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as technology such as calculators and computers, have made the typical mathematics curriculum studied by their parents outdated.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.
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 K-4 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.
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.
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.
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.
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.
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