Reclaiming Numeracy: Reexamining Montessori's Concrete-to-Abstract Math Model in the Age of Digital Dependence
By Sean David Taylor
Abstract
The COVID-19 pandemic reshaped classrooms across the globe, accelerating reliance on digital platforms, screen-based learning, and worksheet-driven instruction. While these tools were deployed as emergency solutions, their prolonged use has had unintended consequences—particularly in the domain of early mathematics education. A critical developmental foundation, number sense and conceptual subitizing, has been compromised by the overuse of abstract algorithms, one-size-fits-all workbooks, and rote practice devoid of understanding. This article revisits Dr. Maria Montessori’s concrete-representational-abstract (CRA) methodology, arguing for a pedagogical renaissance that prioritizes hands-on, exploratory math instruction. Drawing on cognitive science, developmental psychology, and the legacy of Montessori education, we argue that the over-digitization of math instruction is not just ineffective—it is damaging.
The Crisis: A Generation Adrift in Abstraction
COVID-era schooling normalized emergency teaching practices, including the increased use of published workbooks, isolated skill drills, and app-based learning systems. In the rush to “simplify” and standardize instruction, students were pushed prematurely into abstract problem solving, often before developing the underlying number sense needed to support deep mathematical reasoning. As a result, many students today can perform steps of an algorithm by rote but cannot explain why those steps work, nor when and how to apply them flexibly. The Swiss cheese-like gaps in conceptual understanding are now evident in declining national numeracy scores and widespread math aversion among students.
Memory and cognition research confirm that without meaning-making and repeated use across contexts, learned procedures decay quickly. Cognitive overload, working memory failure, and lack of schema development all hinder student progress when mathematics is taught as disembodied steps rather than as logical, interconnected structures grounded in experience.
Numeracy and the Lost Art of Subitizing
At the heart of early math competency lies conceptual subitizing—the intuitive grasp of quantity without counting. It is a cornerstone of number sense, pattern recognition, and mathematical fluency. Subitizing is not taught through worksheets or iPad apps; it is cultivated through rich sensory and perceptual experiences with real-world quantities. Unfortunately, many students today are deprived of this critical foundation due to premature abstraction and passive learning environments.
When we ask children to memorize algorithms without visual, tactile, and contextual grounding, we deny them the chance to build meaningful numerical relationships. Consequently, they are left with shallow procedural fluency but no deep understanding—mathematical "house built on sand."
A Better Way: The Montessori Math Model
Before the term "heuristics" was widely used in education, Dr. Maria Montessori understood its essence. Her concrete-to-representational-to-abstract (CRA) instructional sequence is a developmental blueprint for how children naturally come to understand mathematics. Through materials like the golden bead chains, small and large bead frames, stamp games, and racks and tubes, Montessori learners are guided through hands-on explorations of numeration, place value, operations, and problem solving.
What makes Montessori math profoundly effective is not only the manipulatives themselves but the sequence of experiences and the freedom to explore at one’s own developmental pace. Students spend years moving through increasingly complex mathematical concepts, mastering each through multisensory engagement and repeated application. This model doesn't “teach to the test”—it prepares students to think mathematically.
Digital Tools and the Myth of Rigor
Despite good intentions, many digital math platforms and rigor-driven workbooks bypass essential cognitive development. They assume that drill equals mastery, or that more challenging worksheets somehow develop deeper thinking. In truth, for students lacking conceptual understanding, such tools exacerbate anxiety, boredom, and disengagement. Students are left asking, “Why am I doing this?”—and increasingly, they are concluding that math just isn’t for them.
Apps and software can support practice once understanding is in place, but they are no substitute for the rich, embodied, collaborative, and exploratory nature of early math learning. A screen cannot replicate the cognitive benefits of manipulating physical objects, engaging in peer discourse, or discovering mathematical relationships through guided inquiry.
Reclaiming Hands-On Math in a Post-Pandemic Era
If we hope to reverse the damage done during the pandemic and build a math-competent generation, we must return to time-tested pedagogies that honor child development. The Montessori model, aligned with research from the Science of Learning, offers a roadmap:
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Concrete before abstract: Students must touch math before they symbolize it.
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Paced through mastery: Students should not be rushed to the next skill until deep understanding is demonstrated.
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Freedom within structure: Choice and exploration lead to lasting learning when supported by knowledgeable teachers and clear developmental sequences.
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Low-tech, high-touch: We must resist the allure of screen-based solutions that offer efficiency but fail to deliver depth.
Conclusion: Old School is the New Frontier
We don’t need a new app. We need an old revolution. Dr. Maria Montessori understood that children need concrete, experiential learning to form strong mathematical minds. Her legacy offers us the tools and structure to do what modern systems have failed to do: build thinkers, not just test-takers. In our rush toward modernization, we have neglected the very methods that produce lifelong numeracy. It’s time to rethink our relationship with screens, revalue the tactile world of manipulatives, and reclaim math as a subject of joy, logic, and discovery.
Keywords: Montessori, numeracy, subitizing, concrete-representational-abstract, post-pandemic education, digital learning, math trauma, math manipulatives, screen-free learning, number sense
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