1
The Interaction of Problem-Based Learning with “Murder” Strategy,
Educational Background, and Mathematical Prior Knowledge toward
Mathematical Creative Thinking Ability and Disposition of Preservice
Elementary School Teacher
Maulana*
Didi Suryadi**
Utari Sumarmo***
Jarnawi Afgani Dahlan****
Abstract
Similar to creative thinking ability, the development of high order mathematical ability is
demanded in the mathematics learning activity in preservice elementary school teacher study
program (called PGSD). For that purpose, the selection of appropriate approach and strategy
will make the objective of learning activity be achieved. However, only few people who try
to recognize the other factor beside the learning approach/strategy which is possible in giving
contribution in the development of the creative thinking ability, for example the students’
educational background factor (science and non-science) and mathematical prior knowledge
that is acquired before. In addition, affective aspect, which accompanies creative thinking
ability (creative disposition), is a study that is rarely found. This paper is made to give a brief
description about the selection of learning approach and strategy type that is the problem
based learning “MURDER” strategy and the interaction with the educational background and
mathematical prior knowledge toward the enhancement of thinking ability and PGSD
students’ mathematical creative disposition. The research subject consist of three treatment
groups, those are the classes which is given : (1) problem based learning with “MURDER”
strategy, with the learning material that is made from the result of didactical design research,
(2) problem based learning with “MURDER” strategy, and (3) conventional learning.
Keywords: Problem-based learning, “MURDER” strategy, educational background,
mathematical prior knowledge, mathematical creative thinking ability, mathematical creative
thinking disposition.
*
Student of Mathematics Education Doctoral Program, Indonesia University of Education. Postal address:
PGSD UPI Kampus Sumedang, Jalan Mayor Abdurrahman No. 211 Sumedang, West Java, Indonesia. Postal
Code: 45322. Email: maulana@upi.edu
**
Professor of Indonesia University of Education, Department of Mathematics Education. Postal Address: Jalan
Dr. Setiabudhi No. 229 Bandung, West Java, Indonesia. Postal Code: 40154. Email: ddsuryadi1@gmail.com
***
Emerita Professor of Indonesia University of Education, Department of Mathematics Education. Jl. Dr.
Setiabudhi No. 229 Bandung, West Java, Indonesia. Postal Code: 40154. Email: utari.sumarmo@yahoo.co.id
****
Doctor in Mathematics Education, Indonesia University of Education, Department of Mathematics
Education. Postal Address: Jl. Dr. Setiabudhi No. 229 Bandung, West Java, Indonesia. Postal Code: 40154.
Email: afgani_lan@yahoo.com
2
1 Introduction
How to develop high order thinking ability and to make it into significant objective
that must be achieved in learning mathematics is an actual issue in learning mathematics
nowadays. The mathematical high order thinking ability which is non-algorithmic, complex,
involving the autonomy of thinking, often involving uncertainty that needs consideration and
interpretation, involving the various criteria and occasionally trigger the conflict emergence,
and creating open solution, also need the considerable effort in performance (RESNICK,
1987; ARENDS, 2004).
One of the thinking ability that include in high order thinking ability is creative
thinking ability. There are four exhortations about the need of critical thinking ability
development, those are: (1) demand of the era which desires the citizens to be able to find,
select, and use information for the social and state life, (2) every citizen always deals with
various problems and choices so that he/she is demanded to be able to think critically and
creatively, (3) the ability to see anything with different ways in solving the problem, and (4)
critical thinking is an aspect in solving the problem creatively in order that students are able to
compete fairly and able to cooperate with the other nation (WAHAB, 1996; MAULANA,
2007).
Creative thinking is a process of thinking various ideas in dealing the issue or
problem, playing with the ideas or the elements in mind, finding the new relationship or
relevance to see the subject from the new perspective, and creating the new combination from
two or more concepts in mind (EVANS, 1991).
This creative thinking ability is definitely able to develop through mathematics
learning at school or college, which emphasizes the system, structure, concept, principal, and
the tight relevancy among one element and the others. The essence of mathematics as a
structured and systematic science, as a human activity through active, dynamic, and
generative process, and as a science that develops the critical, objective and open thinking,
becomes very important to be mastered by the student in facing the rapid change of science
and technology.
In fact, as stated by Maier (1985) and Dahrim (2004), it cannot be denied that the
belief of some students which develops nowadays is about mathematics as a subject that is
hard and dislike. Even according to Çatlioğlu, Gürbüz, and Birgin (2014), the preservice
elementary school teachers sill feel greatly mathematics anxiety. Only a few students who are
able to dip into and comprehend the mathematics as a science that can drill the higher order
3
thinking ability, especially the creative thinking. Whereas, they know that mathematics is
important for their life. Other than the students’ bad judgment about mathematics, Slettenhaar
(2000) stated that in the present learning model, generally students’ activity is only listening
and watching the teacher in doing mathematical activity, and then the teacher solves the
problem with one solution, and it ends by giving the exercise to students to be finished by
themselves. That learning activity, as stated by Rif’at (2001) is called as rote learning, which
is learning activity that only makes the students tend to memorize without comprehend or
understand what is taught, while the teacher often does not realize it. In line with the
statement, Abdi (2004) stated that some of students feel the difficulty in absorbing and
comprehending mathematics, but the difficulty in comprehending mathematics is probably
related to the way of the teacher teaching in the classroom that does not make students feel
enjoy and sympathy to mathematics, and the approach that the teacher used is generally less
varied.
Jenning and Dunne (1998) stated that most of students have difficulty to apply
mathematics in daily life because in mathematics learning, real world is just a place for
applying the concept. The other thing that creates the difficulty in mathematics for students is
because mathematics seems to be less meaningful. Teacher in the lesson does not relate
students’ prior knowledge that they are already acquired and they are given less opportunity
to reinvention and self-construct mathematics ideas. Wahyudin (1999) stated that one of the
cause of students are poor in mathematics is they have less ability to comprehend the
knowledge, to identify the basic concept of mathematics that relates to the subject under
discussion.
Based on those reasons, it is clear that students’ creative thinking ability is very
important to be developed. Therefore, teacher or lecturer should investigate and improve
teaching practices that have been implemented, which might just mere routine.
It is true that at this time mathematics learning are quite emphasize on changeoriented approach and introduce the importance of the involvement of students in the use of
mathematics through an active process. In the process of mathematics learning, there are
enough teachers/ lecturers who create condition that allows the students to develop the
mathematical creative thinking ability. Siswoyo (2004), Pomalato (2005), and Wardani
(2009), carry out a study of creative mathematical thinking abilitys to high school students,
both junior and senior high school. Aspects of creative thinking that they are reviewed are
originality, fluency, flexibility, and sensitivity, either on some aspects of five as well as the
4
whole. Meanwhile, the research on mathematical creative thinking disposition is still very
rare.
Apart from that problem, all studies on thinking ability and creative disposition that
have been carried out at the level of secondary school and college, has yet show how
successful the creative thinking ability and disposition to students, the preservice elementary
school teachers (students of PGSD). If the creative thinking ability and investigative students,
prospective elementary teachers, are not developed during undergraduate education, it is not
impossible that after they graduate and become primary school teachers, they are also difficult
to develop students’ thinking abilitys and critical disposition. In fact, students of preservice
elementary school teachers study program (PGSD) are students who are prepared to become
professional homeroom teacher in primary school, who should be able to develop thinking
abilitys and creative disposition of students as mandated by the curriculum in Indonesia.
The situation is ironic because in one side of the creative thinking ability of students
is very important to be owned and developed, but on the other hand the creative thinking
ability of students is still lack. It can be seen from the results of preliminary study conducted
by Maulana (2011) against PGSD students who have diverse latest educational background.
The students come from senior high school (SMA), vocational school (SMK), and special
class dual modes (SPG). For the courses taken are Natural Science, English, Social Studies,
Management, and Engineering. If the students were grouped into large groups, then there are
two major groups that are students with science and non-science background. In preliminary
studies that have been carried out, critical thinking ability test is given and the result is the
average score is less than 50% of the maximum score for both groups (Maulana, 2011).
All the information found in the field-on the low of mathematical creative thinking
ability of students, the prospective teachers, especially PGSD-should not be left. However, it
needs an effort to follow in order to improve; one alternative is to implement a strategy and a
more innovative learning approach.
Along with creative thinking abilitys that must be developed, it could not be
separated from the ability; there is mathematical disposition that should be trained
simultaneously. In mathematics, the guidance of affective component sort of mathematical
disposition will form the desire, awareness, dedication and a strong tendency to students to
think and act mathematically in a positive way and based on faith, piety and good values
(SUMARMO, 2011). The meaning of mathematical disposition as above is basically in line
with the meaning contained in the culture education and nation’s character. Thus the
development of culture and character, mathematical disposition and thinking abilitys basically
5
can be grown themselves together. Mathematical disposition related to creative thinking
ability, in this case is termed as creative disposition.
The research’s result of Sumarmo, et al. (HULUKATI, 2005) suggests that
mathematics instruction today, among others, has the following characteristics: learning is
more centered on the teacher, the approach is more expository, teacher dominates the
classroom activities process more, routine exercises are given more. While the curriculum
requires a learning process which is student-centered, developing students' creativity, creating
a fun but challenging conditions, developing value ability, providing a diverse learning
experience and learning by doing. Therefore, it needs extra hard efforts of all parties
concerned with the educational process to jointly strive to improve the learning process that
occurs at this time.
Realizing the importance of a strategy and learning approach to develop the thinking
abilitys of students, it is absolutely necessary for mathematics learning involve students more
actively in the learning process itself. This can be realized through an alternative form of
learning that is designed in such way that reflects the involvement of students actively and
constructively. Students as learners need to get used to be able to construct their own
knowledge and able to transform them into more complex situations so that that knowledge
will become the learner's own belongings that inherent forever. The process of constructing
knowledge can be carried out by the students themselves based on the experience that has
been owned previously, or it can also be a result of the discovery that involves environmental
factors.
Based on the view of constructivism, a learning strategy must have characteristics are
as follows: using more time to develop an understanding that can improve the ability of
learners to converting knowledge, involving the students in the learning process so that the
abstract concept can be presented more concrete, implementation of discussion in small
groups, and the presentation of the problems are not routine.
One of the mathematics learning approaches which based on the view of
constructivism is problem-based learning (PBL). In the process, this learning presents a
learning environment with the problem as a base. The problem is raised so that the students
need to interpret the problem, gather the needed information, assess alternative solution, select
and present the solution that have been chosen. When students try to develop a procedure to
resolve the problem, then in fact they are integrating conceptual knowledge with their own
ability. Therefore, overall in this case the students who build their knowledge, and be
supported by the presence of teacher who plays a major role as a facilitator of learning.
6
Problem-based learning provides a learning environment that gives many
opportunities for students to develop their mathematical thinking ability. By a problem-based
learning, they try to explore, adapt, change the resolution procedures, and also verify the
appropriate solution to the new situation of the obtained problem. Problem-based learning is
also have the metacognitive atmosphere, which focuses on learning activities, helps and
guides students who face difficulties, and helps to develop their metacognitive awareness,
both in terms of selecting, remembering, recognizing, organizing the faced information, up to
how to resolve problem (SUZANA, 2003). Problem-based learning that is based on the view
of constructivism can trigger the growth of students’ metacognitive ability, because the
learning process begins with cognitive conflict and it resolves by the students themselves
through self-regulation which is finally in the learning process the students construct their
own knowledge through the experience of the result of interaction with the environment.
One of the strategies used in implementing problem-based learning is by using the
"MURDER" strategy. The "MURDER" strategy emphasizes that the interaction and
collaboration with others is an important part of learning (SANTYASA, 2008). The term
"MURDER" is an acronym of the words Mood-Understand-Recall-Detect-Elaborate-Review.
In the Mood phase, the learning is more directed to set the mood appropriately by relaxation
and focusing on learning tasks. The second phase, Understand, students are invited to
understand the particular material of the script without memorizing it. In Recall phase, one
member of the group gives the oral performance by repeating the material being read. Then
Detect phase, the members examine and criticize the emergence of errors, the omission of
notes, or the different views. The fifth phase, a fellow mate Elaborate the 2nd, 3rd, 4th, and 5th
step, and it is repeated for the next material. Finally, in the Review phase, the students review
the results of their work and transmit it to another couple in their group. Through problembased learning "MURDER" strategy, students are expected to (students of PGSD) develop
thinking ability and mathematical disposition.
Based on the description that has been stated above, a study of mathematics learning
alternatives that can develop thinking ability and mathematical creative dispositions is
necessary to be carried out. In this case, the research that implements problem-based learning
with "MURDER" strategy to improve creative mathematical thinking ability and disposition
of PGSD students that are estimated in accordance with the needs of students in developing
thinking ability and creative disposition of the different backgrounds (in this case the level of
prior knowledge, as well as the origin of schools/educational background).
7
2 Method
2.1 Design and research procedure
This study was conducted in two stages: 1) the preparation stage, and 2) the
implementation stage. At the preparation stage, the developmental research of problem-based
learning teaching materials with "MURDER" strategy by using didactical design research
(DDR) is conducted. As stated by Suryadi (2010), DDR is a research methodology that was
developed from the tacit didactical and pedagogical knowledge.
Suryadi (2010) explains that DDR has three stages, those are:
a) Didactical Situation Analysis (DSA) is carried out by the lecturer in the development of
teaching materials before it is tested in a learning event. DSA is a synthesis of the
lecturer’s ideas on students’ various possible responses that are predicted to appear on
learning events and the anticipated steps.
b) Metapedadidactical Analysis (MA) is carried out by lecturer before, during, and after the
trials of teaching materials. MA is the ability of the lecturer to be able to view at learning
events comprehensively, identifying and analyzing the important things that happened, as
well as do a prompt action (scaffolding) to overcome learning obstacles so that the stages
of learning can be run smoothly and students’ learning outcomes become optimal.
c) Retrospective Analysis (RA), is carried out by the lecturer after performing a trial
teaching materials. RA of teaching materials that have been previously developed that will
produce an ideal teaching materials is revised, i.e. teaching materials that suit the needs of
students, can predict and anticipate any learning barriers that arise, so that the stages of
learning can be run smoothly and student learning outcomes can be optimal.
The end of this preparatory stage is by obtaining: (1) teaching materials for problembased learning with "MURDER" strategy with DDR, problem-based learning "MURDER"
strategy without DDR, and conventional learning; (2) a set of mathematical prior knowledge
test, mathematical creative thinking ability test that have met the requirements: validity,
reliability, level of difficulty, and discrimination index; and (3) the mathematical creative
thinking disposition scale.
If the preparation stage has been completed, then proceed with the implementation
stage of research using quasi-experimental method with non-equivalent control group design.
The use of quasi-experimental method is because it is not possible to perform full control of
8
the research sample, so that the subject is not grouped randomly, and the condition of the
subject is accepted as it is (RUSEFFENDI, 2003).
Based on the mathematical prior knowledge test result, students in each grade are
grouped into three categories: high, middle, and low achiever. Grouping of students’
mathematical prior knowledge is determined based on the categorization with grouping
criteria based on the average combined score of all students and the standard deviation. Thus,
quasi-experimental research with the non-equivalent control group design briefly described as
follows (FRAENKEL; WALLEN, 1993; RUSEFFENDI, 2003).
0
0
0
X1
X2
0
0
0
Description:
X1 : Problem-based learning with "MURDER" strategy by the results
of didactical design research teaching materials (PBMM-DDR).
X1 : Problem-based learning with "MURDER" strategy (PBMM)
0
: The distribution of test and non-test at the beginning and end of
the learning.
2.2 Population and sample
The population of the study is students of preservice elementary school teacher study
program (PGSD) at the State University who sign Mathematics Education II up in the scope
of the province of West Java and Banten. Of the population, a number of samples in this study
is taken, 119 of people were distributed into 3 classes. Of the three classes, two classes are
selected as an experimental class and the other class as the control class. In the experimental
class, the lecture with problem-based learning approach "MURDER" strategy is conducted,
while students in the control class obtain conventional learning activities.
2.3 Variables and operational definitions
a) The independent variable is denoted by X. The independent variables in this study are
problem-based learning with "MURDER" strategy with teaching materials based on
the results of DDR (X1), and the regular problem-based learning with "MURDER"
strategy (X2).
9
b) The control variables in this study are the educational background (science and nonscience), as well as the students’ initial level of mathematical ability, which consists of
three categories: high, middle, and low.
c) The dependent variable is denoted by Y. The dependent variables in this study are
creative thinking ability and mathematical creative disposition.
d) Problem-based learning (PBL) is a learning that begins with preparation for the
orientation of the real problems or issues that are simulated to gain an understanding
of concepts, relationships between concepts, application of concepts, communication
of the concept, as well as to find, define, evaluate and present the solution of the
problem according to the invention itself. In general, problem-based learning consists
of five kinds of activities: (1) orientation or preparation, (2) organization, (3)
exploration, (4) negotiation, and (5) integration (developed from Barret, 2005 and
Karlimah, 2010).
e) "MURDER" strategy is an acronym of Mood, Understand, Recall, Detect, Elaborate,
and Review (adopted from Santyasa, 2008).
f) The mathematical creative thinking ability is the ability of high level mathematical
thinking which covers the aspects: (a) sensitivity, (b) fluency, (c) flexibility, (d)
elaboration, and (e) originality.
(1) Sensitivity is the ability to capture and locate the problem in response to a
situation, or ignore the facts that were not appropriate (misleading fact).
(2) Fluency is the ability to build ideas to resolve the problems relevantly or to
provide answers in the form of examples related to specific mathematical concept.
(3) Flexibility is the ability to use a variety of completion strategies, or the ability to
try different approaches in solving the problem, or the ability to switch from one
approach to the other approaches in solving problems.
(4) Elaboration is the ability to explain in detail, in order and coherent to a procedure,
an answer; or specific mathematical situation. These explanations use the concept,
representation, term, or the appropriate mathematical symbol.
(5) Originality is the ability to use strategies that are new, unique, or unusual to solve
the problem; or give examples that are new, unique, or unusual.
g) Mathematical creative thinking disposition is a tendency to think and act in a creative
way to mathematics, which include: (a) Feeling of problems and opportunities, as well
as willing to take risks. (b) Sensitive to the environmental situation, and appreciate the
creativity of others. c) Be more oriented to the present and the future than the past. (d)
10
Have self-confidence and autonomy. (e) Have a great curiosity. (f) Declare and
respond to the feelings and organize emotions. (g) Creating a various considerations.
(h) Respect fantasy, rich in initiatives, has the original idea. (i) Persistent and not
easily being bored; be not desperate to solve the problem (adopted from Sumarmo,
2011).
All data that is netted from the device of mathematical prior knowledge test,
mathematical creative thinking ability test, and mathematical creative disposition scale of the
students and then are analyzed quantitatively, or quantified in the hope to answer the problem
of research related to the interaction of independent, control, and dependent variables in this
research.
The data analysis is carried out in stages, starting from the assumptions of normality
and homogeneity test, t-test (Student) to test the difference of mean of two independent or
dependent samples and One-Way Anova to test the difference of mean of more than two
independent samples. If the assumptions of normality and homogeneity are unfulfilled, then
the U-test (Mann-Whitney) for two independent samples and Kruskal Wallis for more than
two independent samples is conducted. Meanwhile, in testing the interaction, the Two-Way
Anova is used.
3 Results and discussion
3.1 Mathematical prior knowledge analysis
The whole sample group was divided into three groups of mathematical prior
knowledge (MPK), the categories of MPK are high achiever (𝑥 ≥ 70, middle achiever (50 ≤
𝑥 < 70), and low achiever (𝑥 < 50). The assessment is based on an agreement of the
mathematics lecturers of Campus Sumedang PGSD relates to the method of determining the
minimum completeness score of Mathematics Education II subject. In the first experimental
group, of 40 students were known as 9 high achiever, 22 middle achiever and 9 low achiever.
In the second experiment class, of 40 students, there were 9 high achiever, 22 middle achiever
and 9 low achiever. Then in the control class, there were 7 high achiever, 23 middle achiever
and 9 low achiever.
Shapiro-Wilk test provided information that PBMM-DDR and PBMM class showed
normal distribution (the value of its each opportunity are 0.707 and 0.509), whereas the
11
conventional class did not show normal distribution (opportunity value 0.004). Since one of
them is did not normally distribute, then Kruskal-Wallis test was used to test the mean score
difference among the three groups.
Table 1 – Ranks of mathematical prior knowledge score
Research Class
Score_MPK
N
Mean Rank
PBMM-DDR
40
59.12
PBMM
40
59.34
Conventional
39
61.58
Total
119
Source: developed by the authors
Table 2 – Test statisticsb,c
Score_MPK
Chi-Square
.122
Df
2
Asymp. Sig.
Monte Carlo Sig.
.941
.939a
Sig.
95% Confidence Interval
Lower Bound
.935
Upper Bound
.944
a. Based on 10000 sampled tables with starting seed 2000000.
b. Kruskal Wallis Test
c. Grouping Variable: Research Class
Source: developed by the authors
From the Kruskal Wallis test, Asymp.Sig value was known at 0.941. This showed that
at the 5% significance level, there was no difference among the students’ mathematical prior
knowledge at the 1st experiment class, 2nd experiment class, and control class, so prior to the
treatment of PBL-MURDER was carried out, the three groups’ mathematical prior knowledge
were significantly same.
3.2 Mathematical creative thinking ability based on the class
Based on the results of Kruskal Wallis test on the pretest data, it was found that the
value Asymp.Sig = 0.802 > 0.05. This showed that there was not any difference of PGSD
students’ initial creative thinking ability. Or in other words, students in the third grade PGSD
before PBMM-DDR, PBMM, and PK learning was implemented have the same initial
creative thinking ability.
Judging from the posttest result through the One-Way Anova, it was found that there
were mean score differences in the final creative thinking ability (after learning) of all three
groups (PBMM-DDR, PBMM, and conventional learning). The advanced Scheffe test showed
12
that these differences occurred to the three groups, which was the final creative thinking
ability of PGSD students who acquired PBMM-DDR learning was significantly better than
who acquired PBMM and the students’ final creative thinking achievement that followed
PBMM learning was significantly better than students who acquired conventional learning.
Similarly, the creative thinking ability data gain processing, based on the results of
One-Line Anova, p-value = 0.000 was obtained. Thus it can be explained that there were
mean score differences increase of creative thinking ability in those three grades (PBMMDDR, PBMM, and conventional learning). The Scheffe follow-up test showed that these
differences occurred to the three groups, which was an increase in the PGSD students’
mathematical creative thinking ability who acquired PBMM-DDR learning that significantly
better than who acquired PBMM. Then the increase of students’ creative thinking ability that
followed PBMM learning was also significantly better than students who acquired the
conventional learning. The complete computation is in the table below.
Table 3 – Anova
Sum of Squares
Posttest_creative
Gain_creative
Between Groups
df
Mean Square
F
5454.617
2
2727.308
Within Groups
12415.562
116
107.031
Total
17870.179
118
Between Groups
.821
2
.410
Within Groups
1.399
116
.012
Total
2.220
118
Sig.
25.482
.000
34.019
.000
Source: developed by the authors
Table 4 – Multiple comparisons
Scheffe
95% Confidence Interval
Dependent Variable (I) Research Class
(J) Research Class
Posttest_Creative
PBMM
PBMM-DDR
PBMM
Conventional
Gain_Creative
PBMM-DDR
PBMM
Conventional
Mean
Difference (I-J) Std. Error
Sig.
Lower
Bound
Upper
Bound
5.93750*
2.31334
.041
.2011
11.6739
Conventional
16.42803*
2.32812
.000
10.6550
22.2011
PBMM-DDR
-5.93750*
2.31334
.041
-11.6739
-.2011
Conventional
10.49053
*
2.32812
.000
4.7175
16.2636
PBMM-DDR
-16.42803*
2.32812
.000
-22.2011
-10.6550
PBMM
-10.49053*
2.32812
.000
-16.2636
-4.7175
PBMM
.06832*
.02456
.024
.0074
.1292
Conventional
.20072
*
.02472
.000
.1394
.2620
PBMM-DDR
-.06832*
.02456
.024
-.1292
-.0074
Conventional
.13241*
.02472
.000
.0711
.1937
PBMM-DDR
*
-.20072
.02472
.000
-.2620
-.1394
PBMM
-.13241*
.02472
.000
-.1937
-.0711
*. The mean difference is significant at the 0.05 level.
Source: developed by the authors
13
Table 5 – Scheffe test (homogenous subsets) for creative thinking ability
Subset for alpha = 0.05
Research Class
N
1
Conventional
39
PBMM
40
PBMM-DDR
40
2
3
44.9787
55.4693
61.4068
Sig.
1.000
1.000
1.000
Means for groups in homogeneous subsets are displayed.
Source: developed by the authors
Table 6 – Scheffe test (homogenous subsets) for normalized gain of creative thinking ability
Subset for alpha = 0.05
Research Class
N
1
Conventional
39
PBMM
40
PBMM-DDR
40
Sig.
2
3
.3427
.4751
.5434
1.000
1.000
1.000
Means for groups in homogeneous subsets are displayed.
Source: developed by the authors
3.3 Creative thinking ability of science and non-science educational background
From the pretest data through the Mann-Whitney test, the value of Asymp.Sig. =
0.335 > 0.005 was obtained. This showed that the hypothesis which stated that there was no
mean score difference of initial creative thinking ability was accepted. That means the
students’ initial mathematical creative thinking ability between the science and non-science
group was not significantly different.
From the posttest data, the normality assumption was filled through the Shapiro-Wilk
test. From Levene test computation, with p-value = 0.329, the information was obtained that
both of data group was homogeneous. Then based on two independent sample t-test, p-value
= 0.002 < 0.005 was obtained. This indicates that the null hypothesis (H0) was rejected. That
is, the student's final mathematical creative thinking ability between the science and nonscience group were significantly different. In this case, the mean score of science group with
57.45 was better than mean score of non-science group with 50.43.
Then Levene test computation on the data gain gave p-value = 0.278. Thus, the
homogeneity of variance of the two groups was fulfilled. Then based on two-sample of
independent t-test p-value = 0.002 <0.005 was obtained. This indicates that the null
hypothesis (H0) was rejected. This means that the increase in the mathematical creative
thinking ability between the science and non-science group were significantly different. In
14
this case, the average increase in the science (0.4919) was better than the non-science group
(0.4155).
Table 7 – Mann-Whitney test for creative thinking ability pretest
Educational
Background
Pretest_Creative
N
IPA
Non-IPA
Total
Mean Rank
Sum of Ranks
61
62.95
3840.00
58
56.90
3300.00
119
Source: developed by the authors
Table 8 – Test Statisticsa
Pretest_Creative
Mann-Whitney U
1589.000
Wilcoxon W
3300.000
Z
-.964
Asymp. Sig. (2-tailed)
.335
a. Grouping Variable: Education Background
Source: developed by the authors
Table 9 – Independent Samples Test
Levene's Test for
Equality of
Variances
F
Pretest_ Creative Equal variances
assumed
Sig.
.042
.838
Equal variances
not assumed
Postes_ Creative
Equal variances
assumed
.961
.329
Equal variances
not assumed
Gain_Creative
Equal variances
assumed
Equal variances
not assumed
1.187
.278
t-test for Equality of Means
t
Sig. (2tailed)
df
Mean
Diff.
Std.
Error
Diff.
95% Confidence
Interval of the
Difference
Lower
Upper
1.109
117
.270
1.187
1.069
-.93
3.30
1.109
116.702
.270
1.186
1.069
-.93
3.30
3.230
117
.002
7.014
2.172
2.71
11.32
3.240
116.419
.002
7.014
2.165
2.73
11.30
3.152
117
.002
.076
.024
.028
.124
3.166
115.106
.002
.076
.024
.029
.124
Source: developed by the authors
3.4 Creative thinking ability based on mathematical prior knowledge (high, middle, and
low achiever)
After fulfilling the assumptions of normality and homogeneity, based on the pretest
data processing, it was found that p-value = 0.000. Since the value was less than 0.005, it
means that H0 was rejected; so that there were mean score differences of students’ initial
creative thinking ability with high, middle, and low achiever groups. To see where the
15
difference was laid, then post hoc test which use Scheffe test was conducted. Based on
Scheffe test result, the information was obtained that initial creative thinking ability of high
achiever students was better than middle and low achiever students. Thus, the ability of high
achiever students in creative thinking before learning was already better than middle and low
achiever students.
Based on the pretest data processing result, it was found that p-value = 0.001. This
probability value indicates that the final ability as the students’ achievement after learning
was different. To see the difference, Scheffe test was conducted, and the information were
obtained that the final creative thinking ability of high achiever students better than middle
and low achiever students. Or, as when before learning, high achiever students' ability in
creative thinking after learning was still better than middle and low achiever students.
Based on the results of gain data processing, it was found that p-value = 0.015 which
less than 0.05. Thus the decision was made that H0 was rejected, which means that there was a
difference among high, middle, and low achiever students in terms of improving the
mathematical creative thinking ability. To see the difference, Scheffe test was conducted. The
results of this further tests showed that there were significant difference between the increases
of high and low achiever students’ creative thinking ability, where the increase of high
achiever group was much better. While the difference of creative thinking ability increase
between students of high and middle achiever, and between students of middle and low
achiever, was not significant. The complete calculation is shows in the tables below.
Table10 – Anova
Sum of Squares
Pretest_Creative
Between Groups
Mean Square
F
2
382.256
Within Groups
3254.141
116
28.053
Total
4018.653
118
Posttest_ Creative Between Groups
Gain_ Creative
df
764.512
2113.920
2
1056.960
Within Groups
15756.259
116
135.830
Total
17870.179
118
.156
2
.078
Within Groups
2.064
116
.018
Total
2.220
118
Between Groups
Sig.
13.626
.000
7.782
.001
4.384
.015
Source: developed by the authors
Table 11 – Multiple comparisons
Scheffe
Dependent Variable (I) MPK
Pretest_ Creative
High
(J) MPK
Middle
Mean
Difference (IJ)
5.32781
*
95% Confidence Interval
Std. Error
1.24130
Sig.
.000
Lower
Bound
2.2498
Upper Bound
8.4059
16
Middle
Low
Posttest_ Creative
High
Middle
Low
Gain_ Creative
High
Middle
Low
Low
7.32487*
1.47007
.000
3.6795
10.9702
High
-5.32781*
1.24130
.000
-8.4059
-2.2498
Low
1.99706
1.20735
.259
-.9968
4.9909
High
*
1.47007
.000
-10.9702
-3.6795
Middle
-1.99706
1.20735
.259
-4.9909
.9968
Middle
7.73637*
2.73139
.021
.9633
14.5094
Low
12.63892*
3.23480
.001
4.6176
20.6602
High
-7.73637*
2.73139
.021
-14.5094
-.9633
Low
4.90255
2.65670
.187
-1.6853
11.4904
High
-12.63892*
3.23480
.001
-20.6602
-4.6176
Middle
-4.90255
2.65670
.187
-11.4904
1.6853
Middle
.06294
.03126
.136
-.0146
.1405
Low
.10923
*
.03702
.015
.0174
.2010
High
-.06294
.03126
.136
-.1405
.0146
Low
.04629
.03041
.317
-.0291
.1217
High
-.10923*
.03702
.015
-.2010
-.0174
Middle
-.04629
.03041
.317
-.1217
.0291
-7.32487
*. The mean difference is significant at the 0.05 level.
Source: developed by the authors
Table 12 – Scheffe test for creative thinking ability pretest based on MPK
Subset for alpha = 0.05
MPK
N
1
2
Low
27
13.4259
Middle
67
15.4230
High
25
20.7508
Sig.
.317
1.000
Means for groups in homogeneous subsets are displayed.
Source: developed by the authors
Table 13 – Scheffe test for creative thinking ability posttest based on MPK
Subset for alpha = 0.05
MPK
N
1
2
Low
27
48.6115
Middle
67
53.5140
High
25
61.2504
Sig.
.240
1.000
Means for groups in homogeneous subsets are displayed.
Source: developed by the authors
Table 14 – Scheffe test for normalized gain of creative thinking ability pretest based on MPK
Subset for alpha = 0.05
KAM
N
1
2
Low
27
.4057
Middle
67
.4519
High
25
Sig.
.4519
.5149
.378
Means for groups in homogeneous subsets are displayed.
Source: developed by the authors
.167
17
3.5 Three Approaches is Significant in Enhancing Mathematical Creative Thinking
Ability
Based on the two-sample paired t-test, in the experimental class-1 which used PBMMDDR, the result of calculation p-value = 0.000 was obtained. Therefore, it is clear that
learning by using PBMM-DDR was significantly able to improve PGSD students’ creative
thinking ability.
Then for experimental class-2 that was given PBMM, based on two-sample paired ttest, the result of calculation p-value = 0.000 was obtained. Thus, it is clear that learning by
using PBL was significantly able to improve PGSD students’ creative thinking ability.
Similarly, in the control group with conventional approach, through a two-sample
paired t-test, the calculation results p-value = 0.000 was obtained. Therefore, it is clear that
learning by using any conventional approach can significantly improve the PGSD students’
creative thinking ability.
3.6 Mathematical Creative Thinking Ability Increase of Science and Non-Science Group
in Each Classroom
In the experimental class-1, which used PBMM-DDR, since both of data group were
not normally distributed, then U-test (Mann-Whitney) was administered, in order to obtain pvalue = 0.010 that indicates that there was a difference in the increase of students’
mathematical creative thinking ability in the science and non-science group in the PBMMDDR class, where the science group (0.5779) increased better than the non-science group
(0.5012).
Table 15 – Mann-Whitney test for normalized gain of creative thinking ability in PBMMDDR class
Educational
Background
Gain_Creative_PBMMDDR
N
Mean Rank
Sum of Ranks
Science
22
24.80
545.50
Non Science
18
15.25
274.50
Total
40
Source: developed by the authors
18
Table 16 – Test Statisticsb
Gain_Creative_PBMMDDR
Mann-Whitney U
103.500
Wilcoxon W
274.500
Z
-2.570
Asymp. Sig. (2-tailed)
.010
Exact Sig. [2*(1-tailed Sig.)]
.009a
a. Not corrected for ties.
b. Grouping Variable: LatBel_PBMM_DDR
Source: developed by the authors
Meanwhile for the experimental class-2 (PBMM), after the normality and
homogeneity assumption were fulfilled, based on the results of independent samples t-test, the
p-value = 0.000 was obtained which indicates that there was a difference in the increase of
mathematical creative thinking ability of students in the science and non-science group in
PBMM class, where science group (0.5325) increased better than non-science group (0.4117).
Then in the control class that used the conventional approach, after the assumption of
normality and homogeneity were fulfilled, based on the results of the t-test for two
independent samples, p-value = 0.875 was obtained which indicates that there was no increase
difference in students’ mathematical creative thinking ability in the science group (0, 3395)
and the non-science group (0.3454) in the conventional class.
Table 17 – Independent samples test for normalized gain of creative thinking ability in
PBMM and conventional class
Levene's Test
for Equality of
Variances
F
Gain_Creative_
PBMM-DDR
Equal
variances
assumed
Sig.
.758
.390
Equal
variances not
assumed
Gain_Creative_
PBMM
Equal
variances
assumed
.029
.866
Equal
variances not
assumed
Gain_Creative_
PK
Equal
variances
assumed
Equal
variances not
assumed
.178
.675
t-test for Equality of Means
t
Sig. (2tailed)
df
Mean
Diff.
Std. Error
Diff.
95% Confidence
Interval of the
Difference
Lower
Upper
2.614
38
.013
.07671
.02934
.01731
.13611
2.633
37.312
.012
.07671
.02913
.01770
.13571
3.873
38
.000
.12078
.03118
.05765
.18390
3.881
37.841
.000
.12078
.03112
.05777
.18379
-.158
37
.875
-.00591
.03744
-.08178
.06996
-.158
36.365
.875
-.00591
.03737
-.08167
.06984
Source: developed by the authors
19
In addition, there were other findings, that was the increase in the creative thinking
ability of non-science group in PBMM-DDR (0.5012) and PBMM class (0.4117) which was
better than science group in the conventional class (0.3395), although it was not significant.
3.7 Students’ Mathematical Creative Thinking Ability in Each Class Based on the InterSame Mathematical Prior Knowledge Level
Based on the One-Way Anova, p-value = 0.001 was obtained. This indicates that there
was a difference in the increase of creative thinking ability on PBMM-DDR, PBMM, and
conventional class, based on high mathematical initial ability. Then, Scheffe test showed that
the increase of creative thinking ability of high achiever students in PBMM-DDR (0.6125)
and PBMM class (0.5537) was equally good, and the increase in both classes was
significantly better than high achiever students in conventional class (0.3395).
Based on the One-Way Anova (p-value = 0.000) the information that there was an
difference of creative thinking ability increase in PBMM-DDR, PBMM, and conventional
class, based on students’ mathematical prior knowledge who are at middle levels. Then
Scheffe test showed that the increase of creative thinking ability of students was significantly
different at the three classes. The increase of creative thinking ability of middle achiever
students who were being in PBMM-DDR class (0.5385) was significantly better than who
were being in PBMM class (0.4648), and the increase in PBMM class was decisively better
than middle achiever students in conventional class (0.3568).
Based on One-Way Anova (p-value = 0.004) the information that there was a
difference in the increase of creative thinking ability of lower achiever students in PBMMDDR, PBMM, and conventional class.. Then further tests showed that an increase of creative
thinking ability of low achiever students at PBMM-DDR class was significantly better than at
the conventional class. The difference of creative thinking ability increase of low achiever
students between PBMM-DDR and PBMM class, also between PBMM and conventional
class, was not significant.
3.8 Increase Differences of Mathematical Creative Thinking Ability in Each Class in
Each Category of Mathematical Prior Knowledge
In PBMM-DDR class, based on Levene test, it was found that the variance of high,
middle, and low achiever group on PBMM-DDR class was homogeneous (p-value = 0.282).
20
Based on the One-Way Anova (p-value = 0.020) the information that there were differences in
the increase in the creative thinking ability of student groups with high, middle, and low
achievement at PBMM-DDR class. Then Scheffe test showed that in PBMM-DDR class, the
increase of creative thinking ability of high achiever students was significantly better than the
students with lower one. Meanwhile, the difference of creative thinking ability increase
between the group of high and middle achiever students, also between students with middle
and low ones, the difference was not significant.
Then in the PBMM class, based on the Levene’s test, it was found that the variance of
high, middle, and low achiever groups at PBMM class was homogeneous (p-value = 0.484).
Based on the One-Way Anova (p-value = 0.037) the information that there were the increase
differences of creative thinking ability of high, middle, and low achiever students at PBMM
class. Then Scheffe test showed that in PBMM class, the increase of creative thinking ability
of high achiever students was significantly better than the students with low ones. Meanwhile,
the difference of creative thinking ability increase between the groups of high and middle
achiever students, also between middle and low achiever students, was not significant.
As in a conventional class, based on the Levene’s test, it was found that the variance
of high, middle, and low groups on PBMM class was homogeneous (p-value = 0.104). Based
on One-Way Anova (p-value = 0.584) the information that there was no difference of creative
thinking ability increase of group of students with high, middle, and low mathematical prior
knowledge achievement at conventional class. In other words, conventional learning was
equally good in increasing PGSD students’ mathematical creative thinking ability.
3.9 PGSD Students’ Mathematical Creative Thinking Disposition
Based on the initial scale, it was found that in the PBMM-DDR, BPM, and
conventional class, the mean score of initial creative disposition of PGSD students
respectively were 62.62; 62.46; and 63.60. For the mean score scale of the final creative
disposition was found at 68.43; 67.70; and 65.19. While the gain for the three classes were
0.151; 0,139; and 0.043.
For initial and final creative disposition, it was found that all of them distributed
normally, while for creative disposition gain of all three classes did not normally distributed.
Based on the Kruskal-Wallis test, the result was found that there was a significant difference
in mathematical creative disposition increase of PGSD students at the three classes.
21
Table 18 – Kruskal-Wallis test for normalized gain of creative thinking disposition
Research Class
Gain_SD_Kf
N
Mean Rank
PBMM-DDR
40
72.65
PBMM
40
70.75
39
36.00
Conventional
Total
119
Source: developed by the authors
Table 19 – Test statisticsa,b
Gain_SD_Kf
Chi-Square
Df
Asymp. Sig.
28.147
2
.000
a. Kruskal Wallis Test
b. Grouping Variable: Research Class
Source: developed by the authors
To find out where the difference it was, the further test using the Multiple
Comparisons Between Treatments (SIEGEL; CASTELLAN, 1988) was administered by
testing the couple of PBMM-DDR, PBMM, and conventional class. The null hypothesis will
be rejected at the 0.05 significance level if the calculated value is more than the critical value.
From the calculation, it was found that there was no difference between PBMM-DDR and
PBMM class, while between PBMM-DDR and conventional, and PBMM and conventional,
there was significant differences. Thus, an increase in creative disposition between PBMMDDR and PBMM group were same, and it turned decisively better than the conventional
class.
3.10 Interaction between PBL-MURDER, Educational Background, and Mathematical
Prior Knowledge toward PGSD Students’ Mathematical Creative Thinking Ability and
Disposition
By using the Two-Way Anova at a significance level of 5%, some interesting findings
were found related to the interaction between problem-based learning approach (PBL) with
"MURDER" strategy, the educational background, and students’ mathematical prior
knowledge. The summary of the results of Two-Way Anova test are presented in the
following table.
22
Table 20 – PBL-MURDER interaction, educational background (EB), and mathematical prior
knowledge (MPK), toward final achievement of PGSD students’ mathematical creative
thinking ability (MCA) and disposition (MCD)
Type III Sum of
Squares
Mean
Square
Source
Dependent Variable
Corrected Model
MCA Gain
1.279a
17
MCD Gain
.417
b
17
MCA Gain
17.876
1
MCD Gain
1.053
1
MCA Gain
.624
MCD Gain
.198
MCA Gain
Intercept
Class
MPK
EB
Class * MPK
Class * EB
Error
Total
Corrected Total
df
.075
F
Sig.
Partial Eta
Squared
8.074
.000
.576
.025
2.198
.008
.425
17.876
1.918E3
.000
.950
1.053
94.316
.000
.616
2
.312
33.464
.000
.399
2
.099
8.861
.000
.308
.055
2
.028
2.975
.056
.056
MCD Gain
.024
2
.012
1.089
.340
.009
MCA Gain
.108
1
.108
11.543
.001
.103
MCD Gain
.000
1
.000
.009
.923
.002
MCA Gain
.059
4
.015
1.571
.188
.059
MCD Gain
.016
4
.004
.353
.841
.001
MCA Gain
.141
2
.070
7.564
.001
.130
MCD Gain
.008
2
.004
.356
.702
.009
MCA Gain
.941
101
.009
MCD Gain
1.128
101
.011
MCA Gain
26.820
119
MCD Gain
3.022
119
MCA Gain
2.220
118
MCD Gain
1.545
118
a. R Squared = .576 (Adjusted R Squared = .505)
b. R Squared = .270 (Adjusted R Squared = .147)
Source: developed by the authors
Table 20 above indicates several findings as follows.
a) The learning approach had a significant influence to the increase of thinking abilitys and
creative dispositions of PGSD students.
b) Meanwhile, mathematical prior knowledge was not too affecting to the increase of
mathematical creative thinking ability and disposition of PGSD students. From the partial
eta squared value, it can be known that the influence of this mathematical prior knowledge
differences is just 5,6% for mathematical creative thinking ability, and 0,9% for
mathematical creative thinking disposition. In other words, each approach which was
conducted increased thinking abilitys and creative dispositions in the relative same range.
c) Educational background gave significant influence on the increase of PGSD students’
mathematical creative thinking ability asa much as partial eta squared = 10,3%. But the
educational background will not give effect to the increase of the mathematical creative
thinking disposition (the influence is just 0,2%). It means that students who are interested
in science will have a tendency to achieve the increase of mathematical creative thinking
23
ability than non-science group. Meanwhile, the attitude pattern to tend to be creative
between science and non-science group increased equally at the same range.
d) There was no signficant interaction (combined effects) between the approaches and
mathematical prior knowledge toward thinking ability and mathematical creative
disposition mathematical of PGSD students. The influence is 5,9% and 0,1%. It means
that an increase of thinking ability and mathematical creative disposition of the students
had a similar increase at each level in both variables (approaches and mathematical prior
knowledge).
e) Although there was no interaction between the learning approaches and educational
background toward the increase of mathematical creative thinking disposition (it’s
influence is 0,9%), but the interaction was significantly visible toward the increase of
mathematical creative thinking ability (it gave influence as much as 13%).
Graph 1 – The interaction between the approaches and educational background toward the increase of PGSD
students’ mathematical creative thinking ability
Through a series of hypothesis testing, it was found that the raw input of PGSD
students who become research subjects had initial ability or mathematical prior knowledge
which is relatively same. In other words, it is understandable that all new students who passed
the PGSD admission in the place of study passed through the same steps so the generic
ability, especially mathematical prior knowledge was not different. Similarly, after the
distribution of higher, middle, and lower subgroups, the proportion of three subgroups of the
study in those three classes relatively even.
24
The mathematical creative thinking ability of PGSD students at the end of the study
showed a difference, from the point of view of learning activities were used. From the results
of data analysis and statistics hypothesis testing, a valuable information was obtained that
PBL-MURDER learning that using teaching materials which is the DDR study result was
better than learning regular PBL-MURDER and conventional, and the regular PBLMURDER learning was better than conventional learning in achieving mathematical creative
thinking of PGSD students. This can be seen clearly that students’ learning outcomes would
be optimal if the teaching materials are designed in such a way so that students’ learning
barriers can be minimized (SURYADI, 2010). Beside the teaching material optimization
through a series of DDR studies, it was also found that the mathematical prior knowledge
affected enough to the final achievement of PGSD students' mathematical creative thinking
ability. This is in line with findings of Suryadi (2005), Maulana (2007), and Ibrahim (2011)
that students who initially have a superior prior knowledge would have a tendency to achieve
a higher mathematical creative thinking ability.
From the point of view of the approaches and learning strategies, a significant
influence toward PGSD students’ mathematical creative thinking ability achievement was
found. It means that PBL-MURDER learning provides the opportunity for PGSD students to
be able to achieve higher mathematical creative thinking ability than conventional learning.
Related to the increase in mathematical creative disposition, all the approaches that
had been successfully used to significantly increase the disposition, and in fact all of that
approaches can increase disposition with a relatively small difference (or it can be said that
each approach provides a disposition increase that relatively equal). All students’
mathematical creative thinking disposition averagely increased by 11%. If it seen from the
benchmark of gain achievement, it was perceived that it was still a slow improvement. This
can only be understood as a theory of conditioning by Pavlovian (RUSEFFENDI, 1992), that
in order to develop attitude aspects (affective) of students can not be carried out instantly, but
habitually in a long time.
Related to the interactions between the types of approaches that had been selected and
educational background behind the students in learning can be considered more, that although
from the beginning the students have a keen interest in science major, but when conventional
approach was used in learning, that interest that initially was great will be eroded and the
increase was not higher than the students who was interested in the non-science field which
both of them used conventional learning.
25
4 Conclusion
a) Raw input shows that PGSD students had initial ability or mathematical prior
knowledge which was relatively same. Mathematical prior knowledge (high, middle,
and low) greatly affected on the increase of PGSD students’ mathematical creative
thinking ability. Students who initially had a higher achievement would have a
tendency to achieve a high creative thinking ability.
b) The increase of PGSD students’ mathematical creative thinking ability followed the
problem-based learning with mood-understand-recall-detect-elaborate-review strategy
using teaching materials from the result of didactical design research (PBL-MURDER
and DDR) was better than regular PBL-MURDER and conventional learning, as well
as the general PBL-MURDER learning was better than conventional learning in
increasing the mathematical creative thinking ability of PGSD students. Thus, an
attempt to minimize learning barriers (learning obstacles) through a good teaching
materials design and more appropriate for the needs of students will be able to
optimize the learning outcomes of those students.
c) In terms of the educational background, a significant effect on the increase of the
mathematical creative thinking ability of PGSD students could not be ignored. Since
the potential of students at science group was significantly better than the potential of
non-science group in increasing mathematical creative thinking ability. In other words,
students who came from the science group had a tendency to equip and prepare
themselves to face the problems that deplete their creative thinking ability.
d) Learning sets with PBL approach with "MURDER" strategy, whether using DDR
teaching materials or not, gave a better effect than conventional learning in terms of
increasing mathematical creative disposition of PGSD students. It is very possible to
happen, since the PBL process with "MURDER" strategy requires students to be more
active and creative in learning activities, compared to conventional learning which
makes students be more "notified" rather than "finding out".
e) The interaction between the type of the selected approach and educational background
which became the background of students in learning, it was found to have a
significant effect in increasing PGSD students’ creative thinking ability. There was a
tendency that the students who from the beginning had a keen interest in majoring in
science, but because of the conventional approach that was used in learning, their
26
creative thinking ability was not higher than the students that were interested in nonscience which used the same conventional learning.
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