Add to favoritesWritten by: Lori French, LDCSB Math Facilitator
Why Instructional Approaches Matter
Students are at the heart of everything we do as educators, leaders, and community members. They are our "why" — the reason schools exist, and the inspiration behind every lesson plan and instructional decision we make. Their dreams, challenges, and successes remind us of our profound responsibility to guide, support, and nurture their growth. For decades, conversations about math instruction have highlighted different perspectives, at times framed as “the math wars,” — a term widely attributed to mathematics professor David Klein, referring to the tension between inquiry-based and explicit approaches to teaching mathematics. However, when we refocus on what is best for students, we uncover the true beauty of this discussion: it is not about choosing one method over the other to use exclusively, but about understanding when each approach best meets the needs of our learners. This shifts the dialogue from “if teachers should use it” to “when — and in what context — teachers should use it”, fostering unity and collaboration, allowing us to come together to create the most effective learning experiences for all students. Acknowledging the importance of focusing on students’ needs leads us to explore how evidence-based approaches in reading can help shape our thinking about effective math instruction.
What the Science of Learning Tells Us About the Brain
In a classroom rooted in the Science of Learning, we use what we know about how the brain learns, combined with evidence-based instructional practices like systematic, explicit instruction, to improve student understanding and achievement. First, we need to have a clear understanding of how learning happens in the brain. John Sweller developed the cognitive load theory in 1988. In this theory he explains how the human brain processes information and he emphasizes the importance of designing learning experiences that minimize unnecessary mental effort to optimize understanding and retention (Sweller, 1988, as summarized in Swain, 2025). Offering clear instructions, chunking tasks, minimizing distractions in the classroom and using worked examples are all ways classroom teachers reduce cognitive load. When teachers understand the cognitive load theory and recognize its implications for student learning, the positive impact on students is significant. Foundational knowledge of this theory naturally paves the way for incorporating explicit instruction as a powerful, evidence-based strategy to further enhance student learning outcomes.
What We’ve Learned from the Science of Reading
The Science of Reading (SOR) — a robust body of research that offers evidence-based insights into how people learn to read and how we can improve literacy instruction, and how reading difficulties can be addressed—has gained significant recognition in the education community. The Right to Read report, released by the Ontario Human Rights Commission in 2022, emphasized the necessity of adopting evidence-based literacy instruction to ensure the highest degree of success in teaching all children to read. A natural progression of this discussion is to reflect on our current practices in math classrooms across the province. The Science of Math (SOM) exists but has not yet achieved the same level of familiarity or prominence, despite its equally important insights. According to Nathaniel Swain, author of Harnessing the Science of Learning, we can also think in a broader sense about the science of learning which “refers to a multidisciplinary field of research that incorporates child neuroscience, psychology, sociology, behavioral development, and cognitive learning” (Chambers, 2020, as cited in Swain, 2025).
What is the Science of Math?
In December 2020, Sarah Powell — Professor in the College of Education at The University of Texas at Austin and Associate Director of the Meadows Center for Preventing Educational Risk — hosted the first Science of Math meeting, marking the beginning of a collaborative effort to promote evidence-informed math instruction. Inspired by the momentum of the Science of Reading movement, the Science of Math brings together researchers such as Amanda VanDerHeyden, Robin Codding, and Corey Pelletier, all of whom have contributed significantly to our understanding of effective practices that support math learning.
The Science of Math is defined by Codding, Pelletier and Campbell (2023) as a “movement focused on using objective evidence about how students learn math to make educational decisions and to inform policy and practice” (Codding et al., 2023). As educators are presented with different theories of learning or teaching, it becomes increasingly important to ask what evidence supports these practices, and whether that research is both rigorous and relevant. Drawing on the lessons from the Science of Reading, the Science of Math encourages us to pursue similarly research-informed approaches in numeracy, helping ensure that all students have equitable access to effective instruction.
Understanding Mathematical Proficiency
Similar to the interconnectedness of strands that comprise skilled reading in literacy, becoming proficient in mathematics includes a blending of concepts, procedures, strategies, reasoning, and disposition (Kilpatrick, 2001). According to the National Research Council (NRC, 2001), mathematical proficiency encompasses a range of abilities. Students who are proficient in math demonstrate an understanding of fundamental concepts, fluency in basic operations, can access a repertoire of strategic problem-solving skills and have the capacity to reason both clearly and adaptively. Additionally, they approach mathematics with a positive attitude and confidence in their abilities.
This aligns closely with the Ontario mathematics curriculum across grades, which emphasizes the importance of developing a strong foundation in mathematical concepts, reasoning, and communication. Both the elementary and Grade 9 de-streamed curriculums prioritize equitable learning opportunities, evidence-based instruction, and the use of strategies like explicit teaching and culturally responsive pedagogy. Together, these curricula aim to build students’ confidence and competence in math by supporting understanding, skill development, and meaningful application (Ontario Ministry of Education, 2020; Ontario Ministry of Education, 2021).
Why Conceptual Understanding and Fluency Must Coexist
Research highlights the importance of building conceptual understanding alongside procedural fluency, ensuring students not only perform calculations but also understand the "why" behind them (Kilpatrick, 2001). Research also emphasizes the importance of linking prior knowledge with new concepts, promoting connections that enhance learning retention (Brown et al, 2014).
One important component of fluency is developing automaticity with basic math facts, which frees up cognitive resources for higher-order problem-solving and reasoning. When students can recall math facts efficiently, they are better equipped to engage more confidently with more complex tasks — supporting the integration of both procedural and conceptual knowledge. This also fosters deeper learning over time.
For students who are still consolidating foundational skills, evidence suggests that combining explicit instruction with strategies informed by how the brain learns can improve access and equity. As with literacy, some students in mathematics may struggle when instruction relies too heavily on exploratory approaches before core skills are in place. Without sufficient scaffolding, these students risk falling further behind — a pattern observed in both math and reading research (Kirschner et al., 2006).
Applying Cognitive Science Across Grades
Cognitive science shapes math instruction across grade levels by informing teaching methods that align with how students process, retain, and apply knowledge. For younger students, in addition to using explicit instruction, strategies such as using concrete materials and visual aids tap into their developmental stage, where learning is grounded in tangible experiences. Techniques like spaced repetition (reviewing material at increasing intervals) and active retrieval (recalling information without prompts) help build foundational skills by reinforcing memory pathways (Agarwal & Bain, 2019).
As students progress to higher grades, cognitive science supports the use of explicit instruction, scaffolding, worked examples and active retrieval — strategies that align closely with Ontario’s High-Impact Instructional Practices in Mathematics, such as direct instruction, deliberate practice, and small-group instruction, applied thoughtfully and responsively to student needs.
Understanding Explicit Instruction
Anita Archer is highly regarded as a leading authority on explicit instruction and the author of Explicit Instruction: Effective and Efficient Teaching (2011). In her book, she explains this highly effective teaching method as one that includes three parts. The first, modelling, involves the teacher demonstrating and verbalizing the concept, skill or thinking process. Lessons are intentionally planned with students' current skill level at the forefront. This explicit modelling is followed by guided practice, where students and teachers work together, with the teacher prompting students as they work to put their new learning into practice. Finally, guided practice is followed by independent practice. Independent practice occurs when the students have demonstrated success with guided practice. During independent practice, teachers walk the room checking in with students who may be having difficulty and offering prompts and scaffolding if needed. A common misconception is that independent practice is the same as homework, but that is not the case. Having the teacher monitor, give corrective feedback, and scaffold students as they practice on their own are all critical at this stage. Active engagement is woven through all components, ensuring that students have many opportunities to respond in order to receive affirmative or corrective feedback about their learning. This approach can be especially impactful when paired with a broader instructional repertoire that is responsive to the diverse needs and experiences of math learners.
When to Use Explicit vs. Inquiry-Based Approaches
Deciding whether a student needs explicit instruction or inquiry-based instruction depends on where the student lies on the novice-to-expert continuum for a particular concept or skill. Learners who are in the process of developing foundational knowledge greatly benefit from explicit instruction as their minds concentrate on absorbing and structuring new information. This group includes both students who have faced difficulties in mathematics and those who are exploring these concepts and skills for the first time. Techniques like clear explanations, step-by-step instruction, and worked examples help reduce cognitive load, enabling them to process and retain core concepts (Kirschner & Hendrick, 2024).
On the other hand, learners who have already developed a strong understanding of the subject thrive with inquiry-based instruction. Their brains are primed for deeper exploration, creative problem-solving, and connecting ideas in meaningful ways (Kirschner & Hendrick, 2024).
By tailoring instruction to the learner's cognitive stage, educators can create a balanced approach that supports growth and maximizes learning potential. This reinforces the idea introduced earlier in this article: effective instruction is not about choosing between explicit and inquiry-based methods, but about knowing when — and in what context — to use each. A responsive math program provides space for both, supporting all learners in building deep understanding, confidence, and the ability to think critically with mathematics.
Honouring the Evidence, Serving All Learners
The focus of our work is clear—everything we do is centered around students and their achievement and well-being. This reinforces the importance of using instructional practices that are responsive to students’ current levels of understanding — whether they are building foundational knowledge or extending and refining their thinking.
A growing body of research suggests that when used in isolation, minimal guidance approaches — such as inquiry- or problem-based learning — may not provide enough support for students who are still developing core skills (Kirschner et al., 2006). In contrast, approaches that include explicit, systematic instruction have been shown to be particularly effective in supporting deep and sustained learning.
As educators, our goal is to draw on the best available evidence to design learning experiences that are inclusive, intentional, and impactful for all learners.
About the Author:
Powered by LDAO:
This article was reviewed and edited by LD@school, a signature initiative of the Learning Disabilities Association of Ontario (LDAO), a registered charity and provincial leader in inclusive and accessible education.
For over 60 years, LDAO has supported children, youth, and adults with learning disabilities (LDs) and related conditions such as Attention-Deficit/Hyperactivity Disorder (ADHD).
Designed for educators, LD@school provides evidence-based and practice-informed resources, along with classroom-ready strategies to help Ontario teachers create equitable, accessible learning environments for students with LDs, ADHD, and other learning differences.
LDAO also offers other signature platforms, including TA@l’école (for French-language educators), LD@home (for families), and LD@work (for adults and employers), supporting the LD community across the lifespan.
References:
Agarwal, P.K. and Bain, P.M. (2025) Powerful teaching: Unleash the science of learning. Jossey-Bass, a Wiley Brand.
Archer, A.L. and Hughes, C.A. (2011) Explicit instruction: Effective and efficient teaching. The Guilford Press.
Brown, P.C., Roediger III, H.L. & McDaniel, M.A. (2014) Make it stick: The science of successful learning. Harvard University Press.
Codding, R.S., Peltier, C. & Campbell, J. (2023) Introducing the science of math, TEACHING Exceptional Children, 56(1), pp. 6–11.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. National Academy Press.
Kirschner, P.A., Hendrick, C. and Caviglioli, O. (2024) How learning happens: Seminal works in educational psychology and what they mean in practice. Routledge.
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), pp. 75–86.
National Research Council (2001). Adding it up: Helping children learn mathematics. National Academy Press.
Ontario Ministry of Education. (2020). The Ontario curriculum, grades 1–8: Mathematics. Queen’s Printer for Ontario. https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics
Ontario Ministry of Education. (2021). The Ontario curriculum, grade 9: Mathematics (de-streamed course), 2021. Queen’s Printer for Ontario. https://www.dcp.edu.gov.on.ca/en/curriculum/secondary-mathematics
Swain, N. (2025). Harnessing the science of learning: Success stories to help kickstart your school improvement. Routledge.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.
