Introduction

Teaching for conceptual understanding in mathematics focuses on helping students build a deep, connected understanding of mathematical principles, allowing them to solve problems more flexibly and transfer their knowledge to new situations. This approach contrasts with procedural understanding, which emphasizes mastering step-by-step processes or algorithms without necessarily understanding the underlying concepts. There has been a growing consensus in mathematics education that conceptual understanding should precede procedural skills, as this provides a stronger foundation for long-term learning, problem-solving, and knowledge retention.

Pressing Students Conceptually

Research in mathematics education has supported the shift toward emphasizing conceptual understanding before procedural learning. This perspective moves instructional practices beyond simply teaching algorithms and procedures. Instead, teachers are encouraged to press students to explore the relationships between different mathematical models, formalize their informal methods, and engage in higher-order thinking (Carpenter & Lehrer, 1999; NRC, 2001; Siegler & Alabali, 2004). Students can generalize new situations by comparing and connecting various solution methods, leading to deeper learning.

A key element of this instructional strategy is the creation of classroom experiences that help students integrate new knowledge into their existing mathematical framework, often referred to as schema theory in cognitive psychology. For example, when students first encounter multiplication, they can connect it to their previous understanding of addition, patterns, and area. These connections help create a web of knowledge that allows students to approach new problems more effectively. From a social learning perspective, teachers can use classroom discourse to encourage students to reflect on their mathematical models and those of their peers, which helps them build a more coherent and organized mathematical framework (Hiebert & Carpenter, 1992).

This mathematical discourse guides students to think critically about how they model problems, moving from concrete, enactive representations to more formalized iconic and symbolic models. For example, first graders might initially solve a problem with cubes, then transition to using bar models or number lines, and finally work with symbolic notations. These opportunities to connect multiple representations foster deeper understanding and move students toward fluency with abstract mathematical concepts (Brendefur & Frykholm, 2000; Moschkovich, 1999)

The Importance of Conceptual Understanding Before Procedural Knowledge

Several key reasons have been identified for prioritizing conceptual understanding before procedural knowledge in mathematics education:

Better retention and transfer: Conceptual understanding allows students to retain their knowledge more effectively and apply it to new situations. Students with strong conceptual foundations can reconstruct procedures they may have forgotten, unlike those who only learned procedural steps.

Flexible problem-solving: Students who understand the underlying concepts can approach novel problems flexibly, applying their knowledge in various ways rather than relying solely on memorized procedures.

Stronger learning foundation: Research consistently shows that gains in conceptual knowledge lead to gains in procedural knowledge more reliably than the reverse (Rittle-Johnson et al., 2001).

Deeper engagement: Teaching for conceptual understanding encourages students to make sense of mathematical concepts rather than simply memorizing procedures, leading to more meaningful engagement with math.

Alignment with expert problem-solving: Mathematicians and scientists rely on their conceptual understanding to solve complex problems. Teaching for conceptual understanding prepares students to think like experts in these fields.

Support for procedural fluency: The National Council of Teachers of Mathematics emphasizes that procedural fluency should be built on a foundation of conceptual understanding. Without this foundation, procedural knowledge can become fragile and disconnected.

Research on Conceptual and Procedural Understanding in Mathematics Education

Research strongly supports the prioritization of conceptual understanding in mathematics education. Rittle-Johnson et al. (2001) show that students with a solid conceptual understanding can better adapt their knowledge to different problems and perform well in routine and novel mathematical situations. These studies demonstrate the iterative nature of learning, where initial conceptual understanding facilitates the acquisition of procedural skills. In contrast, teaching procedural knowledge without the underlying conceptual framework can lead to a superficial understanding of mathematics, where students may struggle to remember or apply their knowledge in different contexts.

Another critical factor highlighted in the research is the role of teachers’ conceptual understanding. Many studies have shown that teachers who lack a deep understanding of mathematical concepts tend to overemphasize procedural instruction, perpetuating a cycle of superficial learning (Hussein, 2022). Teacher preparation programs and professional development focusing on improving educators’ conceptual knowledge can profoundly impact student learning, ensuring that future generations of students are better equipped to engage meaningfully with mathematics.

Cognitive Psychology Perspective

The cognitive psychology framework supports teaching conceptual understanding first. Schema theory describes how learners organize and integrate new knowledge into their existing mental frameworks. It outlines three key processes: assimilation, accommodation, and equilibration, essential for understanding how students learn new mathematical concepts.

Assimilation: Students incorporate new information into their existing schemas when it fits their prior knowledge. For example, when learning multiplication, students may connect it to their understanding of repeated addition.

Accommodation: When new information does not fit with their existing schemas, students must adjust or create new schemas to accommodate this information. This might occur when a student encounters a concept that challenges their understanding, such as the transition from whole numbers to fractions.

Equilibration: This is the process by which students balance assimilation and accommodation to maintain cognitive stability. Equilibration drives cognitive development as students continuously refine their mental frameworks to accommodate new knowledge.

In mathematics education, schema theory explains why conceptual understanding should come before procedural learning. When students first build a solid conceptual framework, they are better able to assimilate new procedures and apply them meaningfully. If students are taught procedures without the underlying concepts, they may struggle to accommodate new information or adapt their procedures to new problems.

Conceptual understanding also plays a crucial role in developing procedural fluency. Research by Rittle-Johnson et al. (2016) emphasizes that conceptual and procedural knowledge develop iteratively. Initial conceptual knowledge facilitates procedural learning, which in turn reinforces conceptual understanding. This iterative process aligns with schema theory and underscores the importance of teaching for understanding rather than simply focusing on procedural accuracy.

Moreover, studies on working memory and cognitive load suggest that students with a strong conceptual understanding can better manage the cognitive demands of complex mathematical procedures (Gilmore et al., 2017). A solid conceptual foundation frees up cognitive resources, allowing students to focus on problem-solving rather than memorizing steps

Conclusion

Mathematics education and cognitive psychology research strongly support the notion that teaching conceptual understanding before procedural understanding leads to more effective and meaningful learning. By fostering a deep understanding of mathematical principles, students are better equipped to engage with complex problems, transfer their knowledge to new situations, and develop the procedural fluency necessary for success in mathematics. Prioritizing conceptual understanding in mathematics education enhances students’ cognitive development and promotes long-term retention, flexible problem-solving, and deeper engagement.

References

Brendefur, J., & Frykholm, J. (2000). The role of representation in the development of mathematical understanding. Mathematics Education Research Journal, 12(3), 5-18. https://doi.org/10.1007/BF03217363

Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennell (Ed.), Mathematics education: A global perspective (pp. 1-12). National Council of Teachers of Mathematics.

Gilmore, C., Attridge, N., & Cragg, L. (2017). The role of working memory in the development of mathematical skills. Cognitive Development, 43, 1, 12. https://doi.org/10.1016/j.cogdev.2017.06.002

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97).

Macmillan. Hussein, A. (2022). Conceptual knowledge and its importance in teaching mathematics. Middle Eastern Journal of Research in Education and Social Sciences, 3(1), 1, 12. https://doi.org/10.47631/mejress.v3i1.445

Moschkovich, J. (1999). A situated and sociocultural perspective on learning mathematics. Mathematical Thinking and Learning, 1(3), 257-277. https://doi.org/10.1207/S15327833MTL0103_4

Moschkovich, J. (2012). Supporting students in learning mathematics through discourse. Mathematics Teacher, 105(5), 354-359. https://doi.org/10.5951/MT.105.5.0354

National Research Council (NRC). (2001). Adding it up: Helping children learn mathematics. National Academy Press.

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362. https://doi.org/10.1037/0022-0663.93.2.346

Rittle-Johnson, B., & Star, J. R. (2016). The importance of conceptual knowledge for learning mathematics. Educational Psychologist, 51(2), 145, 157. https://doi.org/10.1080/00461520.2016.1155467

Siegler, R. S., & Alabali, M. W. (2004). Development of numerical understanding. In D. H. Clements & J. Sarama (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 3, 22). Lawrence Erlbaum Associates.

Introduction

Addressing misconceptions in mathematics education is crucial for effective teaching and learning. Educators can significantly enhance conceptual understanding, problem-solving abilities, and mathematical reasoning across various contexts by understanding and addressing students' misconceptions. This research overview integrates insights from mathematics education and cognitive psychology, emphasizing the importance of identifying, understanding, and addressing misconceptions to foster deeper comprehension (Borasi, 1987; Borasi, 1994).

The Importance of Addressing Misconceptions

Misconceptions in mathematics are often deeply rooted and persist even after direct teaching. They arise from students’ informal strategies or incomplete understandings (Kapur, 2014). When not addressed, misconceptions can obstruct learning and should be treated as opportunities for growth rather than failures. A systematic approach incorporating students’ mistakes into instruction can improve learning outcomes and cultivate a more resilient understanding of mathematics (Borasi, 1987).

Borasi’s work highlights the importance of using students’ misconceptions as learning tools. Mistakes provide valuable insights into students’ thought processes and can be used as a diagnostic tool to correct misconceptions, ultimately leading to a deeper understanding of the subject matter (Borasi, 1994)..

Misconceptions and Cognitive Psychology

In cognitive psychology, misconceptions are viewed as part of the natural learning process. The concept of "productive failure" illustrates how engaging students in problem-solving before formal instruction can help them confront their misconceptions (Kapur, 2014). This approach creates a context for meaningful learning once formal instruction occurs, emphasizing the importance of error as a learning tool.

Hiebert and Grouws propose that struggle is essential to learning. When students encounter difficulties, they are forced to engage more deeply with the material, which can lead to a better grasp of complex mathematical concepts (Hiebert & Grouws, 2007). However, educators must carefully manage this struggle to ensure that students do not become discouraged.

Classroom Environment and Mistake Tolerance

The role of classroom culture in addressing misconceptions is critical. The concept of mistake tolerance refers to creating an environment where errors are seen as natural parts of learning rather than something to be feared or punished (Alvidrez, 2019). When students are encouraged to discuss their errors, they become more willing to explore their misconceptions and are more likely to correct them.

Collaborative approaches can be effective in addressing misconceptions. Research shows that teams that value open discussions about errors tend to learn more effectively (Tjosvold & Yu, 2004). By working together, students are more likely to identify and correct misconceptions, as they benefit from the diverse perspectives of their peers.

Cognitive Processes Behind Misconceptions

Misconceptions in mathematics often stem from students’ informal strategies and attempts to make sense of mathematical concepts. These errors are not random but are often rooted in students’ pre-existing cognitive frameworks (Harteis & Kleinknecht, 2008). For example, a student might incorrectly apply a familiar rule or procedure to a new context where it does not apply, revealing a misunderstanding of the underlying concept (Supardi, 2021).

The Role of Teachers in Addressing Misconceptions

Teachers play a pivotal role in guiding students through the process of recognizing and correcting misconceptions. When teachers frame errors as valuable learning opportunities, they create a more supportive learning environment where students feel safe to explore their misunderstandings (Gunderson et al., 2011). Teachers who maintain high expectations while supporting students through their errors can significantly enhance students’ confidence and performance in mathematics.

Instructional Strategies for Addressing Misconceptions

Research shows that instructional strategies focused on errors can effectively address misconceptions. Systematically investigating students’ mistakes provides valuable insights into their conceptual understanding (Muhammad, 2021). By analyzing common errors, teachers can develop targeted interventions that address the root causes of misconceptions rather than merely correcting the symptoms.

Error analysis helps teachers identify misconceptions and promotes a culture of reflection and inquiry. When students discuss their mistakes, they are encouraged to think critically about their learning processes, which can significantly improve their mathematical reasoning and problem-solving abilities.

Broader Implications for Educational Practices

Addressing misconceptions in mathematics has implications beyond individual student learning. Educational institutions should cultivate a broader culture of learning from mistakes (Harteis & Kleinknecht, 2008). When schools and educators embrace the idea that errors are an integral part of the learning process, they create an environment that encourages growth and continuous improvement.

Conclusion

Addressing misconceptions in mathematics education is a complex but essential task that requires understanding cognitive processes, classroom culture, and instructional strategies. By fostering a culture that views errors as a natural part of the learning process, educators can help students develop a deeper understanding of mathematical concepts and improve their problem-solving abilities. Instructional strategies such as productive failure, error analysis, and collaborative learning can significantly address misconceptions, leading to improved learning outcomes and a more positive attitude toward mathematics.

References

Alvidrez, M. (2019). From mistakes we learn: Teachers’ positional framing toward errors in mathematical classrooms. Russian Digital Libraries Journal, 22(5), 287-295. https://doi.org/10.26907/1562-5419-2019-22-5- 287-295

Borasi, R. (1987). Learning from mistakes: A study of the role of errors in the learning of mathematics. Journal for Research in Mathematics Education, 18(2), 134-150. https://doi.org/10.2307/749507

Borasi, R. (1994). Capitalizing on students’ mathematical misconceptions: A teaching experiment. Educational Studies in Mathematics, 26(3), 235-252. https://doi.org/10.1007/BF01274078

Gunderson, E. A., Ramirez, G., Levine S.C., & Beilock S.L., (2011). The role of parents and teachers in the development of gender-related math attitudes. Sex Roles, 66(3-4),153-166 https://doi.org/10.1007/s11199-011-9996-2

Harteis C., & Kleinknecht M., (2008). The culture of learning from mistakes: How employees handle mistakes in everyday work. International Journal of Educational Research,47(3),185- 196 https://doi.org/10.1016/j.ijer.2008.07.003

Hiebert J., & Grouws D.A., (2007). The effects of classroom mathematics teaching on students’ learning. In F.K.Lester(Ed.), Second Handbook of Research on Mathematics Teaching and Learning(pp371-404) Information Age Publishing.

Kapur M.,(2014). Productive failure in mathematical problem-solving. Instructional Science,42(1),1- 24 https://doi.org/10.1007/s11251-009-9093-x

Muhammad A.,(2021)Student errors in completing mathematical story problems based on Watson’s criteria during pandemic COVID-19.International Journal of Scientific and Research Publications,9(6),1https://doi.org/10.29322/ijsrp.9.06.2019.p9053

Supardi S.,(2021) Commognitive analysis of students’ errors in solving high-order thinking skills problems. Turkish Journal of Computer and Mathematics Education,12(6),2373- 2380 https://doi.org/10.17762/turcomat.v12i6.2373

Tjosvold D., & Yu Z.,(2004) Team learning from mistakes: The contribution of cooperative goals and problem solving. Journal of Management Studies,41(7),1163-1186 https://doi.org/10.1111/j1467-64862004

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We are thrilled to present our latest research overview on addressing misconceptions in mathematics education! This article delves into identifying and correcting misconceptions to enhance students’ mathematical understanding and problem-solving skills. We explore how leveraging students’ mistakes can become powerful learning opportunities, fostering deeper comprehension and resilience in mathematical thinking. The overview highlights the importance of creating a mistake-tolerant classroom environment, the cognitive processes behind misconceptions, and effective instructional strategies for addressing them. Discover how embracing errors in learning can transform student engagement, boost confidence, and cultivate a more positive attitude toward mathematics. Learn about the impact of collaborative approaches and productive struggle in overcoming misconceptions and building a stronger foundation for mathematical success!

Introduction

Effective math learning is not simply the absorption of procedures, it is the process of encoding, organizing, and flexibly retrieving mathematical knowledge through the interplay of cognitive mechanisms. Drawing on cognitive science, learning begins within working memory's constraints and, with purposeful instruction, develops into durable, adaptable long-term memory. This overview brings together foundational principles from theorists such as Jerome Bruner, David Ausubel, Jean Piaget, and John Sweller, illustrating how schema theory, assimilation, accommodation, cognitive load, and retrieval practice underpin powerful mathematics instruction for lasting understanding and flexible application (Bruner, 1966; Ausubel, 1968; Piaget, 1965; Sweller, 1988).

What Is the Psychological Foundation for Learning Math?

How do students’ mental structures grow and adapt as they encounter new mathematical ideas, and what does this mean for effective teaching?

Learning mathematics is fundamentally a process of building and reorganizing mental representations, schemas, through experience and reflection (Ausubel, 1968; Piaget, 1965). According to schema theory, every learner enters the classroom with existing networks of prior knowledge, which act as frameworks for understanding and integrating new ideas (Ausubel, 1968; Baddeley & Hitch, 1974). Piaget's concepts of assimilation and accommodation provide further insight: assimilation occurs when new information fits within a learner’s existing schema, strengthening and expanding their network effortlessly. When a concept resists easy categorization-say, when a student encounters non-standard models of multiplication accommodation is required; the schema itself must be revised or expanded to integrate this fresh insight (Piaget, 1965).

Instruction that leverages these natural processes scaffolds learning with intention. Bruner’s theory guides teachers in sequencing learning from enactive (action-based, physical manipulation) to iconic (visual representation) to symbolic (abstract formalism). At every stage, teachers can provide opportunities for both assimilation, by linking new content to familiar procedures, and accommodation, by challenging students to resolve contradictions between their prior conceptions and novel ways of thinking. Ausubel’s meaningful learning involves explicit efforts to connect, reorganize, and strengthen schemas through advance organizers and discourse that surface and address misconceptions, making mathematics more logical and memorable (Ausubel, 1968; Bruner, 1966; Piaget, 1965).

How Does a Math Concept Move from First Encounter to Mastery?

How do math ideas transition from fleeting exposure in working memory to robust, operating knowledge, and how do assimilation and accommodation play a role in this journey?

Early mastery begins in working memory, where a student can hold and manipulate only a few pieces of information at a time (Baddeley & Hitch, 1974; Cowan, 2001). Effective instruction reduces cognitive load by using enactive representations manipulatives and hands-on activities that allow students to “offload” thinking onto physical objects (Bruner, 1966; Sweller, 1988). As learners’ skills deepen, iconic models such as diagrams and area models help them assimilate mathematical relationships by “chunking” details into unified, existing schemas (Piaget, 1965).

However, true mastery demands more than assimilation alone. When a learner encounters problems that cannot be resolved by their current schemas such as an unfamiliar algorithm or a counterintuitive approach accommodation becomes necessary. In these moments, students must modify existing schemas or create new ones, thereby advancing their understanding and preparing for higher-level abstraction (Piaget, 1965). Teachers nurture this process by fostering reflection, encouraging explanation, and sequencing challenges that require rethinking and reorganization. Over time, these adaptive changes result in a more flexible and integrated network of mathematical knowledge, ready for transfer and creative problem-solving (Ausubel, 1968; Bruner, 1966; Roediger & Karpicke, 2006).Persistent Challenges and Their Solutions.

Why Is a Psychological Approach Effective?

Why are instructional methods that prioritize schema growth, assimilation, and accommodation proven to yield lasting mathematical mastery?

When teaching supports both assimilation (connecting new ideas to known structures) and accommodation (adapting or restructuring those structures), students develop deep, interconnected schemas. Mental networks enable students to access mathematical knowledge through multiple pathways retrieving a fact via familiar visuals, an analogy, or prior experience. This versatility translates directly to flexible problem-solving and adaptability in new or novel contexts (Bruner, 1966; Ausubel, 1968).

Supporting these mechanisms also reduces cognitive overload and fosters positive dispositions toward mathematics. By grounding abstract mathematical concepts in concrete experiences and familiar schemas, teachers enable confident assimilation and minimize confusion (Sweller, 1988; Piaget, 1965). When new information requires accommodation, targeted guidance and reflective learning experiences ensure that schema expansion happens deliberately, not randomly. Cognitive science further emphasizes the importance of retrieval practice, spaced repetition, and interleaved practice techniques that have been shown to solidify schema changes, reinforce long-term retention, and maintain active learning (Roediger & Karpicke, 2006).

What Are the Disadvantages and Challenges?

What obstacles must educators face when seeking to foster assimilation, accommodation, and schema growth in mathematics classrooms?

The shift to intentional,cognitively based instruction requires rich lesson design, time for scaffolded experiences, and opportunities for both assimilation and accommodation to occur. Still, the pressure to cover content rapidly or prepare for tests may undermine the slow, recursive nature of schema transformation (Sweller, 1988; Rohrer et al., 2015). Traditional assessments often prioritize memorized procedures, leaving little incentive for students to reflect or restructure their schemas when they encounter contradictory approaches (Bruner, 1966; Piaget, 1965).

Another hurdle is the need for sustained pedagogical knowledge and professional development. Teachers must be able to identify when a student is simply assimilating (adding to current knowledge) and when accommodation (schema change) is required, offering targeted supports for each. Without system-level curriculum alignment and formative assessments that reward deep conceptual transformations, the powerful benefits of schema theory and adaptive learning can remain underutilized. Cultivating a classroom and school culture that values not just “right answers,” but the process of cognitive growth including errors, reflection, and schema revision is vital to unlocking genuine mathematical mastery (Ausubel, 1968; Bruner, 1966; Sweller, 1988).

What Are the Implications for Curriculum and Teaching Practice?

How can curriculum and classroom practice be designed to foster both assimilation and accommodation, building robust schemas for mathematics learning?

Instruction should sequence learning from concrete to visual to abstract, giving students repeated opportunities to assimilate familiar ideas and then challenging them to accommodate new, unfamiliar ones. This means incorporating advance organizers, conceptual scaffolding, productive struggle, and explicit discourse at every stage, tools that surface students’ thinking and make schema growth visible (Bruner, 1966; Ausubel, 1968; Piaget, 1965).

Assessment should recognize and foster both mechanisms, valuing not only rapid retrieval, but the ability to revise, connect, and reorganize knowledge when needed. Frequent, formative checks for understanding, ongoing practice with multiple representations, and retrieval exercises help solidify schema changes and maintain both assimilation and accommodation. At the curriculum level, materials and pacing must allow time for recursive and reflective learning, with opportunities for students to both consolidate their existing knowledge and transform their thinking when encountering foundational challenges. Only then can the full power of cognitive psychology be brought to bear cultivating mathematical thinkers who assimilate, accommodate, and contribute creatively (Bruner, 1966; Piaget, 1965; Ausubel, 1968; Sweller, 1988).

Summary

In summary, the psychology of mathematics learning is grounded in schema theory and driven by the dynamic interplay of assimilation and accommodation. By designing instruction that enables students to both expand and modify existing schemas when confronted with new ideas, educators ensure that mathematical knowledge is flexible, durable, and deeply meaningful. By harnessing the psychology of durable memory schema building, assimilation, accommodation, and retrieval educators unlock mathematics as a lasting, empowering discipline for all learners(Bruner, 1966; Piaget, 1965; Ausubel, 1968; Roediger & Karpicke, 2006).

References

Ausubel, D. P. (1968). Educational psychology: A cognitive view. Holt, Rinehart & Winston.

Baddeley, A. D., & Hitch, G. (1974). Working memory. Psychology of Learning and Motivation, 8, 47, 89.

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24(1), 87, 185. https://doi.org/10.1017/S0140525X01003922

Piaget, J. (1965). The child’s conception of number. W. W. Norton.

Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249, 255. https://doi.org/10.1111/j.1467-9280.2006.01693.x

Rohrer, D., Dedrick, R. F., &Stershic, S. (2015). Interleaved practice improves mathematics learning. Journal of Educational Psychology, 107(3), 900, 908. https://doi.org/10.1037/edu0000001

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257, 285. https://doi.org/10.1207/s15516709cog1202_4

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Are We Teaching Math for the Test or for a Lifetime?

Cramming procedures might boost short-term scores, but cognitive science reveals how to build mathematical understanding that endures. It is not about more practice; it is about better, more brain-aware practice.

Our latest Research Overview breaks down the science of learning into actionable strategies for educators and leaders. Discover:

· How to leverage working memory's limits, instead of being defeated by them.· Why the sequence of concrete → visual → abstract (Bruner's Modes) is non-negotiable for deep encoding.· How "desirable difficulties" like spaced and interleaved practice combat the forgetting curve and build fluency that lasts for years.· The role of schema theory in creating flexible problem-solvers who can apply knowledge to novel situations.· This is not just theory, it is a blueprint for transforming math instruction.

What is one strategy you use to help make math "stick" for your students? Share in the comments below!