The Concrete-Representational-Abstract (CRA) Approach has become one of the most widely recognised instructional sequences in mathematics education. For decades, educators have used the CRA approach to help students develop conceptual understanding by progressing from concrete experiences, to visual representations, and finally to abstract mathematical symbols.
Rather than asking students to memorise mathematical procedures before they understand them, the CRA approach helps students develop meaning first. This is particularly valuable when teaching concepts such as place value, addition, subtraction, multiplication, division and fractions, where strong conceptual foundations are essential for future learning.
But why does the CRA approach work so well?
One of the most influential explanations comes from Cognitive Load Theory.
Developed by educational psychologist Professor John Sweller, Cognitive Load Theory explains how the human brain processes new information and why carefully sequenced, explicit instruction can significantly improve learning.
Understanding how Cognitive Load Theory relates to the Concrete-Representational-Abstract (CRA) Approach helps educators appreciate not only what to teach, but why this instructional sequence supports deeper mathematical understanding. It also highlights why the I-CRAVE Maths® Methodology, developed by Esther White, builds upon both the CRA (Concrete-Representational-Abstract) and CPA (Concrete-Pictorial-Abstract) instructional sequences to provide educators with a complete instructional methodology for teaching mathematics.
What Is Cognitive Load Theory?
Cognitive Load Theory is based on a simple but powerful idea: working memory has limited capacity.
When students are learning something new, they can only process a relatively small amount of unfamiliar information at one time. If too many ideas, instructions, symbols or procedures are introduced simultaneously, working memory becomes overloaded. Rather than supporting learning, this unnecessary cognitive demand can make mathematics feel confusing, frustrating and difficult.
By contrast, long-term memory has an enormous capacity to store knowledge.
As students learn, they organise mathematical ideas into schemas - mental frameworks that connect concepts, relationships and procedures. As these schemas become stronger through meaningful learning and practice, students are able to retrieve mathematical knowledge more efficiently and solve increasingly complex problems with less mental effort.
The goal of effective mathematics instruction is therefore not simply to present more information. It is to design learning experiences that manage cognitive load, support schema development and allow students to build genuine conceptual understanding.
How the CRA Approach Supports Cognitive Load Theory
The Concrete-Representational-Abstract (CRA) Approach aligns closely with the principles of Cognitive Load Theory because it introduces mathematical concepts in manageable stages rather than overwhelming students with abstract notation from the outset.
Instead of asking students to simultaneously understand mathematical symbols, procedures, vocabulary and concepts, the CRA approach allows understanding to develop progressively.
Concrete
Students begin by exploring mathematical ideas using carefully selected hands-on manipulatives.
Whether students are building place value with integer blocks, exchanging tens and ones, modelling multiplication or exploring fractions, concrete materials make invisible mathematical relationships visible. Rather than expecting students to imagine mathematical ideas mentally, they experience them physically.
This reduces unnecessary cognitive demand and provides meaningful experiences upon which future learning can be built.
Representational
Students then draw accurate mathematical representations of what they have built. These visual representations become an important bridge between concrete experiences and abstract mathematical thinking.
Rather than simply copying pictures, students learn to create proportionally accurate mathematical representations that preserve mathematical structure and strengthen the schemas developing within long-term memory.
Abstract
Only once conceptual understanding has been established do students move to mathematical symbols, equations and written algorithms. At this point, symbols represent mathematical ideas students already understand.
Instead of memorising procedures without meaning, students are connecting abstract notation to concepts they have already experienced concretely and visually.
This progression helps students develop mathematical understanding while managing the demands placed on working memory.
Why Explicit Instruction Matters
Cognitive Load Theory also highlights the importance of explicit instruction, particularly for novice learners.
When students are introduced to unfamiliar concepts, they benefit from clear teacher demonstrations, worked examples, carefully sequenced explanations and guided practice.
Rather than expecting students to discover mathematical relationships independently, explicit instruction provides the clarity needed for students to focus on understanding the mathematics itself.
When combined with the CRA instructional sequence, explicit instruction allows students to develop confidence because each new idea builds logically upon previous understanding.
CRA and CPA Are Powerful Instructional Sequences
Both the Concrete-Representational-Abstract (CRA) Approach and the Concrete-Pictorial-Abstract (CPA) Approach describe effective instructional sequences that support conceptual understanding.
The primary difference lies in terminology, with CRA emphasising accurate representations and CPA using the term pictorial.
Both approaches encourage students to move from concrete experiences towards abstract mathematical thinking.
However, neither approach, on its own, provides educators with a complete methodology for planning, assessing and delivering mathematics instruction.
Important questions still remain.
- Where should instruction begin?
- How do educators identify learning gaps?
- How should concepts be sequenced to build mastery?
- How can mathematical language remain consistent throughout instruction?
- How do educators know when students are ready to progress?
These questions extend beyond the CRA or CPA instructional sequence itself.
How the I-CRAVE Maths® Methodology Builds Upon CRA and CPA
At Maths Australia, the I-CRAVE Maths® Methodology, developed by Esther White, incorporates the strengths of both the Concrete-Representational-Abstract (CRA) and Concrete-Pictorial-Abstract (CPA) instructional sequences while extending them through diagnostic assessment, explicit instruction, mastery learning, mathematical language, verbal reasoning and multi-sensory teaching strategies.
Rather than viewing CRA or CPA as complete teaching approaches, the I-CRAVE Maths® Methodology integrates these principles into a complete instructional methodology that supports educators from assessment through to long-term mastery.
The methodology includes:
- Identify – determining each student's current level of mathematical understanding before instruction begins.
- Concrete – building conceptual understanding through carefully selected manipulatives.
- Representation – creating accurate mathematical representations that preserve mathematical structure.
- Abstract – connecting understanding to mathematical symbols, equations and algorithms.
- Verbal – strengthening mathematical language through purposeful discussion and explanation.
- Explicit – delivering clear, sequential and transferable instruction that reduces unnecessary cognitive load while supporting schema development.
Together, these elements help educators manage cognitive load, strengthen conceptual understanding and support students in developing genuine mathematical confidence.
The Maths Australia Difference
Everything at Maths Australia is underpinned by the I-CRAVE Maths® Methodology.
While the Concrete-Representational-Abstract (CRA) Approach, the Concrete-Pictorial-Abstract (CPA) Approach and Cognitive Load Theory each contribute valuable insights into how students learn mathematics, the I-CRAVE Maths® Methodology brings these principles together within one practical, evidence-informed instructional methodology.
Developed by Esther White, the methodology provides educators with a structured approach that supports assessment, explicit instruction, mastery learning, mathematical language, verbal reasoning and multi-sensory teaching practices.
The result is mathematics instruction that reduces unnecessary cognitive load, strengthens conceptual understanding and helps students build mathematical schemas that support long-term learning rather than short-term memorisation.
Ready to Transform the Way You Teach Maths?
Understanding Cognitive Load Theory, the Concrete-Representational-Abstract (CRA) Approach and the Concrete-Pictorial-Abstract (CPA) Approach is an excellent place to begin.
Learning how to implement these principles through the I-CRAVE Maths® Methodology is what transforms mathematics teaching.
Through Maths Australia's Multi-Sensory Numeracy Intervention Training, developed by Esther White, teachers, tutors, intervention specialists and school leaders learn how to identify learning gaps, manage cognitive load, explicitly teach mathematical concepts and build genuine conceptual understanding through a complete instructional methodology.
Whether you're supporting a whole class, delivering targeted intervention or working one-to-one with students, you'll gain practical, evidence-informed strategies that can be implemented immediately to help every learner develop confidence, understanding and long-term success in mathematics.
Explore our Multi-Sensory Numeracy Intervention Training today and discover how the I-CRAVE Maths® Methodology can transform the way you teach maths.