Ainsa, T. (1999).
Success of using technology and manipulatives to introduce numerical
problem solving skills in monolingual/bilingual early childhood
classrooms. Journal of Computers in Mathematics and Science
Teaching, 18(4), 361369.
This study tested the effectiveness of using manipulatives and
technology to teach numerical problem solving skills, including
counting, identifying shapes, matching colors and numbers, addition,
and subtraction. Children (ages 46) in five early childhood
classrooms were studied using M&M candies and Skittles (for
those allergic to chocolate) as manipulatives. Of the 101 students,
41 were monolingual and 60 were bilingual. The candy was used
as a handson tool to supplement the M&M Counting Book
(McGrath, 1995), which was used as part of the mathematics curriculum.
The software used in this study included: KidsMath (Great
Wave Software), Stickybear's Math Town (Optimum Resources,
Inc.), and Stickybear Shapes (Optimum Resources, Inc.).
Students were given M&M candies (or Skittles) and a counting
sheet. The teacher read the M&M Counting Book and
made observations as students performed the activities presented
in the book. Students counted aloud, displayed answers on the
sheets, and made shapes using the candy. After completing the
activities, students used the computers to practice the concepts
covered in the book. Teachers documented data while observing
students. In this study, pre and posttests were not administered.
Success was measured through frequency of correct student responses,
observations, and teacher report. The frequency of successful
responses indicates that the combination of manipulatives and
technology leads to higher student achievement, regardless of
linguistic background.
Baker, J. D., & Beisel, R. W. (2001). An
experiment in three approaches to teaching average to elementary
school children. School Science and Mathematics, 101(1),
2331.
Summary: The researchers studied 22 students in grades
46, in three multiage groups. The students were attending a mandatory
summer session at the Indiana University of Pennsylvania lab school.
Three instruction styles were used to teach the concept of average.
The three styles were traditional lecture, concrete/handson activities,
and visual/spreadsheet activities. Using various methods of comparing
pre and posttest data, and interviews, the researchers concluded
that there are some advantages to the visual/spreadsheet approach.
Official Abstract: The types of experiences children should
encounter to best understand average were investigated in this
study. Using a traditional approach with problem solving, a concrete
approach with manipulatives, or a visual approach with computer
spreadsheets, similar lessons on the arithmetic mean were taught
to 22 children in grades 46, in three multiage groups. Differences
among pretest, posttest, and interview performances suggest some
advantage in the use of a visual instructional style. Continued
gains in performance were found after 4 months without further
instruction. An algorithmiclike definition of average corresponded
to better longterm performance than less precise definitions.
Collaborative deliberations resulted in positive implications
for the researchers' teaching.
Balka, D. S. (1983). Mathematics manipulatives in a prevocational
program: Teacher inservice and classroom research.
(ERIC Document No. ED237739). Washington, DC: U.S. Department
of Education.
This article presents the results of action research conducted
by four high school teachers who volunteered for this study.
Participants: The teachers collecting data chose a total
of 42 students (Sowell, 2) to participate in the study. It was
noted that the entire class participated in the activities. Additionally,
all of the students were high school level and classified as "mildly
mentally handicapped" (Sowell, 10).
Manipulatives: "TryATile cards", "Math Match cards",
"Tangle Tables", "Pathway Activities" (Sowell, 4) tangrams, and
polyhedra dice were used in this study.
Math concepts taught: Manipulatives were used daily (1015
min.) to reinforce concepts of place value and basic computational
skills.
Methodology: Quantitative methods were used as a pretest,
posttest design with the data from each teacher analyzed separately.
Results: Increased computational achievement in whole
numbers was reported in three of the four data sets. The analysis
on the fourth data set showed not gain or loss in achievement.
Overall, the data produced significant results (p<.05) in division
and total gains on three of the data sets and on multiplication
in one data set. The researcher suggested that although some results
are considered significant, they "do not lend support to the notion
that the use of mathematics manipulatives with slow learners can
improve computational skills." (Sowell, 8).
Ball, S. (1988). Computers, concrete materials and teaching fractions.
School Science and Mathematics, 88, 470475.
Ball (1988) found that fourthgrade students using both virtual
and physical manipulatives scored significantly higher on conceptual
understanding of fractions than students using no manipulatives.
Barclay, Jennifer. (1992). A study of a manipulative approach
to teaching linear equations to sixth grade students. Unpublished
master's thesis, Texas Woman's University, Denton, TX.
The sevenweek study conducted by sixth grade teacher, Jennifer
Barclay investigated the effectiveness of using the algebra manipulative,
HandsonEquations to teach the concept of solving linear equations.
Ninetytwo of the students were randomly divided amongst 4 different
heterogeneously grouped sixth grade classes. The remaining 31
students were enrolled in a gifted and talented class. The ages
of the students ranged from 11 to 13 years.
All groups received a pretest; approximately one week of instruction
with the manipulative; a posttest; threeweek retention test;
and a sixweek retention test. The results from the test assessed
the level of concept mastery and retention. The range of scores
for the tests follow: pretest 23.66% to 35.5%, posttest 93.6 to
96.8%, 3week retention 87.8 — 96.8%, and 6 week retention 92.1
— 98.3%.
The results of the study reveal that students appear to have
successfully mastered the concepts of solving linear equations
taught in level one of this program. Recommendations for future
studies suggest using three different methods of teaching solving
linear equations: pictorially, with manipulatives, and the traditional
symbolic approach.
Battista, M.T. (1999). Fifth graders' enumeration of cubes
in 3D arrays: conceptual progress in an inquirybased classroom.
Journal of Research in Mathematics Education, 40 (4), 417448.
This is a case study that involves observing and interviewing
three pairs of fifth graders over a 4week teacherdirected instructional
unit where students are involved in constructing an understanding
of volume of rectangular solids. The purpose of this study was
to examine the cognitive connections that developed and how those
connections change in an inquirybased problemcentered based
classroom. Participants in the study used cubes as a physical
manipulative in an instructional task designed allow students
to develop and refine their mental models about finding the volume
of a rectangular solid through enumeration of the cubes required
to fill the box. This study was shaped by the premise that change
occurs as an accommodation to a perturbation and that perturbations
arise through interactions with the physical world and in communications
with other people.
The task or problem is, given a picture of a net for a box on
grid paper, the picture of a box built from grid paper, or the
verbal description of a box predict how many cubes it would take
to fill the box. Participants worked collaboratively in pairs
to build the box from the given information and check their prediction
about how many cubes it would take to fill the box with the actual
cubes. The discrepancy between the participants' prediction
and the actual number of cubes required to fill the box provided
an opportunity for the pairs to reflect on their thinking or enumeration
schemes and to make adjustments in their method. This process
was carried out over six iterations. An interviewer sat with
each casestudy pair to monitor the evolution of their thinking
through observation the pairs and by asking them clarifying questions.
Pre and posttreatment interviews of the casestudy pairs as well
as transcriptions of the videotapings of each observation session
were also used. The findings from this study are generalizable
based on the comparisons of the findings in the three casestudy
pairs to the similar findings based on the observations and notes
made by the teacher and another researcher on other members of
class.
Immediately after the treatment the only one student was unable
to use a mental model to enumerate the number of blocks it would
take to fill a box. On a postinterview four months after the
treatment all of the participants except the one who did not attain
a good mental model during the treatment were still able to use
their mental model to determine the number of cubes required to
fill a box.
From the implementation of the study and the results of the treatment
on the students several conclusions were drawn. The first is
that in a constructivist classroom students' construct, refine
and revise their conjectures to accommodate conflicts that arise
from discrepancies carrying out a task or through communication.
However, students' theory building is incremental and there is
a need for multiple and varied opportunities for students to build
an understanding of difficult concepts and the process needs to
be mediated by teachers and by instructional material. The researcher
expressed his belief that the treatment would not have been so
effective if students had not have been able to selfcheck their
predictions the manipulative. I believe the most powerful statement
from the study and its' implications for preservice and inservice
teacher education is that, "Only by thoroughly understanding the
pedagogical approach and the usual paths students take in learning
particular mathematical ideasincluding stumbling blocks and
learning plateauscan teachers know when to intervene."
Battista, M.T., Clements, D.H., Arnoff, J.Battista, K., &
Van Auken Borrow, C. (1998). Students' spatial structuring of
2D arrays of squares. Journal for Research in Mathematics
Education, 29 (5), 503532.
This is a case study carried out over one year that involves
interviewing twelve second graders, a third grader, and a fourth
grader. The purpose of this study was to examine in detail how
students structured and enumerated 2D rectangular arrays of squares
in order to gain a better understanding of the mental processes
being used. In this study spatial structuring is defined as the
"mental operation of constructing an organization or form for
an object or set of objects." In a prior study, Battista and Clements
found that before students could count or enumerate the number
of blocks accurately in a 3D rectangular solid the student needed
a mental model or spatial structuring in order to organize the
information for counting. In the earlier study the most effective
spatial structuring model was found to be when a student held
a mental model which allowed them to see the rows and columns
as well as the layers that made up the solid. In the results
of the earlier study it was noted that students often had difficulty
in their mental model of a single layer. Spatial structuring
with a 2D rectangular array is important in the development of
the concepts of area and multiplicative thinking and led this
research team to further study.
Prestudy interviews were carried out with students to establish
the protocol for interviewing students and to establish distinctions
in the way they went about the task.Using this information the
research team developed descriptions for the three levels of sophistication
in student's structuring of 2D arrays as well as descriptions
of student's thinking during the activity. All students were
interviewed at the beginning of the year and the end of the year.
Three times during the year 4 of the original group of twelve
were interviewed again. Particular attention was given to students
as they transition from one level to another. Students received
no instruction on counting squares in a rectangular array during
the year that while study was being conducted. The interviews
were videotaped, transcribed, and analyzed. The task used in
the interview was made up of seventeen different rectangular shapes.
Each shape was scored with squares in a different way and two
of the seventeen were blank on the inside. Students were asked
to predict how many squares it would take to cover the rectangle
and then they were asked to draw in the squares then predict again
how many squares it would take. Participants were given tiles
with which to cover the rectangles.
One conclusion drawn from the study was that students progressed
to a more sophisticated level of when they experienced perturbations
or difficulties in predicting the correct number of squares required
to cover the square. This study informs curriculum and instructional
designers that the mental model for the rowbycolumn matrix arrangement
is not in the array itself, students must construct a mental model
for this arrangement. Therefore, before an array can be used
as a tool to develop concepts such as area and multiplication
we must ensure that student's have attained an understanding of
rowbycolumn structure of the array.
Belcastro, F. (1993). Teaching addition and subtraction of whole
numbers to blind students: a comparison of two methods. Focus
on Learning Problems in Mathematics, 15 (1), 1422.
Belcastro conducted an experimental study of five blind firstgrade
students. The students were split into a group of three and a
group of two. The larger group used Belcastro rods to study addition
and subtraction of whole numbers, while the smaller group used
buttons and other traditional materials. The Belcastro rods are
similar to Cuisenaire rods in size and shape, but replace the
colors of the Cuisenaire rods with specific textural clues such
as longitudinal grooves, horizontal grooves, and holes. Rods
of base length 2, 4, and 8 have longitudinal grooves. Horizontal
grooves are found on rods of base length 3, 6, and 9. Rods that
represent 5 and 10 have holes drilled through them. Rods for
1 and 7 have no markings but are distinguishable because of the
large difference in their lengths. The Belcastro rods were employed
in a way similar to the way Cuisenaire rods are employed to teach
whole number addition and subtraction.
In the fall of 1990, the five students were given a verbal pretest.
The next day, intervention began in the form of instruction utilizing
either the Belcastro rods or traditional materials. Cuisenaire
rods were not used as they had been previously found to fail with
blind students. Once the instruction was completed, the children
were administered a posttest. All instruction and testing were
concluded by January 1991.
All students missed all the questions on the pretest so only
the posttest was considered. Students using the Belcastro rods
did better. Their mean score on the 10question test was 9.67
while the traditional group's mean was 7.5.
The author concedes that the sample size was too small to make
any generalizations to all blind students. He suggests, however,
that they are sufficiently promising to warrant additional testing
of the rods with more blind students as well as sighted students.
In addition to the small sample size, I wondered about the author's
bias, given that he was the inventor of the manipulative that
he was testing.
Berlin, D., & White, A. (1986). Computer simulations and
the transition from concrete manipulation of objects to abstract
thinking in elementary school mathematics. School Science and
Mathematics, 86, 468479.
Berlin and White (1986) found no statistically significant differences
between second and third grade students using physical manipulatives,
virtual manipulatives, and both treatments on measures of spatial
sense and patterning.
Bishop, Joyce Wolfer. (1997, March). Understanding of Mathematical
Patterns and Their Symbolic Representations. Paper presented
at the Annual Meeting of the American Educational Research Association,
Chicago, IL. (ERIC Document Reproduction Service No. ED 410 107)
This study explores seventh and eighthgrade students' thinking
about mathematical patterns. Interviews were conducted in which
students solved problems about sequential perimeter and area problems
modeled with pattern blocks and tiles, generalized the relationships
related to the patterns and represented the relationships symbolically,
identified other valid symbolic expressions of the pattern, and
encountered equationevoking situations. Research questions pertained
to the strategies middle school students use to reason when solving
pattern problems, symbolic representations the students develop,
the students' interpretations of equationevoking situations.
The results of this study support the use of mathematical patterns
to promote algebraic reasoning and provide descriptions of middle
school students' reasoning as they engage in solving a specific
type of pattern problem. Findings also suggest that experience
exploring the relationships in sequential perimeter and area patterns
may help students develop an appreciation for the meaning of expression.
Contains 16 references.
Video and audio taping, and examples of student work were collected
and coded for strategies, accuracy of outcomes, and implications
for student understanding. Students used 5 distinct strategies
that were not modeled for them; 3 main strategies for identifying
alternative symbolic expressions; and 8 different strategies for
equationevoking situations.
Three clusters of students were identified: Verbal and SingleOperational
Expressions and Equations, Transition from Verbal and SingleOperational
to Symbolic Expressions, and Equations, and Symbolic Expressions
and Equations.
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Chassapis, Dimitris. (19981999). The Mediation
of Tools in the Development of Formal Mathematical Concepts: The
Compass and the Circle as an Example. Educational Studies
in Mathematics. 37(3), 27593.
This study focuses on the process by which children develop a
formal mathematical concept of the circle by using various instruments
to draw circles within the context of a goaldirected drawing
task. Particular attention was given to the transition from using
tracers and templates to using a compass for drawing circles and
to the extent to which the use of different drawing instruments
may contribute to the formation of a formally defined mathematical
concept of the circle. The critical difference considered in the
study is that the compass, in contrast to circledrawing tracers
or templates, induces by its ph7sical structure and its functional
use the generative features of formal mathematical concepts of
the circle, that is, the centre and the radius. Analysis of the
empirical data indicates that the use of the compass in circle
drawing structures the circledrawing operation in a radically
different fashion than circle tracers and templates, and brings
into play an actionbound practical thinking. Such thinking has
an overall positive influence on the construction of analytical
concepts by children that are analogous to the formally defined
mathematical concepts of the circle.
The use of circle tracers and templates, providing regulation
and control of the humanhand movement in doing the same practical
actions as those a freehand circle drawing, (thus not qualitatively
transforming the circledrawing operation), seems to influence
although not to radically change the children's spontaneous concepts
of the circle. On the other hand, the use of the compass, which
structures the circledrawing operation in a radically different
fashion than circle tracers and templates, creates the preconditions
which may give rise to concepts constructed in the realm of actionbound
practical thinking, because it is a functional meaning of circledrawing
that emerges when using a compass.
ChattinMcNichols, J. (1992). Montessori Programs in Public
School. (Report No. EDOPS927). Champaign, IL: University
of Illinois at UrbanaChampaign Children's Research Center. (ERIC
Document Reproduction Service No. 348165).
Montessori was one of the pioneers of manipulative use. The
Montessori program relies on student participation in different
activities. Teacher presentation is minimal. Students "create"
their own learning. Students work individually or in small groups
for three to four hours each day. Students cooperate rather than
compete with each other. Montessori programs show increased achievement
test data. Regrettably, most children do not have access to Montessori
education due to lack of money, class spaces, teacher training,
and program availability. Traditional programs that use manipulatives
can take the general Montessori philosophy and cater it to meet
the needs of their curricula and classroom restraints. Manipulatives,
classroom interactions, and studentcentered learning combine
to create a beneficial learning environment for children both
socially and academically.
Chester, J., Davis, J., & Reglin, G. (1991). Math manipulatives
use and math achievement of thirdgrade students. Charlotte,
North Carolina: University of North Carolina at Charlotte.
Summary: The researchers studied two thirdgrade classes
in western Iredell County, North Carolina. Each class contained
26 students. Manipulatives (types were not specified) were used
with the experimental group to teach a geometry unit. The same
unit was taught to a control group using only the text and traditional
lecturestyle instruction. The study was conducted over a period
of two weeks. Using analysis of covariance, the researchers concluded
that the experimental group scored significantly higher on the
posttest than the control group.
Official Abstract: Recent reports indicate that although
17yearold high school students know some basic addition and
subtraction facts, few of the students are capable of solving
multistep mathematics problems. A nonequivalent pretestposttest
control group design study examined the effects of a teaching
method emphasizing manipulative use on the mathematics achievement
of thirdgrade students. Two thirdgrade classes with 26 students
each were selected to participate in the study. Reported demographic
data indicated that the control group class from western Iredell
County was composed of 10 (38%) white male students, 3 (12%) black
female students, and 13 (50%) white male students, and that the
experimental group class from southern Iredell County was composed
of 10 (38%) white male students and 16 (62%) white female students.
A 2week geometry unit from the Silver Burdett textbook was administered
in both classes. The experimental group teacher used mathematics
manipulatives to teach the concepts presented in the unit, and
the control group teacher used only drawings and diagrams to teach
concepts. Analysis of covariance revealed that the experimental
group using mathematics manipulatives scored significantly higher
in mathematics achievement on the posttest scores than the control
group. Further study is recommended to see if this finding is
generalizable beyond the two classes studied or the subject of
geometry. The pretest and the posttest are attached.
Cobb, P. (1995). Cultural tools and mathematical learning: a
case study. Journal for Research in Mathematics Education,
26 (4), 362385.
The investigator did a case study of four pairs of second graders
who were beginning to learn about place value, specifically tens
and ones. The researcher was interested in the transition that
children make from counting by ones to counting by tens and ones.
The manipulative used was the hundreds board, and multilink cubes
in bars of ten were also available to the children. Videotapes
were made of all math lessons in the entire class for one whole
year. They were also made of the eight children in the study
over a tenweek period during that year. In the classroom, typically
there was small group problemsolving followed by whole class
discussion. One camera was used for the latter, while two were
focused on the small group work.
It was found that children's use of the hundreds board did not
support their transition from counting by ones to counting by
tens and ones. However, the hundreds board did appear to support
their ability to reflect on their mathematical activity once they
had acquired the concept. The investigator observed that there
seemed to be an allornothing quality to this ability, as if
the children made a sort of quantum leap to it.
The author suggests that the hundreds board does not facilitate
the acquisition of the concept of counting by tens and ones, because
of its specific prestructure. He suggests that an empty number
line might be better, especially if children are encouraged to
discuss their solutions to wellselected tasks that facilitate
rich imagery.
The author poses an interesting analogy between architecture
and math. An architect of a building organizes our experience,
physical and otherwise, within the building. The architecture
of our math notational system, including the manipulatives that
we choose to use to convey it, organize our math experience, both
constraining it and supporting it in ways that we are often unaware
of.
The author also used the investigation to look for evidence to
support learning theories. Specifically, he contrasted the constructivist
point of view emblemized in Piaget with sociocultural theory emblemized
in Vygotsky. While constructivists emphasize individual diversity,
the sociocultural theorists emphasize homogeneity within the cultural
group. Piaget focuses on conceptual reorganization while Vygotsky
would emphasize the need to enculturate children into established
math practices. The sociocultural theorists would propose that
the tools we pick drive the concepts that we teach, while the
constructivists would say that concept construction precedes symbols.
The author, though a constructivist, found evidence in the study
that the two theories are complementary. My own interpretation
of his finding could be stated thus: Act locally to link into
the global mathematics community.
Conroy, L. M., Tracy, D. M., & Eckart, J. A. (1994). The
differential effects of Miras and mirrors on eighthgrade females'
and males' ability to learn principles of plane mirrors. School
Science and Mathematics, 94(8), 395400.
Summary: The researchers studied 101 eighthgrade physical
science students at a Midwestern, suburban, uppermiddle income,
junior high school. There were five classes total. To teach the
five principles of plane mirrors, two classrooms used Miras only,
two classrooms used mirrors only, and two classrooms used Miras
and mirrors. The same unit was taught to all five classrooms.
The study was conducted over a period of 18 weeks. Using analysis
of variance methods, the researchers concluded that both male
and female students benefited from instruction using both Miras
and mirrors. Other findings were that male and female ability
to learn these concepts may be differentially effected by the
manipulatives used.
Official Abstract: Study of (n=101) eighthgrade physical
science students learning principles of plan mirrors using mirrors
and Miras found that males scored significantly higher than females
on a chapter test, but that all students benefited when both Miras
and mirrors were used throughout the learning process.
Cotter, J. A. (2000). Using language and visualization to teach
place value. Teaching Children Mathematics, 7(2), 108114.
Summary: The researcher studied 32 firstgrade students
at a rural Minnesota elementary school during the 199495 school
year. There were two classes of 16 students each. To teach place
value, the experimental classroom used the "Asian" method,
using language patterns and visualization with abacuses and base10
blocks, while the control classroom used a traditional approach.
Using interviews with the two teachers and the students, the researcher
concluded that the students taught in the "Asian" method
exhibited a better understanding of place value.
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Dairy, L. (1969). Does the use of cuisenaire
rods in kindergarten, first and second grades upgrade arithmetic
achievement (Report No. PS002132)? Colorado Springs, CO:
Department of Research and Special Studies. (ERIC Document Reproduction
Service No. ED032128)
A threeyear study was conducted on students in kindergarten,
first grade, and second grade to determine the usefulness of incorporating
Cuisenaire rods into the mathematics program. The study comprised
control groups from Columbia School and experimental groups from
Whittier, both schools comparable in demographics. Number of
participants varied due to enrollment changes over the threeyear
period. In the experimental groups, kindergartners received individual
instruction on the use of the rods, first grade students completed
teachercreated worksheets based on the use of rods in conjunction
with the Laidlaw workbooks, and second grade students performed
tasks directly from the workbooks using Cuisenaire rods. Use
of Cuisenaire rods was ongoing throughout the threeyear period,
except during the geometry and measurement unit when geoboards
were utilized and when money was used to introduce money concepts.
At the end of each year, Test 5 (Numbers) of the Metropolitan
Readiness Test was administered to both kindergarten groups.
First graders took the same test in the fall of the last two years
of the study. Both first and second grade students completed the
Metropolitan Upper Primary Test (Arithmetic) each spring. Consistently,
all three experimental groups (K2 at Whittier) performed at a
higher level than the control groups (Columbia). Using endofyear
norms for the final year's testing, the scores were all above
the 80%ile, indicating that the utilization of Cuisenaire rods
does enhance mathematical achievement of primary students.
Davis, B. and Shade, D. (1994). Integrate, Don't Isolate!—Computers
in the Early Childhood Curriculum. (Report No. EDOPS9714).
Champaign, IL: Children's Research Center. (ERIC Document Reproduction
Service No. ED 376991).
This study looks at the effectiveness of integrating curricula
in computer labs and with computers in the classroom.
Advances in technology make integrated computer use in the classroom
possible. Unfortunately, computers are mostly used as an isolated
tool for specific skill repetition and premade "quests". Children
need to take charge of their learning while using computers as
a resource tool, organizational instrument, and presentation device.
When computers are used as a drill and practice tool, they negate
the positive attributes computers can provide.
If computers are kept in a computer lab, it further isolates
the impact of technology on education. Children may only go to
the lab for one hour once a week. Computers become a separate,
disjointed part of education.
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Ernest, Patricia S. (1994, Nov). Evaluation
of the effectiveness and implementation of a Math manipulatives
project. Paper presented at the Annual Meeting of the Midsouth
Educational Research Association, Nashville, TN. Available:
Eric Document 391 675
The study consisted of 40 high school teachers from 26 schools.
The teaching experience ranged from 142 years with a mean of
17.5 years. The teachers taught the following courses: Math
7, Math 8, General Math, PreAlgebra, Consumer Math, Technical
Math, Algebra I, Algebra II, Geometry, Trigonometry, and PreCalculus.
The teachers attended a weeklong intensive training workshop in
the use of manipulatives, implemented the teaching strategies
discussed during the workshop in their classroom instruction during
the following year, then attended a follow up session to discuss
strategies and problems identified during the implementation phase
of the study.
The manipulative utilized for this study was the Mathematics
Manipulatives Kit consisting of dice, polyhedra dice, spinners,
Pattern Blocks, circular counters, color cubes, attribute blocks,
geoboards, fraction bars, Algebra Tiles, protractors, compasses,
geometric models, graphing calculators.
Data was gathered to evaluate the weeklong teacher training workshop
and the implementation of manipulatives in classroom instruction.
Onsite observations were conducted to record utilization by course
and manipulative, student participation, student attitudes toward
the manipulatives, and interaction with the content. Evaluation
of the workshop revealed that the teachers found the quality of
instruction to be excellent to very good. Evaluation of the Math
Manipulative Observation lessons revealed that students enjoyed
using the manipulatives and that "on task" involvement was very
high. Students exhibited confidence, eagerness, and a desire
for other experiences. They often employed discovery and problem
solving strategies beyond the assignment. Teachers reported that
students enjoyed and were more interested in assignments when
manipulatives were used. Teachers also reported that more planning
time and class time was needed for lessons involving manipulatives.
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Flanagan, Robin. (1996, April). Unintended
Results of Using Instructional Media: A Study of Second and ThirdGraders.
Paper presented at the Annual Meeting of the American Education
Research Association, New York, NY. (ERIC Document Reproduction
Service No. ED 394 514)
Much of the research on classroom use of educational media has
been hampered by difficulties in isolation a single element of
the medium—television programming, for instance—that influences
behavior in a reliable way. Still, each medium facilitates a particular
type of learning environment, and the collective characteristics
of those environments must be examined for possible effects. The
learner in the televisionbased learning is often passive, and
some experts would suggest that such learners exhibit learned
helplessness. This refers to behavior observed in situations where
a person's actions have no effect on outcomes. This report describes
a study which updates the author's previous work in this area.
This study tries to replicate an earlier finding that 15 minutes
of a mediated learning experience, like a math video, would more
often lead to less persistence or propensity for challenge, than
a more active learning environment would. The study focused on
90 second and thirdgraders in four classrooms from three different
schools. Students in two of the classrooms were from a small city
in upstate New York. One of these classes was bilingual. Two of
the classrooms were from suburban New York. Using tangram puzzles
of varying difficulty, the researcher found that students who
viewed a video gave up on hard puzzles and opted for easier ones
sooner than students who has previously been engaged in more active
treatments of the same topic. Five figures and three tables illustrate
the results.
 second and thirdgrade students
 90
 tangrams
 problem solving
 five sessions for 40 minutes
 2x2 matrix: video or nonmediated activity as one dimension
of the matrix and subject matter as the other dimension, either
scale models or mental arithmetic.
The students answered a questionnaire following the initial
activity. (Very hard Very easy; Very fun Very Boring). Students
watching 15 minutes of television would be less persistent in
working on hard math puzzles than they would be following 15 minutes
of an activity on the same topic.
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Garrity, C. (1998). Does the Use of HandsOnLearning,
with Manipulatives, Improve the Test Scores of Secondary Education
Geometry Students? An Action Research Project Submitted at Saint
Xavier University (Chicago, Illinois). Available ERIC Document
ED 422 179
This study documented the difficulty of high school students
to visualize and understand geometry problems and sought to improve
this ability by implementing a constructivist approach which included
manipulatives, cooperative learning, and reallife problem solving.
The study was conducted with 47 sophomore students enrolled in
two high school geometry classes. One group was considered the
control group and was initially taught using the traditional teacher
lecture method and the second group was the experiential group
which was taught using manipulatives, cooperative learning groups,
and reallife problem solving situations. Research methods included
students and parent surveys, teacher created quizzes and tests,
teacher observations, and interviews. Specific concepts and manipulatives
used to teach the classes were listed. They included: geoboards
(points, lines, segments), graphing calculators (angles), plastic
straws (lines, transversals, polygons), and toothpicks (diagonals).
The researcher concluded after the initial part of the study that
the scores of the experiential group were higher than those of
the control group, thus, the traditional teaching method is less
effective than using manipulatives, cooperative groups, and reallife
examples. The researcher also noted students favored group learning
and reallife problems and exhibited positive changes in attitude
and enthusiasm.
Gibson, H. L., Brewer, L. K., Magnier, J. M., McDonald, J. A.,
& Van Strat, G.A. (1999, April). The impact of an innovative
userfriendly mathematics program on preservice teachers' attitudes
toward mathematics. Paper presented at the Annual Meeting
of the American Educational Research Association, Montreal, Quebec,
Canada.
The Impact of an Innovative UserFriendly Mathematics Program
on Preservice Teachers' Attitudes Toward Mathematics is a
study that was a collaborative project between the University
of Massachusetts at Amherst, and Springfield Community Technical
College. It was conducted at the University of Massachusetts School
of Education in 1998 by researchers Dr. Helen L. Gibson, Laura
K. Brewer, JeanMarie Magnier, James A. McDonald, and Dr. Georgena
A. Van Strat. The primary goal of this study was to determine
if a constructivist approach to learning of preservice teachers
in their college mathematics program would improve their attitudes
toward mathematics. The researchers wanted to see how they could
enhance preservice teachers' attitudes towards math as the progressed
in the college level mathematics course sequence. This document
promotes instructional strategies that use manipulatives and handson
learning experiences in order to explore reallife situations
that relate to students' everyday life.
Participants: The participants of this study were 52 paraeducators
enrolled in the UPDATE program.
Manipulatives: A variety of manipulatives were used in
this study. The examples that were given were Cuisenaire rods
and pattern blocks (Gibson et. al, 14).
Math concepts taught: Algebra
Methodology: Between June 1998 and December 1998, two
questionnaires were administered. Students completed the Revised
Teacher Attitudinal Survey (RTAS) and Instructional Strategies
Survey. The revised survey contained 44 statements to which the
students responded on a scale of 15 with 1 correlating to strongly
agree and 5 — strongly disagree. The 44 items were used to compute
four subcategories: "Views about Mathematics, Being Good at Mathematics,
Learning Mathematics, and Teaching Mathematics" (Gibson et. al,
9) that were intended to measure attitudes about mathematics.
The two surveys provided both qualitative and quantitative information
about the program. The RTAS was administered twice per course.
The Instructional Strategies survey was only administered once
at the end of the course.
Results: The results indicated that the attitudes toward
mathematics did not change during any of the three courses that
were taken. The qualitative data indicated that the methods used
helped them more than a more traditional approach would have with
the added benefit was that the subjects now understood the use
of manipulatives.
Gresham, G., Sloan, T., & Vinson, B. (1997). Reducing Mathematics
Anxiety in Fourth Grade "AtRisk" Students. Available ERIC Document
ED 417 931.
This paper examined whether fourth grade mathematics anxiety
could be decreased by employing mathematical instructional strategies
based on National Council of Teachers of Mathematics Standards
(NCTM). The study was conducted for six months with 17 fourth
grade students and one teacher. Pre and posttest anxiety scales
were given to the students and a journal of instructional strategies
was kept by the teacher. Instructional practices included cooperative
learning, reallife problem solving, manipulatives, calculators,
and computers. Specific class activities which included numeration
and number sense, geometry and measurement, as well as computation
and estimation are listed. Also included are manipulatives that
were used during these activities such as geoboards, rulers, pattern
blocks, and computers. The researcher concludes that students
anxiety is decreased when instructional methods are implemented
based on the NCTM Standards.
Groves, Susie. (1994, April). Calculators: A Learning Environment
to Promote Number Sense. Paper presented at the Annual Meeting
of the American Educational Research Association, New Orleans,
LA. (ERIC Document Reproduction Service No. ED 373 969)
The Calculators in Primary Mathematics Project in Australia was
a longterm investigation into the effects of the introduction
of calculators on the learning and teaching of primary mathematics.
The Australian project commenced with children who were in kindergarten
and grade 1 in 1990, moving up through the schools to grade 4
level by 1993. Children were given their own calculators to use
when they wished, while teachers were provided with some systematic
professional support. Over 60 teachers and 1,000 children participated
in the project. This paper describes some critical number sense
and reports on the results of interviews with 4^{th}grade
children (n=58), approximately half of whom had longterm experience
with calculators. Children with longterm experience with calculators
performed better on the 12 mental computation interview items
overall, the 24 number knowledge items overall, and the 3 estimation
items taken individually. Overall, their performance was better
on 34 of the 39 items, with the greatest differences in performance
in mental computation generally occurring on the most difficult
items. Their pattern of use of standard algorithms, leftright
methods, and invented methods for mental computation items did
not vary greatly from that of the noncalculator children.
A written test, a test of calculator use and two different interviews
were used. The first interview focused on different aspects of
children's understanding of the number system, together with their
choice of calculating device for various computational tasks and
their solutions to "real world" problems based on division and
multiplication. The second interview, which focused on number
sense, was designed to complement the two tests and the first
interview.
This paper and other (Groves, 1993a; submitted) show that children
with longterm experience of calculators performed better than
children without such experience on a range of computation and
estimation tasks and some "real world" problems; exhibited better
knowledge of number, particularly place value, decimals and negative
numbers; made more appropriate choices of calculating device;
and were better able to interpret their answers when using calculators,
especially where knowledge of decimal notation or large numbers
was required.
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Hartshorn, R. and Boren, S. (1990). Experiential
Learning of Mathematics: Using Manipulatives. (Report No.
EDORC905. ) Charleston, WV: Appalachia Educational Laboratory.
(ERIC Document Reproduction Service No. ED 321967).
This study is a compilation of results from previous statewide
studies on the use of manipulatives in the classroom. Using data
from surveys, Hartshorn and Boren found: Primary teachers generally
accept the use of manipulatives Manipulatives are useful in the
transition from concrete to abstract when taught in steps (semiconcrete,
semiabstract) Experienced teachers use manipulatives less than
inexperienced teachers Teaching with manipulatives is effective
only when the proper manipulative and activity is used Longterm
use of manipulatives is more effective than short term use. Teachers'
training influences the effectiveness of manipulatives Manipulatives
are used infrequently at the secondary level even though many
students need ideas introduced at the concrete level.
Hatfield, M. M. (1994). Use of Manipulative Devices: Elementary
School Cooperating Teachers SelfReport. School Science of
Mathematics, 94, 303309.
The article discusses the use of manipulatives in the elementary
setting (K6). 87 teachers were obtained for the research based
on a survey that was mailed to 106 (K6) teachers with 5 or more
years in teaching. Those who responded to the survey were used
in the study. This quantitative study shows the familiarity,
availability, and use of eleven different manipulatives. The
manipulatives used in this study were: pattern blocks, cuisenaire
rods, geoboards, flexicounters*, base 10 blocks, ropygrams* number/math
balance, bundleable materials, tangrams, fraction bars, and attribute
blocks. Note: those manipulatives marked with an * are not manipulatives
but were used to determine response bias. 23.8% of those that
responded said they were familiar with the flexicounters and
1.2% said they were familiar with the ropygrams but that neither
of the manipulatives were available.
The results of the study show that there is a decline at the
intermediate grades (46) in terms of use of manipulatives. It
further shows the need for universities to have more say as to
where and with whom their preservice teachers will conduct their
experience.
Haughland, S. (2000). Computers and Young Children.
(Report No. EDOPS004). Champaign, IL: University of Illinois.
(ERIC Document Reproduction Service No. ED 438926).
"Computers have an impact on children when the computer provides
concrete experiences, children have free access and control the
learning experience, children and teachers learn together, teachers
encourage peer tutoring, and teachers use computers to teach powerful
ideas."
Although theory suggests a constructivist philosophy for children's
computer use, most teachers use technology in traditional ways
(basic skills and instructional games). When computers are used
effectively, children have significantly greater developmental
gains than children without computer use. Computer use enhances
children's selfconcept. Young children demonstrate increased
levels of spoken communication and cooperation when working with
a group and discussing their experiences while using the computer.
Teachers with proper computer training (defined by practical experience,
workshops, models and mentors, and supervisory followup) effectively
integrate computers into their lessons when provided with adequate
classroom resources.
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JohnsonGentile, K., Clements, D.H., & Battista,
M.T. (1994). Effects of computer and noncomputer environments
on students' conceptualizations of geometric motions. Journal
Educational Computing Research, 11 (2), 121140.
This is a quantitative study that included interviews, which
involved 223 fifth and sixth grade students from 9 different teacher's
classrooms during the spring semester. The teachers were veteran
teachers in both urban and suburban settings. Participants were
using either Miras or the Logo MIRROR program to identify lines
of symmetry and paper or acetate sheets or the Logo Geometry
MOTIONS microworld to determine congruence, slides, flips and
turns. The purposes of this study examine students' conceptualization
of geometric motions and the effects of presenting the curriculum
via a computer with computerbased manipulatives or via paper
and pencil with hand held manipulatives. Possible impact caused
by gender differences and students' levels of thinking, based
on the van Hiele taxonomy, in the domain of geometric motions
were also investigated.
Two fifth grade classes and onesixth grade class were assigned
to one of three treatment groups for an eightday motions unit.
The LOGO group received all of their instruction using the Motions
strand of the Logo Geometry curriculum, the nonlogo group
received instruction in the identical curriculum using noncomputer
manipulatives rather than the Logo tasks. A third nontreatment
group participated in the regular mathematics program
including a twoday textbook lesson on symmetry. A pretest of
general achievement in geometry was administered to all students.
Immediately upon completion of the unit a posttest on motion geometry
was administered to all students and it was readministered one
month later. Two boys and two girls were randomly selected from
each classroom for an individual thirtyminute structured interview.
An ANOVA on the pretest showed not significant differences but
the ANOVA on the posttest and delayed posttest showed a significant
treatment effect. Both the Logo and nonLogo posttest scores were
higher than the control group. The immediate posttest did not
show a significant difference in the Logo and nonLogo group, however
the delayed posttest scores were significantly higher for the
Logo group.
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Karp, K. (1990). Manipulative materials in
the primary level mathematics lesson: Are there viable alternatives
(Report No. SE051515)? Garden City, NY: Adelphi University.
(ERIC Document Reproduction Service No. ED320774)
This study investigated student achievement differences between
three mathematics programs: Explorations (Addison Wesley),
Mathematics (Silver Burdett), and the Comprehensive
School Mathematics Program (CSMP). Manipulative use
varied among the programs, ranging from a highly handson manipulative
approach (Addison Wesley) to an abstract focus incorporating no
manipulatives (CSMP). Mathematics (Silver Burdett)
was a combination of both approaches. Five elementary schools
in a predominantly white middleclass district were the subjects
for this study. After sampling the three series, 18 teachers
were voluntarily assigned the pilot programs: five working with
Addison Wesley resources, five incorporating Silver Burdett materials,
and eight using the CSMP.
The three mathematics programs were implemented in first grade,
with an average class size of 21 students, over the course of
one school year. Pretesting was administered during a oneday
window in October and posttesting was completed in the spring
during a twoday period. Data from the tests, teacher questionnaires,
and structured interviews were used to determine effectiveness
of the programs. An analysis of covariance (ANCOVA) was completed
to evaluate whether students performance was above or below the
expected scores. Results show the CSMP as the most effective
in raising students to a higher achievement level. The Explorations
cohort showed the least gains. Teachers in this program reported
the following concerns: excessive time spent creating resources,
complexity in managing the classroom, and the need for extra time
to complete lessons.
Kieran, C., & Hillel, J. (1990). ?It?s tough when you have
to make the triangles angle?: Insights from a computerbased geometry
environment. Journal of Mathematical Behavior, 9, 99127.
Kiernan and Hillel (1990) found that sixthgrade students using
the computermicroworld virtual manipulative made significant
gains in understanding the nature of isosceles triangles.
Kim, S. (1993). The relative effectiveness of handson and computersimulated
manipulatives in teaching seriation, classification, geometric,
and arithmetic concepts to kindergarten children. (Doctoral dissertation,
University of Oregon, 1993). Dissertation Abstracts
International, 54(09), 3319.
Kim (1993) found no statistically significant differences between
kindergarten students who viewed or used physical manipulatives
and those using virtual manipulatives on measures of addition,
geometric classification, and counting skills.
Kjos, Ruth, Long, K. (1994). Improving Critical Thinking and
Problem Solving in Fifth Grade Mathematics. An Action Research
Project Submitted at Saint Xavier University (Chicago, Illinois).
Available ERIC Document 383 525
This research describes an intervention to improve the critical
thinking and problem solving ability of fifth grade students.
The study was conducted with 171 fifth grade students from two
public schools in Illinois. The research methods included student
math autobiographies, teacher created tests, teacher surveys,
and student surveys. The instructional interventions implemented
to improve critical thinking skills and problem solving were student
journal writing about metacognitive processes, direct instruction
to students on how to think critically about and solve problems,
and the use manipulatives to improve instruction. Specific manipulatives
that were used were tangrams (shapes and area), unifix cubes (area
and perimeter), colored counters (probability), base ten blocks
(place value and decimals), pattern blocks (fractions and percents),
and calculators (percents). The study concluded that the implementation
of the above mentioned teaching strategies improved student attitudes,
increased the students' ability to write about their own thinking,
and increased student problem solving abilities.
Kohler, M., Kohler, E. (1996). Improving Mathematics Education
in Grades 69 through the Integration of Content, Technology,
and Manipulatives: Formal Cumulative Evaluation Report. National
Science Foundation Grant ESI9155296. Available ERIC Document
ED 401 129
This report described the findings of a threeyear project in
Alabama which focused on improving the teaching behaviors, knowledge,
and attitudes of 58 mathematics teachers in grades 69. Research
methods used were pre and posttests, grades, focus groups, questionnaires,
interviews, observations, and evaluations. This paper reported
whether or not the participants felt the project was successful
rather than describing the actual methods used over the three
year study to improve teaching behaviors, knowledge, and attitudes.
Nevertheless, the research concluded that participants felt that
their mathematics knowledge was increased and they were more skilled
at using manipulatives and computers in their instruction and
felt they did so more frequently and effectively after participating
in the study. Teachers also noted improved student performance
and attitudes in their classrooms.
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Lackey, B. and Reglin, G. (1991). Manipulatives
and Achievement of Subtraction Basic Facts for Rural Second Grade
Students. Journal of Research in Education, 1, 5356.
In this qualitative research, the study investigates the effects
of a manipulative instructional approach and traditional instruction
on the achievement of subtraction facts for 4 AfricanAmerican
and 26 white second graders in a rural North Carolina public school.
The 30 subjects were below/average, average, and above average
in ability. There was no correlation between the race and ability
of the subject. The 30 students were broken down into two groups.
One group used a traditional approach to the subtraction facts.
The other group used a manipulative approach. The data was collected
through tests, and the ability to communicate their understanding
of subtraction. It was concluded that greater gains in achievement
were of subtraction basic facts occurred with the manipulative
instruction approach.
LaraAlecio, R., Parker, R., Aviles,
C., Mason, S., & Irby, B. J. (1998). A study of the use of
manipulatives in the assessment of mathematics instruction with
ESL Hispanic students. Bilingual Research Journal, 22(24),
215235.
Summary: I'm including this reference and abstract because
it seems like an ideal research article for our group. Unfortunately,
the report is written in Spanish only. If anyone is able to translate,
I'd be interested in reading it. The follow abstract was the only
part available in English.
Official Abstract: As an alternative form of mathematics
assessment for use with limitedEnglishproficient students, 14
mathematics tasks using manipulatives were administered to 45
Hispanic students in grades 13 and readministered 23 weeks later.
Test reliability and validity, task difficulty, and the relationship
among test subscales across grades were examined.
One final note: I found abstracts of dissertations that
sounded very interesting and informative on the topic of education
research with math manipulatives, but I learned that acquiring
doctoral dissertations from other universities is very difficult
and/or costly. In some cases, the abstracts included detailed
information about the research and findings that could be useful
to us in our own dissertation research. If you are interested
in investigating this, check the library databases on the GMU
library web site, and search in the topic "Education" to find
the "Dissertation Abstracts" database.
Leinenbach, Marylin; Raymond, Anne M. (1996). A twoyear
collaborative action research study on the effects of a "Handson"
approach to learning algebra. Paper presented at the annual
meeting of the North American Chapter of the International Group
for the Psychology of Mathematics Education, Panama City, FL.
Available: Eric Document 398 081
The twoyear study consisted of two phases: phase one involved
instruction with algebra manipulatives and phase two was a follow
up on participants from phase one. The first phase consisted
of five eighth grade classes, approximately 120 students, age
13. The second phase was a follow up on the same students regarding
their retention of the manipulative "algebra learning strategies"
during their 9^{th} grade math course. The manipulative
used was "HandsonEquations" developed by Dr. Henry Borenson.
The manipulative uses pawns, number cubes and a balance to teach
the concept of solving linear equations.
The first phase consisted of three parts. The first nine weeks
involved instruction taught in a nonmanipulative style using
the adopted textbook. The 26 lessons, Handson Equation manipulative
program were then implemented. After the completion of the manipulative
lessons, the instruction returned to a nonmanipulative style
with the adopted textbook.
Data collection methods consisted of surveys, student reflections,
work samples, test scores and interviews. Students were encouraged
to use manipulatives during quizzes and tests that were designed
in a format that paralleled the manipulative instruction. All
students took a mandatory standardized algebra test at the end
of the school year.
The results of the first phase revealed that the class averages
during the textbook phases were lower than the manipulative phase.
The teacher noted that students were better able to show understanding
of algebraic concepts with the manipulatives. The teacher's
concern was that she had weakened the students' abilities to work
algebraic problems without manipulatives, but the results of standardized
exam revealed that 80% of the students scored 60% or better. This
far exceeded the expectations of the administration and colleagues
and led Leinenbach to believe that she had successfully helped
students bridge the gap between concrete and the more abstract
algebra.
The second phase consisted of a survey of all students who participated
in the 8^{th} grade study, and only nine responses were
received for the second phase of the study. No results were reported
for this phase.
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McClung, Lewis W. (1998). A study on the
use of manipulatives and their effect on student achievement in
a high school algebra one class. Unpublished master's thesis,
SalemTeikyo University, Salem, WV. Available: Eric Document
425 077
McClung's nineweek study investigates the use of Algeblocks
in a high school Algebra I class. There are 2 classes, 49 students
in total, of sophomore and junior level students, ranging in age
from 1517 years. The manipulative studied is Algeblocks and
the topic of the lessons taught during this nine week study is
polynomials.
The study consists of a pretest, treatment and posttest. The
control group, Group A receives a traditional teaching method
of lecture, homework, and inclass worksheets. The treatment group,
Group B receives traditional teaching method of lecture, homework,
but instead of inclass, worksheets the students work with the
manipulative Algeblocks.
McClung uses a twosample ttest to analyze the data. The pretest
data reveal that there is no significant difference between the
two groups but the posttest analysis reveals that there is a significant
difference in the achievement levels of the two groups. A comparison
of the group means shows that Group A mean = 77 while Group B's
mean = 52. These results would seem to indicate that the use
of manipulatives in algebra at the high school level is not beneficial.
McClung suggests several key factors that may have influenced
the results. The students were out of the range of concrete operational
stage and into the formal operational stage of development. The
students were not allowed to use manipulatives on the posttest.
The instructor was new to the concept of using manipulatives
and did not acquire sufficient knowledge of the manipulatives
before the study began which resulted in the manipulatives not
being properly incorporated into the curriculum.
McCoy, L. P. (1989). Perceptual Preferences of Mathematically
Deficient Elemntary Students: Implications for Instruction.
U.S. Indiana: National Center for Research on Teacher Learning.
(ERIC Document Reproduction Service No. ED305 379)
Subjects in this study were eleven students from two public schools
enrolled at a university remedial mathematics clinic. Another
group consisted of eight average/above average students experiencing
slight or no difficulties in mathematics. The students were in
grades 3 through 6. The focus of the qualitative study was on
assessing the use of concrete materials in mathematics instruction,
comparing the perceptual preferences of mathematicallydeficient
and average/above average elementary school students, and using
the information to make recommendations for instruction.
Results concluded that the students in the average/above average
group preferred an auditory or visual mode of learning, while
the remedial students preferred a kinesthetic mode. There was
no difference in preference for tactile mode. The final conclusion
is that the remedial students would benefit from more diverse
instructional activities. The results strongly support the use
of concrete manipulatives.
Meira, L. (1998). Making sense of instructional devices: the
emergence of transparency in mathematical activity. Journal
for Research in Mathematics Education, 29 (2), 121142.
The investigator explored the idea of transparency, explaining
it as an index of the learner's access to mathematical knowledge
and activities. He tried to discern, through this study, whether
transparency resides in the manipulative itself, or whether transparency
emerges from the user's interaction with the manipulative, given
his or her background.
Nine pairs of eighth graders, aged 1314, participated in the
study on a volunteer basis after school. All investigated the
concept of linear functions. Three pairs each were randomly assigned
to use winches, springs, or number machines. The winches had
rollers of different circumferences around which were wound cords
with objects tied to their ends. The springs could hold weights
of various sizes. The number machines were computers with input/output
displays.
The investigator observed the classes of the participants for
three weeks prior to their two after school 1 1/2 hr problemsolving
sessions, which were videotaped.
The manipulatives that were used were ranked by their epistemic
fidelity, that is, by which should inherently show the concept
of linear functions most clearly. They were judged to be ordered
as follows: winch, spring, computer display. The videotapes were
analyzed to see if the students found transparency in the same
order.
It became clear that it is not the manipulative itself that "contains"
the concept, that is, transparency does not reside in the object.
Rather, transparency emerges in the process of the objects being
used by students who come to the task with prior knowledge and
who participate in discussion that ensues in their use. It was
found that the winch and spring, judged most transparent inherently,
were the least transparent to the students. While these two manipulatives
were supposed to make math concepts apparent, the students had
to expend much effort, instead, including employing math, to make
sense of the manipulatives. On the other hand, students readily
made mathematical inferences about linear functions from the input/output
computer display.
Moore, J. L., and Schwartz, D. L. (1994). Visual Manipulatives
for Proportional Reasoning. U.S. Tennesse: National
Center for Research on Teacher Learning. (ERIC Document Reproduction
Service No. ED376 200)
The goal of the qualitative research was to design a learning
environment that facilitates a move from implicit to a more explicit
understanding of proportionality. 49 high ability sixth grade
mathematics students using the Jasper Adventure Series of problems
participated in the research. The Jasper Adventure Series was
developed by the Cognition and Technology Group at Vanderbilt
in 1992. The research was conducted based on prepost test and
being able to extrapolate and visually prove answers. Students
were more successful using manipulatives.
It was concluded that the potential of a manipulable visual
representation for highlighting the structural invariances within
a proportion and the proportional invariances between domains
leads to an understanding that transfers to more complex proportional
problems.
Moyer, Patricia S. (2001). Are we having fun yet? How teachers
use manipulatives to teach mathematics. Educational Studies in
Mathematics: An International Journal, 47(2), 175197.
Teachers often comment that using manipulatives to teach mathematics
is "fun!" Embedded in the word "fun" are important notions about
how and why teachers use manipulatives in the teaching of mathematics.
Over the course of one academic year, this study examined 10 middle
grades teachers' uses of manipulatives for teaching mathematics
using interviews and observations to explore how and why the teachers
used the manipulatives as they did. An examination of the participants'
statements and behaviors indicated that using manipulatives was
little more than a diversion in classrooms where teachers were
not able to represent mathematics concepts themselves. The teachers
communicated that the manipulatives were fun, but not necessary,
for teaching and learning mathematics.
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Neiderhauser, D.; Stoddart,
T. (2001). Teachers' Instructional Perspectives and Use of Educational
Software. Teaching and Teacher Education, 17(1).
1. Age/grade level: elementary
school (K6) teachers
2. Number of participants:
1093 teachers
3. Manipulatives used: educational
software
4. Math concept taught:
basic math skills and openended problems
5. Duration of the study:
survey data
6. Research methods/procedures:
written survey (questionnaire)
7. Results: Eightyfive
percent of the teachers surveyed used only skillbased software.
Teacher surveys reflected a learnercentered orientation and a
constructivist view of learning, but those values were not reflected
in their children's computer use. Virtual manipulatives,
characterized as openended software, would promote a constructivist
approach to computer learning. Unfortunately, teachers view
virtual learning as different then classroom learning.
Noble, T., Nemirovsky, R., Wright, T., &Tierney, C. (2001).
Experiencing change: the mathematics of change in multiple environments.
Journal for Research in Mathematics Education, 32 (1),
85108.
The investigators carried out a case study, observing how two
fifthgrade boys explored the concepts of the mathematics of change
across several embodiments of change. The essential concepts
are rate of change, which in calculus would be the derivative,
and accumulation, which in calculus would be the integral. The
class that the boys were in was participating in a fourweek unit
on the topic. The investigators filmed the whole classroom and
also focused in on one group of two boys.
The students in the class had taken trips across the classroom,
proceeding at different rates, marking their progress by dropping
bean bags at specified time intervals. It was, therefore, possible
for them to see that the space intervals between the bean bags
differed depending on their rate of movement across the room.
The study focused on the boys' interaction with three additional
embodiments of the mathematics of change. The first embodiment
was taking trips with Cuisenaire rods of two lengths along meter
sticks. The second was a similar exploration on a written table
of values. The third was a computer software program called Trips ©.
The investigators explored two concerns: 1) where the mathematics
reside, that is, whether in the manipulative or in the students,
and 2) how students make connections across embodiments.
On the first point, they argue that the mathematics emerges from
the students' process of making the environment into a livedin
space for themselves rather than in the manipulative materials
themselves. While the designer of an activity may have certain
expectations for what the student will experience, the way students
act and make sense of their actions can vary widely from the designers'
expectations. A space is considered "livedin" when
the students' interactions with it are relational, intentional,
and creative. "Relational" refers to how the changes
affect the space as a whole, "intentional" describes
a space in which students do things and accomplish purposes, and
"creative" spaces are those in which the space is constantly
being recreated as it is experienced.
On the second point about students making connections, the investigators
describe students as finding family resemblances among the embodiments
together with their own background of experience. The strength
of the concept results from the overlapping of many fibers, as
in a thread.
In the specific investigation of the two boys, the researchers
observed that the boys brought to the three embodiments their
previously owned concept of racing, even though the curriculum
developers had deliberately avoided terminology of racing in the
design. Nevertheless, this allowed the boys to make the space
their own, by enabling them to become engaged and to interact
with the environments on their own terms. The boys got similar
numerical results in the table and computer environments. However,
because of some difficulty with manipulating the Cuisenaire rods,
the numbers in this environment did not match the results in the
other two environments. Nevertheless, the boys were able to crisscross
their experiences and find "family resemblances" among
the various embodiments of trips, with their underlying concepts
of rates of change and accumulation. The boys found similarities
among the trips, while each trip retained its own identity.
Noss, Richard. Healy, Lulu. Hoyles, Celia. (July 1997). The
Construction of Mathematical Meanings: Connecting the Visual with
the Symbolic. Educational Studies in Mathematics. 33(2),
20333.
In this paper, we explore the relationship between learners'
actions, visualisations and the means by which these are articulated.
We describe a microworld, Mathsticks, designed to help
students construct mathematical meanings by forging links between
the rhythms of their actions and the visual and corresponding
symbolic representations they developed. Through a case study
of two students interacting with Mathsticks, we illustrate a view
of mathematics learning which places at its core the medium of
expression, and the building of connections between different
mathematisations rather than ascending to hierarchies of decontextualisation.
This is a qualitative case study observation between a pair
of students with one computer. They needed to program a computer
to complete the task presented.
The students empirical solution emerged from their expressions
of the invariant structures, rather than preceding them. Second,
with Mathsticks the means of expressing actions is firmly soldered
to the activity. The students were responsible for placing the
matches in such a way that the colourchange occurred, and for
establishing the rhythms of action which led to their becoming
expressed symbolically.
Nute, N. (1997). The impact of engagement activity and manipulatives
presentation on intermediate mathematics achievement, timeontask,
learning efficiency, and attitude. (Doctoral dissertation, University
of Memphis, 1997). Dissertation Abstracts International, 58(08),
2988.
Nute (1997) found no statistically significant differences between
fourth, fifth, and sixthgrade students who viewed or used physical
manipulatives, virtual manipulatives, or both on measures of patterning
and geometric transformations. However, all groups scored higher
than those students with no manipulative exposure.
Nute, N. (1997). The impact of engagement activity and manipulatives
presentation on intermediate mathematics achievement, timeontask,
learning efficiency, and attitude. Dissertation Abstracts
International, 58(08), 2988.
This study examined the effect of engagement activities and manipulativetype
presentations on students' math achievement, timeontask, learning
efficiency, and attitude. The participants were 241 intermediate
students (grades 4, 5, and 6).
Students were randomly assigned to groups. Six groups received
a combination of instructional strategies using manipulativesboth
concrete and computer. One control group had no manipulatives.
Data was collected in three ways: students took a posttest measuring
their achievement of patterns content, completed a timeontask
measurement, and filled out an attitude questionnaire.
The results indicated that the computer only presentation took
more time than the concrete manipulative only presentation. With
regard to grade level effects, timeontask was equal for fourth
and fifth graders. Efficiency was higher for sixth graders than
for fourth and fifth graders. Manipulative groups showed higherlevel
recognition and application achievement performances than the
control groups. Overall, manipulative instruction strategies
showed more effective for higherorder tasks than did no manipulatives
instruction.
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Outhred, L.N. & Michelmore, M.C. (2000) Young
children's intuitive understanding of rectangular area measurement.
Journal of Research in Mathematics. 31(5). pp 602625.
A sample of 115 children was randomly selected from 40 Grades
1 to 4 classes in four schools serving a range of cultural groups
in a medium socioeconomic area of Sydney. The focus of this research
was to analyze the strategies young children use to solve rectangular
covering tasks before they have been taught area measurement.
Research Questions:
1. What strategies do young children use to find the number of
unit squares that cover a rectangle?
2. Can children's strategies be classified into a sequence of
developmental levels?
3. What operational principles underpin this developmental sequence?
Information concerning the strategies that children used to solve
a variety of arraybased tasks was collected in individual interviews
conducted early in the school year. The interviewer (the first
author) inferred children's strategies from a combination of observation
and careful questioning as the children worked through tasks involving
drawing, counting, and measurement
Children's solution strategies were classified into 5 developmental
levels; Level 0: Incomplete covering, Level 1: Primitive covering,
Level 2: Array covering, constructed from unit, Level 3: Array
covering, constructed by measurement, and Level 4: Array implied,
solution by calculation.
Four Principles Underlying Rectangular
Covering
1

The rectangle must be completely
covered by the units, without overlaps or gaps.

2

The units must be aligned
in an array with the same number of units in each row.

3

Both the number of units
in each row and the number of rows can be determined from
the lengths of the sides of the rectangle.

4

The number of units in
a rectangular array can be calculated from the number of
units in each row and in each column.

Crucial learning leaps occurred when children start thinking
in terms of rows. Initially, rows are recognized as geometrically
equivalent; the fact that the number of units in each row is constant
emerges later. Finding the number of rows is the next problem
to be solved; when this problem is solved, a child is only a short
step from being able to calculate the total number of units.
An important implication is that students need to link area measurement
to both linear measurement, and multiplicative concepts before
the area formula can be meaningfully learned.
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Parham, J. L. (1983). A metaanalysis of the
use of manipulative materials and student achievement in elementary
school mathematics. Dissertation Abstracts International 44A,
96.
Park, J. (1993). Time studies of fourth graders generating
alternative solutions in a decisionmaking task using models and
computer simulations. Journal of computing in childhood Education,
4 (1), 5776.
This study is a quantitative study completed with 240 fourth
graders from 12 classrooms in two public school districts. The
students all had prior experience using the computer. The purpose
of this study was to determine any differences in the time required
to complete a decision making task presented in four different
ways. Participants in the study manipulated bags of real jellybeans
as well as images bags of jellybeans on the computer screen. Variables
looked at in this study were the time between when the task was
given and the student begin to show his or her response, the estimated
timepermove while developing the first response, and the time
required if another response could be produced. The computer
kept track of the data generated during the computer simulation
and research cues on an audiotape were used to collect times during
the manipulatives were used. This study was conducted outside
of the regular classroom; a researcher sat beside the student
and read all instructions and questions from the computer monitor.
In addition, the researcher provided scripted prompts. These
were recorded in a notebook or on an audiotape.
The researcher uses the definition for decisionmaking as "the
process of making reasoned choices among two or more alternatives
based on judgments that are consistent with the knowledge and
value of the decisionmaker."; The task in this study is a computer
simulation in which two children have tied while playing a video
game and they must show a fair way to divide nine bags of different
color jellybeans between the two players. Stratified random sampling
was used to assign the 240 fourth graders to one of four treatment
groups. During a 1520 minute session each group was given one
of the four simulations modes. The simulations were 1) a micro
simulation of the task where the student entered their selection
through keyboard input, 2) a micro simulation of the task where
the student entered their selection through keyboard input and
had bags of jelly beans present for reference, 3) a micro simulation
of the task where the student entered their selection through
light pen input, and 4) a manipulation mode where students use
real bags of jelly beans for distribution.
Multivariate and univariate analysis were carried out on the
three time variables. The results of the analysis indicates that
the mode of presentation makes a difference in the decision making
time in this simulation. It appears from these results that it
takes longer to make a decision when using computer simulations
of concrete situations than when using the concrete manipulative
even when the concrete manipulative is near by for reference or
when using a light pen rather than the keyboard to enter responses.
An analysis of other data collected seems to indicate that students
think differently in the different presentation mode. Also, in
all four simulations, it took students longer to generate the
second alternative than it did additional alternatives.
Perry, L., & Grossnickle, F. (1987). Using selected manipulative
materials in teaching mathematics in the primary grades
(Report No. SE047844). Long Beach, CA: California State
University. (ERIC Document Reproduction Service No. ED280684)
In this report, 75 primary teachers in 11 Southern California
schools participated in a survey regarding the availability and
usage of specific mathematics manipulatives. Questionnaires were
distributed to two teachers at each grade level from kindergarten
through third grade. Teachers were asked to estimate the degree
of utilization for each of the following manipulatives: abacuses,
base ten blocks, Cuisenaire rods, and unifix cubes. The fourcategory
scale for usage was: often, some, seldom,
and not used.
Analysis of the completed questionnaires showed a correlation
between availability and usage. Unifix cubes were the most available
(92%) and the most highly used manipulative, with 75% of teachers
reporting usage some of the time or often. Cuisenaire
rods were found in 71% of the schools, but only 7% of the schools
used them often and 33% used them some. Base ten
blocks and abacuses were available in 45% of the schools, with
63% of teachers reporting not used. All 75 teachers in
this study reported using manipulatives at some point during mathematics
instruction. Several recommendations were included in this study:
continue research on the effectiveness of manipulatives, encourage
school districts to identify manipulatives for the development
of pertinent concepts and skills at each grade level, and urge
school systems to train teachers on effectively incorporating
manipulatives.
Pesek, D. & Kirschner, D. (2000). Instrumental instruction
in subsequent relational learning. Journal for Research in
Mathematics Education, 31 (5), 524540.
Instrumental instruction emphasizes rote memorization of formulas
and skills while relational learning emphasizes the intent to
have students derive meaning from their math experience. The
authors express concern about a prevalent practice in these days
of high stakes testing in which teachers dedicate a significant
portion of class time to instrumental instruction with the intention
of preparing students for these tests, sometimes before students
have had the opportunity for relational learning on the topic.
The authors investigated whether interference to learning with
meaning is set up by having students perform rotely before understanding.
The topic selected was area and perimeter of squares, rectangles,
parallelograms, and triangles. The manipulatives employed were
students' hands, 1inch square tiles, geoboards, grid paper, and
plain paper. The investigators studied six classes of fifth graders,
taught by two teachers. Each class was split into two groups.
The study was experimental. One group received instrumental
instruction for five days prior to relational learning for three
days. This group was called IR. The second group received only
relational learning for three days. This group was referred to
as RO, but because it received relational learning during the
same three days as the IR, it seemed to me that it should be
the OR group. An assortment of tests were administered to the
students. All students took a pretest, a posttest after the relational
learning, and a 2week retention test. In addition, the IR students
received a test after their instrumental instruction.
Another component of the study was a series of three interviews
with six students from each group, a stratified random sampling.
All of the interviews were recorded on audiotape and the final
interview was also captured on videotape.
The quantitative results showed no significant difference. However,
the qualitative results from the interviews gave another picture.
The IR group experienced cognitive, attitudinal, and metacognitive
interference.
Evidence for cognitive interference was that IR students tended
to confuse area and perimeter. While they knew that area was
an appropriate measure for carpeting, when faced with wallpaper,
their concepts failed them. Since walls go around a room, many
incorrectly applied the concept of "around" that they
had associated with perimeter. The IR students used more formulas
and were fixed in their approaches, while the OR students used
conceptual and flexible methods. In addition, they applied their
relational knowledge in practical ways. In contrast, the IR
students said that their knowledge might be useful for tests and
college.
Attitudinal interference was observed in the IR students. The
possession of prior attitudes and belief about area and perimeter
prevented their full engagement in the relational learning on
the topics.
Finally, metacognitive interference occurred. While the OR
students had explanations that showed reasoning, that made sense,
and were more grounded in the concrete, the IR students applied
formulas randomly and either could not explain the meaning of
formulas or gave confused explanations of them. The new relational
learning seemed to disrupt what they were hanging on to from the
instrumental instruction.
Peterson, S., Mercer, C., Tragash, J., & O'Shea, L. (1987).
Comparing the concrete to abstract teaching sequence to abstract
instruction for initial place value skills (Report No. EC301777).
Gainesville, FL: Florida University Department of Special Education.
(ERIC Document Reproduction Service No. ED353744)
In this study, place value skills were taught to 24 learning
disabled elementary and middle school students (ages 813). Two
approaches were compared. The first approach introduced skills
in a concrete, semi concrete, abstract sequence. The second method
presented skills only at the abstract level. Subjects were screened
prior to the study to ensure limited knowledge of place value
skills. Students scoring 70% or lower were included. Control
groups and experimental groups were created and students were
assigned to a group at random.
Both groups received instruction during nine lessons. Only one
lesson was taught per day. The experimental group received three
lessons at each of the three levels: concrete, semi concrete,
and abstract. The manipulatives used were unifix cubes, place
value sticks, and place value strips. The control group was instructed
only at the abstract level. After the nine lessons were completed,
students were given a posttest. On the acquisition measures,
students in the experimental group identified one and tens significantly
higher than the control group. In comparing scores between the
screening and retention generalization measures, the experimental
group performed better, although some students were not yet able
to generalize. Results show that manipulatives enhanced skill
acquisition and retention. The overall gain in scores among
the experimental group denotes the concrete to abstract teaching
sequence as the more effective approach.
Pleet, L. J. (1990). The effects of computer graphics and mira
on acquisition of transformation geometry concepts and development
of mental rotation skills in grade eight (Doctoral dissertation,
Oregon State University, 1990). Dissertation Abstracts International,
52(06), 2058.
Pleet (1990) found no statistically significant differences between
eighth grade students using physical manipulatives, virtual manipulatives,
or no manipulatives on measures of geometric transformations.
Pleet, L. J. (1990). The effects of computer graphics and Mira
on acquisition of transformation geometry concepts and development
of mental rotation skills in grade eight. Dissertation Abstracts
International, 52(06), 2058.
This study compared the use of the Motions computer program versus
the Mira manipulative. Its purpose was to examine whether the
Motions program was a more effective tool for helping eighth graders
acquire transformation geometry concepts and develop mental rotation
skills. The study also looked for sex differences.
Participants included 15 teachers at 15 different schools and
involved 560 students in 30 classes. Sixteen classes comprised
the experimental group: eight classes used the Mira manipulative
and eight classes used the Mirrors program. Eight teachers taught
one of each. Fourteen classes comprised the control group; seven
teachers taught two of these classes each.
The students took a pre and posttest and were given a questionnaire.
There were three weeks between the pre and posttest. Data was
analyzed using analysis of covariance. With regard to transformation
concepts, the results showed no significance difference in the
means of the Mira or Motions groups, of the females in the Mira
or Motions groups, and of the males in the Mira or Motions groups.
A significant difference (.05 level) was found between the means
for males in the Mira and Motions groups. With regard to rotation,
the results showed no significant difference in the groups or
in the sexes.
Pratt, D. (2000). Making sense of the total of two dice. Journal
of Research in Mathematics Education, 31(2). pp. 144167.
Making Sense of the Total of Two Dice 16 Children ages 1011
years old used Chance maker virtual manipulative for 2 and 2.5
hour sessions exploring math concepts of probability.
Research Questions
1. What are the internal resources that children use to make
sense of the total of two dice?
2. When children make sense of the total of two dice, what new
internal resources do they forge through the interaction of their
internal and external resources?
3. In this interactive process, what features determine the extent
to which these new resources become tools for the forging of further
connections in related activity?
Research methods and procedures
The sessions were conducted as clinical interviews during which
the researcher acted as a participant observer, interacting with
the children to probe the reasons behind their actions and later
interpreting these reasons in the light of observations based
on other children's work. In general, the aim was to allow the
children to be in control of their explorations by making decisions
and moving in directions of their own choice. Students were involved
in probability experiments involving two spinners and two dice.
The actions of the children within the computer environments were
videotaped and the discussions were transcribed. The transcripts
were analyzed and the researcher identified consistencies and
differences.
Results
Students entered the study with an equiprobability bias with
respect to the total (sum) of two dice. Through the interaction
of the chance maker tool, they were able to recognize that some
totals were represented more often than others. When they moved
from the two spinner activity to the two dice activity, the lesson
realized in the first session was not automatically cued in different
situation. Students again articulated the equiprobability bias.
The researcher concluded that although the newly formed construct
was available for learners to use in different contexts, it did
not have as high a priority in their mental scheme as their prior
construct.
Rust, A. (1999) A study of the benefits of math manipulatives
versus standard curriculum in the comprehension of mathematical
concepts. Dissertation Paper. ERIC document 436395.
This study involved twenty one first grade students with a range
of abilities. Their ages ranged from 67. The study began about
the end of September and carried though about 8 weeks. The four
math concepts taught for the research were addition, subtraction,
measurement, and fractions. The class was divided into two groups
and worked on two concepts simultaneously.
This study attempted to determine which teaching method, manipulatives
or the standard curriculum, allowed the students to learn first
grade math concepts. The manipulatives used were unifix cubes,
personal chalkboards, and work mats. The standard curriculum
used was the Mathematics Plus workbook by Harcourt Brace Jovanovich.
The Knox County Math skills test was the first test given, and
the second test was a Teacher Checklist Manipulatives
Results: Statistics showed that teaching using the book and testing
with the Knox County Math Skill test showed more learning than
the teaching with the manipulatives and testing with the Teacher
Checklist Manipulative Evaluation. The research proposed that
students could be used to testing with paper and pencil than with
the manipulative objects. Some seemed to learn better by manipulating
the objects, where as others did not need the hands on help.
Even though students were able to learn the material no matter
which way it was taught, there were definite differences in student
enjoyment.
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Schultz, K. A. (1984). The Average Ability
Middle School Student and Concrete Models in Problem Solving:
A Look at SelfDirection. U.S. Georgia: National Center of
Research on Teacher Learning. (ERIC Document Reproduction Service
No. ED244 836)
This qualitative study investigated several aspects of a teaching
experiment which focused on five seventh grade students' performance
on nonroutine number theory problemsolving with a close look
at the role of heuristics, concrete models, and the relationship
between these variables. The focus was on those students who
did not qualify for a remedial or gifted program in the school
system. In the experiment, students were given a series of lessons
on number theory, heuristics, concrete models, and the microcomputer.
The microcomputer was used to present the problem, facilitate
hint selection, and to record student work. This study was conducted
for one school year in a suburban middleclass Atlanta public
school. In all there were 30 subjects out of 78 seventh graders.
The data was gathered in questionnaires, Cold ProblemSolving
Test and Post ProblemSolving Test.
The results indicate that all students showed improvement in
problemsolving ability and increased use of concrete models.
Compared to aboveaverage and belowaverage students, the average
ability students showed the greatest gain in demonstrated problemsolving
ability and the greatest use of concrete models.
Sharp, Janet M. (1995, Oct). Results of using Algebra Tiles
as meaningful representations of algebra concepts. Paper
presented at the annual meeting of the MidWestern Research Association,
Chicago, IL. Available: Eric Document 398 080
The study conducted by Janet Sharp was a twoyear study of high
school students and the use of Algebra Tiles. The study consisted
of two three weeklong experiments over a two year period. The
manipulative used in both experiments was Algebra Tiles.
The first experiment studied factoring algebraic expressions
and was conducted in a rural high school, with 37 algebra one
students ranging in age from 15 to 18. There was one treatment
group (n=11) and two control groups (n=13, n=13). The same teacher
taught the treatment group and one control group. The treatment
group used math Algebra Tiles during their instruction. All groups
took the same departmental test at the end of the threeweek period.
The second experiment, a full year experiment on the use of manipulatives,
was conducted in a suburban high school with 20 algebra students,
ranging in age from 13 to 16, and one 9 year old. The experiment
had a treatment group (n=10) and a control group (n=10). The treatment
group was exposed to one manipulative type problem each week during
a twentytwo week period before the experiment. Both groups
were instructed for a threeweek period with the use of manipulatives
during their study of addition, subtraction, multiplication and
factoring algebraic expressions.
Student tdistribution tests revealed that there was no significant
difference (alpha = .025 level, two tailed) between group means
in either experiment. Students were classified as "unusual" if
their test score was within a calculated chapter test score confidence
interval, but their previous semester's grades were below the
calculated confidence interval to predict semester grades. Further
analysis was conducted on their writing and interview responses.
Sharp concluded that students learn algebra through several modes
of representation. Narrative data revealed that the greatest
power of the Algebra Tiles was not in increasing test scores,
but the alternative representation system that was provided.
Many students mentioned that they were able to "visualize the
problems" which is a skill that is unlike memorized facts or rote
manipulations.
Slack, J. B., & St. John, E. P. (1998). A model for measuring
math achievement test performance: A longitudinal analysis of
nontransient learners engaged in a restructuring effort.
San Diego, CA.
Summary: The researchers studied 62 "nontransient"
elementary school students in Louisiana over a period of four
years. Five "innovative" instructional approaches were
used, one of which was math manipulatives and technology. Using
logistic regression, the researchers attempted to determine if
any particular instructional approaches were significant predictors
for standardized test score improvement. One finding was the students
who had received four years of the math manipulatives and technology
instructional approach were more likely to improve their math
scores than those who had less years with that approach.
Official Abstract: This study investigated the mathematics
achievement test performance of 62 nontransient elementary school
learners in accelerated schools using a longitudinal design. Both
the California Achievement Test (CAT) and the Louisiana Educational
Assessment Program (LEAP) test were included in this investigation.
In particular, this study sought to determine whether accelerated
schools with distinct contextual features experienced significantly
different test performances. A logistic regression was used to
explore the relationship of several variables to the schools'
performances. The variables were related to individual background,
school environment, and curriculum and instruction factors. The
researchers developed two logistic regression models to fit the
uniqueness of the CAT and LEAP tests. Each model used a sequential
analysis to examine the association of specific factors to test
score improvement. The most consistent, significant finding across
both models revealed that higher ability students were less likely
to improve than lower ability students. This finding is consistent
with the Accelerated Schools philosophy that "disadvantaged"
students stand the most to gain from innovative teaching approaches.
Additional findings showed the significant impact of age, gender,
school environment, and curriculum and instruction on improvement.
In particular, observations related to the latter factor revealed
that students who were provided with math manipulatives/technology
for longer periods were more likely to improve their standardized
math scores than those who were provided with such instruction
for shorter periods.
Smith, J. P. (1995). The effects of a computer microworld on
middle school students? use and understanding of integers. (Doctoral
dissertation, Ohio State University, 1995). Dissertation Abstracts
International, 56(09), 3492.
Smith (1995) found that sixth and eighthgrade students using
the virtual manipulatives scored significantly higher on tests
of integer addition and subtraction than both those students who
worked with physical manipulatives and those who used both treatments.
Sowell, E. J. (1989). Effects of manipulative materials in mathematics
instruction. Journal for Research in Mathematics Education,
20(5), 498505.
This is a metaanalysis of the results of 60 studies combined
to determine the effectiveness of manipulatives on mathematics
instruction.
Participants: Included in the 60 studies in this meta
analysis were 17 studies in grades k2, 17 in grades 34, 9 in
grades 56, 11 in grades 79 and 6 at the post secondary level.
The studies used in the analysis included 38 journal reports,
3 unpublished reports, and 19 dissertations.
Manipulatives: The manipulatives used in the studies included
beansticks, Cuisenaire rods, and geoboards in addition to others
not identified in the report.
Math concepts taught: A broad range of concepts were
included. However, this study focused on commonalities in concept
retention, transfer, and attitude.
Methodology: A metaanalysis was done using quantitative
measures.
Results: The analysis indicated that the mean effect for
retention was not significant (p<.05). The studies that included
data on transfer was also considered to be nonsignificant. However,
the data on attitudes about mathematics varied in significance
based on random assignment to study groups and length of the treatment.
Steele, D. F. (1994, April). Helping preservice teachers confront
their misconceptions about mathematics and mathematics teaching
and learning. Paper presented at the Annual Meeting of the
American Educational Research Association, New Orleans, LA.
In the study Helping Preservice Teachers Confront Their Conceptions
about Mathematics and Mathematics Teaching and Learning, Diana
F. Steele explored the questions that guided her research were:
"Could I, through modeling constructivist teaching, effect a change
in these students conceptions about mathematics?"(Steele, 4)
and "What are the conceptual changes?"(Steele, 4).
Participants: The participants for this study consisted
of 19 preservice students enrolled in a course "Teaching Mathematics
in the Elementary School" (Steele, 6) at the University of Florida
at Gainesville.
Manipulatives: Steele used a variety of manipulatives
in her study including pattern blocks, fraction circles, and fraction
squares.
Math concepts taught: Fractions
Methodology: Steele used "observation, interviewing and
collection of artifacts"(Steele, 5) as the research approach.
Data was collected within a controlled environment over a significant
duration of time using both qualitative and quantitative methodology.
Results: Steele analyzed the data collected using the
procedures described as the "developmental research sequence"
(Steele, 6). The qualitative data indicated that the students
conceptions about teaching and learning of mathematics began to
change over the course of the study. The quantitative analysis
consisted of analyzing the Mathematics Beliefs Scales. The students
entered the course with beliefs that students should be learning
mathematics through learning number facts by drill and practice.
The answers on the post assessment focused on children constructing
their own knowledge of math concepts with the teacher acting as
a facilitator. The descriptive statistics reflected scores converging
to zero with zero representing the most constructivist score possible.
The range on the pretest was significantly higher than on the
posttest, demonstrating that the students' conceptions of teaching
and learning have shifted. Using a "single factor analysis of
variance" (Steele, 29) Steele tested whether the differences in
the means for the pre and post tests were statistically significant.
She observed that there was a significant difference; therefore
she accepted her alternative hypothesis.
Steele, D. (1993). What Mathematics Students Can Teach Us About
Educational Engagement: Lessons from the Middle School. Paper
presented at the Annual Meeting of the American Educational Research
Association (Atlanta, Georgia, April 13, 1993). Available ERIC
Document ED 370 768
The purpose of this paper was to study the attitudes and understandings
that middle school students have about mathematics. The research
was conducted over a four month period with 53 students in two
seventh grade classes taught by the same teacher. Students were
observed, formal and informal interviews were conducted, and data
was analyzed from homework and tests. The researcher discovered
that students were more engaged and motivated when actively involved
in the learning process, when using manipulatives, and when working
in cooperative groups. The researcher concludes that math instruction
needs to move away from the traditional lecturing, rote memorization,
and computation out of context and more toward student centered
activities.
Stellingwerf, B.P., & Van Lieshout, E.C.D.M. (1999). Manipulatives
and number sentences in computer aided arithmetic word problem
solving. Instructional Science 27(6), 459476.
This is a two factorial pretestposttestcontrol quantitative
study that was carried out in the Netherlands with 122 students
with mean age of 11.3 years. These students had learning problems
or were mildly mentally retarded. The purpose of the study is
to gain more information about instruction methods that will improve
how well children with learning problems are able to solve word
problems. Three instructional methods or treatments were examined
as a part of this study; using external representation with manipulatives
only, using mathematical representation with number sentences
only, or using a combination of both. The treatments are embedded
in a computer program because the computer has the capability
to provide direct feedback and is able to diagnose students abilities
For this study the manipulatives are icons on the computer screen.
The 122 participants in the study were randomly placed into four
treatment groups and one control group. One group learned to
solve word problems by writing open and closed number sentences
only. A second group learned to solve word problems using manipulatives
only. The manipulatives were combined with writing open and closed
number sentences for the third group. A fourth group was taught
to solve number sentences without manipulatives or without writing
number sentences. In the four treatment groups participants were
given corrective feedback when errors occurred during the lesson.
A fifth or control group received no treatment at all. Since
the literature review supported that both using manipulatives
and writing number sentences improve students problem solving
ability, one hypothesis for the study is that students in the
groups receiving the manipulatives only, number sentence only,
or a combination of the two treatments will outperform students
who are taught to solve the word problems with out benefit of
either. A second hypothesis is that students in any of the four
treatment groups will out perform students in the control group.
The experiment consisted of four stages. In the pretest stage
participants were given a paper and pencil test to assess their
reading level, nonverbal intelligence, ability to write number
sentences, and ability to solve word problems. Word problems
were categorized using the semantic structure scheme developed
by Heller and
Greeno. In the second or the computer training stage where students
were individually instructed on how to use the computer program.
The third or treatment stage consisted of 12 individual sessions
of up to 30 minutes per session on the computer. The problem
was read to the students and each treatment group was limited
in the time they could work before entering an answer to the problem
on the screen. Feedback was provided if the solution was incorrect.
At the posttest stage the four treatment groups were administered
two performance tests via the computer and a third paper and pencil
post test was administered to all five groups.
A factorial repeated measurement ANOVA was carried out to test
the first hypothesis. From this analysis there was some evidence
that high competent children were better off with the writing
number sentence only treatment than with the combination of writing
number sentences with using manipulatives. An ANCOVA was carried
out to test the second hypothesis and partial support indicating
that participants who received the manipulatives only treatment
as well as those who learned to solve the word problem without
manipulatives or without writing down the number sentences did
better than the control group that received no treatment.
The conclusion of the study is that children with learning disabilities
benefit from computer aided instruction for solving simple arithmetic
problems. In addition, the implication for designing instruction
is that using manipulatives, writing number sentences, and teaching
children to use mental methods to solve word problems can each
have a positive effect on improving students ability to solve
word problems. The method to be used and the transition within
the curriculum from one stage to the next will be different for
different students as well as for different kinds of word problems.
Suydam, M.N. (1986). Manipulative materials and achievement.
Arithmetic Teacher, 33(6), 10, 32.
Suydam, M.N. (1985). Research on Instructional Materials forMathematics.
ERIC Clearinghouse for Science, Mathematics, and Environmental
Education, Columbus, OH. (ERIC Document Reproduction Service
No. 276 569).
Suydam, M.N., & Higgins, J.L. (1977). ActivityBased Learning
in Elementary School Mathematics: Recommendations from Research.
ERIC Center for Science, Mathematics, and Environmental Education,
Columbus, OH.
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Terry, M. K. (1996). An investigation of differences
in cognition when utilizing math manipulatives and math manipulative
software. Dissertation Abstracts International, 56(07),
2650.
This study investigates the effectiveness of manipulatives and
manipulative software on computational skills and spatial sense.
The students were in grades two, three, four, and five. There
were three treatment groups: manipulatives only, manipulative
software, and both manipulatives and manipulative software. Students
participated in a three week unit focusing on computation (addition
in grades two and three and multiplication in grades four and
five) and a one week unit on spatial sense. Base Ten Blocks were
used in the computation unit and attribute blocks in the spatial
sense unit. Both were used in concrete and software form.
A three way analysis of variance was used to interpret the data.
In four of the six ANOVA's for computation, there was a significant
difference for the group using both concrete manipulatives and
software. With regard to spatial sense, there was no significant
difference detected. There was on significant gender interaction
effect.
Teachers reported a preference for using computer software.
They felt software made it easier to manage instruction, improved
timeontask, and increased student enthusiasm.
Terry, M. K. (1996). An investigation of differences in cognition
when utilizing math manipulatives and math manipulative software.
(Doctoral dissertation, University of MissouriSt. Louis, 1996).
Dissertation Abstracts International, 56(07), 2650.
Terry (1996) found that students in Grades 25 using both the
physical and virtual manipulatives scored significantly higher
on tests of addition, multiplication, and spatial sense than students
using either of the treatments alone.
ThreadgillSowder, J. A., & Juilfs, P. A. (1980). Manipulatives
versus symbolic approaches to teaching logical connectives in
junior high school: An aptitude x treatment interaction study.
Journal for Research in Mathematics Education, 11(5), 367374.
The purpose of this study was to examine interactive effects
between mathematics achievement and manipulative versus symbolic
instruction. Participants were 147 seventh graders at two junior
high schools. Students were randomly assigned to manipulative
or nonmanipulative groups and participated in six minilessons
teaching logical connectives such as conjunction, disjunction,
negation, etc. The manipulatives group used colorcoded cards
and attribute blocks.
Students were given an aptitude test about one month before the
treatment. The treatment lasted three days. After, students
were given a 25 item multiplechoice test and a 20 item transfer
test. A oneway analysis of variance was conducted on the data.
No significant difference was found between the manipulatives
and symbolic treatment groups. However, generally higher mathematics
achievers performed a little better with traditional methods.
Students who were generally lower mathematics achievers performed
a little better with manipulative methods.
Thompson, P. W. (1992). Notations, conventions and constraints:
Contributions to effective uses of concrete materials in elementary
mathematics. Journal for Research in Mathematics Education,
23(2), 123147.
In the study Notations, conventions and constraints: Contributions
to effective uses of concrete materials in elementary mathematics,
Patrick Thompson investigated the way in which students' engagement
with manipulatives contributed to their construction of meaning
for decimal numeration and operations (Thompson, 123).
Participants: Twenty fourth grade students enrolled in
a university lab school participated in this study (Thompson,
130).
Manipulatives: Base ten blocks and Blocks Micro World.
Math concepts taught: Decimal operations.
Methodology: Quantitative methods were used during this
study. A pretest/post test design was used with two treatment
groups. One group used concrete decimal blocks and the other used
Blocks Micro World a computer based decimal block program
created by the author.
Results: The results of the posttest were analyzed in
two parts. First, the students' computational accuracy between
the pretest and posttest showed no significant change in both
groups. However, the students' responses to questions dealing
with new content introduced during the study on decimal numeration
and calculation with decimal numerals was examined and found that
the Blocks Micro World students were more likely to give
answers (although often inaccurate) that suggested that they were
based on principals of numeration.
Thompson, P. W. (1992). Notations, conventions, and constraints:
Contributions to effective uses of concrete materials in elementary
mathematics. Journal for research in mathematics education, 23,
123147.
Thompson (1992) found that fourthgrade students using the virtual
manipulatives made greater gains in understanding addition and
decimal addition than those students who worked with the physical
manipulatives.
Tracy, D.M. & Panelli, B.H. (2000) Teaching money concepts:
Are we shortchanging our kids? Research Report. ERIC Document
451065
72 First and Second graders used Proportional money model, coins
for four half hour sessions over 8 weeks. The math concepts taught
were:
 Number of each coin equaling one dollar
 How many cents each coin was worth
 How to skip count with one dollars worth of the same coin
 Used the proportional and visual models to construct money
concepts.
 Research methods and procedures
Two first grade classes used the concrete and visual money model.
A third first grade classroom used traditional teaching method
and were randomly pre and post tested. The second grade classes
were selected randomly doe pre and posttests. The objective for
the money models and traditional groups were
 Recognizing coins
 Knowing coin names
 Knowing coin values, skip counting
 Counting on with coin combinations and
 Determining how many of each coin are needed to equal one
dollar.
In first grade, students show near mastery of learning coin names
but lower ability to match coin with value. Both the traditional
method and money model group had an overall post test score of
less than 50%. Thus, First grade teachers should not expect mastery
of money concepts. In skip counting, first graders were good
with pennies but had difficulty with nickels, dimes, and quarters.
Only 7/35 students were successful in counting a set of mixed
coins cognition and value. Posttest revealed that among second
graders in the money group 100 % could correctly skip count dimes,
nickels, pennies in a set, with quarters at 67 %. Data indicates
that students â€˜ ability to count on with a variety of coins is
developmentally appropriate among second grader suburban students.
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Vinson, B.M., Haynes, J., Brasher, J., Sloan,
T. & Gresham, R. (1997) A comparison of preservice teachers'
mathematics anxiety before and after a methods class emphasizing
manipulatives. Paper presented at the Annual Meeting of the
Midsouth Educational Research Association, Nashville, TN.
This study investigated changes in mathematics anxiety levels
among future teachers in two different math materials and methods
courses.
Research Method
It included 87 novice teachers who took elementary or intermediate
level mathematics teaching classes Two strategies were used to
gather data at the beginning and the end of each quarter. 1)
98 item questionnaire, the mathematics Anxiety Rating Scale (MARS).
The treatment was a hands on approach to teaching mathematics
with manipulatives in the methods and material courses. After
the 10^{th} week, they took the MARS again. 2) Questionaire
guided narrative interview about factors influencing levels of
math anxiety.
Results:
A multivariate analysis variance revealed a statistically significant
reduction in mathematics anxiety level between the fall and winter
quarters. The change in their anxiety could be the function of
using a) Bruner's framework of developing conceptual knowledge
before procedural knowledge, and b) manipulatives to make mathematics
concepts more concrete.
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