Mathematics and Computers in Contemporary Science
Using Game-like Operational Virtual Manipulatives to Assist Afterschool Remedial Instruction: a Case Study of Third Graders Tutoring
in the “Subtraction” Unit
WEN-CHUNG SHIH*, TSUNG-CHIH CHEN
Department of Applied Informatics and Multimedia
AsiaUniversity
500, Lioufeng Rd., Wufeng, Taichung 41354, Taiwan
TAIWAN
[email protected]://dns2.asia.edu.tw/~wjshih/
Abstract: - Thelearning of subtraction is essential for elementary students. However, common subtraction
misconceptions hinder mathem Department of Applied Informatics and Multimediaatics underachievers from
learning subtraction. These underachievers usually need repeated explanation in many times and in different
ways to understand some concept. Actually, teachers have to consider the progresses of all students in the class
and thus cannot spend a lot of time in explaining some concept for underachievers.After-school volunteer
tutoring emerges as one kind of possibility to solve this problem. Although nonprofessional tutors have time to
explain concepts for underachievers, they do not know how to teach.As more and more undergraduate students
act as voluntary tutors for rural pupils after school, these nonprofessional tutors need a supporting environment
to facilitate the tutoring process. In this study, a simple tutoring method based on misconception diagnosis is
proposed, and supporting game-like manipulatives are prepared for voluntary tutors to use. First, the tutor
identifies the tutee’s misconception by conducting a diagnostic test. Then, the tutor applies several kinds of
game-like manipulatives to make the tutee understand the concept. Further, a case study is conducted to explore
theeffects and limitations of the proposed approach. Evaluation results indicate that despite the need for further
improvement, the two-phased tutoring is effective for correcting subtraction misconceptions. Finally,
recommendations for future research are proposed.
Key-Words: -Game-based Learning, Virtual manipulatives, Subtraction, Misconception diagnosis, Voluntary
tutoring, Remedial instruction
hinderthe comprehension and application of
subtraction. Vanlehn’s study of students' subtraction
systematic error reported similar results [31].
Misconceptions in Subtraction have been
investigated in the literature. Chang et al. reported
on misconceptions in mathematics which showed
that studentsalways had misunderstandings in
processing problem solving, even for simple skills
such asaddition or subtraction. The forming of
misunderstandings may be due to experience inreal
life or come from their learning experiences in the
classroom [27].Brown and Burton [3] have found
that some misconceptions occurred in the
solvingprocedures of the subtraction exercises.
Among these misconceptions, there are six types to
befrequently found in the students.The students
always subtract a smaller digit from a larger
digit,but the smaller one may be a minuend and the
larger one a subtrahend. For example, thestudents
think that the answer of (32 – 19) is 27.
Most of previous studies of remedial instruction
have been focused on self-learning methods, and
1 Introduction
Computational skills in subtraction are an important
component in any mathematics curriculum
forchildren. However, a typical activity for them
inclassrooms involves doing repetitive arithmetic
calculation exercises[5][11][12][19][21-22]. Also,
the ability to solve basic subtraction questions is the
foundation for developing theability to tackle
complicated questions. Taiwanese students who
experienced
mathematical
learning
difficultiesbecame more and more frustrated when
they reached the middle grades because the
difficulty and range ofmathematical courses
intensified and expanded[7]. Hwang et al.[13] also
recommended thatteachers design problem solving
activities to improve students’ mathematical
representation skills.
However, helping elementary students to develop
subtraction literacy is not easy. Subtraction
misconceptions are common in elementary students.
These misconceptions conflict with scientifically
accepted subtraction concepts and seriously
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Mathematics and Computers in Contemporary Science
Then, the tutor applies several kinds of virtual
manipulatives to make the tutee understand the
concept. Further, a formative evaluation is
conducted by using a case study to explore
theeffects and limitations of the proposed approach.
Evaluation results indicate that despite the need for
further improvement, the two-phased tutoring is
effective for correcting subtraction misconceptions.
Finally, recommendations for future research are
proposed.
Recent
studies
have
proposed
virtual
manipulatives
for
helping
students
learn
mathematics. The learner can move the mouse to
operate the dynamic visual objects [1][24-25].
Virtual manipulatives mentioned in the literature are
not simply physical copies of the teaching aids.
Virtual manipulatives and physical teaching aids are
different in functions and effects, such as
interactivity
and
availability[11][25][26][29].Despite the recognized
potential of virtual manipulatives for helping
students to understand mathematics, the designand
development of virtual manipulatives for learning
mathematics still demand further study. For
instance, misconception in subtraction is a
majorobstruction for students to comprehend
subtraction. However, only a few virtual
manipulatives have been designed to correct
misconception in subtraction.Besides, an effective
learning technology requires learning models based
on appropriate theory. However, earlierstudies lack
of theoretical background for designing and
developing virtual manipulatives to enhance the
understanding of subtraction.Finally, applications of
virtual manipulatives for learning in other
fieldsreveal limitations such as passive learning and
cognitive overload.Therefore, the design of virtual
manipulatives
for
correcting
subtraction
misconceptions should maximize the advantages of
virtual
manipulatives
and
minimize
its
disadvantages.
researchers believed that low-achievers can do it
well. Chen [6] proposed a personalized diagnosis
and remedial learning system, and showed learners
who received personalized remedial learning
guidance achievedimproved learning performance.
Huang et al. [12] have developedan computerassisted mathematics learning systemto serve as a
supplementary tool thathelps teachers with remedial
instruction. Heh et al. [10]proposed a full-loop
learningsystem to identify the misconceptions of
students by using a knowledgemap, and then
selected suitable learning materials according to
misconceptions forindividual students to do
remedial learning. However, these works neglect the
inability of underachievers to learn by themselves in
primary school Mathematics. Furthermore, they
usually need repeated explanation in many times
and in different ways to understand some concept.
Actually, teachers have to consider the progresses of
all students in the class and thus cannot spend a lot
of time in explaining some concept for
underachievers.
Volunteer tutoring emerges as one kind of
possibility to solve this problem. The instructional
models of tutoring in small groups and in one-to-one
settings have been investigated in the literature
[2][32]. The tutoring model has been characterized
by the main advantage of taking care of the tutee’s
learning needs. Gaustad[8]indicated this viewpoint
in the following aspects: (1) the teaching is adaptive
to the tutee's pace and learning style; (2) the tutee
can provide immediate feedback to the tutor; (3) the
tutor can identify the learner’s misconception and
solve it. Also, Gaustadac knowledged the emotional
benefits brought by the tutoring process.For
example, tutees can learn in their own pace without
being compared withfaster peer learners.Besides,
learners in the private tutoring environment feel
more comfortable when they make mistakes. Tutors
can provide tutees with specialcare, support and
immediate encouragement. Thus, tutees will
gradually become confident of their success in
learning.
Mostly voluntary tutors are not professional in
Mathematics and education. They have time to
explain concepts for underachievers, but they might
not know how to teach.As more and more
undergraduate students act as voluntary tutors for
rural pupils after school, these nonprofessional
tutors need a supporting environment to facilitate
the adaptive instruction. In this study, a simple
tutoring method is proposed, and supporting virtual
manipulatives are prepared to assist voluntary
tutors. First, the tutor identifies the tutee’s
misconception by conducting a diagnostic test.
ISBN: 978-960-474-356-8
2 Related Work
The proposed tutoring method integrates a tutoring
approach and a computer-assisted self-learning
approach. Actually, the tutor follows a pre-defined
work-flow to help the tutee learn Mathematics.
Also, the theory of multiple representations is
employed to develop virtual manipulatives for
remedial instruction. This section introduces related
work about an instruction model for tutoring, a
learning model for simulation-based computer
assisted learning and the theory of Mathematical
representation.
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Mathematics and Computers in Contemporary Science
2.1 Karsenty’s Instructional
Training Tutors
Model
of each phase and the progression between each
phase aredescribed below.
Externalization phase: Making students aware of
their
preconceptions
before
instructional
intervention is the firststep of the cognitive conflict
approach. Thus, the objective of this phase is to
enable students be aware of their ownideas about the
statistics concepts they are learning. This phase
achieves this by posing a question in the context
ofdaily life so as to make it meaningful to the
students.
Reflection phase: This phase prompts students to
reflect on their own ideas that were externalized in
theExternalization phase. To achieve this, the
simulation-based learning environment displays a
series of questionsrelated to the concepts raised in
the Externalization phase in order to guide them to
manipulate the DLMRs and toobserve the results
progressively.
Construction phase: This phase introduces
exploratory activities to help students construct their
concepts aboutcorrelation. In this phase, clear
learning guides (including definitions of the target
concept and clear manipulatingprocedures) are used
to prompt students to manipulate the DLMRs, to
observe
and
identify
the
relationships
amongdifferent representations and, then, to
construct their own concepts about correlation.
Students who do notunderstand the concept at the
end of this process are asked to repeat the entire
Construction phase.
Application phase: This phase provided students
with two problem situations in different contexts
and with differentsolution paths. When solving the
problems, the students were given minimal support.
One objective was to allowstudents to elaborate on
their newly constructed concepts by applying them
to solve novel problems. The second wasto evaluate
the ability of the students to transfer these concepts.
This model and its corresponding system is a
good approach for students to learn by themselves.
However, most of underachievers need teachers or
tutors to accompany them and provide adaptive
instruction. Moreover, if the system is not intelligent
enough, human supervisors are needed to standby
and provide suitable support to the tutees.
for
Karsenty [16] has proposed an instructional model
for preparing tutors. The instruction model consisted
of a sequence of stages. In the first two stages,
mathematical concepts and representations were
discussed.Since most tutors were familiar to a
certain degree with the mathematical content, this
partof instruction served to refresh their memory
and assist them in making connectionsbetween
different concepts and representations, which some
tutors remembered as isolatedfacts or procedures. In
the next stage of instruction, the counselor discussed
with tutorspossible difficulties that students might
encounter with these concepts. Discussion wasbased
on two resources: the professional knowledge of the
counselor and the tutors'recollections of their own
experience as students or that of their peers.
Although most tutorswere successful in high school
mathematics, some of them could recall having
difficultiesin understanding certain content, and
these were elicited and explored by the group. In
thefourth stage of the instruction sequence, tutors
were introduced to teaching approaches thatmight
assist students in overcoming the difficulties
discussed. As expected, this part wasnew to all
tutors and formed the lion's share of the instruction
time. The four stages were repeated, ifnecessary, for
more mathematical concepts or procedures included
in the tutoring period'sgoal. Finally, tutors were
presented with “toolkits” of strategies and models to
confront gapsin prior knowledge.
This model describes the four phases to train a
tutor. However, the work-flow is presented in
conceptual level. The implementation of each phase
needs more details for tutors to know what to do in
the real tutoring process. For example, tutors might
need some manipulatives in their tutoring process.
Therefore, they should be taught where to get
appropriate manipulatives for specific students and
situations.
2.2 Liu’s Learning Model for Simulationbased Computer Assisted Learning
Liu [20] has developedsimulation-based computer
assisted learning to correct students’ statistical
misconceptions based on cognitive conflict theory.
The cognitive conflict approach was used to develop
a learning model with four phases (Externalization,
Reflection,Construction, and Application). The aim
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2.3 Mathematical representation:theoretical
background for learning model
Mathematical representation means that learners
represent an external concept in different cognitive
symbols when solving a problem. Mathematical
representation plays an important role in
mathematics learning to construct knowledge[30].
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Mathematics and Computers in Contemporary Science
teaching methods demanded by tutors can be illstructured, ill-organized or even unwritten. Thus,
classic information retrieval cannot be simply
applied to retrieve teaching methods.
A two-phased tutoring model is proposed to be
followed by tutors in the tutoring session. In order to
be able to achieve the purpose of after-school
remedial instruction, the researchers must first
knowwhat types of mistakes the students make
when solving problems. Therefore, this study will
put forward the concept of simple and rapid
diagnostic tests for tutors to use. In Phase 1, the
tutor identifies the tutee’s misconception by
conducting a diagnostic test. In Phase 2, the tutor
applies several kinds of game-like manipulatives to
make the tutee understand the concept.
The game-like manipulatives are stored in a
repository, organized according to a taxonomy tree
of common misconceptions. We have constructed
the taxonomy tree, in a tabular form, for common
misconceptions of the third-grade subtraction unit.
Bruner [4] proposed that the progress of learning
consists of three stages: operations, images and
symbols. Heddens[9] divided the students' learning
process into four stages: Concrete, Semi-concrete,
Semi-abstract andAbstract, advocating learners must
establish a good link between the real world and the
abstract world, thereby improving the difficulties in
the understanding of mathematical concepts.
Kaput[15] presented four representations to
model the relationship between Mathematical
representation and Mathematical learning, including
three internal and one external representations. Lesh
et al.[17] pointed out five external representations
used in mathematics education including realworld
object representation, concrete representation,
arithmetic
symbol
representation,
spokenlanguagerepresentation, and picture or graphic
representation. Among them, the last three are more
abstract and higher levelof representations for
mathematical problem solving.
Willis and Fuson [33] explore the use of pictures
characterization for elementary school sophomore,
to help solve addition and subtraction of the
application found to enhance the effectiveness of
their solution through pictures characterization.
Based on the theory of Mathematical representation,
Hwang et al.[14]developed a Virtual Manipulatives
and Whiteboard (VMW) system, which allowed
users to manipulate virtual objects in 3D space and
find clues to solvegeometry problems. In this work,
we apply Mathematical representation to remedial
instruction by providing operational virtual
manipulatives.
3 Introduction to
Tutoring Method
the
4 Methodology
There are three purposes in this study: (1) proposing
a simple tutoring method and designing virtual
manipulatives to assist voluntary tutoring, (2)
understanding the effects of and students’
perceptions toward the proposed tutoring method,
and (3)discussing factors underlying tutoring effects
and students’ perceptions. To achieve these goals, a
supporting tutoring environment and virtual
manipulatives are developed. In addition, this study
collectedboth quantitative and qualitative data from
the tutoring sessions and from related feedback of
students.The research questions are as follows:
1. What are the effects of the proposed tutoring
method with regard to the diagnostic phase and the
remedial phase?
2. What are the perceptions that the students have
with regard to the diagnostic phase and the remedial
phase?
This study uses the case-study method[18][23],
which is considered “an empirical inquiry that
investigates acontemporary phenomenon within its
real-life context, especially when the boundaries
between phenomenon andcontext are not clearly
evident” [34]. That means, a case study is an
empirical and holistic inquiry thatexplores a social
unit, a single instance, or a phenomenon within a
natural setting [23].Therefore, this approach is
particularly appropriate for exploring the possible
effects that the proposed diagnostic and remedial
methods have on the tutoring sessions[18][30].
Proposed
Shih et al. [28], a tutor supporting environment is
illustrated, which is focused on the aspect of social
network support. In fact, the component of the
tutoring resource repository is also important for a
tutor supporting environment. When a tutor fails to
make the pupil understand a concept, the tutor can
ask for help by retrieving tutoring resources from
the tutoring resource repository. Therefore, the tutor
supporting environment is enhanced. A tutoring
resource repository is significantly different from
existing learning object repositories. While learning
objects are designed to be operated in a learning
management system, tutoring resources are teaching
methods which can be used by tutors to make the
pupil understand a concept. A tutoring resource
repository is intended to help tutors find relevant
teaching methods for teaching a concept. However,
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Mathematics and Computers in Contemporary Science
subtraction concepts in both students. All three
misconceptionsheld by Student A were corrected
while three of the four held by Student B were
corrected. To further explore how the proposed
approach benefited students, the following sections
present the results anddiscussions about the effects,
contributions and limitations of this tutoring model.
Two students and two tutors participated in this
case study during October 2011. The subjects of this
research are the third grade students of an
elementary school which is located in Taichung,
Taiwan. They are selected from a class which has
eight students. The performance ranking of the two
case children falls betweenbottom 25% to 40%,
according to the midterm assessment of the school.
The selection criteria for students consider
agreement from parents to participate in this study.
In addition, students with distinct learning
disabilities and violent records are excluded from
this study. Case A is a boy and case B is a girl.
During the tutoring sessions, each tutor sat
withhis/her students in a separate area. Tutoring are
conducted according to a concept of a “fresh start”,
where the key was not students' previous knowledge
but rathertheir willingness to think and keep an open
mind about the material discussed. Sessionswere
usually guided by the game-like manipulatives. The
procedure of this study is composed of five stages:
1. Preparation. In this stage, common
misconceptions of third graders’“subtraction” are
collected according to related literature. Next,
diagnostic items and remedial materials are
designed and implemented in terms of these
conceptions. In addition, test items for a pre-test are
designed.
2. Case Selection. A misconception test is
administered to the class as a pre-test. The students
who are underachievers and willing to participate in
this tutoring program are selected as the cases. The
behavior of the selected cases is analyzed with
respect to their performance in school and at home.
3. Misconception Diagnosis. Diagnostic tests are
administered to the selected cases. The tutors
interview with the tutees to identify their
misconception.
4. Remedial Instruction. The tutors use game-like
virtual manipulatives to conduct remedial
instruction.
5. Evaluation. A misconception test is
administered to the selected cases as a post-test.
Also, the tutors and the tutees are interviewed by the
researcher to understand the effect of the proposed
approach.
5.1 The effects of the diagnostic phase on
identifying misconceptions
The tutoring model includes two phases: Diagnostic
Phase and Remedial Phase. This section describes
the tutoring processesobserved in the misconception
diagnostic phase. In addition, the subsequent
interview results are presented to show how the
diagnostic method canbenefit the students. For
brevity, the following results and discussion focus
on the unit “subtraction of three-digit numbers”. The
items of pre-test are designed to reflect
misconceptions in the options. For example, an item
has four options. One is the correct answer, and the
other three represent three misconceptions
respectively.
According to the pre-test results, case A fails in
the unit of “subtraction of three-digit numbers”.
First, the tutor needs to verify that case A really has
misconceptions in this unit. Hence, the tutor
downloads a diagnostic item from the repository and
helps case A to do it. In the first item, case A drags
the option “156” to the answer area. In the second
item, case A drags the option “233” to the answer
area. These two actions show that case A has the
misconception of “using a large number to subtract
a small number”.
5.2 The effects of the remedial instruction
phase on correcting misconceptions
This
section
describes
the
tutoring
processesobserved in the remedial instruction phase.
In the misconception diagnostic phase, the tutor
found that case A has the misconception of “using a
large number to subtract a small number”.
Therefore, the tutor downloads an operational
virtual manipulative about the concept of
“borrowing” from the repository. Then case A
practices it to conduct remedial instruction. This
operational virtual manipulative guides the tutee to
use the mouse to show the correct sequence of
subtract three-digit numbers.
5 Results and Discussion
The current study utilizes a case study to evaluate
whether the proposed approach can correct
subtraction misconceptionsand what the cases’
perceptions are. A comparison of pre-test and posttest scores on the subtraction unit showed that this
tutoring process could reduce misconceptions about
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5.3 The perceptions of the tutees
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Mathematics and Computers in Contemporary Science
The researcher has interviewed the two cases to
understand their perceptions about the proposed
approach. For the diagnostic process, the tutees
agree that the process can help them identify their
misconception. Case A said, “I can quickly know
my status of learning subtraction by using the
diagnostic tool.” Case B said, “The diagnostic tool
can identify my misconception and it is fun.”
Regarding the remedial instruction phase, the
two cases also agree the game-like virtual
manipulatives are interesting. Case A said, “I got
more impressed by following the guide to conduct
the subtraction steps.” Case B said, “I think the
interactive tool is like video games, which attracts
me to learn mathematics.” In addition, they think the
remedial approach can help them understand
mathematic concepts. In the tutoring process, they
feel focused. More important, mathematics seems
not so difficult as it usually was. That is, the tutees
are more confident about learning mathematics.
To collect participants’ opinions on this
approach, we have conducted an open-endedquestion interview with ten tutor participants and
ten student participants. The open-ended questions
are listed as follows. What is your main problem
while teaching/learning mathematics? What do you
think of using interactive games as an auxiliary
teaching mechanism? Tutors’ opinions are
summarized as follows.
Certain mathematical concepts are difficult for
students to understand. When students cannot fully
understand after the instructions, one of the tutors
said to students: “Practice more exercisesand you
will understand it someday.” The other tutors said,
“Just memorize the formulas.”These tutors wanted
to help students understand these concepts but they
did not know the appropriate teaching methods.
Oppositely, other tutors will look for useful teaching
materials such as web searching or peer assistance.
The tutors stated that the proposed approach can
help them retrieve useful and valuable teaching
methods. One of the tutors said, “Sometimes I do
not know how to teach, this system can help me call
for assistance.” Another tutor said, “Using this
system can retrieve different teaching methods and
interactive games toward this same topic. By using
those teaching methods, my student can actually
understand mathematical concepts.” One tutor said,
“The first game is more useful than the second
game. With the aid of the “move” manipulation, I
can understand the concept of equivalent factors.”
The case study reported in this article focused on
tutoring of mathematical content to lowachievingstudents. The article examines the role of
mathematics tutors as it emergesfrom situations
where schools might lack adequate resources to
address the needs of low-achieving students. They
may turn to thelow-cost option of nonprofessional
tutoring. It should be emphasized here that this does
notimply we can replace professional teachers with
inexperienced enthusiastic volunteers. Moreover,
the study does not aim to compare the outcomes of
professional andnonprofessional tutoring. It is well
known from the literature that professional
andparaprofessional tutoring both have very positive
effects on students' achievements[16]. The novelty
of the study is thatusing nonprofessional tutoring
can still provide a certain positive effect on
students'achievements in elementary school
mathematics. The results showed that, after tutoring
sessions, students' capability to handle the
mathematical materialincreased considerably. This
finding is consistent withresults reported by [8] in
regard
to
the
positive
effect
of
nonprofessionaltutoring.
Although understanding and applying concepts
of subtraction are essential abilities, students often
have more or less misconceptions.This study
proposed a two-phased tutoring method, consisting
of diagnosis and remedial phases, foreliminating
misconceptions about the basic concept of
subtraction. Two important elements were
considered inthe design and development of the
proposed approach. One is the organization of
game-like manipulatives based on a taxonomy of
common misconceptions, which facilitates efficient
retrieval of manipulatives. Another is the
introduction of operational “games” into the tutoring
sessions, which makes the learning process
interesting. The results of the evaluation indicated
that
students
substantially
reduced
their
misconceptionsafter the tutoring sessions.
This work describes a misconception-based
approach to managinginteractive games as learning
objectsfor remedial instruction. Several issues will
be further investigated in the future work. In this
paper, the performance issue of misconception
indexing has not been addressed. When the size of
the learning object repository grows rapidly, lowlevel indexing technologies can be adopted to
alleviate this issue. Resource sharing and fault
tolerance are interesting issues for cloud
applications. The technology of replica management
will be incorporated into this framework of learning
content retrieval to discuss their effect on content
access. In addition, social agreement is an important
6 Conclusion
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Collaborative Cross Number Puzzle Game to
Develop the Computing Ability of Addition
and Subtraction. Educational Technology &
Society, 15 (1), 354–366.
[8] Gaustad, J. (1992). Tutoring for at-risk
students. Eugene, OR: Oregon School Study
Council.
[9] Heddens, J. W. (1984). Today’s mathematics.
(5th ed.). Chicago: Science Research
Associates.
[10] Heh, J.-S., Li, S.-C., Chang, A., Chang, M., &
Liu, T.-C. (2008). Diagnosis Mechanism and
Feedback System to Accomplish theFull-Loop
Learning Architecture. Educational Technology
& Society, 11 (1), 29-44.
[11] Highfield, K., & Mulligan, J. T. (2007). The
role of dynamic interactive technological tools
in preschooler's mathematical patterning. In J.
Watson & K. Beswick (Eds.), Proceedings of
the 30th annual conference of the Mathematics
Education Research Group of Australasia,
Hobart (Vol. 1, pp. 372-381). Adelaide:
MERGA.
[12] Huang, T.-H., Liu, Y.-C., & Chang, H.-C.
(2012). Learning Achievement in Solving
Word-Based Mathematical Questions through a
Computer-Assisted
Learning
System.
Educational Technology & Society, 15 (1),
248–259.
[13] Hwang, W. Y., Chen, N. S., Dung, J. J., &
Yang, Y. L. (2007). Multiple Repesentation
Skills and Creativeity Effects onMathematical
Problem Solving using a Multimedia
Whiteboard System. Educational Technology
& Society, 10(2), 191-212.
[14] Hwang, W.-Y., Su, J.-H., Huang, Y.-M., &
Dong, J.-J. (2009). A Study of MultiRepresentation
of
Geometry
Problem
Solvingwith Virtual Manipulatives and
Whiteboard System. Educational Technology
& Society, 12 (3), 229–247.
[15] Kaput, J. J. (1987). Representation systems and
mathematics. In Janvier, C. (Ed.), Problems
ofrepresentation in teaching and learning of
mathematics (pp. 159-195). Hillsdale, NJ:
LawrenceErlbaum.
[16] Karsenty, Ronnie. (2010). Nonprofessional
mathematics tutoring for low-achievingstudents
in secondary schools: A case study. Educ Stud
Math, 74, 1–21.
[17] Lesh, R., Post, T., & Behr, M. (1987).
Representations and translations among
representations in mathematics learning
andproblem solving. In C. Janvier (Ed.),
Problems of representation in the teaching and
issue for Wiki-based applications. In recent years,
researches on the convergence process of Wiki
applications have attracted extensive attention, such
as the ontology crystallization problem. These
techniques can be applied in the Wiki-based
teaching material design process.
In sum, the tutoring approach presented in this
paper shows a potential for advancingstudents,
whose low attainments in mathematics could be
attributed to social or behavioralcircumstances.
Subsequent research is needed in order to affirm and
broaden the results ofthis study. Questions about the
long-term effect of such programs, the impact of
group workas opposed to individual preparations of
tutors, and the role of social organizations
inmonitoring tutoring models, are few of the issues
that need to be further investigated.
Acknowledgements
This research was supported by National Science
Council of Republic of China under the number
ofNSC100-2628-S-468-001-MY2, NSC 101-2511S-468-003 andNSC 102-2511-S-468 -001.
References:
[1] Baker, M. (2008). Merging technology &
mathematics instruction: The case of virtual
manipulatives & geometric concepts. In G.
Richards (Ed.), Proceedings of World
Conference on E-Learning in Corporate,
Government, Healthcare, and Higher Education
(pp. 566-573). Chesapeake, VA: AACE.
[2] Bloom, B. S. (1984). The search for methods of
group instruction as effective as one-to-one
tutoring.Educational Leadership, 41, 4–17.
[3] Brown, J. S., & Burton, R. R. (1978).
Diagnostic models for procedural bugs in basic
mathematic skills. Cognitive Science, 2, 155192.
[4] Bruner, J. S. (1966). Toward a theory of
instruction.
Cambridge,
MA:
Harvard
University.
[5] Chang, K.-E., Lin, M.-L., & Chen, S.-W.
(1998). Application of the Socratic dialogue on
corrective learning of subtraction, Computers
& Education,31(1), 55-68
[6] Chen, L. H. (2011).Enhancement of student
learning
performance
using
personalizeddiagnosis and remedial learning
system, Computers & Education, 56(1), 289299.
[7] Chen, Y.-H., Looi, C.-K., Lin, C.-P., Shao, Y.J., & Chan, T.-W. (2012). Utilizing a
ISBN: 978-960-474-356-8
247
Mathematics and Computers in Contemporary Science
for Tutors to Retrieve Relevant Teaching
Methods.Educational Technology & Society,
14(4), pp. 207-221.
[29] Steen, K., Brooks, D., & Lyon, T. (2006). The
impact of virtual manipulatives on first grade
geometry instruction and learning. Journal of
Computers in Mathematics and Science
Teaching, 25(4), 373-391.
[30] Tsui, C. Y., & Treagust, D. F. (2003). Genetics
reasoning
with
multiple
external
representations.
Research
in
Science
Education,33(1), 111-135.
[31] VanLehn, K. (1990). Mind bug: The origins of
procedural misconceptions. Cambridge, MA:
MIT press.
[32] Wasik, B. A., & Slavin, R. E. (1993).
Preventing early reading failure with one-toone tutoring: A review offive programs.
Reading Research Quarterly, 28(2), 178–200.
[33] Willis, G. B. &Fuson, K. C. (1988). Teaching
children to use schematic drawings to solve
additionand subtraction word problems.
Journal of Educational Psychology, 80, 192201.
[34] Yin, R. (1994). Case study research: Design
and methods (2nd Ed.), Beverly Hills, CA:
Sage Publishing.
learning
of
mathematics,
Hillsdale,
NJ:Lawrence Erlbaum, 33-40.
[18] Liu, T. C. (2007). Teaching in a wireless
learning environment: A case study.
Educational Technology & Society, 10(1), 107123.
[19] Liu, T. C., Lin, Y. C., &Tsia, C. C. (2009).
Identifying misconceptions about statistical
correlation and their possible causes among
high school students: An exploratory study
using concept mapping with interviews.
International Journal of Science &Mathematics
Education, 7(4), 791-820.
[20] Liu, T.-C. (2010). Developing Simulationbased Computer Assisted Learning to Correct
Students' Statistical Misconceptionsbased on
Cognitive Conflict Theory, using "Correlation"
as an Example. Educational Technology &
Society, 13 (2), 180–192.
[21] Liu, T. C., & Lin Y. C. (in press). Developing
two-tier diagnostic instrument for exploring
students’
statistical
misconceptions:Take
“Correlation” as the example. Bulletin of
Educational Psychology.
[22] Liu, T. C., Lin, Y. C., & Kinshuk. (in press).
The application of Simulation Assisted
Learning
Statistics
(SALS)
for
correctingmisconceptions
and
improving
understanding of correlation. Journal of
Computer Assisted Learning.
[23] Merriam, Sharan B. (1998) Qualitative
Research and Case Study Applications in
Education. San Francisco: Jossey-Bass.
[24] Moyer, P. S., Bolyard, J.J., & Spikell, M.A.
(2002). What are virtual manipulatives?
Teaching Children Mathematics, 8(6), 372377.
[25] Moyer, P.S., Salkind, G., &Bolyard, J.J.
(2008). Virtual manipulatives used by K-8
teachers
for
mathematics
instruction:
Considering mathematical, cognitive, and
pedagogical fidelity. Contemporary Issues in
Technology and Teacher Education, 8(3), 202218.
[26] 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.
[27] Schoenfeld, A. H. (1987). Cognitive science
and mathematics education. Hillsdale, NJ:
Erlbaum.
[28] Shih,
Wen-Chung,
Tseng,Shian-Shyong,
Yang,Che-Ching and Liang, Tyne, (2011),
Time-Quality Tradeoff of Waiting Strategies
ISBN: 978-960-474-356-8
248