Virtual Manipulatives 1
Running Head: VIRTUAL MANIPULATIVES
The Effects of Using Virtual Manipulatives Versus Physical Manipulatives on Achievement to
Teach Basic Fractions to Third Grade Students
East Stroudsburg University
ELED 570: Introduction to Research
July 11, 2011
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The Effects of Using Virtual Manipulatives Versus Physical Manipulatives to Teach Basic
Fractions to Elementary Students
Many elementary math teachers use manipulatives to assist children with visualizing and
processing mathematical concepts. McClung states that “manipulatives assist students in
bridging the gap from their own concrete sensory environment to the more abstract levels of
mathematics” (Brown, 2007). Physical manipulatives have been used over the centuries to bring
math to life and still play an important role in education. Research has shown that physical
manipulatives enhance the learning experience and are met with positive achievement results.
However, with the rapid growth of technology in the past thirty years, technology devices are
providing other options to use virtual manipulatives in the classroom. Taylor (2001) states that
“progression in technology has increased the boundaries of mathematics and emphasized the
importance of the integrations of technology in the mathematics curriculum” (Brown, 2007).
Virtual manipulatives are widely available through the World Wide Web, which can be accessed
in most classrooms. Current elementary teachers have the opportunity to use physical and/or
virtual manipulatives in their classrooms.
The technology resources that allow the use of virtual manipulatives to be integrated into
the math classroom are becoming relatively easier and more accessible. According to Rosen and
Hoffman, “teachers around the country and the world guide children’s mathematical learning
through the use of manipulatives – pattern blocks, base blocks, geoboards, Unifix cubes,
Cuisenaire rods, coins, clocks, and so on. Manipulatives allow concrete, hands-on exploration
and representation of mathematical concepts. In the past few years, online resources for virtual
versions of these common manipulatives have become available” (Rosen and Hoffman, 2009).
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Furthermore, children are growing up with technology as an integral way of life. It is imperative
for teachers to integrate technology in the classroom to engage students, enhance and promote
active visual learning. Using virtual manipulatives in the classroom is still largely under
researched. However, from personal experience, students are enthusiastic to learn math using a
new and exciting way to visualize learning of mathematical concepts. Virtual manipulatives are a
resource that engages students and have the potential to greatly enhance their math achievement.
The Effects of Using Virtual Manipulatives Versus Physical Manipulatives on Achievement to
Teach Basic Fractions to Third Grade Students
1. What are the gain scores on an instrument measuring achievement of students taught
basic fractions using virtual manipulatives?
2. What are the gain scores on an instrument measuring achievement of students taught
basic fractions using physical manipulatives?
3. How do the scores compare?
Definition of Terms
Manipulatives are defined by Taylor (2002) as “physical objects (e.g., base ten blocks, algebra
tiles, pattern blocks, etc.) that can be touched, turned, rearranged, and collected” (Brown, 2007).
According to Rosen, “manipulatives allow concrete, hands-on exploration and representation of
mathematical concepts” that children can explore (Rosen and Hoffman, 2009).
Physical Manipulatives are described by McClung (1998) as “objects that appeal to several of
the senses. They are objects that students are able to see, touch, handle, and move” (Brown,
2007). Physical manipulatives are also called concrete manipulatives and according to
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Mendiburo, “what is “concrete” to a child may have more to do with what is meaningful and
manipulable than with physical characteristics” (Mendiburo, 2006).
Virtual Manipulatives are defined by Moyer, Bolyard and Spikell as “an interactive, Web-
based visual representation of a dynamic object that present opportunities for constructing
mathematical knowledge” (Moyer, 2005). Moyer (2005) further describes virtual manipulatives
saying “virtual manipulatives are essentially replicas of physical manipulatives placed on the
World Wide Web in the form of computer applets with additional advantageous features”
(Brown, 2007) In another study, virtual manipulatives are defined as “computer based renditions
of common mathematics manipulatives and tools” (Suh, 2007).
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Review of the Literature
Maria Mendiburo and Ted Hasselbring learned that in 1990, fewer than half of the high
school seniors, who took the NAEP Mathematics Assessment, demonstrated successful
performance with problems involving fractions, decimals, percents and simple algebra. Only 14
percent of eighth graders who took the NAEP Mathematics Assessment also demonstrated
successful performance with problems involving fractions, percents and simple algebra. In 2000,
eighth graders were given a test where they had to order three fractions from least to greatest.
The fractions were less than 1 and in reduced form. Only 41 percent of eighth graders did this
successfully. Believing that fractions are the most difficult mathematical concept for elementary
students to learn, Mendiburo and Hasselbring decided to conduct their own study to “advance the
current literature about manipulatives and rational numbers by using a randomized experiment to
compare virtual and physical manipulatives” (Hasselbring, 2011). They also decided to conduct
this research to answer the question of “are there differences in students’ knowledge of fraction
magnitude when they are taught basic fraction concepts using virtual manipulatives compared to
when they are taught basic fraction concepts using physical manipulatives?” (Hasselbring, 2011).
The subjects of this study were 67 fifth grade students at a charter middle school in
Middle Tennessee. There were four fifth grade mathematics classes, with 39 girls and 28 boys
who participated in the study with parent consent. Classes at the school were single-gender. It
should be noted that approximately 98.9 percent of the students in the school were African-
American and that 88 percent of students qualified for free and reduced priced lunch. According
to a comprehensive mathematic benchmark assessment recently administered by a private
assessment company before the study took place, 62 percent of students participating in the study
tested below grade level.
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Due to the school’s preference that classes stay intact and single-gender, the researchers
randomly assigned half of the students within each of the four classes to a virtual manipulative
condition and the other half of the students in the four classrooms to a physical manipulative
condition. The students were grouped according to gender and treatment condition, creating a
2x2 Experimental Factorial design. The quantitative factorial design can be better explained as 2
(treatment: physical vs. virtual) x 2 (gender: girls vs. boys). Before the study started, the
researcher administered a pre-assessment to all participating students to determine prior
knowledge of fifth grade fraction content. The paper-and-pencil assessment was created by the
researcher using software provided by a private assessment company that contracted with the
school to measure and improve student achievement and to predict students’ performance on
state exams. The pre-test was made up of 20 multiple-choice questions about fractions. All of the
questions were validated by fifth grade assessment items. Students did not use manipulatives
when completing the pre-assessment. The results of the pre-assessment showed that most
students had at least some prior knowledge of fractions, while most of those same students fell
short of demonstrating mastery of the fifth grade fraction concepts that would likely be on state
The researcher taught all classes using a script to control for possible teacher effects and
pedagogical differences between treatment conditions. The research was conducted for a total of
10 days. Students who were assigned to the physical manipulative condition were taught basic
fraction concepts using a popular commercial curriculum and fraction manipulatives that the
students made out of colored strips of paper. In comparison, the students assigned to the virtual
manipulative condition were taught basic fraction concepts using Macbook laptops. The laptops
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were loaded with a software program designed specifically for the study that was basically a
virtual copy of the commercial curriculum and included a set of virtual fraction manipulatives.
An assessment was given to all students on day 5, where the students in the virtual
manipulatives group scored marginally higher than students assigned to the physical
manipulative condition. However, when controlling for students’ scores on the pre-assessment,
the main effect of the manipulative treatment condition was not statistically significant. Gender
did have an effect, but there was no interaction effect between manipulative treatment and
On day 10 a post assessment was given that showed the virtual manipulative group
answered an average of 1.78 more questions correctly than students in the physical manipulative
group. The contrast was statistically significant. However, the difference between boys and girls
was not statistically significant and the interaction between gender and manipulative treatment
was also not significant. This study concluded that physical and virtual manipulative share
positive effects on student learning and there are no negative learning gains associated with using
the virtual manipulatives.
With the rapid rise of technology in classrooms, Patricia Moyer conducted a study to
explore the use of several virtual manipulative computer applets for instruction during a fraction
unit in a third grade classroom. The researcher also stated that there is limited research on virtual
manipulatives, mainly due to researchers’ lack of both technology and mathematics mastery. The
research question posed was what impact do virtual fraction manipulatives have on students’
conceptual and procedural understanding of fractions?
To answer the above question, Quasi-Experimental Pretest-posttest, Nonequivalent
Control Group Design research was conducted on 19 third grade students. This was an intact
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class of 25 students, of which only 19 were included since the others were absent and four
children with Autism attended mathematics classes in a self-contained classroom. The school
where the research took place is half an hour away from Washington DC Metro area and had a
diverse student population, including 10 Caucasian, 2 Hispanic, 1 African-American, 3 Asian
and 3 Middle-Eastern students.
The teacher of the class had previously taught the fraction concept and tested the
students. The teacher taught the same fraction concepts again to control for the effect of the
virtual manipulatives. The teacher desired to know if there would be changes in students’ test
scores, favorably or unfavorably, attributed to the virtual manipulatives. Students were given a
pre and posttest before and after the two weeks experiment to measure students’ conceptual
knowledge and students’ procedural computation knowledge. The teacher created four tests, a
pre and posttest to determine students’ understanding of the procedural knowledge and a pre and
posttest to determine the conceptual knowledge.
Week one of the experiments involved the teacher instructing students using virtual
manipulatives by having her laptop displayed through the classroom TV. The teacher taught a
unit on base-10 blocks, and also taught students how to use the virtual applets on the computers
in the computer lab. She purposely did not teach the fraction unit during the first week so that the
students could become familiar with the virtual applets. In the second week, the teacher taught
the unit on fractions in the computer lab. The students worked in the computer lab during math
time, which was 1 hour, for four days using the virtual manipulatives. On day one, students used
the “Fractions – Parts of a Whole” virtual manipulative applet under the Numbers and
Operations strand. On day two, students explored parts of a group using the “Pattern Blocks”
applet under the Algebra strand. On days three and four, students used the “Equivalent
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Fractions” and “Comparing Fractions” applets under Number and Operations. At the end of
week two, students were given posttests on conceptual knowledge and procedural computation
Results showed that students scored significantly higher on the conceptual knowledge
posttest compared to the pretest using virtual manipulatives. On average, the class scored 60
percent on the pretest and scored 69 percent on the posttest. However, the results indicated that
the virtual manipulatives helped 53 percent of the students improve their conceptual
understanding of fractions; while 21 percent of students showed no change and 26 percent
actually had scores decrease.
The procedural knowledge assessment indicated no significant difference between the
pretest and posttest, presumably because on average, the class scored 90 percent on the pretest.
The class on average scored 96 percent on the posttest, but because the pretest scores were so
high, the experiment was limited. It is important to note that even though the pretest scores were
very high, that 74 percent of the students had scores that stayed consistent or increased on the
posttest. This study had a small sample size and with more subjects, the outcomes could be
further generalized. Overall, the majority of the students showed improvement on their posttest
using the virtual manipulative applets.
Sonya Brown wanted to know whether or not students who used virtual manipulatives
would out-perform students who used concrete manipulatives on the researcher and teacher
generated posttest. She conducted this study to investigate the impact of using computer
simulated, virtual, manipulatives and hands-on, concrete or physical, manipulatives on
elementary students’ learning skills and concepts in equivalent fractions. To research this
question, the researcher used a quantitative method, Quasi-Experimental Pretest-posttest,
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Nonequivalent Control Group Design research, administered a pretest to both a control and
experimental groups, and administered a posttest.
The subjects were 49 sixth graders from two mathematic classes in one of Detroit’s
public schools. Students were already assigned to classes, were intact and hopefully the variation
in students’ gender, ethnic background and socioeconomic status reflected the composite to the
greater population in that geographic area. Group A received mathematics instruction with
virtual manipulatives and Group B received mathematics instruction with concrete
manipulatives. Group A is the experimental group and Group B is the control group. The
independent variables were the mathematics instruction with the use of virtual manipulatives and
the mathematics instruction with the use of concrete manipulatives. The dependent variables
were the students’ conceptual knowledge and procedural knowledge as it related to fractions.
The groups each received only 1 day of instruction with their respective manipulative.
The pre and posttest instruments were identical and tested students’ conceptual and
procedural knowledge of equivalent fractions. The instruments were designed by the researcher
and the content of the instruments was based on the curriculum standards outline by the National
Council of Teachers in Mathematics. The researcher used a two-sample, paired-data, t-test with a
0.05 confidence level to analyze the data. The pre and posttest gain score showed that the
concrete manipulative group increased mathematic achievement higher than the virtual
manipulative group. One explanation for this is that the concrete manipulative group’s pretest
scores were generally higher than the virtual manipulative group. Another possible reason is that
the instruction with the use of concrete manipulatives was more effective than that of using
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Some concerns for this research study is that the students may not have been exposed to
their respective manipulative for enough time. A single day is not enough time to conduct
adequate research. Plus, the researcher admits that she was a pre-service teacher with no
experience teaching with physical or virtual manipulatives.
Need for the Study
Based on the conflicting outcomes stated in the three research studies above, there is a
need for more research on the topic of virtual vs. physical manipulatives. There is a lack of
research on the effects of using virtual manipulatives in elementary mathematic classrooms, and
those studies that were conducted conflict with findings. The last study brings some validity and
reliability concerns since the study was only conducted for one day and by an inexperienced pre-
service teacher. Inconsistent research findings compel me to add to the existing knowledge so
that educators can have research at their fingertips before trying something new in their
mathematics classrooms; virtual manipulatives.
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The study will be conducted to determine whether virtual or physical manipulatives
impact math achievement scores of third grade students the most. This quantitative study will be
conducted as quasi-experimental research because it will use two different intact third grade
mathematic classes. One group will be the control group, meaning they will learn basic fractions
using traditional physical manipulatives. The other group will be the experimental group because
they will learn basic fractions using virtual manipulatives. Each class will have 20 students, but
different teachers. Both teachers will teach the same math unit on basic fractions, using their
respective manipulatives. Both groups will take a pretest and posttest to measure prior
knowledge and knowledge gained over the study. The quasi-experimental design is diagramed
G1 O1 X O2 GS1
G2 O3 - O4 GS2
G1 and G2 represent the two math classes that are participating in the study. G1 is the
experimental group because they will be exposed to the virtual manipulative treatment, which is
represented with X. G2 is the control group because they will not be exposed to the experimental
variable, but only the normal tradition teaching style using physical manipulatives. The – is used
to represent that G2 will not have an experimental variable. The mean scores for the pretest
instrument will be represented by O1 and O3. The mean scores for the post test measurement
will be represented by O2 and O4. The gain score, found by subtracting the pretest from the
posttest, will be calculated and shown in GS1 and GS2, which represents the gain score.
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Subjects for this research study will be 40 third grade students at a public middle school
in Monroe County. Students will have a mixed socioeconomic profile and a variety of African-
America, Asian, Hispanic and a majority of Caucasian participants. Gender will be evenly
distributed among the groups and ability levels should be mixed. Students will be randomly
assigned to their classes by school administration and each class will have 20 students. Both
classes meet in the morning at the same time during their block period 3, which is right after
their special for the day.
Each class will have a different teacher instruct, since they are intact classes and already
assigned to their respective teacher. Both classes will be exposed to the same math concepts
during the 10 day study, since both classes use the same curriculum in the form of teacher guide,
student textbook and homework book. However, the way in which the math concept is taught
will differ since the experimental group will be using a projector, smart board and students may
be at the computer lab. The teacher for the experimental group will give homework from the
workbook because not all students have access to the internet or have computers. Plus, without a
school provided math homework site, it would be hard for the teacher to access and assess the
homework online. The control group will stay in the classroom and use the overhead and
physical manipulatives. The control group will also have homework from the workbook. The
study will be conducted during the third quarter in the 2011/2012 year.
All students will take the same pretest on day 1 of the study. Students will have all of the
period to take the test. The pretest will add internal validity to the study because the ability and
knowledge level of basic fractions can be determined for each class. This will help in
determining which group actually had the largest gain score. After the students have taken the
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pretest, they will be introduced to the manipulatives in a fun way that excites children to start
learning basic fractions for the coming days. Formal instruction will not take place as students
are usually drained after taking a test.
Days 2 though 9 will be spent with the students receiving direct instruction from their
respective teacher, using their respective manipulative and same lesson. A typical day consists of
one lesson. In group one, the teacher models, and guides and then allows the students to work
independently to complete a task, using the virtual manipulatives throughout each stage. In group
two, the teacher models, guides and then allows the students to work independently to complete
a task, using the physical manipulatives throughout each stage. The teachers will have 8 days to
complete six lessons in the unit. Teachers cannot go on to other lessons past the stopping point if
they finish early and must teach the six lessons in the 8 days allotted. Due to having two different
teachers, teaching styles will vary, but content and homework will be the same. On day 10, the
posttest will be given to students and they will have all period to complete the posttest. No
manipulatives, either physical or virtual, may be used by the students as they take the test.
The pretest and posttest, called Fraction Fun, look similar with the same kind of fraction
problem, but do ask different questions. They each contain 20 questions, with a correct or
incorrect answer being possible. Students will write their answer under the fraction image and
the scores will be averaged to find the mean scores for the pretest and posttest. Scores will them
be compared and analyzed to find the gain score, for each group.
Since the Fraction Fun pretest and posttest was generated using an online worksheet tool,
the measurement instrument has not been tested for reliability and has no internal reliability. This
instrument has moderate validity because the tests are extremely similar and do test basic
fraction knowledge of third graders in a way that is appropriate for the intended age level. The
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measurement instrument is kid friendly, while having the fraction content and criteria for the
content with 20 questions of the same fraction concept.
Both pretest and posttest means will be calculated for both groups so that the mean gain
score can be calculated. Gain scores will be calculated by subtracting the mean pretest score
from the mean posttest score. Results will be shown in the table below:
Mean Gain Scores on Survey Instrument
Control Group (G2
Mean Score Pretest O1 O3
Mean Score Pretest O2 O4
Gain Score GS1 GS2
The mean gain scores for each group will be compared in order to interpret the outcome of the
study and to verify and experimental effect to due to the independent variable, the virtual
manipulatives. A copy of the measurement instrument follows on the next two pages.
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The proposed study asks three research questions. The first question is “What are the gain scores
on an instrument measuring achievement of students taught basic fractions using virtual
manipulatives?” I predict that the students in the virtual manipulative group will demonstrate
positive gain scores because manipulatives enhance student comprehension and conceptualizing
The second question asked is “What are the gain scores on an instrument measuring
achievement of students taught basic fractions using physical manipulatives?” I predict that
students the control group will demonstrate positive gain scores of significance, since instruction
and manipulatives will enhance learning, therefore the students should do better on the posttest
than the pretest.
The third question asked is “How do the scores compare?” I expect both groups to have
positive gain scores because both use manipulatives as a teaching tool. However, I predict that
the manipulated group with the virtual manipulatives will score slightly higher than the control
group since students are active learners when technology is used.
This study is relevant to educators because if both groups demonstrate positive mean
scores, it could show that there is a positive effect of using virtual and physical manipulatives in
a classroom. Educators would then be provided with research that supports teaching basic
fractions and math in general, with manipulatives. Most educators would not simply teach using
only virtual manipulatives, but may be inspired to use both types of manipulatives in their
classroom when available. The use of both kinds of manipulatives, virtual and physical, can
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reach more children and stimulate children’s minds to become better mathematicians.
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Brown, S. E. (2007). Counting blocks or keyboards? a comparative analysis of concrete
versus virtual manipulatives in elementary school mathematics concepts. Online
submission, Retrieved from EBSCOhost.
Mendiburo, M., Hasselbring, T., & Society for research on educational effectiveness, (2011).
technology's impact on fraction learning: an experimental comparison of virtual and
physical manipulatives. Society for Research on Educational Effectiveness, Retrieved
Reimer, K., & Moyer, P. S. (2005). Third-graders learn about fractions using virtual
manipulatives: a classroom study. Journal of Computers in Mathematics and Science
Teaching, 24(1), 5-25. Retrieved from EBSCOhost.
Rosen, D., & Hoffman, J. (2009). Integrating concrete and virtual manipulatives in early
childhood mathematics. Young Children, 64(3), 26-33. Retrieved from EBSCOhost.
Smarkola, C. (2007). Technology acceptance predictors among student teachers and
experienced classroom teachers. Journal of Educational Computing Research, 37(1), 65-
82. Retrieved from EBSCOhost.
Soft Schools (2005). Fraction fun picture worksheets for third grade. Retrieved July 20, 2011,
Suh, J., & Moyer, P. S. (2007). Developing students' representational fluency using virtual
and physical algebra balances. Journal of Computers in Mathematics and Science
Teaching, 26(2), 155-173. Retrieved from EBSCOhost.