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