Recommendations for Implementing the CPA Sequence
Here we summarize recommendations for incorporating manipulatives and pictorial representations into lessons. In subsequent chapters, we will provide more detailed discussion and examples of ways to use the CPA continuum when introducing specific mathematical content.
- 1. Let students use the manipulatives themselves. Simply having the teacher use manipulatives in a demonstration is insufficient. Students need hands-on practice.
- 2. When introducing new concepts and procedures, follow the CPA continuum. Begin with concrete representation, then progress to pictures, tallies, diagrams, and other two-dimensional representation, and then to abstract words and symbols. All three types of representation can be introduced in the same lesson, but students need several experiences with concrete representation before they are ready to discard the manipulatives and work solely at the pictorial level, and several more experiences before the pictorial representation can be faded and students are ready to rely only on abstract representation. Studies have shown that students with disabilities typically needed three lessons using manipulatives, and three more using pictorial representation, before relying solely on abstract words and symbols. Hudson and Miller (2006) suggest that students should be able to complete ten independent practice problems in order to progress from one level to the next. Interventionists can also assess student understanding by asking them to explain their representations. Students who can explain how various representations illustrate the same mathematical concept are ready to progress to the next level.
- 3. Explicitly link concrete, visual, and abstract representations, because students who have difficulty frequently fail to connect the various forms of mathematical representation. Explicitly linking the various representation systems, using consistent language across systems, and having students explain how the representations are connected, has resulted in higher achievement outcomes. Figure 6.1 provided an example of linking representational systems.
- 4. When selecting manipulatives, choose items carefully to clearly highlight the concept. It is not sufficient to simply give a student an object to move. The manipulative should provide a three-dimensional representation of the mathematical concept or procedure the students are learning.
- 5. Provide opportunities for students to model the same concept using a variety of different manipulatives and visual representations. The ability to represent mathematical ideas in multiple ways is a critical component of quantitative reasoning. For example, fraction circles are frequently used to model fractional parts of wholes, but students should not be limited to thinking of fractions as parts of circles. Their understanding will be enhanced if they also experience other examples, such as finding fractional parts of squares and rectangles, or using fraction bars, towers, and other manipulatives. Decimal values can be modeled with a variety of manipulatives, including using base-ten blocks, DigiBlocks, or metric weights. Graph paper and number lines allow students to create visual representations of decimal numbers. Using a variety of different manipulatives and visual representations to model the same concept deepens students' conceptual understanding.
- 6. Provide opportunities for students to translate among different representations. Students with rich number sense can fluently transition among all types of representations, but students who struggle to represent mathematical ideas may have difficulty making the same connections. For example, given the concrete representation of a mathematical expression, a student may be able to write a numerical expression to describe it. That same student may become confused when asked to reverse the process and, given the numerical expression, represent it with manipulatives. Sometimes, teachers routinely ask students to create one type of representation but neglect others. To develop a rich conceptual understanding, students need opportunities to practice converting among all representational forms. They should have opportunities to practice all of the following:
♦ Given a concrete representation, model it using pictures, diagrams, and other visual representations, as well as with numbers and words.
♦ Given a visual representation, model it using concrete materials, numbers, and words.
♦ Given a numerical expression, represent it concretely and visually and explain it in words.
♦ Given a word problem, represent it using concrete representation, visual representation, and with a numerical expression.
These opportunities will develop students' ability to fluently transition among representations.
7. Provide opportunities for students to explain their thinking. For example, students could share with their classmates the process they followed to obtain their answer, or explain why they selected a particular strategy, or they could explain their thoughts in a math journal or use a diagram to explain how they approached a particular problem. Asking students to explain their work helps consolidate understanding and also allows interventionists to assess students' understanding. The National Council of Teachers of Mathematics (NCTM) recommends that all students be provided opportunities to:
♦ Organize and consolidate their mathematical thinking through communication
♦ Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
♦ Analyze and evaluate the mathematical thinking and strategies of others
♦ Use the language of mathematics to express mathematical ideas precisely. (NCTM,
- 2000, p. 128)
- 8. Use manipulatives judiciously and systematically fade their use. Manipulatives provide an excellent foundation for understanding mathematics, but the goal is that students will become proficient with standard symbolic representation and not remain dependent on concrete supports. If students continue to work with concrete objects, they may not develop the ability to function at the abstract level. Interventionists should systematically fade the use of manipulatives and help students transition to visual and abstract representation. Research with students with disabilities suggests that three experiences with manipulatives is usually sufficient to develop initial understanding. As soon as students are able, fade manipulatives and focus on pictorial and abstract representation.
Following these suggestions when incorporating the CPA continuum into lessons can increase success in tiered interventions.
The ability to represent mathematical quantities in multiple ways is a critical component of quantitative reasoning. Representation allows students to organize mathematical information, describe mathematical relationships, and communicate mathematical ideas to others. Conceptual understanding of quantity follows a developmental sequence, beginning at the concrete level with physical actions and three-dimensional objects. As their understanding deepens, students progress to using pictorial representations such as charts and diagrams to model mathematical relationships. If these concrete and visual representations are linked to more abstract words and symbols, students eventually can use words and symbols meaningfully without needing the concrete and pictorial representations. The CPA continuum is an evidence-based practice recommended for students receiving tiered interventions.
Since textbooks used in the core curriculum usually provide limited concrete and pictorial examples, and seldom show students how the various representations are related, interventionists will frequently need to intensify instruction by adding these components to the lesson. In the next chapters, we provide more detailed descriptions of ways to incorporate concrete and pictorial representation to help students master whole numbers and rational numbers.