Developing Understanding Skill, Mathematical Representation and Belief in
Mathematical Learning Using Cognitive Conflict Strategy
Heru Sujiarto1, Tatang Herman2, Jozua Sabandar3, Jarnawi Afghani Dahlan4
ABSTRACT
The skill of mathematical understanding and representing, and student’s belief to
mathematic are an important elements one student should have, to help students to solve
mathematic problems, along with daily problems. One of the ways to develop this is
through a learning process where a situation, fact, and condition that polarize student’s
cognitive structure are involved. In such situation, conflict between student’s knowledge
and designed situation happened. The main problem of this research is about how the
developing understanding skill, representing, and student’s mathematical belief, are
reviewed based on learning method (cognitive conflict strategy and conventional), and
school rank (high and middle). This research is an experiment with pretest-posttest control
group design. Cognitive conflict strategy is given to the experiment group, while
conventional learning is given to control group. This research is involving 140 seventh
grade students in Kotamadya Bandung that represent schools in high and middle rank.
The instruments of this research is Mathematical Understanding Skill test, Mathematical
Representing Skill test, and Mathematical Belief scale. Data analysis used in the
hipotesist test is t-test, two way Anova, and Scheffe test.
Keywords: cognitive conflict strategy, mathematical understanding, mathematical
representation, mathematical belief.
1
Student of Mathematics Education Doctoral Program, Indonesia University of Education.
Postal address: UNINUS BANDUNG, Jl. Soekarno Hatta No. 530 Bandung.
Email: mathheru@yahoo.com
2
Professor of Indonesia University of Education, Department of Mathematics Education.
Postal Address:
______________________________________________________________________.
Email:
3
Emerita Professor of Indonesia University of Education, Department of Mathematics
Education. Postal Address: ________________________________________________.
Email:
.
4
Doctor in Mathematics Education, Indonesia University of Education, Department of
Mathematics Education. Postal Address: .
Emails:
INTRODUCTION
Realize it or not, cognitive conflict often shows up in mathematic studying and
lucturing activities, it is caused by varied cognitive skills from each individu and the
characteristic of the material that is given. It means that cognitive conflict can happened
in studying process when student’s information and knowledge is unbalanced with the
information they faced when in studying scene.
In mathematical problem situation, student usually facing challenges and they
oftenly facing an impasse. By providing a designed congnitive conflict, it is one of the
efforts to make students accustomed to such situation and giving them an experience on
how to handle an unwanted situation, giving them challenge and opportunity to stabilize
their mathematical knowledge and skill.
Knowledge inventing according to constructivity sees active students creating
cognitive structures in their interaction with environment. With the help from their
cognitive structure, subject arranges the definition of the reality. Cognitive interaction
will happened as far as the reality are arranged through cognitive structure that the
students made their own. Cognitive structure had to be changed constantly and be
adjusted based on environmental pressure that changing. An adaptation process is
happened continually through reconstruction process (Piaget, 1988).
In constructivism theory, the most important thing at the learning process is that
the students themselves had to be active developing their knowledge, and had to be
responsible to their learning result. The emphasizing in this student’s active learning
process is got to be developed. Student’s creativity and liveliness will help them to stand
alone among student’s cognitive life, so that studying could be directed more into
concrete experience, discussing with classmates, then become contemplated idea and new
concept development.
Mathematical understanding is important to learn mathematic in meaningful way,
of course the teachers hope an understanding that student can reach is not limited to an
understanding that only can be connected. According to Ausubel (1978), a studying
process will become meaningful if the information that will be learned by student is
arranged matching to student’s cognitive structure, so that the students can link their new
information to their own cognitive structure. It means, the students can link between
knowledge they have with another situation so they understand while studying.
The importance of understanding is realized in mathematic teaching, that there are
two understandings that can be learnt in mathematic, they are conseptual understanding
and procedural understanding. Both understandings have the same important role and
both understandings need to be taught in school. Hiebert and Carpenter (1992) stated that
understanding process in mathematic means to make a conncetion between ideas, fact, or
procedur that all of them is a part of the system. Therefore, the problem that once was
understood can be solved by understanding the connection between ideas, fact, or
procedure that are there in the system.
In studying mathematic, the important goal in using representation is to be able to
communicate with the others using representing manifestation. Representing means to
make another form from the idea or the problem, e.g. one table is represented into diagram
shape or vice versa. Representing could help student explain concept or idea and make
them easier to get the strategy to solve mathematical problem. E.g, graphic can be used
as a tool to communicate information and understanding and make definition in
mathematical way. By using graphic, student can dig invisible aspect in one context;
representing process could lead a question about the context itself; student can construct
something new and concepting the first context by graphic’s important character; so that
students can elaborate their understanding about graphic and the context through
mathematical representing that they do, Monk (in Wu, 2004).
Goldin (2002) explain that one’s belief is shaped by his attitude to mathematic,
then the belief will form mathematic value in that person. Regarding another role of
beliefs, Greer, Verschaffel, and Corte (2002) emphasized that to be able to do mathematic
is not enough by knowing on how to do it, but had to be along with the belief of concept
truth and its procedure. E.g. when doing a manual counting or counting with a tool, the
belief element is there (Nunokawa, 1998).
In order to create an optimal learning process, rank factor or school qualification
is also necessary to be concern and consider. It has several reasons: (1) the fact that shows
that school rank is very related to student’s mathematical skill in general; and (2) diverse
student’s background is oftenly show varied responses as well. It could be seen from
several research result in middle rank school just like the report result in the research by
Suryadi, D (2005); Herman, T (2006) that stated that school rank is significantly
influential to the improvement of student’s mathematical skill.
It can be understood because the problems the teacher served in mathematic
learning with cognitive conflict strategy needed the teacher’s role as facilitator that will
help student have a more active role in the studying process. Meanwhile, the skill of
mathematical understanding, the skill of mathematical representing, and mathematical
belief inside the students can be improved to be more optimal because student’s role in
class can be proceed even more. Therefore, mathematical learning with cognitive conflict
strategy is potential to be able to interact with the skill of mathematical understanding,
the skill of mathematical representing, and student’s mathematical belief. This is possible
to happen when a studying process that been applied to students gave significant influence
compare to conventional learning approach.
Based on briefing above, the needs to do a study that is focused on applying
mathematical learning with cognitive conflict strategy is allegedly could develop the skill
of mathematical understanding, the skill of mathematical representing, and student’s
mathematical belief, is seen by the authors to be very urgent and prime. In this correlation,
the researchers conduct a research that linked to mathematical learning process with
cognitive conflict strategy. By considering that: (1) the research that relate to such
problem in Junior High School rank is still rare; (2) the skill of mathematical
understanding, the skill of mathematical representing, and student’s mathematical belief
are important to student for a stock to come to higher education; That is why, a research
for Junior High School rank is so important and urgent to be done soon. Therefore, the
title that is proposed for this research is “Developing The Skill of Understanding,
Representing, and Belief in Mathematic on Mathematical Learning Using Cognitive
Conflict Strategy (Eksperimental study to Junior High School student)”.
METHODS
Design and Research Procedure
This research is a quasi experimental research, with research design by pretestposttest control group design, that can be describe as follows:
O X O
--------------O
O
Notes:
X = Mathematical learning with cognitive conflict strategy
O = Measurement of mathematical understanding skill and mathematical representing
skill before and after learning process
----- = Subject is not picked randomly
In this design, every group is given pretest (O), then one group is given a
mathematical learning process with cognitive conflict strategy (X), and one group which
the class control is given a conventional learning approach or any special treatment. After
each attitude is applied to each group, then posttest is held. Pretest and posttest that are
given is a mathematical understanding skill and mathematical representing skill.
Procedure of this research is consist of three steps: preparation step, doing step,
and data analysis step. These three steps is elaborated as follows:
1. Preparation Step
Activities that is done in this step are:
a. Designing learning tool and research instrument, also asking expert’s
valuation.
b. Analysing the validation result of learning tool and research instrument with
a purpose to improve learning tool and research instrument before field test is
held.
c. Socializing mathematical learning design with cognitive conflict strategy to
the teacher and observer that will be involved in the research.
d. Held the field test and observe didactive situation and pedagogical during
testing process is occured.
e. Analysing test result of learning tool and research tool with a purpose to
improve learning tool and research instrument before experiment is held.
2. Doing Step
Activities in this step are:
a. Giving pretest to experiment class and control class. This test is held to
measure mathematical understanding skill, and mathematical representing
skill in student before mathematical learning is occured.
b. Doing the mathematical learning with cognitive conflict strategy to
experiment class (during this activity, an observation about the didactive and
pedagogical situation that happened is held).
c. Doing the conventional learning to control class (during this activity, an
observation about the didactive and pedagogical situation that happened is
held).
d. Giving posttest to experiment class and control class. This test is to measure
mathematical understanding skill and mathematical representing skill in
student after the mathematical learning is held.
3. Data Analysis Step
Activities in this step are as follows:
a. Doing data analysis and hypothesis testing.
b. Doing a session that relates with data analysis, hypothesis test, observation
test, and literature research.
c. Data analysing so that findings can be found and to arrange research report
result.
Population and Sample
Population in this research are all of the Junior High School students in
Kotamadya Bandung. The determination of this research’s sample is previously done by
classifying schools in two rank, which are school with high rank and school with middle
rank, based on the data of Bandung’s passing grade this last three years.
Randomly, SMPN 30 Bandung is chosen to represent the school with high rank,
and SMPN 27 Bandung is chosen as representation of the school with middle rank. In
each school rank, two classes are chosen randomly as the class who have mathematical
skill in relatively general, one class is given the mathematical learning with cognitive
conflict strategy (experiment class) and another class is given the conventional class
(control class). In SMPN 30 Bandung, VII-4 class is chosen to be the experiment class,
with 35 students, meanwhile VII-1 is chosen to be the control class with 35 students. In
SMPN 27 Bandung, VII-J is chosen to be experiment class, with 35 students, meanwhile
VII-I is chosen to be control class with 35 students. In total, there are 140 students that
become the research’s sample.
Variable and Operational Definition
1. There are several variables that have to be concern in this research, which are
independent variable (X) and dependent variable (Y). Independent variable in this
research is the learning variable that consist of mathematical learning with
cognitive conflict strategy (X1) and conventional learning (X2). And the
dependent variable are mathematical understanding skill/ KPM (Y1),
mathematical representing skill /KRM (Y2), and mathematical belief /KYM (Y3).
2. Mathematical learning with cognitive conflict strategy
Cognitive conflict is a consciuous situation where an individu facing an
imbalance. The imbalance is made of the consiousness of informations that are
contradict to the informations that are saved in his cognitive structure. To deal
with it, researchers use cognitive conflict strategy in mathematical learning which
consist of these phases: First phase: exposing alternative frameworks; Second
3.
4.
5.
6.
7.
phase: creating conceptual conflict; Third phase: encouraging cognitive
accomodation.
Mathematical Understanding Skill
Mathematical understanding is a cognitive level that is more complex than
a knowledge to mathematical concept. So an understanding is a definition of a
relationship between factor, concept, causation relationship, and conclusion
deducting. Mathematical understanding skill is measured to student’s skill in: (1).
Re-declaring a concept; (2). Classifying object according to its concept; (3).
Giving example and non example from a concept; (4). Serving concept in
mathematical representation; (5). Developing the needed requirement or adequate
requirement from a concept; (6). Using certain procedure or operation; and (7).
Applying concept or algorhytm of solving problem.
Mathematical Representing Skill
Mathematical representing is a description, translation, disclosure,
symbolization, or even a modelization of ideas, mathematical concept, that are
contained in one construction, or certain problem situation that student shows in
varied form as an effort to get a clear explanation, or finding solutions of the
problems he handled. Mathematical representing skill is measured on student’s
skill in: (1). Interpretating mathematical ideas into visual representation form
(pictures, diagram, graphic, or table); (2). Interpretating mathematical ideas into
symbolic as representation form (mathematical declaration/ mathematical
notation, numeric / aljabar symbol); and (3). Interpretating mathematical ideas
into verbal representation form (written text / words).
Mathematical Belief
Mathematical belief is student’s affective factor that related to mathematic
and mathematical learning. Mathematical belief in this research is a positive belief
that will trigger motivation. The scale of this mathematical belief is consist with
three aspects: (1). Beliefs about mathematics education; (2). Beliefs about the self;
and (3). Beliefs about the social context, i.e., class context.
The instruments of this research are Mathematical Understanding Skill test,
Mathematical Representing Skill test, and Mathematical Belief scale.
Data analysis techniques are:
a. Testing statistical analytic requirements that needed to be the fundamental
of hypothetical tests, which are normality test of each group and varriants
homogenity test in both paired or entirety.
b. Testing all of the proposed hypothesist with stastical test that suit the
problem and the requirement of statistical analysis.
c. Data analysis in hypothesis test is using uji-t, two way Anova, and Scheffe
test.
RESULT AND DISCUSSION
1. Result Analysis of Mathematical Understanding Skill
Data analysis result shows that in general both studied from learning method and
school rank, the average of student’s mathematical understanding skill that is given
the cognitive conflict strategy are better than student’s with conventional learning. For
the school with high rank, on the class that is given cognitive conflict strategy has the
average of 15,56 and on conventional class the average is 15,32. For school with
middle rank, the class with cognitive conflict strategy learning has an average of 15,27
and conventional class has the average of 15,11. This founding is in line with the
research that conducted by Priatna (2002) to students in third grade of Junior High
School at Banding, that found the quality of mathematical understanding skill was not
satisfying, which only about 50% from the ideal score. But if studied from the school
rank, mathematical understanding skill on students with high rank is quite satisfying,
meanwhile for the school with middle and low rank are still unsatisfying. The
description of mathematical understanding skill is served on Table 1.
Table 1
Description of Mathematical Understanding Skill based on Learning Method and
School Rank
School Mathematical
Rank Understanding
Total
Middle
High
High
Medium
Low
Sub Total
High
Medium
Low
Sub Total
High
Medium
Low
Total
Metode Pembelajaran
Cognitive Conflict
Conventional
Strategy
Mean
SD
n
Mean
SD
n
19,75
15,36
11,57
15,56
19,32
15,12
11,36
15,27
19,54
15,24
11,47
15,42
1,49
1,47
0,52
3,48
1,18
1,45
0,79
3,42
1,34
1,46
0,66
3,46
8
20
7
35
7
18
10
35
15
38
17
70
19,50
15,14
11,32
15,32
19,19
14,94
11,20
15,11
19,35
15,04
11,26
15,22
1,43
1,41
0,45
3,29
1,11
1,43
0,72
3,26
1,27
1,42
0,59
3,28
10
20
5
35
7
18
10
35
17
38
15
70
Notes: Ideal score is 25
Before doing the average difference test between each sample group, normality
and homogenity test is firstly conducted. Normality test on mathematical understanding
in school with high rank and middle rank is using Kolmogorov-Smirnov test, served on
Table 2 below.
Table 2.
Tests of Normality
Kolmogorov-Smirnova
Shapiro-Wilk
Statistic
df
Sig.
Statistic
df
Sig.
*
KPM SKK PA
,113
35
,200
,963
35
,289
*
KPM KV PA
,120
35
,200
,952
35
,131
*
KPM SKK PM
,123
35
,200
,957
35
,186
KPM KV PM
,144
35
,064
,954
35
,155
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
Based on the counting result served in Table 4.2 above, shows that SKK class and
conventional class has a significant level or probability value of each group are all bigger
that 0,005, so that it can be conclude that the result data of mathematical understanding
on each class is distributed normally.
Next, homogeneity test to variant’s score of mathematical understanding for two
sample group on each school rank is using the Levene test. The result is served on Table
3 and Table 4.
Table 3.
Test of Homogeneity of Variances
KPM PA
Levene
df1
df2
Sig.
Statistic
1,709
11
57
,095
Tabel 4.
Test of Homogeneity of Variances
KPM PM
Levene
df1
df2
Sig.
Statistic
,801
9
58
,617
Based on the counting result that is served in Table 3, shows that Levene test result
for mathematical understanding score in high rank school is 1,709, and Table 4 with
middle rank school is 0,801, meanwhile the significance value are all bigger than 0,05.
Therefore, it can be conclude that two sample group (cognitive conflict strategy class and
conventional class) in high rank and middle rank are homogenous.
After discovering that the two groups are distributed normally and homogenous,
then for the difference test will be using t test on each school rank. The counting result
with t test is served on Table 5 and Table 6.
Mean
PRE KPM
Pair
PA - POS
1
KPM PA
-4,70000 4,28124 ,51171
Mean
PRE KPM
Pair 1 PM - POS
KPM PM
Table 5
Paired Samples Test
Paired Differences
Std.
Std.
95% Confidence
Deviatio Error
Interval of the
n
Mean
Difference
Lower
Upper
-5,72082
df
Sig.
(2tailed
)
-3,67918 -9,185 69
,000
Table 6
Paired Samples Test
Paired Differences
Std.
Std.
95% Confidence
Deviatio Error
Interval of the
n
Mean
Difference
Lower
Upper
-6,62857 2,81904
,33694 -7,30075
-5,95639
t
t
df
69
19,673
Sig.
(2tailed
)
,000
Based on the counting result that is served in Table 5, the t test result shows
mathematical understanding score in high rank school is thitung = -9,185, and ttabel = t(0,025;69)
= 1,995 with significance value of 0,178. Because of thitung<ttabel then H0 is denied.
Therefore, it can be summarized that in comparation of high rank school with student’s
mathematical understanding in cognitive conflict strategy learning and student’s
mathematical understanding in conventional learning is significantly different. Table 6
with the score of mathematical understanding in middle rank school is thitung = -19,673,
and ttabel = t(0,025;69) = 1,995 with the significance value of 0,001. Because thitung<ttabel then
H0 is denied. Therefore it can be summarized that in comparation of middle rank school
with student’s mathematical understanding in cognitive conflict strategy learning and
student’s mathematical understanding in conventional learning is significantly different.
After knowing that the collage group of school with those ranks is distributed
normally and homogenous, then two way ANOVA test is conducted to discover the role
of learning factor, school rank factor, and the interraction between those two factors. The
result of two way ANOVA test is served on Table 7.
Table 7
ANOVA Counting Result on The Score of Student’s Mathematical Understanding
According to Learning Method and School Rank
Tests of Between-Subjects Effects
Dependent Variable: Mathematical Understanding
Source
Type III Sum
df
Mean
F
of Squares
Square
Corrected Model
467,486a
3
155,829
29,175
Intercept
36612,114
1
36612,114 6854,691
SCHOOL
291,457
1
291,457
54,568
LEARNING
120,714
1
120,714
22,601
SCHOOL *
55,314
1
55,314
10,356
LEARNING
Error
726,400
136
5,341
Total
37806,000
140
Corrected Total
1193,886
139
a. R Squared = ,392 (Adjusted R Squared = ,378)
Sig.
,000
,000
,000
,000
,002
From the ANOVA result that is served on Table 7 above shows that significance
value from Learning factor, School factor, and interaction factor (School * Learning), all
of them is smaller than 0,05 which is 0,000. Therefore it can be concluded that: (a)
student’s mathematical understanding skill with Cognitive Conflict Strategy learning is
significantly different to students with conventional learning; (b) there is a significant
difference of student’s mathematical understanding skill in two school rank groups; (c)
there is an interaction between school rank factor and learning factor to student’s
mathematical understanding skill in general.
Interaction between school and learning factor to student’s mathematical
understanding skill can be seen from Table 7. On the table, it is shown that interaction
between school and learning factor have F value as 10,356 and its significance is smaller
than 0,05 which is 0,002. So it can be conclude that there is a significant interaction
between school factor and learning factor to student’s mathematical understanding skill.
This also shows that cognitive conflict strategy can be applied to high rank school and
middle rank school to improve student’s mathematical understanding in mathematical
learning. This interaction can also be seen graphically in Image 1.
Image 1. Interaction of School Rank and Learning Method
to Mathematical Understanding Skill
2. Result Analysis on Mathematical Representing Skill
Data result analysis shows that whether to be reviewed on learning method or on
school rank, the average of student’s mathematical representing skill with cognitive
conflict strategy given is better than student’s with conventional learning. For high
rank school with the class that is given cognitive conflict strategy, has the average of
15,55, and in conventional class, the average is 14,93. This discovery is in accordance
with research conducted by Rahman (2004), that in the research, he observed about
how the students choose representing way to communicate their problems and
investigate the relation between representing way and achievement level. The
description of mathematical representing skill is served on Table 8.
Table 8
The Description of Mathematical Representing Skill Based on Learning Method
and School Rank
Learning Method
Cognitive Conflict
School Mathematical
Conventional
Strategy
Rank
Representing
Mean SD
n Mean SD
n
Middle
High
High
Medium
Low
Sub Total
High
Medium
Low
Sub Total
19,78
15,15
11,71
15,55
19,27
14,82
11,36
15,15
1,29
1,61
0,55
3,45
0,76
1,52
0,69
2,97
9
20
6
35
9
15
11
35
19,57
15,03
11,56
15,39
19,03
14,61
11,14
14,93
1,32
1,49
0,49
3,30
0,71
1,49
0,67
2,87
9
19
7
35
8
16
11
35
Total
High
Medium
Low
Total
19,53
14,99
11,54
15,35
1,03
1,57
0,62
3,22
18
35
17
70
19,30
14,82
11,35
15,16
1,02
1,49
0,58
3,09
17
35
18
35
Notes: Ideal Score 25
Before testing the average difference between sample groups, normality test and
homogeneity test is conducted firstly. Normality test to mathematical representing in
high rank and middle rank school is using Kolmogorov-Smirnov test, served on Table
9.
Table 9
Tests of Normality
Kolmogorov-Smirnova
Shapiro-Wilk
Statistic
df
Sig.
Statistic
df
Sig.
KRM SKK PA
,140
35
,080
,939
35
,052
*
KRM KV PA
,122
35
,200
,952
35
,128
KRM SKK PM
,148
35
,052
,921
35
,015
KRM KV PM
,139
35
,084
,928
35
,024
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
Based on the counting result in Table 9 above, shows that for SKK Class and
conventional class have significance level or probability value of each group are all bigger
than 0,005, so it can be conluded that the data result of mathematical representing in for
those class is distributed normally.
Following it, homogeneity test to variants’ mathematical representing score for two
sample groups in each school rank is conducted by using Levene test. The counting
result is served on Table 10 and Table 11.
Table 10
Test of Homogeneity of Variances
KRM PA
Levene
df1
df2
Sig.
Statistic
,923
9
57
,512
Table 11
Test of Homogeneity of Variances
KRM PM
Levene
df1
df2
Sig.
Statistic
,545
9
58
,836
Based on the counting result in Table 10, shows that Levene test’s result for
mathematical representing score in high rank school is 0,923, and Table 11 for middle
rank school is 0,545, meanwhile the significance value are all bigger than 0,05. Therefore
it can be concluded that the two sample groups (cognitive conflict strategy class and
conventional class) on each high rank school and middle rank school is homogenous.
After knowing that those two groups are distributed normally and homogenous, then
the difference test is conducted by using t test to each school rank. The t test counting
result is served on Table 12 and Table 13.
Table 12
Paired Samples Test
Paired Differences
t
df Sig.
(2Mean
Std.
Std.
95% Confidence
tailed
Deviatio Error
Interval of the
)
n
Mean
Difference
Lower
Upper
PRE KRM
Pair 1
PA - POS
-5,11429 3,05762 ,36546 -5,84335 -4,38522 -13,994 69 ,000
KRM PA
Mean
Pair 1
PRE KRM
PM - POS
KRM PM
Table 13
Paired Samples Test
Paired Differences
Std.
Std.
95% Confidence
Deviatio Error
Interval of the
n
Mean
Difference
Lower
Upper
-5,77143 2,42075 ,28934 -6,34864 -5,19422
t
df
Sig.
(2tailed
)
-19,947 69
,000
Based on the counting result at Table 12 above, shows that t test result for
mathematical representing score in high rank school is thitung = -13,994, and ttabel = t(0,025;69)
= 1,995 with significance value of 0,000. Because thitung<ttabel then H0 is denied. Therefore
it can be concluded that on high rank school, the result of student’s mathematical
representing with cognitive conflict strategy learning compare to the result of student’s
mathematical representing with conventional learning is significantly different. Table 13
on middle rank school with the t test to show the score of the mathematical representing
is thitung =-19,947, and ttabel = t(0,025;69) = 1,995 with significance value of 0,000. Because
of thitung<ttabel then H0 is denied. Therefore it can be concluded that in middle rank school,
the score of student’s mathematical representing with cognitive conflict strategy learning
compare to the score of student’s mathematical representing with conventional learning
is significantly different.
After knowing that the combination of those school ranks are distributed normally
and homogenous, then two way Anova test is conducted to discover the role of learning
factor, school rank factor, and the interaction between those two factors. The counting
result of two way Anova test is served on Table 14.
Table 14
Counting Result of ANOVA to The Score of Student’s Mathematical Representing
According to Learning Method and School Rank
Tests of Between-Subjects Effects
Dependent Variable: Mathematical Representing
Source
Type III Sum
df
Mean
F
Sig.
of Squares
Square
Corrected Model
687,486a
3
229,162
36,533
,000
Intercept
39111,429
1
39111,429 6235,193
,000
SCHOOL
460,829
1
460,829
73,466
,000
LEARNING
173,829
1
173,829
27,712
,000
SCHOOL *
52,829
1
52,829
8,422
,004
LEARNING
Error
853,086
136
6,273
Total
40652,000
140
Corrected Total
1540,571
139
a. R Squared = ,446 (Adjusted R Squared = ,434)
From the ANOVA counting result on Table 14 above, shows that the significance
value from Learning factor, School factor, and interaction factor (School * Learning), all
of them are smaller than 0,005 which are 0,000. Therefore, it can be concluded that: (a)
student’s mathematical representing skill with Cognitive Conflict Strategy learning is
significantly different to students with conventional learning; (b) there is a significance
different in student’s mathematical representing skill on two groups of school rank; (c)
there is an interaction between school rank factor and learning factor to student’s
mathematical representing skill in general.
Interaction between school factor and learning factor to student’s mathematical
representing skill can be seen at Table 14. On the table, shows that the interaction of
school factor and learning factor has F value of 8,422 and its significance is smaller than
0,005 which is 0,004. Therefore, it can be concluded that there is a significance interaction
between school factor and learning factor to student’s mathematical representing skill.
This also shows that cognitive conflict strategy can be applied on high and middle rank
school to improve student’s mathematical representing in mathematics learning. This
interaction can be seen graphically in Image 2.
Image 2. Interaction of School Rank and Learning Method
to Mathematical Representing Skill
Middle
High
3. Analysis Result of Mathematical Belief
Data analysis result shows that in general, whether reviewed from learning method
and school rank, the average of student’s mathematical belief with cognitive conflict
strategy is better than students with conventional learning. For high rank school with
cognitive conflict strategy learning, the average is 200,27, and in conventional class,
the average is 199,88. For middle rank school, the class with cognitive conflict strategy
has the average of 199,36, and the average in conventional class is 198,44. This
discovery is in line with the research result about belief to mathematics, which was
conducted by Schoenfeld (1992), showed that there is a strong correlation between the
result of mathematical test that students expected and their belief to their skill.
Description of the mathematical belief in table form is served on Table 15.
Table 15
Description of Mathematical Belief According
to Learning Method and School Rank
Learning Method
Cognitive Conflict
School Mathematica
Conventional
Strategy
Rank
l Belief
Mean
SD
n
Mean
SD
n
High
220,42
Medium
Low
Sub Total
High
Medium
Low
198,58
181,82
200,27
219,73
198,08
180,27
6,27
5,57
8,00
19,84
5,64
6,09
8,20
12
12
11
35
11
13
11
220,36
198,46
180,82
199,88
218,91
197,15
179,27
5,61
6,20
8,00
19,81
5,41
5,87
7,73
11
13
11
35
11
13
11
199,36
220,08
198,33
181,05
199,82
Total
Sub Total
High
Medium
Low
Total
Notes: Ideal score 250
19,93
5,96
5,83
8,10
19,89
35
23
25
22
70
198,44
219,64
197,81
180,05
199,17
19,01
5,51
6,04
7,87
19,42
35
22
26
22
70
Normality test for mathematical belief scale in high rank school and middle rank
school is using Kolmogorov-Smirnov test. The counting result for each school rank is
served on Table 16.
Table 16
Tests of Normality
Kolmogorov-Smirnova
Statistic
KYM SKK PA
KYM KV PA
KYM SKK PM
KYM KV PM
df
,095
,095
,086
,097
Shapiro-Wilk
Sig.
Statistic
df
Sig.
35
,200*
,966
35
,351
35
,200*
,966
35
,351
35
,200*
,966
35
,354
35
,200*
,966
35
,347
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
Based on the counting result served in Table 16 above, shows that for Cognitive
Conflict Strategy class and conventional class have significance level or probability
value on each group are all bigger than 0,05, therefore it can be concluded that the
result data for mathematical belief score on each class is distributed normally.
Next, homogeneity test is conducted to discover the mathematical belief variant’s
score to two sample groups on each school rank using the Levene test. The counting
result is served on Table 17 and Table 18.
Table 17
Test of Homogeneity of Variances
KYM PA
Levene Statistic
df1
1,721
df2
16
Sig.
30
,097
Table 18
Test of Homogeneity of Variances
KYM PM
Levene Statistic
1,560
df1
df2
18
Sig.
27
,144
Based on the counting result served on Table 17, shows that Levene test result to
score mathematical belief on high rank school is 1,721, and Table 18 for middle rank
school is 1,560, meanwhile the significance value are all bigger than 0,05. Therefore,
it can be concluded that the two sample groups (cognitive conflict strategy class and
conventional class) in each high rank and middle rank school is homogenous.
After knowing that the combination of these school rank are distributed normally
and homogenous, then two way ANOVA test is conducted to discover the role of
learning factor, school rank factor, and the interaction between those two factors. The
counting result with two way ANOVA test is served on Table 19.
Table 19
The Counting Result of ANOVA to The Score of Student’s Mathematical Belief
based on Learning Method and School Rank
Tests of Between-Subjects Effects
Dependent Variable: Mathematical Belief
Source
Type III Sum of
df
Mean Square
F
Sig.
Squares
107,907a
3
35,969
,122
,947
5575626,579
1
5575626,579
18904,899
,000
SEKOLAH
75,779
1
75,779
,257
,613
PEMBELAJARAN
32,064
1
32,064
,109
,742
,064
1
,064
,000
,988
Error
40110,514
136
294,930
Total
5615845,000
140
40218,421
139
Corrected Model
Intercept
SEKOLAH *
PEMBELAJARAN
Corrected Total
a. R Squared = ,003 (Adjusted R Squared = -,019)
From the ANOVA result served on Table 19 above shows that significance value
of Learning factor, School factor, and interaction factor (School * Learning), are all
bigger than 0,05 which is 0,988. Therefore, it can be concluded that: (a) student’s
mathematical belief with Cognitive Conflict Strategy learning method is not significantly
different with students that learning with conventional way; (b) there is no significant
different on student’s mathematical belief on the two groups of school rank; (c) there is
no interaction between school rank factor and learning factor to student’s mathematical
belief in general.
Interaction between school factor and learning factor to student’s mathematical
belief can be seen on Table 19. On the table, shows that interaction between school and
learning factor has the F value of 0,000 and its significance is bigger than 0,05, which is
0,988. So it can be concluded that there is no significant interaction between school and
learning factor to student’s mathematical belief.
CONCLUSION
1.
2.
3.
4.
5.
6.
The summary of this research is as following.
In general, student’s mathematical understanding skill that is given the cognitive
conflict strategy is significantly different than student that is given the
conventional learning.
There is an interaction between learning method (cognitive conflict strategy and
conventional) and school rank (high and middle) to student’s mathematical
understanding skill.
In general, student’s mathematical representing skill that is given the cognitive
conflict strategy is significantly different than student that is given the
conventional learning.
There is an interaction between learning method (cognitive conflict strategy and
conventional) and school rank (high and middle) to student’s mathematical
representing skill.
In general, student’s mathematical belief that is given the cognitive conflict
strategy learning is not significantly different than student that is given the
conventional learning.
There is no interaction between learning method (cognitive conflict strategy and
conventional) and school rank (high and middle) to student’s mathematical belief.
REFERENCES
Ausubel, D. P. (1978). Educational Phsychology: A cognitive view 2nd. New York: Holt
Rinehart and Winstone.
Goldin, G.A. (2002). “Affect, Meta-Affect, and Mathematical Beliefs Structures”.
Beliefs: A Hidden Variable in Mathematics Education? Editor: Leder, G.C.,
Pehkonen, W., dan Torner, G. London: Kluwer Academics Publisher.
Greer, B., Verschaffel, L., & Corte, E.D. (2002). “The Answer is Really 4,5: Beliefs about
Word Problems”. Beliefs: A Hidden Variable in Mathematics Education? Editor:
Leder, G.C., Pehkonen, W., dan Torner, G. London: Kluwer Academics Publisher.
Herman, T (2006). Pembelajaran Berbasis Masalah Untuk Meningkatkan Kemampuan
Berpikir Matematis Tingkat Tinggi Siswa Sekolah Menengah Pertama.
Bandung:PPS UPI. Disertasi tidak diterbitkan.
Hiebert J & Carpenter, T.P. (1992). Learning and Teaching with Understanding. In D.A
Grouws (Ed). Handbook of Research on Mathematics Teaching and Learning.
NCTM. NewYork : Macmilan Publishing Company.
Nunokawa, K. (1998). “Empirical and Autonomical Aspect of School Mathematics”.
Tsukuba Journal of Educational Study in Mathematics, Vol. 17, 1998, pp. 205-217.
[Online] Tersedia: http://www.juen.ac.jp/g_katei/ nunokawa/ kaita/ empirical.pdf
[1 Juni 2008].
Priatna, N. (2002). Kemampuan Penalaran dan Pemahaman Matematika Siswa Kelas III
SLTP Negeri di Kota Bandung. Disertasi doktor PPS UPI Bandung: tidak
dipublikasikan.
Rahman, A.(2004). Meningkatkan Kemampuan Pemahaman dan Kemampuan
Generalisasi Matematik Siswa SMA melalui Pembelajaran Berbalik. Tesis pada
Sekolah Pasca Sarjana UPI.: Tidak Diterbitkan.
Schoenfeld, A. H. (1992) Learning to Think Matematically: Problem solving,
metacognition, and sense making in mathematics. Electronic Journal: International
Journal For Mathematics Teaching and Learning. [Online]. Tersedia:
http://www.intermep.org.
Suryadi, D. (2005). Penggunaan Pendekatan Pembelajaran Tidak Langsung Serta
Pendekatan Gabungan Langsung dan Tidak Langsung dalam Rangka
Meningkatkan Kemampuan Berpikir Matematik Tingkat Tinggi Siswa SLTP.
Disertasi, PPS-UPI. Tidak diterbitkan.
Wu, Z. (2004). The Study of Middle School Teachers’ Understanding and Use of
Mathematical Representation in Relation to Teachers’ Zone of Proximal
Development in Teaching Fractions and Algebraic Functions.