CRA vs CPA: What’s the Difference and Is Either Approach Enough?
If you’ve been researching evidence-based mathematics instruction, you’ve likely come across both the Concrete-Representational-Abstract (CRA) Approach and the Concrete-Pictorial-Abstract (CPA) Approach. At first glance, they look almost identical. Both describe a progression from hands-on learning to visual representation before students work with abstract mathematical symbols.
So are CRA and CPA different teaching approaches? Or are they simply different names for the same instructional sequence?
The short answer is that CRA and CPA describe the same underlying sequence, and the difference is largely one of terminology and geography rather than educational philosophy. What matters far more to educators is a different question: is that sequence being delivered as a stand-alone strategy, or as part of a complete instructional methodology that also covers assessment, explicit teaching, mathematical language and mastery? That distinction is what separates the CRA and CPA sequence from the I-CRAVE Maths® Methodology, developed by Esther White at Maths Australia.
What Is the CRA Approach?
The Concrete-Representational-Abstract (CRA) Approach is an instructional sequence that develops conceptual understanding by guiding students through three connected stages:
Concrete – learning with hands-on manipulatives.
Representational – drawing accurate mathematical representations.
Abstract – working with mathematical symbols and algorithms.
The sequence has its roots in the work of Harvard psychologist Jerome Bruner, who proposed in Toward a Theory of Instruction (1966) that learners move through enactive (action-based), iconic (image-based) and symbolic (language- and symbol-based) modes of representation. Special education researchers later adapted this theory specifically for mathematics: studies by Witzel, Mercer and Miller (2003) and Witzel (2005) found that middle-school students with learning difficulties who were taught algebra through a concrete-to-abstract sequence outperformed peers taught with abstract instruction alone.
This research base is reflected in national guidance. 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 visual and concrete representations of mathematical ideas - a practice the panel rated as backed by moderate evidence.
Rather than asking students to memorise procedures, the CRA approach helps them understand why mathematics works.
If you’d like a complete explanation of the CRA approach, including how it supports conceptual understanding and how the I-CRAVE Maths® Methodology extends it, read our guide: The Concrete-Representational-Abstract (CRA) Approach Explained: How the I-CRAVE Maths® Methodology Extends CRA.
What Is the CPA Approach?
The Concrete-Pictorial-Abstract (CPA) Approach follows the same three-stage sequence. The only real difference is the language used to describe the second stage: instead of “Representational”, CPA uses the word “Pictorial”.
The CPA label is most closely associated with Singapore. When Singapore’s Ministry of Education began developing its own primary mathematics curriculum in the early 1980s, it built the teaching sequence directly on Bruner’s enactive-iconic-symbolic model, describing the three stages as concrete, pictorial and abstract. The approach became internationally known as “Singapore Math” after Singaporean students achieved consistently high rankings in international assessments such as TIMSS and PISA from the mid-1990s onward, prompting educators worldwide to study and adopt the Primary Mathematics textbooks and their underlying pedagogy.
In practice, many educators today use “CRA” and “CPA” interchangeably, and both describe the same evidence-informed progression from physical experience to abstract mathematics via visual representation.
Why Maths Australia Uses “Representational” Rather Than “Pictorial”
At Maths Australia, we deliberately use the term Representational rather than Pictorial.
A picture does not necessarily represent mathematical structure accurately. A quick sketch, a set of dots, or a ten-frame can visually resemble a concept without preserving the proportional relationships that make the underlying mathematics true.
Within the I-CRAVE Maths® Methodology, representations are constructed to preserve mathematical meaning. Students learn to draw proportional, accurate mathematical representations that connect directly to the concrete materials they have used - so that, for example, a bar representing ten is genuinely twice the length of a bar representing five. These representations become a bridge to abstraction rather than simply an illustration of it.
CRA and CPA Are Instructional Sequences - Not Complete Methodologies
Whether educators refer to CRA or CPA, both describe an instructional sequence: a research-informed explanation of how students move from concrete experience toward abstract mathematical thinking.
What they don’t provide is a complete methodology for teaching mathematics. The research literature on effective mathematics intervention makes this distinction clear. The IES practice guide referenced above, for instance, treats the concrete-to-representational-to-abstract progression as just one of several recommended practices, sitting alongside separate, independently evidenced recommendations for universal screening, explicit and systematic instruction, and structured, cumulative review.
In other words, the CRA or CPA sequence answers the question of how a concept should be introduced. It does not answer:
- Where should instruction begin?
- How do we identify learning gaps?
- How do we know when a student has achieved mastery?
- What role does explicit instruction play?
- How do we use mathematical language consistently?
- How do we reduce cognitive load?
These questions extend beyond the CRA or CPA sequence itself, and answering them is what separates an instructional sequence from an instructional methodology.
How the I-CRAVE Maths® Methodology Builds on CRA and CPA
The I-CRAVE Maths® Methodology, developed by Esther White, CEO and Founder of Maths Australia and a specialist in multi-sensory mathematics instruction and dyscalculia, incorporates the strengths of the CRA and CPA sequence within a complete, six-part instructional framework:
Identify: Before teaching begins, educators use diagnostic assessment to determine a student’s current level of mathematical understanding, identifying which foundational concepts are secure and which require attention. 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 using specific, colour-consistent manipulatives that accurately represent mathematical structure.
Representation: Students draw their own proportionally accurate representation of what they have built, preserving mathematical meaning rather than simply illustrating it.
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, teaching the lesson back to their instructor. This deliberate checking for understanding echoes Rosenshine’s Principles of Instruction (2012), which identifies frequent questioning and having students restate material in their own words as some of the most effective, evidence-supported classroom practices.
Explicit: Instruction is clear, sequential, consistent and purposeful, aligning models, representations, symbols and verbal explanation so every lesson reduces unnecessary cognitive load. Explicit, systematic instruction is the single most strongly evidenced recommendation in the IES practice guide on mathematics intervention.
Rather than treating Concrete, Representation and Abstract as a stand-alone strategy, the I-CRAVE Maths® Methodology integrates the sequence with diagnostic identification, verbal reasoning and explicit instruction - informed by a wider body of research that includes Cognitive Load Theory, Universal Design for Learning (UDL), Rosenshine’s Principles of Instruction, and Multi-Tiered Systems of Support (MTSS) and Response to Intervention (RTI) frameworks.
The Broader Evidence Base Behind I-CRAVE Maths®
Because I-CRAVE Maths® treats CRA/ CPA 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. The I-CRAVE Maths® Methodology applies this directly by using colour-consistent manipulatives and a single, congruent model across the Concrete, Representation and Abstract stages, reducing 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, ensuring instruction is adapted to a student’s starting point and checked through spoken explanation, not just written work.
Multi-Tiered Systems of Support (MTSS) and Response to Intervention (RTI): The IES practice guide referenced above 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.
Together, these frameworks help explain why a methodology built around CRA/ CPA, but not limited to it, is positioned to produce more consistent outcomes than the instructional sequence alone.
Beyond Manipulatives
A 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 and consistently connected to a matching visual representation, precise mathematical language, and symbolic notation. Students are not simply building models. They are building durable, transferable mathematical understanding.
Why This Matters
Mathematics is cumulative. A small misunderstanding in place value affects addition and subtraction. Weak understanding of addition and subtraction affects multiplication and division. These difficulties compound into fractions, decimals, percentages and algebra.
This is why identification and mastery matter as much as the concrete-to-abstract sequence itself. The I-CRAVE Maths® Methodology encourages educators to identify gaps, strengthen foundations, and ensure genuine conceptual understanding before progression - creating learning that is more transferable, more durable, and more meaningful than procedural memorisation alone.
Which Approach Should Teachers Use?
Whether you encounter the term CRA or CPA, the underlying principle is the same: students learn mathematics most effectively when conceptual understanding develops before abstract procedures are introduced.
The more important question is not whether you use the term CRA or CPA. It is whether those instructional stages sit inside a complete methodology that also supports diagnostic assessment, explicit teaching, mathematical language and mastery learning. That is the role of the I-CRAVE Maths® Methodology.
Ready to Go Beyond CRA and CPA?
Understanding the CRA and CPA instructional sequences is an excellent starting point. Learning how to implement them through a complete instructional methodology is what transforms mathematics teaching.
Discover how the I-CRAVE Maths® Methodology equips educators with practical, evidence-informed strategies through Maths Australia’s Multi-Sensory Numeracy Intervention Training, helping teachers confidently build strong mathematical foundations and improve student outcomes.
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), 121–131.
Witzel, B. S. (2005). Using CRA to Teach Algebra to Students with Math Difficulties in Inclusive Settings. Learning Disabilities: A Contemporary Journal, 3(2), 49–60.
Rosenshine, B. (2012). Principles of Instruction: Research-Based Strategies That All Teachers Should Know. American Educator, 36(1), 12–19, 39.
Sweller, J. (1988). Cognitive Load During Problem Solving: Effects on Learning. Cognitive Science, 12(2), 257–285.
CAST (2024). Universal Design for Learning Guidelines. https://udlguidelines.cast.org/
Wong, K. Y., & Lee, N. H. (2009). Singapore education and mathematics curriculum. In Mathematics Education: The Singapore Journey (Vol. 2, pp. 13–47). World Scientific Publishing.
Maths Australia. How It Works: The I-CRAVE Maths® Methodology. https://mathsaustralia.com.au/how-it-works/
Maths Australia. The Concrete-Representational-Abstract (CRA) Approach Explained: How the I-CRAVE Maths® Methodology Extends CRA. https://mathsaustralia.com.au/the-concrete-representational-abstract-cra-approach-explained-how-the-i-crave-maths-methodology-extends-cra/