Notions of equivalence through ratios: Students with and without learning disabilities

• The author studied three third graders’ strategy levels and multiplicative thinking.
• One student with LD showed increased strategy levels and multiplicative thinking.
• Yet, this student evidenced misconceptions about ratios even after instruction.
• Qualitative analyses revealed that a focal student labeled as having a learning disability (LD) used similar strategies to solve typically and low achieving students and improved his ability to think multiplicatively and use viable strategies to solve problems.
• The student labeled as having LD had misconceptions about ratio equivalencies that persisted even after the intervention was completed.

Students with learning disabilities (LD) specific to mathematics historically underperform in foundational content such as rational number equivalence. This study examined the strategy usage and multiplicative thinking of three third grade children (i.e., Bill, a child identified as having a learning disability specific to mathematics, Carl, a child labeled as low achieving in mathematics, and Albert, a child labeled as typically achieving) before, during, and after participating in tutoring sessions consisting of student-centered pedagogy and equivalence tasks presented through an underutilized interpretation of rational number: namely, the ratio interpretation. Constant comparison analysis of the children's work during the tutoring sessions as well as responses to tasks during two clinical interviews seemed to indicate that all three children increased their use of viable strategies, with notable differences in the sophistication of the strategies as well as the level of multiplicative thinking utilized before and after the ratio-based tutoring sessions. Yet, Bill's continued use of rudimentary strategies reflects a need for continued research to investigate why the use of such strategies persists and how supporting the development of more sophisticated strategies (especially among children with LD) can be achieved.