word problems to represent different operations, and ultimately to create their own word problems for the various operations.

Though we saw that students in Grades 7–8 had some difficulty with choosing the correct operation to solve the problem, Solve It! Instruction generally provided sufficient support for them to improve their problem-solving performance. However, for students in 5th and 6th grades, their difficulty with the four operations (including what each operation means and when to apply them) was significant enough that even with explicit Solve It! instruction, students did not initially improve their problem solving.

Exinn: So, how do teachers accommodate 5th and 6th grade students who have difficulty with these operations?

Dr. Krawec: It was only after we developed a lesson specifically on the concepts of operations (including definitions, visuals, and application) that preceded Solve It! instruction that students gained proficiency in problem solving. Early iterations of the concepts lesson focused on verbal repetitions of the definitions with opportunities for students to create word problems based on these definitions.

After implementing the lessons with students, we saw that students needed more. We created visuals of the definitions, general schematic representations of each operation using a number line format, activities to reinforce the concepts/definitions, and an operation decision tree that students could reference to support their application of the lesson during actual problem solving. These tools and supports are built into the Solve It! approach for Grades 5 and 6.

Exinn: Were these supports sufficient for all students?

Dr. Krawec: For some students, particularly for students with learning disabilities, an entire lesson dedicated to concepts of operations may not provide them with a sufficient foundation for problem solving. In these cases, teachers should revisit these concepts of operations frequently and integrate them into problem-solving (as well as general math) instruction. Discourse in math classes should reflect the concepts that underlie so much of the curriculum, and these definitions provide a good starting point that gives students, particularly those with learning disabilities, common language to connect each of the operations. 

Dr. Krawec is a research assistant professor at the University of Miami, where she has been conducting research on the Solve It! approach. Contact Dr. Krawec at krawec@miami.edu