Concrete, Representational, Abstract

The C-R-A Learning Progression

Why is this beneficial to our students? Research-based studies show that students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations Harrison & Harrison, 1986; Suydam & Higgins, 1977

C→ R → A Concrete → Representational → Abstract Concrete: The “doing” stage using concrete objects to model problems Representational: The “seeing” stage using representations of the objects to model problems Abstract: The “symbolic” stage using abstract symbols to model problems

C→ R → A Based on work of Jerome Bruner in the 1960s “To instruct someone… is not a matter of getting him to commit results to mind. Rather, it is to teach him to participate in the process that makes possible the establishment of knowledge. We teach a subject not to produce little living libraries on that subject, but rather to get a student to think mathematically for himself, to consider matters as an historian does, to take part in the process of knowledge-getting. Knowing is a process not a product.” (1966: 72)

C→ R → A Sequencing matters A gradual and systematic approach which builds on a child’s existing understanding Concrete  Show students that math can be used to solve real life problems Use of manipulatives or hands-on activities

C→ R → A Concrete and pictorial representations should be used at all grade levels. By using cognitive strategies such as CRA, teachers provide students a technique for tackling mathematics problems rather than just searching for an answer.

Concrete The concrete stage is the “doing” stage. During this stage, students are using concrete objects or hands-on activities to act out and model problems. Students use manipulatives to show how they make sense of and how to solve the problem.

Concrete What are new concepts in our grade level that lend themselves to concrete experiences? What manipulatives will students choose to use to show their thinking?

C→ R → A The representational stage is the “seeing” stage. Students need to understand how the representational or pictorial examples relate to the concrete examples Representational.  Show the visual representation of the concrete Help students visualize the mathematics operation

Representational When should we plan for students to use the representational stage? As students begin to master concrete work As a way of recording concrete work When students have learned the concept at an earlier point and you are looking to reactivate that concrete knowledge When students are struggling with the abstract

Representational What are some examples of representations that students know from previous grade levels? What are some examples of representations that are used at your grade level? How do you connect these representations to students’ concrete experiences?

Abstract Abstract. The abstract stage is the “symbolic” stage. Show how symbols provide a shorter, efficient way to represent numerical operations During this stage, students move to using abstract symbols to model problems. Students need to understand how the abstract symbols relate to both the concrete and representational examples.

Abstract What are some examples of abstract concepts that students know from previous grade levels? What are some examples of abstract concepts that are used at your grade level? How do you connect these abstract concepts to students’ concrete or representational experiences?

C→ R → A Concrete and pictorial representations should be used at all grade levels. By using cognitive strategies such as CRA, teachers provide students a technique for tackling mathematics problems rather than just searching for an answer.