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The Influence of Independent Learning and
Structure Sense Ability on Mathematics
Connection in Abstract Algebra
Junarti
IKIP PGRI Bojonegoro
YL. Sukestiyarno
Universitas Negeri Semarang
junarti@ikippgribojonegoro.ac.id
Mulyono
Universitas Negeri Semarang
Nur Karomah Dwidayati
Universitas Negeri Semarang
Abstract---This study aimed to reveal the influence of
independent learning and structure sense ability on
mathematics connection in abstract algebra through
mentoring modules as an initial step to overcome the
students’ difficulties in learning and to habituate
students to recognize the structure sense and
mathematics connections through weekly tasks. This
research was conducted at the fifth-semester students
of the Mathematics Education Study Program for 7
weeks. A quantitative research design with two
independent variables (independent learning and
structure sense ability) and one dependent variable
(mathematics connection ability) was employed in this
study. This study took 26 students for the sample. The
data to measure independent learning was gathered
through a questionnaire; the data to recognize the
structure sense ability was gathered through weekly
tasks, and the students’ mathematics connections were
measured through a test. The results of simple
regression and multiple regression analyses,
simultaneously with independent learning and
structure sense variables, affect students’ mathematics
connection ability in abstract algebra course. The
results indicate that independent learning is more
dominant than structure sense ability in influencing
students’ mathematics connection ability. Thus, in
order to achieve high mathematics connection abilities,
students should, first, have high independent learning,
and then develop the ability to recognize structure
sense.
Keywords: independent learning, structure sense,
mathematics connection, abstract algebra
I. INTRODUCTION
It is expected that authors will submit
carefully The importance of building the character
of independent learning is to meet the demands of
Presidential Regulation Number 87 in the year 2017,
and the demands of the 21st-century in learning
mathematics. One of the central character values that
originate from Pancasila, which are prioritized for
the development of character building, is
independent learning. Independent learning is an
attitude and behaviour that does not depend on
others and use all energy, thoughts, and time to
realize hopes, dreams and ideals. Students who have
Advances in Social Science, Education and Humanities Research, volume 443
International Conference on Science and Education and Technology (ISET 2019)
Copyright © 2020 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license -http://creativecommons.org/licenses/by-nc/4.0/. 57
independent learning have a good work ethic, robust,
empowered, professional, creative, courage, and
become lifelong learners [1].
There are four mathematical skills developed
in the 21st-century era, i.e., critical thinking skills,
creative thinking skills, communication skills, and
collaboration skills [2]. From these 4 pillars, a
bachelor student should refer to, at least, mastering
theoretical concepts in specific fields of knowledge
and skills in general and theoretical concepts
specifically in the field of knowledge and skills
deeply [3].
Problem-solving and independent learning
skills are two skills that should be owned by students
in learning physics at the college level [4]. It is
based on the characteristics of physics material that
are considered difficult and complicated. Therefore,
they developed a Book entitled ‘Model Physics
Independent Learning’ (PIL) to improve problem-
solving skills and students' independent learning
skills [4].
Several learning approaches have been
carried out in order to develop independent learning
skills through a lecture contract approach and to
improve the students’ independence, as well as
mathematics learning outcomes in the material of
differential calculus with the Snowball Drilling
method has an average score of 81.62% with
minimum completeness of 89.74% [5] [6].
Through independent learning, a student will
be able to determine the steps that must be taken in
learning, able to obtain self-learning resources, and
able to conduct self-evaluation activities and
reflection on learning activities that have been
carried out in abstract algebra course [7]. Further, [8]
states that reciprocal teaching-learning is better than
a facilitator model in fostering students’
independence in learning mathematics. These habits
become one of the reasons to increase other factors
such as the awareness of learning and the sensitivity
to structure sense which becomes an important part
of abstract algebra course.
Yerizon [9] reviewed students’ independent
learning in the Real Analysis course with modified
APOS approach. The result shows the medium
category. Then, Zhang et al., [10] conducted another
study related to the relationship of mathematical
anxiety (MA) on mathematical performance. The
result shows that middle school students in Asian
countries have stronger negative mathematics
anxiety than European students.
Independent learning is positively correlated
with academic achievement in traditional higher
education classroom settings for several samples
[11]. It has a positive influence on students’ learning
outcomes in the algebraic structure course [7].
However, it is not denied that academic
achievement, especially mathematics, in general, is
low. Nevertheless, it is not an obstacle to always try
to improve in terms of various components of
mathematical knowledge including mathematical
objects.
There are several components of
mathematical knowledge as a part of direct
mathematical objects such as the introduction of
structure sense in algebra courses at the university
level. Recognizing the structural sense of the set
elements in the form of numbers is one of the
obstacles in learning abstract algebra [12]. This is
because the students do not master the basic,
intermediate, and secondary mathematics that
underlies the set [13]. Some structural components
in university algebra are analogous components of
high school algebra, so it is recommended to
emphasize the structure sense of high school algebra
[14]. The sensitivity of the mathematical structure
[15] can lead to an intuitive ability to symbolic
expressions, including skills to interpret, manipulate,
manage, and perform symbols in different roles in
learning algebra [16]. Low sensitivity to structure
(structure sense) can also affect students’ algebraic
thinking abilities [17] and the ability to connect
mathematical structures in algebra.
The structure that is described as a structure
in a set and several types of set properties is a
comparison of its properties, and a structure as all
properties that are studied [18]. Besides, the
structure of algebra is also a knowledge that informs
the nature of solving equations, simplifying
expressions, and multiplying polynomials [19].
Abstract algebra is the study and
generalization of the structure of algebraic structures
needed for algebraic reasoning [13]. Abstract
algebra is also an essential part of the preparation of
secondary and middle school teachers [20]. The core
concepts in abstract algebra are binary operations
and functions. The core concept in abstract algebra
has a productive potential to connect to the middle
school level. Binary functions and operations as
initial concepts play essential roles in many abstract
algebra topics [21].
The vital role of the concept is to connect the
school mathematics, thus learning abstract algebra
requires basic mathematics. This interest is intended
to help build connections in understanding abstract
algebra. Some scholars have examined the
relationship between school mathematics and
abstract algebra [19][22][23][24][25][26]. Cook [27]
further emphasizes in his dissertation hypothesis that
the difficulties experienced by students in learning
abstract algebra are due to the lack of an established
connection between university mathematics and
school mathematics.
Some connections are formed when students
try to build formulas through procedures that lead to
the acquisition of concepts in the unit (Evitts in
[28]). Formulas, rules, and algorithms are used for
completing any mathematical tasks [29]. Bass [24]
provides an example of how ideas from abstract
algebra and other fields of mathematics can be
developed from and connected with school
curriculum mathematics. Emphasizing mathematics
connections and helping to understand the operation
or nature of algebra are used to achieve coherence
[15] thoroughly.
Building connections as mathematical
processes or activities throughout mathematics,
students must be involved in building activities or
identifying the connections that are contained
[18][30] are the aim of this study.
The purpose of this study is (1) to examine
the extent to which the students’ independent
learning in abstract algebra through mentoring
modules as a first step to overcoming students’
difficulties can influence, simultaneously, with the
ability to recognize a structure sense on the ability of
mathematics connections, and (2) to assess the
extent to which the students’ habits of independent
learning and the ability to recognize structure sense
through weekly tasks, partially, affects the ability of
mathematics connections.
While the formulation of the problems in this
study is as follows.
(1) Do the students’ independent learning and the
structure sense ability simultaneously affect the
mathematics connections in abstract algebra
material?
(2) Do the students’ independent learning and the
structure sense ability partially affect the
mathematics connections in abstract algebra
material?
(3) Which variable, independent learning or structure
sense ability, is more dominant in influencing the
mathematics connection?
II. RESEARCH METHOD
The method used in this study is quantitative,
with one saturated sample group [31] of 26 students.
This research was conducted for 7 weeks. The
treatment was by giving the students a module to
assist the students in independent learning activities.
Every week, the students were given independent
assignments that are already contained in the
module. The assignment was submitted based on the
Advances in Social Science, Education and Humanities Research, volume 443
58
schedule, so there is no overlap with other students.
The goal is that the lecturer can freely observe and
ask a little about the tasks being done. Then after 7
weeks, a mathematics connection test and an
independence questionnaire were given.
The research variables are two independent
variables = and one dependent variable = .
Independent learning is variable; the structure
sense ability is variable, and mathematics
connection is variable. The series of statistical
analyses were employed, such as assumption test,
linearity test, multiple regression test, the partial test
of multiple regressions, and determination test using
the SPSS program (Sukestiyarno, 2016).
2.1.
Research Instrument
The research instrument used was a questionnaire. It
was used to measure independent learning with 38
questions (20 favourable questions and 18
unfavourable questions) using a Likert scale. The
instrument to measure the ability of structure sense
was using the questions in the module as a weekly
task. The questions in the assignment were arranged
based on indicators of ability to recognize the
structure sense and are equipped with rubrics and
indicator predictions on the questions about structure
sense.
2.2.
Instrument Trials
Meanwhile, to measure the ability of mathematics
connections, test questions consisting of 3 item
descriptions were used. All questions were validated
by experts and tested on students who had taken
abstract algebra. The results of the empirical
validation of the two independent learning
questionnaire instruments and mathematics
connections with SPSS obtained high categories for
the connection test questions by 0.846, and the
reliability of the independent learning questionnaire
was 0.677 with ‘sufficient’ categories and the
validity of all valid items results varied between
sufficient, high and very high. While the validity of
the mathematics connection instrument, item 1 was
0.922, the category was very high, item 2 was 0.685,
the category was quite high, and item 3 was 0.928,
the category was quite high.
III. RESEARCH FINDINGS
The data were analyzed statistically, including
assumption tests, linearity tests, simple regression
tests, multiple regression tests, and multiple partial
tests (Sukestiyarno, 2016). The assumption test
shows that the normality test with the Kolmogorov
Smirnov test shows the significance value sig =
0.001> 0.05 which means the distribution of
variables is normal. Furthermore, the decision is also
strengthened in Figure 1, which shows the diagram
is not far from the normal diagonal line, and then the
data is normally distributed. Overall, the data of the
mathematics connection ability is normally
distributed. So the assumptions are fulfilled.
Figure 1 Plot of Q-Q Diagram
Furthermore, the homogeneity test results, based on
Figure 2, show that because the value of kurtosis = -
0.536, it shows a negative value, so the data tends to
be blunt, but the value is not far from zero so it can
be said to be homogeneous data. Next, by looking at
all three quartile values, they indicate values that are
not too wide. If it is seen from the box plot in Figure
2 and Figure 3, it does not show a significant slope
and because the normality test has been met, it can
be concluded that the assumption of homogeneity is
met.
Figure 2 Output Statistic
Figure 3 Homogeneity test results of mathematics
connection ability
Statistics
Mathematics Connection Variable
N
Valid
26
Missing
0
Mean
55.0000
Std. Error of Mean
3.56133
Skewness
-.963
Std. Error of Skewness
.456
Kurtosis
-.536
Std. Error of Kurtosis
.887
Percentiles
25
39.2500
50
64.0000
75
68.0000
Advances in Social Science, Education and Humanities Research, volume 443
59
The linearity test through testing simple linear
regression to with the Model
shows a linear equation or has a linear
relationship with . Similarly, testing simple linear
regression to with the Model
shows a linear equation or has a linear
relationship with y.
Because the assumption test and the linearity
test are fulfilled, further tests, namely the multiple
regression test and the multiple partial regression
test, to examine the effect of the two independent
variables on the dependent variable are presented in
the following explanation.
3.1 Multiple Regression Test
Regression model:
a) Forms of Linear Model Hypotheses:
(the equation is non-linear or
there is no relation between and )
(the equation is linear or there is a
relation between and )
b) The analysis design formulation: the multiple linear
model estimator is = with a two-
party test and a significance level of 5%. The
regression equation based on the sample can be seen
in table 1. It was obtained values
and , so the regression equation is
=
c) Testing the values of a, b, and c by accepting or
rejecting the hypothesis can be seen in table 2. It was
obtained value F = 30,006, sig = 0,000. Because the
value of sig = 0,000 <0.05 then is rejected and
is accepted. So, the equation is linear or
simultaneously have a linear relationship to y or
simultaneously have a positive effect on y.
d) Analysis of the coefficient of determination can
be seen in table 3. The summary obtained value
. This value indicates that the
independent learning variable can explain the
variation of the mathematics connection variable y
and the sense structure variable simultaneously
by 72.3%. In other words, and simultaneously
affect the mathematics connection variable y by
72.3%, and there are still 27.7% of the y variable
that is influenced by other variables.
Table 1. Output Coefficients
Model
Unstandardized
Coefficients
Standard
ized
Coeffici
ents
t
Sig.
B
Std.
Error
Beta
1
(Constant)
-
56.313
17.285
-3.258
.003
Independe
nt
Learning
Variable
1.024
.460
.386
2.227
.036
Structure
Sense
Variable
.807
.271
.516
2.976
.007
Table 2. Output ANOVA
Model
Sum of
Squares
df
Mean
Square
F
Sig.
1
Regression
5959.847
2
2979.924
30.006
.000a
Residual
2284.153
23
99.311
Total
8244.000
25
Table 3. Output Model Summary
Model
R
R Square
Adjusted R
Square
Std. Error of
the Estimate
1
.850a
.723
.699
9.96549
3.2 Partial Test of Multiple Regressions
Form of the hypothesis proposed:
: Regression coefficient is not significant (there is
no effect)
: Regression coefficient is significant (there is an
effect) The results of the analysis can be seen in
table 1, on the sig value of the t distribution. It was
obtained the independent learning variable sig =
0.036 <0.05, so is rejected, and is accepted,
meaning that independent learning affects the
variable of mathematics connections. Whereas for
structure sense variables, the value of sig = 0.007
<0.05 so is rejected and is accepted, meaning
that the structure sense variable influences the
mathematics connection variable.
At last, the regression test, in this case, can be
concluded that both partial and multiple regressions
of simultaneously affect the y variable.
Advances in Social Science, Education and Humanities Research, volume 443
60
3.3 Investigation of Dominant Influence
Factors
The variable influences y variable by
61.3%, after involving the variable is only able to
increase R2 by 72.3% - 61.3% = 11%. On the other
hand, the variable affects y variable by 66.3%, by
involving the variable, it can increase the value of
R2 by 84.9% - 66.3% = 18.6%. So the variable
gives more dominant contribution to the y variable.
3.4 Multicollinearity, Autocorrelation, &
Heteroscedasticity Checks
Table 4. Output Coefficients
Model
Collinearity Statistics
Tolerance
VIF
1
Independent Learning
Variable
.401
2.493
Sense Structure
Variable
.401
2.493
From table 5 below, it can be seen that the
value of tolerance and VIF are far from 1, so, it can
be concluded that there is multicollinearity
disturbance. Then, the table also shows that the
correlation between independent learning and
structure sense is above 0.5, i.e., -0.774. This shows
a high degree of correlation. It means that there is an
intersection of indicators between independent
learning and structure sense.
Table 5 Output Coefficient Correlations
Model
Structure
Sense
Variable
Independent
Learning
Variable
1
Corre-
lations
Structure
Sense
Variable
1.000
-.774
Independent
Learning
Variable
-.774
1.000
Cova-
riances
Structure
Sense
Variable
.074
-.096
Independent
Learning
Variable
-.096
.211
To check the autocorrelation, the Durbin-Watson
value can be seen from table 6 below. It shows the
value of 1.419. This value is in the interval -2 <DW
<2, meaning that it is in an area that there is no
autocorrelation. It means that the assumption of each
observation measurement from one observation to
the next is to meet the requirements to have a
homogeneous variant.
Table 6 Output Model Summary
Model
R
R
Square
Adjusted
R
Square
Std. Error
of the
Estimate
Durbin-
Watson
1
.850
.723
.699
9.96549
1.419
For heteroscedasticity checks, it can be seen
in the scatter plot diagram between the errors that
occur (the difference between the prediction of the
dependent variable with the dependent variable
observational data): it appears that the points that
occur are entirely spread around the zero lines, some
are above the zero lines and there are which is below
the zero line. In this case, it does not form a specific
pattern. So the assumption that the variant error is
identical is fulfilled.
Figure 3 Plot Diagram
The general conclusion of simple regression
analysis and multiple regression analysis dealing
with the effect of independent learning and
understanding of structure sense in abstract algebra
caused by mentoring process of using structure
sense-based modules affect the achievement of
mathematics connection ability.
Table 7. Output Residuals Statistics
Minimum
Maximum
Mean
Std.
Deviation
N
Predicted Value
20.8066
86.3867
55.0000
15.44001
26
Std. Predicted Value
-2.215
2.033
.000
1.000
26
Standard Error of
Predicted Value
1.993
5.134
3.248
.974
26
Adjusted Predicted
Value
20.7455
90.4157
55.1751
15.66519
26
Residual
-17.28585
20.64243
.00000
9.55856
26
Std. Residual
-1.735
2.071
.000
.959
26
Stud. Residual
-1.829
2.218
-.008
1.016
26
Deleted Residual
-19.41575
23.66117
-.17509
10.74445
26
Stud. Deleted Residual
-1.935
2.446
-.003
1.066
26
Mahal. Distance
.038
5.674
1.923
1.718
26
Cook's Distance
.000
.263
.042
.070
26
Centered Leverage
Value
.002
.227
.077
.069
26
Advances in Social Science, Education and Humanities Research, volume 443
61
To sum up, it turns out that the dependent
variable tends to be normally distributed and
homogeneous. This shows that the mentoring
strategy through a structure sense-based module can
raise students’ independent learning in building
mathematics connection skills that are almost equal
to the mean of 55,0000 (see table 7). Based on the
test, it shows that the independent learning variable
has a dominant effect compared to the structure
sense variable on the mathematics connection ability
variable. It means that the variation in students’
mathematics connection ability is explained more by
the independent learning variable than the structure
sense variable. Thus, in order to achieve high
mathematics connection abilities, students should,
first, have high independent learning, and then
develop the ability to recognize structure sense.
IV. CONCLUSIONS
Based on the analysis of statistical test data,
the following conclusions are obtained.
1) There is an influence of students’ independent
learning and the ability to recognize a structure sense
simultaneously on the ability of mathematics
connections in abstract algebra material;
2) There is an influence of students’ independent
learning and the ability to partially recognize
structure sense on the ability of mathematics
connections in abstract algebra material; and
3) The more dominant variable that influences the
mathematics connection is independent learning
compared to structure sense.
To sum up, it turns out that the dependent
variable tends to be normally distributed and
homogeneous. This shows that the mentoring
strategy through a structure sense-based module can
raise students’ independent learning in building
mathematics connection skills that are almost equal
to the mean of 55,00 (see table 7). Based on the test,
it shows that the independent learning variable has a
dominant effect compared to the structure sense
variable on the mathematics connection ability
variable. It means that the variation in students’
mathematics connection ability is explained more by
the independent learning variable than the structure
sense variable. Thus, in order to achieve high
mathematics connection abilities, students should,
first, have high independent learning, and then
develop the ability to recognize structure sense.
Acknowledgments. The sincere gratitude goes to
the Rector of IKIP PGRI Bojonegoro for giving
permission to study and giving time for this
research.
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