Interpreting the Results 

To make meaningful sense of this literature, it is necessary to recognize that mathematics instruction involves not only different instructional approaches, but also distinct forms of mathematical knowledge. Most researchers agree that mathematical knowledge can be broadly divided into two primary domains: procedural knowledge and conceptual knowledge (Rittle-Johnson & Schneider, 2015).

​

Procedural knowledge refers to the ability to accurately and efficiently carry out mathematical procedures and algorithms. This includes knowledge of steps, rules, and symbol manipulations used to solve problems. Conceptual knowledge, in contrast, refers to an understanding of the underlying principles, relationships, and structures that explain why mathematical procedures work.

​

For example, dividing fractions procedurally involves knowing to multiply by the reciprocal of the divisor. Conceptually, however, division represents the question “how many of this quantity fit into that quantity,” and multiplying by the reciprocal functions as a way of counting how many units of the divisor fit within the dividend (Rittle-Johnson, 2011). While procedural knowledge enables correct execution, conceptual knowledge provides meaning and justification for the procedure.

​

While this two-domain distinction is well established, it can be further refined. Procedural knowledge can be usefully divided into:

​

1. Computational knowledge – knowledge of basic facts and numerical combinations (e.g., multiplication facts, number bonds), and
    

2. Operational knowledge – knowledge of multi-step algorithms and procedures (e.g., long division, fraction operations).
    

Similarly, conceptual knowledge can be divided into:

1. Concept knowledge – understanding of mathematical principles and relationships (e.g., place value, equivalence, proportionality), and
    

2. Application knowledge – the ability to flexibly apply conceptual and procedural knowledge to novel or non-routine problems that differ from those explicitly taught.
    

Importantly, application knowledge is not a separate domain of mathematics, but rather a performance outcome that depends on the integration of both conceptual and procedural knowledge. Successful transfer requires that procedures are sufficiently fluent to avoid cognitive overload and that concepts are sufficiently understood to guide problem interpretation.

​

In general education, mathematics instruction has often emphasized the development of application knowledge as the primary goal of learning. This emphasis has contributed to instructional approaches that prioritize complex, open-ended problems and student-led inquiry, frequently accompanied by a reduced emphasis on direct procedural instruction over the past two decades (Boaler & Dweck, 2016; Liljedahl, 2021; Small, 2009).

​

These approaches are commonly associated with constructivist instructional models, which emphasize meaning-making, inquiry, and problem solving over memorization and explicit teaching of procedures. A similar shift occurred in literacy education, where Whole Language and Balanced Literacy approaches prioritized reading comprehension and authentic texts while de-emphasizing explicit instruction in foundational skills such as phonics (Pressley, 2002).

​

In contrast, research traditions rooted in special education, behavioral science, and cognitive psychology have placed greater emphasis on explicit instruction and systematic procedural teaching, particularly for students with learning difficulties. These perspectives have informed policy papers and research syntheses produced by groups associated with the “Science of Math” movement (Codding, Peltier & Campbell, 2023).

​

Taken together, contemporary debates in mathematics education largely center on two interrelated questions:

​

1. Conceptual versus procedural instruction, and

2. Explicit versus implicit instruction.
    

Regarding the first question, the meta-analytic evidence summarized in Table 1 does not provide direct guidance, as no quantitative meta-analyses were identified that isolate conceptual-only versus procedural-only instruction. However, a landmark narrative review by Rittle-Johnson and Schneider (2015) concluded that mathematics achievement is highest when conceptual and procedural instruction are integrated. Their review further demonstrated a bidirectional and synergistic relationship, in which gains in one domain support growth in the other. Current evidence therefore does not support prioritizing one form of knowledge to the exclusion of the other.

​

The second question—explicit versus implicit instruction—can be addressed more directly through meta-analytic evidence. Three meta-analyses examining explicit mathematics instruction were identified in the ERIC database. Gersten et al. (2009) reported a large mean effect size (g = 1.22) across 11 studies involving students with learning disabilities. Baker, Gersten, and Lee (2002) examined four studies of low-achieving mathematics students and reported a mean effect size of g = 0.58. Stockard et al. (2018), analyzing 70 studies of Direct Instruction mathematics programs, reported a mean effect size of g = 0.55. Across studies, effect sizes ranged from moderate to very large, with the strongest effects observed among students with the highest instructional needs.

​

By comparison, two meta-analyses examined inquiry-based mathematics instruction. Xie, Wang, and Hu (2018) analyzed 26 studies of inquiry-based instruction with Chinese elementary students and reported a mean effect size of g = 0.52. However, Chinese mathematics education is typically characterized by strong procedural foundations and extensive explicit instruction, which may have enabled students to benefit more from inquiry-based approaches. Lazonder and Harmsen (2016) conducted a meta-analysis of five mathematics studies and reported a smaller mean effect size of g = 0.28.

Problem-based learning, often associated with constructivist instruction and characterized by minimally guided group problem solving, was also examined by Xie et al. (2018). Across 21 studies, they reported a mean effect size of g = 0.58. Notably, both Xie et al. (2018) and Lazonder and Harmsen (2016) found that inquiry-based approaches produced larger effects when accompanied by additional instructional guidance, suggesting that the effectiveness of inquiry depends substantially on the presence of structured support.

Overall, this body of research suggests that explicit instruction tends to be more effective in mathematics, especially for students who struggle or have significant learning gaps. For students with fewer difficulties, the advantage is less clear. That said, there are important limitations in this research that teachers should be aware of before drawing strong conclusions.

​

First, these meta-analyses do not adequately account for the types of tests used to measure learning. They do not distinguish between different strands of math, different kinds of knowledge (such as procedural versus conceptual), or whether the assessments were closely aligned with instruction or designed to measure broader transfer. They also do not separate norm-referenced tests from classroom-based or curriculum-aligned measures. This matters because what we choose to test strongly influences what “works”, and combining very different kinds of assessments can make results look clearer than they actually are.

​

Second, many of these studies only examine one instructional approach at a time, such as inquiry-based learning or direct instruction, rather than directly comparing the two. Because studies with positive findings are more likely to be published, both approaches often appear effective, even though they represent very different ways of teaching. In addition, many studies compare a new instructional approach to “business as usual,” rather than to a clearly defined alternative. This makes it difficult to determine whether an approach is truly more effective, or simply different from typical classroom practice.

​

Third, and perhaps most importantly, these studies rarely account for the size of the groups being studied and this turns out to matter a great deal. In a meta-analysis I currently have under peer review, about 80% of the variation in reported effect sizes was explained by sample size alone. Studies with more than 100 students showed effect sizes that were, on average, seven times smaller than those found in smaller studies. In practical terms, this means that small studies often make teaching approaches look far more powerful than they really are. Smaller samples are more likely to produce unstable and exaggerated results, especially when only positive findings are published. Larger studies tend to give more realistic estimates of impact.

​

This does not mean that instruction doesn’t matter. It means that very large effect sizes should be treated with caution, particularly when they come from small or tightly controlled studies. For teachers, the takeaway is simple: impressive numbers do not always translate into equally impressive classroom results.

​

Because these limitations are not addressed in most existing meta-analyses, it is hard to know whether their conclusions would hold up under more careful analysis. A more rigorous synthesis one that compares instructional approaches directly, accounts for assessment differences, and gives greater weight to large, well-designed studies, would likely produce more modest but more trustworthy estimates. (Indeed, I would love to conduct such a meta-analysis this year. If you are interested in collaborating on such a project, please contact me.)

​

At present, the strongest evidence comparing explicit and constructivist approaches comes from reading research. Multiple large reviews have found that systematic, explicit instruction produces roughly twice the impact of less explicit, constructivist-oriented approaches (National Reading Panel, 2000; Stuebing et al., 2008; Hansford, 2025). Importantly, these reviews did not merely conclude that explicit instruction was more effective; they also emphasized that instruction should be systematic—that is, organized around a carefully designed scope and sequence rather than delivered in an ad-hoc or purely responsive manner.

​

While systematic instruction has been studied far less directly in mathematics, there are strong reasons to believe it is especially important in this domain. Mathematics involves a large number of interdependent concepts and algorithms, many of which rely on prior knowledge for successful application. Gaps early in instruction can therefore compound quickly. Although reading and mathematics are distinct subjects, both operate under the same fundamental learning constraints: limited working memory, the need for practice to build fluency, and the importance of mastering foundational skills before engaging in more complex tasks. For this reason, it is unlikely that students would benefit from fundamentally different instructional principles in reading and mathematics.

What Are the Pillars of Effective Math Instruction?

When educators think about reading research, they often reference the five pillars of reading: phonemic awareness, phonics, fluency, vocabulary, and comprehension. Although this framework has been critiqued for oversimplifying the complexity of reading, it has nevertheless proven useful for teachers as an organizing structure. Each pillar represents a broad strand of instruction, and each is supported by a substantial body of meta-analytic evidence demonstrating its contribution to reading achievement.

​

A similar organizing framework is useful in mathematics. Based on the research reviewed in Table 1, four broad components of mathematical knowledge emerge as foundational: math facts, operations, concepts, and application. These components correspond closely to the refined knowledge distinctions discussed earlier and capture both the content and performance demands of mathematics learning.

​

To identify instructional pillars within mathematics, I reviewed the meta-analyses summarized in Table 1 and examined which pedagogical approaches were (a) clearly aligned with one or more of these four domains and (b) associated with moderate to large effect sizes. This process informed the instructional framework presented below. **Please note that the below pillars are coded on html for PC. It will work far better on a personal computer than a tablet or phone.