NCSM Position Paper

NCSM: Leadership in Mathematics Education mathedleadership.org in research and thoughtful commentary on the teaching of mathematics. The National Research Council’s Adding It Up (2001) concludes its review of research on the role of manipulatives with the following statement: “The evidence indicates, in short, that manipulatives can provide valuable support for student learning when teachers interact over time with the students to help them build links between the object, the symbol, and the mathematical idea they represent” (p. 354). Numerous studies have examined the effectiveness of specific manipulatives to teach specific topics. For example, the Milken Family Foundation analysis of NAEP data suggests that the use of hands-on materials is highly effective. The findings note that “when students are exposed to hands- on learning on a weekly rather than a monthly basis, they prove to be 72% of a grade level ahead in mathematics” (Wenglinsky, 2000, p. 27). Additionally, Sowell (1989) conducted a meta-analysis of studies focused on teaching with manipulatives and found them to have a positive impact on mathematics learning. Cramer et al. (2002) compared the performance of 1,600 fourth and fifth grade students studying fractions using both manipulative-based curricula and non- manipulative based curricula. They found that students in the manipulative-based program had higher mean scores at the end of the unit as well as higher retention scores. Manipulatives are also considered an important element of teacher preparation. For example, the Conference Board of Mathematical Sciences’ 2012 report, the Mathematical Education of Teachers II, includes numerous references to the use of manipulatives in classroom instruction and the importance of teacher preparation for this use. The authors continue by pointing out that teachers must work to help students see the connections between the manipulatives or other tools and the mathematical concept being taught. A number of studies cited in Van de Walle et al. (2012) suggest that manipulative instruction which follows a pattern of “do as I do” is one of the most widespread misuses of manipulatives. Stein and Bovalino (2001), for example, suggest three key features of successful manipulative lessons that avoid this pitfall. They conclude that 1) teachers have extensive training in the use of manipulatives; 2) teachers prepare by using manipulatives to complete the same instructional activities they would ask of their students; and 3) teachers prepare the classroom for activities by organizing students in groups, preparing materials, and thinking through the logistics of the lesson. Similar findings on the importance of effective instructional strategies when teaching with manipulatives appear in the 2009 Institute for Education Sciences report on response to intervention in mathematics (Gersten et al., 2009). The report states that “research shows that the systematic use of visual representations and manipulatives may lead to statistically significant or substantively important positive gains in math achievement” (pp. 30–31). The report goes on to discuss the importance of transitioning from concrete objects to visual representations and then to abstract notation. It provides a comprehensive summary of the evidence supporting the use of manipulatives, including evidence supporting the Concrete—Representational—Abstract (CRA) method of instruction. This method, grounded in Bruner’s (1966) constructivist discussion of enactive/iconic/ symbolic progression in learning, provides a basis for an effective framework for teaching with manipulatives. Under this framework, teachers begin with concrete manipulative experiences, transition students to using visual representations (drawings), and finally transition to using abstract mathematical notation. Hattie (2012) states “when teachers see learning occurring or not occurring, they intervene in calculated and meaningful ways. In particular, they provide students with multiple opportunities and alternatives for developing learning strategies based on the surface and deep levels of learning some content or domain matter, leading to students building conceptual understanding of this learning, which the students and teachers then use in future learning” (p. 15). Hattie later cites research on the power of balance in the classroom: “There is a balance between teachers talking, listening, and doing; there is a similar balance between students talking, listening, and doing” (p. 76). Manipulatives provide a foundation around which teachers and students can talk, listen, and do. Other research from Hattie (2009) concludes that, more often than not, when students do not learn, they do not need “more;” rather, they need “different” (p. 83). Again, to ensure that every student learns mathematics, a wide range of different strategies are needed for teaching and both physical and virtual manipulatives are a critical part of this toolkit. Witzel et al. (2003) describe an example of successful implementation of the CRA approach in teaching algebra to middle grades students. The Association of Middle Level Education’s research summary, Manipulatives in Middle Grades Mathematics (Goldsby, 2009), provides further information about this and other studies.