Key ideas
Core concepts
- The maths wars present a false dichotomy between explicit and non-explicit teaching; effective mathematics education sequences explicit teaching of foundations before problem-solving applications.
- Progressive approaches that lead with real-world contexts overwhelm novice learners who lack foundational knowledge.
- Genuine explicit teaching is a whole system of planning, teaching, and assessment, not token instruction added to a discovery lesson.
- Meta-analytic evidence supports systematic, explicit instruction (Stockard et al., 2018).
Connected to
Reading Wars | Explicit Teaching | Non-Explicit Teaching | Logical Fallacies | Cognitive Load Theory | Knowledge-Based Curriculum | Problem-Solving
The term “maths wars” refers to the ideological divide between two primary approaches to mathematics teaching: explicit instruction (systematic teaching of discrete skills) and non-explicit teaching (discovery through meaningful contexts).
This debate often presents a false dichotomy. Effective mathematics education does not require balance between approaches but rather appropriate sequencing: explicit teaching must lay the foundational knowledge before students can meaningfully engage in problem-solving.
Historical development
New Math era (1950s-1970s)
Following the Soviet Union’s 1957 Sputnik launch, the U.S. National Science Foundation funded development of new mathematics curricula as part of a broader science education initiative (Phillips, 2015). The 1958 National Defense Education Act provided additional funding for mathematics education reform.
New Math curricula emphasised conceptual understanding of mathematical principles and de-emphasised computational skills. The approach featured set theory concepts and terminology as foundational elements, and transformation geometry replaced traditional Euclidean geometry. By the mid-1960s, over half of U.S. high schools had adopted some form of New Math curriculum, rising to approximately 85% of all schools (K-12) by the mid-1970s.
Parents and teachers opposed New Math on the grounds that the curriculum was too abstract and removed from students’ ordinary experience. The material placed new demands on teachers, many of whom were required to teach content they did not fully understand. A 1967 five-year study showing American students lagging in mathematics skills among Western nations damaged New Math’s credibility. By the early 1970s, opinion polls indicated Americans favoured a “Back to Basics” approach emphasising computation, formulas, and mathematical laws.
Back to Basics movement (1970s-1980s)
The Back to Basics movement emerged as a direct reaction to New Math’s perceived failures. Some states and school districts introduced minimum competency testing as requirements for grade promotion or graduation. Schools devoted more classroom time to language arts and mathematics, reintroduced phonics instruction, and required regular homework with emphasis on neatness and decorum.
By the mid-1970s, New Math had fallen out of favour, as students’ performance on standardised mathematics tests did not improve as expected. However, a late-1970s study of U.S. schools concluded that “there appears to be little change in mathematics instruction in grades K-12” and “the single textbook is still the primary source of mathematics curricula,” suggesting limited implementation of either approach.
NCTM Standards era (1989-present)
In 1986, the National Council of Teachers of Mathematics (NCTM) Board of Directors commissioned the Curriculum and Evaluation Standards for School Mathematics, published in 1989. This document launched the current era of standards-based educational reform in mathematics. The NCTM employed a consensus process involving classroom teachers, mathematicians, and educational researchers.
The 1989 Standards emphasised that students need to learn more mathematics, learn new kinds of mathematics, and learn mathematics differently, with problem-solving as the major thrust. The standards promoted equity and mathematical power for all students, encouraged calculator use and manipulatives, and de-emphasised rote memorisation.
Education officials supported these standards, and the National Science Foundation funded several curriculum development projects aligned with the standards’ recommendations. By the late 1990s, though, the 1989 standards had generated considerable controversy, focused on specific instructional materials developed in alignment with the standards. These debates became known as the modern “math wars.”
California controversy (1997)
In 1997, the group Mathematically Correct was created by educators, parents, mathematicians, and scientists concerned about reform mathematics curricula based on NCTM standards. Although the organisation had a national scope, much of its focus was on opposing mathematics curricula prevalent in California in the mid-1990s, making California the centre of the Math Wars.
Mathematically Correct opposed most programmes developed from National Science Foundation-funded research projects. The group objected to the integrated approach taken by California’s draft mathematics standards, arguing there was little research to support it and advocating for “only a proven, research-based curriculum.” On 11 December 1997, the California State Board of Education approved the California Mathematics Academic Content Standards for Grades K-12.
Contrasting approaches
The two approaches differ in philosophy, sequencing, and cognitive impact. Non-explicit teaching follows a progressive, discovery-based philosophy in which novice mathematics learners encounter mathematics in meaningful contexts such as real-world scenarios: contexts first, foundational skills afterwards. Explicit Teaching follows a systematic, knowledge-building philosophy that teaches discrete knowledge and skills before asking students to problem-solve, establishing foundational skills first and then applying them to contexts.
Non-explicit teaching creates predictable cognitive overload because students must simultaneously grapple with unfamiliar mathematical concepts whilst navigating complex contextual information. Explicit teaching manages working memory limits by separating skill acquisition from application. Non-explicit approaches disadvantage struggling students who lack compensatory support outside school, whilst explicit instruction benefits students across the attainment range, especially those from disadvantaged backgrounds.
There is broad consensus in the literature that mathematics knowledge must be explicitly and systematically taught. Where it is not, student outcomes suffer across the board, with the impact falling most heavily on children from disadvantaged backgrounds who lack compensatory support outside school.
The more recent iteration of non-explicit teaching is “launch, explore, summarise”, which incorporates token amounts of explicit teaching. This superficial inclusion allows advocates to claim they use explicit instruction whilst maintaining progressive structures that undermine systematic knowledge building. Genuine explicit teaching requires a whole system that incorporates planning, teaching, and assessment over the short, medium, and long term. The curriculum is knowledge-based, not resource-based or activity-based, driven by what students need to learn rather than what materials or activities seem engaging.
Research evidence
Research on standards-based reform mathematics curricula has found that students using these curricula perform as well as students in traditional programmes on measures of computational skill and procedural knowledge. Students using reform curricula perform better on assessments of conceptual understanding and ability to use mathematics to solve problems (Schoenfeld, 2004). Communities that adopted reform curricula generally saw increases in students’ mathematics test scores, but correct implementation was essential to achieving these outcomes.
Rittle-Johnson and Alibali (1999) found an iterative relationship between conceptual understanding and procedural skill: children’s initial conceptual knowledge predicts gains in procedural knowledge, and gains in procedural knowledge predict improvements in conceptual knowledge. This suggests that procedural skill and fluency develop more effectively with a sound grounding in conceptual understanding.
The 2008 National Mathematics Advisory Panel examined the debate and recommended a balanced approach between reform and traditional mathematics teaching styles, rather than continuation of ideological conflicts between proponents of the two approaches (National Mathematics Advisory Panel, 2008).
Scientific versus political resolution
The maths wars, like the reading wars, suffer from treating educational disputes as political rather than scientific questions (Stanovich & Stanovich, 2003). Educational practice has suffered because its dominant model for resolving disputes has been more political (with corresponding factions and interest groups) than scientific. The field’s failure to ground practice in the attitudes and values of science has made educators susceptible to fads and gimmicks that ignore evidence-based practice.
The “anything goes” mentality that characterises much progressive mathematics teaching threatens professional autonomy. It provides a fertile environment for untested educational remedies that are not supported by an established research base. Teachers need quality control mechanisms to evaluate claims about teaching methods, and peer-reviewed research journals in educational psychology and mathematics education provide those mechanisms (Gersten, 2001).
Converging evidence from multiple methodologies (experimental studies, quasi-experimental designs, meta-analyses, and classroom research) consistently supports systematic, explicit instruction in mathematics. Stockard et al. (2018) conducted a meta-analysis of a half century of research on direct instruction curricula, finding strong positive effects (effect size = 0.59) across diverse student populations and settings. This convergence gives educators confidence that explicit teaching is research-based practice rather than ideological preference.
When research findings are amalgamated through meta-analysis, clear patterns emerge that are obscured when looking at individual studies. Meta-analysis is useful for ending disputes that seem to be nothing more than a “he-said, she-said” debate, providing a way of dampening contentious disputes about conflicting studies that plague education (Stanovich & Stanovich, 2003).
References
Gersten, R. (2001). Sorting out the roles of research in the improvement of practice. Learning Disabilities: Research & Practice, 16(1), 45-50.
Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75-86. https://doi.org/10.1207/s15326985ep4102_1
Klein, D. (2003). A brief history of American K-12 mathematics education in the 20th century. In J. M. Royer (Ed.), Mathematical cognition (pp. 175-225). Information Age Publishing.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.
Phillips, C. J. (2015). The new math: A political history. University of Chicago Press.
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175-189. https://doi.org/10.1037/0022-0663.91.1.175
Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253-286. https://doi.org/10.1177/0895904803260042
Stanovich, P. J., & Stanovich, K. E. (2003). Using research and reason in education: How teachers can use scientifically based research to make curricular and instructional decisions. National Institute for Literacy.
Stockard, J., Wood, T. W., Coughlin, C., & Rasplica Khoury, C. (2018). The effectiveness of direct instruction curricula: A meta-analysis of a half century of research. Review of Educational Research, 88(4), 479-507. https://doi.org/10.3102/0034654317751919
Sweller, J., van Merriënboer, J. J. G., & Paas, F. (2019). Cognitive architecture and instructional design: 20 years later. Educational Psychology Review, 31(2), 261-292. https://doi.org/10.1007/s10648-019-09465-5