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Eğitimde Kuram ve Uygulama

2008 , 4 (2):

239-252

Journal of Theory and Practice in Education Articles/ Makaleler

ISSN: 1304-9496 http://eku.comu.edu.tr/index/4/2/jrrains_cakelly_rldurham.pdf

© Çanakkale Onsekiz Mart University, Faculty of Education. All rights reserved.

© Çanakkale Onsekiz Mart Üniversitesi, Eğitim Fakültesi. Bütün hakları saklıdır.

THE EVOLUTION OF THE IMPORTANCE OF

MULTI-SENSORY TEACHING TECHNIQUES IN

ELEMENTARY MATHEMATICS: THEORY AND

PRACTICE

1

İLKOKUL MATEMATİĞİNDE ÇOKLU-DUYUMA DAYALI ÖĞRETME

TEKNİKLERİNİN ÖNEMİNİN EVRİMİ: KURAM VE UYGULAMA

Jenny R. RAINS

2

Catherine A. KELLY

3

Robert L. DURHAM

ABSTRACT

In recent years, partially because of federal legislation, there have been increases in demand for

accountability in all educational venues. Performance in elementary mathematics is no exception. In this

paper we review the relevant parts of the learning theories of Piaget, Bruner, and Vygotsky and address

the difficulties teachers may face when introducing mathematical concepts. The review of theories, along

with a review of previously published empirical studies, supports the use of multi-sensory teaching

techniques in the elementary, specifically kindergarten through third grade, classrooms. Since students

(both regular and special needs) develop and learn at different rates, it is unlikely that all will be

developmentally prepared to assimilate new mathematical concepts at the same time. Multi-sensory

techniques allow many students, by assimilation, to grasp elusive concepts and keep up with their peers.

Keywords: Mathematics teaching, multi-sensory teaching techniques.

ÖZ

Kısmen yasal gelişmeler nedeniyle, bütün eğitim alanlarında performansla ilgili sorumluluk alma eğilimi

giderek yaygınlaşmaktadır. İlkokul düzeyinde matematik performansı da bu konuda bir istisna değildir. Bu

çalışmada Piaget, Bruner ve Vygotsky’nin kuramlarının ilgili bölümlerini ve öğrencileri matematik

kavramlarıyla tanıştırırken öğretmenlerin karşılaşabilecekleri zorlukları gözden geçirdik. Yayınlanmış

olan kuramsal ve deneysel çalışmalar, özellikle anaokulu seviyesinden üçüncü sınıf sonuna kadarki

dönemde, çoklu-duyuma (multi-sensory) dayalı öğretim tekniklerinin kullanımını desteklemektedir.

Normal ve engelli çocukların gelişim ve öğrenme hızları farklılık gösterdiğinden, çocukların tümünün

yeni matematik kavramlarını idrak etmeye aynı anda hazır olma olasılığı düşüktür. Çoklu-duyum

teknikleri, birçok çocuğun anlaşılması zor matematik kavramlarını asimile etme yoluyla öğrenmesini

sağlayarak akranlarından geri kalmamasını sağlamaktadır.

Anahtar Kelimeler: Matematik öğretimi, çok duyuma dayalı öğretim teknikleri

1

Research supported, in part, by funds from Innovative Learning Concepts, Inc.

2

This review was part of the first author's thesis in partial fulfillment of a Masters degree in Psychology

3

Correspondence should be directed to the second author.

University of Colorado at Colorado

Springs. E-mail: ckelly@uccs.edu

*

Special gratitude to Benek ALTAYLI, Psy.D., for translating the title and article abstract from English to

Turkish.

The evolution of the importance of multi-sensory teaching techniques in elementary mathematics:

theory and practice

Egitimde Kuram ve Uygulama / Journal of Theory and Practice in Education

http://eku.comu.edu.tr/index/4/2/jrrains_cakelly_rldurham.pdf

240

INTRODUCTION

In recent years, there has been an increase in the accountability of student

performance across all educational venues. This has been manifested in the

appearance of performance indicators (i.e., student achievement tests) of student

learning and growth and has forced educators to reexamine didactic methods to

ensure that all students have the opportunity to learn. The learning area of

elementary mathematics is no exception. In this paper, we review the relevant

aspects of learning theories that impact mathematics learning among young

children. We argue that the theories (Piaget, 1958; Bruner, 1973; Vygotsky,

1978) indicate the necessity of employing multiple methods of presenting

mathematical concepts because children of the same chronological age are not

necessarily at the same stage of mental readiness (Van de Walle, 2007; Kamii and

Rummelsburg, 2008). We further suggest that these methods should include

multi-sensory teaching techniques across a variety of classrooms (Clements,

1999).

When introducing new mathematical concepts to elementary students it is

important that teachers help all students understand the material. Since different

students may have different learning styles (see below), multiple methods of

presentation can facilitate their comprehension of new concepts. Teachers today

are not only driven by their teaching values to reach all children, but by

legislation as well. The No Child Left Behind Act (2002) has greatly increased

districts', principals', and teachers' accountability by requiring schools to report

their students' achievement test results to the public. If a school does not meet

federal standards, parents have the right to relocate their children to another

school (United States Department of Education, n.d.). Because of this increase in

accountability, it is more important than ever for teachers to find ways to reach

every child and help him master the material to be learned in the classroom.

This may be challenging since, as proposed by Piaget (e.g., 1965), not all

students may be capable of grasping certain mathematical concepts at the same

time as others (e.g., Kamii and Rummelsburg, 2008). As a result, students at

varying developmental levels and with differing learning preferences can be left

behind while the teacher continues on to the next lesson. This leads to the next

problematic issue. Research by Hiebert (1988) has shown that students must learn

and understand the earlier foundations of mathematical concepts before they will

be able to comprehend the next level of processes. This introduces the possible

need for additional teaching methods and materials to mediate students' learning

when addressing certain mathematical concepts. For example, teachers can

introduce the concept of addition by demonstrating with beans, have the class

repeat the process verbally and manipulate them physically. What one child does

Jenny R. RAINS Egitimde Kuram ve Uygulama

Catherine A. KELLY Journal of Theory and Practice in Education

Robert L. DURHAM 2008, 4 (2):239-252

© Çanakkale Onsekiz Mart University, Faculty of Education. All rights reserved.

© Çanakkale Onsekiz Mart Üniversitesi, Eğitim Fakültesi. Bütün hakları saklıdır.

241

not comprehend by hearing and saying it, they have a chance of grasping it by

visual and tactile-kinesthetic means.

Some students may not have reached the mental maturity yet (addressed in

detail later) by which they are ready to learn the mathematical concepts being

taught in class. They may require supplemental learning materials, outside the

main curriculum, to enhance their understanding. Multi-sensory supplements,

such as math manipulatives, support the child’s use of visual, tactile, and/or

auditory interactions with the material. These types of materials can help to

bridge the gaps that most elementary teachers will encounter when trying to teach

young children novel and abstract mathematical concepts (Bullock, 2003).

Multi-sensory learning, as the name implies, is the process of learning new

subject matter through the use of two or more senses. This may include

combining visual, auditory, tactile-kinesthetic, and/or even olfactory and taste

(Scott, 1993). The place for multi-sensory teaching techniques in elementary

mathematics classrooms can be illustrated through a brief review of relevant

aspects of the theories by Piaget, Bruner and Vygotsky. These theories concern

stages of learning and readiness and can provide a better insight into students’

development. These theories also can anticipate some of the frustrations students

may encounter when learning mathematics. Stage theories suggest that as humans

develop, they progress through different stages of cognitive development (i.e., a

toddler is not simply a small adult, although adults can develop through stages as

well, e.g., Perry, 1970). As children develop through different stages, they are

prepared to assimilate different types of material at each stage, but not before.

Now, while these stages are related to age, they are not age dependent. These

theories, outlined below, for several different reasons, support the use of multi-

sensory teaching techniques. These techniques can help students at slightly

different stages of mental development interact with to-be-learned material and

become familiar with those concepts.

In addition to the discussion of these theories with focus on their relation

to the current topic, a review of prior empirical studies will be presented. By

reviewing the influence of multi-sensory teaching techniques across both

different student populations and academic disciplines, including elementary

mathematics, its success in reaching students' understanding can be demonstrated.

THEORIES OF LEARNING

In the fields of psychology and education there have been many theories of

learning and development advanced over the years. The theoretical positions

involving stages of development are seen as most pertinent to the question of

readiness-to-learn, and the stage theories seen as most relevant to early classroom

The evolution of the importance of multi-sensory teaching techniques in elementary mathematics:

theory and practice

Egitimde Kuram ve Uygulama / Journal of Theory and Practice in Education

http://eku.comu.edu.tr/index/4/2/jrrains_cakelly_rldurham.pdf

242

learning are those of Piaget, Bruner, and Vygotsky. Relevant aspects of these

theories will be briefly summarized.

Piaget

Jean Piaget’s (e.g., 1958; 1965) theory of stage development describes

children’s progress through certain stages of development. He gives a loose time

frame, in which the children enter each stage, but it must be understood that the

ages he gives are approximate; it is the order of the stages that he believes to be

universal. For example, studies with mentally retarded children have revealed that

these children progress through Piaget’s stages of development in the same order,

though not at the same ages, as other children. Piaget’s stages described below, in

chronological order, are: sensorimotor, pre-operational, concrete operational, and

formal operational. Furthermore, there is no reason to believe that all "normal"

children will progress through stages at exactly, or even roughly, the same

chronological ages. The reader should also note the broad range of ages included

in each stage.

Sensorimotor: This stage encompasses most infants from birth to about

two years of age. As the name suggests, the child in this stage is occupied only

with his own motor activity and his mental activities are strictly limited to what

his senses detect. Because the child is dependent on solely what his senses detect,

he has no concept of object permanence; he has no idea that an object exists once

it is no longer being sensed (Bybee, 1982; Thorne and Henley, 1997).

Pre-operational: This stage ranges from approximately two years to seven

years of age. In this phase of life a child does not yet have the ability to perform

the mental activities Piaget called “operations.” One of these operations, most

applicable to learning math concepts, is conservation of quantity; the concept that

the quantity of something remains the same even though its physical appearance

changes (explained further later). Also, children in this phase have an egocentric

perspective of thinking. These children view themselves as the center of

everything and have difficulty accepting any views but their own (Bybee, 1982;

Thorne and Henley, 1997).

Concrete operational: Between the ages of seven and eleven the child

enters the concrete operational stage. The child becomes less egocentric. And it is

here that the child develops the concept of conservation. Although he has not

fully developed the cognitive skills needed to handle abstract problems that

require mental manipulations, he can now deal with tangible problems (Bybee,

1982; Thorne and Henley, 1997).

Formal operational: From approximately 12 or 13 years of age through

adulthood, Piaget considers a person to be in the formal operational stage. A

Jenny R. RAINS Egitimde Kuram ve Uygulama

Catherine A. KELLY Journal of Theory and Practice in Education

Robert L. DURHAM 2008, 4 (2):239-252

© Çanakkale Onsekiz Mart University, Faculty of Education. All rights reserved.

© Çanakkale Onsekiz Mart Üniversitesi, Eğitim Fakültesi. Bütün hakları saklıdır.

243

person is now capable of abstract thought, he can think reflectively, and can test

hypotheses either systematically or hypothetically (Bybee, 1982; Thorne &

Henley, 1997).

Piagetian Theory and Elementary Teaching: The two stages most relevant

to K-3 children, and therefore of most concern to their elementary teachers, are

the pre-operational and concrete operational stages. It is not the details of these

stages that are so important for teachers to understand, but the differences

between these two stages in regards to a child’s conceptual abilities. In the

preoperational stage, the child is not yet able to complete mental operations.

Mental operations may include a task such as adding numbers in their head. Once

they reach the concrete operational stage they can complete mental operations

and therefore no longer need to use overt trial and error methods; trial and error

can now be done in their heads (Bruner, 1973). It should be emphasized that

“normal” children reach the concrete operational stage between the ages of seven

and eleven. However, that is a fairly broad target for educators to use. For the

child to be considered in the concrete operational stage their mental operations

must be reversible (Bruner). Reversibility is the child’s ability to think in two

opposite directions at the same time. The lack of reversibility in a child in the

preoperational stage can be demonstrated with any one of the many Piagetian

conservation tasks.

One conservation task that is easily related to problems early elementary

teachers face with their students is the conservation of number task described by

Kamii (1982). A teacher lines up eight foam sticks in a row in front of her, and

the child lines up a row of eight pieces to match the ones that the teacher has

placed down. But if a child’s, who is in the preoperational stage, set of pieces is

spread out, as demonstrated in Figure 1, she now thinks that she has more pieces

than the teacher. A child that does not have mental operations that are reversible

is not able to understand conservation of number. They cannot grasp the concept

that the row can be brought back to its original state and is therefore the same

number as before (Bruner, 1973).

Figure 1. Demonstration of a Conservation Task

The evolution of the importance of multi-sensory teaching techniques in elementary mathematics:

theory and practice

Egitimde Kuram ve Uygulama / Journal of Theory and Practice in Education

http://eku.comu.edu.tr/index/4/2/jrrains_cakelly_rldurham.pdf

244

According to Figure 1, in situation A the child knows that he has the same

number of pieces as his teacher. In situation B, when the child’s pieces are spread

out in front of him, he believes that he has more pieces than his teacher.

Another example of reversibility in a mathematical task is filling in the

missing-addend.

3 + ? = 7

This task demonstrates the child’s ability to look at the 3 as part of the whole, 7.

Children that are not yet in the concrete-operational stage may be able to answer

this problem due to rote memorization but not due to comprehension (Kamii,

Lewis, and Booker, 1998).

As Bruner (1973) states in a discussion regarding Piaget’s stages of

development, teachers are extremely limited in conveying concepts to children in

the preoperational stage. But, the child’s development, according to Bruner, is

responsive to the learning environment. It is just a matter of finding the right

method of delivering the material to help a child progress through the stages of

learning. Multi-sensory approaches allow children to receive the information in a

variety of ways. These can facilitate development in general and math in specific

by providing tools for the students to relate to until the concept is fully embraced.

If any impression is made on a teacher from this theory it should be that if

the child has not progressed into at least the beginning of the concrete operational

stage of development, then this theory suggests that he may not be capable of

understanding the abstract mathematical concepts. Examples of these concepts

are linking symbols to the ideas they represent or manipulating items in their

heads, and as a result may not be able to advance to where he is expected to be.

Bruner

Bruner’s theory (e.g., 1973) (alluded to earlier) is based on the foundation

set by Piagetian Theory. A summary of Bruner's theory indicates that Piaget's

stages may not be all or nothing; there may be a "degree" of stage development

within each stage. For example, a child is not in sensorimotor, and then out of it;

he progresses through the stage. This progression through one stage to the next

may be facilitated with appropriate teaching techniques. The following is a brief

description of Bruner’s theory and how it is applied to elementary learning and

teaching. The following two sections (how children learn and the teacher’s place

in learning) are taken from this source.

How children learn: Jerome Bruner is a strong advocate for constructive

learning. Constructive learning involves hands-on activities in which the child

can create and test his own hypotheses. Bruner believes a child is an active

participant in learning and should be encouraged to participate in the learning

Jenny R. RAINS Egitimde Kuram ve Uygulama

Catherine A. KELLY Journal of Theory and Practice in Education

Robert L. DURHAM 2008, 4 (2):239-252

© Çanakkale Onsekiz Mart University, Faculty of Education. All rights reserved.

© Çanakkale Onsekiz Mart Üniversitesi, Eğitim Fakültesi. Bütün hakları saklıdır.

245

process. Bruner’s theory is founded on the idea that children construct new ideas

based on their previous knowledge. They use their current knowledge to create

hypotheses and to help them solve problems and discover relationships. This idea

emphasizes the importance of ensuring all children's understanding of a concept

before moving on to the next. He also emphasizes that different forms of

representation of a single concept may be more appropriate for children at various

ages and/or stages of learning than others (discussed more in the next section).

From Bruner’s theory, in regards to how children learn, it is most important for a

teacher to understand that it will be difficult for a child to grasp new concepts

when he has no knowledge on which to base the new information. Also important

is that children may require different means of learning; one child’s

understanding may not come from the same source as another child.

The teacher’s place in learning: Although Bruner considers the child to be

an active participant in the knowledge gathering process, he emphasizes the

importance of the instructor as a “translator” of mathematical material to the

child. Children may be in different stages of development and as a result have

different abilities. Therefore the needs of individual children differ and learning

must be made appropriate for each child in a way that he can understand during

his particular stage. Bruner's theory supports the idea that there is at least one way

in which to reach any child and help her understand a concept. He expresses the

availability and the importance of giving children “multiple embodiments of the

same general idea” in order to increase their understanding of it. A teacher should

address the child’s predisposition towards learning, determine the best way to

present the material to be learned, and establish the best sequence in which to

present it. With a classroom full of children, probably at different stages of

development (or at different levels of sophistication within the same level), it is

obvious why a teacher would have difficulty individualizing her method of

translation for each student. For this reason, knowledge of supplemental materials

that approach the children in a variety of ways in addition to the core curriculum

might be helpful for the teacher to have. Materials that teach students via multiple

means can address the needs of more students in a classroom.

Vygotsky

The aspect of Lev Vygotsky’s (e.g., 1978) theory of development most

applicable to education is his theory of the “zone of proximal development"

(ZPD). This theory is similar to the above theories in that it is a stage theory.

However, Vygotsky emphasizes how a child transitions from one stage to the

next as a function of social interactions rather than on specific stages. The ZPD is

defined as the difference between the stage of development at which the child is

The evolution of the importance of multi-sensory teaching techniques in elementary mathematics:

theory and practice

Egitimde Kuram ve Uygulama / Journal of Theory and Practice in Education

http://eku.comu.edu.tr/index/4/2/jrrains_cakelly_rldurham.pdf

246

currently and the potential stage to which he can reach with the proper assistance

from adults or more capable peers (Tudge, 1990; Vygotsky, 1978). Vygotsky’s

ZPD implies a stage development through which children pass with the aid of

social interaction. He suggests that cognitive and linguistic skills appear two

times to children; first they appear socially, between two people, and then

internally within the child (Gallimore and Tharp, 1990). The time for this

transition differs for each child. As suggested earlier, it is not all or none.

A teacher’s part, according to Vygotsky’s (1978) ZPD theory, is to assist

the child’s performance through the zone into the next phase. Vygotsky’s theory

suggests that what children can do today with assistance, they will be able to do

tomorrow proficiently on their own. According to this theory, with the

appropriate guidance, performance can precede competence (Cazdan, 1981).

Moll’s (1990) summary of Vygotsky’s theory suggests that rote drill and practice

instruction is not the type of assisting in ZPD that Vygotsky would suggest. He

would recommend that teachers should assist in basic activities to learn skills as

opposed to teaching the basic skills without activities. According to this

perspective, learning materials act as a tool to aid the child in problem solving

until the ability to completely comprehend a concept is developed. It is important

here to emphasize the distinction between comprehension and performance. The

child can “do” the activity and be correct before she grasps to concept

completely.

Summary of Theories

The three theorists summarized above offer complementary views on the

mental development of children. Piaget (e.g., 1965) introduces the idea of stages

of mental readiness in how a child interacts with the world. Bruner (e.g., 1973)

builds on this idea but adds the notion that there may be levels within those stages

that the teacher can utilize. And Vygotsky closes the circle by introducing the

idea that social interaction can facilitate transition from one stage to the next and

that a child’s performance can be affected independently of the transitions.

Elementary teachers should take into account the following important

ideas that are supported by the above theories. 1) A child’s stage in development

may be responsible for his inability to understand abstract concepts, which, in

turn, may affect his progress in the class’s mathematical curriculum. 2) A child

builds his knowledge off of his understanding of prior concepts. 3) Teachers play

an important role in assisting children via the use of appropriate methods. 4)

Teachers may need to approach different children in different ways with the

material to be learned. 5) Transition between stages may be different for each

child and may be gradual as different "operations" are assimilated.

Jenny R. RAINS Egitimde Kuram ve Uygulama

Catherine A. KELLY Journal of Theory and Practice in Education

Robert L. DURHAM 2008, 4 (2):239-252

© Çanakkale Onsekiz Mart University, Faculty of Education. All rights reserved.

© Çanakkale Onsekiz Mart Üniversitesi, Eğitim Fakültesi. Bütün hakları saklıdır.

247

If a child needs extra attention to acquire the knowledge needed to move

on the next concept or process, and he does not receive it, he may be left behind

and will continue to digress as the teacher, without the correct teaching tools,

continues on in the curriculum without him. Introducing supplemental multi-

sensory materials may facilitate a child to interact with math concepts by two or

more means in addition to the core curriculum. As supported by the above

theories, these materials may be helpful in teachers' efforts to reach every child’s

understanding. In addition to the theoretical foundation for multi-sensory teaching

techniques, there is some empirical support as well.

EMPIRICAL STUDIES SUPPORT OF MULTI-SENSORY TEACHING

While the theoretical foundations introduced above indicate a need for

multi-sensory approaches to early learning, what kind of empirical support is

there for such techniques? It is to this small but diverse and provocative body of

literature we now turn. There have been only a few studies conducted and they

have employed different types of students in different academic subjects.

In a study executed across classrooms in Queensland, Australia, Thorton,

Jones, and Toohey (1982) implemented a multi-sensory teaching program,

Multisensory Basic Fact Program (MBFP), into remedial classrooms for students’

grades two through six. The program incorporates visual learning through

pictures, as teachers provide oral prompts. Students are also involved

kinesthetically when learning new concepts by tapping or finger-tracing. To test

the usefulness of this multi-sensory teaching program, these students were given

an addition-facts test before beginning the program and again after the 11-week

instruction phase. All of the grade levels except grade two (possibly because they

were not yet at the stage in which the material could be absorbed) showed marked

improvement from the pretest to posttest. And, although the students had not

reviewed the information before the follow-up test, they retained their knowledge

of the concepts after a three-week period.

An example of examining the impact of a multi-sensory approach to

teaching reading is exemplified in a study by Dev, Doyle, and Valente (2002).

They used the Orton-Gillingham technique (Institute for Multi-sensory education,

2000), which involves visual, auditory, and kinesthetic modalities, with first

grade children at the special education level. These children improved enough in

their reading abilities to advance them out of the special education level. The

maintenance of the gains that they achieved with the use of the multi-sensory

approach was evaluated after a two year period. None of the children had returned

to special education classes (Dev, et al., 2002).

The evolution of the importance of multi-sensory teaching techniques in elementary mathematics:

theory and practice

Egitimde Kuram ve Uygulama / Journal of Theory and Practice in Education

http://eku.comu.edu.tr/index/4/2/jrrains_cakelly_rldurham.pdf

248

Another empirical study involving the effect of a multi-sensory approach

involved testing regular (non-special education) fourth and sixth graders’ learning

of spelling. Students completed a series of six spelling lessons, which were

followed by spelling tests in which they wrote down each word that the teacher

pronounced. Each of the students graded their own tests by two different

methods. Half of each student’s tests were graded by hearing the teacher spell the

words aloud. The other half of their tests was graded by comparing their spelling

to a correct spelling list followed by the teacher spelling the words aloud. The

students were given four post tests after a six week period, and the words graded

by the multi-sensory method resulted in significantly higher scores than those

graded by the strictly auditory method (Kuhn and Schroeder, 1971).

Multi-sensory learning techniques have also proven to be helpful in the

development of a foreign language. Drills that contain visual, auditory, and even

tactile involvement by students improve their comprehension of the foreign

language (Kalivoda, 1978). Multi-sensory techniques are very helpful when

employed in adult ESL classrooms. These students speak various languages other

than English and many have no formal schooling background in reading or

writing. Teachers must employ activities that involve many, if not all, the senses

in order to teach these non-academic students to communicate (Bassano, 1982).

Manipulatives: A Specific Type of Multi-sensory Math Tool

Most of the literature with regard to multi-sensory learning in elementary

mathematics involves manipulative materials. These materials, while they have a

broad definition, are one way that teachers can incorporate multi-sensory learning

into their elementary classrooms. The term “manipulative” has varying, but

similar, definitions. Reys (1971) defines manipulative materials as objects that the

students can feel, touch, and handle. They are learning tools that appeal to several

senses and are distinguished by physical involvement by the student.

Manipulatives, according to Chester, Davis, and Reglin (1991), are anything that

a student can move either physically or mentally in order to discover the solution

to a problem. Sowell (1989) describes two types of manipulatives, concrete and

pictorial representations. Concrete manipulatives are items that students can work

with directly. And pictorial manipulatives can be audiovisual presentations,

observed demonstrations with concrete manipulatives done by others, or even

pictures in printed materials. Examples of manipulative devices given by the

National Council of Teachers of Mathematics (NCTM) (1973) are colored beads,

blocks, and rods, place value devices, games and puzzles, and measurement

devices. Specific attention to manipulative use was again reiterated in the

Principles and Standards for School Mathematics (NCTM, 2000).

Jenny R. RAINS Egitimde Kuram ve Uygulama

Catherine A. KELLY Journal of Theory and Practice in Education

Robert L. DURHAM 2008, 4 (2):239-252

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© Çanakkale Onsekiz Mart Üniversitesi, Eğitim Fakültesi. Bütün hakları saklıdır.

249

Becoming increasingly available are virtual manipulatives, or

electronically-based materials which are interactive, computer-based tools that

function much like the more commonly used tangible, concrete manipulatives,

but are maneuvered by the computer mouse (Spicer, 2000). Some consider these

dynamic, interactive visual representations (manipulatives) the wave of the future

(Moyer, Bolyare, and Spikell, 2002) and point out that although there are many

computer-generated images being called virtual, they are truly not appropriate for

this category unless they are interactive and dynamic (p. 373). In multi-sensory

mathematics, interaction with and among materials, both concrete and virtual, is

pivotal for deep understanding.

Using concrete materials in class to enhance students understanding is

encouraged by the NCTM Standards (2000), which applies to virtual, interactive

materials as well. In congruence with Piaget (1965), young children can only

sense that which exists in their presence, so concrete objects with which they can

explore are important to have available (NCTM, 1973). According to studies

done by Suydam and Higgins (1984), lessons with manipulatives are more likely

to produce greater achievement by the students than lessons that do not

incorporate manipulatives. However, to be effective, manipulatives do not

necessarily need to be handled directly by the student for all of the lessons.

Hiebert (1989) recommends counting craft sticks as a manipulative device as a

means of developing meaning in numeric symbols for young children.

As a final comment regarding any teaching material, it is not adequate for

teachers to simply incorporate the materials into the mathematics lessons without

specific strategies for their use. In congruence with Vygotsky's (1978) theory of

the ZPD, the literature is showered with warnings to teachers about the

importance of the teacher's role in the use of these materials, specifically

manipulatives (e.g., Baroody, 1989; Hiebert, 1988; Marzola, 1987; NCTM, 1973;

Reys, 1971). For example, when teachers use manipulatives in teaching

mathematical concepts without reflecting on what these concrete objects

represent (one block equals the numeral 1; two tens [base ten] equals the numeral

20), the real mathematics concept may be lost or misconstrued (Clements, 1999).

And, when used appropriately, manipulatives can clarify an otherwise unclear

concept such as dividing up a pizza to help young children mentally and

physically represent the concept of half and quarter of a whole.

CONCLUSION

Based on both theoretical foundation and experimental support, it is

important for teachers to be aware that not all students in an elementary

mathematics class are at equal levels of mental maturity. For this reason, multiple

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modalities of presentation need to be incorporated into math lessons. Multi-

sensory techniques and materials can help satisfy this requirement. If teachers

continually approach new math concepts by only one means of representation,

many of their elementary students will not grasp them. This may leave the

children unprepared for mathematics lessons yet to come.

We know that using multi-sensory materials such as manipulatives can

give a child a tool which he can use until he is truly ready to comprehend difficult

mathematical concepts. It may also be true that these same tools can expedite the

child's transition through these stages. The previously mentioned study with

remedial students in Australia by Thorton, et al (1982) serves to emphasize this

idea. Further, if multi-sensory tools can expedite the developmental process for

remedial level students, then why should it not be generalized to the regular

population as well? While there is only limited support for this from other subject

areas, it seems a logical conclusion and should be the topic of future studies.

It has always been important for teachers to attempt to reach the

understanding of all of their students; however, now legislation has placed even

more urgency upon it. It is encouraging to see through emerging research by

Kelly, Durham, and Rains (2004) that teachers are turning towards the use of

multi-sensory materials, manipulatives in particular, in the elementary

mathematics classroom. By utilizing more avenues of introducing new math

concepts to a class, multi-sensory teaching techniques can assist teachers in

"translating" novel, abstract mathematics concepts to young learners.

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