How do you support your students' mathematical reasoning and communication using CRA or CPA frameworks?

According to me, the Concrete-Representational-Abstract (CRA) framework offers a powerful lens through which to understand and scaffold students' mathematical learning. At its core, CRA recognizes the importance of providing learners with multiple entry points to grasp abstract mathematical concepts. The CRA framework serves as a roadmap for guiding students on their mathematical journey, from concrete exploration to abstract mastery. By leveraging this approach, educators can cultivate a rich learning environment where students not only grasp mathematical concepts but also develop critical thinking skills, problem-solving abilities, and a lifelong love for learning.

Although there is a difference between CRA and CPA, what really counts is using the three stages regardless of the name of the second stage. Teachers decide what goes well in their classes, taking into consideration several elements such as students' age, achievement level, the level of complexity of the taught concept, and time constraints. They can even combine elements of both approaches! The key is to find a method that resonates with students and helps them achieve a deep understanding of mathematical concepts.

I use and promote CRA/CPA in much of my teaching of maths from Foundation to year 9. However, I am more flexible with the way I use them and encourage their use. For me it’s about providing opportunities for students to work with and learn maths in different ways and less about following the exact structure of CRA/CPA. Having said that I am very adamant on students beginning with either the C or the R when learning a new skill before moving to the A.

For me it’s the link between these 3 representations and movement back and forth between them that’s important. Asking in for abstract equation such as 3 + 4, what does the 4 represent and linking back to a concrete context supports the transference of understanding, which is vital. I also prefer to think of the middle term as a ‘mathematically structured’ or ‘iconic representation, it’s more than just a picture but an image that exposes the structure of the mathematics. An example of this is a a bar model where the 4 and 3 are exposed as being equivalent to the 7 and demonstrate the part part whole relationship.

Human beings learn through their senses and experiences which is what makes maths so hard to learn sometimes. With the CRA/CPA models it allows students to apply their tactile and visual senses to help them make sense of the abstract mathematics. It’s the missing link in maths education IMO.

These frameworks are beneficial in the sense that they are a tool to teach a deeper conceptual understanding of mathematical ideas. Students are not just memorizing procedures, but are actively constructing their knowledge and making sense of the underlying concepts. However, it is important to consider that there is a vast array of math teaching techniques available to educators. These frameworks may work with some, but not others, and another tool may need to be explored. Also, to ensure a well-rounded mathematical education, it is crucial to extend beyond frameworks and expose students to a variety of learning opportunities, such as technology, real-world applications, additional types of differentiation, etc.

For me it’s the moment ‘back and forth’ between these 3 modes of representation that’s important for example in the concrete model of 3 red flowers and 4 yellow flowers is 7 flowers altogether and can be expressed in the abstract as 3 + 4 = 7. Asking the question ‘what does the 4 represent?’ links back to the concrete context and supports the transference of understanding, which is vital. I also like to think of the middle term as a mathematically structured image, rather than just a picture of the concrete context. A bar model is an example where the 3 and 4 are lined up against the 7, exposing equivalence and the part part whole relationship.

An essential point to note about both cycles is that they shouldn't be linear, a one-way direction towards abstraction. Moving between the three stages is key. For students encountering challenges at the abstraction level, the ability to "move backward" strengthens understanding, resolves confusion, and bridges knowledge gaps. Even students who grasp the concepts initially benefit from reinforcement. Importantly, these connections often serve as the missing link for students struggling with a particular concept. By intentionally building bridges between stages through questioning, providing examples, and encouraging reviews, teachers can ensure a deeper and more meaningful understanding of math for all students.