EFFECTIVENESS OF CONCRETE AND COMPUTER SIMULATED
MANIPULATIVES ON ELEMENTARY STUDENTS' LEARNING SKILLS AND
CONCEPTS IN EXPERIMENTAL PROBABILITY
FELICIA M. TAYLOR
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
EFFECTIVENESS OF CONCRETE AND COMPUTER SIMULATED
MANIPULATIVES ON ELEMENTARY STUDENTS' LEARNING SKILLS AND
CONCEPTS IN EXPERIMENTAL PROBABILITY
FELICIA M. TAYLOR
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
This dissertation is dedicated to my mother,
Laura Ann Roberts,
whose love and belief I was never without
to my husband,
Reginald Eugene Taylor,
my mate and friend.
My gratitude is extended to the many people who provided a great deal of support
for me before and during the dissertation process.
To Dr. Thomasenia Lott Adams, my chair and mentor who provided support and
educational challenges, I owe my future contributions as a mathematics educator. I am
thankful to my committee members, each of whom provided guidance throughout the
dissertation process. To Dr. Miller, I owe great thanks for his statistical and professional
advice. To Dr. Sebastian Foti, and Dr. Li-Chien Shen, I owe thanks for their academic
I am especially indebted to Dr. Israel Tribble of the Florida Endowment Fund for
Higher Education for supporting me as a McKnight Doctoral Fellow financially and
I am forever grateful to my family, especially my husband Reginald Eugene
Taylor, who was all I needed him to be; my mother, Laura Roberts; my siblings, Carolyn,
Prudence, Eric, and Chantell; and my father-in-law, Eddie Lee Taylor, whom I
considered my father. These very special people have provided me with the love and
support that only family can provide. I owe a continuous effort to improve self and
empower others through education.
TABLE OF CONTENTS
1 DESCRIPTION OF THE STUDY 1
Purpose and Objectives of the Study 12
Rationale for the Study 13
Significance of the Study 14
Organization of the Study 16
2 LITERATURE REVIEW 17
Theoretical Framework 17
Constructivism-Grounded Research 22
Concrete Manipulad ves for Teaching Mathematics 44
Technology and Simulated Manipulatives in the Classroom 56
3 METHODOLOGY 71
Overview of the Study 71
Research Objective 71
Description of the Research Instruments 72
Pilot Study 74
Research Population and Sample 78
Data Analysis 82
Other Findings 89
Limitations of the Study 91
5 CONCLUSION 93
A EXPERIMENTAL PROBABILITY INSTRUMENT 100
B EXPERIMENTAL FRACTION INSTRUMENT 108
C TREATMENT LESSONS 112
D TEACHER CONSENT FORM 140
E PARENTAL CONSENT FORM 142
F STUDENT CONSENT SCRIPT 145
Abstract of Dissertation Presented to the Graduate School of the
University of Florida in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy
EFFECTIVENESS OF CONCRETE AND COMPUTER SIMULATED
MANIPULATIVES ON ELEMENTARY STUDENTS'
LEARNING SKILLS AND CONCEPTS IN
Felicia M. Taylor
Chairperson: Thomasenia Lott Adams
Major Department: School of Teaching and Learning
This study was designed to investigate the impact of using concrete
manipulatives and computer-simulated manipulatives on elementary students' learning
skills and concepts in experimental probability. Of primary interest to the researcher was
students' ability to predict outcomes of simple experiments. A secondary interest was the
incidental fraction learning that might occur as students engage in the experimental
The research sample consisted of 83 fifth-grade students. There were four
treatment groups. Treatment Group I students received computer instruction. Treatment
Group II students received manipulative guided instruction. Treatment Group III
students received both computer instruction and manipulative guided instruction.
Treatment Group IV served as a control group and received traditional instruction. The
researcher used an analysis of covariance (ANCOVA). After controlling for initial
differences, the researcher concluded that students experiencing the computer instruction
only significantly outperformed students experiencing a traditional instructional
environment regarding experimental probability learning skills and concepts.
DESCRIPTION OF THE STUDY
While the curriculum in the United States continues to focus on basic skills well
past the fourth grade, classrooms in Japan and Germany emphasize more advanced
conceptsincluding algebra, geometry, and probability (Riley, 1998). The aim of the
United States is to be first in science and mathematics, teachers may need to introduce
children to more advanced topics at earlier grade levels. Principles and Standards for
School Mathematics, published by the National Council of Teachers of Mathematics
(NCTM, 2000), recognizes probability as an advanced topic that should be learned by
students in the K-12 curriculum. In the Principles and Standards, emphasis is placed on
introducing the same topics and concepts throughout the Pre K-12 curriculum. Young
children need to explore the process of probability. The study of probability in the early
grades provides a stronger foundation for high school students (NCTM, 2000). Bruner
(1960) stated in his discussion of the spiral curriculum,
If the understanding of number, measure, and probability is
judged crucial in the pursuit of science, then instruction in these
subjects should begin as intellectually honestly and as early as
possible in a manner consistent with the child's forms of thought.
Let the topics be developed and redeveloped in later grades.
He also believed,
If one respects the ways of thought of the growing child, if one is
courteous enough to translate material into his logical forms and
challenging enough to tempt him to advance, then it is possible to
introduce him at an early age to the ideas and styles that in later life
make an educated man. (p. 52)
The foregoing reasons seem to point to the teaching of experimental probability concepts
beginning in the elementary grades.
"As with other beautiful and useful areas of mathematics, probability has in
practice only a limited place in even secondary school instruction" (Moore, 1990, p. 119).
The development of students' mathematical reasoning through the study of probability is
essential in daily life. Probability represents real-life mathematics. Probability also
connects many areas of mathematics, particularly numbers and geometry (NCTM, 1989).
"Research in medicine and the social sciences can often be understood only through
statistical methods that have grown out of probability theory" (Huff, 1959, p. 11). Moore
Probability is the branch of mathematics that describes randomness.
The conflict between probability theory and students' view of the
world is due at least in part to students' limited contact with
randomness. We must therefore prepare the way for the study of
chance by providing experience with random behavior early in the
mathematics curriculum, (p. 98)
An understanding of probability theory is essential to understand such things as politics,
weather reports, genetics, state lotteries, sports, and insurance policies. As shown Table
1, the list of questions presented in the different areas shows the need for experimental
probability. These questions require considerations of probabilities and what they mean
Real Life Mathematics Using Experimental Probability
Politics A random survey of voters show that 381 out of 952 are planning
to vote for Candidate McCain in the primary election. What is
the probability that a randomly selected voter will vote for
Weather Reports The probability of rain today is 70%. What are the odds in favor
of rain? What are the odds against rain?
Genetics In genetics, we can use probability to estimate the likelihood for
brown-eyed parents to produce a blue-eyed child. The gene for
brown eyes in humans is dominant over the gene for blue eyes. If
each parent has a dominant brown and a recessive blue gene,
what is the chance that their child will have blue eyes?
State Lotteries A lottery consists of choosing 6 numbers, the first 5 numbers
being one of the digits 0-9 and the sixth number being from 0-
39. How many possible sets of lottery numbers are available for
selection? If the winning numbers are randomly selected, what
is the probability of winning with a single ticket?
Sports Softball statistics (NCTM, 1989, p. Ill)
X ote 1
Above is the record of a player's last 100 times at bat during the
softball season. What is the probability the player will get a
home run? What is the probability that the player will get a hit?
Insurance Policies Insurance companies use the principle of probability to
determine risk groups. One way of figuring out what insurance
you need is by looking at the probability of the event and the
financial loss it would cause. What is the probability of men
between 18-25 years of age being involved in a car accident?
Determine if the cost of each occurrence is high or low.
The inclusion of activities dealing with experimental probability in the elementary
school enhances children's problem-solving skills and provides variety and challenges for
children in a mathematics program (Kennedy & Tipps, 1994). Current and past
recommendations for the mathematics curriculum identify experimental probability as
one of several critical basic skill areas that should occupy a more prominent place in the
school curricula than in the past (National Council for Supervisor of Mathematics
(NCSM), 1989; Mathematical Sciences Education Board [MSEB], 1990; Willoughby,
1990; NCTM, 2000).
From a historical perspective, members of the Cambridge Conference on School
Mathematics (1963) also acknowledged the role probability and statistics played in our
society. The Cambridge Conference was an informal discussion of the condition of the
mathematics curriculum in the United States at the elementary and secondary level. The
members of the conference addressed revisions to the mathematics curriculum. They
recommended that probability and statistics not only be included as part of the modem
mathematics of that day; but also these were recommendations that they considered for
1990 and 2000. Other researchers have also suggested that elements of statistics and
probability be introduced in the secondary school curriculum and possibly at the
elementary level as part of the basic literacy in mathematics that all citizens in society
should have (Schaeffer, 1984; Swift, 1982).
Therefore, changes are being made today to introduce probability into the
elementary school curriculum (NCTM, 2000). Experience with probability can
contribute to students' conceptual knowledge of working with data and chance (Pugalee,
1999). This experience involves two types of probability-theoretical and experimental.
There may be a need for students to be exposed to more theoretical models involving
probability. Theoretical models organize the possible outcomes of a simple experiment.
Some examples of theoretical models may include making charts, tree diagrams, a list, or
using simple counting procedures. For example, when asked to determine how many
times an even number will appear on a die rolled 20 times, students can list the ways of
getting an even number on a die (2, 4, 6) and may conclude that one should expect an
even number one-half of the time when a die is rolled. Then, students can roll the die 20
times, record their actual results and make conclusions based on their experiment.
Experimental models are then the actual results of an experiment or trial. Another
example that involves experimental modeling is the following:
If you are making a batch of 6 cookies from a mix into which you
randomly drop 10 chocolate chips, what is the probability that you will
get a cookie with at least 3 chips? Students can simulate which cookies
get chips by rolling a die 10 times. Each roll of the die determines
which cookie gets a chip. (NCTM, 1989, pp. 110-111)
The researcher's interest was with experimental probability and not on theoretical
probability but used the term probability throughout the study, since there is no
way to teach experimental without theoretical.
A secondary interest with experimental probability involves incidental fraction
learning. Incidental learning is unintentional or unplanned learning that results from other
activities. Proponents of incidental learning believe that effective learning can take place
this way. For example, the incidental learning theory suggests that children learn
mathematics better if it is not methodically taught to them (Clements & Battista, 1992).
In addition, Brownell (1935) suggested that incidental learning could help counteract the
practice of teaching arithmetic as an isolated subject. Studying and solving probability
problems give students many opportunities for practicing and reinforcing previously
learned concepts of basic mathematics skills (Horak & Horak, 1983; NCTM, 2000) that,
according to the Allen, Carlson, and Zelenak (1996), are inadequate. "For example,
fraction concepts play a critical role in the study of probability ... as well as whole
number operations and the relationships among fractions, decimals, and percents"
(NCTM, 1989, p. 111).
There are also opponents of incidental learning. These opponents believe that
although the study of probability may provide incidental learning for fractions, it may not
be enough to develop students' ability and concepts of fractions. They also suggest that
incidental learning does not provide an organization for the development of meaningful
concepts and intelligent skills in which the development of authentic mathematic ability
is possible (Brownell, 1935).
The use of different modes of representation can promote meaningful learning,
retention, and transfer of mathematical concepts (Lesh, 1979). One example includes the
use of manipulatives. Manipulatives may be physical objects (e.g., base ten blocks,
algebra tiles, pattern blocks, etc.) that can be touched, turned, rearranged, and collected.
Manipulatives may be real objects that have social application in everyday situations, or
they may be objects that are used to represent an idea. Manipulatives enhance
mathematics achievement for students at different grade levels (Suydam, 1984). Various
types of manipulatives used for teaching and learning mathematics are presented as
follows: tangrams, cuisenaire rods, geoboards, color tiles, pattern blocks, coins, color
spinners, number spinners, snap cubes, base ten blocks, dice, fraction strips, dominoes.
clock dials, color counters, and attribute blocks. Examples of the mathematics content
for some of the manipulatives are given:
1. Geoboards can be used for activities involving geometric shapes,
symmetry, angles and line segments, and perimeter and fractions in a
2. Tangrams can be used for creating and comparing size and shape,
measurement, properties of polygons, and transformations as well as
developing spatial visualization.
3. Color tiles can be used to teach topics such as fractions, percents, and
ratios; probability, sampling, graphing, and statistics; and
Students' knowledge is strongest when they connect real-world situations,
manipulatives, pictures, and spoken and written symbols (Lesh, 1990).
Manipulatives help students construct meaningful ideas and learn mathematics more
easily (Bums, 1996; Clements & McMillan, 1996).
Support for the use of manipulatives come from Piagetian theory, in
which cognitive development is described as moving from concrete to
abstract, through a series of developmental stages that are roughly age-
related. A concrete-operational child cannot handle abstract concepts
before arriving at the appropriate stage. However, with manipulatives,
it is possible for such a student to take the first steps towards exploring
the concepts; manipulatives are concrete introductions to abstract
ideas. (Perl, 1990, p. 20)
The use of manipulative materials appears to be of definite importance in how well
children understand and achieve in mathematics in different content areas.
Technology takes a special place in the child-driven learning environment as a
powerful tool for children's learning by doing (Strommen & Lincoln, 1992). The NCTM
(2000) technology principle indicated that mathematics instructional programs should use
technology to help all students understand mathematics and prepare students to use
mathematics in a growing technological world. The use of computer technology is an
integral part of the vision of the future of teaching and learning mathematics. One of the
position statements of NCTM (1998) is as follows:
The appropriate use of instructional technology tools is central to the
learning and teaching of mathematics and to the assessment of
mathematics learning at all levels. Technology has changed the ways
in which mathematics is used and has led to the creation of both new
and expanded fields of mathematical study. Thus, technology is the
driving change in the content of mathematics programs, in methods for
mathematics instruction, and in the ways that mathematics is learned
and assessed. [On-line]
Inclusion of technology in classroom projects enhances the authenticity of the task
Technological aids allow greater realism in the classroom (MSEB, 1990).
Progression in technology has increased the boundaries of mathematics and emphasized
the importance of the integration of technology in the mathematics curriculum. At every
grade level, there are ways in which students can experience mathematics with
technology that provide deeper and more substantial understanding (NCTM, 2000).
Children's traditional classroom tools-pencil, notebooks, and texts-are still vital
but inadequate for children to adequately solve problems, completely modify ideas, and
thoroughly extend their learning experience. "Rather than simply listening to teachers
talk, watching the teacher write symbolic procedures on the board, and doing pencil and
paper practice, children should learn through meaningful hands-on activities with
manipulative materials, pictures, and technology" (Weibe, 1988, p. 66). "Computers can
provide an important link in the chain, a connection between the concrete manipulatives
and the abstract, symbolic, paper and pencil representation of the mathematical idea"
(Perl, 1990, p. 21). From a Vygotskian perspective, "appropriately implemented,
technology offers tools to the classroom that may promote high level thinking skills and
support concept development" (Harvey & Chamitski, 1998). Clements (1998) also
believed that the computer can provide practice on arithmetic processes and foster deeper
Computer simulations can help students develop insight and confront
misconceptions about probabilistic concepts. "With the computer and an LCD [Liquid
Crystal Diode] panel such as the PC [Personal Computer] viewer, this process becomes
easier and more powerful" (Perl, 1990, p. 22).
Probability concepts and their meaning depend not only on the level of
theory and on their representation but also the means of working with
them. Tools to represent knowledge or to deal with knowledge have a
sizeable impact on subjects' individual formation of this knowledge.
Clearly, the advent of computers marks a huge change ... the
computer will radically change available means of working on
problems and means of representation. (Kapadia & Borovcnik. 1991,
An advantage of simulation by computer as compared to physical random
generators is the possibility for larger numbers of repetitions and extensive exploration of
assumptions (Kapadia & Borovcnik, 1991). Although this is true, students in the
elementary grades need to work with random events first generated through physical
simulation and then followed by simulations done electronically using available software.
This would lead to deeper understandings as students reach middle and high school
(NCTM, 2000). Moore (1990) stated.
The first step toward mathematical probability takes place in the
context of data from chance devices in the early grades. Computer
simulation is very helpful in providing the large number of trials
required if observed relative frequencies are to be reliably close to
probabilities, (p. 120)
Moore also stated, "Simulation, first physical and then using software can demonstrate
the essential concepts of probability" (p. 126).
The constructivist perspective on learning suggests that knowledge is not received
passively; it has to be built up (Maher, 1991). Constructivists generally agree on the
following basic tenets (Noddings, 1990);
1. All knowledge is constructed. Mathematical knowledge is
constructed, at least in part, through a process of reflective
2. There exist cognitive structures that are activated in the
processes of construction. These structures account for the
construction; that is they explain the result of cognitive
activity in roughly the way a computer program accounts for
the output of a computer.
3. Cognitive structures are under continual development.
Purposive activity induces transformation of existing
structures. The environment presses the organism to adapt.
4. Acknowledgement of constructivism as a cognitive position
leads to the adoption of methodological constructivism.
a. Methodological constructivism in research develops
methods of study consonant with the assumption of
b. Pedagogical constructivism suggest methods of teaching
consonant with cognitive constructivism (p. 10).
Essentially, students construct knowledge from their own experiencespersonal and
academic. One goal from a constructivist perspective is for students to develop
mathematic structures that are more complex, abstract, and powerful than what
students currently possess (Clements & Battista, 1990; Cobb, 1988). Teachers who
adhere to a constructivist perspective emphasize understanding and building on
students' thinking. A comparison of things that teachers with a constructivist
perspective do differently compared to a traditional curriculum is listed in Table 2.
Differences Between Two Teaching Styles
Emphasis is on basic skills
Emphasis is on big concepts
Teachers follow fixed curriculum
Teachers allow student questions
to guide the curriculum.
Activities rely heavily on texts
Activities rely on primary
sources of data and manipulatives
Teachers give students
information; students are
viewed as black slates.
Teachers view students as
thinkers with emerging theories
about the world.
Students mostly work alone.
Teachers look for correct answers
to assess learning.
Students primarily work in groups.
Teachers seek students' points of
view to check for understanding.
Assessment is viewed as separate
from teaching and occurs mostly
Assessment is interwoven with
teaching and occurs through
observation and student exhibits and
Concrete manipulatives and computers can be a powerful combination in the
mathematics curriculum (Perl, 1990). There has been a growing need to investigate how
to integrate the use of computers with manipulative materials to facilitate mathematics
instruction and to meet the needs of different learners (Berlin & White, 1986). Aligning
computer usage and concrete manipulatives with the constructivist perspective may help
to improve students' performance in mathematics. For example, manipulative materials
appeal to several senses and are characterized by a physical involvement of pupils in an
active learning situation (Reys, 1971).
In addition, computers also serve as a catalyst for social interaction with other
studentsallowing for increased communication (Clements, 1998). Social interaction
constitutes a crucial source of opportunities to learn mathematics (Piaget, 1970).
Involving children in the process of doing mathematics and using concrete materials
relates to their development where children are actively thinking about the mathematics.
While children work with objects and discuss what they are doing, they begin to
recognize a sense of relationship among mathematics concepts (Suydam, 1984) that
support their learning.
Purpose and Objectives of the Study
The purpose of this study was to examine the effectiveness of concrete
manipulates and computer-simulated manipulates on elementary students'
experimental probability learning skills and concepts. The objectives of the study were
1. To determine the effectiveness of computer-simulated manipulatives on
elementary students' learning skills and concepts in experimental
2. To determine the effectiveness of concrete manipulatives on elementary
students' learning skills and concepts in experimental probability,
3. To determine the effectiveness of both concrete manipulatives and
computer-simulated manipulatives on elementary students' learning skills
and concepts in experimental probability, and
4. To determine if incidental fraction learning occurs as students engage in the
experimental probability experiences.
Rationale for the Study
The most important use of studying probability is to help us make decisions as we
go through life (Newman, Obremski, & Schaeffer, 1987). For example, in issues of
fairness, students may pose a question based on claims of a commercial product, such as
which brand of batteries last longer than another (NCTM, 2000).
Benefits of Probability Knowledge in Early and Later Grades
The committee for the Goal for School Mathematics, the report of the 1963
Cambridge Conference, recommended introducing basic ideas of probability very early in
the school program. The study of probability allows a learner to make sense of
experiences involving chance.
If Studied in Early Grades. Teachers Need Information
As previously mentioned in the Introduction, Bruner (1960) believed that by
constantly reexamining material taught in elementary and secondary schools for its
fundamental character, one is able to narrow the gap between advanced knowledge and
elementary knowledge. Bruner said that if you wish to teach calculus in the eighth grade,
then begin in the first grade by teaching the kinds of ideas and skills necessary for the
mastery of calculus in later years. This is one of the changes made in the Principles and
Standardsto include the same concepts throughout all grade levels, teaching necessary
skills appropriate at that grade level, and continuing at each grade level. Bruner believed
concepts should be revisited at increasing levels of complexity as students move through
the curriculum rather than encountering a topic only once (Schunk, 1991). This would
also apply to teaching probability. If students are to understand probability at a deeper
level in high school and college, then the skills necessary for its mastery must begin in
the elementary grades (NCTM, 2000). Therefore, it is important that evidence be
available to help teachers develop appropriate topics in the elementary school curriculum.
The kind of reasoning used in probability is not always intuitive, and so it may not be
developed in young children if it is not included in the curriculum (NCTM, 2000).
Lack of Research in Early Grades
In the past, the teaching of probability reasoning, a common and important feature
of modem science, was rarely developed in our educational system before college
(Garfield & Ahlgren, 1988; Bruner, 1960). Currently, probability and statistics are often
included in the secondary school curriculum only as a short unit inside a course
(Shaughnessy, 1992,1993). Recommendations concerning school curricula suggest that
statistics and probability be studied as early as elementary school (MSEB. 1990; NCTM,
2000). However, much of the existing research on probability is on the conception of
beginning college students or secondary school students (Shaughnessy, 1993). Very little
research in the development of children's understanding of probability concepts has been
done with young children (Watson, Collis, & Moritz, 1997). The learning of probability
in early grades will provide students with a stronger foundation for further study of
statistics and probability in high school.
Significance of the Study
Little research has been conducted about how to teach probability effectively in
the early grades. Much of the research done on probabilistic intuitions of students from
elementary school through college level has been conducted in other countries such as
Europe and Germany (Garfield & Ahlgren, 1988). If students in the United States are to
be first in mathematics and science, there must be adequate research with a population of
U.S. students to develop a better understanding of the difficulties these students
encounter in understanding probability and statistics. It is not possible to learn much
about learners unless we leam about learners' learning specific mathematics (Papert,
1980). This study provides information about children's learning of probability.
Many middle and high school students have difficulty understanding how to
report a probability. Inadequacies in prerequisite mathematics skills and abstract
reasoning are part of the problem (Garfield & Ahlgren, 1988). These difficulties may be
due to little or no curriculum instruction for probability given at the elementary school
level. The challenge is to relate to children and engage them in learning experiences in
which they construct their own understanding of probability concepts.
There is a scarcity of literature in terms of teaching and learning probability from
a constructivist perspective at the elementary school level. Recent research efforts
emphasize constructivist approaches to teaching and learning mathematics (Thornton &
Wilson, 1993). One implication for a constructivist theory of knowledge, according to
Confrey (1990), is that students are continually constructing understanding of their
experiences. In addition, this is the first research study conducted using the software
developed by Drier (2000) in whole class instruction. Previous researchers (Clements &
McMillan, 1991; Jiang, 1993) have concluded that computer-simulated manipulatives
enhance students' mathematics learning ability. However, most of those studies involved
older children. This researcher used computer-simulated manipulatives at the elementary
level to teach probability. Because children's thinking processes are much less
developed, there is doubt about whether computer-simulated manipulatives are similarly
effective for young children. Usually when probability is taught, it is done using concrete
manipulatives. Because much of the research on probability learning skills and concepts
has focused on university-level students, additional research is needed to create a
framework from which to understand elementary students' probability learning better.
Organization of the Study
This chapter included the purpose of the study, its rationale, and its significance to
the field of mathematics education. The review of relevant literature presented in
Chapter 2 includes the theoretical framework, studies of probability, studies involving
concrete manipulatives, and studies involving technology with a focus on computer-
simulated manipulatives. Reported in Chapter 3 are the design and methodology of the
study. Chapter 4 contains the results of the analysis and the limitations of the study. A
summary of the results, implications, and recommendations for future research is
presented in Chapter 5.
In this chapter, the researcher presents a review of the relevant literature. The
researcher focuses on studies pertaining to the following areas: constructivist theory,
concrete manipulatives, studies involving technology with a focus on computer-simulated
manipulatives, and studies of probability.
A theory of instruction, which must be at the heart of educational psychology, is
principally concerned with how to arrange environments to optimize learning and transfer
or retrievability of information according to various criteria (Bruner, 1960). Learning is a
process of knowledge construction, dependent on students' prior knowledge, and attuned
to the contexts in which it is situated (Hausfathers, 1996). This definition is what the
constructivist believes. Constructivism focuses our attention on how students learn.
Constructivism implies that much learning originates from inside the child (Kamii &
Ewing, 1996). The students construct their own understanding of each mathematical
concept. Constructivism appears to be a powerful source for an alternative to direct
instruction (Davis, Maher, & Noddings, 1990). Mathematical learning is an interactive
and constructive activity (Cobb, 1988) that Constructivists in mathematics education
argue that cognitive constructivism implies pedagogical constructivism; that is,
acceptance of constructivist premises about knowledge and knower implies a way of
teaching that acknowledges learners as active knowers (Noddings, 1990). Direct
instruction does not suit children's thinking because it is based quite often on a direct-
teaching and practice model (Broody & Ginsburg, 1990). With direct instruction one
finds a relatively familiar sequence of events: telling, showing, and doing approach
(Confrey, 1990). It moves quickly, often overlooking students' development, preventing
assimilation of higher cognitive skills (Broody & Ginsburg, 1990; Confrey, 1990).
We must help students develop skills that will enable them to be more productive
when faced with real world situations. When students do not have the skills to solve
everyday problems, they have difficulty finding a solution. Owen and Lamb (1996) gave
the following example:
John and Terry worked at a fruit stand during the summer. The two ran
the register, which has a scale to weigh the fruit. One day the fruit stand
lost electricity. The customers wanted to buy bananas, apples, and
tangerines. John and Terry were not sure what to do without any
electricity. They suggested waiting until the owner returned. Terry
saw an old scale in the comer of the store. They decided to use the scale
and figure out the cost of the fruit using paper and pencil. John and
Terry tried to remember formulas taught to them in school earlier that
year but were stumped. The bananas cost 39c per pound and the customer
had 3-1/2 pounds of bananas. One boy suggested that when you have
fruit problems, you divide the money by the weight of the fruit. They
did this, came up with 1 lc, and thought that was incorrect because of the
low price. They continued in this manner until they finally thought they
found the right answer. John and Terry's teachers used teaching
techniques that only provided them with minimal skills. Faced with a real
life problem, John and Terry could not rely on skills taught to them by
Students learn through a social process in a culture or a classroom that involves
discovery, invention, negotiation, sharing, and evaluation (Clements & Battista, 1990)
and that actively constructs their learning. Social interactions constitute a crucial source
of opportunities to learn mathematics that in the process of constructing mathematical
knowledge involve cognitive conflict, reflection, and active cognitive reorganization
(Piaget, 1970). Vygotsky (1978) had this to say about students' social interaction:
Any function in the child's cultural development appears twice or
on two planes. First, it appears on the social plane, and then on
the psychological plane. First it appears between people as an
interpsychological category, and then within the child as an
intrapsychological category ... social relations or relations,
among people genetically underlie all higher cognitive function
and their relationships, (p. 57)
Vygotsky also stated,
An essential feature of learning is that it creates the zone of
proximal development; that is, learning awakens a variety of
internal development processes that are able to operate only when
the child is interacting with people in his environment and in
cooperation with his peers, (p. 90)
To promote meaningful learning, teachers must know how to tailor instruction so
that it meshes with children's thinking (Brownell, 1935; Dewey, 1963; NCTM, 1989;
Piaget, 1970). When children have to give up their own thinking to obey the rules of the
teacher (Kamii & Ewing, 1996), they are forced to compromise their own thinking.
Children compromise their inventive, problem-solving methods for mechanical
applications of arithmetic skills (Carpenter, 1985; Resnick & Omanson, 1987).
Eventually, students may be interested only in getting the correct solution. In a
constructivist classroom, emphasis is not on wrong or right answers; students construct
meaning from preexisting knowledge within the student. Completing objectives or
preparing for tests is not the focus in the constructivist classroom. These do not
sufficiently determine if learning occurred or if there are any changes in a student's
Schulte (1996) described a curriculum unit on weathering and erosion taught in a
constructivist classroom. Students make predictions using hands-on exploration activities
with unexpected outcomes that arouse curiosity. Terms have not yet been introduced.
Students discuss the results of the activities with group members. The teacher poses
questions looking for misconceptions, and activities are sequenced to challenge students.
Once students begin to make connections between activities and understand the concept,
the vocabulary is introduced. Questions focus on application and are used to look for
student understanding. Alternative forms of assessment are used. For example, the
students are to make a book on erosion including pictures, drawings, and text using
landforms affected by erosion. Assessment may also focus on issues in society. For
example, students may be asked to research methods that people are using to stop erosion
in certain areas and design a beachfront home to build, considering the concept of
erosion. Schulte believed that learning depends on the shared experiences of students,
peers, and the teacher. He also felt that collaboration with others is important and that
cooperative learning is a major teaching method used in the constructivist classroom.
Using the constructivist premise would imply there are many roads to most
solutions or instructional endpoints (Noddings, 1990). Rote responses do not represent
students' understanding of concepts. Emphasis on instruction forces us to probe deeper
into students' learning and to ask the following questions (Noddings, 1990):
1. How firm a grasp do students have on the content?
2. What can students do with the content?
3. What misconceptions do students entertain?
4. Even if wrong answers are visible, are students constructing
their learning in a way that is mathematically recognizable?
These address only a few questions at the heart of constructivist learning and teaching.
Most curriculums pay little or no attention to the developmental abilities of children.
Consequently, children do not learn much of what is taught to them (Brooks, 1987).
In a constructivist classroom there is consensus that the following things are
noticeable (Anderson, 1996; Glatthom, 1994; Schulte, 1996):
1. Teacher asks open-ended questions and allows wait time for
2. Higher-level thinking is encouraged.
3. Students are engaged in experiences that challenge hypotheses
and encourage discussion.
4. Classes use raw data, primary sources, manipulatives, physical
and interactive materials.
5. Constructivism emphasizes the importance of the knowledge,
beliefs, and skills an individual brings to the experience of
learning; recognizes the construction of new understanding as a
combination of prior learning, new information and readiness to
In order for students to establish a strong foundation in mathematics, teachers
need to provide experiences to students that involve forming patterns and relations, which
are essential in mathematics (Midkiff & Thomasson, 1993). Caine and Caine (1991)
posited the following:
Children live with parallel lines long before they ever encounter
school. By the time parallel lines are discussed in geometry, the
average student has seen thousands of examples in fences, windows,
mechanical toys, pictures, and so on. Instead of referring to the
parallel lines students and teachers have already experienced, most
teachers will draw parallel lines on the blackboard and supply a
definition. Students will dutifully copy this "new" information into a
notebook to be studied and remembered for test. Parallel lines
suddenly become a new abstract piece of information stored in the
brain as a separate fact. No effort has been made to access the rich
connections already in the brain that can provide the learner with an
instant "Aha!" sense of what the parallel lines they have already
encountered mean in real life, what can be done with them, and how
they exist than as a mathematical abstraction, (p. 4)
From a constructivist perspective, mathematical learning is not a process of internalizing
carefully packaged knowledge but is instead a matter of reorganizing activity, where
activity is interpreted broadly to include conceptual activity or thought (Koehler &
The constructivist teaching experiment is a technique designed to investigate
childrens mathematical knowledge and learning experiences (Cobb & Steffe, 1983;
Cobb, Wood, & Yackel, 1992; Hunting, 1983). Several studies in mathematics
education, inspired by the constructivist perspective, provided some information on
childrens thinking in mathematics (Bednarz & Janvier, 1988; Carpenter, Moser, &
Rombert, 1982; Ginsburg, 1983; Schoenfeld, 1987).
Bednarz and Janvier (1988) presented the results of a 3-year longitudinal study
using 39 children as they progressed from the first to the third grade. The main purpose
of the study was to develop a constructivist approach to the concept of numeration and its
learning, leading children to build a meaningful and efficient representation of numbers.
In order to evaluate the effects the constructivist approach had on children's
understanding of numeration and difficulties encountered, Bednarz and Janvier were
challenged to understand the way children were thinking. The researchers developed a
constructivist approach that "helped children to progressively construct a significant and
operational system for the representation of number" (p. 329). Of interest to the
researcher was how students carry out operations involving groupings with concrete
materials, and how students communicate representation of numbers. The constructivist
"approach developed required a knowledge of children's thinking (difficulties,
conceptions encountered by them), and continuous analysis of procedures and
representations used by them in learning situations" (p. 302). Quantitative results
revealed that the abilities and procedures developed by the children were still present.
Most of the children using a constructivist approach were able to identify groupings and
relations between them.
Researchers have recommended different teaching strategies to promote students'
meaningful learning (Anderson, 1987; Anthony, 1996; Confrey, 1990; Minstrell, 1989;
Novak & Gowin, 1984). Constructivism commits one to teaching students how to create
more powerful constructions (Confrey, 1990). Confrey wanted to make a model of the
practices of a teacher committed to constructivist beliefs. A Fundamental Mathematics
Concepts class was chosen for the study. The study took place at a SummerMath
program, an experimental summer program for young girls in high school. The focus of
the study was on the teacher-student interactions. There were 11 students in the study
ranging from 9th to 11th grades. In order to be in this program, students had to score less
than 45% on a multiple choice placement test consisting of 25 items from the high school
curriculum. The students in the study worked in pairs on the curriculum materials
provided each day. The purpose of the work was to suggest that alternative forms of
instruction can exist in mathematics that differ in their basic assumptions from the
tradition of "direct instruction" (p. 122). Data were taken from videotapes of the
interactions. The method used was the idea of reflection in action. From the results,
students were more persistent, more confident, and asked more questions in class.
Follow-up evaluations of the program also indicated that students' scores on the
mathematics part of the Standardized Achievement Test (SAT) improved as a result of
An important tenet of constructivism is that learning is an idiosyncratic, active,
and evolving process (Anthony, 1996). Anthony presented two case studies to
understand the nature and consequences of passive and active learning behaviors in the
classroom. He believed that "the nature of a student's metacognitive knowledge and the
quality of learning strategies are critical factors in successful learning outcomes" (p. 349).
Many believe that mathematics is most effectively learned through students' active
participation. "Learning activities commonly identified in this manner include
investigational work, problem solving, small group work, collaborative learning and
experiential learning" (p. 350). Passive learning activities included "listening to the
teacher's exposition, being asked a series of closed questions, and practice and
application of information already presented" (p. 350). The purpose of the case studies
was to provide a detailed description of how learning strategies are used in the learning
context and to examine the appropriateness and effectiveness of each of the students'
strategic learning behaviors. Data were collected throughout the school year by the
researcher using nonparticipant classroom observations, interviews, students' diaries,
students' work, and questionnaires. For each case study, three lessons were recorded
using two video cameras, creating a split-screen image of the teacher and student on a
single tape. Each student was requested to view segments of the lesson and relive, as
fully as possible, the learning situation. Although the case studies only illustrated
contrasting learning approaches and acknowledged that much of the data reflected
students' abilities to discuss their learning, the studies offered "an interesting perspective
of learning processes used in the classroom and homework context" (p. 363). In each
case study, the students believed they were pursuing learning in mathematics. The
student in the first case study employed learning strategies appropriate for task
completion rather than the cognitive objective for which the task was designed. The
resulting mathematical knowledge and skills that the student acquired tended to be inert
and available only when clearly marked by context. The student's learning was
constrained by a limited application of metacognitive strategies. When faced with
difficulties, the student provided little evidence that he was able to analyze, evaluate, or
direct his thinking.
"Ineffective use of metacognitive control strategies such as planning, reviewing,
reflection, selective attention and problem diagnosis meant that the student depended on
the teacher and text to provide information and monitor his progress" (p. 364). The
student in the second case study was able to adapt to the task the learning situation, and to
maximize his learning opportunities. His learning environment was organized around
goals of personal knowledge construction rather than goals of task performance. In both
case studies, the students contrasting learning approaches served "to illustrate the manner
in which each student strives to cope in their world of experience" (p. 365).
Creating a learning environment where students have opportunities to negotiate;
to challenge; and to question their own ideas, others' ideas, or the teacher's ideas can
promote students' epistemological understanding about science (Tsai, 1998). This also
may be the case for mathematics. Opportunities for children to construct mathematical
knowledge arise as children interact with both the teacher and their peers (Davis et al.,
1990). Many agree that meaningful learning for students could produce better cognitive
outcomes, allow for more integrated knowledge, and foster greater learning motivation
(Anthony, 1996; Cobb, Wood, & Yackel, 1991, 1992; Zazkis & Campbell, 1996). From
a constructivist perspective, interactive learning with others is significant for effective
learning (Bishop, 1985; Bruner, 1986; Clement, 1991; Cobb, Wood, & Yackel, 1990;
Jaworski, 1992). Several researchers have emphasized that studies of children's learning
should focus on the effects of children's interactions with other children (Cobb, Wood &
Yackel, 1991; Tzur, 1999; Wiegel, 1998; Yackel, Cobb, Wood, Wheatley, & Merkel,
Science does not arise from a vacuum; rather, it comes from a complex interaction
between social, technological, and scientific development (Tsai, 1998, p. 486). Tsai
(1998) researched the importance of presenting the constructivist philosophy of science
for students. He conducted a study to acquire a better understanding of the interaction
between scientific epistemological beliefs and learning orientations in a group of
Taiwanese eighth-grade students. Twenty eighth graders in a large urban junior high
school were used in the study. Subjects were chosen using the following criteria: They
were above-average achievers, and they expressed a strong certainty and clear tendency
regarding scientific epistemological beliefs (SEB) based on questionnaire responses. The
questionnaire consisted of bipolar agree-disagree statements on a 5-1 Likert scale,
ranging from empiricist to constructivist views about science. A qualitative analysis
through interviewing the subjects revealed that students holding constructivist
epistemological beliefs about science (knowledge constructivists) tended to learn through
constructivist-oriented strategies when learning science, whereas students having
epistemological beliefs, more aligned with empiricism (knowledge empiricists), tended to
use more rote-like strategies to enhance their understanding. Knowledge constructivist
subjects tended to have a more realistic view about the value of science, and their interest
and curiosity about science mainly motivated them, whereas knowledge empiricist
subjects were mainly motivated by performance on examinations. Although the study
was not conducted using an experimental design, Tsai believed the results strongly
suggested that "students' SEB played a significant role in students' learning orientations
and how they organize scientific information" (pp. 485-486). The growth of science
results from "interaction between science, technology, and society" (p. 486).
Zazkis and Campbell (1996) were concerned with the concept of divisibility and
its relation to division, multiplication, prime and composite numbers, factorization,
divisibility rules, and prime decomposition. They used a constructivist-oriented
theoretical framework for analyzing and interpreting data acquired in clinical interviews
with preservice teachers. The participants were 21 preservice elementary school
teachers. The actions in the study included the following:
1. Examining preservice teachers' understanding of elementary
concepts in number theory.
2. Analyzing and describing cognitive strategies used in solving
unfamiliar problems involving and combining these concepts.
3. Adapting a constructivists-oriented theoretical framework.
Individual clinic interviews were conducted that probed participants' understanding of
number theory concepts. The instrument was designed to reveal the participants' ability
to address problems by recall or construction of connections within their existing content
knowledge. "The results of the study suggested that in the schooling of the participants
involved in this study, insufficient pedagogical emphasis has been placed on developing
an understanding of the most basic and elementary concepts of arithmetic" (p. 562).
Cobb, Wood, and Yackel (1991) conducted a teaching experiment, where their
primary focus was on understanding students cognitive development. The instructional
activities included teacher-directed whole class activities and small group activities. In
1-hour sessions, the children worked in groups for 25 minutes, whole class instruction for
20 minutes, and 15 minutes for students to discuss with the class solutions and answers.
During these instruction times, the teacher constantly moved about the class observing
and frequently interacting with the students as they engaged in mathematical activities.
The noise level was generally higher when the students were working with their partners.
The teacher used a nonevaluative approach when students gave answers and solutions,
even if their answer or solution was incorrect. Two video cameras were used to record
the mathematics lessons for the school year. Analysis of "whole class dialogues and
small group problem solving interactions focused on the quality of the children's
mathematical activity and learning as they tackled specific instructional activities" (p.
161). The teacher's observations and analysis guided the development of instructional
activities. The teacher's purpose for whole class discussions was to encourage the
children to verbalize their solution attempts. "The obligations and expectations mutually
developed during whole class discussions also provided a framework for the children's
activity as they worked in small groups, in that they were expected to solve problems in a
cooperative manner and to respect each others' efforts" (p. 168). The children's level of
conceptual understanding of mathematics influenced the social relationships that the
children negotiated. Cobb, Wood, and Yackel concluded that the teaching experiment
was reasonably successful. The children were able to solve mathematic problems in
ways that were acceptable to them. "It was also apparent that the children's abilities to
establish productive social relationships and to verbalize their own thinking improved
dramatically as the year progressed" (p. 174). The researchers noticed the enthusiasm
and persistence in the children. They also noticed that when the children were given
challenging problems, they did not become frustrated but experienced joy working with
In 1992, Cobb, Wood, and Yackel explored the relationship between individual
learning and group development of three 7-year-old students engaged in collaborative
small group activity. They conducted a quantitative comparison of the control and
treatment of students' arithmetic achievement, beliefs, and personal goals. There was no
individual paper and pencil assignment and no grading of the students' work. Children
began instructional activities by working collaboratively in groups followed by teacher-
initiated discussions of mathematical problems, interpretations, and solutions. The
materials and instruction strategies reflected "the view that mathematical learning is a
constructive, interactive, problem solving process" (p. 99). The students learned as they
interactively designed situations for justification or validation, producing a controlled
A follow-up study was conducted with inner-city students with a predominately
minority population. Cobb, Wood and Yackel found it necessary to conduct a
quantitative comparison of arithmetical achievement, beliefs, and personal goals of the
students. They found favorable results particularly with "students' conceptual
development and problem solving capabilities in arithmetic, their perceptions of the
reality of classroom life, and their personal goals as they engaged in mathematical
activity" (p. 100). The children learned in classroom situations "as they interactively
constituted situations for justification or validation" (p. 119).
Tzur (1999) conducted a constructivist teaching experiment with two fourth-grade
students on the relationship of teaching and children' construction of a specific
conception of their fraction knowledge. In a 3-year teaching experiment that began in the
third grade and ended in the fifth grade, the author addressed the fourth-grade year in this
study. Studying of fractions through teaching and learning contributed to the learning
process in children's learning and the author's learning as the researcher-teacher. "The
children's work during the study fostered an important transformation in their thinking
about fractions" (p. 414). Teaching episodes were videotaped during the students' fourth-
grade year. What generally took place in the teaching episode was children completing
task to solve in Sticks. The researcher defined Sticks as a collection of possible actions
that might be used to establish and modify fraction schemes. Students also were given
opportunities to pose problem tasks to each other. The researcher realized that by
allowing students to work this way enhanced their fraction schemes.
Wiegel (1998) conducted a study to investigate collaborative work with pairs of
kindergarten students while they worked on tasks designed to promote early number
development. The participants were 10 students in a public elementary school. The
students in the study were selected based on the following: ability to work with someone,
willingness to work with the researcher, and counting development skills. The 6-month
study consisted of interviews with individual students and of teaching sessions with pairs
of students. Students were interviewed three times during the study. The purpose of the
interviews was to address student's understanding of standard number-word sequences
and their ability to recognize and represent spatial patterns. The interviews were
videotaped and audiotaped in the hallway outside the classroom. The teaching sessions
involved activities focusing on the order of the number-word sequence, counting, and
visual and auditory patterns. The teacher's role was to pose problems and facilitate the
groups of students. As the sessions progressed, all students made some progress toward
forms of organization requiring more coordination and toward increased involvement in
the partner's actions. "Working in pairs supported and enhanced the students' cognitive
development and promoted more sophisticated ways of social interaction" (p. 223).
Wiegel also noted that the interactive working condition where students were paired
homogeneously instead of heterogeneously provided learning opportunities different
from whole-class, one-to-one teaching situations, or in small heterogeneous group
settings. "For students who were able to reflect on and anticipate their actions, working
in pairs led to cooperative ventures in which they solved counting tasks they were unable
to solve alone" (p. 223).
Leikin and Zaslavsky (1997) discussed effects on different types of students'
interactions while learning mathematics in a cooperative small-group setting. The study
was conducted in four low-level ninth-grade classes consisting of 98 students. The main
mathematics topic in the study was quadratic functions and equations, which is part of the
Israeli secondary school mathematics curriculum. The topic was divided into six units.
Each unit began with a whole class introductory lesson. The unit ended with a seventh
lesson in which a unit test was given to all students. The four classes were divided into
two groups (group la & lb and group 2a & 2b) for comparison:
1. Students in group 1 a learned all the material according to the experimental
cooperative learning method.
2. Students in lb served as a control group, learning all the material the
3. In the second group, students in 2a and 2b learned by both methods,
changing from one to another by the end of each unit.
The researchers collected data by observation, students' written self-reports, and an
attitude questionnaire. Leikin and Zaslavsky noticed the following: (a) an increase in
students' activeness, (b) a shift toward students' on task verbal interaction, (c) various
opportunities for students to receive help, and (d) positive attitudes toward the
cooperative experimental method. The results favored the experimental small-group
A number of studies also have documented that students can draw on their
informal knowledge to give meaning to mathematical symbols when problems
represented symbolically are closely matched to problems that draw on students informal
knowledge (Fennema, Franke, Carpenter, & Carey, 1993; Lampert, 1986, 1990; Mack,
1990, 1995; Saenz-Ludlow & Walgamuth, 1998; Streefland, 1991). Saenz-Ludlow and
Walgamuth (1998) analyzed the interpretations of equality and the equal symbol of third-
grade children who participated in a year-long, whole-class socio-constructivist teaching
experiment. A socio-constructivist classroom teaching experiment was conducted
primarily to understand students' conceptual constructions and interpretations. One of the
major goals of this teaching experiment was to analyze the influence of social interaction
on children's progressive understanding of arithmetical concepts. All of the students at
the elementary school in which the teaching experiment took place were considered at-
risk students. Twelve third-grade students participated in the study. Lessons were taped
daily, and field notes of children's solutions were kept to analyze progressive activity and
changes in the children's cognition. "The researcher's daily presence in the classroom
allowed first-hand observation of the unfolding interaction and meaning-making
processes of the students and teacher" (p. 158). The instruction these students received in
first and second grades was characterized as traditional. The researchers believed that the
teachers in the first and second grades emphasized performance and gave little
consideration to the idiosyncratic ways in which children thought about numbers and
mathematical symbols. "At the beginning of the third grade, even when the students
could perform the addition algorithm correctly, they could give no justification for any of
its steps indicating their lack of understanding of place value" (p. 162). The researchers'
first objective was to remove students' procedural learning of operating with numbers by
supporting their understanding of place value through activities with money exchanges
and the packing of beans. When students were given the freedom to operate with
algorithms, they were able to develop strategies different from conventional ones. At the
completion of the study, Saenz-Ludlow and Walgamuth observed the following:
1. The children symbolized equality in different ways.
2. The dialogues indicate that these children initially interpreted
the equal symbol as a command to act on numbers.
3. The dialogues also indicate the crucial role of classroom
discussions on the children's progressive interpretations of the
4. The dialogues and the arithmetical tasks on equality indicate
these children's intellectual commitment, logical coherence,
and persistence to defend their thinking unless they were
5. The teacher's adaptation to the children's current knowledge
and understandings was a determining factor in sustaining the
children's dialogical interactions.
6. These children's dialogues raise our awareness of the cognitive
effort entailed in the interpretation of and the construction of
mathematical meanings from the conventional symbols, in
particular the equal symbol, (pp. 184-186)
The children in the study expanded their conceptualizations of equality due to their active
role in class discussions, the arithmetical tasks that took into account children's
difficulties, and the teacher's ability to obtain a balance between teaching and the freedom
Mack (1995) conducted a study to examine the development of students'
understanding of fractions during instruction. Four third-grade and three fourth-grade
students received individualized instruction on addition and subtraction of fractions in a
one-to-one setting for 3 weeks. The researcher, as teacher, gave instructions on a one-to-
one setting. Each session lasted 30 minutes during regular school hours. Mack met with
the students six times each for the 3-week period. The sessions, which included clinical
interviews, were audiotaped and videotaped. The author was interested in helping
students make connections mainly by posing problems and asking questions. Students
were given problems verbally and encouraged to think aloud while solving the problems.
The results of the study led to the following conclusions:
1. Although students can draw on their informal knowledge to
give meaning to mathematical symbols, they may not readily
relate symbolic representations to informal knowledge on their
own, even when problems presented in different contexts are
2. Students' ability to relate symbols to their informal knowledge
is influenced by their prior knowledge of other symbols.
3. As students moved from working with symbolic
representations involving only proper fractions to working
with representations involving both whole numbers and
fractions, students explained the whole numbers represented
symbolically in terms of inappropriate fractional quantities.
As students attempted to construct meaning for symbolic representations of fractions,
they overgeneralized the meanings of symbolic representations for whole numbers to
fractions, and they overgeneralized the meanings of symbolic representations for
fractions to whole numbers.
Language and symbols play a role in constructivist accounts of mathematical
development (Bednarz, Janvier, Poirier, & Bacon, 1993; Confrey, 1991; Kaput, 1991;
Pirie & Kieren, 1994; Siegel, Borasi, & Fonzi, 1998). Siegel and others (1998) addressed
the specific function that reading in conjunction with writing and talking can serve in
mathematical inquiries to understand how inquiry experiences can be used in
mathematics education. The participants in the study included a reading and mathematics
educator and a group of secondary mathematics teachers. Data collection included
teachers' written plans, photocopies of instruction materials and students' work, teachers'
anecdotal records of what occurred during selected lessons, and audiotapes and
videotapes of classroom experiences. Based on the results, the researchers suggested that
reading can serve multiple roles in inquiry-based mathematics classes and provide
students with unique opportunities for learning mathematics. They also concluded that
reading can make an important contribution to students' engagement in mathematical
Constructivist perspectives on learning have been central to an empirical and
theoretical work in mathematics education (Cobb, 1995; Simon, 1995; Steffe & Gale,
1995; von Glassersfeld, 1991) and, consequently, have contributed to shaping
mathematics reform efforts (NCTM, 1989, 1991). Understanding learning through
individual and social construction gives teachers a conceptual framework to understand
the learning of their students (Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti. &
Perlwitz, 1991; Simon, 1991).
Cobb (1995) investigated the role that four second graders' use of particular
instructional devices (e.g., hundreds boards) played in supporting their conceptual
development based on reform recommendations of NCTM (1989). Cobb looked at
students' use of the hundreds board for two reasons: (a) to observe the role instructional
devices might have in supporting children's construction of numerical concepts and (b) to
look at theoretical differences between constructivist and socialcultural conceptual
development. "The investigation of children's use of the hundreds board provides an
opportunity to explore the relationship between psychological constructivist and
sociocultural accounts of mathematical development" (p. 363). The investigation took
place in a second-grade classroom in the course of a year-long teaching experiment.
Modes of childrens early number learning were used to develop instructional activities.
The instructional strategy used was "small-group problem solving followed by a teacher-
orchestrated whole-class discussion for children's interpretations and solutions" (p. 365).
Videotapes of the mathematics lessons were done for an entire year. Manipulatives (e.g.,
hundreds boards) were made available to the children in all lessons involving arithmetical
computation and problem solving. Students generally decided what available materials
might help in solving a problem. The analysis Cobb presented focused on "individual
children's cognitive construction of increasingly sophisticated place-value numeration
concepts" (p. 379). The analysis indicated that the children's use of the hundreds board
did not support the construction of increasingly sophisticated concepts of 10. "Children's
use of the hundreds board did appear to support their ability to reflect on their
mathematical activity once they had made this conceptual advance" (p. 377).
Simon (1995) presented data on a whole-class, constructivist teaching experiment
in which problems of teaching practice required the teacher/researcher to explore the
pedagogical implications of his theoretical (constructivist) perspectives. The 3-year
study of the mathematical and pedagogical development of potential elementary teachers
was part of a Construction of Elementary Mathematics (CEM) Project teaching
experiment. Simon was interested in ways to increase these potential elementary
teachers' mathematical knowledge and aid in their development of views of mathematics,
learning, and teaching. There were 26 potential elementary school teachers involved in
the study. Data were collected from a mathematics teaching and learning course, a 5-
week prestudent-teaching practicum, and a 15-week student-teaching practicum taken by
the participants in the study. The researcher took notes and videotaped the classes.
There were no lecture class lessons, only small-group problem solving and teacher-led
whole-class discussions. The mathematical content of the mathematics teaching and
learning course began with exploration of the multiplicative relationship involved in
evaluating the area of rectangles. Simon believed that an advantage of the teaching
experiment design was that time was built into the project to reflect on the understandings
of the students. This teaching episode seemed to emphasize that the teacher did not
The success of such efforts is in part determined by the adequacy
of his model [meaning the teacher] of students' understanding. It
also seems to support the notion that learning does not proceed
linearly. Rather, there seem to be multiple sites in one's web of
understandings on which learning can build, (p. 140)
Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz (1991) also
conducted a study on students' response to constructivism in the classroom. They were
interested in students' computational proficiency, conceptual development in arithmetic,
personal goals in mathematics, and beliefs about reasons for success in mathematics. The
instructional approach used in this teaching experiment was compatible with the
constructivist view. The year-long study consisted of 338 second-grade students. The
treatment group consisted of 187 students, and the remainder of the students were in the
control group. The instructional activities designed to facilitate constructions of thinking
strategies included activities from What's My Rule (Wirtz, 1977) and War (Kamii, 1985).
At the end of the school year, two tests were administered to the study groups. One test
was a state-mandated multiple choice standardized achievement test (I STEP) and the
second was a Project Arithmetic Test developed by the researchers. An ANOVA was
conducted to determine the achievement of the treatment and control groups on the
ISTEP and Project Arithmetic Test. "The comparison of students' performance on both
the ISTEP and Project Arithmetic Test indicated that the treatment students developed a
higher level of reasoning in arithmetic than the control group" (p. 21). There was an
indication that the teachers in the treatment group were successful partly because of their
"development of classroom mathematics traditions in which their students could publicly
express their thinking without risk of embarrassment" (p. 23). Students in the treatment
group believed that success in mathematics stemmed from developing their own methods
for solving problems. In contrast, students in the control group believed success was
determined by using the same solution methods as the teacher or other students.
Nicholls, Cobb, Wood, Yackell, and Patashnick (1990) conducted a study on
children's response to constructivism in the classroom. They were interested in second-
grade children's reasons for success in mathematics. A series of questionnaires were
administered to six second-grade classes to establish whether children's beliefs about the
causes of success in mathematics were meaningfully related to their personal goals in
mathematics. In conjunction with student's theories of success in school mathematics,
the researchers were interested in measures that would determine how motivational
influences of different mathematics teaching practices influenced students' success in
mathematics. Only one class used teaching practices compatible with the NCTM (1989)
recommendation that emphasis should be on establishing a climate that places critical
thinking at the core of instruction. All statements should be subject to question, and both
teacher and children should be open to reaction and elaboration from others in the
classroom. The teaching practice consistent with a constructivist tradition in education
was observed in class one. Class one was the only group consistent with either the
NCTM recommendation or an approach to mathematics teaching compatible with
constructivism. In class one, after tasks were posed, students worked cooperatively in
pairs to complete the tasks.
"One implication of constructivism is that mathematics, including arithmetical
computation, should be taught through problem solving" (Cobb, Wood, & Yackel, 1991,
p. 158). Franke and Carey (1997) conducted a study to investigate first-grade children's
perceptions of mathematics in problem-solving environments. Two questions were
addressed in the study:
1. What are children's perspectives on what it means to do mathematics in
problem solving classrooms?
2. Do children in problem-solving classrooms with different
demographic characteristics hold similar perspectives about what it means
to do mathematics?
Individual interviews were conducted with 36 first-grade children. Six children,
three girls and three boys, were randomly selected from each of six classrooms. The
researchers were interested in determining children's perceptions about mathematics in
problem-solving environments. Open-ended questions were asked to determine
children's perceptions of what doing mathematics entailed. "The combination of open-
ended, context-driven questions and children's ability to talk about their thinking allowed
us more insights into young children's perceptions about mathematics" (p. 22). When
asked about the use of manipulatives when doing mathematics, children responded that
manipulatives were helpful in solving problems. The children recognized and accepted a
variety of solution strategies, with many of the children valuing all solutions equally and
assuming shared responsibility with the teacher and their peers for their mathematics
learning. Children had varying perceptions of what it meant to succeed in mathematics.
The children in the study have helped the researchers to "see how critical it is to begin to
understand children's perceptions of what it means to engage in mathematics" (p. 24).
Wood and Sellers (1996) conducted a class-by-instruction factorial design used to
compare students in problem-centered classes for 2 years with students in problem-
centered classrooms for 1 year, and with students in textbook classes for 2 years on a
standardized achievement test. Also, the authors studied classes using problem-centered
instruction for 2 years and compared them with students in problem-centered classes for
1 year on an instrument designed to assess students' conceptual development in
arithmetic. Six classes received problem-centered mathematics instruction for 2 years in
second and third grades. This instruction was generally reflective of a socioconstructivist
theory of knowing and compatible with recommendations for reform in mathematics
education. The researchers used an instrument designed to examine personal goals and
beliefs about reasons for success in mathematics. Two tests were given to students to
evaluate their arithmetic learning, the Arithmetic Test and the Personal Goals and Beliefs
questionnaire. The results indicated that after 2 years of instruction in reform-based
classes, students scored significantly higher on standardized measures of computational
proficiency as well as conceptual understanding and they understood arithmetic concepts
better than those in textbook-instructed classes. The students in the study believed "that
it is important to find their own or different ways to solve problems, rather than conform
to the method shown by the teacher" (p. 351).
One's representation of space is not a perceptual "reading off' of the spatial
environment but is built up from prior active manipulation of the environment (Battista,
Clements, Amoff, Battista, & Van Auken Borrow, 1998; Clements & Battista, 1992;
Clements, Swaminathan, Hannibal & Sarama, 1999). Battista and others (1998)
examined students' structuring and enumeration of two-dimensional (2D) rectangular
arrays of squares. The authors interviewed 12 second graders during the fall. Neither the
research team nor the students' classroom teacher instructed students on enumerating
rectangular arrays during the school year. The interviews were videotaped. After
demonstrating how a plastic square inch tile fit exactly on a graphically indicated square
given in a rectangle drawing, students were asked to do the following: (a) predict how
many squares were in the rectangle, (b) draw where they thought the squares would be
located on the rectangle, (c) then predict again how many squares would be needed, and
(d) cover the rectangle with square tiles and determine again the number of squares
needed. The results indicated that many students' understanding of the row-by-column
arrays was unclear.
In the traditional view of learning, it is assumed that row-by
column structure resides in two-dimensional rectangular arrays
of squares and can be automatically apprehended by all.
However, as we have seen in the present study, and consistent
with a constructivist view of the operation of the mind, such
structuring is not in the arrays-it must be personally constructed
by each individual, (p. 531)
In a previous study focusing on three-dimensional arrays, Battista and Clements
(1996) were only able to see consequences of a student's ability to perform coordinating
actions. In their study, the researchers not only were able to see the consequences of a
student's ability or inability to perform sequence of coordination actions used to organize
a set of objects but they also were able to observe these actions, which gave them a better
understanding of their characteristics.
Clements, Swaminathan, Hannibal, and Sarama (1999) investigated criteria
preschool children used to distinguish members of a class of shapes from other figures.
The participants in the study were 97 middle-class children ages 3 to 6 from two
preschools and one elementary school with two kindergarten classes. The researchers
investigated young children's methods used to distinguish geometric shapes common in
the social-cultural environment. Data were collected using clinical interviews and paper-
and-pencil tasks in a one-to-one setting. Children identified circles with a high degree of
accuracy. Six-year-olds performed significantly better than did the younger children.
Battista (1999) used psychological and sociocultural components of a
constructivist paradigm to provide a detailed analysis of how the students cognitive
constructions are used to enumerate three-dimensional arrays of cubes developed and
changed in an inquiry-based problem-centered mathematics classroom. There were six
fifth-grade students in the study. The study took place over 4 weeks for 1 hour per day
covering a unit on volume. The students had to find a way to predict correctly the
number of cubes that would fill boxes described by pictures, patterns, or words. Each
student in the study was given individual activity sheets to record answers. The teacher
walked around the classroom, observing students interacting in pairs and encouraging
collaborative communication. Students made predictions and checked their predictions
by making paper boxes and filling them with cubes. "Having students first predict then
check their predictions with cubes was an essential component in their establishing the
viability of their mental models and enumeration schemes" (p. 442). This study showed
how powerful learning can occur in problem-centered inquiry-based instruction. Cobb
noted the following implications:
Using the constructivist theory of abstraction permits an
elucidation of students' need for repeated and varied opportunities
to properly construct difficult concepts. Use of this theory also
provides those insights into students' learning required for
effective curriculum development and instruction because it
describes precise constructive itineraries students follow in
acquiring particular mathematical ideas. Second, instructional
materials designed to promote intrapersonal (vs. interpersonal)
cognitive conflict can be very effective in engendering
accommodations and the resulting mathematics learning.
(Battista, 1999, pp. 447-448)
Concrete Manipulatives for Teaching Mathematics
Many mathematics and science teachers in elementary and middle schools use
manipulatives with varying degrees of success. Educators have argued that concrete
manipulatives can help develop and enhance math problem solving and critical thinking
skills, decrease anxiety, and increase concentration (Cuisenaire, 1992; Davidson, 1990;
Langbort, 1988; Marzola, 1987; Ohanian, 1992; Scheer, 1985; Stone, 1987; Taylor &
"One does not come to know a tool through a description of it, but only through its
activity" (Confrey, 1990, p. 110). Reys (1971) gave the following suggestions on how
concrete manipulatives can enhance the learning of students at all levels:
1. To vary instructional activities
2. To provide experiences in actual problem solving
3. To provide a basis for analyzing sensory data, necessary in
4. To provide an opportunity for students to discover
relationships and formulate generalizations
5. To provide active participation by pupils
6. To provide for individual differences
7. To increase motivation related, not to a single mathematics
topic, but to learning in general, (p. 555)
Bums (1996) suggested other common uses of concrete manipulatives:
1. Manipulatives help make abstract ideas concrete.
2. Manipulatives lift math off textbook pages.
3. Manipulatives build students' confidence by giving them a
way to test and confirm their reasoning.
4. Manipulatives are useful tools for solving problems.
5. Manipulatives make learning math interesting and
enjoyable, (p. 46)
Reys (1971) also suggested the following basic tenets for using manipulative materials
1. Concept formation is the essence of learning mathematics.
2. Learning is based on experience.
3. Sensory learning is the foundation of all experiences and
thus the heart of learning.
4. Learning is a growth process and is developmental in
5. Learning is characterized by distinct, developmental
6. Learning is enhanced by motivation.
7. Learning proceeds from the concrete to the abstract.
8. Learning requires active participation by the learner.
9. Formulation of a mathematical abstraction is a long
process, (p. 552)
In addition to Reys and Bums' list of ways to use manipulatives. Yeatts (1991)
suggested other ways manipulative materials could help:
1. To introduce a new mathematical concept
2. To reinforce previous learning
3. To provide concrete representations of abstract ideas
4. To provide for individual learning styles
5. To foster creative thinking processes
6. To provide experiences in problem solving situations
7. To provide opportunities for students to become active
participants in their own learning experiences
8. To provide opportunities for students to exchange
viewpoints with their classmates
9. To diversify the educational activities
10. To enhance interest and motivation for learning new
concepts, (p. 7)
In a meta-analysis of 60 studies from kindergartners to college-age students,
Sowell (1989) found that long-term use of concrete manipulative material increased
mathematics achievement and improved students' attitudes toward mathematics, when
instruction using concrete material was presented by teachers knowledgeable about their
Many of the studies favoring the use of concrete material took place more than 20
years ago (Allen, 1978; Babb, 1976; Bledsoe, Purser, & Frantz, 1974; Bolduc, 1970;
Bring, 1972; Brown, 1973; Callahan & Jacobson, 1967; Derderian, 1980; Earhart, 1964;
Ekman, 1967; Fuson, 1975; Johnson, 1971; Letteri, 1980; Nichols, 1972; Nickel, 1971;
Purser, 1974; Robinson, 1978; Romero, 1979; Stockdale, 1980; Tobin, 1974; Toney,
1968; Trueblood, 1968; Wallace, 1974; Wheatley, 1979). There have only been a few
current studies found on the use of concrete manipulatives (Baxter, Shavelson, Herman,
Brown, & Valadez, 1993; Burton, 1992; Canny, 1984; Chester, Davis, & Reglin. 1991;
Gardner, Simons, & Simpson, 1992; McClung, 1998). It seems apparent that there is a
gap in the literature on studies on the use of manipulatives and their effects on students'
learning. This may be because manipulatives used to teach mathematics have been
stressed many times and in different contexts, where it is assumed that using
manipulatives in teaching mathematics is apparent (Behr, 1976).
Baxter and others (1993) were interested in the performance assessment and their
effects on students in diverse groups. They developed a study with 105 Latino and Anglo
sixth graders in two types of mathematics curriculumhands-on and traditional. In the
hands-on group of 40 students (13 Latino and 27 Anglo), the curriculum consisted of
problem solving using manipulatives.
The instruction in the traditional group of 65 students (43 Latino and 22 Anglo)
consisted of a textbook and worksheet curriculum. Performance assessment was
comprised of several tasks, including measurement, place value, and probability. Due to
time constraints, the hands-on group received only the measurement and place value
assessment. The traditional group received all three assessments. The performance
assessment consisted of describing objects, measuring length, and finding the area. The
researchers concluded that the mean differences on the two performance assessments,
where students with varying instructional histories could be compared, showed that
students who had received hands-on mathematics instruction scored higher, on average,
than students in the traditional curriculum. When comparing the groups by ethnicity, the
average scores for the Anglo students were higher than Latino students on all
achievement measures. The extent of the difference varied, based on the instructional
history of each student. The researchers concluded that assessments on mathematics
performance relating closely to hands-on instructional activities provided reliable
measures of mathematics achievement.
The manipulatives and strategies that young children would use if given free
choice were of interest to Burton (1992). In the study, Burton explored young children's
understanding of division. The subjects used in the study were 117 second-graders from
five second-grade classrooms, three in one school and two in another school. The testing
environment for the students in the study consisted of individual interviews conducted in
the hallway outside the classroom. Each student was given an opportunity to explore the
testing kit (manipulatives) before the interview began. The researcher was interested in
how students would solve unfamiliar mathematical problems. Twelve problems
presented one at a time were read and shown to the child during interview. The child
decided which manipulative and strategy he or she wanted to use. The child's strategy
and choice of manipulative were recorded for each problem. Each interview, on the
average, lasted approximately 23 minutes. Some students took longer and some students
took less time to answer the questions. All of the manipulatives in the kit were used at
least once by one child, in spite of the 38 children that did not use any manipulatives on
any of the 12 questions. Burton concluded that "second-grade children are capable of
solving some division word problems if manipulatives are supplied. They are especially
successful in this if the objects match the problem of context" (p. 13).
Many classroom teachers focus on covering the material in the textbook from an
abstract approach despite the support in the literature for the use of manipulative
materials. Canny (1984) was interested in determining if manipulative materials had a
greater impact when used to introduce or reinforce a concept or both. The researcher
investigated the role of manipulative materials in improving achievement in computation,
concept-formation, and problem solving in fourth-grade mathematics. There were 123
fourth graders included in the study. The students were divided into four groups. Group
A teachers used manipulative material to introduce the concepts. Group B teachers used
manipulates for reinforcement after the traditional teaching method. Group C teachers
used manipulatives to introduce the concept, practiced, and reinforced the concept using
the textbook and manipulatives. Group D teachers only used the textbook; they did not
use manipulative materials for the mathematics lessons. The classroom teachers
implemented the lessons to avoid as much disruption of the classes as possible. The
students in the study were given The Science Research Associates (SRA) achievement
test, a researcher-designed written achievement test and retention test that matched the
lessons and the textbook. The researcher conducted in-service sessions and demonstrated
teaching lessons because the teachers in the study had little experience with using
concrete objects to aid in learning. An ANCOVA model with multiple contrasts was
used in the study.
Canny concluded that "the group using manipulative materials for only
reinforcement scored the lowest while the group using manipulatives for only
introduction scored the highest" (p. 71). Most of the test results in this study provided
additional support for the research favoring the use of manipulative materials in teaching
and learning mathematics. The overall findings of this study would influence the
researcher to use manipulative materials to introduce concepts because it produced the
highest scores on the SRA test and the retention test.
"Using math manipulatives is one method of teaching that makes learning real to
students" (Chester and others, 1991, p. 4). Manipulatives help students to become
actively involved in doing mathematics instead of just listening to lectures and
completing paper and pencil activities. Chester and others investigated the effect of
manipulatives on increasing mathematics achievement of third-grade students. The
students in the study consisted of two randomly selected third-grade classes with 26
students in each class. Before the start of the study on a geometry unit, the students in the
study were given a multiple choice unit test. The researchers used an ANCOVA and t-
test to analyze the data. Manipulatives were used to teach concepts in the geometry unit
during the 2-week hourly teaching sessions. Drawings and diagrams were the only things
used to teach the concepts in the control group. There was a significant difference
between the pretest and posttest scores of both groups. Chester and others concluded that
"the experimental group, which used the math manipulatives, received higher adjusted
posttest scores than the control group, which used only the textbook" (p. 17).
McClung (1998) conducted a study to determine if the use of manipulatives in an
Algebra I class would make a difference in the achievement of the students. The
researcher used two Algebra I classes mixed with 10th and 11th graders. One group had
24 students and the other group had 25 students. Group A was the control group, and the
students in this group were taught using traditional teaching methods consisting of
lecture, homework, and in-class worksheets. Experimental Group B was also taught
using the traditional teaching method but used manipulative Algeblocks instead of the
worksheets. A pretest was given prior to the beginning of the study to assure
homogeneity. A posttest, identical to the pretest, was given at the end of the study. A
two-sample t-test was used to analyze the data. There was a significant difference
between the two groups. The control group's mean score, using no manipulatives, was
higher than the experimental group's mean score, using manipulatives. This implied that
students using the traditional teaching methods outperformed the students using the
manipulatives. According to Piaget (1970), the concrete operational stage (7 to 12), is
the foundation for the use of manipulatives. One reason students using manipulatives
may have scored lower than students not using manipulatives could be because students
used manipulatives in the class lessons but were not permitted to use them on the posttest.
Although no information was given about the instructor's knowledge on manipulatives, a
second reason may be that the instructor had limited knowledge of the concept of using
manipulatives and did not obtain this information before the study began.
Hinzman (1997) conducted a study to determine the differences in mathematic
scores/grades of middle school students after the use of hands-on manipulatives and
group activities. Thirty-four eighth-grade students in two prealgebra classes participated
in the study. The instruments used in the 18-week study included a pretest, observation
of the students participating in the lessons, and a student attitude survey. Chapter tests
were given to the study groups to determine the effects of manipulative use on test
performance. Notebook tests were also given to determine the effects of homework on
class performance. A T-test was used to compare the test results of the two prealgebra
classes. There was no control group used in the study.
Hinzman concluded that students' performance during in class instruction
improved with the use of manipulates. The study did not show any significant changes
in grades for students who used manipulatives and students who did not use
manipulates. However, based on the results of the student attitude survey, there was a
significant improvement of students' attitude towards mathematics.
There was significant interest in determining whether combining computer-aided
instruction with hands-on science activities would increase students' cognitive and
affective assessment outcome (Gardner and others, 1992). Gardner and others used a
"weatherschool" meteorology program to determine the effect of computer-aided
instruction in the elementary school classroom. The instruments used to measure
students' cognitive and affective domains were paper and pencil tests. The first test
contained 13 questions and measured the knowledge and application level. The second
test contained 15 questions and measured students' attitudes toward science and
computers. There were three treatment groups. The first treatment group with 47
students included hands-on activities. The second treatment group with 46 students
included a combination of hands-on activities and the weatherschool program. The third
treatment group with 21 students used text based learning only. All three treatment
groups met for 10 days for two class periods each day. The results of the study indicated
that students' attitude and knowledge toward science increased with hands-on activities.
The pretest and posttest scores from Treatment Group II showed higher gains than
Treatment Group I. Treatment Group III scored higher on the pretest for conceptual
assessments. However, they scored lower than both groups on the posttest. "Hands-on
activities, CAI, or both apparently lead to increased understanding and more positive
attitudes" (p. 336).
Not all researchers would agree that concrete manipulatives have significant
effects on students' learning (Labinowicz, 1985; Resnick & Omanson, 1987; Thompson,
1992). Thompson was interested in the effects of concrete manipulatives on student's
understanding of whole number concepts. He investigated students' engagement in tasks
involving the use of base-ten blocks and their contributions to students' construction of
meaning of decimals. The study consisted of 20 fourth-grade students, 10 boys and 10
girls. Students were assigned to one of two treatment groups based on result of their
pretest scores. Students with similar scores were put in opposite groups. One treatment
group, taught by their regular classroom teacher, used Blocks Microworld in instruction.
The other treatment group, taught by a research assistant, used wooden base-ten blocks.
The microworld instruction was videotaped. The wooden base-ten blocks instruction
could not be videotaped because the class had two special education students in
attendance who were not part of the study. The teacher in the microworld experiment
group used a computer connected to a large screen projector for whole-class discussion.
The teacher in the wooden base-ten blocks experiment group used the overhead projector
and plastic blocks for whole-class instruction. Both teachers in the treatment groups used
a highly detailed script for instruction, which restricted their actions. This type of
instruction was used to get more responses from the students. At the end of the 9-day
study, students were administered a posttest containing items from the pretest and
additional items on decimal concepts. Thompson concluded that there were no
significant changes in either group regarding whole number concepts.
Tanbanjong (1983) conducted a study on the effects of using manipulative
material in teaching addition and subtraction to first-grade students. The 6-week study
consisted of 350 first-grade students. Half of the students were assigned to the treatment
group and the other half were assigned to the control group. The instructional activities
covering content in addition and subtraction of one-digit and two-digit numbers included
activities from a textbook provided by the Ministry of Education of Bangkok. Thailand,
for first-grade students. Students in both groups received mathematics instruction for 40
minutes per day for 6 weeks. In the treatment group, the instructor used the following
manipulatives: a pocket chart, a set of trading chips, a bundle of sticks, and an abacus.
In the control group, there were no manipulative materials used, only traditional
teaching methods. The researcher observed each group every week. A 20-item posttest,
used to measure students' achievement, was given at the conclusion of the study. A one
way ANOVA was used to analyze the data. The children in the treatment group scored
significantly higher on the achievement test than the children in the control group. The
researcher concluded that the manipulative approach was more effective than the
traditional teaching method in facilitating achievement gains.
Kanemoto (1998) also conducted a study on the effects of manipulatives on
students learning. The researcher was interested in the effects of using manipulatives
on fourth-grade students in learning place value. The 4-week study consisted of 51
fourth-grade students from two different classes. The treatment group contained 26
students, and the control group contained 25 students. A pretest was given to both
groups to make sure there were no significant differences in the groups. The treatment
group was taught place value using a combination of manipulatives and textbook
instruction. The control group was taught place value using textbook instruction only
but no manipulative material. There were 11 one-hour mathematics lessons conducted
during the study. A 20-item posttest was given at the completion of the study. A chi-
square test was used to analyze the data. The researcher concluded that the results of the
posttest showed that the use of manipulatives did not provide a difference in
mathematics achievement between the treatment and control groups when studying
Harris (1993) also conducted a study to determine the effects of the use of
manipulative materials on the development of mathematical concepts and skills. The
12-day study consisted of 331 seventh-grade students. The classrooms were already
intact; therefore, random assignment was not possible. The control group consisted of
160 students, and the remainder of students were in the treatment group. Both groups
were given The California Achievement Test as a pretest to measure broad concepts of a
specific content area. All classes in the study used the same mathematics textbook.
Eleven lessons designed by the researcher, were developed for both groups. The lessons
covered objectives relating to perimeter and area. The control group received traditional
lecture, demonstration, and paper and pencil instruction. The treatment group received
the same instructional lessons except their instruction included the use of the following
manipulatives: geoboards, rubber bands, rulers, tape measures, trundle wheel, overhead
tiles, unifix cubes, square tiles, dot paper, and grid paper. At the end of the study, two
tests were administered to the two groups. One test was the California Achievement
Test (CAT), and the second test, consisting of multiple-choice and open-ended
questions, was a teacher-made unit test developed by the researcher. An ANOVA was
conducted to determine the achievement of the treatment and control group on the CAT
and the teacher-made unit test. The comparison of students' performance on both the
CAT and the teacher-made unit test indicated that the students in the treatment group
developed a better understanding of perimeter and area than the control group. Students
in the treatment group posited that manipulatives helped them to understand perimeter
and area and to apply the formulas correctly. No response was given for the control
Technology and Simulated Manipulatives in the Classroom
An instructional model for teaching mathematics to young children should
emphasize active learning and problem solving with concrete materials (NCTM, 1989).
"Technology must support and enhance this model" (Campbell
Technology education at the elementary level begins with three things in mind: the child,
the elementary school curriculum, and the appropriate technology activity (Kirkwood.
1992). Implementing technology in the mathematics education classroom helps "promote
high level thinking skills and support concept development" (Harvey fe Chamitski, 1998,
p. 157). Campbell and Stewart (1993) stated that "perspective for using technology with
young children is that these tools permit teachers and children to investigate the richness
of mathematics" (p. 252).
An important role for computers is to assist students with exploring and
discovering concepts, with the transition from concrete experiences to abstract
mathematical ideas (NCTM, 1989). "In computer environments, students cannot overlook
the consequences of their actions, which is possible to do with physical manipulatives"
(Clements & McMillan, 1996, p. 274). Clements and McMillan gave the following
advantages to using the computer instead of physical manipulatives:
1. They avoid distractions often present when students use physical
2. Some computer manipulatives offer more flexibility then do their non
3. Another aspect of the flexibility afforded by many computer manipulatives is
the ability to change an arrangement of the data. Primary Graphing and Probability
Workshop (Clements, Crown, & Kantowski, 1991) allows the user to convert a picture
graph to a bar graph with a single keystroke.
4. Students and teachers can save and later retrieve any arrangements of
computer manipulatives. Students who had partially solved a problem can pick up
immediately where they left off.
5. Once a series of actions is finished, it is often difficult to reflect on it. But,
computers have the power to record and replay sequences of actions on manipulatives.
This ability encourages real mathematical exploration.
6. The computer connects manipulatives that students make, move, and change
with numbers and words.
7. Computer manipulatives dynamically link multiple representations. For
example, many programs allow students to see immediately the changes in a graph as
they change data in a table.
8. Students can do things that they cannot do with physical manipulatives. For
example, students can expand computer geoboards to any size or shape.
Computers also serve as a catalyst for social interaction (Clements, 1998). Social
interaction constitutes a crucial source of opportunities to learn mathematics (Piaget,
Connell (1998) conducted a study to determine the impact for technology use in a
constructivist elementary mathematics classroom. Students were randomly assigned to
one of two conditions. Twenty-five of the students were assigned to the technology-
aligned classroom (TAC), where the teacher used the computer as a student tool for
mathematics exploration. Twenty-seven students were assigned to the technology-
misaligned classroom (TMC), where the teacher used the computer as a presentation tool.
Daily teacher instruction was the treatment in the study. "Technology was present in
both classrooms, but only used as an integral portion of the mathematics instruction in the
TAC" (p. 325). Both classrooms used a similar constructivist mathematical pedagogy
with similar set of materials. Technology "was used in a constructivist fashion" (p. 331),
not as an afterthought. Both groups were successful in learning the required grade level
mathematics. "It is clear that the constructivist approaches to instruction as utilized in
this study were effective" (p. 331).
There is an assumption that computer software produces the same cognitive effect
as touching physical mathematics manipulatives (Terry, 1995). Terry conducted a study
to determine the effects that mathematics manipulatives (MM) and mathematics
manipulative software (MS) had on students' computation skills and spatial sense. The 4-
week study consisted of 214 students in Grades 3, 4, and 5. Sixty-eight students were in
the MM group; 68 students were in the MS group; and the remainder of students were in
the mathematics manipulative with mathematics manipulative software (BOTH) group.
Base ten blocks and attribute shapes, both in concrete manipulative and computer
manipulative form, were used for the computation and spatial sense lessons. At the end
of the study, two tests, designed by the researcher, were administered to the three groups.
One test covered computation, and the second test covered spatial sense. A three-way
ANOVA was conducted to determine the achievement of the three groups on the
computation and spatial sense test. The comparison of students in the three groups on the
spatial sense test indicated that there was no significant difference. The comparison of
the three groups on the computation test indicated that the students using both concrete
manipulatives and computer manipulatives developed a better understanding of
computation than the other two groups. The researcher felt that the use of both was the
most effective way to teach computation.
Kim (1993) also was interested in the effects of concrete manipulatives and
computer-simulated manipulatives on students' learning. Kim conducted a study on
young children to determine the effects computer-simulated and concrete manipulatives
would have on their learning of geometric and arithmetic concepts. The 6-week study
consisted of 35 kindergarten children. Seventeen students were in the hands-on group,
and the remainder of students was in the computer simulation group. The instructional
activities for both groups included activities with geoboards, attribute blocks, and
Cuisenaire rods. There was no control group in the study. The researcher was
concerned with possible differences in initial intellectual ability and maturity, because
the two treatment groups were not randomly assigned. There were 12 lessons, with each
lesson lasting 25 minutes. At the beginning of the three units a pretest was
administered, and at the end of each unit, a posttest was given to determine students'
progress in the three units. The first unit covered geometry; the second unit covered
classification; and the third unit covered seriation, counting, and addition. An
ANCOVA was conducted to determine the achievement of the treatment groups on the
unit tests. The results indicated that hands-on and computer-simulated manipulatives
were similar in their effectiveness. Both groups made significant gains from the pretest
to the posttest on all three unit tests. The researcher concluded that using either concrete
or computer-simulated manipulatives helped children learn mathematical concepts
relating to geometry, seriation, counting, and addition. The researcher also believed that
hands-on and computer-simulated manipulatives were equally effective for teaching
mathematics concepts to young children.
Berlin and White (1986) studied the effects of combining interactive
microcomputer simulations and concrete activities on the development of abstract
thinking in elementary school mathematics. The sample of students was randomly
selected from two elementary schools. Three levels of treatment were administered to the
113 students in the study: (a) concrete only activities, (b) combination of concrete and
computer simulation activities, and (c) computer-simulation only activities. Two paper
and pencil exams requiring reflective abstract thought were administered to all the
subjects. The researchers felt it was still unclear what effects concrete manipulation of
objects and computer simulations had on students' understanding. The researchers
concluded that effects of concrete and computer activities produced different results,
depending on the socio-cultural background and gender of the child. Berlin and White
posited more research was needed to determine the effect of concrete and computer
activities in the learning of elementary school mathematics.
Researchers have also found computer simulations useful in the science
curriculum (Bork, 1979; Lunetta & Hofstein, 1981). Choi and Gennaro (1987) conducted
a study that compared the effects of microcomputer-simulated experiences with a parallel
instruction using hands-on laboratory experiences for teaching volume displacement
concepts. The subjects consisted of 128 eighth-grade students in earth science classes.
The method used in the study was an experimental design with a control group. The
students were randomly assigned to one of two treatment groups-the microcomputer-
simulated group (the experimental) and the hands on laboratory group (the control). The
experimental group used a computer program which included graphics, animation, and
color for visualizing concepts; blinking words to help students focus on important
instructions; and immediate feedback to enhance learning. The control and the
experimental groups conducted parallel experiments, but the experimental group used
worksheets and hands-on experiments. The experiments took 55 minutes in the control
group and only 25 minutes in the experimental group. After the 2-day experiment, a 20-
item multiple choice posttest was administered to students in both groups to measure the
students' understanding of concepts related to volume displacement. There was no
pretest given. The results of the study showed that computer-simulated experiments were
as effective as hands-on laboratory experience. Forty-five days later a retention test was
given to the students in both groups. A three-way ANOVA was used to interpret the
data. Choi and Gennaro found that the computer-simulated experiences were as effective
as the hands-on laboratory experiences. Also, there was not a significant difference in
retention scores of the two groups.
One implication of this study was that "computer simulations could be used for
the teaching of some concepts without the extra needed effort and time of the teacher to
prepare make-up material" (p. 550). Another implication was that the computer-
simulated experience "enabled the students to achieve at an equal performance level in
approximately one-fourth the time required for the hands-on laboratory experiences (25
minutes compared to 95 minutes)" (p. 550).
Because of the emphasis on probability learning in the school curriculum, there
is a need for further on-going research on the learning and teaching of probability
(Shaughnessy, 1992). The study of statistics and probability stress the importance of
questioning, conjecturing, and searching for relationships when formulating and solving
real-world problems (NCTM, 2000). We live in a society where probabilistic skills are
necessary in order to function. Probability describes the world in which we live. Many
everyday functions depend on knowing and understanding probability. Milton (1975)
suggested the following reasons for introducing probability as early as the primary level:
1. The basic role which probability theory plays in modem
society both in the daily lives of the public at large, and
the professional activities of groups within the society, e.g.
in the sciences (natural and social), medicine and
2. Probability theory calls upon many mathematical ideas
and skills developed in other areas of school coursed, e.g.
set, mapping, number, counting, and graphs.
3. Students are able to work in a branch of mathematics,
which is relevant to current activities in life. (p. 169)
Jones, Langrall, Thornton, and Mogill (1999) conducted a study to evaluate the
thinking of third-grade students in relation to an instructional program in probability.
The focus of the evaluation was on student learning. The design of the instructional
program was consistent with a constructivist orientation to learning. "Opportunities to
construct probability knowledge arise from students attempt to solve problems, to build
on and recognize their informal knowledge, and to resolve conflicting points of view"
(p. 492). There were two third-grade classes involved in the study: one was an early-
instruction group, in the fall, and the other one was a delayed-instruction group, in the
spring. The instructional program consisted of 16 40-minute sessions with two sessions
occurring each week for 8 weeks. The probability problem task was based on the
following essential constructs: "sample space, probability of an event, probability
comparisons, and conditional probability" (p. 494). Problems were chosen so they
would be accessible to students functioning at different levels. The most important
feature of the data was the number of students in both groups that increased their
informal quantitative level of probabilistic thinking following instruction. An important
finding was that 51% of the students who began instruction below an informal
quantitative level reached this level after instruction. A repeated measures ANOVA was
used to evaluate the overall effect. This analysis "demonstrated that both the early and
delayed instruction groups showed significant growth in performance following
instruction" (p. 517).
Analyses of the students' learning during the instructional program, which
revealed a number of patterns that appear to have critical ramifications for children
understanding of probability, are related to the following:
1. children's conception of sample space
2. their use of part-part and part-whole relationships in
comparing and representing probabilities
3. their use of invented or conventional language to describe
their probability thinking, (p. 515)
Many of the students revealed higher level sample-space thinking following instruction.
The results of this study showed that the probabilistic thinking framework could be used
to develop an effective instructional program in probability. A further conclusion of this
study was that the use of part-part reasoning provided a threshold level for dealing with
probability situations that incorporated some need for quantitative reasoning. The
researchers suggested that further research is needed to help classroom teachers use the
probabilistic thinking framework to enhance student learning.
Fishbein and Gazit (1984) analyzed the effect teaching of probability indirectly
has on intuitive probabilistic judgments. The researchers defined intuition as "a global,
synthetic, non-explicitly justified evaluation or prediction" (p. 2). The study evolved
around the indirect effect of a course on probability on typical, intuitively-based
probabilistic misconceptions. The teaching program designed for students in Grades 5, 6,
and 7 was intended to determine the capacity of their ability to assimilate, correctly and
productively, basic concepts and solving procedures in probability. The students were
given various situations that gave them opportunities to be active in the following areas:
1. calculating probabilities
2. predicting outcomes in uncertain situations
3. using operations with dice, coins and marbles for
watching, recording and summing up different sets of
outcomes, (p. 3)
Two questionnaire assessments were used to determine the teaching effects. The first
questionnaire was used to determine to what extent pupils in the experimental class had
assimilated and were able to use concepts and procedures taught. The purpose of the
second questionnaire was to determine the indirect effect of instruction on children's
intuitively based misconceptions. The first questionnaire was too difficult for fifth
graders. Nevertheless, for sixth and seventh graders, progress was evident. Because of
the low negative results for fifth-grade with questionnaire one, the second questionnaire
was not reliable and relevant enough for this grade level. However, "in grades six and
seven, the teaching programme has had an indirect positive effect on their intuitive
biases" (p. 22).
Vahey (1998) was interested in middle school students' understanding of
probability. In his dissertation, he used a Probability Inquiry Environment (PIE)
curriculum. The PIE curriculum consisted of computer-mediated inquiry-based
activities, hands-on activities and whole-class discussion. Vahey used a quasi-
experimental design with four seventh-grade classes. Two of the classes used the PIE
curriculum for 3 weeks and the other two classes used a teacher-designed curriculum for
3 weeks. The same teacher taught the 3 week curriculum for both groups. The students
in both groups were given the same quantitative pretest and posttest. There was no
significant difference between the groups on the pretest. However, a significant
difference was found between the two groups on the posttest. The students using the PIE
curriculum out-performed the students in the comparison group on the posttest. Vahey
posited that students using the PIE curriculum could successfully build on their existing
understandings by engaging in activities before formal introductions of concepts,
definitions, and terms.
Drier (2000) was also interested in children's probabilistic understanding. In her
dissertation, she conducted a 2-month exploratory teaching experiment with three 9-year-
old children at the end of their third-grade year. Each child in the study participated in 10
hours of teaching sessions using a computer microworld program designed by the
researcher. The computer microworld program, Probability Explorer, was designed for
students to explore with probability experiments. A pretest and posttest interview
assessment was given to all three children. All teaching sessions were videotaped and
audiotaped. The children's development of probabilistic reasoning and their interaction
with the computer tools varied during the study. The results of the study indicated that
open-ended computer tools could represent children's development of probability
concepts. Drier posited that further research was needed with Probability Explorer in a
variety of small groups and classroom situations with different task and technological
versus nontechnological learning on students' reasoning and understanding of probability.
She also recommended that further research be done to help students connect their
"understanding of probability with their understandings of other mathematics concepts
and their everyday experiences" (p. 359).
Jiang (1993) also developed a simulated computer program, CHANCE. The
program was used to eliminate the use of physical materials when teaching and learning
probability. Without being able to generate large number experiments, it is difficult for
students to believe that certain events have the same possible chance of occurring (Jiang
& Potter, 1994). Jiang and Potter conducted a teaching experiment to evaluate the
performance of CHANCE. The researchers were interested in knowing if the use of
CHANCE could effectively help students achieve conceptual understanding of
probability. There was no control group due to time constraints. Like the
aforementioned study, the teaching experiment was exploratory, and a small sample was
used to obtain information that was more detailed. Three boys and one girl participated
in the study. They were in 5th, 6th, 8th, and 11th grades, respectively. The students in
Grades 5 and 6 were in group one and the students in Grades 8 and 11 were in group two.
The teaching experiment consisted of two 1-hour sessions per week for 5 weeks. The
children in the study took the same pretest and posttest. The pretest was administered to
determine students' previous knowledge. The posttest was used to determine both
students' knowledge and students' ability to solve problems using simulations and
modeling. They found CHANCE useful in making probability instruction meaningful,
stimulating and increasing students' learning difficulties, and in helping students
overcome their misconceptions about probability. Generalizability was limited due to the
small sample used in the study. Also, because the students were above average, there
was no way to determine if students below average would benefit or if the positive effect
of CHANCE was due partly because of the beginning academic level of the students.
Fischbein, Nello, and Marino (1991) conducted a study on students' difficulties
with probabilistic concepts. For example, the concept of certain events is more difficult
to comprehend than that of possible events. They used 618 students in three groups. In
the first group, there were 211 students in Grades 4 and 5. In the second group, there
were 278 students in Grades 1, 2, and 3 with no prior instruction in probability, and in the
third group, there were 130 students in Grades 1,2, and 3 with prior instruction in
probability. Students answered one of two parallel 14-item questionnaires (A or B) of
probability problems. The items on the questionnaires required the subjects to explain
their answers. On many of the problems, the subjects had to determine how the sample
space related to a certain event. The authors indicated that students had difficulty
answering the probability problems on the questionnaire because of students' lack of
skills for rational structures.
Fast (1999) also noticed that students overcame misconceptions in probability
after instruction. Fast investigated the effectiveness of using analogies to effect
conceptual change in students' alternative probability concepts. He wanted to determine
if using analogous anchoring was an effective approach to overcoming probability
misconceptions. Anchors are analogies or examples used to simplify difficult concepts.
It was difficult generating anchoring situations that would be effective for the students.
The investigation centered on high school students' probability concepts. Most of the
students in the study felt analogies were useful in reconstructing their thinking.
Generally, misconceptions are "highly resistant to change" (p. 234). Thus, "the results of
the study showed that the use of analogies was effective for reconstructing high school
students' probability misconceptions" (p. 234). Fast concluded that the use of analogies
could make a valuable contribution to overcoming probability misconceptions.
There is much support on children learning concepts of probability as early as
elementary school. Many important documents, such as Principle and Standards
(2000), have been revised to include the teaching and learning of probability as early as
kindergarten. Although few studies exist at the elementary level, researchers recognize
the importance of introducing probability at an earlier age (Shaughnessy, 1992; Vahey,
In the review, many of the studies (e.g., Fishbein & Gazit, 1984; Baxter and
others, 1993; Jiang, 1993; Vahey, 1998) approached the concepts of probability and
statistics starting in the middle grades and higher through computer simulations and
concrete manipulatives. It was evident in a few of the studies (e.g., Choi & Gennaro,
1987; Kim, 1993) that the use of computer simulations was helpful for learning
mathematics and science and providing more learning time for students. Some of the
studies (Jones and others, 1999; Drier, 2000) provided evidence that students may learn
probability concepts earlier than the age predicted by Piaget and others. At this
particular time, there are very few studies reported on the investigation of how computer
simulations and concrete manipulatives can benefit elementary students in learning
probability concepts. It is not clear at this point the effects that concrete manipulatives
and computer simulations have on elementary students' understanding probability
concepts at an early age.
In many of the studies (Mack, 1995; Anthony, 1996; Frank & Carey, 1997;
Wiegel, 1998; Drier, 2000, interviews and case studies were used to obtain information.
Some researchers (e.g., Choi & Gennaro, 1987; Jiang, 1993; Kim, 1993) did not establish
control groups or validate their instruments. Many of the instruments were used to
provide information on students' probabilistic thinking and misconceptions about
probability. In some quantitative studies (e.g., Confrey, 1990), researchers used
nonrandom samples which were often a consequence of school-based research. In
addition, generalizability was not possible due to the small number of students used in
these studies, which may have weakened statistical results. It is still important to
consider the results of these studies because they can serve as a starting point for further
investigation. In some studies, researchers designed their own computer program (e.g.,
Jiang, 1993; Drier, 2000) for teaching probability with computer simulations because
they posited there was inadequate software available. One program found to be the most
promising to use for teaching probability using computer simulations was Probability
Explorer. However, there has been no research conducted on its effect on whole-class
Some studies revealed that when examining children about specific concepts in
probability, many misconceptions regarding children's understanding of probability were
evident. For example, some students may have misconceptions about a coin toss being
fair if they believed that getting a tail was best.
The constructivist theory has been prominent in recent research on mathematics
and science learning. The constructivist learning and teaching theory has influenced the
instructional environment. Because of this theory, many studies (e.g., Bednarz & Janvier,
1988; Confrey, 1990; Anthony, 1996; Schulte, 1996) have stressed the importance of
students taking an active role in their learning process. The constructivist learning theory
emphasizes the importance of the knowledge, beliefs, and skills an individual brings to
the experience of learning. It also provides a basis for recent mathematics education
reform efforts. There is a need for further research for teaching and learning probability
from a constructivist perspective, especially at the elementary level.
Overview of the Study
This chapter describes the research objectives, the development of the research
instruments, and the participants for the study. It outlines the procedures for the
research design and the data analysis.
The purpose of this study was to investigate the impact of using computer-
simulated manipulatives and concrete manipulatives on elementary students' probability
learning skills and concepts of experimental probability. Of interest to the researcher
was students' ability to predict outcomes of simple experiments and to discuss and
describe the likelihood of events using words such as certain, likely, unlikely, and
impossible. The researcher was also interested in incidental fraction learning that might
occur as participants engage in probability experiences.
The following null hypotheses were tested in the study:
1. There will be no significant difference between students
who use computer-simulated manipulatives and students
who do not use computer simulated manipulatives
regarding students' learning skills and concepts in
2. There will be no significant difference between students
who use concrete manipulatives and students who do not
use concrete manipulatives regarding students' learning
skills and concepts in experimental probability
3. There will be no interaction between students who use
concrete manipulatives and computer-simulated
manipulates regarding students' learning skills and
concepts in experimental probability.
Description of the Research Instruments
Experimental Probability Instrument
The researcher designed the Experimental Probability Instrument (EPI) (see
Appendix A) and administered it as a pretest and posttest. The EPI was used to measure
students' probability learning skills and concepts of experimental probability. The
research was conducted to examine the effects of employing computer-simulated
manipulatives and concrete manipulatives on the above factors.
To insure that the content of the instrument reflected content domain of
elementary students regarding the concept of probability, the researcher used several
methods to determine the content validity of the instrument. First, the researcher used
the literature as a guide to locate items specific to probability. Secondly, the researcher's
teaching experience was used to examine the items for appropriate content. In addition,
items used to create the EPI were adapted from Konold, Pollatsek, Lohmeier, and
Lipson (1993), Vahey (1998), and Drier (2000). The objectives of the EPI are given in
Table 3. Fourthly, the researcher submitted the instrument to two mathematics
educators for their review and analysis of content and later received written comments
from them regarding the content and objectives of the instrument. They provided
feedback on the wording and numbering of some of the items and drawings of the
objects the researcher incorporated into the design of the instrument.
Objectives of Experimental Probability Instrument
# of items
Objective: Test the students understanding, learning skills, concepts
and ability to ...
decide whether a game is fair or unfair by specifying the estimated
probability of an event, given the data from an experiment
discuss the degree of likelihood by distinguishing whether an event is
an instance of certainty, uncertainty, or impossible
test predictions and identify estimates of true probability given a set
of data from an experiment, by counting the number of outcomes of
predict the probability of outcomes by specifying the probability of
simple events or an impossible event
specify the probability of an impossible event
interpret that the measure of the likelihood of an event can be
represented by a number from 0 to 1 by specifying the outcome of an
event between 0 and 1
describe events as likely or unlikely by identifying the most likely
event of two unequally likely events
find outcomes of probability experiments and decide if they are
equally likely by identifying two equally likely events as being
Experimental Fraction Instrument
To measure incidental fraction learning, the researcher also administered a
parallel Experimental Fraction Instrument (EFI) (see Appendix B) as a pretest and
posttest. The role of the EFI was to measure students incidental fraction learning skills
after working with experimental probability.
To insure that the content of the instrument reflected content domain of
elementary students regarding the concept of fractions, the researcher's teaching
experience and the literature were used as a guide. The researcher incorporated items
for the EFI similar to those found in elementary mathematics textbooks.
The researcher conducted a pilot study during the Spring 2000 term using two
fifth-grade elementary classes in a school district in the southeastern United States. The
purpose of the pilot study was to obtain reliability data for the instruments and examine
the application of the treatment.
After the implementation of the EPI and EFI in one class, the researcher made
several changes regarding the content of the instrument. First, objects representative of
manipulatives were redrawn so that the values were a true representation of the size.
These objects included marbles and spinners. Second, more specific directions were
given. Third, the researcher eliminated two questions because they were repetitive.
The researcher observed that the success of the EPI for students was probable
only when students were given an opportunity to try experiments using the computer-
simulated probability software. Because students were not familiar with some of the
terminology used in the lessons, the researcher designed the lessons to provide time for
whole group discussion prior to the activities for all groups in the study. This eliminated
many questions except a few that related to the activities. For example, students
discussed the term least and gave examples of least likely events. All groups continued
in this way until all unfamiliar terms were understood.
An internal consistency reliability estimate for the final version of the EPI was
estimated using an EXCEL program. The reliability estimate for the EPI was given as
.89. An item analysis was done to determine if items had a reasonable level of
discrimination. The difficulty and discrimination indices for the individual items of the
EPI are reported in Table 4. Item discrimination for the EPI ranged from 0 to .26 with a
median of .21.
An EXCEL program was also used to estimate the EFI. The reliability estimate
for the EFI was also .89. The difficulty and discrimination indices for the individual
items of the EFI are reported in Table 5. Item discrimination for the EFI ranged from 0
to .27 with a median of. 15.
The discrimination index is used to describe the validity of a test in terms of the
persons in contrasting groups who pass each item. The researcher obtained the
discrimination index by subtracting the proportion of students in the lower half of the
group who answered the item correct from the proportion of students in the upper half of
the group who answered the item correct. The item-total correlation as a measure of
discrimination for EPI and EFI was given as .55 and .32, respectively.
The item difficulty index is used to describe the percentage of persons who
correctly answer a particular test item. The difficulty index was obtained by the number
of subjects who answered the item correctly, divided by the total number of subjects
taking the test.
The first administration of the study was conducted during September 2000. The
study included four classes taught by the classroom teachers. The study began with the
administration of the EPI and the EFI in all classes. After implementation of the two
Difficulty and Discrimination Indices for Experimental Probability Instrument
Difficulty and Discrimination Indices for Experimental Fraction Instrument
instruments, the researcher decided to make a change to the format of the instruments.
The researcher attached the EFI onto the EPI to encourage students to complete the
The classroom teachers were instructed to teach the probability lessons to
students following the administration of the covariates. Several changes were made to
the lessons based on the researcher's observations. The lengths of the lessons for the
manipulative and the computer/manipulative group were shortened to provide enough
time for discussion. Manipulatives were presorted and labeled before the treatment
lessons for affective use of time. All classes except the control group were assigned to
small groups within the classes prior to the start of the lessons. Another major change
based on the researcher observation was for the researcher to facilitate the lessons
instead of training the teachers to facilitate the lessons. The reason was that although the
teachers were trained and given materials to use for the lessons, they did not follow
through with their roles in the study. One classroom teacher did not use any of the
concrete manipulatives provided. Another classroom teacher did not follow the lessons,
which were designed according to the constructivist theory. The teacher in this
classroom taught using more traditional teaching methods. For example, the teacher
would have the students conduct a simulation using marbles on the computer and then
would ask how many of each did students obtain. There would be no further discussion
of the experience. This continued for 5 minutes, and then the teacher concluded the
lesson. Students were not able to discuss what they learned from their results of the
simulations. The researcher concluded that the study could not be conducted properly
without adherence to the theoretical framework.
Research Population and Sample
The population for this study consisted of students in the fifth grade at public
elementary schools. The research sample consisted of 83 fifth-grade students enrolled in
elementary school in the southeastern United States. The subjects were of various
abilities, mixed gender, and racial background. The students in the classes represented
characteristics of the population of students in these elementary schools. There were
four classes of students: three treatment classes and one control class. Because the
students were already in their assigned classes, true random assignment was not
possible. First, students in one class participated with concrete manipulatives. A second
class experienced instruction with the computer and probability software. A third class
was taught using a combination of concrete manipulatives and the computer. The control
group participated in the traditional setting.
Prior to the study, the researcher obtained permission from the University of
Florida Institutional Review Board for the investigation to take place. The students were
informed and had to obtain parental permission to participate in the study (see Appendix
E). The classroom teachers administered the pretest and posttest instruments in class. A
schedule for the procedures is provided in Table 6.
Schedule of Administration of Instruments and Treatment
January 8, 2001
January 9-11, 2001
January 12, 2001
January 8, 2001
January 22-25, 2001
January 26, 2001
January 8, 2001
January 22-25, 2001
January 26. 2001
January 8, 2001
January 16-18 2001
January 19, 2001
** manipulative group; *computer & manipulative group; ***computer group
Treatment consisted of the use of concrete manipulatives and computer-simulated
manipulates during in-class lessons. There were three preplanned lessons, each lasting
one 60-minute class period. The lessons were presented in a constructivist perspective.
According to information provided in the review of the literature, this included student
discovery, collaboration with other students in groups, and doing activities that relied
heavily on sources of data and manipulatives. Rather than correct students' initial
analysis, the researcher let the experimental results guide and correct students' thinking
(Van de Walle, 1998). This one instructional method gave students an opportunity to
develop and build upon their own mathematical ideas. Rather than the teacher being the
dispenser of knowledge, students were able to use their own inquiry to investigate events.
The concrete and computer-simulated manipulatives used in this study included
coins, dice, spinners, and marbles. These manipulatives were used as simulations of real
situations. These manipulatives can illustrate impossible and certain events. All of these
manipulatives can be used to teach basic concepts of probability. Students need to see
that simulations aid in learning fundamental concepts of probability.
The researcher taught all classes. Each group had its own version of the lessons.
This was due to the type of materials used for each group. Although the lessons were
designed according to the nature of the group, the objectives for the lessons for each
group were parallel. All treatment groups were put into small cooperative groups of four
to six students. The lessons used during the treatment phase are found in Appendix C. A
brief outline follows:
Objective: To have students explore concepts of chance and to decide whether a
game is fair or unfair using concrete manipulatives or computer generated manipulatives.
Procedure: Students worked in cooperative groups of four and six. Students in
each group had a role in their group. The roles in the group included two students in
charge of the manipulatives, one student to record the outcomes and a student to report
the information to the whole class on the performance of their group. The instructor
assessed students' prior knowledge through inquiry about chance and luck. Also, through
whole class discussion, students described what fair meant to them. The students spent
the remaining class time working on activities and reflecting on their outcomes through
group and whole class discussion. The activities are described in detail in the
motivational sections of the lessons in Appendix C.
Objective: To have students work with likely or unlikely events.
Procedure: Students worked in cooperative groups. The instructor checked for
understanding of terms by first having students discuss in their groups what the terms
certain, impossible, likely, or unlikely meant to them. In whole class setting, students
discussed the meaning of these terms. Students worked on activities in cooperative
groups. Following the activities, students responded to questions regarding their
Objective: To have students make and test predictions for simple experiments.
Procedure: Students spent the first part of class time working on activities in their
cooperative groups and remaining class time reviewing concepts of chance, and simple,
likely and unlikely events with the instructor.
According to the NCTM (2000), the study of chance done with the use of
concrete manipulatives should be followed by computer-simulated manipulatives using
available software. This would gradually lead to a deeper understanding of these ideas
through middle and high school. Therefore, the researcher chose to use a computer
software program, "Probability Explorer," developed by Drier (2000). Drier designed
the program to generate simulations of manipulatives for large trials in a few minutes.
The software displays results as a table that presents the experimental results as a
frequency (count) or relative frequency (fraction, decimal, or percent), a pie graph that
displays the relative frequencies, or as a bar graph that shows the frequency distribution.
These added features of this newly designed software program give teachers an
opportunity to reinforce previously taught skills. According to many researchers (e.g.,
Weibe, 1988; Perl, 1990; Strommen & Lincoln, 1992; Clements, 1998; Greening, 1998),
students' experiences with technology provide deeper and more substantial
A nonequivalent control group design was used for the study. There were four
groups with one class of students in each. Treatment Group I was allowed to use
concrete manipulatives for the lessons. Treatment Group II participated in computer
simulations with the lessons, and Group III used a combination of both concrete
manipulatives and computer simulations for the lessons. The Control Group was not
provided with manipulatives or the use of the computer for use with the lessons. The
quasi-experimental research design involved a 2x2 matrix: the level of concrete
manipulative use and the level of computer simulation. The objective was to determine
the effect of the two independent variables, individually and interactively, on the
dependent variable (posttest scores). The researcher used an analysis of covariance
(ANCOVA). The advantage of using an ANCOVA was to control for any initial
differences that may have existed between the four groups and to determine if there was
a significant treatment effect. The EPI and the EFI were used as pretests (covariates)
and as posttests.
In addition to the quantitative analyses of the instrument, the teachers in the study
classes observed the experimental classes to record students' reaction to the use of
concrete manipulatives and computer-simulated manipulatives. These observations
were used to examine students' problem-solving process.
This chapter contains the descriptive statistics and the results of the analysis
pertinent to the null hypotheses of this study. Participants for the study consisted of 83
fifth-grade students from four classes at three elementary schools in Southeast U.S. The
instructional phase consisted of three lessons that tested students understanding, learning
skills, concepts and ability of experimental probability.
Descriptive statistics for the pretest and posttest results for the probability
instrument are presented in Table 7, which includes the number (n) in each group, mean
(m), and standard deviation (SD).
Pretest and Posttest Means and Standard Deviations for EPI
Computer & Manipulative
The researcher employed an ANCOVA to control for initial differences that may
have existed in the pretest results among the four groups. The ANCOVA results revealed
that when considering the dependent variable, posttest scores on the Probability
Instrument, there were no statistically significant difference among students using
concrete manipulatives based on the results of the EPI (see Table 8). Therefore, the
following null hypotheses could not be rejected:
There will be no significant difference between students who use concrete
manipulatives and students who do not use concrete manipulatives regarding
students' learning skills and concepts in experimental probability.
There will be no interaction between students who use computer-simulated
manipulatives and concrete manipulatives regarding students' learning skills
and concepts in experimental probability.
These findings indicated that the interaction terms did not account for a significant
proportion of the variance in posttest scores (adjusted R2 = .3861).
Analysis of Covariance: EPI
Type III SS
Pretest x Manipulative
Pretest x Computer
Pretest x Computer/Manipulative
Note: *significant for p= .05
After these terms were removed, there was minimum change in the observed
variance (adjusted R2 = .3833). However, there was a statistically significant variable for
students' use of computer-simulated manipulatives on the EPI (see Table 8). Therefore.
the following null hypotheses could be rejected:
There will be no significant difference between students who use computer-
simulated manipulatives and students who do not use computer simulated
manipulatives regarding students' experimental probability learning skills
The parameter estimate (see Table 9) indicates how the treatment groups faired
compared to the control group. Results of the interaction analyses show that achievement
in mathematics can significantly interact with computer instruction. Students with very
low scores on the EPI received higher scores on the EPI when instruction included
computer instruction. On average, students in the computer group scored 13 points
higher on the posttest than students in the control group given their individual pretest
scores. This was possible only when students in the computer group had low pretest
EPI: Comparison of the Parameter Estimate for the Treatment and Control Groups
Pretest x Manipulative
Pretest x Computer
Pretest x Manipulative-Computer
Note: *significant for p = .05
scores on the measure of probability skills. However, this advantage disappears when
comparing students who have higher pretest scores (see Figure 1). Low-achieving
students were obviously not benefiting as much from the usual mode of instruction
employed by their teachers, and even though a computer approach was new to them, it
proved to be more beneficial than the more familiar type of instruction. This indicates
that students who have little previous probability knowledge would benefit from
participation in the computer group (see Figure 1). It is also an indication that computer
use shows positive results as indicated by Connell (1998), Kim (1993), and Terry (1995).
In addition, the same interpretation applies when comparing the computer group
to the manipulative group and the manipulative-computer group (see Figures 2 and 3).
Figure 1. Interaction between computer and control groups.
Interaction between Computer and
Figure 2. Interaction between computer and manipulative groups.
Figure 3. Interaction between computer and manipulative-computer groups.
Students in the computer group who have low pretest scores on probability skills tend to
do better than students in the manipulative group and the manipulative-computer group.
This is an indication that neither the manipulative group, nor the manipulative-computer
group was significantly different from the control group. However, like the control
group, the manipulative group and the manipulative-computer group were similarly
different from the computer group.
Incidental learning, which is not planned, is often overlooked. The study of
probability gives students an opportunity to revisit and practice previously learned
concepts and skills. Therefore, the researcher included an EFI to measure incidental
learning of fractions. Descriptive statistics for the pretest and posttest results for the EFI
are presented in Table 10, which includes the number (n) in each group, mean (m), and
standard deviation (SD).
Pretest and Posttest Means and Standard Deviations for EFI
Computer & Manipulative
Significance tests of the parameter estimates for group affiliation indicate that
there was no detectable difference between the mean adjusted posttest scores of students
in the control group and the mean adjusted posttest scores of students in the
computergroup or the computer-manipulative group using a significance level of p = 0.05
given the students fraction pretest scores (see Table 11). The only significant difference
was in the scores of the students in the manipulative group (see Table 11). Even with the
interaction terms removed, there was still a relatively significant change in the scores of
the students in the manipulative group (see Table 12). Controlling for fraction pretest
scores, students in the manipulative group still scored on average, 3.30 points lower on
the fraction posttest than students in the control group (see Table 12). Students in the
computer group and the
Analysis of Covariance: Fraction Instrument
Pretest x Manipulative
Pretest x Computer
Pretest x Manipulative-Computer
Analysis of Covariance: Fraction Instrument
manipulative-computer group still did not appear to have different posttest scores than
students in the Control group.
Not all unplanned learning is effective. Incidental learning is generally not
recognized or labeled as learning by the learner or others; therefore, it is difficult to
Limitations of the Study
Several limitations must be considered when interpreting the results of this study.
Due to the time of year, the classes were intact and nonrandomly assigned to treatment
and control groups. This is not uncommon among studies conducted in school settings.
There may have been preexisting differences in the intact classes to begin with. For
example, the computer/concrete manipulative group was a higher scoring class to begin
with. Because of intact classes, it was not possible to take several classes of students and
include them in each group. Therefore, only one class of students, which included 16-25
students, was involved in each group. This is the reason for the small sample size.
The Probability Explorer was new software that had never been used in whole
class instruction. Therefore, the researcher was unable to compare the results to that of
students who used the Probability Explorer software in whole class instruction. In
addition, the software was only designed for use in IBM computers, which limited where
it could be used in the school district.
The duration of the study was only 4 days. The question might be raised whether
4 days is sufficient time to develop the concepts completely and effectively with
computers and concrete manipulatives. The researcher believes that because traditional
teaching practices are still used in many classrooms, students may not be used to a
constructivist learning environment where students are actively engaged in their learning.
Thus, it may take students a few trials to become adjusted to different teaching and
learning styles. However, because of the schools curriculum, school administrator and
teachers limited outside activities to a few days.
Finally, the researcher has been a mathematics teacher in the southeastern United
States for more than 10 years and was aware of the possibility that the researcher
influencing results may introduce a threat to validity. Using specific preplanned lessons
for students in all the study classes minimized this.
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INGEST IEID EDOAL5H14_8RR8OY INGEST_TIME 2014-06-11T22:29:34Z PACKAGE AA00022179_00001
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