International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 03, March 2019, pp. 795-806. Article ID: IJCIET_10_03_077
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=03
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication
Scopus Indexed
RESTRUCTURING OF STEM-BASED STUDENT
THINKING IN CONSTRUCTING THE CONCEPT
OF DEFINITION A FUNCTION
Ukhti Raudhatul Jannah*
Doctoral Program Mathematics Education, Universitas Negeri Malang
Mathematics Education Department, Universitas Madura
Toto Nusantara
Mathematics Education Department, Universitas Negeri Malang
Sudirman
Mathematics Education Department, Universitas Negeri Malang
Sisworo
Mathematics Education Department, Universitas Negeri Malang
Faisal Estu Yulianto
Civil Engineering Department, Universitas Madura
*Corresponding Author
ABSTRACT
This study is qualitative research. It is to restructure the STEM-based students
thinking in constructing the concept of formal definition of a function based on the
assimilation and accommodation process through the provision of scaffolding. The
research subjects are two students chosen from 18 students by a consideration that the
students meet the determined criteria. The results show both the first and the second
subjects experience errors in relation, the general definition of a function, and
algebraic representation. Errors in Cartesian product and numerical representation
are also experienced by the second subject. Scaffolding by using questions and
instructions in the process of assimilation and accommodation are used to reconstruct
the concept of formal definition of a function. It is essential for the teachers to create
assimilation and accommodation in their thinking process in order to help the students
to understand formal definition of a function in appropriate ways.
Keywords: Definition of a function, constructing the concept of formal definition of a
function, restructuring, student thinking, STEM education.
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Restructuring of Stem-Based Student Thinking in Constructing the Concept of Definition a
Function
Cite this Article: Ukhti Raudhatul Jannah, Toto Nusantara, Sudirman, Sisworo and
Faisal Estu Yulianto, Restructuring of Stem-Based Student Thinking in Constructing
the Concept of Definition a Function, International Journal of Civil Engineering and
Technology, 10(3), 2019, pp. 795-806.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=03
1. INTRODUCTION
The most important part to be studied in Mathematics is the concept of function (Harel &
Dubinsky, 1992). Generally, the function is an important topic in Mathematics’ curriculum and
it also relates to other lessons such as astronomy, engineering, and physics (Kjeldsen & Lützen,
2015; O’Shea, Breen, & Jaworski, 2016; Rossouw, Hacker, & de Vries, 2011; Salas-Morera et
al., 2013; Sánchez & Llinares, 2003; Steele, Hillen, & Smith, 2013). One characteristic of the
concept of a function is that it can be represented in various ways in form of tables, graphs,
symbolic equations, and verbally. Moreover, the important thing of understanding the concept
of a function is an ability to use various representations and translate them from one form to
another (Lin & Cooney, 2001; Sajka, 2003). Consequenly, the concepts of function take an
important role in the curriculum because it is closely related to not only in Mathematics itself
but also to other fields of science.
Students have difficulties connecting to their own previously function concepts with their
new knowledge. Thus, they made some mistakes in constructing the formal definition of a
function. To connect and construct those concepts, the teachers use Science, Technology,
Engineering, and Mathematics (STEM). The term "STEM" refers to teaching and learning in
the fields of science, technology, engineering, and mathematics (Gonzalez & Kuenzi, 2012).
The linking of observation, analysis, experiment as well as processes to form or to explain a
new knowledge is called Science (White, 2014). In this study, Science is the concept of the
formal definition of a function. Technology is a branch of knowledge related to the use of tools
to achieve learning goals. It helps the students to achieve the formal definition of a function,
while Engineering is an application of knowledge to construct concepts. Engineering in this
research is an assimilation and accommodation. This allows students to build upon their own
experiences and to provide opportunities constructing new science and math knowledge
(Kelley & Knowles, 2016). Mathematics is a group of science that contains algebra, calculus,
and geometry. In Mathematics, formal definitions of a functions are contained in algebra and
calculus. The constructions of STEM require declarative, operational and conditional
specifications about abstract concepts and their relationships among those concepts (Reif,
2008). STEM associates to scientific inquiry by formulating questions and answers through
investigation to inform the students before engaging the engineering design process to solve
problems (Kennedy & Odell, 2014).
STEM can improve the structure of students' thinking when they have made some mistakes
in constructing formal definitions of a functions. Their thinking process in form of a problem
solving can be described into the structure of thinking which is usually done by a person when
they want to solve a problem (Kumalasari, Nusantara, & Sa’dijah, 2016). STEM can help the
next generation of students to solve real-world problems by applying concepts cutting across
disciplines as well as capacities of critical thinking, collaboration, and creativity because
STEM including conceptual understandings and procedural skills (Burrows & Slater, 2015;
Bybee, 2010). By applying STEM, the teachers hope that the students are able to define formal
definition of a function; however, they are still have some difficultiies in defining formal
definition of a function. Students although the teachers use STEM to increase sudents
understanding in defining formal definition of a function, they are bringing or recalling the
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concept of function in their senior high school. Students are familiar with the concepts
informing the formal definition of a function, namely set, Cartesian product, relation, functions,
examples, and non-examples of a function, and function representation. The formal
mathematics can reveal new embodied and symbolic ways of interpreting mathematics’ (Tall,
2013). The definition is commonly understood by them infuenced by the definition they have
known in the senior high school, that is a function 𝑓 from a set 𝐴 into a set B is a rule of
correspondence that assigns to each element x in A a uniquely determined element f(x) in B
(Bartle & Sherbert, 2011). It means that the students have some difficulties and errors to
interpret the expression "a rule of correspondence". The students who have had the initial
knowledge about concepts of a formal definition of a function have to relate them according to
the scientific concept. They did some mistakes in relating these concepts to formal definitions
of a function. Figure 1 shows the students’ error in defining formal definiton of a function.
Figure 1 The Students’ Error in Defining Formal Definiton of a Function
The concepts in figure 1 describe the prior concepts of students.The quoation of students
means that a function is a relation pairing a set to another set provided that all of the group of
a domain is pairing a set exactly one element with another. It shows that the students are not
able to describe the concepts of relation and their definition are not appropriate to formal
definition of a function. It is the appropriate concepts to define formal definition of a function:
Let A and B be set. A function from A to B is a set f of ordered pairs in 𝐴 × 𝐵 such that for
each 𝑎 ∈ 𝐴 there exist a unique 𝑏 ∈ 𝐵 with (𝑎, 𝑏) ∈ 𝑓. (In other words, if (𝑎, 𝑏) ∈ 𝑓 and
(𝑎, 𝑏’) ∈ 𝑓 then 𝑏 = 𝑏’) (Bartle, 2011). Bartle also formulates the way to form scientific
concept of the formal definition of a function as in figure 2.
Set
Cartesian
Product
Relation
Function
Example and
Non-example
of a Function
Function
Representation
Formal
Definition of a
Function
Figure 2 Scientific concept in forming the formal definition of a function
Figure 2 shows the scientific concepts in forming a formal definition of a function
containing the forming concepts of a formal definition of a function. These concepts are used
when the students construct the concept of the formal definition of a function.
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Restructuring of Stem-Based Student Thinking in Constructing the Concept of Definition a
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Figure 3 Structure of student thinking in constructing the concept of formal definition of a function
Figure 3 shows the structure of students’ thinking in constructing the concept of a formal
definition of a function based on its scientific concept. The definition is let A and B be sets.
Then a function from A to B is a set f of ordered pairs in 𝐴 × 𝐵 such that for each 𝑎 ∈ 𝐴 there
exists a unique 𝑏 ∈ 𝐵 with (𝑎, 𝑏) ∈ 𝑓.
The problems of the students in defining definition of a function are caused by the STEM
applied by the teachers. The scaffolding in STEM used by the teachers is not appropriate to the
students’ need; the use STEM in the teachers’ scaffolding are not exploring the students’
ability, they are not in steps of discovering the scientific concepts. Therefore, it is important to
restructure the concepts of formal definition of a function. Thus, the purpose of this study is to
restructure the students’ thinking (restructuring) by STEM-based in constructing the concept
of formal definition of a function based on the process of assimilation and accommodation by
scaffolding so they are able to understand the formal definition of the function as a whole,
critical and creative in solving problems especially related to the problem about function in
daily life.
2. METHODOLOGY
This study is a qualitative research. It is to restructure (restructuring) of student thinking by
STEM-based in constructing the concept of a formal definition of a function based on the
assimilation and the accommodation through scaffolding. It is helping students to have
complete understanding the formal definition of a function, and to have critical and creative in
solving the problem. The subjects of the research are the students of Mathematics Education
program at Madura University who have prior knowledge about scientific concepts forming
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formal definition of a function. Eighteen students are asked to define the function formally
according to the concepts. The subjects are chosen based on some errors in constructing the
concept, so they were not considered making the formal definition of the function,
communicative ability, and collaborative ability. In addition, their concepts of formal definition
of a function are still the concepts of senior high school students, without inviting the concepts
of scientific. Then, they are tested and interviewed. The tests are given to check the structure
of student thinking in constructing the concept of formal definition of a function. The interview
is conducted to restructure of STEM-based students thinking in constructing of function
definition concept based on the assimilation and accommodation process through the provision
of scaffolding. To get the confidence of the data, this study compares the data to the peers and
experts relevant to the study to have cross examination and make the disinterestedness of the
data as well as the participants of the study. This research also uses another method to have
trustworthiness of the data.
3. RESULTS AND DISCUSSION
In this study, the researcher gave a test to students who already had had scientific concepts of
formal definition of a function. They were asked to define about function formally. This stage
was conducted to know the structure of student thinking whether it had been based on the
concept of a formal definition of a function or not. When they made an error in constructing
the formal definition of a function, the researcher then restructured their thinking process, so
they would have a whole concept about it.
Figure 4 shows the thinking structure of S1 (first subject) when he made some errors in
constructing formal definition of a function. He got some difficulties in defining the function
formally. The yellow colour indicates the type of errors, namely; concept of the relation, the
function definition in general and the function representation.
Test:
Formal Definition of a
Function
As
A is a set
A
Ø
As
B is a set
B
Ø
As
AS
As
A ×B
Ak
As
Relation A to B
As
As
Ak
As
( a, b)
As
ⱯaЄA
ꓱ!bЄB
Ak
(a, b) Є f
Ak
Ak
Ak
Ɐ a Є A, ꓱ ! b Є B Э (a, b) Є f
Ak
Ak
Non Example of a Function
Example of a Function
Ak
Verbal
Representation
Numeric
Representation
Ak
Ak
Visual
Representation
Ak
Algebraic
Representation
Equilibration
Let A and B be sets. Then a function from A to B is a set f of ordered pairs in A×B such that for each a Є A
there exists a unique b Є B with (a, b) Є f
Figure 4 The fault of concept construction in formal definition of a function on S1
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After the researcher identified his errors in constructing the formal definition of a function,
he reorganized the thinking structure as shown in Figure 5.
Test:
Formal Definition of a
Function
As
Create an example of
sets A and B
As
As
A is a set
A Ø
Mapping the set A to B
with the arrow diagram
(Cartesian product)
B is a set
B Ø
As
As
AS
As
Determining
members of relation
A to B
A×B
Ak
As
As
As
Relation A to B
As
Writing in
ordered pairs
Ak
As
Ak
( a, b)
ⱯaЄA
ꓱ!bЄB
Ak
(a, b) Є f
Ak
As
Constructing relation A
to B with A × B:
relation A to B is subset
A×B
As
Determine the
members of the
relation where each
member in A has the
exact match one in B
Ak
Ak
Ɐ a Є A, ꓱ ! b Є B Э (a, b) Є f
Ak
Ak
Ak
Non Example of a Function
Ak
Example of a Function
Ak
Verbal
Representation
Hint:
If f: A B, then
algebraic form:
f(a) = b, Ɐ a Є A, ꓱ !
bЄB
Ak
Numeric
Representation
Ak
Visual
Representation
Represents the
algebraic shape
according to the
created example
Ak
Algebraic
Representation
Ak
Equilibration
Let A and B be sets. Then a function from A to B is a set f of ordered pairs in A×B such that for each a Є A
there exists a unique b Є B with (a, b) Є f
Figure 5 Restructuring thinking of S1 in constructing the formal definition of a function
Figure 5 shows the restructuring of students' thinking in constructing the definition based
on assimilation and accommodation through scaffolding to help S1 in defining a correct formal
definition of a function. When the student made some errors in constructing relations, they
would be restructured by using scaffolding about a Cartesian product. Scaffolding was applied
in form of commands and questions by creating examples of sets A and B, mapping the set A
to B with the arrow diagram, determining members of relation A to B, writing in ordered pairs,
connecting relation A to B with 𝐴 × 𝐵: relation subset 𝐴 × 𝐵. The general errors they made
was corrected by determining the master of the relation where each member in A has the unique
𝑏 ∈ 𝐵 with (𝑎, 𝑏) ∈ 𝑓. The errors on defining formal definition of a function provide them
scaffolding function representation and provide scaffolding with hints of the algebraic form of
function, then they would define the function according to the instructions.
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Test:
Formal Definition of a
Function
As
A is a set
A Ø
As
B is a set
B Ø
As
AS
As
A×B
Ak
As
Relation A to B
As
As
Ak
As
( a, b)
As
ⱯaЄA
ꓱ!bЄB
Ak
(a, b) Є f
Ak
Ak
Ak
Ɐ a Є A, ꓱ ! b Є B Э (a, b) Є f
Ak
Ak
Non Example of a Function
Example of a Function
Ak
Verbal
Representation
Numeric
Representation
Ak
Ak
Visual
Representation
Ak
Algebraic
Representation
Equilibration
Let A and B be sets. Then a function from A to B is a set f of ordered pairs in A×B such that for each a Є A
there exists a unique b Є B with (a, b) Є f
Figure 6 The fault of concept construction in formal definition of a function on S2
Figure 6 shows the thinking structure of the second subject (S2) when he made some errors
in constructing the formal definition of a function. He also had some difficulties like the first
subject (S1), but it is in different types. Those are the concept of the Cartesian product, relation,
definition of a function in general, and function representation. While figure 6 shows the
restructuring of students thinking in constructing the definition based on assimilation and
accommodation through scaffolding in the form of questions to help the S2 achieving the
formal definition of a function.
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Restructuring of Stem-Based Student Thinking in Constructing the Concept of Definition a
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Make an
example of two
sets A and B
Test:
Formal Definition of a
Function
As
Determing members
of relation from A
to B
As
As
As
Mapping two sets
A to B through the
arrow diagram
A is a set
A Ø
Write down the
members of the
relation in the form
ordered pairs
B is a set
B Ø
As
As
Ak
AS
Definition of
A×B
As
Associates relation
with A × B:
relation A to B is
subset A × B
A×B
Ak
As
Ak
As
Relation A to B
As
Ak
Classify members of
relation as members
of a function
Ak
As
( a, b)
ⱯaЄA
Ak
(a, b) Є f
Ak
As
As
ꓱ!bЄB
Define the
function
Ak
Ak
Ak
Ak
Ɐ a Є A, ꓱ ! b Є B Э (a, b) Є f
Create an arrow
diagram as an
example of a function
Ak
Ak
Non Example of a Function
As
Example of a Function
Ak
Ak
Ak
Ak
Ak
Create a table with
components A, B, and
f(a) based on the
arrow diagram
Ak
Verbal
Representation
Numeric
Representation
Visual
Representation
Algebraic
Representation
Equilibration
Represents the
algebraic shaape
according to the
created example
Ak
Let A and B be sets. Then a function from A to B is a set f of ordered pairs in A×B such that for each a Є A
there exists a unique b Є B with (a, b) Є f
Hint:
If f: A B, the
algebraic form are:
f(a) = b, Ɐ a Є A, ꓱ !
bЄB
Figure 7 Restructuring thinking S2 in constructing the formal definition of a function
The student (S2) also made some errors in constructing the concept of the formal definition
of a function. it can be seen on their thinking structure. When they made errors on the Cartesian
product concept, they would be restructured by giving them instruction to make an example of
two sets A and B, map the set A to B with the arrow diagram, and define of 𝐴 × 𝐵. To
restructure the errors of the concept on relation could be improved by scaffolding. This step
mean giving them some questions and commands to determine members of Relation from A
to B, writing down the members of the relation in the form of ordered pairs, and associating
relation with A × B: Relation A to B is a subset 𝐴 × 𝐵. The wrong concept in defining function
in general could be restructured through scaffolding in the form of classifying members of
relations as members of a function, defining the function. The way of scaffolding restructure
in representing numerical error could be done by creating an arrow with components A, B, and
𝑓 (𝑎) based on the arrow diagram. The last, the error construction on the algebraic
representation could be restructured by creating a table with components A, B, and 𝑓 (𝑎) based
on the arrow diagram and representing the algebraic shape according to the created example.
Students of university should have good representations and improve them. Representing
definition formal of a function are verbal, numeric, visual and algebaric. Ordered pairs,
equations, and Cartesian graphs are the ways to represent a function (Cho, 2013). According to
Rockswold (2009) formal definition of a function can be represented by diagrams, tables, and
verbal descriptions. The features of formal definition function concepts are in a formal
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symbolic way, and it is nearly without using words (Sierpinska, 1992). When the students
understand the concepts of a function, uncover relationships between various representations,
decide the appropriate representation in solving problems, and transfer between representations
with relative ease, it indicates that the students have ability to define definition formal of a
function (Metcalf, 2007).
The process of restructuring in students thinking can be used to solve the students’ errors
in structuring concepts of formal definition of a function. The goal of restructuring process on
students' thinking in constructing the formal definition of STEM-based functions is to make
students think critically and creatively in solving daily problems related to a function because
the concept of a function is related to many other sciences and our daily life. There are several
transferable skills can be gained by the students from STEM study as identified by National
Governors Association (NGA). To recognize and evaluate the problems, they are using critical
thinking as well as using math, science, technology, and engineering concepts (Barakos, Lujan,
& Strang, 2012). STEM is crucial in modern conception since it contains practical purposes on
various disciplines as well as real-world problems solving (Labov, Reid, & Yamamoto, 2010;
Sanders, 2009). STEM activities provide students with rich learning experiences, and the
students of all educational backgorund have posibilities to access. Therefore, the teachers
should be professional in developing the ways to be design experiences supporting the learners
in the classroom (Brenneman, Lange, & Nayfeld, 2018).
STEM education consist of Science, Technology, Engineering, and Mathematics. The main
parts of STEM Education are Science and Mathematics because most people can recognize and
relate to them (White, 2014). In this research, Science is the concept definition formal of a
function. Science is discussing the origin of formal definition of a function, but Mathematics
is about the ways to inquire the natural science , social sciences, engineering, and technology
(Michelsen, 2006). Mathematics comprise algebra and calculus, consisting a function. It needs
a certain ability and critical thinking to understand the concept of a function by using the
algebraic approach. However, calculus and other practical sciences invite the students to have
changes in understanding as the effect of instruction (Malik, 1980). STEM is not only includes
the teaching mathematics in isolation, but also invite other disciplines. It builds on the content
knowledge and understanding developed in and across the four disciplines while
acknowledging that all STEM learning activities are underpinned by Mathematics (Bruton,
2017)
Technology and Engineering support the restructuring thinking process of students in
structuring thinking of formal definition of a function. In this study, scaffolding is as the
representation of technology to achieve learning objectives. Scaffolding is a process that
includes tasks and questions to help students in solving problems (Wood, Bruner, & Ross,
1976). Scaffolding is applied through several questions, providing examples and giving clues
related to student concept errors in constructing formal definition of a function. The strategies
in scaffolding is questioning, modeling, instructing (Tharp & Gallimore, 1991). However,
engineering in this research is the process of assimilation of accommodation. The structure of
students thinking structures can be assessed by assimilation and accommodation of their
frameworks of the previous concepts. This is in line with Piaget statement that a person
interacts with the environment or prior knowledge, there will be a cognitive process called
assimilation and accommodation (Subanji & Supratman, 2015). The assimilation process is the
process of integrating new stimulus into the scheme. The accommodation process is the process
of new stimulus integration through the modification of the old scheme or through the
formation of the new scheme to adapt to the new acquired stimulus. In solving the problem,
the process of assimilation and accommodation will continue to take place until the existence
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of balance (equilibration) (Subanji & Supratman, 2015). When they reach the equilibration, the
students will recall the previous knowledge to define formal definition of a function by using
scaffolding, assimilation and accommodation process.
Transferability by comparing the finding to experts, peers, and previous studies related to
participants, scope, data, richness, and the depth of information is asserted that constructs and
results of the study relevant to the context of students and it gives effect on the students
concepts in defining formal definition of a function, so that they are able to solve their problem
in social life. Appropriate scaffolding given to the students offers positive effects on their
concepts to define formal definition of a function. The findings of this study facilitate the
development of STEM in recent years; it can create a new strategy that is unusual and original
to change the development of education that can be used in practical teaching learning process.
It also helps the students to get easier jobs and associate themselves in nature.
4. CONCLUSION
Based on the result and analysis of this study, it can be concluded that,
a. The three errors of constructing concept made by student (S1) are relations, the
definition of a function in general, and function representation.
b. The restructuring process of S1 can be done by the following steps; errors in relation
can be solved by scaffolding of Cartesian product, errors on the general definition of
the function can be solved by scaffolding of the relation, while errors on function
representation can also be solved by scaffolding on the concept of the function example,
c. The four errors of the construction made by student (S2) are the Cartesian product,
relation, definition of the function in general, and function representation.
d. The restructuring process on S2 can be done by the following steps; errors on Cartesian
product can be solved by scaffolding identifies the set, error on the relation with the
scaffolding of Cartesian product, the general definition of the function by scaffolding
about the relation, and scaffolding on the concept of the function example can be used
to solve errors about the representation of function.
e. Scaffolding helps students to construct the concept of formal definition of a function.
It facilitates the students in particular students with dominant errors in constructing
formal definition of a function. The more scaffolding is given to the students, the more
construction of thinking of students is getting better.
This study gives contributions to students and to the development of STEM education in
modern period. Its facilities students to have critical and creative thinking in overcoming the
problems associated to the function as well as to create a new and more innovative STEM
education appropriate to the context.
ACKNOWLEDGEMENT
The authors are very grateful to the Editor and anonymous referees for the constructive
comments and suggestions which led to the final presentation of this paper. They also thank
Madura University for providing them with research facilities and supporting the
administration. Special thanks go to the students and lecturers for their involvement and
helping this study running smoothly.
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Ukhti Raudhatul Jannah, Toto Nusantara, Sudirman, Sisworo and Faisal Estu Yulianto
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