Building Math Fluency in Middle School Classrooms

Connecting intervention to grade-level instruction

This is part 1 of our series on strengthening math fluency across core instruction and intervention in middle school math. Join our mailing list to follow along. 

Middle school math is critical for students’ high school readiness and sets the stage for future coursework and long-term opportunities. Yet, we know many middle school students aren’t getting what they need out of math instruction—in 2024, only 28% of 8th-grade students performed at or above NAEP Proficient.

Core instruction (Tier 1) and intervention (Tier 2) are meant to work together to help students succeed with grade-level mathematics. But in practice, they often create two separate learning experiences for students, focused on disconnected concepts and skills. In many schools, this disconnect is shaped by practical realities—intervention is based on a separate instructional program, often taught by a different educator.

This lack of alignment can make it difficult for middle school students to build confidence and momentum toward grade-level understanding.

Consider this example:

A student is working on division with decimals in their intervention class, but the focus for their core instruction is on graphing. The student excels when practicing division with decimals, but struggles to apply that learning in the context of grade-level graphing. Over time, they feel frustrated and unsuccessful because success in one setting doesn’t translate to success in another. Add to that different language, routines, and models across settings, and the student begins to experience math as a series of unrelated tasks, rather than a coherent progression of learning.

To address this fragmentation, instructional leaders and teachers need to deeply understand how mathematical fluency develops across grade levels. This shared understanding can help create a bridge between Tier 1 and Tier 2, ensuring that the work students do in intervention moves them closer to success in core instruction.

What is mathematical fluency?

Mathematical fluency is the ability to solve familiar types of problems accurately, efficiently, and with understanding. It develops through experiences that build conceptual understanding, strategy use, visual models, meaningful practice, and mathematical discussion (Cartwright 2018; Cartwright 2023; Hattie et al., 2017; NCTM 2023).

Whether students are engaging with grade-level content during core instruction or earlier concepts during intervention, we can strengthen fluency by:

• Developing mathematical reasoning, so students understand the “why” alongside the “how.”
• Using familiar models, routines, and language across tiers to create coherence in students’ learning experiences.
• Supporting progress along a continuum that fosters accuracy, flexibility, and confidence over time.

How does building math fluency help students succeed in core instruction and intervention?

Rather than a collection of discrete skills, learning math is more like building a network of big ideas that connect facts and skills across standards and grade levels—creating a framework that continually deepens students’ conceptual understanding over time (Wiggins & McTighe 2005).

Teachers use a variety of models to develop students’ understanding of math ideas:

• Visual: Illustrate, show, or work with mathematical ideas using diagrams, pictures, number lines, graphs and other math drawings.
• Symbolic: Record or work with mathematical ideas using numerals, variables, equations, tables, and other symbols.
• Verbal: Use language to interpret, discuss, define, or describe mathematical ideas, bridging informal (concept-based) and formal mathematical language.
• Contextual: Describe mathematical ideas in everyday, real-world, imaginary, or geometric situations and contexts.
• Physical: Use concrete objects or gestures to show, study, act upon, or manipulate mathematical ideas.

Guiding students to flexibly move between different representations of the same math idea (e.g., equations, tables, and graphs) provides them more opportunities to connect and make sense of math concepts. As the National Council of Teachers of Mathematics (NCTM) notes, “flexibility is a major goal of fluency, because a good strategy for one problem may or may not be as effective for another problem.”

When planning instruction, it’s important for teachers to understand which models students are already familiar with so they can build on those experiences and make connections to grade-level content. Consider this example:

A 7th-grade teacher was working on ratio reasoning problems in core instruction and intervention. In intervention, her students had been using “little dots” and she’d had no idea what they were doing. She’d been trying to get the students to use the 6th- and 7th-grade models.

Looking back at earlier grade-level representations, she realized that the “little dots” were arrays—a visual model from 3rd-grade math. Her students did have the beginnings of ratio reasoning. And now, she had a way in. She could build on her students’ understanding of arrays to help them make connections in core instruction to the work they were doing in intervention, supporting them in progressing to grade-level models and reasoning.

By leveraging familiar models and guiding students to connect them to new strategies, teachers can support fluency development across tiers—helping students to build on what they already know and experience math as a more coherent learning experience. ​​Of course, individual skill development remains critical. As students progress and move across conceptual representations and strategies, they also benefit from meaningful practice that strengthens accuracy and efficiency with the underlying skills and facts embedded in this work.

Through coordinated teacher support and aligned Tier 1 and Tier 2 instruction, instructional leaders and teachers can build students’ fluency and guide them along a path to success in grade-level math.

Over the coming months, Instruction Partners will release additional blogs and tools to help middle school math leaders support teachers in this work. These resources will explore how fluency progressions can be used in instructional planning to differentiate core instruction and target gaps in intervention.

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