Student Worksheet in Mathematical Creative Thinking
Retno Marsitin and Nyamik Rahayu Sesanti
Universitas Kanjuruhan Malang, Jl. S. Supriyadi 48 Malang, Indonesia
Keywords: Student worksheet, mathematical creative thinking.
Abstract: This study aimed to develop student worksheet on a calculus with APOS to improve mathematical creative
thinking. The development model includes: identifying learning objectives, conducting learning
analyzes, formulating specific objectives, developing assessment instruments, developing learning
strategies, developing and selecting learning materials, designing and executing evaluation,
revising learning materials, designing and carry out evaluation. The research is done by validating the
student worksheet on calculus with APOS and the subject of the test was the mathematics education student.
The results show that the trial of student worksheets on the calculus with APOS showed very good and very
effective results and appropriate to be used in learning. This is because the student worksheet got positive
responses from student and learning by using student worksheet is very interesting and students understand
the calculus material more easily and student’s ability of mathematical creative thinking improved with 83%
achievement. The student worksheets on the calculus material with APOS has impact on the student's
academic ability which has improved very well.
1 INTRODUCTION
One of the innovation efforts in order to improve
students' creative thinking ability is by developing
student worksheet. Student worksheet is developed
by using APOS with stages that include action,
process, object, and scheme. The developed student
worksheet using APOS is on the calculus material.
The development student worksheet with APOS is
expected to improve students’ creative thinking
ability
APOS stands for action, process, object, and
schema. APOS requires the ability of students to think
creatively in solving mathematical problems.
(Dubinsky and Mcdonald, 2002) argue that APOS is
a mathematical learning approach at the college level,
which integrates computer use, small group
discussion, and considering the mental constructions
by students in understanding a mathematical concept.
These mental constructions are APOS which includes
action, process, object, and scheme. (Weller, Clark
and Dubinsky, 2003) states that APOS can be used to
improve thinking ability that are particularly suitable
for advanced mathematical materials, including:
functions; abstract algebra topics such as binary,
group, subgroup, normal group; discrete
mathematical topics such as mathematical induction,
permutation, symmetry; statistical topics such as
averages, standard deviations, and central calculus
theorems; the topic of number such as place value in
the n-base number, division, multiplication and the
conversion of numbers from one base to another;
topics of calculus such as calculus, chain rules,
graphic derivations, and infinite sequences. In
relation to the previous research on the of APOS and
creative thinking (Arnawa, Kartasasmita and
Baskoro, 2007) which showed that the proof ability
in the APOS group is significantly better, thus it is
strongly recommended to apply the APOS in abstract
algebra. In addition, the results of the study (Cetin,
2015) were the first attempt to use the APOS
framework in the context of educational
programming with the results of the study indicating
that the APOS with frameworks is useful for
providing students with understanding of concepts.
The results of the study (Marsitin, 2017b) indicated
that the calculus learning module with developed
APOS was very effective and feasible to be used for
students. Students more easily understand the
material calculus and the achievement of students'
academic ability had improved very well.
In mathematics learning there are abilities that
have to be achieved those are the understanding of
mathematics, mathematical connections,
Marsitin, R. and Sesanti, N.
Student Worksheet in Mathematical Creative Thinking.
In Proceedings of the Annual Conference on Social Sciences and Humanities (ANCOSH 2018) - Revitalization of Local Wisdom in Global and Competitive Era, pages 321-325
ISBN: 978-989-758-343-8
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
321

mathematical reasoning, mathematical problem
solving, and mathematical communication
mathematics (National Council Of Teachers Of
Mathematics, 2000). Creative thinking has an
important role in the process of mathematical
thinking. Creative thinking has the meaning that a
person has diversity in solving a problem. Mumford,
Medeiros and Partlow (2012), Harris (2002) states of
creative thinking development is the awareness of
thinking, thinking observation, thinking strategy and
thinking reflection. Improvements in developing the
next level of mathematical creative thinking are based
on students' creative thinking patterns that include:
fluency, flexibility and novelty in problem solving
and problem presenting. In the assessment of
students’ creative thinking, the student must have the
ability to think creatively that includes fluency,
flexibility and novelty and flexibility (Siswono, 2005;
Marsitin, 2017a). Students use a method of planning
solution, in solving mathematical problems so that it
can implement through constructing the solution of
the problem.
The process of creative thinking with indicators in
the level of creative thinking that describes the
creative process by understanding the problem,
generating ideas / ideas and planning actions.
Understanding the problems are finding goals, data
(facts) and problems as target questions. Generating
ideas / ideas is to find options to answer the problem.
Planning action is finding a solution. At this stage, the
individual analyzes, refines or develops the choice of
appropriate ideas and then prepares a choice or
alternative to increase the support and value (Isaksen
and Aerts, 2011; Lithner, 2004; Maharani, 2014). In
reality, students are accustomed to accept the material
presented by the lecturers only, so that the mindset of
the students tend to be passive and the lecturers need
to make the learning innovation so that the students
are active and have creative thinking ability in solving
mathematical problems. The process of mathematical
thinking in the students appears when students
construct their knowledge in mathematical solution
(Marsitin, 2016). The exposure and phenomenon
above, is very interesting to be discussed related to
mathematics learning, APOS and creative thinking
ability, with the aim of developing a student
worksheet with APOS on the calculus to improve
mathematical creative thinking.
1.1 APOS
APOS is a in a mathematics learning approach at the
college level by integrating the use of computers,
learning in small groups, and paying attention to
mathematical connections and creative thinking to
understand a mathematical concept, namely: action,
process, object and scheme abbreviated as APOS
developed (Dubinsky and Mcdonald, 2002;
Parraguez and Oktaç, 2010). Learning with APOS
includes: (a) Action is a transformation of mental
objects that a person is said to experience an action if
the person is focusing his mental processes on
understanding a concept; (b) Process is a person who
has experienced a process of a concept, if his thinking
is calculus to the mathematical idea encountered, it is
characterized by the emergence of the ability to
reflect on the mathematical idea; (c) Object is a
person who has the object conception of a
mathematical concept, if he has been able to treat the
idea or concept as a cognitive that includes the ability
to act on the object, and provide a reason or
explanation of its characteristics and be able to do the
decomposition of an object into a process when the
characteristics of the object in question will be used;
(d) Scheme is when a person has the ability to
construct examples of a mathematical concept in
accordance with the characteristics of the concept..
The of APOS is a constructivist of how one learns a
mathematical concept, thereby helping to develop the
mathematical thinking process within itself
(Parraguez and Oktaç, 2010; Arnawa, Kartasasmita
and Baskoro, 2007).
1.2 Creative Thinking
Creative thinking has the meaning that a person has
diversity in solving a problem. Mumford, Medeiros
and Partlow (2012) states the definition of creative
thinking development that is the awareness of
thinking, thinking observation, thinking strategy and
thinking reflection. Improvements in felling the
ability of the next level of mathematical creative
thinking are based on students' creative thinking
patterns that include: fluency, flexibility and novelty
in problem solving and problem presenting. In the
assessment of students' creative thinking, the student
must have creative thinking ability which includes
fluency, flexibility and novelty and flexibility (Hu,
Wu and Shieh, 2016). Students use a method of
planning solution, in solving mathematical problems
so that it can implement through constructing the
solution of the problem. Creative thinking process
with indicators in the level of creative thinking that
describes the creative process by understanding
problems, building ideas and planning actions.
Understanding the problem includes the stages of
finding goals, finding data or facts and finding
problems as target questions. Generating ideas
ANCOSH 2018 - Annual Conference on Social Sciences and Humanities
322

includes decreasing options for answering open
issues. Planning actions includes finding solutions
and finding support. At this stage, the individual
analyzes, refines or develops the choice of
appropriate ideas and then prepares a choice or
alternative to increase support and value (Harris,
2002; Runco and Acar, 2012) Any text or material
outside the aforementioned margins will not be
printed.
2 METHOD
This research is a development research. The research
design of this module development uses development
design (Richey, Klein and Tracey, 2011) with the
model (Dick, Carey and Carey, 2009). The
development model includes: identifying learning
objectives, conducting learning analyzes, formulating
specific objectives, developing assessment
instruments, developing learning strategies,
developing and selecting learning materials,
designing and executing evaluation, revising learning
materials, designing and carry out evaluation. The
research subject is student of mathematics education
of Kanjuruhan University of Malang who take
calculus course in the third semester. The research
subjects are form class A for small group trials is 9
students and class B for large group trials is 30
students. Student worksheets with APOS are
validated by material validators, learning validators
and student worksheet product design validators. After
being validated, calculus field trials are conducted on
small group and extensive field trials for large group
to find out the lecturers’ responses and students'
responses to student worksheet. The effectiveness of
student worksheet is analyzed qualitatively, while the
ability of mathematical creative thinking is analyzed
quantitatively. The instruments used for the collection
of research data are: response questionnaire, test of
mathematical creative thinking ability.
3 RESULTS AND DISCUSSION
The results of the research include several stages of
analyzing, planning, developing, and evaluating. The
first stage is the analyzing; analyzing the plan for
lecturing implementation on calculus material,
reviewing the learning objectives of calculus. Phase
Two is planning, by developing instruments and
designing product specifications and the structure of
the student worksheet content. The third stage is
developing, analyzing and developing the
components of the developed student worksheet,
validating the student worksheet for calculus with
APOS by material validation, learning validation and
product design validation and then conducting trials
on small groups, as well as large group tests. The
fourth stage is evaluating, evaluating and analyzing
the effectiveness of developed student worksheet by
conducting creative thinking ability tests. The results
include four stages of analyzing, planning,
developing, and evaluating. Prior to the research
trials, validation of research instruments and
validation of student worksheet products are done by
material validators, learning validators and design
validators. The validation result shows that the
validated instrument and student worksheet product
has been declared valid. Furthermore, trials are
conducted for small group trials with 9 student’s form
class A and large group trials with 30 student’s form
class B. The results are shown in table 1.
Table 1: Students’ worksheet results in trials.
Description
Action
Process
Object
Scheme
Students’
Worksheet-1
85
85
80
80
Students’
Worksheet -2
85
80
80
80
Students’
Worksheet -3
80
80
80
80
Students’
Worksheet -4
85
85
85
85
Students’
Worksheet -5
85
80
80
75
Total
420
410
405
400
Average
84
82
81
80
The results of the student worksheet in table 1, the
obtained data is for the action the average score is 84,
the average score of the process is 82, the average
score of object is 81 and the average score of scheme
is 80. It can be said that the students still have
difficulties in the object and scheme stages, which
means that students are still difficult to conclude from
Student Worksheet in Mathematical Creative Thinking
323

the problem solving and it is difficult to conclude the
formula used in the solution of the problem so that the
students need the assistance in experiencing the
difficulty, according to the opinion (Sutawidjaja and
Afgani, 2015; Bikmaz et al., 2010; Speer and
Wagner, 2009) that the main role of lecturers in class
discussions is as a facilitator/indirect
intervention/scaffolding.
Student worksheet is very helpful for students to
change old mindsets that just accept the material but
can change themselves to be able to understand the
mathematical concept itself. Through the student
worksheet, the students learn to transfer the concept
they have understood among friends and students to
learn to improve if there is a misconception that he
understands. The tasks in the student worksheet done
in groups give concepts strengthen of constructed
mathematics and apply the concepts or theorems of
mathematics that have been studied and discussed to
think about the concepts that have not been studied.
This is in line with opinion (Speer and Wagner, 2009;
Koçak, Bozan and Işik, 2009; Zakaria, Chin and
Daud, 2010) which state that learning is an active and
constructive process in which students try to solve
problems by actively participating in training of
mathematics during the learning process, so that
creative thinking is a person's ability to generate new
ideas that are effective and ethical.
Table 2: Recapitulation of research result.
Description
Achievement
Test of mathematical
creative thinking ability
83%
Students’ response toward
the questionnaire
85%
Lecturers’ response toward
the questionnaire
90%
The results of the student worksheet in table 2, the
achievement of mathematical creative thinking
ability test is 83%, thus it can be concluded that the
student worksheet with APOS can improve the
creative thinking ability. Mathematical creative
thinking ability test with indicators in the creative
process by understanding problems, building ideas
and planning actions with opinion (Lince, 2016;
Williams, 2007). The result of students’ response
toward the questionnaire of the student worksheet
with APOS shows 85% students is agree, therefore it
can be concluded that the student worksheet with
APOS get good responses from the students, student
worksheet calculus is in accordance with the needs of
the students that is easy, interesting and useful for
students. The result of lecturers’ response toward the
questionnaire of the student worksheet with APOS
shows 90% is agree, thus it can be concluded that the
student worksheet with APOS is in accordance with
the needs of the student, it also can be said that the
student worksheet provides practical, effective and
feasible in learning. Mathematics learning not only
provides concepts related to the definition of but must
apply examples as solutions to mathematical
problems with learning strategies that support
learning achievement. Learning that involves the
students then provide the opportunity for students to
work independently by constructing their knowledge,
thus can achieve maximum academic ability.
4 CONCLUSIONS
The conclusions of this development research on
student worksheet with APOS on the calculus material
to improve creative thinking are: the achievement of
creative thinking ability test in experiment is 83%,
thus it can be concluded that the student worksheet with
APOS can improve the creative thinking ability. The
result of students’ response toward the questionnaire
of the student worksheet with APOS shows 85%
students is agree and 90% lecturers’ is agree, thus it
can be concluded that the student worksheet with
APOS is in accordance with the needs of the student
is easy, interesting and useful for students. It also can
be said that the student worksheet provides practical,
effective and feasible in learning. Suggestions that
can be given are lecturers who teach calculus courses
should use teaching materials in the form of student
worksheet to make the students master the
mathematics concepts and academic quality of
mathematics is better.
ACKNOWLEDGEMENTS
This research activity is supported by Mathematics
Education Study Program, Faculty of Science and
Technology, Kanjuruhan University of Malang.
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324

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