manipulatives: when are they useful?

Manipulatives: when are they useful?

Constance Kamii*, Barbara A. Lewis, Lynn Kirkland

School of Education, University of Alabama at Birmingham, Birmingham, AL 35294-1250, USA

Abstract

This article examines the usefulness of manipulatives in light of Piaget’s theory of how children

acquire logicomathematical knowledge. It argues that since children construct logicomathematical

knowledge through their own thinking, manipulatives are desirable when they encourage children

to think (i.e., to make relationships through constructive abstraction) in problem solving. A specific

object can therefore be beneficial if used in certain ways but not in others. The same object can

also be useful at a certain time in the child’s development but not at others. We conclude by

pointing out that mathematical relationships do not exist in objects and that children do not acquire

these relationships through empirical abstraction from objects. D 2001 Elsevier Science Inc. All

rights reserved.

Keywords:Manipulatives; Manipulatives and thinking; The use of fingers; The use of counters; The value of base-

ten blocks; The value of Unifix Cubes

1. Introduction

To counteract the traditional practice of symbol manipulation by rote, mathematics

educators strongly recommended the use of manipulatives. However, we have not heard

much about how children learn from or with manipulatives, and teachers have simply

assumed that children learn abstract concepts by touching and moving these objects.

The purpose of this article is to examine which manipulatives are good to use, how they

are best used, and why. We do this by arguing that manipulatives are useful to the extent that

they encourage children to think in problem solving. To explain why children’s thinking is

0732-3123/01/$ – see front matter D 2001 Elsevier Science Inc. All rights reserved.

PII: S0732 -3123 (01 )00059 -1

* Corresponding author.

E-mail address: [email protected] (C. Kamii).

Journal of Mathematical Behavior

20 (2001) 21–31

important, it is necessary to review the distinction Piaget (1971) made between physical and

logicomathematical knowledge and between empirical and constructive abstraction.

2. Physical and logicomathematical knowledge

Physical knowledge is knowledge of objects in the external world. Examples of physical

knowledge are our knowledge of the weight and color of an object and of the fact that a ball

usually comes to rest after rolling. This kind of knowledge has its source partly in objects in

external reality, and children acquire physical knowledge empirically by observation. (Our

reason for saying ‘‘partly’’ will be explained shortly.)

Logicomathematical knowledge has a very different source. It consists of mental relation-

ships, which each child creates from within. For example, we can look at a red chip and a blue

one and think that they are different or similar. ‘‘Different’’ and ‘‘similar’’ are relationships

we create between the chips. The color of each chip is physical knowledge, but the difference

between them does not exist in the external world. The chips become different only when we

think about them as being different. If we think about them as being similar, they become

similar. Relationships are not empirical knowledge because they originate in each person’s

mind. A third example of a mental relationship we can create between the two chips is two.

Each chip is observable with our eyes, but the number two is not.

It is thus possible to say theoretically that physical knowledge has a source in objects and

that the source of logicomathematical knowledge is in each child’s mind. However, in the

psychological reality of young children, the two kinds of knowledge exist inseparably. For

example, physical knowledge plays an important role in the child’s construction of number

because if objects behaved like two drops of water that combine to become one, it would be

impossible for the child to construct the relationship two. Conversely, logicomathematical

knowledge is essential for the children’s construction of physical knowledge because it would

be impossible to know that a red chip is a red chip if we could not make categories such as

‘‘colors’’ and ‘‘chips.’’

Physical and logicomathematical knowledge are thus impossible to separate in early

childhood, but logicomathematical knowledge progressively becomes independent. For

example, when we say ‘‘four apples,’’ we are still talking about apples (physical knowledge).

By the time we get to ‘‘4 + 4,’’ ‘‘5� 4,’’ and ‘‘5 + 4x,’’ however, this logicomathematical

knowledge is independent of physical knowledge.

Piaget (1971) also made a distinction between two types of abstraction: empirical

abstraction and constructive abstraction. (Empirical abstraction is also known as simple

abstraction, and constructive abstraction is also known as reflective or reflecting abstraction.)

The abstraction of color or weight (physical knowledge) from objects is an example of

empirical abstraction. In empirical abstraction, we focus on one or more properties that are in

objects (such as color) and ignore the others (such as weight and the fact that the object is

made of plastic).

In constructive abstraction, by contrast, we create mental relationships such as ‘‘two,’’

‘‘different,’’ ‘‘the same,’’ and ‘‘more.’’ We thus construct logicomathematical knowledge

C. Kamii et al. / Journal of Mathematical Behavior 20 (2001) 21–3122

through constructive abstraction. The child makes the number ‘‘four’’ by constructive

abstraction and later puts two ‘‘fours’’ into a relationship through constructive abstraction

when he or she goes on to 4 + 4. When the child then calculates 5� 4, this multiplication is

constructed out of repeated addition (4 + 4 + 4 + 4 + 4) through constructive abstraction.

With these theoretical distinctions that Piaget and others verified with 60 years of research

all over the world, it becomes clear that each child must construct mathematics through

constructive abstraction. It also becomes clear that what we commonly call ‘‘thinking’’ or

‘‘reasoning’’ is constructive abstraction.

The distinction between physical and logicomathematical knowledge and between

empirical and constructive abstraction suggests a dichotomy we can make between manip-

ulatives with which children can learn mathematics and those from which children are

expected to learn. We begin our discussion of manipulatives by focusing on the first category.

3. Manipulatives with which children can learn mathematics

Three examples will be discussed with respect to this group: Tangrams, counters, and

playing cards used in games.

3.1. Tangrams

Tangrams can be very useful for spatial reasoning if children are encouraged to think to

solve problems such as the one in Fig. 1 (Educational Teaching Aids). To solve this problem,

children have to figure out which two, four, or five of the seven pieces to use. They also

engage in trial and error to figure out why a piece that looked promising does not work. They

thus learn from the relationships they made unsuccessfully and go on to make better ones.

While Tangrams can be an excellent teaching tool, it is easy for a teacher to reduce their

value. For example, if the teacher ‘‘helped’’ struggling students by saying, ‘‘I do not think that

piece will work,’’ ‘‘Try turning it,’’ and so on, children will be deprived of the possibility of

doing their own thinking. Some teachers are afraid of frustrating children and ‘‘help’’ them

too much. If a student is frustrated, a better intervention is to suggest an easier problem.

Children use the knowledge they already have to solve harder problems. Therefore, when a

problem seems too hard, a good intervention is to help children make the lower-level

relationships they need to make higher-level relationships (constructive abstraction).

3.2. Counters

When young children are introduced to addition and subtraction problems and have no

idea what to do, counters can be very helpful. Teachers often offer counters to children as

pretend cookies to do four cookies plus four cookies, for instance.

While counters can be very helpful, they can also be used in overly prescriptive ways that

interfere with children’s thinking. For example, one of us once wondered why some first

graders were getting answers like 3 + 5 = 5. The reason soon became apparent: The children

C. Kamii et al. / Journal of Mathematical Behavior 20 (2001) 21–31 23

were putting out three counters for the first addend and five counters for the second addend

including the three that were already out. They then counted all the counters as they had been

instructed to do. In adding two numbers, children at a higher level of abstraction put two

wholes together (3 and 5) to make a higher-order whole (8) in which the previous wholes

become two parts (see Fig. 2a). Children who cannot yet make this part–whole relationship

through their own thinking often make two sets that are not disjoint (see Fig. 2b).

Children are sometimes encouraged to use counters when they are no longer useful.

Olivier, Murray, and Human (1991) made the following observation about young children in

a constructivist program they developed in South Africa:

Although informal writing materials as well as counters are always available, it seems that

students seldom use counters to model a problem. Rather, the problem context is drawn in

greater or lesser detail, and then solved by further drawing in the actions needed (p. 17).

One of us (Kamii, 2000) asked many kindergarten and first-grade teachers in various parts

of the United States and Japan to conduct research in their classrooms to test the validity of

Fig. 1. An example of a Tangram problem.


Olivier et al.’s (1991) statement. The teachers were asked to remind their students frequently

that they (the students) were free to use the counters, paper, pencil, or anything else in the

classroom to solve word problems.

The teachers reported that, most of the time, children preferred to draw rather than to use

the counters that were equally accessible. The only exception, they said, was at the beginning

of the school year, when the students were unfamiliar with word problems. The other

observation the teachers made was that young children prefer to use their fingers. When the

numbers got bigger than 10, however, the students switched to tally marks, a different kind

of symbol.

The term ‘‘symbol’’ requires clarification. In Piaget’s terminology, examples of symbols

are counters used as pretend cookies, fingers, and tally marks. Symbols bear a resemblance to

the objects being represented and are invented by each child. For conventional forms of

representation such as spoken and written numerals and mathematical signs (+ and =), Piaget

used the term signs. When children solve word problems, they should be allowed to choose

the symbols or signs that best help them think.

Fig. 3 illustrates the drawings three first graders made early in September to answer the

question, ‘‘How many feet are there in your house?’’ Fig. 3a shows great details of physical

knowledge such as people’s heads, arms, hands, clothes, and hair. Fig. 3b, on the other hand,

represents only the child’s logicomathematical knowledge of number. When the child is at a

higher level of constructive abstraction, the physical knowledge of heads, arms, and clothes

becomes irrelevant to the question about how many feet there are. Fig. 3c can be categorized

in between. This child represented her physical knowledge of feet but focused only on the

body parts that she thought were relevant to the problem.

Drawings thus permit children to use their own representations of their ideas at their

own levels of constructive abstraction. Counters do not allow this kind of personal

representation and have properties that interfere with the child’s representation of his or

her ideas. Fingers are likewise highly personal symbols that children use with their

mental images.

Some teachers forbid the use of fingers and make children use counters instead. This is

another example of an undesirable way of using counters. Fingers permit children to

represent their numerical ideas more directly than counters, thereby facilitating their

Fig. 2. (a) The addition of two wholes that are disjoint and (b) the counting of two overlapping wholes.


thinking. Besides, children can take their fingers much more easily than counters when they

go shopping!

A good use of counters is as game pieces in games like Cover-Up. This two-person game

is played with two dice and the board shown in Fig. 4. The players sit on opposite sides of

the board, take turns rolling the dice, and use a counter to cover the number corresponding to

the total of the two numbers rolled. If a number has already been covered, the turn is wasted.

The player who covers all the numbers on his or her side first is the winner.

Fig. 3. Three drawings made by first graders to answer the question ‘‘How many feet are there in your house?’’

Fig. 4. A gameboard for Cover-Up.


In short, the mathematics is not in the manipulatives. The value of the manipulative

depends on how it is used by the child to solve problems.

3.3. Cards used in games

The game called Leftovers can be played with the cards shown in Fig. 5. In this game,

cards numbered 1–9, 10, 11, or 12 are aligned as shown. The first player rolls two dice and

turns over the card(s) that make the same total. For example, if the player rolled a 4 and a 2,

the cards that can be turned over are the 6, the 5 and the 1, or the 4 and the 2. The player then

rolls the dice again and continues to play until he or she is stuck without cards to turn down.

This player’s score is the total of all the points on the leftover cards.

The turn passes to the next player, and the winner is the person with the lowest total score

when everybody has had three (or more) turns.

This game can thus be played with cards, but the commercially made game called Shut the

Box or Wake Up, Giant is more appealing to most children because they can flip wooden

pieces instead of turning cards down. However, the cards and wooden pieces are not

indispensable because children can write all the numbers on a piece of paper at the beginning

of each turn and cross them out as the game progresses. The value of the cards or wooden

pieces lies in the fact that they facilitate the child’s thinking such as the addition of two

numbers rolled (4 + 2, for instance), the partitioning of the total (into 5 and 1), and the

strategy of using numbers that are advantageous to use first (6 in this case). Having to write

numerals and cross them out slows children’s thinking. This is why we say that these

manipulatives are not indispensable but desirable.

In this game, too, the teacher can give too much ‘‘help,’’ thereby interfering with the

children’s thinking. If a child rolls a 6 and a 4, for instance, he or she may want to turn down a

2 and an 8. Some teachers at this point teach the advantage of using a large number like 10 (to

save smaller numbers for future use). Such teaching may be helpful in the short run, but it

deprives the child of a chance to do his or her own thinking.

We have so far been discussing manipulatives with which children can learn mathematics.

However, some manipulatives were invented by educators for the specific purpose of

teaching certain aspects of mathematics. We now turn to these didactic materials from which

children are expected to learn.

4. Manipulatives from which children are expected to learn mathematics

Two examples will be discussed: A balance designed to teach addition and base-ten blocks

and Unifix Cubes intended to teach tens, ones, and so on.

Fig. 5. The arrangement of cards to play Leftovers.


4.1. A balance

The catalog describing the balance in Fig. 6 states that children learn that 3 + 5 = 8 by

hanging weights on ‘‘3’’ and ‘‘5’’ on one side and on ‘‘8’’ on the other side.

This balance uses weight, which is physical knowledge. Aside from the fact that balances

ordered through catalogs are often not accurate enough to balance, balance is a physical

phenomenon. If we put 3 + 5 people on one side of an accurate, big balance and 8 people on

the other side, we can be fairly sure that the two sides will not balance. Children would

fortunately not learn that 5 + 3 6¼ 8 in this situation because no one learns addition from a

balance. An elephant is one, and a mouse is also one.

As stated earlier (see Fig. 2a), addition is a mental operation in which we combine two

wholes (3 and 5 in this situation) to make a higher-order whole (8), in which the original

wholes become two parts. As can be seen in Fig. 2a, the 3 and the 5 stay in the total (8). On a

balance, however, the total is on the right-hand side, separate from the 3 and the 5. Balance, a

physical phenomenon, is not the same thing as the logicomathematical relationship of

equality. Balances may therefore be useful to teach the measurement of weight, but they

are completely useless for teaching addition.

4.2. Base-ten blocks and Unifix Cubes

Base-ten blocks, Unifix Cubes, and bundles of 10 toothpicks are examples of manipu-

latives that give the impression of being useful when teaching place value, ‘‘carrying,’’ and

‘‘borrowing.’’ However, they are usually not useful. We begin by reviewing the conservation-

of-number task to show that children do not abstract ones empirically from objects, and they

do not abstract tens empirically from objects either.

In the conservation-of-number task, 4-year-olds can usually make a one-to-one corre-

spondence to put out the same number of counters as the interviewer has aligned. However,

when one of the rows is spread out and the other row is pushed together, most 4-year-olds

think that the longer row has more. By the age of 6–7, however, most children have

developed their logic sufficiently to deduce, with the force of logical necessity, that the two

Fig. 6. A balance designed to teach addition.


rows have the same number. The conservation task is a test of children’s logicomathematical

knowledge, which results from constructive abstraction.

It follows that there is no such thing as a ‘‘concrete number.’’ Two cookies are concrete and

observable, but the number ‘‘two’’ is neither concrete nor observable.

When first graders say ‘‘34,’’ the number they usually have in mind is 34 ones, and the

structure of their thinking is illustrated in Fig. 7a. When adults say ‘‘34,’’ on the other hand,

they have three tens and four ones in their minds as illustrated in Fig. 7c. Fig. 7c also shows

that when adults think ‘‘one ten,’’ they are simultaneously thinking about ‘‘ten ones.’’ The key

word here is ‘‘simultaneously,’’ and first graders often reveal their inability to think

simultaneously at two hierarchical levels when they count by tens as follows.

If we ask first graders to put ten beads into each cup as shown in Fig. 8 and ask them to

count all the beads by tens, they often say ‘‘10, 20, 30’’ as they count the cupfuls and go on to

count the loose ones by saying ‘‘40, 50, 60, 70.’’ Adults and older children know when to

shift to ones in this situation because they are thinking about the ones while counting by tens.

The structure of base-ten blocks, Unifix Cubes, and bundles of 10 toothpicks is illustrated

in Fig. 7b. This structure shows that these manipulatives are made by merely partitioning the

system of ones. Note that there are no higher-order units (tens) in this structure.

Adults, who can think about ‘‘one ten’’ and ‘‘ten ones’’ simultaneously, can look at a long

base-ten block and see ‘‘one ten’’ and ‘‘ten ones’’ in it simultaneously. First graders, who have

not constructed a system of tens out of a system of ones, can see ‘‘one ten’’ and ‘‘ten ones’’

only successively in time. ‘‘One ten’’ has to be constructed by the child from within, by

constructive abstraction, out of his or her own system of ones. Therefore, it is not possible for

Fig. 7. The structure of (a) 34 ones, (b) 34 ones partitioned into tens, and (c) 3 tens and 34 ones function-

ing simultaneously.


children to acquire a system of tens empirically from base-ten blocks, Unifix Cubes, or

bundles of straws or toothpicks.

We say that these manipulatives are ‘‘usually not useful’’ because they can be useful (a) to

children who do not have an intuitive feel for the approximate magnitude of ‘‘ten,’’ ‘‘a

hundred,’’ and ‘‘a thousand’’ and (b) to those who are at a fairly high level of constructive

abstraction and on the verge of constructing tens. The teachers with whom we work, however,

never use base-ten blocks, Unifix Cubes, or bundles of toothpicks. The children they work

with construct tens out of their own system of ones that are in their heads.

How teachers can encourage the construction of tens is beyond the scope of this article, but

we would like to say that this construction occurs when children are encouraged to think

(constructive abstraction). For example, the teacher can ask, ‘‘What is a quick and easy way

to do 9 + 6?’’ Most second graders respond with (9 + 1) + 5, thinking about a ten and the ones

simultaneously. Likewise, when second graders are asked to invent a way to deal with

Fig. 8. The way many first graders count three cupfuls and four loose beads.

Fig. 9. The board used in the Towers Game.


19 + 12, they invent a variety of ways such as (19 + 1) + 11, (10 + 10) + (9 + 2), and

10 + 10 + 9 + 2. When children are asked to invent ways of adding two-digit numbers, they

struggle to deal with tens and ones simultaneously. This struggle (constructive abstraction)

constitutes the process of construction, and the result is solid ideas of tens (Kamii, 1989;

Kamii & Joseph, 1988).

Although we do not recommend Unifix Cubes for teaching tens and ones, these can be used

beneficially for other purposes. For example, some teachers use Unifix Cubes as markers in

path games. They also use them in a game called the Towers Game (source unknown).

In this game, Unifix Cubes are used to make towers of various heights as shown in the

gameboard in Fig. 9. The numbers in this figure indicate the numbers of Unifix Cubes to

stack (only one cube on ‘‘1,’’ two cubes on ‘‘2,’’ and so on). It can be seen that the tall towers

are generally in the middle of the arrangement. The players take turns rolling two dice and

collecting towers to equal the total rolled. For example, if a player rolls a 5 and a 3, he or she

can take two towers (4 + 4), three towers (3 + 3 + 2), four towers (2 + 2 + 2 + 2), five towers

(1 + 1 + 2 + 2 + 2), or six towers (1 + 1 + 1 + 1 + 2 + 2). The person who has collected the most

towers at the end is the winner.

5. Conclusion

What is important for children’s construction of logicomathematical knowledge is that they

think (constructive abstraction). We recommend the use of Tangrams because these objects

encourage children to think and make spatial relationships. Card games such as Leftovers

stimulate numerical thinking. In short, the mathematics we want children to learn does not

exist in manipulatives. It develops as children think, and manipulatives are useful or useless

depending on the quality of thinking they stimulate.

We recommend games such as Leftovers and the Towers Game for first and second

graders, but there comes a point when these games become too easy and of little value.

Teachers’ theoretical understanding is therefore essential, as well as their ability to infer what

is taking place in individual children’s heads.

References

Educational Teaching Aids. Tangram cards. Chicago: Educational Teaching Aids.

Kamii, C. (1989). Young children continue to reinvent arithmetic 2nd grade. New York: Teachers College Press.

Kamii, C. (2000). Young children reinvent arithmetic (2nd ed.). New York: Teachers College Press.

Kamii, C., & Joseph, L. (1988). Teaching place value and double-column addition. Arithmetic Teacher, 35 (6),

48–52.

Olivier, A., Murray, H., & Human, P. (1991). Children’s solution strategies for division problems. In: R. G.

Underhill (Ed.), Proceedings of the 13th annual meeting, North American chapter of the International Group

for the Psychology of Mathematics Education (vol. 2, pp. 15–21). Blacksburg: Virginia Polytechnic Institute.

Piaget, J. (1971). Biology and knowledge. Chicago: The University of Chicago Press (B. Walsh, Trans.; original

work published 1967).


manipulatives: when are they useful?

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