High-Quality Instructional Materials: Tools for Successful Learning Outcomes

Consider the relationship between an artisan and their tools.  

It is the artisan’s skillful use of appropriate tools that leads to the desired results. Without the artisan, the tools themselves will not produce anything. The artisan is indisputably the crux of the relationship. Tools must be applied with skill and attention to produce what they are capable of yielding, and two artisans with distinct levels of skill will produce different results even while using the same tools. However, tools are still essential in yielding the desired output. Without quality tools, the artisan will have a difficult time producing quality results. While a particularly resourceful artisan might find alternate approaches or even repurpose less appropriate tools, the artisan will achieve the best results with high-quality tools designed specifically to yield the desired outcomes.  

Although this metaphor simplifies the many factors that influence student learning, it captures the essence of the relationship between a teacher (i.e., the artisan) and their instructional materials (i.e., their tools): both the skill of the teacher and the quality of the materials shape student learning outcomes. In the same way that the artisan’s tools have limited utility until they are in the hands of a skilled artisan, instructional materials cannot yield positive student learning outcomes without implementation by an effective teacher. This claim is backed by decades of research which consistently points to the teacher as the most important school-based factor in yielding student achievement.^1-5 Teachers ultimately use instructional materials to shape students’ daily learning experiences, leading to three distinct curricula: intended (by developers) curriculum, planned (by teachers) curriculum, and enacted (by students and teachers) curriculum.^6,7  The enacted curriculum, which is dependent on decisions made by the teacher throughout planning and implementation, will ultimately determine student learning outcomes. But effective teachers need high-quality instructional materials to support maximum student growth and achievement. Research indicates that students learn more when teachers use high-quality instructional materials, even when not implemented to their full potential. ⁶ Learning outcomes are greatest when effective teachers skillfully use high-quality instructional materials to yield the intended results. 

At Big Ideas Learning, we believe in empowering teachers by providing them with the highest-quality tools to achieve the greatest possible outcomes in student learning. This is achieved through intentional design of instructional materials proven to support the desired student learning outcomes.

Research and best practices point to several foundational pillars of high-quality instructional materials in mathematics, including: Focus & Coherence, Rigor, Conceptual Foundation, Procedural Fluency, Problem-Solving, Mathematical Modeling, Mathematical Discourse, Assessment, Student Supports, and Teacher Supports.

Focus & Coherence

Any tool is only as worthy as the outcomes it is designed to yield. In education, academic standards identify the desired learning outcomes that effective implementation of high-quality instructional materials should yield. The Common Core State Standards for Mathematics,⁸ for example, identified “shifts in mathematics” to describe qualities of the mathematical understanding students should develop over the course of their K-12 mathematics experience. Two of these qualities, focus and coherence, emphasize the importance of a narrower and deeper curriculum to yield rich and transferrable student learning.  

Focus highlights the importance of emphasizing the major work of each grade level. Focused instructional materials identify big ideas of each grade level and ensure students have opportunities to build a deep understanding of those ideas. Focused instructional materials support an approach that allows for in-depth coverage of truly foundational mathematical ideas versus a hurried, shallow approach to cover a long list of topics.

Coherence refers to how multiple topics connect both within a grade level and across multiple grade levels. Coherence leans on the structural nature of mathematics to ensure that topics build in a meaningful way, showing connections to prior grades and ensuring that all content within a grade supports the big ideas (i.e., the focus) rather than existing as isolated topics.

When effective teachers implement high-quality instructional materials built to support focus and coherence of big mathematical ideas, students build knowledge of mathematics as a meaningful and connected subject. 

Rigor

In addition to focus and coherence, the Common Core State Standards for Mathematics also describe rigor as an important quality of mathematical understanding.⁸ So what does rigor mean and what are the features of instructional materials designed to support rigor?

While we often equate rigor with meaning something hard or rigid, in the mathematics classroom rigor describes understanding that allows students to use mathematics flexibly and nimbly. This is because rigor describes mathematical understanding that integrates three aspects of understanding, namely conceptual understanding, procedural fluency, and application. Rigorous mathematics instruction is often touted as instruction that balances the three aspects of rigor, providing ample time and attention for students to build a conceptual foundation, to build procedural skill and strategies, and to apply their knowledge while problem-solving. It is important to note, however, that merely balancing the three aspects of rigor does not establish the flexible thinking that is a hallmark of rigor. It is also important that these aspects connect in rich and meaningful ways. As a result, high-quality instructional materials move beyond simply ensuring all three aspects of understanding are covered, but also focus on opportunities for students to reflect and connect these aspects to see how they fit together as part of an overarching whole.

When effective teachers implement rigorous instructional materials, students build a deep and flexible understanding of mathematics. 

Conceptual Foundation

Conceptual understanding has long been recognized as a vital component of mathematical proficiency. It serves as a reminder that the goal of mathematics instruction is not simply to teach students to follow step-by-step algorithms, but to ground facts and strategies in a larger, connected web of mathematical understanding that can be applied to make sense of and solve even unfamiliar problems.^9-11 Conceptual understanding describes a “big picture” understanding of mathematics as a coherent and sensible body of knowledge. Conceptual understanding requires students learn why mathematical processes work, not just how they are applied. This involves reasoning about how different strategies are related to each other and to the contexts in which they were learned.  

So how might instructional materials support students’ development of conceptual understanding of mathematics? Guidance is provided by the National Council of Teachers of Mathematics in one of Eight Effective Mathematics Teaching Practices,¹⁰ which states that mathematics instruction should build procedural fluency from conceptual understanding. Not to be misinterpreted as implying that conceptual understanding is somehow more important than procedural fluency, this research-based teaching practice highlights how beginning with a conceptual foundation allows students to develop reasoning and flexible use of appropriate strategies in different problem-solving contexts.¹² High-quality instructional materials can support student development of a conceptual foundation by providing opportunities to engage with concepts prior to formalizing procedures. Task sequences that build from students' current understanding to support sense-making about new approaches allow students to connect their conceptual foundation with procedural fluency in a way that strengthens both imperative types of mathematical understanding. 

When effective teachers implement high-quality instructional materials that establish a conceptual foundation before formalizing procedures, students gain an appreciation for why mathematics works, not just how to solve problems. 

Procedural Fluency 

With a strong conceptual foundation in place, students are ready to develop procedural fluency. So, what is procedural fluency? And why is conceptual understanding an important foundation upon which it is built? Let us start with what procedural fluency is not. Procedural fluency is not rote memorization of step-by-step procedures or mindless memorization of facts. A student who mimics a step-by-step process learned during class is not demonstrating the full intent of procedural fluency, even when solving with accuracy. Procedural fluency describes the ability to apply procedures efficiently, flexibly, and accurately.^9,11-12 This definition implies the need for students to have a repertoire of accessible strategies, an understanding of the mathematical contexts in which each strategy applies and is efficient, and flexibility to choose and move between the various strategies. The understanding of why each strategy works and the connections between them requires a conceptual foundation as a starting point. 

High-quality instructional materials support procedural fluency by allowing students to build strategies for solving problems. Unlike algorithms, strategies are not rigid, step-by-step procedures. Rather, strategies are flexible, adaptable approaches to solving a problem with understanding.¹² High-quality instructional materials also ensure that students build multiple strategies for solving problems. This empowers students to be mathematical decision-makers by allowing them to select the most appropriate strategy for a given problem, encouraging them to solve problems using more than one strategy, and embedding opportunities for reflection both during and after problem-solving. When students have opportunities to reason about the strategies available and choose the most appropriate strategy for solving a problem, they are in turn strengthening their conceptual understanding. In this way, building a conceptual foundation both precedes and coincides with supporting procedural fluency.¹²

When effective teachers implement high-quality instructional materials that support procedural fluency, students gain confidence in their mathematical solutions and demonstrate accuracy, efficiency, and flexibility in using strategies. 

Problem-Solving  

One of the most valued skills in learning mathematics is the ability to apply and problem-solve. Problem-solving is not only imperative in mathematics; it is a skill used in nearly every aspect of our lives. So, what is problem-solving and how do high-quality instructional materials promote problem-solving opportunities? 

Students are engaged in problem-solving when they make sense of a problem to determine an entry point, monitor and evaluate their progress while solving, and reflect on their answer to ensure it makes sense within the mathematical or real-world context.⁸ Promoting problem-solving requires instructional materials that engage students in “intellectual challenges” through truly problematic tasks.¹³ These tasks may or may not include real-world contexts, but always engage students in thinking meaningfully about the problem context to make sense of an approach. High-quality instructional materials promote problem-solving through tasks that provide opportunities for students to engage with mathematics in meaningful ways. Mathematical tasks that promote this type of engagement can be classified as worthwhile tasks¹⁴ or as high-cognitive demand¹⁵ tasks. Regardless of what framework is used to classify tasks, high-quality instructional materials incorporate tasks that enhance students’ understanding and mathematical development. ¹³  

So how can we spot tasks that promote problem-solving? Specific task features, such as ‘word problems,’ tasks that require explanations, or tasks that are open-ended are often misinterpreted as guaranteeing opportunities for problem-solving. While tasks with these features may be great tasks for promoting problem-solving, they also might fall short. It is important to dive into the type of thinking required while engaging in these tasks to decipher whether tasks support mathematical connections, encourage multiple approaches, and facilitate mathematical discourse.  

 When effective teachers implement tasks that engage students in problem-solving, students learn to make sense of new problems, apply strategies to find solutions, and monitor their progress along the way. 

Mathematical Modeling  

A primary goal of mathematics instruction is to prepare students to make sense of the world and apply mathematical thinking to answer questions encountered in our daily lives. Within his blog post titled “Fake-World Math”, Dan Meyer points out the clear dichotomy between “real-world” problems in school mathematics and real-world problems that are actually encountered in the, well, real world.¹⁶ This distinction largely comes down to mathematical modeling. School mathematics frequently engages students in using mathematical models, including manipulatives, graphs, and equations. But just using models falls short of mathematical modeling. How so? 

Mathematical modeling requires using mathematics to “represent, analyze, make predictions or otherwise provide insight into real-world phenomena”.^8(p.8) High-quality instructional materials can embed mathematical modeling by engaging students in the elements of a six-step cycle that begins when students (1) make sense of a problem and (2) formulate a model to describe relationships. From here, students (3) compute answers through application of the model and (4) interpret what the results mean in terms of the problem. Next, students (5) validate that their answer makes sense mathematically and in the real-life context, and if so, (6) report the findings so that others can understand their thinking. Implicit in this cycle is the potential to return to the formulation of the model itself during the validation, depending on the findings.¹⁷ 

Instructional materials that emphasize mathematical modeling put students at the center of reasoning and decision-making throughout the modeling process rather than asking them to step in only at the compute and interpret steps of the process. As a result, high-quality instructional materials can support learning experiences that build mathematical thinkers capable of using mathematics “to analyze empirical situations, to understand them better, and to improve decisions.” ^17(p.72)  

When effective teachers implement high-quality instructional materials that engage students fully in mathematical modeling, students learn to use mathematics to represent and solve problems that matter to them and their world. 

Mathematical Discourse

Facilitate meaningful mathematical discourse is one of NCTM’s 8 Effective Mathematics Teaching Practices.¹⁰ A necessary starting point for implementing this teaching practice is utilizing instructional materials that support mathematical discourse. Simply put, instructional materials must provide learning experiences that give students “something to talk about.” 

Much more than just talking, mathematical discourse involves the “purposeful exchange of ideas through classroom discussion, as well as through other forms of verbal, visual, and written communication.”^10(p.29) This definition of discourse reminds us that (1) students may be talking in the mathematics classroom but still not engaging in mathematical discourse, and (2) mathematical discourse doesn’t always need to be heard (i.e., not always verbal communication). We can ensure classroom conversation is mathematical discourse by focusing on what students are communicating. Students engage in mathematical discourse when they “express their ideas about mathematical topics and clarify mathematical understanding.”^18(p.145) Therefore, it is important to notice what students are communicating in the mathematics classroom—true mathematical discourse requires a focus on expressing mathematical ideas and thinking—not just a focus on answers and steps used to solve a problem. 

An important starting place for instructional materials to support discourse is including mathematical tasks that support multiple problem-solving approaches. Mathematical tasks with a prescribed approach or limited opportunities for interpretation reduce opportunities for students to share their thinking and build connections to alternate approaches to solving the problem. Tasks that allow for varied approaches and multiple mathematical representations provide more opportunities for students to communicate their thinking and to make connections to approaches used by their classmates. In this way, tasks that promote mathematical discourse may also encourage students to use and connect mathematical representations, another of NCTM’s 8 Effective Mathematics Teaching Practices¹⁰.  

When effective teachers implement high-quality instructional materials that encourage mathematical discourse, students learn to express their mathematical thinking and to critique the reasoning of others’ mathematical arguments. 

Assessment

High-quality instructional materials recognize that assessment is not merely a tool for judging student performance but, when used appropriately, is a powerful tool that advances student learning.¹⁹ Leveraging assessment as a tool for learning requires that high-quality instructional materials incorporate frequent, varied assessments that make student thinking visible.  

Frequent assessments are important if results are to influence learning. Formative assessments throughout each lesson allow teachers and students to continually monitor whether the lesson learning goals are being met. Teachers can use formative assessment to determine whether changes to the instructional plan are warranted, and students can self-assess whether they may need additional learning supports. Ultimately, teachers must infer what the results of any assessment imply about student understanding.  

To support teachers in making inferences about student understanding, high-quality instructional materials incorporate varied assessments that emphasize opportunities to demonstrate student thinking, not just provide answers. Varied assessments ensure that all students have opportunities to demonstrate their understanding. Some learners might prefer to demonstrate their thinking on traditional paper-and-pencil assessments, while others might do well with performance tasks that allow for creativity. Even within a single assessment, varying types of questions and mathematical representations can provide additional insight to the teacher about each student’s strengths and weaknesses with the content. Assessment prompts that allow students to explain their thinking, justify their approach, or solve a problem in more than one way can reveal more about student thinking compared to prompts that ask only for an answer or require students to solve a problem in a prescribed way. In turn, teachers can draw better inferences and more effectively use the instructional materials in future lessons.

When effective teachers implement high-quality instructional materials that use frequent, varied assessments that reveal student thinking, teachers and students are better able to monitor and adapt throughout the learning cycle.  

Student Supports 

Teachers implement curricula in classrooms with diverse student populations, various time constraints, and wide-ranging resources. High-quality instructional materials recognize the importance of each school and classroom context in shaping student learning and prioritize usability across settings.   

High-quality instructional materials support students by incorporating Universal Design of Learning (UDL) principles to minimize barriers and maximize learning for all students.²⁰  UDL principles ensure high-quality instructional materials are designed with diverse needs in mind, rather than considering only how the materials might be adapted to meet the needs of diverse learners or contexts when needs arise. UDL begins by clearly defining learning goals and building intentional instructional experiences while considering representations, actions and expressions of student understanding, and student engagement for all learners.²⁰  

Incorporating multiple representations in each lesson makes the content accessible for a wider range of learners and contexts. High-quality mathematics instructional materials use different modalities throughout each lesson and focus on using and connecting mathematical representations¹⁰ of a single mathematical concept. These materials also provide vocabulary support as a regular part of instruction to ensure accessibility regardless of students’ language backgrounds. High-quality instructional materials vary learning activities, including use of concrete mathematical manipulatives and visuals. Incorporating varied actions and expressions of student understanding ensures all students have access points into a lesson and have opportunities to demonstrate their understanding. High-quality instructional materials motivate and engage all students by using photographs and illustrations that empower diverse populations and by embedding applications of the content relatable and understandable by all students.  

When effective teachers implement high-quality instructional materials that leverage principles of UDL, more students across more settings will achieve the desired learning outcomes.   

 Teacher Supports

While all the features highlighted above are important characteristics of high-quality instructional materials, the eventual teacher and student implementation of the materials will ultimately determine the impact on student learning.^6,7 As a result, high-quality instructional materials must also focus on supporting teachers in implementation that upholds the potential embedded within the materials themselves. Just as our classrooms are filled with diversity, our K-12 mathematics teachers are also a diverse group of educators with different backgrounds and experiences. Those who develop instructional materials cannot assume that all teachers will come with the same set of tools and expertise to deliver instructional materials. And even highly trained and experienced teachers face a multitude of in-the-moment instructional decisions in every lesson. High-quality instructional programs use teaching supports to aid teachers in implementation by providing pertinent background information for the planning process and offering in-the-moment instructional supports during live instruction.  

Teacher supports embedded throughout instructional materials empower teachers to leverage the materials in the most effective way for their classroom contexts. Teacher supports should consider effective use of materials broadly (i.e., across the program) and narrowly (i.e., lesson or task-specific). Examples of teacher supports across these scopes include, but are not limited to, the following:

• Program-wide supports: Program pacing guidelines, assessment system definitions, strategies for using tiered support
• Chapter-level supports: Chapter pacing guidelines, how lessons fit together, chapter-level coherence and correlations
• Lesson-level supports: Point-of-use instructional strategies, lesson-specific supports for diverse student learning needs, lesson-specific tiered instructional supports, lesson-specific evidence of student understanding
• Task-level supports: Point-of-use question prompts, likely strategies and misconceptions, alternative explanations and pathways  

When effective teachers implement high-quality instructional materials infused with specific and meaningful teacher supports, the enacted curriculum is well-aligned with the intended curriculum, and student learning outcomes are maximized.    

Supporting the Artisan

As we have seen, high-quality instructional materials can be powerful tools in the artisan's toolbox. But how does the artisan become highly skilled? By learning, practicing, and dedicating to their craft. Teachers become effective teachers by seeking and utilizing high impact teaching strategies, pursuing professional development, and collaborating with other educators. They engage in continued growth through reflective practice, learning from their past successes and seeking to improve on areas for growth. By being lifelong learners, teachers continue to grow and become more skillful users of the tools available to them, learning to build the desired student learning outcomes.

At Big Ideas Learning, we believe in teachers and are dedicated to supporting their efforts to build mathematical thinkers. We are committed to leveraging research to build impactful learning materials and are passionate about supporting teachers to use those materials to transform student learning in their classrooms. 

About the Author

Dr. Courtney Nagle joined Big Ideas Learning in 2023 as the Vice President of Teaching & Research. Dr. Nagle has nearly two decades of teaching experience, including both mathematics and mathematics education courses at the undergraduate level at Penn State Behrend in Erie, PA. Her teaching accolades include the 2022 Penn State University Eisenhower Award for Distinguished Teaching and the 2018 MAA Allegheny Mountain Section Award for Distinguished College or University Teaching of Mathematics.

Dr. Nagle holds a B.A. and M.A. in Mathematics, a Ph.D. in Mathematics Education, and her Pennsylvania Secondary Mathematics Teaching certificate. In addition to teaching, Dr. Nagle has established a strong record of research, serving as PI or Co-PI on four NSF grants totaling nearly $2.5 million and publishing her work in journals such as Educational Studies in Mathematics, The Journal of Mathematical Behavior, The International Journal of Science and Mathematics Education, and The Mathematics Teacher.

Dr. Nagle is passionate about helping students develop their confidence in mathematics while building a deep conceptual understanding of the mathematics they study. She works with preservice and inservice teachers on a variety of professional development topics including rich mathematical tasks, NCTM's 8 Effective Teaching practices, and building conceptual understanding.

About Big Ideas Learning

Big Ideas Learning, founded in 2008 by Dr. Ron Larson—an award-winning and influential math textbook author—has been at the forefront of educational publishing. With over 35 million copies of Larson’s titles sold in the U.S., his impact on math education is profound. Building on Dr. Larson’s legacy, which began with his first edition of Calculus in 1978, Big Ideas Learning publishes rigorous, relevant, and conceptually rich K-Higher Ed mathematics programs.  Our vision is to shape a future where education transcends boundaries, curiosity is nurtured, and individuals are inspired to dig deeper.

References

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² Cantrell, Steven, and Thomas J. Kane, Ensuring Fair and Reliable Measures of Effective Teaching: Culminating Findings from the MET Project’s Three-Year Study, Seattle, Wash.: Bill & Melinda Gates Foundation, Policy and Practice Brief, Measures of Effective Teaching project, January 2013. 

³ Hattie, John. 2023. Visible Learning: The Sequel. London, England: Routledge. 

⁴ Yu, Rongrong, and Kusum Singh. "Teacher support, instructional practices, student motivation, and mathematics achievement in high school." The Journal of Educational Research 111, no. 1 (2018): 81-94. 

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⁸ National Governors Association Center for Best Practices. Common Core State Standards for Mathematics. Washington, DC: NGA Center, 2010 

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¹¹ Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26. 

¹² National Council of Teachers of Mathematics. (2023). Procedural Fluency: Reasoning and Decision-Making, Not Rote Application of Procedures Position. https://www.nctm.org/Standards-and-Positions/Position-Statements/Procedural-Fluency-in-Mathematics/ 

¹³Cai, J. & Lester, F. (2010). Why is teaching with problem solving important to student learning? Problem Solving Research Brief. Quander, J. R. (Ed.). National Council of Teachers of Mathematics. https://www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Problem-Solving/ 

¹⁴ National Council of Teachers of Mathematics. (1991). Professional Standards for Teaching Mathematics. NCTM.  

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¹⁶ https://blog.mrmeyer.com/2015/fake-world-math-when-mathematical-modeling-goes-wrong-and-how-to-get-it-right/ 

¹⁷ Consortium for Mathematics and Its Applications and Society of Industrial and Applied Mathematics. GAIMME: Guidelines for Assessment and Instruction in Mathematical Modeling Education. Second edition. Garfunkel, Sol and Montgomery, Michelle, eds. Philadelphia, PA: 2019.  

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²⁰ https://udlguidelines.cast.org/