Creative Education
2012. Vol.3, No.7, 1231-1240
Published Online November 2012 in SciRes (
Copyright © 2012 SciRes. 1231
The Effects of Teaching Mathematics Performed with the Help of
CSCM on Conceptual Learning
Ramazan Gürbüz1, Emrullah Erdem1, Selçuk Fırat2
1Department of Elementary Mathematics Education, Faculty of Education, Adiyaman University,
Adiyaman, Turkey
2Department of Computer Education and Instructional Technology, Faculty of Education, Adiyaman University,
Adiyaman, Turkey
Received September 3rd, 2012; revised October 5th, 2012; accepted October 19th, 2012
This paper explores the effect of teaching mathematics performed with the help of Computer-Supported
Concept Maps (CSCM) on the conceptual learning. To achieve this end, CSCM were developed and used
in the process of teaching probability subject. Within the true-experimental research method, a pre- and
post-test control groups study was conducted with 39 seventh graders—20 in experimental group, and 19
in the control group. Each group was taught three times/week, 40 min/session, for 2 weeks. A 12-item in-
strument was used to collect data. After the teaching intervention, the same instrument was re-adminis-
tered to both groups as post-test. The results suggested that students in the experimental group performed
significantly better than those in the control group, in terms of conceptual learning.
Keywords: Teaching Mathematics; Computer-Supported Concept Maps (CSCM); Conceptual Learning;
Cooperative Learning; Probability
The main purpose of education is to teach students the nature
of knowledge based on concepts. Concepts can be defined as
generalised and symbolised aspect of an encountered event or a
learned object. In showing concepts and relationships between
them, Concept Maps (CM) are among very important materials.
CM strategy, which shows how people learn and make sense of
knowledge, was developed as a result of the research project
carried out with students studying in University of Cornell by
Joseph Novak in the year 1974. CM are defined as tools re-
vealing the connections between the concepts in the form of
hierarchical and cross-linking (Willerman & Harg, 1991). CM
strategy is an educational strategy aiding students in compre-
hending subjects, in integrating old and new knowledge, in
improving their perceptional levels, and in increasing their suc-
cess (Heinze-Fry & Novak, 1990). In this sense, it is thought
that CM strategy which has been frequently used recently
(Anderson-Inman & Zeitz, 1993; Jonassen, 1996; Anderson-
Inman, Ditson, & Ditson, 1998; Novak & Cañas, 2006) can be
an effective teaching strategy. But it is more difficult to deter-
mine all the probable relationships in a large content concept
map and to put them on a limited space on a paper (Novak &
Gowin, 1984; Gürbüz, 2006a). For this reason, in recent years,
Computer-Supported Concept Maps (CSCM) that have unlim-
ited space are employed as teaching materials in learning envi-
Computer-Supported Concept Maps (CSCM)
CSCM, which are one of the most effective tools in con-
creteization of abstract concepts, are important teaching materi-
als (Anderson-Inman & Ditson, 1999), because in the CSCM;
concepts, connections, and formulas which have been placed
into the gaps in the map are shown on a virtual platform. In this
way, students have the opportunity to select among these con-
cepts, connections, and formulas (Chang, Sung, & Chen, 2001;
Tsai, Lin, & Yuan, 2001). The importance of CSCM is also
because they give the opportunity to construct knowledge visu-
ally (Gürbüz, 2006a), to construct their concept maps via feed-
backs given in computer environment (Chang et al., 2001) and
to make learners engaged (Novak & Gowin, 1984; Anderson-
Inman & Zeitz, 1993). In addition to these, CSCM provide a lot
of advantages such as being able to save and to print documents,
to make changes if required, to create big maps, to make con-
nections and to zoom (Rautama, 2000). Moreover, practicality,
dynamic linking, digital communication, and digital recording
can be considered to be the other advantages of the CSCM.
The literature mentioned the positive effects of combining
computer software with concept maps in learning environments
(Anderson-Inman & Zeitz, 1993; Anderson-Inman & Horney,
1996; Anderson-Inman & Ditson, 1999; Simone, Schmid, &
McEwen, 2001; Chang, Sung, & Chen, 2002; Baki & Mandacı-
Şahin, 2004; Kwon & Cifuentes, 2009; Chiu & Hsiao, 2010;
Huang et al., 2012). For example, Anderson-Inman & Horney
(1996) indicated that computer-based visualization made learn-
ing process more accessible to students, and it alleviates the
frustration felt by students during the process of constructing
and revising concept maps using paper and pencil. Anderson-
Inman & Ditson (1999) stated that CSCM ensured the con-
creteization of abstract concepts and thus they could be used in
learning environments effectively. In another study, Kwon and
Cifuentes (2009) pointed out that the computer-based concept
mapping was facilitative of knowledge construction.
The cooperative goal structure is also an important factor in
order for enhancing the effectiveness of CSCM strategy. This is
because the cooperative learning environment requires students
to work in small, mixed-ability learning groups (Slavin, 1987).
The cooperative learning approach increases students’ motiva-
tion to learn mathematics while it also enables them to have fun
doing that (Johnson & Johnson, 1989) and it helps the students
focus on the subject or task by enabling them to work with each
other and by providing a comfortable environment to work in
(Slavin, 1996). In conjunction with increasing use of computer
technology, this cooperative learning approach can be per-
formed in computer-supported learning environments more
effectively. According to Ledesma (2010), computer technol-
ogy makes it possible for the students using it to develop skills
that are not developed by other students using pencil and paper
and blackboard. In this sense, Matin (2012) points out that
computer-supported learning environments will engage stu-
dents, give positive interdependence, face-to-face interaction,
and will help them develop interpersonal skills and individual
accountability in better understanding, critical thinking and
judgment. Thus, more effective learning can be ensured by
employing the CSCM strategy together with cooperative groups.
Recent studies also discussed the positive effects of the in-
structtions conducted in cooperative groups with the CSCM
(Stoyanova & Kommers, 2002; Brown, 2003; Kwon & Cifuen-
tes, 2009).
Literature Review Regarding CSCM
When searching literature, it is possible to encounter with a
great deal of studies mentioning positive effects of CSCM. For
example, in their study, Anderson-Inman and Zeitz (1993) con-
cluded that computer-based concept mapping using Inspiration
TM encouraged students to revise or change their maps more
(compared to the maps they drew using paper and pencil) and
fostered knowledge representation and construction. In another
work, Sturm and Rankin-Erickson (2002) examined the effects
of two forms of concept mapping; hand-drawn and computer-
generated, on the descriptive essay writing of middle level stu-
dents with learning disabilities. Results showed that students’
hand-written descriptive essays and those produced in com-
puter-mapping conditions demonstrated significant increases. It
was also found that students’ attitudes towards writing were
significantly more positive in computer-mapping condition
compared to no-mapping and hand-mapping conditions. Baki
and Mandacı-Şahin (2004) carried out a study to determine the
pre-service elementary teachers’ misconceptions about the sub-
ject “set” through CSCM. It was suggested that the students’
concept mapping process using the Inspiration® package pro-
gramme could be used as an effective assessment technique.
Royer and Royer (2004) investigated the difference between
hand drawn and computer generated concept mapping with 9th
and 10th graders. They found that the group using the com-
puters created more complex maps than the others did. Also,
they theorised that computers enabled students to communicate
more clearly, to add and revise concept maps more easily, and
to discover relationships between sub-concepts more readily.
Yavuz (2005) examined the effectiveness of conceptual change
instruction accompanied with demonstration and computer
assisted concept mapping on seventh grade students’ under-
standing matter concepts. The results indicated that this teach-
ing process provided a better acquisition of scientific concep-
tions related to matter concepts and produced more positive
attitudes toward science as a school subject than traditionally
designed science instructions. Kwon and Cifuentes (2009) in-
vestigated the comparative effects of individually-constructed
and collaboratively-constructed computer-based concept map-
ping on middle school science concept learning. They found
that computer-based concept mapping facilitated knowledge
construction and the students had a deeper understanding by
working collaboratively rather than by working individually.
Chiu and Hsiao (2010) studied how elementary school students
generated concept maps in computer-supported collaborative
learning. They found that almost 70% of the collaborative
groups were classified as passive or reticent and frequently off-
task. These student groups were poorly functioning collabora-
tive groups and produced poor quality discourses and products.
Also, it was concluded that there was a great need for methods
such as training or intervening approaches that could enhance
the interaction and improve the quality of the discourse in the
computer-mediated collaboration for elementary school stu-
dents. Huang et al., (2012) explored the effect of multidimen-
sional concept mapping instruction on students’ learning per-
formance in a web-based computer course. The 103 fourth
graders were divided into three groups: multidimensional con-
cept map (MCM) instruction group, Novak concept map (NCM)
instruction group, and traditional textbook (TT) instruction
group. The experimental results suggested that subjects in the
MCM group performed significantly better than those in the
NCM group which in turn performed significantly better than
those in the TT group.
As understood from the reviewed literature, CSCM ensure
learners perform more effective learnings especially when com-
pared to no-mapping and/or hand-mapping conditions. There-
fore, in the current study, we examined the effect of teaching
mathematics performed with the help of CSCM on the con-
ceptual learning.
Within true-experimental research method, this study was
conducted with a total of 39 seventh-grade students from an
elementary school in Turkey. These students studying in the
same class were divided into two groups according to whether
their school numbers were even or odd (experimental group (E)
= 20 and control group (C) = 19). Students in experimental
group were organized as 2-student cooperative groups (see
Figure 1(b)).
Computer-Supported Concept Maps (CSCM)
The CSCM material was developed by NetBeans’ editor by
using the Java language. While constructing CSCM material,
extra attention was paid to ensure that it included all concepts
on probability subject taught at 7th grade level and that rela-
tionships between these concepts were given in a concise way.
For example, the relationship between “Experiment” and “Re-
sult” is stated as “what is gathered from experiment is a result”.
However, all these concepts and relationships were not given
directly; rather, users were given some cues in the system to
find them by themselves. For instance, some cues such as “it
takes values ranging from 0 to 1” and “it is a concept related to
chance” were given for the space in which “probability” con-
cept was to be placed. If the user puts a wrong concept, link or
Copyright © 2012 SciRes.
Figure 1.
Reflections from experimental group.
formula into the system, “you have entered wrong info” feed-
back is shown on the screen and, thus, the first cue that will
lead the user to correct answer is provided. If the user enters
wrong info despite this cue, system gives another cue with a
more explanatory feedback. For example, when the user writes
“Not-Mutually Exclusive Event” upon seeing the cue “the
probability of a randomly selected geometric shape from the
screen to be red or hexagon, is related to … concept”, system
gives the feedback “Congratulations! Correct Answer”. If the
user writes “Mutually Exclusive Event” upon seeing the same
cue, system presents another cue (Cue 2) such as “do these two
incidents have common units or not?” When, however, the user
who cannot find the correct answer despite all these cues, clicks
on the Cue 2 in Figure 2, the system will give another more
detailed explanation to the user such as: “Let us calculate the
probability of a geometric shape selected randomly from the
screen in Figure 2 to be red or square together. After analyzing
the screen, it is possible to see that when the geometric shape is
both red and square, we call this Not-Mutually Exclusive
Event. Therefore, the probability of these two incidents can be
calculated by this formula: P(RUS) = P(R) + P(S) P(RS).
The reason why we subtract P(RS) is that we use red squares
both when calculating red shapes and squares”. This material
improves students’ self-confidence both because it motivates
them and also because it helps create a learning environment in
which they can construct information by themselves. Moreover,
since this material creates an environment in which students
discover the relationships between concepts by themselves, it
gives users opportunity to build new relationships they do not
This material was pilot-studied with 18 seventh-grade stu-
dents who did not participate in the real study and were divided
into nine groups each consisting of two students. By using the
pilot study, the probable deficiencies of the material and the
problems, which could be encountered during the application
process, were determined and necessary corrections were made.
A sample interface from the designed material is illustrated
in Figure 3.
The instrument used (Conceptual Learning Test-CLT) was
composed of 12 questions (some of them were given in Appen-
dix) which were in open-ended format. Some of the questions
were developed by the researchers, and some were developed
with the help of related literature (Fast, 1997; Pratt, 2000;
Baker & Chick, 2007; Nilsson, 2007, 2009; Gürbüz, 2006b;
The validity of the instrument was confirmed by two mathe-
matics educators and two mathematics teachers. Furthermore,
the pilot test was performed with 34 seventh-grade students
who did not participate in the real study. The pilot study re-
vealed that questions on the subject of probability were under-
standable and clear for seventh-grade students. In this study, the
Kuder-Richardson formula 20 (KR-20) reliability coefficient of
the instrument was found to be 0.88.
The CLT was administered to both groups as a pretest before
instructions. Both groups were encouraged to answer all the
questions. A researcher managed the experimental group, while
the maths teacher in the study managed the control group by
conducting the application process simultaneously. The remain-
ing two researchers participated in the study as observers for
experimental and control groups by rotating. Each group was
taught three times/week, 40 min/session, for 4 weeks. After the
instructions, CLT was re-administered to both groups as a post-
In experimental group, some questions related to daily life
such as “what does scoring a goal through a penalty shoot de-
Figure 2.
A sample cue and explanation given to users by the system.
Copyright © 2012 SciRes. 1233
Copyright © 2012 SciRes.
Figure 3.
An interface from CSCM material
pend on?”, “the probability of having rain in Adiyaman is 1/4.
How do you think I got it?” were asked to students. After these
questions had been discussed with students for a while, teach-
ing of probability concepts were performed with materials
(spinners, dice, dart, board etc. (see Figure 1(a)) by an experi-
enced researcher. Then, CSCM was employed in teaching pro-
cess after students had been informed about CM and CSCM. In
this process, the students were requested to run the computer
animations, and they were asked to share their thoughts with
their partners. Moreover, the student groups asked questions to
each other such as “why are you doing that?” and “how did you
get that?” or made statements such as “oh no, that is not right,
because…” and “…but that is wrong, because…” during the
implementations. This process was ensured by devoting con-
siderable class time to solving problems, proposing and justi-
fying alternative solutions, critically evaluating alternative cour-
ses of action, leading to different methods of solving problems.
Students are expected and encouraged to make conjectures,
explain their reasoning, validate their assertions, discuss and
question their own thinking and the thinking of others, and
argue what is mathematically true. Also during this process, the
researcher acted as a counselor, cooperator, and supervisor. As
a result, students became more active, improved their know-
ledge, questioned the knowledge they received, and were able
to explain what they had just learned instead of behaving as
merely passive receivers. In other words, it could be stated that
students in this group learnt subjects the way they wanted. For
instance, they reached solution of any given question by con-
ducting limitless experiments, by using tree diagrams and by
discussing among themselves as they wished.
In the control group, the instructions were performed on a
teacher-centered basis and delivered verbally, according to the
book. The teacher (usual teacher) would note down the ne-
cessary points on the chalkboard (talk-and-chalk type instruc-
tion). While writing on the board, the teacher framed the
important parts using colored chalk. During the process the
students sat in their seats silently and listened to the teacher.
Then, the teacher gave them some time to take notes from the
board. The teacher also asked if they had any questions about
the subject. Meanwhile, he walked around the class and ans-
wered their questions. In brief, 70% - 75% of the probability
subject was composed of only the teacher’s talk. At the end of
the lesson, the teacher asked the students to answer the ques-
tions at the end of the unit. In the control group, questions such
as “suppose that there are some balls numbered from 1 to 8 in a
glass jar. When you close your eyes, mix the balls and choose a
ball in the jar, what is the probability of getting a ball num-
bered with an odd number?” were generated and solved. To
sum up, the experiments were done and the results were
obtained by imagination without using any other materials or
animations. Students in this group learnt based on rules esta-
blished by the teacher. When a student asked teacher a question
while solving a question, the teacher solved it by reminding
students of rules he showed before. For example, when a stu-
dent asked a question such as: “Teacher! Why are we adding
the scores? Should not we multiply them instead?” the teacher
responded: “Let us remember our rule, if two events are
discrete, probability of them are added”.
During the study, most of the students in the control group
wondered how their peers in the experimental group were doing.
Therefore, they frequently checked the computer laboratory to
figure out what was going on there. After hearing from their
friends in the experimental group about the applications there,
they asked researchers/teachers to add them to this group, as
Data Analysis
The effect of teaching with the help of CSCM and of tradi-
tional teaching was investigated by the CLT. Students’ answers
have been classified according to the levels in a Rubric (see
Table 1) developed by Gürbüz (2007, 2010). According to the
scores presented in Table 1, statistical comparisons of concep-
tual learning levels of groups were made. To achieve this end,
the mean scores gathered from the questions in CLT were cal-
culated. Scores gathered were analyzed through SPSS statistical
package program. Data were analyzed using independent sam-
ples t-test, and analysis of covariance (ANCOVA). In fact,
covariance analysis was applied in order to observe any poten-
tial difference between the means of the post-test scores of the
groups. A Bonferroni pairwise comparisons test was used to
determine the direction of differentiation.
Results and Discussion
Descriptive statistical results of the pretest and posttest for
the experimental and control groups are given in Table 2. In
this study, the independent samples t-test was performed to
compare the pretest scores of the groups. The results of the
independent samples t-test showed that no significant differ-
ence was found among the pretest scores of the groups [t(37)
= .487, p = .629]. Therefore, it can be said that both groups had
the same level prior to the instructional process.
In order to compare the effects of the instructional strategies
Table 1.
Criteria used in order to assess CLT.
Levels Score Content Students’ sample responses
Level A
Correct Answer
The explanations which are
accepted as scientifically true
are included in this group
Q1 The area of red section is 9π, of the green section π(52 – 42) = 9π and yellow
section π(4 2 – 32) = 7π. So the probability of hitting yellow section is the lowest
because P(R) = 9/25; P(Y) = 7/25 and P(G) = 9/25
Q3 The sixth baby may be boy or girl. Because, for each child, the proba
ility o
being boy or girl is equal and is 1/2.
Spinner 2
12 3 4 5
Spinner 1
5D D D D S
There are 5 cases of having the same numbers and 20 cases of having different
numbers. Thus, the probability of having different numbers is higher (D:
Different, S: Same).
Level B
Correct Answer
Explanations are true but com-
pared to the correct answers;
some parts are missing, so it
is included in this group.
Q1 π(32) = 9π Red area
π(52 – 42) = 9π Green area
π(42 – 32) = 7π Yellow area
Q2 The probability of choosing a blue shape is 10/30 and of choosing a rectangle
is 6/30. Thus, the probability of a randomly chosen geometric shape to be blue o
rectangle is 10/30 + 6/30 = 16/30.
Level C
Wrong Answer Type
The explanations, which con-
tain partially correct statements
but are connected to the right
reasons or don’t give reasons
are included in this group.
Q1 3. 32 = 27 Red section
3. 12 = 3 Yellow section
3. 12 = 3 Green section
The probability of stopping at red section is the highest
Q2 The probability of chosing a blue shape is 10/30 and of chosing a rectangle is
Level D
Wrong Answer Type
Expressions that contain whol-
ly wrong or irrelevant expla-
nations are in this group.
Q2 Small geometric shapes have higher chance to be chosen
Q3 The sixth baby would more likely be girl, because the first five are all boys.
Q4 Game is a chance. The one in his luck day wins the game
Level E
Uncodeable 1
Incomprehensible explanations
or explanations that have no
connection to the question are
in this group.
Q1 It depends on the ability of the shooter.
Q4 Whoever turns the spinner first, has the highest chance
Q5 Blue because blue balls are placed on the upper part of the basket.
Level F
Unanswered 0
Those that made no expla-
nations and those who wrote
the question itself in the expla-
nation part are in this group.
Q2 What is the probability of a randomly chosen geometric shape to be blue o
rectangl e?
Q3 Their first five children are males
Q6 (123 45) (123 45)
Note: Qa: Some question items used in CLT.
Copyright © 2012 SciRes. 1235
implemented on the groups in the post-test scores using AN-
COVA, the tests of homogeneity within the group regression
slopes were conducted. As a result of the analysis the slopes
were found to be homogenous, as in Group*Pre-test [F(1 - 35)
= .609, p = .440], within the groups. Therefore, a covariance
analysis was applied in order to observe any potential differ-
ence between the means of the posttest scores of the groups.
The result of the one-way ANCOVA is given in Table 3.
As shown in Table 3, the analysis of the posttest score data
indicates significant overall intervention effects, controlling the
pretest [F(1, 36) = 35.946, Partial η = .500, p < .01]. Regarding
the posttest scores, the students in the experimental group bene-
fitted significantly more than those in the control group (Mean
difference = 1.031, p < .01). From the results of the pair-wise
test, it can be stated that the CSCM strategy was more effective
than the traditional teaching methods in terms of improving
conceptual learning. In this study, the effect sizes (partial eta
sequared) were calculated to be .500. It can be stated that the
CSCM strategy had a high effect on the conceptual learning
according to Cohen (1988). These outcomes corroborate the
results of Anderson-Inman & Zeitz (1993), Anderson-Inman et
al., (1998), Simone et al., (2001), Chang et al., (2002), Stoyanova
& Kommers (2002), Brown (2003), Baki & Mandacı-Şahin
(2004), Kwon & Cifuentes (2009) and Huang et al., (2012).
It could be asserted that researcher teacher factor has also
affected this process along with CSCM, because, teacher sel-
dom invited students to present their work to other students and
never discussed or allowed students to share unsuccessful at-
tempts. In the contrast, researcher encouraged students to solve
problems in any way they desired and to discuss with the whole
class their methods as well as their unsuccessful attempts. He
also encouraged solution of problems in alternative ways.
Clearly, teacher and researcher’s students were offered prob-
ability theory of different natures. The nature of probability
theory made available to learn in teacher’s classes was charac-
terized by a domain that deals with final results only, where
ways employed to reach these results are not important, and
problems are solved by simply following rules developed by
experts. In contrast, by participating in the researcher’s classes,
students were exposed to a different nature of probability the-
ory. The nature of probability theory made available to learn in
researcher classes was characterized by a domain that deals
Table 2.
Pretest and posttest scores of the groups.
Pre-test Post-test Estimated post-test*
Group n
Experimental 20 1.57 .66 3.12 .73 3.092.120
Control 19 1.47 .59 2.02 .63 2.060.123
Table 3.
Results of the covariance analysis on post-test scores of the groups.
(Post-test) F df p Partial η2Direction
Overall 35.946 1-36 .000 .500
(Bonferroni) Mean Difference
Experimental (I)
versus Control (J) 1.031 .000 I > J
with final results as well as with ways of reaching these results;
a domain in which examining mistakes is important and con-
structive and could help in achieving correct solutions and un-
derstanding. In parallel with this, Boaler (1997), Even and
Kvatinskt (2010) and Gürbüz, Birgin and Çatlioğlu (2012) also
focused on the fact that teachers and different teaching ap-
proaches adopted by teachers had a significant influence on
students’ learning. The researcher referred to the role of prob-
ability theory on mathematics and in other domain. Also, he put
emphasis on probability theory depending on predictor’s know-
ledge. But, teacher didn’t mention any of these cases. What the
teacher did not address in class was basically not required ex-
plicitly for the examinations and was not included in the text-
books they used, which were closely connected to the examina-
tions. Some related research on teacher anxiety (Black & Wil-
lam, 1998; Ayres, Sawyer, & Dinham, 2004) also showed that
central exams had an effect on learning environments and that
these exams encouraged test-based teaching.
It is thought that providing students’ pre-test and post-test
answers will more clearly show the effect of this intervention.
For this reason, explanations related to some questions in CLT
made by some of the students of experimental group in the pre-
and post-test were examined carefully.
It was found that the students who had mistakes regarding
question Q1 had different justifications for their wrong answers
in pretest. For example, “Green because green is on the narrow
side of dart”, Red has the highest probability since it is on
center”, The probability of targeting on yellow and green col-
ored sections is the lowest because they have smaller radii. As
stated by Kahneman (2003), Gürbüz (2007), Gürbüz, Çatlıoğlu,
Birgin and Erdem (2010) and Gürbüz and Birgin (2012), these
students gave their answers based on their visual intuitions
rather than on their logical reasoning. Some of the students
were found to give non-mathematical answers such as “It de-
pends on the ability of the shooter or It depends on chance,
so, no comment can be made. Such approaches of students
who gave wrong answers to question Q1 are in line with the
student approaches in the studies of Jones, Langrall, Thornton
and Mogill (1997), Gürbüz (2007, 2010), Gürbüz et al., (2010),
Erdem (2011) and Gürbüz and Birgin (2012). In posttest, when
comparing to pretest, the students had more true justifications
regarding question Q1. For example, “The area of red section is
9π, of the green section π(52 42) = 9π and yellow section π(42
32) = 7π. So the probability of hitting yellow section is the low-
est because P(R) = 9/25; P(Y) = 7/25 and P(G) = 9/25”, You
need to find the area each section covers. π(32) = 9π Red area;
π(52 – 42) = 9π Green area and π(42 32) = 7π Yellow area.
It was observed that, in Q2, students’ knowledge related to
mutually exclusive and not-mutually exclusive events were not
sufficient in pretest and thus they gave wrong or incomplete
answers for different reasons. For example, responses such as
The probability of getting blue is 10/30 while getting rectan-
gle is 2/30”, “The probability of getting blue and rectangle is
the same because there is blue color, too, in rectangular
shapes The probability of getting blue and rectangle is 2/10”
were given by the students. On the other hand in posttest, it was
found out that students grasped the difference between discrete
and indiscrete events. However, it cannot be said that all stu-
dents understood that there was an intersection set in indiscrete
events. For example, P(AUB) = P(A) + P(B) – P(AB), the
probability of getting blue is 10/30, the probability of getting
rectangle is 6/30 and the probability of getting both blue and
Copyright © 2012 SciRes.
rectangle is 2/30, so, P(AUB)= 10/30 + 6/30 2/30 = 14/30”,
the probability of getting blue is 10/30, the probability of get-
ting rectangle is 6/30”, P(AUB)= 10/30 + 6/30 = 16/30”.
It was also found out that the students, who made mistake in
Q3 in pretest, also gave different justifications for their wrong
answers. For example, The sixth baby would more likely be boy
because the first five are all boys”, The sixth baby would more
likely be girl because the first five are all boys. Kahneman and
Tversky (1972) stated that for a family with 6 children, it’s
believed that the order of genders will more likely be MFF-
MFM (M: Male; F: Female) instead of MMMMMM or MMM-
FFF. It’s possible to find similar results in the studies by Fast
(1997) and Gürbüz and Birgin (2012). However, it can be sug-
gested that these mistakes disappeared considerably in the post-
test. Students gave more correct answers in post-test such as
the probability of a baby being boy or girl is equal and it is
1/2”, The fact that the first five children are boys does not
affect whether the sixth baby will be a boy or a girl”, The fact
that the first five children are boys does not necessitate the
sixth baby to be a boy, so they are equal.
In Q4, students either gave wrong answers or did not give
answers at all as they confronted different dice. They gave re-
sponses such as “I am not answering that question as this dice
is different from the dice we are familiar with”, “Ali will win as
there are more even numbers”, “Veli will win because the
probability of total score to be 7 or 9 is higher”. Students’ such
kind of misconceptional answers can be argued to have stem-
med from students’ lack of sufficient knowledge in sample
space concept. Within the same context, Baker and Chick
(2007), Bezzina (2004), Chernoff (2009), Fischbein, Nello and
Marino (1991), Gürbüz (2007, 2010), Keren (1984), Nilsson
(2007), Polaki (2002) and Gürbüz and Birgin (2012) showed in
their studies that students’ knowledge about sample space con-
cept played an important role in their answers to questions re-
lated to probability subject. Some students, on the other hand,
gave illogical answers without any mathematical thinking and
gave responses such as “Whoever starts earliest will win”, Ali
will win because 10 steps they have to take is also an even
number”, Whoever is on his/her luck day will win. Polaki
(2002) names this type of thinking as subjective probabilistic
thinking. However, in posttest, most students corrected the
mistakes they made in pretest and gave more logical answers
such as “Ali and Veli has equal chance because cases of getting
even numbers are (2,4) or (2,6); cases of getting odd numbers
are (3,4) or (3,6)”, we can show this on a table (see Figure
4(a)). There are equal numbers of even and odd numbers,
therefore, the chances of winning are equal for both of them.
Question Q5 is the question that most students answered cor-
rectly. Though there were a few wrong answers given, in pre-
test, students showed overall a good performance in answering
it. Explanations related with Q5 given by some of the students
in pretest are as follows: “It is green because green ones are on
the top of the basket”, since the number of green balls in the
basket is higher than others, the probability of getting green is
the highest. Here, since the students focused on the location of
the balls in question Q5, they made some mistakes. This finding
is in line with the findings gathered from studies carried out by
Jones et al., (1997), Gürbüz (2007), Gürbüz et al., (2010), Er-
dem (2011) and Fırat (2011). However, it was seen that stu-
dents gave correct answers to this question in post-test such as
it is green because the number of green balls is the highest”,
Since the number of green balls in the basket is higher than
others, the probability of getting green is the highest.
Numerically, P(G) = 4/9”.
Question Q6 can be argued to be one of the questions for
which the students revealed the highest number of mistakes.
For example, there were responses such as “since there are 2
spinners and 5 numbers, probability is 2/5”, because location
of numbers are different in 1st and 2nd spinner, the probability
of getting different numbers is higher”, the probability of
getting different numbers is high because it is generally impos-
sible to get even numbers. These mistakes stem from the fact
that students perceive these cases as independent, they lack
sufficient knowledge about the concept of sample space. Fis-
chbein et al., (1991), Lecoutre (1992), Batanero and Serrano
(1999), Baker and Chick (2007), Nilsson (2007, 2009), Gürbüz
(2010), Erdem (2011), Fırat (2011) and Gürbüz and Birgin
(2012) reported similar conclusions in parallel with this in their
studies. It was observed that students gave correct answers to
Q6 in posttest. Explanations related with Q6 provided by some
of the students in posttest are as follows: “the probability of
getting different numbers is higher because there are fewer
cases of getting the same numbers [(1,1) or (2,2), (3,3), (4,4),
(5,5)]”, the probability of getting the same numbers is P(S) =
5/25 and the probability of getting different numbers is P(D) =
20/25, so, the probability of getting different numbers is higher
(S: Same, D: Different)”, as can be observed on Figure 4(b),
there are 5 cases of getting the same numbers and 20 cases of
getting different numbers. Thus, the probability of getting
different numbers is higher.
These findings can be summarized that the teaching per-
formed with the help of CSCM showed a positive impact on
conceptual learning of probability. This positive effect is thought
to be provided by CSCM accompanied by a student-centered
learning environment. The findings of the present study suggest
that different teaching approaches and different teachers are
also effective in students’ level of learning the subject.
+4 44 6 6 6
6 6 6 8 8 8
6 6 6 8 8 8
6 6 6 8 8 8
7 7 7 9 9 9
7 7 7 9 9 9
7 7 7 9 9 9
Spinner 2
12 3 4 5
Spinner 1
5D D D D S
Note: D: Different, S: Same.
Figure 4.
Some examples of students’ answers.
Copyright © 2012 SciRes. 1237
General Discussion, Conclusion and Proposals
In this research, we determined the effect of teaching mathe-
matics performed with the help of CSCM on 7th graders’ con-
ceptual learning of probability. From the findings, it may be
suggested that the posttest scores of both groups show a sig-
nificant increase when compared to the pretest results. Thus,
both instructional methods can be argued to improve the stu-
dents. Yet, when the improvements in the groups are compared
it can be said that the intervention in the experimental group
was more effective in terms of conceptual learning. When
monitoring the students in the experimental group, it was ob-
served that they enjoyed the process, were interactive, and had
the opportunity to construct knowledge visually. These effects
of CSCM on learning corresponded with the results of a great
deal of studies (Anderson-Inman & Zeitz, 1993; Anderson-
Inman et al., 1998; Simone et al., 2001; Chang et al., 2002;
Stoyanova & Kommers, 2002; Brown, 2003; Baki & Mandaci-
Şahin, 2004; Kwon & Cifuentes, 2009; Huang et al., 2012).
Contributions of groups’ members to each other indicated
that one of the other factors ensuring this process to be effective
is cooperative learning strategy, because almost all positive
effects of cooperative learning were effectively used during this
process. During the instructions, it was seen that especially
students whose learning motivations were low benefitted more
from the process, through their partners. This result was in ac-
cordance with previous studies (Stoyanova & Kommers, 2002;
Brown, 2003; Kwon & Cifuentes, 2009) that discussed the
positive effects of working with the CSCM in cooperative
groups. Moreover, feedbacks given in the system depending on
students’ answers enhanced the effectiveness of CSCM. In their
study, Chang et al. (2001) referred to similar effect of feed-
backs in CSCM environment.
Furthermore, it could be claimed the fact that different
teachers conducted the instructions in each group was also ef-
fective in groups’ different level of learning probability subject.
That teachers decided on which approach to employ during the
process and that they used it effectively influenced students’
learning. Effective guidance by teacher (researcher) in experi-
mental group, opportunities to talk and correct feedback given
by researcher as much as possible, having students use CSCM
effectively in cooperative groups and using dice, spinners and
darts made the process effective. That the researcher followed
the process in order to teach effectively could be explained by
his/her professional competence. From this point of view, it can
be stated that educatories’ professional competence is one of
the other important factors affecting this process. In this sense,
many previous academic studies (Shulman, 1986, 1987; Ball,
1988, 1990; Hill, Rowan, & Ball, 2005; Davis & Simmt, 2006;
Tchoshanov, 2011) highlighted the importance of this compe-
tence in order for performing effective instructions.
Regarding suggestions for further research, it is advisable to
look into what should be done for the students’ adaptation to
work in CSCM. The effect of the CSCM on determining and
remedying misconceptions should be investigated. These types
of CSCM can be used as assessment tools besides teaching
materials. By observing probability teaching process of differ-
ent teachers (both through video-camera and participative ob-
servation), approaches employed by them and their effects on
learning could be compared.
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Appendix: Some Question Items in CLT
In the dartboard, whose radii lengths are shown above, “r
represents red, “g” represents green and “y” represents yellow.
As each shot targets at any yellow, green or blue color, the
probability of targeting at which color is the lowest when a
random shot occurs? Why? Could you use numerical expres-
sions to support your ideas?
In the board shown above, “R” represents red, “B” represents
blue and “Y” represents yellow. What is theprobability of a
randomlychosengeometricshapeto be blueorrectangle? Could
you express your ideas numerically?
Family Gül areexpectingtheirsixthbaby. Their first five chil-
dren are males. Having a male or female baby is more probable
in this case? Why?
Ali and Veli will play a game by using toy cars on a 10 step
long road. Each player will roll the two dice designed such as
(222 333) and (444 666) at the same time. If the sum of the
outcomes is even, Ali will move his toy car one step further. If
the sum of the outcomes is odd, then Veli will do the same. The
one who completes the 10-step long road earlier will win. In
your opinion, who wins the game? Why?
On the balls,
“R” r epres ents re d
“B” repr esents bl u
and “G” re pr esents
reen res
There are 4 green, 3 red and 2 blue balls, in total 9 balls in
this basket. When you close your eyes, mix the balls and
choose a ball in the basket, the probability of getting which
colored ball is the highest? Why? Could you use numerical
expressions to support your ideas?
Spinner 1Spinner 2
Do you think the probability of getting the same numbers or
different numbers is higher when spinners above are turned
together? Why? Could you express your ideas numerically?
Copyright © 2012 SciRes.