The Concrete-Representational-Abstract (CRA) Approach Explained
The Concrete-Representational-Abstract (CRA) Approach is widely recognised as one of the most effective instructional sequences for teaching mathematics conceptually. Rather than asking students to memorise procedures before they understand them, the CRA approach guides learners from hands-on experiences, to visual representations, and finally to abstract mathematical symbols.
For decades, educators have used the CRA approach to help students develop deeper mathematical understanding, particularly when introducing new concepts such as place value, addition, subtraction, multiplication, division and fractions. It is a genuinely evidence informed instructional sequence, and in this article we unpack exactly where that evidence comes from.
However, while the CRA approach explains how students learn a mathematical concept, it does not, on its own, provide educators with a complete instructional methodology for teaching mathematics from assessment through to mastery. This is where the I-CRAVE Maths® Methodology differs.
Developed by Esther White, CEO and Founder of Maths Australia and a leading multi-sensory maths and dyscalculia specialist, the I-CRAVE Maths® Methodology incorporates and extends the strengths of the CRA approach into a comprehensive methodology that guides educators through every stage of mathematics instruction, from diagnostic assessment to long term mastery.
What Is the Concrete-Representational-Abstract (CRA) Approach?
The Concrete-Representational-Abstract (CRA) Approach is an instructional sequence that supports students in moving from concrete experience to abstract mathematical thinking. Instead of introducing symbols first, students develop understanding through three connected stages.
Concrete: Students begin by physically building mathematical ideas using carefully selected manipulatives. They can see quantities, compare amounts, exchange place value units and experience mathematical relationships before attempting a single written calculation.
Representational: Students then draw or interpret accurate representations of what they have built. These visual models bridge the gap between physical experience and abstract mathematics, helping students internalise mathematical structure rather than simply copy a picture.
Abstract: Only after understanding has been established do students move into mathematical symbols, equations and written algorithms. At this stage, the symbols represent ideas students already understand, which makes learning more meaningful and reduces reliance on memorisation.
This progression helps students build confidence because they understand why mathematics works, rather than simply remembering the steps of a procedure.
Where the CRA Approach Comes From
The CRA sequence is not a marketing invention. It is grounded in the work of Harvard psychologist Jerome Bruner, who argued in Toward a Theory of Instruction (1966) that learners move through three modes of representation when acquiring new knowledge: enactive (learning through physical action), iconic (learning through images and models), and symbolic (learning through language and abstract notation). Bruner proposed that this progression holds true across subjects and even across age groups, including adult learners.
Special education researchers later adapted Bruner's framework specifically for mathematics instruction. In a series of studies through the early 2000s, Brad Witzel and colleagues found that middle school students with learning difficulties who were taught algebra using a concrete to representational to abstract sequence significantly outperformed peers who were taught the same content using abstract instruction alone, both immediately after teaching and on follow up assessments some weeks later.
This body of research is reflected in national guidance for educators. The U.S. Institute of Education Sciences' practice guide Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools (Gersten et al., 2009) recommends that intervention materials give students extensive opportunities to work with concrete and visual representations of mathematical ideas, and that interventionists be proficient in using them. The expert panel rated this recommendation as backed by moderate evidence, meaning it is supported by a number of well designed studies, though not yet the volume of large scale trials required for the panel's strongest rating.
Why the CRA Approach Works
One of the greatest strengths of the CRA approach is that it develops conceptual understanding rather than surface level procedural fluency.
Many students can successfully complete a worksheet by following a memorised procedure, yet struggle to explain what they are actually doing. When the numbers become larger or the question changes slightly, confidence often disappears because understanding was never fully established in the first place.
The CRA approach addresses this by allowing students to experience a mathematical concept before they are expected to work with it abstractly. For example, rather than telling students to carry the one during addition, students physically exchange ten ones for one ten using concrete materials. They see why regrouping occurs before they are ever asked to record it symbolically.
Likewise, instead of teaching students to borrow during subtraction, students exchange one ten for ten ones and observe the place value relationship directly. The written algorithm becomes a record of understanding, not a rule to memorise.
This aligns with a well established principle in cognitive science: knowledge that is built on a genuine mental model transfers more readily to new and unfamiliar problems than knowledge that is memorised as an isolated procedure.
Why the CRA Approach Is Not, On Its Own, Enough
At Maths Australia, we consider the Concrete-Representational-Abstract (CRA) Approach to be one of the most valuable instructional sequences available to educators.
However, effective mathematics teaching requires more than moving students through three stages. Teachers also need to know:
- Where should instruction begin?
- What foundational concepts are already secure?
- Which mathematical understandings are missing?
- How can learning be sequenced to build mastery?
- How should concepts be explained clearly and consistently?
- How do we ensure students truly understand before moving forward?
The CRA approach does not answer these questions on its own, and nor does the research literature suggest it should try to. The IES practice guide referenced above treats concrete and visual representation as just one of several independently evidenced recommendations, sitting alongside separate guidance on universal screening, explicit and systematic instruction, and structured, cumulative review. In other words, CRA explains how a concept should be introduced. It was never designed to explain where to start, how to assess progress, or how to sequence an entire curriculum.
The I-CRAVE Maths® Methodology was developed to provide educators with a complete instructional methodology that incorporates the strengths of CRA while extending it through a structured teaching framework.
How the I-CRAVE Maths® Methodology Extends CRA
The methodology includes six connected components.
Identify: Before teaching begins, educators identify the student's current level of mathematical understanding. Rather than assuming the problem lies with the current topic, educators determine whether earlier concepts require strengthening first. This mirrors the IES practice guide's own recommendation, also rated moderate evidence, that all students be screened to identify those at risk of mathematics difficulty.
Concrete: Students build mathematical understanding through carefully selected hands on materials that accurately represent mathematical structure.
Representational: Students create accurate visual representations that bridge concrete experience with abstract thinking, rather than loosely illustrative pictures.
Abstract: Students connect their understanding to mathematical notation, equations and algorithms only once conceptual understanding has been established.
Verbal: Students explain their reasoning using precise mathematical language, including teaching a step back to their instructor. This helps educators assess genuine understanding while strengthening mathematical communication. It also echoes Rosenshine's Principles of Instruction (2012), which identifies frequent questioning and having students restate material in their own words among the most effective, evidence supported classroom practices.
Explicit: Instruction is clear, sequential, consistent and purposeful. Every explanation, representation and demonstration aligns to reduce confusion and build confidence. Explicit, systematic instruction is the single most strongly evidenced recommendation in the IES practice guide on mathematics intervention.
Rather than replacing the CRA approach, the I-CRAVE Maths® Methodology expands it into a complete instructional framework that supports planning, teaching, assessment and long term mastery.
The Research Behind the Wider Methodology
Because I-CRAVE Maths® treats CRA as one component within a larger system, it also draws on several additional, well established bodies of research.
Cognitive Load Theory: John Sweller's foundational 1988 research demonstrated that working memory is limited, and that poorly designed instruction can overload it with irrelevant detail before a concept is even understood. Using colour consistent manipulatives and a single, congruent model across the Concrete, Representational and Abstract stages reduces the instructional clutter that competes with learning.
Universal Design for Learning (UDL): CAST's UDL framework recommends offering multiple means of representation, action and expression so that a wider range of learners can access the same content. The Identify and Verbal components of I-CRAVE Maths® build this in from the outset, adapting instruction to a student's starting point and checking understanding through spoken explanation, not written work alone.
Multi-Tiered Systems of Support (MTSS) and Response to Intervention (RTI): The IES practice guide recommends structuring mathematics support in tiers, with universal screening at Tier 1 and increasingly intensive, explicit intervention at Tiers 2 and 3. I-CRAVE Maths® is designed to be delivered across all three tiers, from whole class teaching through to individualised intervention.
Beyond Manipulatives
One common misconception is that using manipulatives alone constitutes multi sensory mathematics instruction.
In reality, manipulatives are only one part of effective teaching. Without explicit instruction, purposeful questioning and carefully sequenced learning, manipulatives can become an activity rather than a teaching tool.
Within the I-CRAVE Maths® Methodology, every concrete experience is intentionally connected to visual representation, mathematical language and symbolic notation. Students are not simply building models. They are building understanding.
Why This Matters
Mathematics is cumulative. A small misunderstanding in place value can affect addition and subtraction. Weak understanding of addition and subtraction influences multiplication and division. These difficulties continue into fractions, decimals, percentages and algebra.
This is why mastery matters. The I-CRAVE Maths® Methodology encourages educators to identify gaps, strengthen foundations and ensure students develop conceptual understanding before progressing.
Rather than teaching students to remember procedures, educators help students understand the mathematical relationships behind them. This creates learning that is more transferable, more durable and more meaningful.
The Maths Australia Difference
Everything at Maths Australia is underpinned by the I-CRAVE Maths® Methodology.
Developed by Esther White, the methodology combines evidence informed teaching practices with practical classroom application to help educators confidently teach mathematics through understanding rather than memorisation.
While the Concrete-Representational-Abstract (CRA) Approach remains an essential instructional sequence, the I-CRAVE Maths® Methodology extends it into a complete framework that integrates assessment, mastery learning, explicit instruction, verbal reasoning and multi sensory teaching. The result is mathematics instruction that is structured, sequential and designed to build genuine mathematical confidence.
Ready to Transform the Way You Teach Maths?
Understanding the Concrete-Representational-Abstract (CRA) Approach is an excellent place to begin. Learning how to implement it through the I-CRAVE Maths® Methodology is what transforms mathematics teaching.
At Maths Australia, our Multi-Sensory Numeracy Intervention Training, developed by Esther White, equips teachers, tutors, intervention specialists and school leaders with the practical knowledge, confidence and instructional framework to teach maths conceptually, explicitly and systematically.
Through the I-CRAVE Maths® Methodology, you will learn how to identify learning gaps, build strong mathematical foundations, reduce cognitive load and develop genuine conceptual understanding through evidence informed, multi sensory teaching practices.
Whether you are supporting a whole class, delivering targeted intervention or working one to one with students, you will gain practical strategies that can be implemented immediately to help every learner build confidence, develop mathematical understanding and experience long term success.
Explore our Multi-Sensory Numeracy Intervention Training today and discover how the I-CRAVE Maths® Methodology can transform the way you teach maths.
References & Further Reading
Bruner, J. S. (1966). Toward a Theory of Instruction. Harvard University Press.
Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools (NCEE 2009-4060). Institute of Education Sciences, U.S. Department of Education.
Witzel, B. S., Mercer, C. D., & Miller, S. P. (2003). Teaching Algebra to Students with Learning Difficulties: An Investigation of an Explicit Instruction Model. Learning Disabilities Research & Practice, 18(2), 121 to 131.
Witzel, B. S. (2005). Using CRA to Teach Algebra to Students with Math Difficulties in Inclusive Settings. Learning Disabilities: A Contemporary Journal, 3(2), 49 to 60.
Rosenshine, B. (2012). Principles of Instruction: Research-Based Strategies That All Teachers Should Know. American Educator, 36(1), 12 to 19, 39.
Sweller, J. (1988). Cognitive Load During Problem Solving: Effects on Learning. Cognitive Science, 12(2), 257 to 285.
CAST (2024). Universal Design for Learning Guidelines. https://udlguidelines.cast.org/
Maths Australia. How It Works: The I-CRAVE Maths® Methodology. https://mathsaustralia.com.au/how-it-works/
Maths Australia. I-CRAVE Maths® Methodology. https://mathsaustralia.com.au/icrave-maths-methodology/