Math Manipulatives
by Louise Johnson1
Students presently in the primary grades will spend most of their
lives in the twenty-first century. New research findings as to how
child learn, as well as technology such as calculators and
computers, have made the typical mathematics curriculum studied
by their parents outdated.
We cannot be certain what mathematics today’s children will need
in order to function effectively in their futures and to make the
choices necessary for them to follow God’s plans for their lives.
However, to prepare them for a future that will likely make
greater demands on their mathematical knowledge, changes will
be needed in how they are taught mathematics as well as in what
mathematics they are taught.
Children must have the opportunity to gain the knowledge and
attitudes needed to become lifelong learners of mathematics. To
achieve this goal, educators are concerned that more time be
spent on understanding math concepts and less on how to do paper-and-pencil computations. The implementation
of this trend is beginning in the early elementary grades.
Base 10 block
Why Use Manipulatives?
Top math authorities agree that young children learn to understand math concepts best with the use of
manipulatives. The Curriculum and Evaluation Standards for School Mathematics, published by the National Council
for Teachers of Mathematics (U.S.A.) in 1989, is considered a state-of-the-art document by other disciplines
responding to the trend toward national standards. One specific assumption underlying the standards for grades K4 is that children should be actively involved in doing mathematics.
Educational research indicates that the most valuable learning occurs when students actively construct their own
mathematical understanding. One way to facilitate this is to provide opportunities for children to explore, develop,
test, discuss, and apply ideas. Extensive and thoughtful use of physical materials, particularly in the primary
grades, is conducive to the concrete kinds of learning that lay a satisfactory foundation for the development of this
mathematical understanding.
Although the value of manipulatives has been recognized for many years, some teachers and parents have been
reluctant to include them in their lessons. They may lack confidence in their ability to use them successfully or
underestimate their value to real mathematical understanding.
The following suggestions can be valuable in promoting the success of
a lesson that involves manipulatives.
Manipulatives in Support of Objectives
First, choose manipulatives to support the lesson’s objectives. Have the learning
objective clearly in mind before choosing the activity and the manipulatives that will best
help the student attain the desired objective. Some of the manipulatives mentioned later in
this article may not be familiar to you, and space does not permit detailed descriptions.
However, some teacher’s manuals for recent textbooks contain good descriptions and have
detailed directions on how the selected manipulatives are meant to be used in the
lessons.
Pattern blocks
For example, Real Math (by Open Court) [now SRA] and The Math Experience (by
Houghton Mifflin, 1992) [now Houghton Mifflin Math] are programs that emphasize
appropriate use of manipulatives. Choose a text and use the manipulatives suggested in
the teacher’s manual.
Introducing Manipulatives
Second, provide opportunity to become acquainted with manipulatives before they are used in the
lesson. This initial free play is important, and certain basic concepts can often be gained from this exploration.
Children should not be expected, however, to discover the lesson objectives without a specific plan designed to use
the manipulatives.
Learning Styles
Third, consider the child(ren) when planning the lesson and choosing the appropriate manipulatives.
Not all children learn in the same way or react to physical materials in the same way. Researchers have found that
children frequently benefit from extensive exploration of numerical relationships with various manipulatives. Some
children, however, feel more secure and are helped most when only one or two kinds are used. The manipulatives
that are effective for one child may not be effective for another. The amount of time required for a child to go from
a concrete representation to a more abstract idea also varies from one child to another.
Questions and Discussion
Fourth, encourage the child(ren) to talk and write about what they have learned. It is important that
each child be allowed to explain in his or her own words what he or she has discovered. Asking appropriate
questions frequently helps a child learn to communicate effectively. Most children benefit from discussing their
work; it helps them to clarify, perfect, and organize their thinking. It also provides information to the teacher on
the child’s mathematical progress.
From Manipulatives to Paper-and-Pencil
Finally, help the child(ren) to move from manipulatives to paper-and-pencil operations. We have found
that a child may know how to write a correct answer to an addition problem on paper without understanding
addition. The reverse is also true. Although the child may have an understanding of the mathematical concept
when manipulating physical objects, he or she may have to be shown how to express it in the desired way on
paper.
Examples of Manipulatives
Many good manipulatives could be suggested. Some commonly used ones are mentioned here, along with sample
tasks that children can be asked to perform and questions they can answer that will help them develop the
mathematical concept.
Manipulatives such as buttons, dried beans, cubes, and animal counters can help young people develop
a rich understanding of “number” which forms the basis for counting, arithmetic, and real-world
applications. They can be used for comparing as well as sorting and classifying. For example, children can use
any of these objects to make, count, and order collections of objects. (E.g. Give me four buttons. How many beans
are there? Put these groups in order from smallest to largest.)
Sets of attribute logic blocks with their different shapes and colors are particularly effective for
classifying, sorting, and ordering. (E.g. Find all the pieces that belong in the set labeled “small and square.”
Look at the three shapes in the loop. How are they alike? How are they different?) Attribute blocks can also be
used to explore size, fractional relationships, and area.
Base ten blocks, Cuisenaire® rods, Unifix® cubes, and currency can be used to help children learn a
wide range of number concepts, including place value, addition, subtraction, and multiplication. With
the base ten block manipulatives, cubes represent ones, rods represent tens, flats represent hundreds, and blocks
represent thousands; so a combination of manipulatives can be used to represent “twenty-five” or “two groups of
ten and five ones,” etc. (E.g. Can you trade ten cubes for a rod? Two rods and five cubes is the same as one rod
and how many cubes?)
By building number combinations with these materials, children are helped to understand the logic of
the concept of carrying or borrowing or regrouping as used in the paper-and-pencil computations for
addition and subtraction.
The cost of manipulatives varies. Common items such as beans, buttons, and money can be just as effective as the
more expensive, purchased items. With adequate time, children can make some items, and much learning can take
place in the process. For instance, a child could make base-ten manipulatives by gluing beans on cardboard.
Some texts include cardboard manipulatives in the back of the student books (K-2). If you choose these texts, look
at what is included before you purchase additional items.
Reprinted from the May 1993 issue of Parents Teaching Overseas. Permission to copy, but not for commercial use.
1
Dr. Louise Johnson, Ed.D., is a retired mathematics educator and former university dean.
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Web site for Math Munipulatives
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