Manipulatives

Manipulatives There are many positions on the use of manipulative materials in the mathematics classroom at every level. Gagnon and Maccini (2001) define manipulatives as objects that students physically manipulate to represent mathematical concepts and relationships. A broader definition of manipulatives might take into account a differentiation between physical, computer, and virtual manipulatives. Computer manipulatives can be defined as computer representations of physical manipulatives via computer software (Clements & McMillen, 1996). Clements and McMillen recognize non-traditional and non-standard teaching aides as manipulatives, such as computer spreadsheets and databases, under this definition. Virtual manipulatives, similar to computer manipulatives, can be thought of as computer manipulatives that are available from the internet without the restrictive use of purchased software (Moyer, Bolyard, & Spikell, 2002). Examples include web-based applets that allow users to interact through the computer with tangrams, rods, geometry and algebra tiles, and many other computer versions of physical manipulatives. For the purposes of this discussion, the term manipulative can refer to any of the three designations mentioned above although the traditional definition of manipulative as a physical object will be most prevalent. The use of manipulatives is often viewed by teachers as a behavior reward rather than a vital tool for many students’ conceptual development (Moch, 2001). Moyer (2001) warns against teachers at any level equating the use of manipulatives and fun. Moyer argues that if teachers present manipulatives as a diversion, “they send a message to students that explorations and conjectures with representations are not connected to real mathematical learning” (p. 191). Although the temptation may be present for teachers to encourage students to enjoy math class and using manipulatives may seem more like fun and games than learning mathematics, the usefulness of manipulative materials as mathematical representations is diminished and devalued by teachers focusing on the fun of manipulatives (Moyer, 2001). Moyer and Jones (2004) argue that if teachers do not understand the mathematical benefits of manipulatives and, instead, see manipulatives as “secondary to the serious work of learning mathematics will inadvertently encourage their students to use these materials for play, rather than for mathematical learning or understanding” (p. 29). Moyer and Jones (2004) investigated middles grades mathematics teachers’ uses of manipulatives and how it related to teacher’s classroom control. They found that when teachers were apprehensive about how to use manipulatives, the teachers were less likely to use manipulatives in their classroom. Moyer and Jones also found that students will use manipulatives to review previously learned math

concepts. Moyer and Jones did not, however, discourage the use of manipulatives; instead, they discouraged the motives for which many secondary mathematics teachers use them. If middle and secondary teachers view manipulatives this way, as a reward, they may be less likely to use them in their classroom. As the primary use of manipulatives occur in the elementary classrooms followed by some use in the middle grades and very little at the high school level, there is little research on the effectiveness of manipulatives at the secondary level (Sowell, 1989). Howden (1986) studied the use of algebra tiles and their effects on students transitioning from concrete to abstract algebraic conceptual understanding and found that careful attention must be paid to a teacher’s choice of manipulatives else students run the risk of not making a transition from concrete to abstract concepts in mathematics. As research has shown (e.g., Tooke, Hyatt, Leigh, Snyder, & Borda, 1992), teachers at the secondary level feel they do not have the time or support to make this necessary careful choice of manipulatives. Also, Tooke and colleagues indicate that secondary level teachers are not willing to learn about using manipulatives. Ball (1992) cautions, “although kinesthetic experience can enhance perception and thinking, understanding does not travel through the fingertips and up the arm” (1992, p. 47). “Manipulatives are not necessarily transparent,” (Moyer, 2001, p. 177) in that the mathematical ideas behind the device are not automatically conveyed to students without some intervention from a teacher or peer. This is not to say that manipulatives cannot be quite effective as a teaching aide for mathematics at all levels, but that teachers need to take care that they are not letting the manipulatives speak for themselves. Reform mathematics educators argue that manipulatives provide a developmentally appropriate bridge regardless of grade band. At every grade level there exist many students that need help developing conceptualization at the concrete level. With untracked, heterogeneously grouped classes, there are those students in every class that need remediation on concepts that were previously taught. These same students will, likely, have a further distance to go to get from concrete to abstract. The National Council of Teachers of Mathematics (NCTM) recommends students “develop and deepen their understanding of mathematical concepts and relationships as they create, compare, and use various representations” (2000, p.280). Manipulatives can serve to guide students on a pathway from concrete to abstract representations of mathematics. Not all concepts in mathematics build upon previously learned concepts. The learning of these concepts may benefit from a strong foundation in prerequisite

knowledge, but the concept may appear to the student as a new idea. For these concepts, whose categorization differs for each learner, traversing the entire spectrum from concrete to abstract may be necessary. The evidence in the research on the effectiveness of manipulatives points to positive influence of using manipulatives to aid instruction and learning of mathematics. Ball (1988) found that fourth grade students that used both virtual and physical manipulatives showed conceptual understanding of fractions on a level higher than those students who did not use manipulatives. Sowell (1989) asserts that long-term use of manipulatives benefit student achievement and attitude when their teacher is knowledgeable about the use of manipulatives to aid mathematics instruction. The use of manipulatives has a strong foundational background in the suggestions of Piaget (1952) and Bruner (1960), among others. Piaget asserted that young children cannot grasp abstract mathematical ideas from only words and symbols and need concrete experiences with these concepts in order to learn them. Bruner’s first of three stages of representation is the enactive stage where the role of physical objects is especially important in advancement. Von Glasersfeld (1995) asserts that students are active participants in their learning, constructing knowledge through assimilating from and adapting to new experiences. Skemp (1987) argues that students early experiences with physical objects help create a strong base on which abstract knowledge can be built. If teachers are not willing to take the time to choose manipulatives carefully and learn how and when to use them, then those teachers should probably not use manipulatives. However, teachers who are willing to use manipulatives in careful and thoughtful ways will likely find that their instruction is more effective for more of their students and mathematical concepts become more meaningful and understood with the aid of manipulatives.