Identifying High-Quality Instructional Materials in Math Education

Avoiding historical extremes in math education

Today it feels like our society is out of balance and pushed to extreme arguments on a variety of issues. Unfortunately, mathematics education is not immune from these phenomena. Classroom teachers and leaders hear from advocates for math instruction who emphasize direct instruction over inquiry (or the opposite) and skills over conceptual understanding (or the opposite). While these debates dominate mathematics education discussions today, it is important to realize the arguments are not new.

The extreme arguments in mathematics education are as old as the country itself. In 1788, the same year the U.S. constitution was ratified, Nicolas Pike published the first major U.S. mathematics textbook, Arithmetic. Pike advocated direct instruction, recommending that mathematics instruction proceed by the teacher stating a rule, the teacher working an example, and students practicing problems similar to the example.

In the 1820s, arguments were first made for a very different approach to mathematics instruction. In 1826, Warren Colburn published Colburn’s First Lessons: Intellectual Arithmetic, Upon the Inductive Method of Instruction. Colburn advocated teachers use a series of carefully sequenced questions and concrete materials so students could discover mathematical rules for themselves (inquiry-based instruction). By the 1830s, Colburn received pushback with demands for mathematics to be taught the “good old-fashioned way” (Botham, 1832).

The history of mathematics education in the United States is an ongoing pendulum swing between the extremes Pike and Colburn established. In more recent history, the 1960s were characterized by “new math,” which emphasized inquiry teaching and conceptual understanding. New math gave way to the “back to the basics” movement of the 1970s and 1980s, characterized by a pendulum swing back to an emphasis on direct instruction and procedural skills. “Back to the basics” in turn gave way to the era of standards-based reform in the 1990s, coinciding with the National Council of Teachers of Mathematics, or NCTM, publishing Curriculum and Evaluation Standards for School Mathematics in 1989, which advocated a return to more inquiry-based instruction and conceptual understanding. 

By the late 1990s, pushback to NCTM’s standards led to the “math wars” of the early 2000s. States joined together to develop and adopt the Common Core State Standards for Mathematics (NGA & CCSSO, 2010), a version of standards-based reform, and today the 50 states continue to adopt and implement their own standards-based mathematics education. In the midst of these extreme pendulum swings, it is natural for anyone charged with selecting mathematics curriculum materials to ask which extreme defines high-quality instructional materials, or HQIM. The answer is neither.

The reality is that both the direct instruction vs. inquiry and the procedural skills vs. conceptual understanding extremes represent false dichotomies in mathematics education. High-quality instructional materials in mathematics education avoid these extremes and include both approaches and content emphases, but it’s not as simple as including both sides and balancing them. 

What is HQIM in math education?

High-quality instructional materials in mathematics education intentionally align the instructional approach, either direct or inquiry-based instruction, to the nature of the learning task. HQIM leverage direct (or explicit) instructional approaches when the learning objective requires students to acquire new skills, learn prerequisite information, or solve novel problems, and they leverage inquiry-based approaches when the learning objective is deep conceptual understanding. Most important, high-quality instructional materials in mathematics intentionally and appropriately integrate both approaches in complementary ways, depending on the context, to maximize student learning. 

There is a time and place for both instructional approaches. It is not a question of which approach is more effective, it is question of how elements of each instructional approach can most effectively be integrated in different contexts (de Jong et al., 2022). High-quality instructional materials guide teachers in knowing when each approach is most appropriate and how they can be integrated to maximize student learning. 

High-quality instructional materials also reflect the fact that inquiry-based instruction does not “abandon” students to flounder or become confused (Hattie, 2023). Rather, they provide teachers appropriate scaffolding, discussion questions, and other monitoring tools to ensure students have the appropriate guidance necessary to successfully complete inquiry-based learning tasks (de Jong et al., 2022; Kirschner & Hendrick, 2020).

Characteristics of high-quality instructional materials

In mathematics, this means that high-quality instructional materials integrate content intentionally. Over two decades ago the National Research Council (2001) promoted a multifaceted and interwoven definition of mathematical content learning: “Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy. . . . Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding and using procedures can help strengthen and develop that understanding” (p. 122).

Ideally, students develop procedural fluency from and alongside a foundation of conceptual understanding (NCTM, 2014/2024; NRC, 2001) and learn to draw on both their conceptual understanding and procedural skills to solve novel problems. Students not only learn how to do something, but also why it works and when to apply their knowledge and skills in context so they can be empowered by mathematics in their lives.

The impact of high-quality instructional materials in math

Both direct (explicit) instruction and inquiry-based instructional approaches can have significant positive impacts on student learning, and that is why both are leveraged in integrated ways in high-quality instructional math materials. Based on John Hattie’s meta-analyses research (2023), the average effect size of direct instruction is very high at 1.2, and the average effect size for inquiry learning is moderate at 0.5. The key characteristic of high-quality instructional materials is that the two approaches are intentionally aligned with the instructional objective. The average effect size when there is instructional alignment between the instructional approach and the learning objective is strong at 1.1 (Hattie, 2023).

The impact of high-quality instructional materials in mathematics that include conceptual understanding, procedural skill, and problem-solving is so significant that the National Mathematics Advisory Panel recommended that mathematics curricula simultaneously develop all three (NMAP, 2008).

Tips for selecting high-quality instructional math materials

It is a long-standing myth that highly effective teachers don’t use published or district curricula in favor of creating their own curricular materials (Chan, 2022). This simply is not the case. In fact, researchers have argued that only the implementation of high-quality instructional materials at the classroom level has the potential to meaningfully reform classroom teaching and improve student learning at scale (Polikoff, 2021). 

There is evidence to suggest that many teachers spend 10 hours or more per week looking for or creating instructional materials (Chan, 2022). Providing teachers with an HQIM curriculum can free up their time to focus on other high-impact teaching activities like building relationships with students, differentiating instruction to meet individual student needs, and participating in professional learning (CPRL, 2021). 

Therefore, it is critical when seeking to improve teaching and learning in mathematics to select high-quality instructional materials due to both their impact on student learning and their potential to reduce teachers’ workload. Here are some questions to consider when selecting high-quality instructional materials in mathematics:

• Do the instructional materials include both inquiry-based and direct (explicit) instructional strategies?
• Are the instructional strategies in the materials aligned with the learning objective? In general, teachers should use inquiry-based instruction for conceptual understanding and direct instruction when the learning objective is more skill-based.
• Are inquiry-based and direct instructional strategies integrated, depending on the context and learning objective?
• Do the instructional materials emphasize conceptual understanding, procedural skill, and problem solving?
• Do the instructional materials interweave conceptual understanding, procedural skill, and problem solving, or treat them as separate learning domains?
• Do the instructional materials provide support for teachers to implement inquiry-based and direct instructional strategies in appropriate and meaningful ways?
• Does the structure and design of the high-quality instructional materials include sufficient technology components to eliminate much of the lesson and assessment preparation time associated with traditional instructional materials?

Tips for implementing high-quality instructional materials

It does not matter how high-quality the instructional materials are if teachers can’t or won’t use them. Therefore, teachers must have the time and support necessary to learn and implement the materials. Here are some ways teachers can be supported in implementing high-quality instructional materials:

• Teachers need to be given significant, regular, and yearlong collaborative time to work with one another across a course or grade level as they implement high-quality instructional materials for the first time.
• During the implementation phase, teachers must be provided feedback as they implement the new instructional materials in the classroom so they view the implementation process as one of continuous improvement.
• The structure and design of the high-quality instructional materials need to support teachers in learning how to utilize the materials as they implement them. I.e., the high-quality instructional materials must be teacher friendly.

Math teachers play such an important role. What they teach is deeply connected, both within math and to other subjects, and the ability to solve math problems fearlessly can result in long-lasting, deep rewards throughout a student’s life. When the instructional materials are high-quality, they combine procedural fluency with conceptual understanding and give students powerful ways to think about modeling problems with math and solving them. With the right resources, math teachers not only provide an “HQIM education,” they set students up for lifelong success.

The views expressed in this article are those of the author and do not necessarily represent those of HMH.

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Discover how Into Math for Grades K–8, and the Spanish version HMH ¡Arriba las Matemáticas! raises student achievement and builds productive perseverance.

Works cited

Botham, P. E. B. (1832). The common school arithmetic. Henry Benton.

Center for Public Research and Leadership [CPRL], Chu, E., Clay, A., & McCarty, G. (2021). Fundamental 4: Pandemic learning reveals the value of high-quality instructional materials to educator-family-student partnerships. https://cprl.law.columbia.edu/content/fundamental-4-pandemic-learning-reveals-value-high-quality-instructional-materials-educator

Chan, J. (2022, March 17). ­­­3 common misconceptions about high-quality instructional materials. EdReports. https://edreports.org/resources/article/3-common-misconceptions-about-high-quality-instructional-materials

Colburn, W. (1826). Colburn’s first lessons: Intellectual arithmetic, upon the inductive method of instruction. Boston: William J. Reynolds & Co.

de Jong, T., Lazonder, A. W., Chinn, C. A., Fischer, F., Gobert, J., Hmelo-Silver, C. E., Koedinger, K. R., Krajcik, J. S., Kyza, E. A., Linn, M. C., Pedaste, M., Scheiter, K., & Zacharia, Z. C. (2023). Let’s talk evidence – The case for combining inquiry-based and direct instruction. Educational Research Review, 39, 100536. https://doi.org/10.1016/j.edurev.2023.100536

Hattie, J. (2023). Visible learning: The sequel: A synthesis of over 2,100 meta-analyses relating to achievement. Routledge.

Kirschner, P. A., & Hendrick, C. (2020). How learning happens: Seminal works in educational psychology and what they mean in practice. New York: Routledge.

National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and evaluation standards for school mathematics. NCTM.

National Council of Teachers of Mathematics [NCTM]. (2014/2024). Principles to actions: Ensuring mathematical success for all. NCTM.

National Governors Association [NGA] Center for Best Practices & Council of Chief State School 

Officers [CCSSO]. (2010). Common Core State Standards for mathematics. NGA Center for Best Practices & CCSSO. http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf

National Mathematics Advisory Panel [NMAP]. (2008, March). Foundations for success: The final report of the National Mathematics Advisory Panel. US Department of Education.

National Research Council [NRC]. (2001). Adding it up: Helping children learn mathematics. (J. Kilpatrick, J. Swafford, & B. Findell, Eds.). National Academy Press. https://doi.org/10.17226/9822

Pike, N. (1788). A new and complete system of arithmetic, composed for the use of the citizens of the United States. John Mycall.

Polikoff, M. (2021). Beyond standards: The fragmentation of education governance and the promise of curriculum reform. Harvard Education Press.