How do we effectively bridge the gap between tangible, real-world experiences and the often-abstract world of mathematical concepts? The Concrete-Representational-Abstract (CRA) approach provides a powerful and well-researched framework for achieving this. Both Maths Australia’s I-CRAVE pedagogy and AERO’s review, “Supporting Students Significantly Behind in Literacy and Numeracy,” strongly advocate for the use of the CRA approach in mathematics education.
- Benefits of the CRA Approach
AERO’s review provides compelling evidence of the benefits of the CRA approach across a wide spectrum of mathematical skills. The research highlighted in the review demonstrates that using CRA leads to significant improvements in students’ basic recall of number facts, their understanding of algebra, their proficiency in arithmetic, and their overall general mathematics skills. This is a powerful endorsement of the CRA approach’s versatility and its capacity to support students’ mathematical development from the foundational level to more advanced topics.
- The Three Stages of CRA
Let’s break down what the CRA approach actually entails:
- Concrete: This stage involves using tangible, physical objects, often called manipulatives, to represent mathematical concepts. Think of using blocks to understand addition, fraction bars to visualise fractions, or base-ten blocks to grasp place value.
- Representational: In this stage, students move from handling concrete objects to using pictorial representations, such as drawings, diagrams, or even tally marks, to represent the mathematical ideas.
- Abstract: This is the final stage, where students work with abstract symbols, such as numbers and mathematical notation, to solve problems.
- CRA and I-CRAVE
Maths Australia’s I-CRAVE pedagogy also recognises the power of the CRA approach. In fact, the I-CRAVE methodology, developed by Esther White, explicitly incorporates the concrete and representational stages. I-CRAVE emphasises the importance of building a strong foundation with manipulatives (the concrete stage) and then progressing to pictorial representations. Maths Australia’s I-CRAVE pedagogy provides a great example of the concrete stage in action when teaching fractions. By using physical overlays to divide a whole into equal parts, students can see and touch the concept of fractions, making it far more accessible than simply looking at the symbols “1/2” or “2/4”.
The beauty of the CRA approach lies in its ability to create a gradual and meaningful progression of understanding. By grounding mathematical concepts in concrete experiences, students develop a deeper and more intuitive grasp of the underlying principles. This, in turn, sets them up for success when they move on to the abstract stage of working with symbols and equations. In conclusion, both AERO’s review and Maths Australia’s I-CRAVE pedagogy champion the CRA approach as a cornerstone of effective mathematics instruction. By systematically moving students from the concrete to the representational to the abstract, educators can build bridges to understanding and empower all learners to achieve mathematical proficiency.
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