Concrete-to-Representational-to-Abstract (CRA) Sequence of Instruction

1. C = learning through concrete or hands-on instruction using actual manipulative objects.

2. R = learning through pictorial representations of the previously used manipulative objects during concrete instruction.

3. A = learning through abstract notation such as Arabic numbers and operational symbols.

The CRAMATH strategy:

CRAMATH includes seven steps to help math instructors implement CRA successfully with secondary mathematical skills.

1. Choose the math topic to be taught.

2. Review procedures to solve the problem.

3. Adjust the steps to eliminate notation or calculation tricks.

4. Match the abstract steps with an appropriate concrete manipulative.

5. Arrange concrete and representational lessons.

6. Teach each concrete, representational, and abstract lesson to student mastery.

7. Help students generalize what they learn through word problems.

Once fraction concepts are understood as “part of a whole,” students can practice the steps involved in reading and writing fractions. A variety of physical materials can be used to show the meaning of a fraction as “part of a whole.” For example, fraction cubes, counters, fraction bars, or geometric shapes can indicate a fraction (e.g., 3 red cubes (part) out of the 5 cubes (whole, the total number of cubes). Representations and numeric symbols of the fraction can develop the skills of reading and writing fractions. The abstract stage is developed by writing a numeric symbol of the number of squares or parts of the whole in correct fraction form. This step involves the order in which digits should be read or written. For a fraction, which number is (represented) written on the top? Which number is (represented) written on the bottom? An example of reading and writing fractions in the correct order follows.

Goal: To develop the spatial organization, visually and kinesthetically, to read and write fractions correctly.

Materials : Red squares and larger black squares are displayed to help with sequencing and number placement.

Teacher: “Today we are going to write and say fractions.”

Concrete: Teacher points to the squares arranged on a table. “What colors are the squares?” (Student says black and red. ) “Count the total number of squares (whole).” (Student points and counts to 8. Student says 8.) “How many red squares are there?” (Student points and counts to 3. Student says 3.)

Represent: “When we talk about fractions, we say the ‘part of the whole.’ (Say together ‘part of the whole’). We can write a fraction showing the part of the whole, as shown above. The number for the part is written on the top and the number for the whole is written on the bottom. (Say part on top and whole on bottom.) What was the total number of squares?” (Student says 8.) “Let’s call that the whole.”

Abstract: “Write the total number of squares or the whole on the bottom where the word ‘whole’ is shown.” (Student writes 8.)

Represent: “How many red squares are there?” (Student says 3.) “Let’s call the red squares ‘part’ of the whole.”

Abstract: “Write the number of red squares on the top where the word ‘part’ is written. (Student writes 3.)

Summary: “From this example, what did you write for the fraction?” (Student says 3 and 8.) “We say 3 out of 8 or three-eighths.” Practice several different examples with the squares, writing and reading the fractions.

Adapted from Teaching Mathematics to Students with Learning Disabilities, Bley & Thornton, p. 296.

Benefits of using CRA:

First, the CRA instructional sequence provides multiple opportunities for students to acquire mathematical skills through multimodal forms of learning. Learning through multimodal forms includes seeing, hearing, muscle movement, and touch. Visual, auditory, kinesthetic, and tactile learning also aids memory and retrieval (Engelkamp & Zimmer,1990; Nilsson, 2000).

Second, students with various learning preferences are able to interact in multiple ways that may increase engagement and improve attitude toward content (Oberer, 2003).

Third, meaningful manipulations of materials in the concrete and representational phase allow students to rationalize abstract mathematics into understandable steps and definitions (Demby, 1997; Noice & Noice, 2001).

Fourth, generalization developed thorough the CRA instructional sequence may provide a more meaningful and contextually relevant alternative to shallow and rote memorization of algorithms and rules taught in isolation of the purpose of the computation. By designing lessons based on the CRA instructional sequence, teachers encourage students to take personal ownership of abstract concepts by first making personalized concrete, meaningful connections through the use of appropriate manipulatives in meaningful directed activities. Interactions with concrete materials and connected personalized language allow students to tie relevance to the manipulation and realize the contextual relevance of mathematical concepts and procedures.

Finally, for testing and working outside of class, learning to manipulate pictorial versions of concrete objects enables students to solve abstractly worded problems without actually reasoning at the concrete level. In other words, if a student who thinks in terms of physical objects is asked an abstract problem such as multiply 1/3 and 1/2, the student may first draw an apple being cut apart rather than immediately multiplying numerators and denominators.

Sources: Witzel, B., Riccomini, P., & Schneider, E. (2008, January 1). Implementing CRA with Secondary Students with Learning Disabilities in Mathematics. Intervention in School and Clinic, 43(5), 270-276.