Starting the school year with three evidence-based instructional practices for mathematics
For decades, K–12 mathematics education has been framed as a debate between procedural fluency and conceptual understanding. However, effective mathematics instruction is not an either-or proposition. Evidence-based teaching intentionally connects concrete experiences, visual representations, and strategic problem-solving to build both conceptual understanding and procedural fluency (Witzel et al., 2008; Powell & Fuchs, 2018). While there are many evidence-based instructional practices for mathematics, let’s begin by focusing on three practices to set the foundation for yearlong success for all students.
1. Explicit instruction and the CRA Framework
Research identifies explicit instruction as a critical practice for introducing new concepts and supporting struggling learners (Aceves & Kennedy, 2024; Archer & Hughes, 2011). Hattie’s Visible Learning research reports an effect size of 0.59, exceeding the 0.40 “hinge point” associated with accelerated student learning (Hattie et al., 2017). Through systematic instruction, including modeling, guided practice, immediate feedback, independent application, and active engagement, explicit instruction minimizes ambiguity and supports mastery. (Follow this link for more information on explicit instruction )
Explicit instruction pairs naturally with the Concrete-Representational-Abstract (CRA) framework. First, students build understanding through concrete experiences using manipulatives such as base-ten blocks or algebra tiles (PaTTAN, 2020). Then, instruction progresses to visual representations before connecting students to abstract mathematical symbols. This sequence helps students link concrete, representational, and abstract thinking, deepening conceptual understanding and reducing reliance on memorization (PaTTAN, 2020). When the concrete and representational stages are skipped, students often learn procedures without understanding the underlying concepts. (Follow this link for more on CRA)
2. Schema-based strategies for problem solving
A common practice in problem-solving instruction is teaching students to rely on keywords. For example, students may assume that the word altogether always signals addition. Research shows this approach becomes increasingly unreliable as problems grow more complex (Powell et al., 2022). In a review of 214 routine word problems, keywords correctly identified the operation less than 50% of the time in single-step problems and less than 10% of the time in multi-step problems (Powell et al., 2022).
Instead, research supports combining explicit instruction with Schema-Based Instruction (SBI), which teaches students to identify the underlying mathematical structure, or schema, of a problem (Powell et al., 2022; What Works Clearinghouse, 2021). Common schemas include additive structures (i.e., change, group, compare) and multiplicative structures (i.e., equal groups, ratios). Students learn to connect problem information to schema-based graphic organizers before writing equations, helping them focus on relationships rather than keywords (What Works Clearinghouse, 2021). These organizers build transferable problem-solving frameworks that support success within and beyond mathematics (Powell et al., 2022). (Follow this link for more on schema-based instruction)
3. Mathematical discourse
Effective mathematics classrooms also prioritize mathematical discourse, or “math talk.” Through discussion, students explain their reasoning, defend solutions, and make connections across mathematical concepts (Kersaint, 2015). Verbalizing thinking helps clarify understanding while revealing misconceptions that can be addressed through discussion. Listening to peers’ strategies exposes students to multiple solution pathways and promotes flexible thinking (Kersaint, 2015). Ultimately, structured discourse helps students view mathematics as a language of reasoning rather than a collection of rules (Van Dine, 2024). Teachers can support productive discourse by explicitly teaching Accountable Talk sentence frames. These provide students with a structure for initiating and sustaining meaningful mathematical conversations while offering needed scaffolds for participation. (Follow this link for more on Accountable Talk)
As schools begin a new year, focusing on a few high-impact, evidence-based practices can yield meaningful gains for learners at every level. Starting with these three foundational approaches can help ensure that mathematics instruction is both accessible and effective for all students.
References
Aceves, T., & Kennedy, M. (Eds.). (2024). High-leverage practices for students with disabilities (2nd ed.). Council for Exceptional Children and CEEDAR Center.
Archer, A., & Hughes, C. (2011). Explicit instruction: Effective and efficient teaching. Guilford Press.
Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Corwin Mathematics
Kersaint, G. (2015). Orchestrating mathematical discourse to enhance student learning. In TTAC Online. https://ttaconline.org/Document/zxbIhX_YCJNP0qvIYsAjT0x-qdzE3VlX/WP-Curriculum_Associates%20Orchestrating_Mathematical_Discourse.pdf0.pdf
PaTTAN. (2020). Concrete-Representational-Abstract: Instructional sequence for mathematics. PaTTAN. https://www.pattan.net/Publications/Concrete-Representational-Abstract-CRA-Instructi
Powell, S. R., & Fuchs, L. S. (2018). Effective word-problem instruction: Using schemas to facilitate mathematical reasoning. TEACHING Exceptional Children, 51(1), 31–42. https://doi.org/10.1177/0040059918777250
Powell, S. R., Namkung, J. M., & Lin, X. (2022). An investigation of using keywords to solve word problems. The Elementary School Journal, 122(3), 452–473. https://doi.org/10.1086/717888
Van Dine, D. (2024). How mathematical discourse can help students succeed. In Ed.gov. https://ies.ed.gov/learn/blog/how-mathematical-discourse-can-help-students-succeed
What Works Clearninghouse. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. Institute of Education Sciences. https://ies.ed.gov/ncee/wwc/Docs/PracticeGuide/WWC2021006-Math-PG.pdf#page=47
Witzel, B. S., Riccomini, P. J., & Schneider, E. (2008). Implementing CRA with secondary students with learning disabilities in mathematics. Intervention in School and Clinic, 43(5), 270–276. https://doi.org/10.1177/1053451208314734
For more information, contact Leslie Murphy Brown ([email protected]), Program Specialist, T/TAC at VCU.
Categories Inclusive Practices, Math