Professional competency: Pre-service mathematics teachers’ understanding toward probability concept

To cite this article: Sulfiah, S. K., Cholily, Y. M., & Subaidi, A. (2021). Professional competency: Pre-service mathematics teachers’ understanding toward probability concept. JRAMathEdu (Journal of Research and Advances in Mathematics Education), 6(3), 206-220. https://doi.org/10.23917/jramathedu.v6i3.13779 Professional competency: Pre-service mathematics teachers’ understanding toward probability concept


Introduction
Probability as one of mathematics lessons which is fundamental in mathematics curriculum in the world (Elbehary, 2020) is used for the learning material from primary education to higher education as the fulfillment of students' needs when students confront with a problem (Batanero, 2020). Additionally, it becomes one of content standards which should be mastered by students (NCTM, 2000). In addition, it is one of contexts examined in PISA to assess mathematical literacy (OECD, 2016). Probability is a challenging lesson for students especially in compound events. They tend to make incorrect predictions of events http://journals.ums.ac.id/index.php/jramathedu (Abrahamson, 2014) so that many students have difficulty learning probability. They find it tricky to connect theoretical and empirical contexts (Nilsson, 2014).
In general, there are some difficulties confronted by students when solving the problem of probability. These are combination contents, binomial probability comparisons, product rules, and general problems of binomial probability (Sánchez & Landín, 2014). Several aspects leading to students' difficulties in resolving probability problems are due to the lack of concept understanding, patterns, and motivation for doing probability lessons (Putridayani & Chotimah, 2020;Sánchez & Landín, 2014). Based on the previous research conducted by Williams dan Nisbet (2014) that there is a positive correlation among attitudes, beliefs, and students' probability concept understanding. The fundamental problem of understanding probability concepts that students should master is obtained in its understanding and the relationship between probability lessons and other mathematical aspects (Brase, Martinie, & Castillo-Garsow, 2014;Savard, 2014).
One of the factors influencing students' learning is teachers. However, teachers' competency still lacks at solving the problems of probability (Batanero, Contreras, Díaz, & Sánchez, 2015), even though, according to the latest research conducted by Estrada and Batanero (2020) that a teacher had performed a positive attitude towards probability lessons and his instruction was acceptable. To perform effective learning, teachers need to master general and specific content knowledge of probability (Groth, 2010). Hence, based on some previous research, it is fascinating to research anew to achieve more credible results concerning pre-service teachers' understanding of concept based on the current research result.
There have been several previous studies regarding the ability to understand mathematical concepts of prospective teachers such as the relationship between student final grades and understanding of mathematical conceptsconducted by Faradillah (2018), and investigation of students' errors in completing probability lessons by Indriani (2020). However, Indriani's studies only focus and discuss student mistakes in solving a problem in probability. She did not discuss the cause of student mistakes and how the students' understanding of mathematical concepts affects them to make the mistakes. Besides, the understanding of mathematical concepts is very important for pre-service mathematics teachers..
Based on the problem analyzed, this research aims to describe pre-service teachers' professional competency in probability lessons as vessels that will enter into school to educate and teach students since teachers in education 4.0 should possess professional competency (Zulfitri, Setiawati, & Ismaini, 2019). Besides, teachers' profile implicates in building students' concepts (Yuliyanti, Winarso, & Misri, 2019), especially the ability of understanding concepts now that it will influence other competencies of teachers (Yilmaz, 2020). However, the competency that will be researched is limited to the understanding concept in probability lessons as the quality of concept understanding determines the learning effectiveness that needs to be considered (Afriyani, Sa'dijah, Subanji, & Muksar, 2018). Furthermore, the fundamental basis for solving the problem is that apprehending the concept underlying a problem can adopt an appropriate resolution strategy (Sundary, Jatmiko, & Widyastuti, 2020). The probability lesson, especially in combinatorics, is vital to be optimized or mastered by all grades (Ihsan & Kosasih, 2018

Research Method
This research employs a qualitative research method with a descriptive approach since it aims to describe the pre-service teachers' professional competency of understanding concepts towards probability lessons observed based on mathematical ability. The subjects of this research are three college students of mathematics education who have passed discrete mathematics. Each of them has advanced, intermediate, and basic mathematics ability in a college in Madura. The three subjects are mathematics preservice teachers who will enter into mathematics later on and teach students, especially probability lessons. The determination of subjects utilizes the purposive sampling technique. Several things considered in naming the subjects are the GPA (Grade-point Average) of the last semester and information from the in-charge lecturer since it is unlikely to administer a subject election test due to online learning applied at the college; hence it is improbable to face to face.
These research instruments are worksheets and interview guidance. The worksheet was used to know the pre-service teachers' professional competency towards combinatorics lessons observed based on mathematical ability. The used worksheet contained one question of probability. It was validated by two mathematics experts (Discrete Mathematics lecturer) and an experienced mathematics teacher who teaches in one of schools in Madura and It is deemed valid to become a research instrument. The validation used included content validity, language and question writing, and the suitability of the questions with indicators of the ability to understand mathematical concepts. Each validator performs the same validation according to these points to be more sure of the instrument's validity. Meanwhile, the guidance of a semi-structural interview was employed to determine more thorough information and more relied on the credibility of data result to obtain a conclusion.
Methodological triangulation is used to check data validity, namely giving written tests and interviewing with the subject to get valid data. The phases employed in data analysis correspond to the steps of qualitative research. These are data reduction, data presentation, and concluding. Afterward, the test results conducted by the subjects and the interview transcript are reduced, the data which are reputedly imperative are presented in the research result and discussion, then the conclusion obtained. To determine the preservice teachers' professional competency, the indicator used in this research refers to Jin and Wong (2015) that is an individual concept, connecting some concepts and relating the concept to its operation. In Table 1, the studious description is provided. It corresponds to the research purpose that is to know pre-service mathematics teachers' understanding of the concept.

Results and Discussion
The result of the written test and interview for the three subjects regarding the understanding concept ability as one of the teacher's professional competencies are presented as follows.
Subject with advanced mathematics ability  Figure 1 is the answer sheet of the written test results of subjects with advanced mathematicsability regarding concept understanding of the probability lessons, especially on combination content. Based on the written test result in Figure 1, subject with advanced mathematics ability (ST) could do the written test of probability lessons correctly. Moreover, he was able to name a commensurate strategy to finish it. The excerpt of the interview between the researcher (R) and ST is presented as follows. Based on the interview result, ST could choose an appropriate strategy to answer, searching the number of the probability of the occurrence regarding 4 yellow balls removal from box A and 4 blue balls from box B by using combination concept. ST could also comprehend the reason why he utilized the combination concept in answering the question. He had been able to determine example and non example from the concept used. However, he had not been able as yet to present any reasons concerning the general use of the combination concept. When he was asked further why not employing another formula such as using permutation concept, ST could not correct answers since he had not apprehended the concept completely. It is discussed in the following interview result: Besides determining an appropriate strategy, ST could also connect one concept to other concepts, after having determined the number of event and determining sample space by using the combination. He continued using probability concept towards each box to seek the probability of 4 yellow balls removal from box A and the probability of 4 blue balls removal from box B at the same time. It is described in the following interview result. However, at the end of answering the question, ST could not wholly explain the excuse why seeking the probability of ball removal from box A and box B, why it should be multiplication, and why it did not use other operations such as addition, subtraction, or division. Therefore, even though the answer given was correct and asked further questions regarding the probability category he answered belongs, he could not answer. Furthermore, the last concept understanding is connecting symbol with a concept. ST had been able to answer accurately. It is described in the following interview result.

P (A) means that the Probability of occurrence A and P (B) is the Probability of occurrence B. Symbol of a combination
In a thorough going interview with ST regarding the definition of the concept, ST did not comprehend its definition itself. He states that the concept he employed in answering the question is only a combination concept because probability is not a concept, but it is a theory. According to ST, a concept is a technique or a complete strategy to solve a problem. The excerpt of the interview is presented below:  Figure 2 is the answer sheet for the written test results of subjects with intermediate mathematical ability regarding concept understanding of the probability material, especially on combination content. Based on the written test result in Figure 2, subject with intermediate mathematics ability (SS) could not do the written test of probability lessons correctly. Based on the interview result with the subject, however, he was able to name an appropriate strategy to finish the question. It is discussed as the following interview result. However, when asked further questions concerning why he employed combination in finishing the question, he could not give any correct answer. He did not understand why the combination was used. Finally, he stated that he utilized combination due to random removal as presented in the excerpt of the interview as follows.  However, when asked further questions concerning why he employed combination in finishing the question, he could not give any correct answer. He did not understand why the combination was used. Finally, he stated that he utilized combination due to random removal, as what is quoted from the interview below: SS's incomprehension towards combination concept induces him false in determining probability towards box B. He did not use combination in determining either sample point or sample space to decide the probability of 4 blue balls in box B since the available balls in box B are 4 blue balls and 4 red balls. As a result, the number of balls which will be taken out equals to the number of available balls. Therefore, he perceived that balls which are going to be taken are 4 desired blue balls. It is discussed as the following interview. Besides the lack of comprehension of combination concept, he also lacked the understanding of probability concept. He could not able to comprehend probability definition. He was a bit short of apprehending either sample point or sample space in probability even though he understood the procedure of doing probability and had been able to do combined with a correct procedure as presented in the excerpt of the following interview. At the end of answering the question, SS could not also seek the probability of 4 yellow balls removal from box A and 4 blue balls removal from box B. He added the outcome of probability removal in box A and box B and he also did not know why he used addition, and he was not able to determine to what category the probability that he answered belongs. It is discussed in the following interview.

R : The question in the question is what is the chance of taking 4 yellow balls from box A and the chances of taking 4 blue balls from box B SS
: the results are Summed, Furthermore, the previous concept understanding is connecting symbol with a concept. SS had been able to answer correctly as presented in the excerpt of the interview as follows.

R
: What C and P mean? SS : C is the symbol of combination, P is the symbol of Probability http://journals.ums.ac.id/index.php/jramathedu Figure 3 is the answer sheet for the written test results of subjects with basic mathematical ability regarding concept understanding of the probability material, especially on combination content. Based on the written test result in Figure 3, the subject with basic mathematics ability (SR) could not do the written test of probability lessons accurately. SR could name an appropriate completion strategy to answer it, that is seeking combination to determine the sample point and the sample space of the probability of the ball removal from the box. However, it is arguable that he only memorized the procedure without comprehending what he was answering. SR himself was not able to clarify the result of his answer. He did not apprehend what combination is and how it is used. In addition, he could not determine the example and the non-example of combination concept itself, as discussed in the following interview:

Figure 3. The written test result of subject with basic mathematics ability
Besides the incomprehensibility of concept definition used, the procedure employed in answering the question was also incorrect, like the use of combination procedure, factorial, and probability. He was not able to understand what he was writing. It is discussed in the interview as follows. : 5! Is the number of yellow balls and 4! of the number of yellow balls that will be taken Furthermore, for the last concept, understanding concept and its symbol, SR indicates could not correlate them. He did not understand the meaning of P(A) which he wrote because he could not answer correctly when He asked about it. He did not know and forgot about what he was writing. The excerpt of the interview with SR is presented below.

R
: What is C? SR : The symbol of the combination R : What is P(A)? SR : I don't know The ability of concept understanding viewed based on mathematics ability Based on the analysis result of the three research results towards the subjects, it is grouped as presented in Table 2. -Based on this indicator, the subject has not been able to understand the definition of concept. However, he has been able to explain correctly concerning the use of the concept he employed in answering the question. He could give the example and the non-example of the concept such as in determining the use of combination and seeking Probability of an event.
-The subject has been able to employ the correct procedure of every completion step, such as the use of pattern, symbol, and numeral like factorial numbers, combination procedure, and the procedure of answering probability.
-Based on this indicator, the subject has not been able to comprehend the definition of concept and explain the concept he used in answering the question, for instance, he has not understood the reason of the used combination concept and to determine Probability of the ball removal in 2 boxes so that it caused an error in determining probability of the ball removal in the second box.
-The subject has been able to employ the correct procedure of every completion step, such as the use of pattern, symbol, and numeral like factorial numbers, combination procedure, and the procedure of answering probability.
-Based on this indicator, the subject was not able to comprehend the definition of concept and the concept he used in answering the question.
-The subject was not able to employ the correct procedure of every completion step, such as the use of pattern, symbol, and numeral like factorial numbers, combination procedure, and the procedure of answering probability.
http://journals.ums.ac.id/index.php/jramathedu The subject has been able to connect one concept to other concepts. He was able to determine the finishing strategy correctly so that the solution of the answer he did was also correct even though he did not comprehend what concept he employed in the last completion.
The subject was lack of ability to connect some concepts. Even though he employed the accurate strategy, in the last completion, he lacked the understanding of connecting one concept to other concepts so that it caused an error in determining the solution of the completion.
The subject was not able to connect some concepts. The strategy he used was not accurate so that it caused an error in determining the solution of the completion.
Connecting concept with its operation -The subject has been able to connect a concept with its operation, for instance P(A); the Probability of event A, n(A); the number of events in A, n(S); the sample space of an event and Concept understanding ability is an ability which is imperative to be possessed especially by pre-service teachers who will enter into the education sphere. It is also a professional competency that should be resided in teachers. Teachers knowledge and competency of mathematics have a crucial role towards teaching practice (Stahnke, Schueler, & Winter, 2016). A teacher's good competence will influence teaching and learning process in a class as a good instruction conducted by teachers to students induces the lesson delivered easier to be comprehended by students (Priskila & Helti Lygia, 2019). Thus, the component mastery of content knowledge and teachers' mathematical pedagogy is more essential than other components (Qian & Youngs, 2015). In the research result, the concept understanding ability of the three subjects, the subject with advanced, intermediate, and basic mathematics ability has been analyzed.
The subject with advanced mathematics ability could employ the concept and the correct procedure of every completion step and connect one concept to other concepts. The solution of the answer he did was also correct, and he could also connect a concept with its operation. Subject with intermediate mathematics ability could employ the concept correctly and the procedure of every completion step. However, he could not connect one concept with other concepts so that it caused an error to determine the solution of the answer. Meanwhile, he had also been able to connect a concept with its operation. Subject with basic mathematics ability could not correctly employ the concept and the procedure of every completion step and he was also not able to connect some concepts so that it http://journals.ums.ac.id/index.php/jramathedu caused an error in determining the solution of the answer. In addition to that, he was not able to connect a concept to its operation. Meanwhile, concerning the indicator of concept understanding improvement, the three subjects have not reached this phase since they have not understood the definition of concept correctly, particularly probability concept, even though the subject with advanced mathematics ability is procedurally capitalized.
Based on the previous research result conducted by Afriyani et al. (2018), students' concept understanding ability is grouped into three stages. First, students' can perform one-concept interpretation, but they cannot perform a correct completion since they only employ one concept in creating a conclusion. Second, they can perform interpretation by employing some concepts, but they enable an error in determining the completion because information synthesis or interim conclusion was not executed. Third, they can perform the interpretation of some concepts in completion and provide a correct conclusion based on a completion process. Based on the previous research by Hutauruk and Panjaitan (2020), pre-service teachers lacked competence in mastering mathematics learned at school, especially in understanding the concept. The difference in research result level can occur due to factors such as more extensive area, the research field, purpose, and focus of the research conducted (Chick, 1998).
Should it be viewed from the previous discussion, the competency of teacher candidates needs to be explicitly advanced concept understanding towards probability lessons. There are five strategies can enhance concept understanding ability that can be implemented either at school or college. First, the strategy selects learning strategy and considering the number of learning hours since they can impact learning effectiveness (Tsubaki, Ogawara, & Tanaka, 2020). Second, it provides scaffolding by entering into dialogue with students. It can also enhance students' conceptual understanding (Kazak, Wegerif, & Fujita, 2015). Third, it administers assignments or exercises repeatedly since students can also broaden their knowledge based on their previous experience (Kazak & Confrey, 2007). Fourth, it gives the first stimulus by apprising the philosophy and the history of mathematical theory, specifically probability in order that students' motivation improves learning the material (Kuznetsova, 2019). Fifth, it conducts cognitive guidance analysis, concept clarification, logical multiplication and student ratio (Nunes, Bryant, Evans, Gottardis, & Terlektsi, 2014). To enhance teachers' professionalism, however, teachers should employ an intuitive approach followed by a subjective approach, frequency, and classic (Vásquez & Alsina, 2019).

Conclusion
Based on the research result and the previous discussion, it is obtained a conclusion that the pre-service teachers' concept understanding ability remains varied based on their ability level of mathematics. The subject with advanced mathematics ability could employ the concept and the correct procedure of every completion step and connect one concept to other concepts so that the solution of the answer he did was correct. Furthermore, he could connect concept with its operation. The subject with intermediate mathematics ability could employ the concept correctly and the procedure of every completion step. However, he could not connect one concept to other concepts to induce an error in determining the solution of the answer. The subject had been able to connect a concept with its operation. The subject with basic mathematics ability could not employ a concept and a procedure accurately towards every completion step and not able to connect some concepts, so that it induced an error in determining the solution the answer. The subject could not also be able http://journals.ums.ac.id/index.php/jramathedu to connect a concept with its operation. Concerning the indicator of concept understanding advancement, the three subjects have not reached this phase. They had not been able to comprehend the definition of concept fabulously specifically probability concept, even though the subjects with mathematics ability are procedurally capital.
Hopefully, this research result will be a reference for the subsequent researchers in order that it becomes more enhanced or the research subject is more widened and either for high school students or higher education students. Thus, it can either improve the concept understanding ability they have possessed or enrich the students' concept understanding.