By Constance Hallemeier
“I didn’t think the Science of Reading had anything to do with me. I was wrong. This blog is for my secondary friends (teachers, administrators, and coordinators)–the ones who hear ‘Science of Reading’ and immediately think ‘elementary.’. That was me.” You’ve read this before if you have read Dr. Natalie Fallert’s blog five part series (Blog 1) on the Science of Reading. This post explores how these principles apply to secondary mathematics instruction. Check out Dr. Sherri Lorton’s examination of the intersection of the Science of Reading and elementary mathematics in a companion piece, What Math Can Learn from Reading Research.
Integrating the Science of Reading (SoR) research into mathematics instruction is not about copy-pasting literacy frameworks into a math class. And, most secondary math teachers don't know what the Science of Reading says to know how to incorporate the ideas—but it’s useful information. Here’s a brief explanation to help summarize the core lesson of cognitive science around learning reading: students thrive when instruction is intentional, varied, and honors the interwoven ways the human brain builds understanding.
Just as reading proficiency requires weaving together word recognition and language comprehension, mathematical proficiency requires connecting conceptual understanding, procedural fluency, and problem-solving strategies.
As non-ELA secondary teachers we don’t focus on the literacy world as much, but we can learn from the depth of research that has been gathered about how learning happens. One of the most powerful ideas from literacy research is that reading development is not built through focusing on a single skill or instructional method. When instruction becomes too narrowly focused, only on isolated skills or only on meaning-making, students often struggle to transfer their learning. When instruction is more comprehensive, students are more likely to develop flexibility and independence.
Mathematics works in much the same way. Students do not become mathematically proficient through memorizing procedures alone, nor through exploration without structure. They need opportunities to build understanding, develop efficient strategies, make sense of relationships, and apply reasoning in increasingly complex contexts.
Comprehensive instruction does not mean equal time for every activity every day. It means that over time, instruction is intentionally designed to develop the full range of mathematical thinking students need.
For example, in algebra, students often learn procedures for solving equations before they fully understand equality as a relationship. When students instead develop a strong conceptual understanding of equality—seeing both sides of an equation as balanced expressions—they are more likely to reason flexibly when equations become more complex.
This example is not about choosing one approach over another. It is about ensuring that procedures and understanding are developed together so that each strengthens the other.
Much of the current conversation in mathematics education becomes polarized around instructional “either/or” thinking:
In practice, these are not competing approaches. They are complementary elements of strong instruction.
Explicit instruction helps students access new ideas and build an accurate understanding. Inquiry and exploration help students make sense of patterns and relationships. Procedural fluency allows students to solve problems efficiently, while conceptual understanding ensures that those procedures are meaningful and adaptable. Productive struggle helps students develop persistence and reasoning, but it is most effective when it is supported—not left unsupported.
Strong mathematics instruction is not defined by choosing one side of these pairs. It is defined by knowing when and how to use each approach in the service of student understanding.
The Science of Reading has also outlined a very clear pathway to understanding, to help students become fluent readers with strong comprehension—build word recognition through phonemic awareness and phonics along with language comprehension through vocabulary, oral language and background knowledge. While mathematics lacks a formal "Rope" analogy, students require a similarly cumulative progression, building foundational number sense in elementary years to support advanced mathematical development.
Reading comprehension doesn’t just happen, just like graphing functions doesn’t just happen. Comprehension is a byproduct of word recognition, vocabulary, and background knowledge. Because math knowledge is built in logical, step-by-step chains, gaps in background knowledge cause complete comprehension collapse. Math educators must use learning progressions to explicitly connect prior mathematical concepts to new abstract learning.
For example, students can relate graphing transformations of quadratic functions to transformations of exponential functions because the concept of the vertical and horizontal shifts are similar, even though the functions have different shapes. But, students need to have the background knowledge of reading functions, graphing points, graphing linear functions, and graphing quadratic functions. Expecting students to learn each individual skill in every math class as if it is not related to anything else creates a lack of coherency in understanding. By linking prior knowledge, instruction becomes part of a larger developmental story.
To utilize a comprehensive list of instructional strategies and help students build understanding, the National Research Council’s framework for mathematical proficiency offers insight that is the same for elementary and secondary teachers. Rather than defining success in mathematics as speed or accuracy alone, this framework describes five interconnected strands:
These strands provide a more complete picture of what it means to be mathematically proficient. Importantly, they are not separate goals that can be checked off independently. They are interwoven dimensions of a single learning process. So, maybe math does have a “Rope” or we could make one.
In strong mathematics classrooms, the five strands are not isolated instructional targets. They show up together in meaningful learning experiences.
For example, consider a middle school lesson on proportional relationships. Students might begin by exploring real-world situations such as recipes or map scaling. As they work, they develop a conceptual understanding of ratios and multiplicative relationships. They use tables or double number lines to represent their thinking, building strategic competence as they decide how to organize information.
As patterns emerge, students begin to generalize rules and develop procedural fluency in solving proportion problems. When asked to explain their reasoning, they engage in adaptive reasoning, justifying why their methods work and comparing strategies. Over time, as students experience success and see the relevance of mathematics, their productive disposition strengthens—they begin to see mathematics as meaningful rather than arbitrary.
No single strand stands alone. Each supports and strengthens the others.
While instructional decisions happen in classrooms, coherence is built at the system level. In math, similar to reading, clear standards help define what students should know and be able to do, but standards alone are not enough. Educators also need a shared understanding of how learning develops over time. Grade level teams should determine what is acceptable evidence for proficiency for each standard.
This is where learning progressions become especially important. When teachers understand how mathematical ideas build within and across grade levels, they are better able to design instruction that connects prior knowledge to new learning. Instead of seeing each lesson as an isolated event, instruction becomes part of a larger developmental story.
In coherent systems, standards and learning progressions support—not replace—professional judgment. They help educators make informed decisions about pacing, emphasis, and instructional design while maintaining focus on developing all strands of mathematical proficiency.
A skateboard is originally priced at $120. A store is running a sale for '25% off all skateboards.' However, a customer also has a coupon for 'an additional 10% off the sale price.' What is the final price of the skateboard?
In middle school, students would need the conceptual understanding of what a percent is and how it relates to the whole; the procedural fluency to convert percents to decimals; the strategic competence to identify key information and represent the problem mathematically; the adaptive reasoning to apply the sale and the discount in the correct order; and productive disposition to approach the multi-step problem and justify the reasonableness of their answer.
A local phone repair shop charges a flat diagnostic fee of $40 to look at a broken screen, plus an additional $20 per hour for the labor required to fix it. Write a linear equation that models the total cost of a repair, graph the equation, and determine how much it would cost for a 3-hour repair.
In high school, students would need the conceptual understanding of variables and slope or rate of change; the procedural fluency for writing and evaluating equations; the strategic competence to scale the graph correctly; the adaptive reasoning to identify an appropriate domain for the real world situation; and the productive disposition to explain the difference between the slope and the y-intercept for this situation as well as how it applies to other real world situations.
Across all grades, the goal is the same: students develop deep understanding alongside flexible, efficient, and confident use of mathematics.
Sustaining comprehensive mathematics instruction and building understanding is not just a matter of individual teacher practice. It requires coherent systems that support educators in making thoughtful instructional decisions to build understanding.
When systems are aligned in this way, teachers are not asked to choose between competing priorities. Instead, they are supported in developing instruction that reflects the full complexity of mathematical learning.
As mathematics education continues to evolve, one of the most important questions we can ask is not “Which instructional approach is best?” but rather:
Comprehensive instruction is not a compromise. It is a commitment to the full development of mathematical thinking to build student understanding.
When mathematics instruction is intentionally designed to develop understanding, skill, reasoning, strategy, and productive engagement together, students are more likely to experience mathematics as coherent, meaningful, and accessible. And when systems support that kind of instruction, the impact extends well beyond individual classrooms. Math instruction that models this commitment to student understanding is not just applicable to the secondary classroom. Check out the Dr. Sherri Lorton’s look at the application to the elementary classroom, What Math Can Learn from Reading Research.
Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. Routledge.
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academy Press.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.
Shanahan, T. (2019). Reading research and reading instruction: Introducing the science of reading. American Educator.