Vol. 7, No. 2, November 2022, pp. 154 – 163 e-ISSN: 2550-1461
https://ijeisr.net
MATHEMATICAL CONNECTION SKILL OF PROSPECTIVE VOCATIONAL ENGINEERING TEACHERS IN MATHEMATICS
APPLICATIONS
Dedi Rohendi1 I Made Arnawa2
1Universitas Pendidikan Indonesia, Bandung, Indonesia (E-mail: [email protected])
2Universitas Andalas, Sumatera Barat, Indonesia (E-mail: [email protected])
Abstract: This study attempts to discover the mathematical connection skill of prospective vocational engineering teachers especially the concept of mathematical applications in engineering. Mathematical connection skill which was observed includes the ability of prospective teachers to link mathematical ideas inside the context of mathematics and mathematical ideas outside the context of mathematics. The study was conducted on 30 prospective vocational engineering teachers. The results reveal that the mathematical connection skill of the prospective vocational engineering teachers is categorized as good in both indicators. Averagely, the prospective vocational engineering teachers have been capable to use their mathematical connection skills in solving mathematical application problems in engineering.
Keywords: Mathematical Connection Skill, Prospective Teacher, Vocational Engineering, Mathematics Application Engineering.
Abstrak: Penelitian ini bertujuan untuk mengetahui kemampuan koneksi matematis mahasiswa calon guru vokasi teknik terutama pada konsep aplikasi matematika dalam teknik.
Kemampuan koneksi matematis yang diamati meliputi kemampuan mahasiswa dalam mengaplikasikan keterkaitan antar ide dalam matematika dan ide matematika dalam konteks di luar matematika. Penelitian dilakukan terhadap 30 mahasiswa calon guru vokasi teknik.
Hasil penelitian menunjukkan bahwa kemampuan koneksi matematis mahasiswa calon guru vokasi teknik untuk kedua indikator tersebut dikategorikan baik. Secara rata-rata mahasiswa calon guru vokasi teknik sudah mampu menggunakan kemampuan koneksinya dalam menyelesaikan permasalah aplikasi matematika dalam bidang teknik. Abstrak dibuat dalam 2 bahasa, yaitu bahasa Indonesia dan bahasa inggris. Edisi bahasa Indonesia merupakan terjemahan dari abstrak dalam bahasa inggris dengan format sama seperti abstrak dalam bahasa inggris.
Kata Kunci: Kemampuan Koneksi Matematis, Mahasiswa Calon Guru, Kejuruan Teknik, Aplikasi Matematika, Bidang Teknik.
1. INTRODUCTION
Mathematics is one of the important fields of study to be learned by students. Learning mathematics can improve students’ thinking abilities; in addition, it has relevance to various other disciplines (Patton et al., 1997), (Fabiyi, 2017). With mathematics, we can solve various phenomena that we encounter in life (Utubaku & Elizabeth, 2011), (Bergner & Neubauer, 2011), (Adolphus, 2011). Therefore, mathematics is taught at all levels of education (Utubaku
& Elizabeth, 2011). Moreover, mathematics can be used as a tool to learn science, humanism, and technology(Sriyanto, 2004). Mathematics also plays an important role in education, from primary to higher education in every country worldwide (Utubaku & Elizabeth, 2011).
The implementation of the learning process of mathematics does not always happen without difficulties. Most students dislike mathematics and still think that it is one among other difficult subjects (Patton et al., 1997). Many students complain that learning mathematics is stressful particularly if the mathematics teacher is frightening, short-tempered, punitive, monotonous, hypercritical, and teaching too fast. Furthermore, if the teacher does not give enough motivation to students to learn mathematics, they will face difficulties in understanding the learning materials. Those problems may due to the teachers’ limited skill, government’s inadequate attention towards teachers, teachers’ low qualifications (Owolabi & Adedayo, 2012), and students’ low interest in learning mathematics (Gafoor & Kurukkan, 2015).
The mathematics learning process at every level of education, from primary to higher education, has not yet optimized all its supporting factors. The learning process has not been in line with curriculum demands, learning methods that actively involve students (student- centered), and has not optimized technology-based innovative media. Many students still experience difficulties in applying mathematical concepts to real-life (Sue, J. R, 1999).
Students also feel that the learning process of mathematics is less meaningful which makes mathematics difficult. Teachers have not linked the complexity of the materials with students’ thinking schemes in the process of learning. Furthermore, students are not allowed to rediscover and construct their mathematical ideas (Evi, 2011); in fact, the mathematical characteristics between every topic affect each other as well as between every concept and between other concepts of knowledge.
Based on observation, it is found that the teaching and learning method of mathematics is still dominated by teachers, and the use of innovative learning media is still limited. This kind of method will only burden students’ memory and make them reluctant to learn mathematics which also can cause a decrease in their learning motivation. Later, this may result in students’ low mathematical skills. Learning mathematics in several majors in tertiary institutions such as engineering, social, and economics, involves many applied concepts.
Applied concepts can be resolved when they have been converted into mathematical models.
Most students, however, have difficulty in constructing these mathematical models.
Engineering mathematics which is learned in engineering majors in university is a supporting course since mathematical concepts are required in other engineering courses. Some engineering fields that require mathematical concepts include thermodynamics, mechanical engineering and fluid mechanics, kinematics, automotive, production and design, modeling and simulation, and so on. (Universitas Pendidikan Indonesia, 2018).
Mathematical connection skill is the ability to connect a mathematical concept with another mathematical concept, mathematical concepts with other concepts outside mathematics, and mathematical concepts with daily life matters (Keller et al., 2001). The mathematical connection between topics in mathematics is structured and interrelated science.
Thus, concepts and operations in mathematics will be connected one with another. For example, the connection of a proposition with another proposition, a theory with another theory, a topic with another topic, and among mathematics’ branches (Ruseffendi, 2006). The connection of mathematical concepts with other concepts outside mathematics and with daily life matters is frequently found in people’s real life. Indicators of mathematical connections are: 1) looking for relationships between various representative concepts and procedures; 2) understanding the relationship between mathematical topics; 3) using mathematics in other disciplines or daily life; 4) comprehending the equivalent representative of the same concept;
5) finding connections of one other procedure in equivalent representation; and 6) using connections between mathematical topics and between mathematical topics with other topics (Sumarmo, 1987).
Mathematical connection skill forms mathematics as one entity. One mathematical concept is related to another concept which means that, if students are going to learn one particular mathematical concept, they have to learn other related concepts as the prerequisite before moving on with that one particular concept. For instance, when students are learning the concept of sets, they need to learn the concept of numbers; when students are learning the concept of limit, they need to learn the concept of sets, numbers, functions, and so on.
To improve students’ mathematical connection skills, solving mathematical connection problems is essential to be practiced. Since every concept in mathematics is interrelated, for example, one proposition with another proposition, a theory with another theory, and a topic with another topic, students should be given more opportunities to explore and create those mathematical connections to successfully master mathematical connection skills.
Mathematical connection skills and ability to explain that skill is not necessarily mastered by students who learn and comprehend mathematical concepts (Lembke & Reys, 1994). Students who can connect mathematical ideas may have a deeper understanding of the concepts and memorize them longer because they can see the relationship between mathematical topics inside and outside mathematics, and with daily life matters (Keller et al., 2001).
Connectivity theory proposes that every concept, principle, and skill in mathematics is interrelated. Mathematics is a coherent science and not partitioned into various branches (Bruner & Kenney, 1963). As an illustration, branches of mathematics such as algebra, geometry, trigonometry, and statistics cannot be separated from each other because they are interrelated; moreover, learning mathematics also means learning each branch of mathematics (Noto et al., 2019).
Mathematics is not a solitary and perfect science because of itself. It is developed to assist humans in understanding and mastering social problems (Kline, 1973). Mathematics is considered to be the queen of science because mathematics is widely used in many fields and daily life. Mathematical concepts are also extensively used in engineering fields such as fractal mathematics in the field of Soil Water Retention Estimation (Tyler & Wheatcraft, 1989). A study from (Rutledge & Cote, 2003) furthermore states that mathematics is used as the
quantitative kinetic PCR and the application of standard curves. In the study, the mathematics of quantitative PCR is examined in detail, from which several fundamental aspects of the threshold method and the application of standard curves are illustrated.
In addition, mathematical concepts in engineering are used in thermodynamics, fluid mechanics, physics which is ranging from vectors to differential equations; in the automotive field which is in the process of brake thinning, friction torque, lathing, welding, and so on. In engineering mechanics, kinematics, and dynamics, mathematical concepts are used in the calculation process to explain the concepts of force on an object. Thus, mathematical concepts have a vital role in engineering.
Insufficient skills in basic mathematics cause problems for those who are majoring in engineering in university. Most students seem to be able to find the correct solution of exam questions using familiar steps and procedures. Some of the most important skills which are required by engineering students are problem solving and creative thinking but they still have some difficulties working on these skills (Adams et al., 2010).
All essential mathematical concepts in engineering have to be conveyed to prospective vocational engineering teachers because they have to be able to explain engineering concepts, which require mathematical concepts, to their students at school later when they become teachers. Several studies focusing on the ability of prospective teachers and teachers have been conducted by (Erdik, 2019), (Alrajeh & Shindel, 2020), (Minor et al., 2002). However, the prospective vocational teachers’ mathematical connection skill has not been explored.
Therefore, this present study aims to reveal the mathematical connection skill of prospective vocational engineering teachers. Mathematical connection skill which is examined are the ability to understand problems, create plans, make mathematical modeling, solve the mathematical problems and evaluate the answers of engineering application problems that have been solved.
2. METHOD
This study attempts to reveal the mathematical connection ability of prospective vocational engineering teachers specifically on the concept of mathematical applications in engineering.
The prospective teachers took a test instrument contains engineering-related questions that could only be solved using mathematical concepts. The questions given to prospective vocational engineering teachers discussed the depreciation of the volume of oil and the cooling process of the object that had been lathed which was inserted into a liquid as shown in Figure 1.
The mathematical concepts that were required to be used to solve the questions include the concepts of numbers, functions, derivatives, integrals, and differential equations. The respondents were asked to solve these problems. The answers from the respondents were then analyzed to investigate how they connect mathematical ideas and apply them in contexts outside mathematics.
The focus of this study is to examine the mathematical connection ability of prospective vocational engineering teachers. The subjects of this study were 30 prospective vocational engineering teachers. The collected data was the prospective teachers’ answers to engineering concept questions which must be solved by utilizing mathematical concepts. All prospective
teachers’ answers to each question were collected and then analyzed. Respondents’ answers to the observed mathematics application problems were following the technical indicators presented in Table 1.
Figure 1: An Example Of Mathematical Connection Skill Questions
Table 1: The Indicators of Achievement Of Prospective Vocational Engineering Teachers’ Mathematical Connection Skill
Aspect Code Technical Indicator
1. Using interrelated ideas in mathematics
IT-1 IT-2 IT-3
Using the relationship between facts, concepts, and mathematical principles
Identifying the relationship of mathematical principles between one another
Using the relationship of mathematical principles between one another to establish new principles or formulas which are needed
2. Applying mathematics ideas in the contexts outside mathematics
IT-4 IT-5
Identifying facts, concepts, and mathematical principles in the contexts outside mathematics
Using the relationship between concepts, procedures, and arithmetic operations for problems/contexts outside mathematics
The respondents’ answers were analyzed whether they met the technical indicators of connection skill or not. Then, all the answers of the respondents were converted into the percentage. It is performed to unveil the percentage of prospective teachers who have mathematical connection skills according to these indicators.
3. RESULTS AND DISCUSSION
The questions in Figure 1 are tested to the prospective vocational engineering teachers to reveal their mathematical connection skill based on the indicators which are displayed in Table 1 i.e.
to unveil whether they can determine the appropriate relationship between facts, concepts, and mathematical principles, then to determine the appropriate relationship between mathematical
principles, and to determine the appropriate mathematical principle relationships to construct the correct mathematical model to solve the oil volume and temperature problems in the questions. Furthermore, the study intends to see the way the respondents identify facts, concepts, mathematical principles from contexts outside mathematics, the way they use the relationship of the concepts with the procedures and arithmetic operations to solve problems/contexts outside mathematics so that the problems in the questions can be solved.
Based on the analysis of the answers given by the respondents, the results are shown in Table 2 below.
Table 2: The Recapitulation Of The Respondents’ Answers Based On The Indicators In Question 1 And Question 2
Indicator Percentage
Question 1 Question 2
IT-1 80% 85%
IT-2 75% 78%
IT-3 82% 80%
IT-4 82% 83%
IT-4 75% 78%
In Table 2, it was revealed that 80% of the prospective vocational engineering teachers can apply the relationship between facts, concepts, and mathematical principles to solve question 1 which discusses the volume of the oil. It means that 80% of the prospective vocational engineering teachers or 24 out of 30 respondents have been able to determine the fact of the question including the oil volume is 3000 ml when the “time equals zero” (t = 0), it then becomes 2950 ml after it is used for 4500 km (or t = 10 months), and the condition which is asked in the question is the oil volume when it has been used for 12 months (or 5000 km).
Furthermore, they have also been able to determine the appropriate mathematical principles to solve these problems that are using the principle of variable separation in differential equations. The illustration of the respondents’ answers is shown in Figure 2.
Figure 2: The Illustration Of The Respondents’ Answer To Question 1
For IT-2 of question 1, 75% of the prospective vocational engineering teachers have been able to find the interrelation of mathematical principles to solve the problem. It can be seen from the example of the answers in Figure 2, 75% of the prospective vocational
engineering teachers answer that the principle of volume reduction against time is proportional to its mass so the suitable equation to be used is 𝑑𝑣
𝑑𝑡 = 𝑘𝑉 . Hence, they can determine the relationship of the problem with the appropriate mathematical principles.
For IT-3 of question 1, it is found that 82% of the prospective vocational engineering teachers can use the relationship of one mathematical principle to another to construct new principles or formulas which are required to solve the question. They state that to solve the differential equation by separating the variables they must separate the variables that are related to their operators, or they may separate the variable V with t and classify the dv operator with the variable V and the dt operator with a constant or variable t. Afterward, they conduct the integration process to discover the desired formulation to solve the problem.
For IT-4 of question 1, 82% of the respondents are found to be able to identify facts, concepts, and mathematical principles in contexts outside mathematics. It can be seen from their answers in which they mention that this concept is related to concepts in automotive engineering and understand the process of oil volume shrinkage over time.
For IT-5 of question 1, 75% of the respondents can use the association of concepts with procedures and arithmetic operations to solve problems/contexts outside mathematics. It means that they can utilize the facts of the existing problems and are able to solve them. As shown in Figure 2, it can be noticed that they can determine the general equation from the obtained differential equation i.e. 𝑉 = 𝐴𝑒𝑘𝑡. Subsequently, by substituting V = 3000 when t = 0, it is obtained that the coefficient A = 300 and the equation becomes 𝑉 = 3000𝑒𝑘𝑡. It is then followed by substituting V = 2950 at t = 10 months, so that the constant is k = -0.00168 or k = -0.0017 is obtained, so that the equation 𝑉 = 3000𝑒−0,0017𝑡) is obtained. From these equations, the answer of question 1 is acquired i.e. when t = 12 then V = 2940 ml.
Next, the answers to questions 2 are shown in Figure 3. Question 2 asks about the temperature drop of an object that has heat due to the lathing process which is then put into a coolant with a certain temperature.
Figure 3: The Illustration Of The Respondents’ Answer To question 2
In Table 2, it can be seen that the percentage of the prospective vocational engineering teachers who can use the relationship between facts, concepts, and mathematical principles in
question 2 (IT-1) i.e. determining the temperature drop in an object causes by lathing is reaching 85%. It indicates that 85% of the prospective vocational engineering teachers or 26 people out of 30 respondents have been able to determine the appropriate facts from the question, in this case, the temperature of the object when the “time equals zero” (t=0) which is 3500 C and its temperature after dipping it into a solution for 30 minutes which becomes 2000 C (or t=30 minutes) and the condition that is asked is the temperature of the object when it has been dipped for 60 minutes (T=?). Afterward, they have also been able to determine suitable mathematical principles to solve these problems which are using the principle of differential equations with variable separation or integrated factors.
For IT-2 of question 2, 78% of the prospective vocational engineering teachers can find the connection of one mathematical principle to another to solve the problem. It can be seen from the example of the answers in Figure 2 that 78% of respondents answer that the principle of the temperature drop of the object in is proportional to the duration it is dipped in a liquid cooler than the object, so the concept of the differential equation used by the formula is 𝑑𝑇
𝑑𝑡 = 𝑘(𝑇 − 𝑀) or Newton’s Third Law. Hence, they have been able to determine the connection between the problems and their appropriate mathematical principles.
For IT-3 of question 2, 80% of prospective vocational engineering teachers can use the relationship of mathematical principles to construct new principles or formulas to solve problems. They state that to solve the differential equation they must separate the variables that are related to their operators, or they can also separate the variables T and t by grouping the dT operator with the variable T and the dt operator with the constant or variable t. Subsequently, they do the integration process to find the appropriate formulation to solve the problem.
For IT-4 of question 2, 83% of the respondents are seen to be able to identify facts, concepts, and mathematical principles from contexts outside mathematics. It is signified by their answers towards the question which states that this concept is related to the concept in the field of engineering i.e. thermodynamics, and understanding the process of the temperature drop along with the time of use.
For IT-5 of question 2, 78% of the respondents can implement their understanding regarding the interrelation between concepts, procedures, and arithmetic operations to solve the problems/contexts outside mathematics. They can use facts to solve the problems. As an illustration, in Figure 3 it can be seen that the respondents can determine the general equation from the resulting differential equation, which is 𝑇 = 𝐴𝑒𝑘𝑡+ 25. Afterward, they have to substitute T = 35000 C when t = 0 to obtain the coefficient value A = 3250 C and make the equation become 𝑇 = 325𝑒𝑘𝑡+ 25. It is then followed by substituting T = 200 when t = 30 minutes to obtain the constant k = -0.02 so that the equation 𝑇 = 325𝑒−0,02𝑡 is obtained. From the equation, the answers can be attained i.e. when t = 60 then T = 122.50 C.
Based on these results, it can be concluded that the prospective vocational engineering teachers averagely have good connection skills which means that they already have the characteristics of mathematical connection skills. The measured connections skill includes the ability to use the connection between ideas in mathematics and applying mathematics to other contexts outside mathematics.
In engineering, many concepts involve mathematics so students need to have a thorough understanding to be able to relate all mathematical concepts in solving engineering-related problems. For that reason, this study observes the mathematical connection skill of prospective vocational engineering teachers based on existing indicators. It does not mean their skill is only based on these indicators; it is also viewed from their overall skill because the mathematical connection is not separated in various topics but based on its essence that mathematical skill is a unity (Bruner & Kenney, 1963). Moreover, mathematics also cannot be separated from other science and problems that happen in the environment. Without mathematical connections, humans must continue to learn and remember separate mathematical concepts and procedures (Keller et al., 2001). In the case given in this study, the respondents have been able to integrate principles, facts, theories, propositions, concepts, and their relationship with other fields outside mathematics, so that it can be said that their mathematical connection skill is good.
Mathematical concepts used by the respondents in solving problems faced in this study are derivation, integrals, differential equations, and other calculations which are considered as essential unity in this case.
If a prospective teacher can connect mathematical ideas, their comprehension of mathematics will become deeper. In addition, their understanding will also last longer since they can see the relationship between topics both inside and outside mathematics and in daily life (Keller et al., 2001).
Connectivity theory views that every concept, principle, and skill in mathematics is related to other concepts, principles, and skills. Connectivity is vital because mathematics is a coherent and non-partitioned science. Branches of mathematics such as algebra, geometry, trigonometry, and statistics are interrelated (Bruner & Kenney, 1963). Mathematical connection skill plays a very important role in mathematics. With that ability, prospective teachers can understand mathematics thoroughly and deeply.
4. CONCLUSION
According to the results of the study, it shows that the mathematical connection skill of prospective vocational engineering teachers is categorized as good in both indicators.
Averagely, the prospective vocational engineering teachers have been able to use their connection skills in solving mathematical application problems in engineering.
ACKNOWLEDGMENTS
Acknowledgment is due to the Department of Mechanical Engineering Education, Faculty of Technology and Vocational Education, Universitas Pendidikan Indonesia, which has provided the opportunity for the authors to conduct this research. This research is supported by the Directorate of Research and Community Service Directorate General of Research and Development Strengthening the Ministry of Research, Technology, and Higher Education following the Research Funding and Community Service Agreement Fiscal Year 2019-2020.
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