Analyzing Unexpected Data after a Novel Mathematics Lesson Using the Critical Friend Process

Providing all students with equitable treatment and access has been a cornerstone of mathematics education reform (Association of Mathematics Teacher Educators, 2015; National Council of Teachers of Mathematics, 2014). Studies in mathematics classrooms show that using hands-on and interactive materials (i.e., "manipulatives") support students' conceptual understanding (Carbonneau et al., 2013), even for those who have learning difficulties and disabilities (Jitendra et al., 2017; Marita & Hord, 2017). Unfortunately, instructional approaches that incorporate manipulatives have not been implemented on a regular basis in high school mathematics classrooms (Swan & Marshall, 2010), likely due to the misconceptions of the effectiveness of manipulatives with older students (Carbonneau et al., 2013) and teachers' belief that manipulatives can be distracting (Corkin et al., 2019). The concrete-representation-abstract instructional approach, which involves the use of manipulatives, provides a promising instructional method for use in high school mathematics classrooms.