Feature Article from NUMEL NEWS

Feature Article from NUMEL NEWS (October 2000)

Manipulatives

The definition generally used for manipulatives is that they are any concrete objects that can be moved about and handled. They can be everyday objects or commercially prepared objects specifically designed to teach mathematical concepts. In this article I will share with you the department’s position on the use of manipulatives, some research on manipulative use, and specific recommendations for each grade level.

SED Position

The State Education Department has had a position on the use of manipulatives for many years. In the last years of the third and sixth grade PEP tests we provided a list of "allowable items" that students could be allowed to use when taking the tests. The new state mathematics assessments also allow the use of manipulatives. I would like to share with you our official position on manipulatives and their use by quoting from two publications on manipulatives published by the department.

"The State Education Department’s Bureau of Mathematics Education strongly recommends that, when ever appropriate, manipulative materials be used throughout the mathematics education program, K-12, to develop understanding, concepts, and skills, to improve student attitudes and interest, and to assist with necessary remediation.

Following are some Postulates of Manipulatives:

Manipulatives help students understand mathematics.
Manipulatives are "objects" that appeal to several senses and that can be touched, moved about, rearranged and otherwise handled by children.
Manipulatives are "objects" that represent mathematical ideas that can be abstracted through physical involvement with the objects.
Research supports the use of manipulatives at all school levels.
Manipulatives are an integral part of instruction; not a supplement to it.
Manipulatives result in improvement in motivation, involvement, understanding, and achievement.
Manipulative materials promote problem-solving and computational skills.
A proper use of manipulatives may remove the need for later remediation.
Manipulatives selected for use should be appropriate for students’ developmental level and learning style.
Selecting manipulatives for use in mathematics instruction is an important responsibility of teachers."

-from Using Manipulatives to Teach Mathematics, K-3 printed by NYSED in the late 1980’s.

A quote from a later NYSED publication, M ⁶= Manipulative Materials Make Math More Meaningful published in the early 1990’s follows:

"Strong evidence supports the claim that the use of manipulative materials generates improvements in motivation, involvement, and achievement while helping children to understand mathematical concepts. Also, learning theories suggest and research supports the fact that children whose mathematical learning is firmly grounded in manipulative experiences will be more likely to bridge the gap between the world in which they live and the abstract world of mathematics.

Such beliefs, though not always practiced, have long been the prominent focus of primary mathematics programs. Also manipulative materials have long been used successfully with students with handicapping conditions….

Children demonstrate their need for the use of manipulative materials when they count on their fingers. Instead of recognizing this need, an adult often responds to the child with a scolding!

When choosing which objects to use as manipulative materials, there are certain criteria to consider. Some important considerations are:

Cost

Accessibility

Adaptability or versatility

Ease of use and storage

Durability

Attractiveness

…

Using manipulative materials in math class need not be at all expensive. …In most cases, …easily obtainable ‘odds and ends’ serve adequately. Ordinarily, these are either free or can be ‘scrounged’. They include such things as egg cartons, beans, pasta, containers of all kinds, bottle caps, buttons, shells, Popsicle sticks, Q-tips, clocks. The list is endless. So are their uses.

Various games of a mathematical nature can also be classified as manipulative materials. Games are also an excellent, as well as enjoyable, medium for developing logical thinking and providing practice in arithmetic operations.

One more important fact about the use of manipulative materials needs to be stated. That is, the use of manipulatives in and of themselves can be practically useless. Because the use of concrete materials very often forces students to obtain a correct answer, it is important that the representation of the same concept be expressed in several different ways-with manipulatives, pictorially, and symbolically, so that a student can see the manipulative/abstract connection.

MANIPULATIVE® PICTURES® OTHER SYMBOLS® ABSTRACT

The chart below gives a list of easily obtainable materials that NYSED approved for use with the third and sixth grade PEP tests. Also included are examples of how each could be used. These are not allowed on the new state mathematics assessments however they are provided here to give you ideas of objects that are possible for even the highest need districts to make available to their students.

Item

Suggested Material

Examples of Use

Counters

Buttons, bottle caps, beans, chips, Popsicle sticks with rubber bands, pasta, cubes, interlocking cubes, fraction bars

	To assist and validate computation
	To represent place value and grouping
	To compare numbers for probability investigations
	To represent fractions (part of a group)

Graph paper

Centimeter, inch, dot

	To show base ten grouping
	To show fractional relationships
	To explore geometric relationships
	To assist and validate computation
	To make and use hundreds charts
	To construct number lines

Paper

Lined, blank, construction, greeting cards

	To make illustrations
	To construct geometric shapes
	To make counters, play money, fractional representations, etc.
	To make tangrams

Scissors

Left handed, right handed

	To construct geometric models
	To construct fraction pieces
	To make counters, play money paper models of rulers, protractors

Crayons/colored markers

6 or 8 basic colors

	To construct models
	To color illustrations
	To represent fractional parts
	To assist in probability investigations

Straightedge

Ruler, cardboard/tagboard strips

	To compare numbers
	To make number lines
	To construct geometric shapes
	To measure

Protractor and/or compass

Paper model, commercial compass

To construct and measure geometric figures

Money

Paper models, commercial, real

	To assist with computation and problem solving
	To compare amounts of money

Clock

Paper model, commercial, real

To assist with computation and problem solving

String

Twine, yard, thread

	To measure
	To construct geometric shapes
	To group objects
	To make Venn diagrams

Cube

Paper/tagboard model, commercial, containers, children’s’ blocks

To examine surfaces, edges, vertices, spatial relations

Round paper plate

Commercial

	To examine circle graphs
	To assist in probability investigations
	To make models for fractional parts and operation algorithms
	To assist with geometry investigations.

Research Reference

Evelyn Sowell conducted a meta- analysis of the results of 60 studies on the effectiveness of mathematics instruction with manipulative materials (1989). The students ranged in age from kindergartners to college age and a variety of mathematics topics were included. Results showed that

	mathematics achievement is increased through long-term use of concrete instructional materials. (Long term means a school year or longer.)
	students’ attitudes toward mathematics are improved when they have instruction with concrete materials provided by teachers knowledgeable about their use.
	Instruction with pictures and diagrams did not appear to differ in effectiveness from instruction with symbols. (The category of pictorial includes animated audiovisual presentations, observed demonstrations with concrete materials by the teacher, or pictures in printed materials.)

In the following sections we will take a look at manipulative materials that can be used at different levels of

education. We will point out some specific research studies and some materials that are appropriate to the content and developmental levels of the students with suggestions of how many of them to get for a standard sized class.

Elementary (Pre K-4)

Commercially available Manipulative Materials

The commercially available manipulatives in the chart below are basic to a developmentally based mathematics program. The quantities listed are for a classroom of 20 students. These quantities give the teacher and students ready access to materials for whole class activities as well as allow for the establishment of learning stations so that students may use the materials as they proceed at their own pace. This is only a sample list and not intended to include all possibilities.

*Grade*	*Manipulative*	*Amount*	*Concepts*
Kindergarten	Pattern blocks	5 tubs	Patterns, one-to-one correspondence, sorting, classification, size, shape, color, spatial visualization
	Unifix cubes	1000 cubes	Number concepts, counting, classification, sorting colors, patterns, spatial visualization.
	Balance beam	1	Greater, less than and equal to
	Balance scale	1	Weight, mass, equality, inequality, measurement, estimation
	Color cubes	2 sets	Number concepts, counting, classification, sorting, colors, patterns, graphs
	Building materials such as Legos, Lincoln Logs	1 set	Spatial visualization, estimation
Grade 1	Pattern blocks	5 tubs	As in kindergarten plus geometric relationships, problem solving, logical reasoning, symmetry
	Unifix cubes	1000 cubes	As for kindergarten plus fact strategies, equality, inequalities, operations on whole numbers, even and odd number, graphing (bar graphs), measure length
	Geoboards	20	Size, shape, counting, estimation, spatial visualization, logical reasoning, symmetry
	Balance beam	1	As for kindergarten plus operations on whole numbers, open sentences, equations, fact strategies, measurement, logical reasoning
	Balance scale	1	Same as for kindergarten
	Color cubes	3 sets	Same as for kindergarten plus fact strategies, equality, inequalities, compare lengths or amounts, symmetry, probability
	Attribute blocks	6-8 sets	Sorting, classification, investigations of size, shape, color, logical reasoning, sequencing, patterns, symmetry, similarity, thinking skills, geometry
	Place value models (bean cards, coffee stirrers, digi-blocks	Classroom set	Addition and subtraction facts, place value
	Building materials	1 set	Spatial visualization, estimation
Grade 2	Pattern blocks	5 tubs	Same as grade 1 plus fraction concepts
	Multi-link cubes	1000 cubes	Same as Unifix cubes plus introduction to averages, commutative and associative properties, fractions, measure length
	Tangrams	Bag of 20 sets	Geometric concepts, spatial visualization, logical reasoning, fraction concepts, classification, sorting, patterns
	Geoboards	20	Same as grade 1
	Balance beam	1	Same as Grade 1
	Balance scale	1 with metric weight set	Same as grade 1
	Color tiles	3 sets	Color, shape, patterns, estimation, counting, number concepts, equality, inequality, operations on whole numbers, probability, measurement, even and odd numbers, probability, spatial visualization
	Attribute blocks	20 sets	Same as Grade 1 plus organization of data
Grade 3	Pattern blocks	5 tubs	Same as Grade 2 plus ratio, geometry, tessellation
	Multi link cubes	1000 cubes	Same as Grade 2 plus plane and solid geometry, perimeter, area and volume, prime numbers, composite numbers, square numbers
	Tangrams	Bag of 20 sets	Same as Grade 2
	Geoboards	20	Same as Grade 2 plus area, perimeter, circumference, symmetry, coordinate geometry, square numbers, polygons,
	Balance Beam	1	Same as Grade 2 plus equality, inequality, equations, multiplication and division facts, open sentences, relationship between addition and subtraction and between multiplication and division.
	Balance scale	1 with metric weight set	Same as Grade 2 plus equations, multiplication of whole numbers estimation
	Color tiles	3 sets	Same as Grade 2 plus area, perimeter, prime & composite numbers, ratio, percent, integers, square numbers.
	Attribute blocks	20 sets	Same as Grade 2 plus sequencing
	Base-ten blocks	2 sets with lines	Place value, operations on whole numbers, decimals, comparing, ordering, classification, sorting, number concepts of square numbers, area, perimeter, metric measurement.
Grade 4	Pattern blocks	5 tubs	Same as grade 3 plus similarity, congruence, angles
	Tangrams	25 sets	Same as Grade 3 plus congruence, angles
	Geoboards	25	Same as Grade 3 plus angles, circle concepts
	Color Tiles	3 sets	Same as Grade 3 relationship between multiplication and division
	Attribute blocks	1 set for every 2 students	Same as Grade 3 plus similarity, congruence
	Base-ten blocks	2 intermediate classroom sets	Same as Grade 3 plus decimal-fraction-percent equivalencies, percent
	Cuisenaire rods	12 trays of 74 rods	Classification, sorting, ordering, counting, number concepts, comparisons, fractions, ratio, place value, patterns, even & odd numbers, prime & composite numbers, logical reasoning, estimation, operations on whole numbers
	Balance beam	10 with metric weight sets	Same as Grade 3
	Metric beaker set for volume	1 set	Measurement, capacity, volume, estimation

Manipulatives provided with the Elementary Mathematics Assessment are: Square Counters, Pattern Blocks, Ruler with both customary and metric measurements. Students are only assessed with the metric measure. These manipulatives are provided as tools for students to use to answer questions and solve problems. It is essential that they have experiences with these manipulatives in order to be able to utilize them in the testing situation.

Intermediate (Grades 5-8)

Commercially Available Manipulative Materials

The quantities listed are for a classroom of 25 students. At the intermediate grades it would be ideal to have all the manipulative materials available in each classroom. If this is not possible, an adequate supply, as listed below, could be shared at each grade level. At these grade levels, it would also be beneficial to have a classroom set of Fraction Bars or the Fraction Factory, as well as a classroom set of Decimal Squares. This is only a sample list and not intended to include all possibilities.

Grades	Manipulative	Amount	Concepts
Grades 5 & 6	Pattern blocks	5 tubs	Patterns, sorting, classification, geometric relationships, symmetry, similarity, congruence, area, perimeter, reflections, rotations, translations, problem solving, logical reasoning, fraction operations, spatial visualization, tessellation, angles, ratio, proportion,
	Tangrams	25 sets	Geometric concepts, spatial visualization, logical reasoning, fractions, similarity, congruence, area, perimeter, ratio, proportion, angles, classification, sorting, patterns, symmetry, reflections, translations, rotations
	Geoboards	25	Area, perimeter circumference, circle concepts, symmetry, fractions, coordinate geometry, angles, estimation, percent, similarity, congruence, rotations, reflections, translations, classification, sorting, square numbers, polygons, spatial visualization, logical reasoning
	Pentominoes	25 sets	Logical reasoning, spatial visualization, reflections, translations, rotations
	Color Tiles	3 sets	Patterns, estimation, fraction operations, probability, area, perimeter, surface area, even & odd numbers, prime & composite numbers, ratio, proportion, percent, integers, square numbers, spatial visualization
	Base Ten Blocks	2 intermediate classroom sets	Decimals, decimal-fractional-percent equivalencies, comparing, ordering, number concepts, square and cubic numbers, area, perimeter, metric measurement, volume
	Cuisenaire Rods	12 trays of 74 rods	Number concepts, comparisons, fractions, ratio, proportion, patterns, even & odd numbers, prime & composite numbers, logical reasoning, estimation, operations on whole numbers, percent
	Balance scales	1 for every 3 students with metric and customary weight sets	Weight, mass, equality, inequality, equations, estimation, measurement
	Graduated cylinders and beakers	Sets for each 3 students	Measurement, capacity, volume, estimation, proportion
Grades 7 & 8	Geoboards	25 sets	Same as Grades 5 & 6 plus Pythagorean Theorem
	Color tiles	3 sets	Integers, fractions, probability, surface area, prime & composite numbers, ratio, proportion, percent, square numbers, combinations, algebra
	Algebra tiles	1 set for every 3 students	Integers, equations, inequalities, polynomials, similar terms, estimation
	Compasses	1 per student (Triman compasses or safe compasses)	Constructions, angle measurement
	Decimals Squares	1 per student	Decimals (place value, comparing, ordering, operations) percent,
	Geometric solids	1 set	Shape, size, relationship between area & volume, volume, classification, sorting, measurement, spatial visualization
	Miras	1 per student	Symmetry, similarity, congruence, reflections, rotations translations, angles, parallel & perpendicular lines, constructions
	Polyhedra models	1 set	Shape, classification, sorting, polyhedra, spatial visualization, probability
	Spinners	Various sets	Fractions, mental math, probability, generation of problems
	Thermometers	1 for every 3 students	Temperature, integers, measurement
	Two-color counters	20 per student	Number concepts, fractions, integer operations, probability, proportion
	Protractors	1 per student	Construction, angle measurement

Manipulatives provided with the Intermediate Math Assessment are: Protractor and Ruler with both customary and metric measurements. Students could be assessed with either the metric or customary measure. These manipulatives are provided as tools for students to use to answer questions and solve problems. It is essential that they have experiences with these manipulatives in order to be able to utilize them in the testing situation.

High School (Math A/B)

Commercially Available Manipulatives

It would be ideal to have all the manipulative materials available in each classroom. If this is not possible, an adequate supply, as listed below, could be shared at each grade level.This is only a sample list and not intended to include all possibilities.

Assessment	Manipulative	Amount	Concepts
Math A	Algebra tiles	1 set per student	Integers, equations, inequalities, polynomials, similar terms, factoring, estimation
	Dice, spinners	1 per student	probability
	Geometric models	3 sets	Study of solids, volume
	Tessellation tiles	Sets for each student	transformations
	Mirror or miras	1 per student	Transformations, symmetry
	Geoboards	1 per student	Area, perimeter, circumference, circle concepts, symmetry, fractions, coordinate geometry, slopes, angles, Pythagorean Theorem, estimation, percent, similarity, congruence, rotations, reflections, translations, polygons
	Conic section models	1 set	Circles and parabolas
	Volume demonstration kits	1 set	Volume, area, solids
	Compass/ruler	1 per student	Circles, constructions
Math B	Algebra tiles	1 set per student	Same as Math A
	Geoboards	1 per student	Same as Math A
	Tessellation tiles	Set for each student	Same as Math A
	Compass/ruler	1 per student	Same as Math A
	Conic section models	1 set	Same as Math A plus hyperbola, ellipse

For both Math A and Math B students should have available a straight edge and compass. A triman compass combines both of these tools.

Some vendors of Manipulatives

	Creative Publications, Customer Service, 5623 W. 115^th Street, Alsip, IL 60803
	Cuisenaire/Dale Seymour, P.O. Box 5026, White Plains, NY 10602-5026
	Delta Education, P.O. Box 3000, Nashua, NH 03061-300
	Didax, 395 Main Street, Rowley, MA 01969-1207
	ETA, 620 Lakeview Parkway, Vernon Hills, IL 60061-9923
	Spectrum Educational Supplies, 7711 Welborn Street, Suite 111, Raleigh, NC 27615
	Trican Publishing, 6400 S. Crawford Rd, MT. Pleasant, MI 48858

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This page was last updated on 09/03/01
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