Experiential Learning of Mathematics: Using Manipulatives

ERIC Educational Reports by Robert Hartshorn, Sue Boren

Experiential education is based on the idea that active involvement enhances students' learning. Applying this idea to mathematics is difficult, in part, because mathematics is so "abstract." One practical route for bringing experience to bear on students' mathematical understanding, however, is the use of manipulatives. Teachers in the primary grades have generally accepted the importance of manipulatives. Moreover, recent studies of students' learning of mathematical concepts and processes have created new interest in the use of manipulatives across all grades.

In this Digest "manipulatives" will be understood to refer to objects that can be touched and moved by students to introduce or reinforce a mathematical concept. The following discussion examines recent research about the use of manipulatives. It also speculates on some of the challenges that will affect their use in the future.DEVELOPMENT OF MANIPULATIVES FOR TEACHING MATHEMATICSBoth Pestalozzi, in the 19th century, and Montessori, in the early 20th century, advocated the active involvement of children in the learning process. In every decade since 1940, the National Council of Teachers of Mathematics (NCTM) has encouraged the use of manipulatives at all grade levels. Every recent issue of the "Arithmetic Teacher" has described uses of manipulatives. In fact, the entire February 1986 issue considered answers to the practical questions of why, when, what, how, and with whom manipulative materials should be used.

Research suggests that manipulatives are particularly useful in helping children move from the concrete to the abstract level. Teachers, however, must choose activities and manipulatives carefully to support the introduction of abstract symbols. Heddens divided the transitional iconic level (the level between concrete and abstract) further into the semiconcrete and semiabstract levels, in the following way:

The semiconcrete level is a representation of a real situation; pictures of the real items are used rather than the items themselves. The semiabstract level involves a symbolic representation of concrete items, but the pictures do not look like the objects for which they stand. (Heddens, 1986, p.14)

Howden (1986) places specific manipulatives on this continuum. These manipulatives rank from the concrete to the abstract. In place value, for example (going from concrete to abstract), they include pebbles, bundled straws, base-ten blocks, chip-trading, and the abacus. Howden cautions that building the bridge between the concrete and abstract levels requires careful attention. She notes that, even if children can solve a given problem at the concrete level, they may not be able to solve the same problem at the abstract level. This problem occurs if the bridge has not been structured by a careful choice of manipulatives.

Suydam and Higgins (1977), in a review of activity-based mathematics learning in grades K-8, determined that mathematics achievement increased when manipulatives were used. Sowell (1989) performed a meta-analysis of 60 studies to examine the effectiveness of various types of manipulatives with kindergarten through postsecondary students. Although these studies indicate that manipulatives can be effective, they suggest that manipulatives have not been used by many teachers.

IMPLEMENTATION OF MANIPULATIVES IN GRADES K-12

The reasons that teachers do not use manipulatives are beyond the scope of this Digest. Several related issues are, however, relevant within the scope of this Digest. Sowell (1989) found that long-term use of manipulatives was more effective than short-term use. Even so, when manipulatives are used over an extended period of time, teachers' training critically influences effectiveness. Gilbert and Bush (1988) examined the recognition, availability, and use of 11 manipulatives among primary teachers in 11 states. Results indicated that inexperienced teachers tended to use manipulatives more often than experienced teachers. (A possible explanation is that experienced teachers lack the training that more recent graduates have had.) Directed-inservice training with manipulatives, however, increases use among all teachers.

Availability is probably the most important factor affecting the use of manipulatives. Certainly, if manipulatives are unavailable, teachers cannot use them. Nonetheless, many manipulative materials--such as buttons and spools--can simply be collected by teachers. Others, such as beansticks and attribute blocks, are easy to make.

Consideration of these factors has led to the appropriate use of manipulatives at specific grade levels. The Middle Grades Mathematics Project (Lappan, Fitzgerald, Phillips, Shroyer, & Winter, 1986), for example, is an activity-based mathematics program in which such manipulatives as tiles, cubes, geoboards, dice, and counters are used. Here the students continually explore by building, drawing, and discussing various "challenge situations."

Manipulatives have, unfortunately, been implemented more slowly at the secondary level. As a result, research on their effectiveness at this level is minimal. One example, however, is Howden's (1986) application of tiles to help ease the transition to the abstract level in algebra. The tiles model the basic concepts of polynomials, from definitions to multiplying and factoring polynomials. The program's focus on connecting geometry to algebra allows students to apply previous knowledge to new topics. As they use the tiles, students are continually encouraged to draw pictures and to see mental images.