Investigating Differentiated Instruction and Relationships between Rational Number Knowledge and Algebraic Reasoning in Middle School


The research goals of this project are to investigate how to differentiate mathematics instruction for middle school students with different ways of thinking, and to understand how students’ rational number knowledge and algebraic reasoning are related. In years 3-6 of the project we also investigated how classroom teachers learn to differentiate instruction.

The educational goals of this project are to enhance the abilities of prospective and practicing teachers to teach diverse students, to improve doctoral students’ understanding of relationships between students’ learning and teachers’ practice, and to form a community of mathematics teachers committed to on-going professional learning about how to effectively differentiate instruction.



Amy taught middle school and high school students for 9 years prior to doctoral work in mathematics education. The origins of this project come from Amy’s desires to communicate better with more students in the same classroom. As Amy came to better understand students’ different ways of thinking in her doctoral studies, her desires to differentiate instruction only grew.


Three broad, interrelated reasons underlie the project.

  • First, today’s middle school mathematics classrooms are becoming more diverse (National Center of Educational Statistics, 2018; U.S. Census Bureau, 2015). Traditional responses to diversity are tracked classes that contribute to opportunity gaps (Flores, 2007) and can result in achievement gaps. Differentiating instruction is a pedagogical approach to manage classroom diversity in which teachers proactively plan to adapt curricula, teaching methods, and products of learning to address individual students’ needs in an effort to maximize learning for all (Heacox, 2002; Tomlinson, 2005). Thus, DI involves systematic forethought rather than only reactive adaptation.
  • Second, broadly speaking, students enter middle school at three different levels of reasoning that have significant implications for how they build mathematical knowledge in middle school, including their rational number knowledge (e.g., Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009; Steffe & Olive, 2010) and their algebraic reasoning (Hackenberg, 2013; Hackenberg & Lee, 2015; Olive & Caglayan, 2008).
  • Third, students continue to struggle to learn algebra (ACT, 2010; National Mathematics Advisory Panel [NMAP], 2008), which is now a middle school course for many students (Chazan et al., 2007). Although rational number knowledge is implicated in many aspects of students’ algebraic reasoning, little is known about how students’ rational number knowledge can support students’ algebraic reasoning—and vice versa.

This project is supported by the National Science Foundation through Award DRL-1252575. The views expressed here are not necessarily those of the NSF.