ERIC Identifier: ED321967
Publication Date: 1990-06-00
Author: Hartshorn, Robert - Boren, Sue
Source: ERIC Clearinghouse on Rural Education and Small Schools
Experiential Learning of Mathematics: Using Manipulatives.
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 MATHEMATICS
Both 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.
The closest thing to an integrated K-12 activity-based mathematics program
based on manipulatives has been developed by Mortensen. This five-year-old
program, Mortensen More Than Math (1988, Hayden Lake, Idaho), uses manipulatives
and workbooks to teach five strands: algebra, arithmetic, calculus, problem-solving,
and measurement (including both geometry and trigonometry). Some school
systems and states have approved all or parts of the program, but formal
research on its effects is not yet available.
NEW DIRECTIONS FOR PUTTING THEORY INTO PRACTICE
Two influences will probably affect the use of manipulatives in the
future. These influences are: (1) schools' efforts to conform to the Curriculum
and Evaluation Standards for School Mathematics (NCTM, 1989); and (2) the
commitment of state resources to transform theory into practice.
The Standards will doubtless be a major influence. They: (1) describe
competencies in broad areas, such as geometry, algebra, statistics, and
probability; and (2) specify instructional strategies for grades K-4, 5-8,
and 9-12. The Standards include a new approach for evaluating specified
competencies. The new approach involves greater use of open-ended questions,
descriptions of problem-solving processes, and drawings to represent geometric
situations. To help students attain the competencies, the Standards (pp.
17, 67-68) explicitly recommend a sample list of manipulatives for each
classroom in grades K-8. No specific list of manipulatives is recommended
for grades 9-12, unfortunately. Many students at this level will, however,
need to have ideas introduced at the concrete level.
Some state and local systems have mandated the implementation of manipulatives
through policy, law, or curriculum documents. Some have also provided various
levels of funding. One of the strongest philosophical positions has been
taken in California. In that state, the use of manipulative devices in
all elementary classrooms is recommended (Gilbert & Bush, 1988). The
state framework, approved in 1985, emphasizes student use, rather than
teacher-directed demonstrations. California, interestingly enough, examined
14 textbook series and compared them to the framework, for possible adoption.
All were turned down. One of the major reasons cited for this action was
the lack of concrete materials to introduce concepts and to provide reinforcement
until understanding occurs.
North Carolina has made a significant commitment to the use of manipulatives
in its schools. Manipulative kits for primary (K-3), intermediate (3-6),
and secondary (7-12) levels have been designed. State funding has provided
for at least one kit in each elementary school (Center for Research in
Mathematics and Science Education, 1988). Yet, in a survey of 964 elementary
teachers in North Carolina, only 11.2 percent reported frequent use of
the kits. Moreover, 9.5 percent were apparently unaware that the kits were
Tennessee's mathematics curriculum guides for grades K-8 state that
new concepts will be introduced through concrete experiences. To facilitate
this goal, the guides' objectives have been written in concrete, iconic,
and abstract terms. The secondary guides encourage, rather than mandate,
the use of manipulatives. The Mathematics Activity Manuals (Center of Excellence
for the Enrichment of Science and Mathematics Education, 1989) for grades
K-8, Algebra I and II, Unified Geometry, and for Selected Topics From Advanced
Mathematics Courses have been developed for use with the guides. They are
being field tested during 1989-1990. The activities for objectives at all
three levels include patterns for inexpensive non-commercial manipulatives.
Tennessee has provided no funding for manipulatives.
Chapter 75 of current Texas law states that new concepts should be introduced
with appropriate manipulatives at the elementary and secondary levels (Peavler,
DeValcourt, Montalto, & Hopkins, 1987). Students are to be actively
involved in structured activities that develop understanding and the ability
to apply skills. Evaluation of mastery at the concrete level is supposed
to include student demonstrations with manipulative materials. Minimum
manipulative kits are described and are to be made available in each classroom,
at each grade level K-8. Local districts, however, must provide funding.
AREAS FOR INCREASED ATTENTION
As the decade of the 1990s dawns, research on the effectiveness of manipulatives
in grades K-8 will doubtless continue. Forthcoming research should also
seek to study the use and effects of manipulatives at the secondary level.
Increasing the use of this experiential strategy at both levels will require
more states to make the type of financial commitment that those in the
forefront have already made. In that vein, studies should be carefully
designed to determine the impact of different funding formulas for effectively
implementing this teaching strategy across the grades.
Center for Research in Mathematics and Science Education. (1988). Keys
to the future. Raleigh, NC: Center for Research in Mathematics and Science
Center of Excellence for the Enrichment of Science and Mathematics Education.
(1989). Mathematics activity manuals (Draft Versions: K-8, Algebra I and
II, Unified Geometry, Selected Topics from Advanced Mathematics Courses).
Martin, TN: The University of Tennessee at Martin.
Gilbert, R., & Bush, W. (1988). Familiarity, availability, and use
of manipulative devices in mathematics at the primary level. School Science
and Mathematics, 88, 459-469.
Heddens, J. (1986). Bridging the gap between the concrete and the abstract.
Arithmetic Teacher, 33(6), 14-17.
Howden, H. (1986). The role of manipulatives in learning mathematics.
Insights into Open Education, 19(1), 1-11.
Lappan, G., Fitzgerald, W., Phillips, E., Shroyer, J., & Winter,
M. (1986). Middle grades mathematics project. Menlo Park, CA: Addison-Wesley.
National Council of Teachers of Mathematics. (1989). Curriculum and
evaluation standards for school mathematics. Reston, VA: National Council
of Teachers of Mathematics.
Peavler, C., DeValcourt, R., Montalto, B., Hopkins, B. (1987). The mathematics
program: An overview and explanation. Focus on Learning Problems in Mathematics,
Sowell, E. (1989). Effects of manipulative materials in mathematics
instruction. Journal for Research in Mathematics Education, 20, 498-505.
Suydam, M., & Higgins, J. (1977). Activity-based learning in elementary
school mathematics: Recommendations from research. Columbus, OH: ERIC Clearinghouse
for Science, Mathematics, and Environmental Education. (ERIC Document Reproduction
Service No. ED 144 840)