3 Strategies to Support Mathematical Reasoning
Too often, math is taught as a series of steps to follow, leaving students to believe that success comes from memorizing rules rather than understanding concepts. But when students are encouraged to recognize patterns, make connections, and reason through problems, they develop a more flexible and lasting understanding of mathematics. By focusing on the ways students naturally think about numbers, teachers can guide them toward deeper mathematical reasoning and more confident problem-solving.
Mathematical reasoning begins with recognizing when and how math applies to a real-world situation. Students must be able to reframe everyday scenarios as mathematical problems and reason through them using the concepts, strategies, and procedures they've learned. This process transforms informal observations into structured problem-solving opportunities, leading to deeper understanding and more flexible thinking.
6 Key Mathematical Understandings
To be mathematically literate, students need opportunities to reason, not just compute. Beyond standards and concepts, there are key mathematical understandings students need that support the development of mathematical reasoning:
- Understanding quantity, number systems and their algebraic properties
- Appreciating the power of abstraction and symbolic representation
- Seeing mathematical structures and their regularities
- Recognizing functional relationships between quantities
- Using mathematical modeling as a lens onto the real world
- Understanding variation as the heart of statistics
The key understandings identified above are not taught in isolation; they are built through meaningful problem-solving experiences and deep engagement with mathematical thinking. When teachers use students’ informal reasoning as a starting point and guide them toward more formal concepts, they help students make sense of complex ideas and build lasting mathematical literacy. This approach creates a strong foundation for mathematical literacy by honoring students’ thinking and helping them connect intuitive strategies to more sophisticated mathematical structures.
3 Instructional Strategies that Build Math Reasoning
Practice Tasks with High Cognitive Demand
Assign problems with multiple solution paths or strategies, where students must explain their thinking rather than just compute an answer. Research by Stein et al. (1996) shows that high-level tasks promote reasoning, representation, and connection-making. These tasks encourage students to explore structure, generalize, and make sense of relationships.
Use Visual Representations and Multiple Modalities
Incorporate number lines, bar models, diagrams, manipulatives, and student-created visuals to support reasoning. Representations help students move from concrete to abstract thinking. Research from Boaler (2016) supports the use of visual models to make mathematical structure visible and accessible. “When students learn through visual approaches, mathematics changes for them, and they are given access to deep and new understandings.”
Connect Informal Strategies to Formal Mathematics
Start with students’ own reasoning or invented strategies, then guide them toward more efficient or formal methods. Research shows (Baek, 2005) that students who used invented strategies before learning standard algorithms showed better understanding of place value and properties of operations.
Supporting mathematical reasoning means valuing how students think, not just what they get right. When instruction begins with students’ ideas and builds toward more formal understanding, learners become confident, creative problem-solvers. By prioritizing reasoning in our classrooms, we open the door to deeper understanding and help students see themselves as capable, empowered thinkers who can use math to make sense of the world.
References
Baek, J. M. (2005). Children’s mathematical understanding and invented strategies for multidigit multiplication. Teaching Children Mathematics, 12(5), 242–247.
Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as understanding: The importance of visual mathematics for our brain and learning. Journal of Applied & Computational Mathematics, 5(5), Article 325. https://doi.org/10.4172/2168-9679.1000325
Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549. https://doi.org/10.2307/749690
About the author
Stephanie Gold brings over two decades of experience in educational leadership, curriculum development, and digital learning to her role as a Learning Designer at Edmentum. With a deep-seated passion for transforming education through technology, Stephanie has held pivotal roles in STEM education, course design, and school leadership, notably influencing digital curriculum development across various educational settings.
With a Master of Arts in Science Education from New York University and now pursuing a Master's program in Instructional Design and Technology, Stephanie is adept at integrating pedagogical expertise with technological acumen to craft educational experiences that resonate with both students and educators. Her journey through the educational landscape includes leadership positions in Independent Schools, Online Schools, and the development of Professional Learning Platforms.
