Using Manipulatives to Teach Elementary
Mathematics
INTRODUCTION
According to the Principles and Standards for School
Mathematics, “the foundation for children’s mathematical development is
established in the early years” (Seefeldt & Wasik, 2006,p. 249). It is
important for children to have a variety of materials to manipulate and the
opportunity to sort, classify, weigh, stack and explore if they are to
construct mathematical knowledge. “In order to have opportunities to learn
math, children need firsthand experiences related to math, interaction with
other children and adults concerning these experiences and time to reflect on
the experiences” (Seefeldt & Wasik, 2006, p. 250). Educational research
indicated that the most valuable learning occurs when students actively
construct their own mathematicalunderstanding, which is often accomplished
through the use of manipulatives.
HISTORY OF MANIPULATIVES
Since ancient times, people of several different
civilizations have used physical objects to help them solve everyday math
problems. The ancient civilizations of Southwest Asia
used counting boards, which were wooden or clay trays covered in a thin layer
of sand. The counting board users would draw symbols in the sand to tally
inventory or whatever else they may need to count. The ancient Romans created
the first abacus based on counting board. The abacus wasmade of beans or stones
which moved in grooves in sand or on tables of wood, stone, or metal.“The
Chinese abacus, which came into use centuries later, may have been an
adaptation of theRoman abacus” (“Research on the” n.d.). The Mayans and the
Aztecs both had counting devices that were made of corn kernels strung on
string or wires that were stretched across a wooden frame. The Incas also had
their own counting tool, which was knotted strings called quipu (“Research on
the”, n.d.).“The late 1800s saw the invention of the first true
manipulative-maneuverable objects that appeal to several different senses and
are specifically designed for teaching mathematical concepts” (“Research on
the” n.d.). In 1837, German educator Friedrich Froebel introduced the world’s
first kindergarten. “He designed the educational play materials known as
Froebel Gifts, or Frobelgaben, which included
geometric building blocks and pattern activity blocks” (“Friedrich Froebel”,
2009). Then in the early 1900s, Italian educator Maria Montessori
continued with the idea that manipulatives are
important to education. She designed several materials to help elementary
students learn the basic ideas of math. “Since the 1900s, manipulatives have
come to be considered essential in teaching mathematics at the elementary
school level” (“Research on the,” n.d.). In fact, the National Council of
Teachers of Mathematics (NCTM) has
recommended the use of manipulatives in teaching
mathematical concepts at all grade levels.
MANIPULATIVES DEFINED
Manipulatives can come in a variety of forms and they
are often defined as “physical objects that are used as teaching tools to
engage students in the hands-on learning of mathematics” (“Using
manipulatives,” 2009). Manipulatives can be purchased at a store, brought from
home, or teacher and student made. The manipulatives can range from dried beans
and bottle caps to Unifix cubes and base-ten blocks. They are used to
introduce, practice, or remediate a math concept. “A good manipulative bridges
the gap between informal math and Journal of Instructional Pedagogies Using
manipulatives to teach, Page 3 formal math. To accomplish this objective, the
manipulative must fit the developmental level of the child” (Smith, 2009, p.
20). Kindergarten children should have individual counters, whereasolder
students could use colored wooden rods that represent different numbers. The
manipulative must fit the mathematical ability of the child or it is useless.
WAYS TO USE MANIPULATIVES
Manipulatives can be used in teaching a wide variety
of topics in mathematics, including the objectives of: problem solving,
communicating, reasoning,
connections, and estimation. The materials should “foster
children’s concepts of numbers and operations, patterns, geometry, measurement,
data analysis, problem solving, reasoning, connections, and representations”
(Seefeldt & Wasik, 2006, p.93). Teachers could use counters, place-value
mats, base-ten blocks, and fraction strips while teaching from the numbers and
operations standard. The counters could be used to teach one-on-one
correspondence, ordinal numbers, and basic addition and subtraction. The
fraction strips could be used to add and subtract fractions or to show
equivalent fractions. Pattern blocks, attribute blocks and scales could be used
to assist students in the learning basic algebra. Student could use geoboards
when trying to identify simple geometric shapes. They could also use geometric solid
models when
learning about spatial reasoning. Teachers could use
standard and non-standard rulers and measuring cups to represent length or
volume in measurement lessons. The students could also use tiles when trying to
find the area or perimeter of an object. When it comes to data analysis and
probability, students could use spinners to find the probability of landing on
a designated area. They could also use number cases or dice to find the
probability of rolling a certain
number or combination of numbers (“Using
manipulatives”, 2009). The numbers of ways that manipulatives can be used are
limitless. In fact, some schools use math manipulatives as a way to get parents
involved. Stephen Currie, math specialist for grades Kindergarten through
fourth grade at Poughkeespsie
Day School in New York, created
‘mathtubs’ to pique math interest for both kids and their parents. Each Friday
several students are selected to receive a mathtub,
which are not due back until the next Wednesday. The
mathtubs are filled with “math games and puzzles, two or more different kinds
of manipulatives such as number cubes or tangrams and math challenges—questions
which required no materials but creative brain power” (Currie, 2005, p. 52).
Feedback from the parents was both positive and helpful. “In general, the
parents appreciated the activities and were please to see their child engaged
in mathematical thinking”
(Currie, 2005, p. 53).
USING MANIPULATIVES CORRECTLY
Manipulatives can be extremely helpful young children,
but they must be used correctly.
Children must understand the mathematical concept
being taught rather than simply moving the
manipulatives around. Smith (2009) stated that there
are probably as many wrong ways to teach
with manipulatives as there are to teach without them.
The math manipulatives should be
appropriate for the students and chosen to meet the
specific goals and objectives of the
mathematical program. “The complexity of the materials
provided will increase as children’s
thinking and understanding of mathematical concepts
increase” (Seefeldt & Wasik, 2006, p. 93).
It is also important for teachers to allow their
students to have free time to play with the
manipulatives. After the students have explored the
manipulatives, “the materials cease to be
Journal of Instructional Pedagogies Using
manipulatives to teach, Page 4 toys and assume their rightful place in the
curriculum” (Smith, 2009, p.17). Carol Seefeldt and Barbara Wasik also think
that teachers should provide children with opportunities to work with materials
with open-ended objectives that have no specific preset goals. These
opportunities allow the children the chance to explore their own questions and
generate a variety of answers. “These experiences help children think about
their world in alternative ways and help them understand that there are
multiple ways to solve problems. Generating multiple solutions to problems in
an essential strategy in mathematics” (Seefeldt & Wasik, 2006, p. 250).
RESEARCH AND BENEFITS OF MANIPULATIVES
The use of manipulatives is recommended by because it
is supported by both learning theory and educational research in the classroom.
“Manipulatives help students learn by allowing them to move from concrete
experiences to abstract reasoning” (“Research on the”
n.d.). When students manipulate objects, they are
taking the first steps toward understanding
math processes and procedures. “The effective use of
manipulatives can help students connect
ideas and integrate their knowledge so that they gain
a deep understanding of mathematical
concepts” (“Research on the, “ n.d.). Over the past
few decades, researchers have studied the use of manipulatives in several
different grade levels and in several different countries. The majority of the
studies indicate that mathematics achievement increases when manipulatives are
put to good use. Many studies also suggest that manipulatives improve
children’s long-term and short-term retention of math. Cain-Caston’s (1996)
research indicates that using manipulatives helps improve the environment in
math classrooms. When students work with manipulatives and then are given a
chance to reflect on their experiences, not only is mathematical learning
enhanced, but math anxiety is also greatly reduced. Kenneth Chang (2008)
examined the work of research scientist Jennifer Kaminski and he found that
children better understand math when they use concrete examples. Puchner,
Taylor, O’Donnell, and Fick (2008) conducted a case study which analyzed the
use of manipulatives in math lessons developed and taught by four groups of
elementary teachers. There four researchers decided to study the way teachers
use the manipulatives rather than studying the outcomes of the students. “The
study found that in three of four lessons studied manipulative use was turned into
an end in and of itself rather than a tool, and that in the fourth lesson
manipulative use hindered rather than helped the student learning” (Puchner,
Taylor, O’Donnell, & Fick, 2008, n.p.). The researchers believe this
occurred because of the “deeply embedded focus in U.S. mathematics teaching on the
procedure and the product” (2008, n.p.). In a few of the lessons, the
manipulative use became an exercise separated from the solving of the problem.
In the second grade lesson, the students simply copied the teacher’s example
and never attached meaning to the manipulatives. The teacher’s manipulative use
and misuse provided the researchers with a focus for further study. The
researchers also realized that “teachers need support making decisions
regarding manipulative use, including when and how to use manipulatives to help
them and their students think about mathematical ideas more closely” (Puchner,
Taylor, O’Donnell, & Fick, 2008, n.p.). Catherine Kelly, a member of the
Montana Council of Teachers of Mathematics, stated that “teachers need to know
when, why, and how to use manipulatives effectively in the classroom as well as
opportunities to observe, first-hand, the impact of allowing learning through
exploration with concrete objects” (Kelly, 2006, p.188). Dave Munger, author of
Researching Online, reported the results of a study
designed to describe the benefits of manipulatives. The sample consisted to two
third-grade classes with Journal of Instructional Pedagogies Using
manipulatives to teach, Page 5 twenty-six students. A two-week geometry unit
from the Silver Burdett textbook was administered in both classes. The
experimental group teacher used mathematical manipulatives to teach the
concepts presented in the unit, and the control group teacher used only
drawings and diagrams to teach the concepts. “Analysis of covariance revealed
that the experimental group using mathematical manipulatives scored
significantly higher in mathematical achievement on the posttest scores than
the control group” (Munger, 2007, n.p.). Additional studies have shown that
students who use “manipulatives in specific mathematical subjects are more
likely to achieve success than students who don’t have the opportunity to work
with manipulatives” (“Research on the,” n.d.). Some children need to use
manipulatives to learn to count, while other students’
understanding of place value increases with the use of manipulatives. Research
also indicates that using manipulatives is especially useful for teaching
low-achievers, students with learning disabilities, and English language
learners.
CONCLUSION
Elementary teachers who use manipulatives to help
teach math can positively affect student learning. Students at all levels and
of all abilities can benefit from manipulatives. Mathematician, Seymour Papert,
believes manipulatives are ‘objects to think with’. “Incorporating
manipulatives into mathematics lessons in meaningful ways helps students grasp
concepts with greater ease, making teaching most effective” (“Research on the,
“ n.d.).
REFERENCES
Cain-Caston, M. (1996). Manipulative queen [Electronic
version]. Journal of Instructional
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Chang, K. (2008, April 25). Study suggests math
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York
Times. Retrieved December 10, 2009, from
http: //www.nytimes.com/2008/04/25/science/25math.html
Currie, S. (2005). The mathtubs are coming! Teaching
PreK-8 35(4), 52-53.
Friedrich Frobel (2009, March 28). In Wikipedia, the
free encyclopedia. Retrieved December
9, 2009, from http://en.wikipedia.org/wiki/Friedrich_Fr%C3%B6bel
Kelly,
C.A.
(2006). Using manipulatives in mathematical problem solving: A performance
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Montana
Mathematics Enthusiast 3(2), 184-
193.
Munger, D. (2007, October 9). Children learn and
retain math better using manipulatives
[Msg.1]. Message posted to
http://scienceblogs.com/cognitivedaily/2007/10/children_learn_and_retain_math.php
Puchner, L., Taylor A., O’Donnell, B., & Fick, K.
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manipulatives: A collective case study about teacher
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