Investigating Differentiated Instruction and Relationships between Rational Number Knowledge and Algebraic Reasoning in Middle School (IDR2eAM) (NSF #1252575)

This project is investigating how to differentiate instruction to meet middle school students’ mathematical learning needs. We are also investigating how students’ ideas about rational numbers and algebra are related. We have studied the PI’s teaching of students in after school classes and then in classrooms, co-teaching with teachers.

Target Audience: 
Middle School Students and Teachers
STEM Discipline(s): 
What Issue(s) in STEM Education is your Project Addressing?: 

Today’s middle school mathematics classrooms are marked by increasing diversity (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 (DI) 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.

In addition, broadly speaking, students enter middle school at three different levels of multiplicative reasoning that have significant implications for how they build mathematical knowledge in middle school, including their fractions knowledge (e.g., Hackenberg & Tillema, 2009; Steffe & Olive, 2010), integers (Ulrich, 2012), and aspects of their algebraic reasoning (Hackenberg, 2013; Hackenberg & Lee, 2015; Olive & Caglayan, 2008). Teaching students at all of these levels is a significant challenge and requires research into student thinking at these levels.

In our project, we are using iterative design experiment methodology to study these issues with middle school students in after school settings and classrooms. We are also creating a community of teachers engaged in these issues through a teacher study group.

What are your Findings?: 

We have developed a theory of differentiating mathematics instruction for middle school students (see below), and we have several papers that report findings on students’ thinking and learning in the domains of rational number knowledge and algebraic reasoning (see below). We have developed a video case of differentiating instruction for prospective secondary teachers that is also represented in a publication for teachers. One thing we have learned is that differentiating mathematics instruction well is very challenging when done thoroughly but very rewarding when done well.

  • Project Website
  • CADRE Project Page
  • Video: 2019 STEM for All Video Showcase
  • Publications
  • Hackenberg, A. J. (in press). Differentiating instruction. In Tabor, P., Dibley, D., Hackenberg, A. J., & Norton, A. H. (Eds.), Numeracy for all: Teaching mathematics to learners with special needs. London: SAGE.
  • Hackenberg, A. J., Aydeniz, F., & Matyska, R. (2019). Tiering instruction on speed for middle school students. In Otten, S., Candela, A., de Araujo, Z., Haines, C., & Munter, C. (Eds.), Proceedings of the Forty-first Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1396-1404). St. Louis, MO: University of Missouri.
  • Hackenberg, A. J. Norton, A. H., Wright, R. J. (2016). Developing fractions knowledge. London: SAGE. [Chapter 13 is about the IDReAM project]
Amy Hackenberg