Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. NCEE 2009-4060.

Manipulatives are usually used in lower grades in the initial stages of learning as teachers introduce basic concepts with whole numbers. This exposure to concrete objects is often fleeting and transitory. The use of manipulatives in upper elementary school grades is virtually nonexistent.117 117. Howard, Perry, and Lindsay (1996); Howard, Perry, and Conroy (1995). RECOMMEnDATIOn 5. InTERvEnTIOn MATERIAlS ShOulD InCluDE OPPORTunITIES fOR ThE STuDEnT ( 33 ) The panel suggests that the interventionist use concrete objects in two ways. First, in lower elementary grades, use concrete objects more extensively in the initial stages of learning to reinforce the understanding of basic concepts and operations.118 Concrete models are routinely used to teach basic foundational concepts such as place value.119 They are also useful in teaching other aspects of mathematics such as multiplication facts. When a multiplication fact is memorized by question and answer alone, a student may believe that numbers are to be memorized rather than understood. For example, 4 × 6 equals 24. When shown using manipulatives (as in example 7, p. 35), 4 × 6 means 4 groups of 6, which total as 24 objects. 118. Darch (1989); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008). 119. Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Powell et al. (2008). Second, in the upper grades, use concrete objects when visual representations do not seem sufficient in helping students understand mathematics at the more abstract level. Use manipulatives expeditiously, and focus on fading them away systematically to reach the abstract level.120 In other words, explicitly teach students the concepts and operations when students are at the concrete level and consistently repeat the instructional procedures at the visual and abstract levels. Using consistent language across representational systems (manipulatives, visual representations, and abstract symbols) has been an important component in several research studies.121 Example 8 (p. 35) shows a set of matched concrete, visual, and abstract representations of a concept involving solving singlevariable equations. 120. Fuchs et al. (2005); Witzel (2005); Witzel, Mercer, and Miller (2003). 121. Fuchs et al. (2005); Butler et al. (2003); Witzel (2005); Witzel, Mercer, and Miller (2003). Example 4. Representation of the counting on strategy using a number line 0 1 2 3 4 5 6 7 8 9 10 RECOMMEnDATIOn 5. InTERvEnTIOn MATERIAlS ShOulD InCluDE OPPORTunITIES fOR ThE STuDEnT ( 34 ) Example 5. Using visual representations for multidigit addition Example 6. Strip diagrams can help students make sense of fractions A group of ten can be drawn with a long line to indicate that ten ones are joined to form one ten: Simple drawings help make sense of two-digit addition with regrouping: 36