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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), 361-369.
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 4-6) 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 hands-on 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),
23-31.
Summary: The researchers studied 22 students in grades
4-6, 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/hands-on activities,
and visual/spreadsheet activities. Using various methods of comparing
pre- and post-test 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 4-6, 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 algorithmic-like definition of average corresponded
to better long-term performance than less precise definitions.
Collaborative deliberations resulted in positive implications
for the researchers' teaching.

Balka, D. S. (1983). Mathematics manipulatives in a pre-vocational
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: “Try-A-Tile 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 (10-15
min.) to reinforce concepts of place value and basic computational
skills.
Methodology: Quantitative methods were used as a pre-test,
post-test 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, 470-475.
Ball (1988) found that fourth-grade 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 seven-week study conducted by sixth grade teacher, Jennifer
Barclay investigated the effectiveness of using the algebra manipulative,
Hands-on-Equations to teach the concept of solving linear equations.
Ninety-two 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; three-week retention test;
and a six-week 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 inquiry-based classroom.
Journal of Research in Mathematics Education, 40 (4), 417-448.
This is a case study that involves observing and interviewing
three pairs of fifth graders over a 4-week teacher-directed 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 inquiry-based problem-centered 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 case-study pair to monitor the evolution of their thinking
through observation the pairs and by asking them clarifying questions.
Pre and post-treatment interviews of the case-study pairs as well
as transcriptions of the video-tapings of each observation session
were also used. The findings from this study are generalizable
based on the comparisons of the findings in the three case-study
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 post-interview 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 self-check 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 ideas---including stumbling blocks and
learning plateaus---can 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), 503-532.
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.
Pre-study 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 row-by-column 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
row-by-column 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), 14-22.
Belcastro conducted an experimental study of five blind first-grade
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 10-question 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, 468-479.
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 eighth-grade 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 equation-evoking 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 equation-evoking 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
equation-evoking situations.
Three clusters of students were identified: Verbal and Single-Operational
Expressions and Equations, Transition from Verbal and Single-Operational
to Symbolic Expressions, and Equations, and Symbolic Expressions
and Equations.
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Chassapis, Dimitris. (1998-1999). 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), 275-93.
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 goal-directed 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 circle-drawing 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 circle-drawing operation in a radically
different fashion than circle tracers and templates, and brings
into play an action-bound 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 human-hand movement in doing the same practical
actions as those a freehand circle drawing, (thus not qualitatively
transforming the circle-drawing 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 circle-drawing 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 action-bound
practical thinking, because it is a functional meaning of circle-drawing
that emerges when using a compass.

Chattin-McNichols, J. (1992). Montessori Programs in Public
School. (Report No. EDO-PS-92-7). Champaign, IL: University
of Illinois at Urbana-Champaign 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 student-centered 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 third-grade students. Charlotte,
North Carolina: University of North Carolina at Charlotte.
Summary: The researchers studied two third-grade 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
lecture-style 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
17-year-old high school students know some basic addition and
subtraction facts, few of the students are capable of solving
multi-step mathematics problems. A non-equivalent pretest-posttest
control group design study examined the effects of a teaching
method emphasizing manipulative use on the mathematics achievement
of third-grade students. Two third-grade 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 2-week 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), 362-385.
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 multi-link 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 ten-week period during that year. In the classroom, typically
there was small group problem-solving 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 all-or-nothing 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 well-selected 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 eighth-grade females'
and males' ability to learn principles of plane mirrors. School
Science and Mathematics, 94(8), 395-400.
Summary: The researchers studied 101 eighth-grade physical
science students at a Midwestern, suburban, upper-middle 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) eighth-grade 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), 108-114.
Summary: The researcher studied 32 first-grade students
at a rural Minnesota elementary school during the 1994-95 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 base-10
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 three-year 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 three-year
period. In the experimental groups, kindergartners received individual
instruction on the use of the rods, first grade students completed
teacher-created 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 three-year 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 (K-2 at Whittier) performed at a
higher level than the control groups (Columbia). Using end-of-year
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. EDO-PS-97-14).
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 pre-made “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 Mid-south
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 1-42 years with a mean of
17.5 years. The teachers taught the following courses: Math
7, Math 8, General Math, Pre-Algebra, Consumer Math, Technical
Math, Algebra I, Algebra II, Geometry, Trigonometry, and Pre-Calculus.
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.
On-site 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 Third-Graders.
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 television-based 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 third-graders 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 third-grade students
- 90
- tangrams
- problem solving
- five sessions for 40 minutes
- 2x2 matrix: video or non-mediated 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 Hands-On-Learning,
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 real-life 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 real-life 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 real-life
examples. The researcher also noted students favored group learning
and real-life 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
user-friendly 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 User-Friendly 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, Jean-Marie 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 hands-on
learning experiences in order to explore real-life 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 1-5 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 “At-Risk” 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, real-life 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 long-term 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 4th-grade
children (n=58), approximately half of whom had long-term experience
with calculators. Children with long-term 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, left-right
methods, and invented methods for mental computation items did
not vary greatly from that of the non-calculator 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 long-term 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.
EDO-RC-90-5. ) 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 (semi-concrete,
semi-abstract) Experienced teachers use manipulatives less than
inexperienced teachers Teaching with manipulatives is effective
only when the proper manipulative and activity is used Long-term
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 Self-Report. School Science of
Mathematics, 94, 303-309.
The article discusses the use of manipulatives in the elementary
setting (K-6). 87 teachers were obtained for the research based
on a survey that was mailed to 106 (K-6) 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, flexi-counters*, 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 flexi-counters 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 (4-6) in terms of use of manipulatives. It
further shows the need for universities to have more say as to
where and with whom their pre-service teachers will conduct their
experience.

Haughland, S. (2000). Computers and Young Children.
(Report No. EDO-PS-00-4). 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 self-concept. 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 follow-up) effectively
integrate computers into their lessons when provided with adequate
classroom resources.
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Johnson-Gentile, 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), 121-140.
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 computer-based 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 one-sixth grade class were assigned
to one of three treatment groups for an eight-day motions unit.
The LOGO group received all of their instruction using the Motions
strand of the Logo Geometry curriculum, the non-logo 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 two-day 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 thirty-minute 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 hands-on 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 middle-class 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. Pre-testing was administered during a one-day
window in October and post-testing was completed in the spring
during a two-day 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 computer-based geometry
environment. Journal of Mathematical Behavior, 9, 99-127.
Kiernan and Hillel (1990) found that sixth-grade students using
the computer-microworld virtual manipulative made significant
gains in understanding the nature of isosceles triangles.

Kim, S. (1993). The relative effectiveness of hands-on and computer-simulated
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 6-9 through the Integration of Content, Technology,
and Manipulatives: Formal Cumulative Evaluation Report. National
Science Foundation Grant ESI-9155296. Available ERIC Document
ED 401 129
This report described the findings of a three-year project in
Alabama which focused on improving the teaching behaviors, knowledge,
and attitudes of 58 mathematics teachers in grades 6-9. 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, 53-56.
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 African-American
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.
Lara-Alecio, 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(2-4),
215-235.
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 limited-English-proficient students, 14
mathematics tasks using manipulatives were administered to 45
Hispanic students in grades 1-3 and readministered 2-3 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 two-year
collaborative action research study on the effects of a “Hands-on”
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 two-year 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 9th grade math course. The manipulative
used was “Hands-on-Equations” 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 non-manipulative style using
the adopted textbook. The 26 lessons, Hands-on Equation manipulative
program were then implemented. After the completion of the manipulative
lessons, the instruction returned to a non-manipulative 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 8th 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,
Salem-Teikyo University, Salem, WV. Available: Eric Document
425 077
McClung’s nine-week 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 15-17 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 in-class worksheets. The treatment group,
Group B receives traditional teaching method of lecture, homework,
but instead of in-class, worksheets the students work with the
manipulative Algeblocks.
McClung uses a two-sample t-test 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 mathematically-deficient
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), 121-142.
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 13-14, 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 problem-solving
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 pre-post 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), 175-197.
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 (K-6) teachers
2. Number of participants:
1093 teachers
3. Manipulatives used: educational
software
4. Math concept taught:
basic math skills and open-ended problems
5. Duration of the study:
survey data
6. Research methods/procedures:
written survey (questionnaire)
7. Results: Eighty-five
percent of the teachers surveyed used only skill-based software.
Teacher surveys reflected a learner-centered orientation and a
constructivist view of learning, but those values were not reflected
in their children's computer use. Virtual manipulatives,
characterized as open-ended 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),
85-108.
The investigators carried out a case study, observing how two
fifth-grade 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 four-week 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 lived-in
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 "lived-in" 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 criss-cross
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),
203-33.
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 colour-change 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, time-on-task,
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 sixth-grade 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, time-on-task,
learning efficiency, and attitude. Dissertation Abstracts
International, 58(08), 2988.
This study examined the effect of engagement activities and manipulative-type
presentations on students’ math achievement, time-on-task, 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 manipulatives--both
concrete and computer. One control group had no manipulatives.
Data was collected in three ways: students took a post-test measuring
their achievement of patterns content, completed a time-on-task
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, time-on-task was equal for fourth
and fifth graders. Efficiency was higher for sixth graders than
for fourth and fifth graders. Manipulative groups showed higher-level
recognition and application achievement performances than the
control groups. Overall, manipulative instruction strategies
showed more effective for higher-order 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 602-625.
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 array-based 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
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1
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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 meta-analysis 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 decision-making task using models and
computer simulations. Journal of computing in childhood Education,
4 (1), 57-76.
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
time-per-move 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 com |