Computers & Education 46 (2006) 105–121
www.elsevier.com/locate/compedu
Development and evaluation of multimedia whiteboard
system for improving mathematical problem solving q
Wu-Yuin Hwang
a,*
, Nian-Shing Chen b, Rueng-Lueng Hsu
c
a
c
Network Learning Technology Institute, Graduate School of Network Learning Technology,
National Central University, Jung-li, Taoyuan 300, Taiwan
b
Department of Information Management, National Sun Yat-Sen University, Taiwan
Department of Information Management, National Central University, Jung-li, Taoyuan 300, Taiwan
Received 24 February 2004; accepted 24 May 2004
Abstract
This study developed a web-based multimedia whiteboard system to help students learning with mathematical problem solving. The purpose is to promote a new online mathematical learning model that students not only use electronic whiteboard to write down their mathematical problem solving solutions but
also use voice recording tool to give oral explanations about their thinking behind the solutions. To cultivate studentsÕ critical thinking capability and encourage collaborative peer learning, the new learning model
also requests students to criticize othersÕ solutions and reply to othersÕ arguments. With the multimedia supporting tools, students can communicate easily with each other about what they think and how they solve
mathematical problems. We have conducted an experiment with sixth grade primary school students for
evaluation. After the experiment, a questionnaire about studentsÕ attitude toward the multimedia whiteboard system for math learning was then held. The results show that students were satisfied with the use
of the multimedia whiteboard system for helping them with learning fractional division. Most students were
interested in studying mathematics with the multimedia whiteboard system and thought this tool is particularly useful for doing collaborative learning. After analyzing the recorded solving processes and discussions content of students, we found that the performance of female students was superior to male
q
The authors thank the National Science Council and Ministry of Education of the Republic of China for financially
supporting this research under NSC91-MOE-S-110-002-X3.
*
Corresponding author. Tel.: +886 3 4227151x4808; fax: +886 03 4237529.
E-mail address: [email protected] (W.-Y. Hwang).
0360-1315/$ - see front matter 2004 Published by Elsevier Ltd.
doi:10.1016/j.compedu.2004.05.005
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students in communications and mathematical problem solving. Additionally, students with higher final
exam grades had better mathematical abilities for doing critiques, arguments and communications.
2004 Published by Elsevier Ltd.
Keywords: Mathematical problem solving; Multimedia whiteboard; Oral explanation; Fractional division
1. Introduction
The problem of mathematics learning is always a major concern for most teachers. While students cannot thoroughly realize the characteristics and meanings of math symbols, it is unreasonable to ask students to simulate or recite their arithmetic calculations. For deep learning, students
need to learn how to make conjectures and provide explanations for their solutions not just learn
how to calculate. Therefore, there is no routine to enable students to solve math problem automatically. Polya (1973) proposed that an effective mathematical solution has four steps: understanding the problem, devising a plan, carrying out the plan, and looking back. However, this
proposed mechanism is only concerned with an individualÕs effort. For web-based learning environments, collaborative learning is an effective approach for improving studentsÕ math problem
solving. However, it is not easy to present and construct mathematical symbols and write down
the process of arithmetic calculation in web-based environment. To make students construct the
process of solving math problems easily and collaboratively in a web-based system, a multimedia
whiteboard system is required to provide students writing down not only math symbols but also
arithmetic calculation with oral explanation. In addition, all students can iterate the process of
solving a math problem collaboratively until a good solution has been reached. The electronic
whiteboard tool applied in this approach is different from the traditional classroom. This approach is designed to stimulate students as well as teachers to collaborate in solving math problems. However, in the traditional classroom, only teachers and few students have the chance to
employ the non-electronic whiteboard.
As information technology continues to progress, teaching mathematic with multimedia is
becoming a new way of instruction (Najjar, 2001). Nevertheless, before this new way of instruction is feasible and practicable, a challenge should be taken into account. The challenge is how
mathematical symbols and processes of writing solutions can be easily expressed on the web. Writing mathematical fractions is not an easy task on the Internet (Wu & Wu, 2002). According to the
systems currently available on the Internet to show mathematical symbols on the web, the symbols must be translated into image format, like GIF or JPG and saved as separated image files.
However it is difficult to align every calculation in the image files, besides those files canÕt be easily
modified and the downloading time is usually time-consuming. Thus, another approach called
electronic whiteboard was proposed to provide an alternative solution for mathematical symbol
writing on the Internet (Jonassen, 1996; Huang, Chen, Ge, Cai, & Wang, 2002). In our approach,
some graphics functions and a virtual pen are supported for easily drawing geometric blocks and
writing mathematical equations. These geometric blocks and equations are stored in vector format instead of image so that it is easy to download and modify on the Internet. Moreover, the
proposed electronic whiteboard is more flexible and accessible to teachers and students not like
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the traditional non-electronic whiteboard, which is mainly used by teachers. In our proposed approach, every student has access to an electronic whiteboard to write down the whole process of a
math problem solving using a virtual pen; or even to record oral explanation of his/her idea and
thinking behind the solution using a voice recording tool.
In traditional teaching, to assess whether students have understood a mathematical problem is
based on whether they could describe the correct arithmetic procedure. However, it is not enough
to evaluate studentsÕ math concepts and abilities of solving math problems merely depending on
their writing. Some oral interpretation and explanation should be considered from multiple
assessment points of view. In our approach, the multimedia whiteboard system provides students
both writing down procedures and recording oral explanations during students engaging in math
problems solving.
Accordingly, the objective of this study is to develop a multimedia whiteboard system and evaluate studentsÕ acceptance and satisfactory use of the system for learning mathematics. In addition,
studentsÕ learning performance and their ability of mathematic problem solving are also
investigated.
2. Literature review
2.1. Multimedia-enabled learning
To make a popular and efficient multimedia system, the general principle for user interface design should be examined (Mayhew, 1992; Smith & Mosier, 1986). Additionally understanding the
way of how people thinking, learning and realizing is also one important factor in designing userfriendly interface for multimedia systems (Najjar, 1997).
Mayhew (1992) and Smith and Mosier (1986) proposed that interface design considerations
could be grouped under e psychology, computer science, graphical design and curriculum design.
Whereas Najjar (2001) suggested the focus should be on improving students learning experiences,
while constructing the multimedia system. Clark and Mayer (2003) combined those cognitive
learning theory and proposed the following principles:
Multimedia: presentations using images and verbal expression simultaneously are clearer than
just verbal expression is selected. According to cognitive theory and results from experiments,
information consisting of images and verbal data (oral or written data or both) can produce
better learning performances. Presentations using multimedia can encourage students to establish linkage between words and graphics, resulting students with stronger willingness to learn.
Formation: the addition of oral explanation with text presentation improves the learner experiences. However, overloading with visual material should be avoided. When reading text students tend to concentrate on complicated words rather than the whole information. Audio
delivery ensures sequential rate of information processing is maintained. This is essential when
the quantity of information is large.
Teachers have used the traditional chalkboard to illustrate the mathematical thinking process.
There are limits to available space that requires frequent erasure. Earlier examples must be
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methodically reproduced if required for revision work. Physical constraints limit the use of the
traditional chalkboard by students. The electronic whiteboard tool enables concurrent use of
teachers and students in conjunction with linked audio explanation. Furthermore all actions
are recorded and can be selectively replayed for remedial or reversionary work.
2.2. Mathematics assessment
The National Assessment of Educational Progress (NAEP) Mathematics Framework is the set
of specifications used to develop the 1990, 1992, 1996, 2000, and 2003 assessments. It describes the
skills and content measured in the assessment. The framework was developed by the National
Assessment Governing Board (NAGB), such that both mathematical abilities and mathematical
power can be measured.
Mathematical abilities include conceptual understanding, procedural knowledge and problem
solving. Conceptual understanding can be regarded as evaluating whether students are ‘‘knowing
what’’, while procedural knowledge is to evaluate whether students are ‘‘knowing how’’. Mathematical power is characterized as a studentÕs overall ability to gather and use mathematical knowledge through exploring, conjecturing, and reasoning logically; solving non-routine problems;
communicating about and through mathematics; and connecting mathematical ideas in one context with mathematical ideas in another context or with ideas from another discipline in the same
or related contexts.
The new assessment items focus on mathematical power by emphasizing reasoning and communication by providing students with opportunities to connect their learning across mathematical content strands. These connections are addressed through individual item designed to tap
more than one content strand or more than one ability, as well as across items throughout item
clusters.
In 1990, NAEP introduced short-answer open-ended items as a means of assessing mathematical communication. The extended open-ended items included in the 1992 assessment further required students to communicate their ideas and to demonstrate the reasoning they used for
solving problems.
Many teachers have an impression that most students are ‘‘not used and able to think’’. In fact,
the reason might be that teachers impart too many topics in one lecture such that students have
no ample time to think about what the teacher imparts. The issue here is how teachers can effectively lead students thinking. Teachers need to help students to discuss, communicate and think
through brainstorming to achieve good math learning. According to VygotskyÕs (1967) point,
higher order of psychology process is being internalized from external social activities. The whole
psychological development structure is started from external social activities and terminated at
individualÕs internal activities. The social activities here, especially interpersonal conversation,
self-examination and thinking development, facilitate oneÕs internalization process through interpersonal conversation. Furthermore, Rorty (1979) also contended that a person will make conviction after contact with environment, but it has to become fact through defense, and then be the
knowledge.
The traditional role of a teacher is to guide students to learn in proper sequences according to
textbooks, moreover to provide exercises to ensure studentÕs familiarization. However, in recent
years, math education advocates the teaching should emphasize: searching for solutions instead
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of memorizing procedures, exploring models instead of memorizing formula, assuming solving
routes instead of exercising only. PolyaÕs math solution model is concerned with individual solving
processes including problem understanding, planning solution strategy, executing solution strategy and looking back. In this research, we extend PolyaÕs model by introducing peerÕs ‘‘comment’’
and ‘‘feedback’’ to enhance math problem solving. Students are requested to ask and comment
othersÕ solutions using the voice recording and hand writing tools provided by the multimedia
whiteboard system. These comments and feedbacks among peers are considered as social activities. Besides individual ability, peerÕs opinion and interaction are also very important to math
problem solving.
3. Research design and implementation
In order to understand whether mathematics learning can be improved by using the multimedia
whiteboard system, an experiment on the topic of fractional division learning was conducted at
Po-I elementary school. The language used for speaking and writing in the class is Chinese. Students were classified into high and low achievement groups based on their final grades in the
exam. After the experiment, a test was examined to see if the two groups yielded significant differences in the quantity of communications by using the multimedia whiteboard system. The
quantity of communications represents the involvement and performance of studentsÕ calculations, comments and feedbacks on learning fractional division which can be used as an indication
of studentÕs capability of problem solving. Some researches have also shown that students who
can explain what they have learned according to their understanding produce superior performances (Chi, 2000; Chi, Bassok, Lewis, Reimann, & Glasser, 1989, 1994).
3.1. Research design
Our research design for peersÕ communications divided into two phases, the first phase was for
peer comment and defending each othersÕ solutions; the second phase was for students to justify a
correct solution from three possible solutions to a math problem and explain why they thought it
was right (Table 1).
Phase I: All math problems solving and assessment processes such as calculations, critiques and
refutations made by students were automatically recoded. Here ‘‘Critiques’’ are comments that
students made about othersÕ solutions by employing the multimedia whiteboard system. Whereas
Table 1
Design activities of peerÕs ‘‘Comment’’ and ‘‘Feedback’’
Phase
Activities
Characteristic
I
Critique and refutation
II
Judgment and explanation
Mathematics
Non-mathematics
Correct judgment
Correct explanation
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‘‘refutations’’ are responses that students replied to othersÕ comments. The content of both critiques and refutations are further classified as Mathematic and Non-mathematic. Mathematic
means the content is related to math problem solving, while Non-mathematic is unrelated. Students can be well facilitated to deduce the correct solving process and solution through these kinds
of communications once the content of critiques and refutations are containing much mathematical meaning.
Phase II: Students were given some math problems, and each problem was provided with several different solutions but only one was correct and others were wrong. Every student was asked
to choose a right answer and used the multimedia whiteboard system to explain why he/she
thought it was right and how it worked. ‘‘Correctly justified’’ means a student can correctly
judge the right answer by doing a correct calculation process, meanwhile ‘‘correctly explained’’
means a student can also explain the reason why he/she thinks the solution should be a correct
one.
3.1.1. Measurement
The 25 items questionnaire for investigating studentsÕ perceived acceptance of using the multimedia whiteboard system is based on the technology acceptance model (Davis, 1989). Also the
perceived satisfaction and perceived improvement in math learning with the multimedia whiteboard system were surveyed. The questionnaire was originally in Chinese. A five-point Likert-type
scale is used. In addition, the content of studentsÕ comments and feedbacks are classified and
quantified into different classes. The collected data will be used to investigate the relationships
among learning achievement and gender through statistical analysis.
3.1.2. Subjects
The subjects in the experiment were 38 sixth grade elementary school students. The learning
topic was mathematical fraction division, and the experiment period was about one semester. Students were taking 1.5 h classes for fractional division learning using the multimedia whiteboard
system every week. Students were first giving two weeks tutorials to learn how to use the multimedia whiteboard system. After that, math problem solving activities were then conducted. During the experiment, instructors gave one math problem every week and asked students to solve it
with the multimedia whiteboard system.
After the experiment, students who were in the top 27% of achievement were classified as the
higher achievement group, and those who were in the bottom 27% of achievement were classified
as the lower achievement group.
3.2. System implementation
LetÕs first describe the user interface design and screen layout of the system. As shown in Fig. 1,
there are three sub windows for the screen layout; the upper window is for the description of a
math problem in text and its multimedia descriptions in image, electronic whiteboard and voice.
The left bottom window is for the list of answers and discussion threads proposed and interacted
by students to this math problem. The corresponding content of each answer/discussion when
being clicked will appear in the right bottom window. Each answer/discussion may include text,
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Fig. 1. UI of multimedia discussion board.
Fig. 2. UI of voice recording.
image, voice and electronic whiteboard. The user interface of voice recording tool is shown in Fig.
2 and the electronic whiteboard is shown in Fig. 3(a).
The electronic whiteboard provides some basic drawing tools and editing functions to allow
users to write down and modify mathematical calculation processes and description. The drawing
tools include line, circle, rectangle and text; the editing functions include copy, paste, cut, move,
undo and redo. The two main purposes for the designed multimedia whiteboard system are to
make studentsÕ collaborative math problem solving much easier, and to support teachers making
comments and suggestions to studentsÕ works more efficient. A studentÕs original calculation process will be automatically loaded into the teacherÕs electronic whiteboard and can be commented as
easily by using different color pens with oral description as shown in Fig. 3(b) and (c). The whole
content including studentsÕ collaborative solving processes and teachersÕ comments/suggestions
can be completely recorded by the system for later reuse and analysis.
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Fig. 3. (a) Student A used electronic whiteboard to write down his solution procedure to this fractional division with
oral narratives. (Oral narratives: eight and one third divided by two fifths, the equation can be written as one hundred
thirty-five fifteenths divided by six fifteenths. The quotient is twenty-two and the remainder is three fifteenths. This is my
answer.) (b) Student B made a comment on one calculation error. (Oral narratives: eight and one third is equal to one
hundred twenty-five fifteenths, not one hundred thirty-five fifteenths. Please correct the error.) (c) Student A gave his
feedback to correct the error. (Oral narratives: Thanks. I have corrected the error by changing one hundred thirty-five
to one hundred twenty-five. And my final answer is twenty and five-fifteenths.)
4. Analysis
4.1. Reliability and validity
The questionnaires were sent to 38 sixth-grade elementary school students in the late period of
the experiment. Thirty-six valid questionnaires were received and used for the analysis. Table 2
shows the coefficient value of Cronbach a of the questionnaire data.
In Table 2, the reliability of usefulness and easy of use, satisfaction on mathematical problem
solving and peerÕs communication in the questionnaire data are all above 0.81, the total Cronbach
a coefficient is 0.95, this implies the questionnaire data have a high reliability.
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113
Table 2
Questionnaire Cronbach a coefficient value
Order
Dimensions
Cronbach a
1
2
Perceived usefulness and ease of use
Satisfaction on mathematical problem solving and peerÕs communication
0.85
0.93
Total Cronbach a
0.95
To develop a valid questionnaire, three experts in multimedia system design, software development and mathematics teaching collaboratively constructed the questionnaire. Some erratum, unclear meaning and unsuitable items were deleted and modified several times before the
questionnaire was ready for this research. Tables 3 and 4 show the results for the two dimensions
(translated from Chinese).
Based on the data collected from the questionnaire, the researchers tried to analyze some
meaningful information and findings from the educational point of view. The results are as
follows:
1. Survey on perceived usefulness and ease-of-use. According to the statistics results shown in
Table 3, the averages of all questionnaire items are higher than 4.2. These are evidences to
show that most subjects agreed the functions and user-interface of the multimedia whiteboard
system were useful and easy to use. Most users also responded positively that the designed
multimedia whiteboard system was very helpful for them to solve math problems. Meanwhile,
majority of subjects agreed that they could become skillful at writing mathematical symbols,
calculation procedures and oral explanations using the tools provided by the multimedia
whiteboard system. However, many subjects also expressed that they were not used to listening to their own voices online.
2. Survey on satisfaction on mathematical problem solving and peerÕs communication. For the satisfaction survey, Table 4 reveals that all average values are higher than 4.3, which implies
most subjects agreed that the multimedia whiteboard system could satisfy the needs of math
problem solving and peerÕs communication. The average values of items 2, 4, 5, 6, 12, 13,
and 14 are all higher than 4.5, which show that the multimedia whiteboard system could help
students to easily express logic reasoning and clearly explain solving processes for their math
solutions. The system can also support peers to do suggestions or comments on each otherÕs
solutions. Based on studentsÕ higher agreements on these items, we are confident that the system can improve studentsÕ mathematical problem solving. Many of them also expressed
higher interests in using the multimedia whiteboard system for study with their classmates
after class.
In the questionnaire, there was one open-ended question to allow students to express their opinions about using the multimedia whiteboard system for math problem solving. The results show
that most of the students are positive and considering the system is very convenient and helpful to
their learning. However, some students reported the disadvantage is the inconvenience of using
the mouse to write down mathematical symbols. The following are some of the critical comments
made by students (originally in Chinese).
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Table 3
Perceived usefulness and ease of use
Item
1
2
3
4
5
6
7
8
9
10
11
Using the multimedia
whiteboard system can improve
the quality of my mathematical
problem solving
Using the multimedia
whiteboard system can increase
the efficiency of my mathematical
problem solving
Using the multimedia
whiteboard system can enhance
the effectiveness of my
mathematical problem solving
Using the multimedia
whiteboard system is easy to get
whatever information I want
Using the multimedia
whiteboard system is easy to me
I can easily become skillful at
using the multimedia whiteboard
system
I can easily describe solution
procedures and share learning
experiences with peers using the
multimedia whiteboard system
I can easily use the electronic
whiteboard function provided in
the multimedia whiteboard
system
I can easily use the voice recorder
function provided in the
multimedia whiteboard system
It is easy and convenient for me
to use the electronic whiteboard
to write down mathematical
symbols for equations
It is easy and convenient for me
to use the drawing and editing
functions of the electronic
whiteboard
Strongly agree
Agree
Unsure
Disagree
Strongly disagree
Average
5
4
3
2
1
25
13
0
0
0
4.66
17
16
5
0
0
4.32
20
16
2
0
0
4.47
27
7
4
0
0
4.61
18
11
9
0
0
4.24
23
11
3
1
0
4.47
25
12
1
0
0
4.63
31
4
3
0
0
4.74
29
5
4
0
0
4.66
24
11
3
0
0
4.55
27
9
2
0
0
4.66
Students A: I am interested in reviewing what I have learned at the school. However, data
downloading is slow due to slow dial-up connection at home; so I can watch the whiteboard content but I cannot listen to and record oral explanations.
W.-Y. Hwang et al. / Computers & Education 46 (2006) 105–121
115
Table 4
Satisfaction on mathematical problem solving and peerÕs communication
Item
1 The electronic whiteboard can help me
to express my mathematic calculation
clearly
2 The voice recording function can help
me to express my methods and ideas in
solving math problems clearly
3 The overall functions in the
multimedia whiteboard system can
help me to express mathematical
solving processes clearly
4 I can grasp various math solution
methods through studying othersÕ
solving processes in the multimedia
whiteboard system
5 I can get helpful comments about my
math solutions from peers in the
multimedia whiteboard system
6 I can get much help for constructing
my math solutions through
cooperating with other classmates in
the multimedia whiteboard system
7 It is helpful and effective for me to use
the electronic whiteboard to mark out
classmatesÕ solution mistakes
8 Replaying mathematical solving
processes in the electronic whiteboard
can help me to understand the solution
steps clearly
9 Recording my oral explanation can
help my understanding mathematical
solving process and concept
10 I will compare and reflect my solving
process with others
11 The electronic whiteboard can help me
to find out the right solving processes
12 It is helpful to math problem solving
using voice recording to explain my
solving process
13 It is helpful to math problem solving
using voice playback to listen othersÕ
oral explanation about their solving
methods
14 It is helpful to math problem solving
using electronic whiteboard to write
down my calculation procedures
Strongly agree Agree Unsure Disagree Strongly disagree Average
5
4
3
2
1
17
17
3
1
0
4.32
24
10
4
0
0
4.53
21
13
4
0
0
4.45
27
8
3
0
0
4.63
22
14
2
0
0
4.53
22
13
3
0
0
4.50
18
16
4
0
0
4.37
21
14
3
0
0
4.47
23
11
4
0
0
4.50
20
13
5
0
0
4.39
18
17
2
1
0
4.37
24
12
2
0
0
4.58
26
9
2
1
0
4.58
23
13
2
0
0
4.55
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Student B: The multimedia whiteboard system is just amusing. It is easy to record oral explanation such that others can easily understand the thinking of your solutions.
Student C: There are something good and something bad with the multimedia whiteboard system. The good thing is that itÕs convenient to get information and communicate with others; the
bad thing is itÕs hard to write words; my written words in the multimedia whiteboard system are
always unsightly. By the way, it is my personal problem as the mouse is in bad condition, so my
written word is ugly.
Student D: EverybodyÕs communication in the multimedia whiteboard system is good. We can
easily pick up othersÕ thinking and methods by listening to their oral explanations. So I would say
the multimedia whiteboard system is really very ‘‘wonderful’’.
4.2. Analysis of gender and learning effect
The results stated above are based on subjectsÕ subjective viewpoints; yet objective evaluation
on learning performance was needed. In order to investigate whether a significant difference in the
quantity of communications using the multimedia whiteboard system with respect to achievement
and gender, the MANOVA method was employed to analyze the data. The results are shown in
Table 5.
From MANOVA analytical results, it was found that with respect to correct explanations in
the multimedia whiteboard system, female students perform better in explaining the process of
mathematical problem solving than male students. Specifically, males perform worse than
females in the sixth grade on oral explanation for mathematical problem solving. It is worth
for further exploration about this interesting phenomenon in sixth grade elementary school
students.
Table 5
MANOVA analyses
Source
Activities
Measure
Mean square
F
Significance
Gender
Calculation
Critique
Correctness
Mathematics
Non-mathematics
Mathematics
Non-mathematics
Correct judgment
Correct explanation
6.664
10.152
22.465
1.254
1.273
4.677
15.646
2.188
2.447
2.995
0.851
0.549
3.122
6.496
0.149
0.128
0.093
0.363
0.464
0.087
0.016*
Correctness
Mathematics
Non-mathematics
Mathematics
Non-mathematics
Correct judgment
Correct explanation
1.178
28.838
14.795
0.579
4.677
12.103
12.720
0.387
6.952
1.972
0.393
2.018
8.079
5.282
0.538
0.013*
0.170
0.535
0.165
0.008*
0.028*
Refutation
Judgment and explanation
Achievement group
Calculation
Critique
Refutation
Judgment and explanation
*
Significance < 0.05.
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117
With respect to achievement, students in the higher achievement group perform better than
those in the lower achievement group in the three factors ‘‘mathematical critique’’, ‘‘correct
judgment’’ and ‘‘correct explanation’’ in the multimedia whiteboard system. That is students
in the higher achievement group are better at replying to othersÕ comments about their math
solving processes. Further, in the ‘‘correct judgment’’ and ‘‘correct explanation’’, the higher
achievement students perform better than those in the lower achievement group. This reveals
that students with higher achievement can easily judge which solution is correct and clearly
explain the reasons.
4.3. Qualitative analysis
During the course, teachers not only asked students to write solutions by themselves but also
encouraged them to comment on othersÕ solutions. Through this kind of iterative communication,
many correct responses to othersÕ comments or queries were derived. Table 6 shows some of
examples about mathematical critiques and refutations.
In addition to critique and refutation are useful for improving studentsÕ learning, oral communication is another important factor for examining performance in this study. Through employing
the electronic whiteboard and voice recording tools in conducting problem solving and oral explanation, it was helpful for students to clarify their thoughts of mathematical problem solving and
share their ideas with others. With ease-of-use of the system and teachersÕ encouragement, most
students were willing to use the system to describe their ideas and solutions clearly and thoroughly. Some samples are shown in Table 7.
Through this kind of solving process and oral explanation described above, teachers are more
likely to know whether students really understand the fractional division concept. However, we
also found the following phenomena in learning fractional division, which may be interesting
and helpful to mathematics teaching.
Table 6
Critiques and refutations among peers
Action
Date
Q1
Mother has 3 l of milk. If the volume of a bottle is 2/5 liter, how many bottles can be
filled up? How many liters are left over?
2
2
4/15
5 ¼ 0:4 3 0:4 ¼ 7 . . . 0:2 (The student orally explained that 5 can be
converted into 0.4. The answer was that 7 cups can be filled up and 0.2 l
was left over)
4/15
Why is 25 equal to 0:4?
4
4
4/15
Because 25 equals 10
; 10
equals 0:4, so 25 equals 0:4
Student AÕs solution
Student BÕs critique
Student AÕs refutation
Q2
Student AÕs solution
Student BÕs critique
Student AÕs refutation
Summary of communication in the multimedia whiteboard system
To pour 10 l of water to the bottle with 1 25 l capacity, how many bottles can be filled up?
How many liters are left over?
7
50
7
1
4/24
10 1 25 ¼ 50
5 5
5 5 ¼ 7 . . . 5 (Here the student incorrectly given an
oral explanation of the fraction number, 7 and 15)
4/24
You made a mistake in saying 7 remainder 15 as 7 and 15, please correct it
5/22
Thanks for telling me this mistake; I have corrected my oral explanations
of 7 remainder 15 instead of 7 and 15
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Table 7
StudentsÕ oral description
Student
Solution (electronic whiteboard)
Q1
To divide 8 pieces of pizza for 3 persons without remnantsÕ how many pizza pieces can each person
be allotted?
8‚3 = 2. . .2
First 8 pieces divided by 3 persons, every person gets 2
2 3 ¼ 23 2 þ 23 ¼ 2 23
full pieces and 2 pieces remain, and then divide the
Answer is 2 23
remaining 2 pieces for 3 persons, everyone gets 23 Piece. 2
full pieces plus 23 are 2 23 that are my answer 2 23 pieces
Student A
Q2
Student B
Mother has 3 l of milk, if she pours
liters are left over?
3 ¼ 21
7
21
2
1
7 7 ¼ 10 . . . 7
Answer is (1) 10 cups (2) 17 l
Oral description
2
7
l into each cup, how many cups can she fill up? How many
Because the denominator of 27 is 7, I convert 3 to 21
7 to
let them have the same denominator, then calculate
21‚2 = 10. . .1, the remainder 1 does not mean 1 but 17.
So the answer should be (1) She can fill up 10 cups (2)
1
7 l is left over
Most students can easily understand and solve the kind of math problems for ‘‘fractional division with the remainder’’.
When students go to solve the fractional division problem 3 25, they used to convert 25 into 0.4
by employing their past concept in solving decimal division. The situation is different from the
teachersÕ expectations that students should use the concept of fraction to solve the problem.
Therefore, teachers modify the math problem by changing the divisor from 25 to 27, pushing student to use the fraction concept to solve it. More than 30 among 38 students can solve the problem successfully. It means most subjects can easily solve the kind of math problems, shown in
Table 8.
It is hard for most students to really understand and clearly explain the kind of math problems for ‘‘fractional division without the remainder’’.
Most of the subjects can solve the problem but only few of them are able to clearly explain their
solution. Obviously, ‘‘if a student can successfully solve a math problem by arithmetic calculation
that does not mean the student really understands it’’. This highlights the importance of using oral
explanation to improve the evaluation of mathematic problems solving. From studentsÕ solving
Table 8
Most students can easily understand and solve the kind of math problems for ‘‘fractional division with the remainder’’
Question
Mother has 3 l milk, if she pours
2
7 l into a cup, low many cups
can she fill up? How many liter
is left over?
To pour 10 l water to the bottle
with 1 25 l capacity, how many
bottles can be filled up? How
many liters are left over?
Solution summary
2
7
21
7
2
7
10 . . . 17
3 ¼ ¼
Answer: 10 cups remains
Description
1
7
7
1
10 1 25 ¼ 50
5 5 ¼ 7...5
Answer: 7 bottles remains
Number
21
7,
Convert 3 into
and then is divided 32
by 27. The answer 10 and the remainder 17
is obtained
l
Convert 10 into
1
5
21
7
l
50
5,
and then solve it
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119
portfolio shown in Table 10, it shows the diversity of approaches to solve the kind of math problems. Some students used the strategy of setting divisor and dividend to have the same denominator and next divide the numerator of dividend by the numerator of divisor to get the
solution (Table 9, Strategy 1). Other students used the strategy of multiplying dividend by reversed divisor to solve the problem (Table 9, Strategy 2). Most of students employing the strategy
2 cannot clearly explain the algorithm of multiplying reversed divisor, that is to say they do not
understand why dividend that is divided by a fraction can convert to multiply the reciprocal of the
fraction. From the oral explanation, we realized that students using the strategy 2 had taken it just
as a rule (as shown in Table 9). It is worth to emphasize here that some students employ the third
approach to solve the math problem by extending the remainder concept. They first get the
Table 9
Most students cannot really understand and explain the kind of math problems for ‘‘fractional division without the
remainder’’
Strategy type
Summary
10 1 25
¼
Description
50
5
7
5
¼
50
7
¼
7 17
50
5
7
5
Number
50
7?
Why ¼
(most cannot explain it
clearly)
6
125
Why 125
5 15 ¼ 6 ? (most cannot explain
it clearly)
Strategy 1: Set divisors and
dividends to have the same
denominator
6
125
5
8 13 25 ¼ 125
5 15 ¼ 6 ¼ 20 6
Strategy 2: Multiply reversed
divisor to get the solution
5
50
1
10 1 25 ¼ 10
1 7 ¼ 7 ¼ 77
1
2
25
5
125
8 3 5 ¼ 3 2 ¼ 6 ¼ 20 56
Why can it multiply the reversed divisor?
(most cannot explain it clearly)
16
15
Strategy 3: Get the quotient and
the remainder first, and next
divide the remainder by the
divisor
7
1
10 1 25 ¼ 50
5 5 ¼ 7...5
1
7
1
1
1
7þ 7 ¼ 77
55¼7
Divide the remainder 15 by 75 and get 17,
then he answer is 7 þ 17 (the student can
explain it clearly)
First obtain the quotient, and then the
divide the remainder 2 (the two students
can explain it clearly)
1
8 3 ¼ 2 . . . 2 2 3 ¼ 23
2 þ 23 ¼ 2 23
8
12
2
Table 10
Most students cannot give correct oral explanations about the implication of multiply reversed divisor
Action
Date
Question
Mother buys 8 13 kg red beans and 25 kg green beans, how many times are red
beans the weight of green grams?
2
25
5
125
5
5/2
8 13 25 ¼ 25
3 5 ¼ 3 2 ¼ 6 ¼ 20 6
5/2
is multiply reversed divisor?
5/2
As I know, the rule of ‘‘multiplying reversed divisor’’ is
Employed to transfer dividing operations into
multiplying operations; the divisor 25 is reversed to 52. As
for others, I will tell you the reasons later after asking
my father when I go home
Student AÕs solution
Student BÕs critique
Student AÕs refutation
Summary of communication in the multimedia
whiteboard system
[Remarks]: ‘‘Multiply reversed divisor’’ is beyond the scope for the grade. Although there are some students using this
strategy for solution under their parentsÕ guidance, actually they donÕt really understand the principles.
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W.-Y. Hwang et al. / Computers & Education 46 (2006) 105–121
remainder and quotient, and then divisor divides the remainder, finally the solution is obtained
with the summation of the quotient and the output produced by dividing the remainder (Table
9, Strategy 3). This indicates these students can really understand and conduct fractional division
by extending remainder approach.
5. Conclusion
In this paper, we have developed and evaluated a multimedia whiteboard system for helping
students in learning mathematical fractional division. Students can share their solving strategies/solutions with classmates by handwriting calculation processes and oral explanations using
the multimedia whiteboard system. It was found that most students are satisfied with the usefulness and ease of use of the multimedia whiteboard system. Moreover, students have strong desires
to use the multimedia whiteboard system to solve math problems, explain how they solve the
problem and provide helpful suggestions to others. Many versatile solving strategies can be obtained through the iterations of critiques and refutations. The multimedia whiteboard system,
supporting a text discussion board with file attachment, an electronic whiteboard and a voice recorder, has demonstrated a useful tool for learning mathematical fraction problems. Students are
interested in and enjoy the discussion in the multimedia whiteboard system as it allows them to
express their thought through text, images, voice and electronic whiteboard. Peer communication
is one of the important issues that enhance studentÕs involvement in the solving process. The multimedia whiteboard system provides electronic whiteboard for writing symbols and voice recorder
for oral explanations to facilitate peersÕ interactions and communications, such that students can
easily and effectively discuss math topics with peers; their mathematical abilities are then
enhanced.
It was found that the learning performance of female students is superior to those male students
in the oral explanation of math problem solving. Moreover students in the higher achievement
group perform better in the mathematical abilities of critique, judgments and explanations than
those in the lower achievement group. About the quality of math problem solving, it was found
that some students cannot correctly provide their oral explanation clearly while they were asked
to explain their solving processes. Further qualitative analysis found that some students can do a
correct arithmetic calculation but cannot understand the real mathematical meanings. Obviously,
if a student can successfully solve a math problem by arithmetic calculation that does not mean
the student really understand it. Thus, asking students to further providing oral explanations can
help teachers to assess whether students really understand the meaning of their solutions. This
concludes that the developed multimedia whiteboard system is important and useful for improving students in learning mathematical problem solving.
Acknowledgement
The authors would like to acknowledge the anonymous reviewers for their valuable suggestions
and comments that made a great improvement of this paper. We also appreciate for the teacher
Hsieh-Fen Hong at Po-I elementary school for her assistance in the experiment.
W.-Y. Hwang et al. / Computers & Education 46 (2006) 105–121
121
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