“Getting” Math:

1. “Getting” Math: The Concrete-Representative-Abstract Teaching Approach

2. Concrete-Representational-Abstract • Instructional sequence • Builds thorough understanding of • Math concepts • Math skills • Developing concrete understanding increases likelihood of • Performing a skill • Recognizing abstract math concepts

3. What is the CRA instructional sequence? • Model each concept/skill with concrete materials • Provide many opportunities to practice and demonstrate mastery using concrete materials • Model the concept/skill using representational techniques by drawing pictures that represent the concrete objects previously used • Provide many opportunities to practice and demonstrate mastery by drawing solutions • Model the concept/skill using only numbers and mathematical symbols • Provide many opportunities to practice and demonstrate mastery at the abstract level

4. What do you need to know? As you move through a concrete-to-representational-to-abstract sequence of instruction, the abstract numbers and/or symbols should be used in conjunction with the concrete materials and representational drawings to promote association of abstract symbols with concrete and representational understanding

5. By using the concrete items with symbols you are increasing the likelihood that students with ASD will have a better understanding of the concept. Before you introduce a manipulative to your class you need to be sure to play with it yourself and problem solve any possible stumbling block for your students with autism. All of the students in your class will benefit from this learning style!

6. What else do you need to know? • Always teach concrete understanding first. • Use appropriate concrete objects to teach particular math concept/skills • Teach representational understanding second. • Use appropriate drawing techniques or appropriate picture representations of concrete objects • Use appropriate strategies for assisting students to move to the abstract level of understanding for a particular math concept/skill

7. How do I implement the strategy? • When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding). Try to use concrete objects that the students are interested in. This goes for students with ASD as well as for typically developing peers. If they are interested you will hold their attention. • Provide students many practice opportunities using concrete objects. Don’t go straight from the concrete to the abstract. In other words… stay away from worksheets!

8. How do I implement the strategy? • When students demonstrate mastery of skill by using concrete objects, describe & model how to perform the skill by drawing or with pictures that represent concrete objects (representational level of understanding). • Provide many practice opportunities where students draw their solutions or use pictures to problem-solve. Some students with ASD might have problems with drawing so be thinking how you might modify this. One way is to have the pictures already drawn so they can move them around. This would be a very good time to incorporate students’ individual interests! • When students demonstrate mastery drawing solutions, describe and model how to perform the skill using only numbers and math symbols (abstract level of understanding).

9. How do I implement the strategy? • Provide many opportunities for students to practice performing the skill using only numbers and symbols. • After students master performing the skill at the abstract level of understanding, ensure students maintain their skill level by providing periodic practice opportunities for the math skills. At any point along the way the task may be moved to an independent practice area.

10. How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems? • Helps passive learners to make meaningful connections • Teaches conceptual understanding by connecting concrete understanding to abstract math process • By linking learning experiences from concrete-to-representational-to-abstract levels of understanding, the teacher provides a graduated framework for students to make meaningful connections. • Blends conceptual and procedural understanding in structured way

11. Research Support For The Instructional Features Of This Instructional Strategy: • Allsopp (1999); Baroody (1987); Butler, Miller, Crehan, Babbit, & Pierce (2003); Harris, Miller, & Mercer (1993);  Kennedy and Tips (1998); Mercer, Jordan, & Miller (1996); Mercer and Mercer (2005); Miller, Butler, & Lee (1998); Miller and Mercer, 1995; Miller, Mercer, & Dillon (1992); Peterson, Mercer, & O'Shea. (1988); Van De Walle (2005); Witzel, Mercer, & Miller (2003).