The study took place in a single middle school in the northeastern United States.
All students in the study had learning problems: 18 of the 22 students were identified by the school as having a learning disability, 1 had a severe emotional disorder, and 3 were at risk for mathematics failure. Participant selection was based on the following criteria: (1) teacher identification of students experiencing substantial problems in mathematics word problem solving and (2) a score of 70% or lower on the pretest, which involved multiplication and division word problem solving.
The schema-based instruction intervention condition included several components. First, students received instruction in recognizing the differences between multiplicative compare and proportion problems. Students were taught to identify the problem type, and use a schematic diagram to represent the problem. Then, students transformed the diagram to a math sentence and solve the problem. The instruction focused on two types of word problems: multiplicative compare problems and proportion problems. Sessions focused first on one type of problem, then on the other, and then the two types were mixed. Students first learned to identify problem types/structures and represent the problem using a schematic diagram, then learned to solve for an unknown quantity in word problems. For each problem type, teachers emphasized the salient features of that problem schema, and students received a prompt sheet that described these features and four strategy steps: (1) reading to understand, (2) identifying the problem and representing it using a schema diagram, (3) transforming the diagram into a math sentence and solving the problem, and (4) checking. Students received instruction in 1-hour sessions, 3-4 times a week, for a total of 12 sessions. The first four sessions focused on multiplicative compare problems, the next four on proportion problems, and the final four on both types.
Students in the comparison group received general strategy instruction. They received the same amount of instruction from the same teachers as the intervention group. Instruction for both groups included explicit instruction, followed by teacher-guided practice and independent student work, with corrective feedback and additional modeling as needed during practice sessions. Both groups were taught a four-step general problem solving procedure of (1) reading to understand, (2) representing the problem, (3) solving the problem, and (4) checking. However, steps 2 and 3 differed for the two conditions. While students in the intervention group were taught in step 2 to identify the problem type and use the schema diagram to represent it, those in the comparison group were taught to use other methods, such as drawing a semi-concrete picture, to represent the problem.
Support for implementation
The first author developed teaching scripts for both conditions and piloted them prior to using them in the study. Instructors received two 1-hour training sessions to familiarize them with lesson formats, the suggested teacher wording, and lesson materials when implementing the two instructional approaches. A checklist of critical instruction steps was developed to assess adherence to instruction strategy. Fidelity of implementation was assessed by a doctoral student for about 30% of lessons. The researchers provided feedback to instructors on fidelity whenever it fell below 85%. Overall fidelity was 100% for the comparison group and 94% for the intervention group.