Probability makes it easy to simply quantify the likelihood of an event, and statistics help rational decision making by reasoning about phenomena and drawing conclusions (Ministry of Education, 2015). Reflecting the emphasis on communication skills i...
Probability makes it easy to simply quantify the likelihood of an event, and statistics help rational decision making by reasoning about phenomena and drawing conclusions (Ministry of Education, 2015). Reflecting the emphasis on communication skills in modern society, probability and statistics are very important subjects in that they help develop communication skills. However, most students have difficulty in the probability and statistics unit (Kim Won-kyung, Kim Hyejin, 1992), and teachers also have difficulty in teaching the probability and statistics subjects (Lee Young-ha, 1992; Kim Won-kyung, Moon So-young, Byun Ji-young, 2006; Lim Ji-eun, 2011). According to the preceding study, the difficulties with the terms in probability and statistical units had the most negative effects on learning (Lee Hye-jin, 1992), and students understood mathematical terms differently depending on how definitions were stated in textbooks (Shin Eun-joo, 2002). Meanwhile, the probability and statistical knowledge of secondary math teachers were highly dependent on textbooks (Byun Ji-young, 2005). Therefore, teachers and students tend to recognize knowledge based on what is presented in textbooks, and it is necessary to analyze whether the definitions of terms and symbols are clearly and consistently presented in each textbook.
According to Cho Young-mi (2002), the definition of terms in school mathematics corresponds to a pedagogical transformation, and the definition appears in various forms in the transformation process. In addition, the term definitions learned by students may differ depending on what term definitions the teacher presents to students. Therefore, it is necessary to compare and analyze the definitions of terms in textbooks and the definitions of terms that teachers present to students. Meanwhile, the mathematical knowledge possessed by prospective teachers may affect students' understanding when they become teachers in the future (Lee Jean, 2015). Therefore, it is meaningful to look at the knowledge of the prospective teacher. Looking at the previous studies on term definition, most of the textbook analysis studies were conducted, and the analysis contents were limited to the geometric domain, the functional domain, and the algebra domain (Woo Jung-ho, Cho Young-mi, 2001; Kang Hong-kyu, Cho Young-mi, 2002; Cho Young-mi, 2002). In addition, there was no study analyzing the definition and definition method of high school probability and statistical textbooks according to the 2015 revised curriculum, and the study of recognition of term definitions was focused on elementary and middle school students. Therefore, study on the perception of prospective teachers was very insufficient. Particularly, in the probability and statistics, the prospective teacher recognition study was partially addressed for a particular term and there was no study that examined the prospective teacher recognition in an integrated manner.
Therefore, the purpose of this study is to analyze the definition and definition method of terms and symbols used in high school probability and statistics textbooks, and to examine the perceptions of prospective teachers in this regard. To this end, we first selected 55 terms and symbols to be analyzed in this study by combining the learning factors in the 2015 revised curriculum and terms and symbols of probability and statistics textbooks. Then, through textbook analysis, the definitions of terms and symbols were identified used in each of the nine textbooks, and the definition methods were divided into connotative definitions, denotative definitions, and synonymous definitions. At this time, in order to analyze the definition method, the sub-type classification for each definition method for terms in the previous study (Cho Young-mi, 2001) was used as an analysis framework. In addition, based on the results of the analysis, 10 probability and statistical terms and symbols were selected, and each term and symbol was described for 3 definition methods and composed of questionnaire questions for prospective teacher's perception survey.
The questionnaire developed conducted surveys and interviews with five prospective teachers to identify if they could correctly describe the definitions of terms and symbols, what types of definitions they were most aware of each term and symbol, and what types of definitions they would choose when guiding students to terms and symbols. This method of study is expected to confirm the understanding of the prospective teacher's mathematical knowledge of the definitions of terms and symbols of high school probabilities and statistics and the prospective teacher's understanding of students.
According to the analysis of textbooks, 12 out of 55 terms and symbols (approximately 22%) showed different definitions. These terms and symbols are 'fundamental event', 'mathematical probability', 'P(A)', 'independence', 'random variable', 'P(X=x)', 'probability distribution', 'discrete random variable', 'continuous probability variable', 'σ(X)', 'a large number of laws' and 'normal distribution'. In addition, as a result of analyzing the definition method by analysis frame, the connotative method accounted for the highest proportion (47%), and the denotative method accounted for the lowest proportion (20%). It seems that the reason why the connotative method is used more than the denotative method and the synonymous method is that the strictness is emphasized in the high school mathematics rather than the elementary and middle school mathematics. In particular, there were a total of 12 terms and symbols showing various definition method, including 'trial', 'complementary event', 'mutually exclusive event', 'independent trial', 'sample survey' and 'sample' with the same definitions presented in each textbook, but various definitions were applied, and 'event', 'independence', 'discrete random variables', 'continuous probability variables', 'normal distribution' and 'estimation' were using different definitions for each textbook.
As a result of confirming the degree of definition of the prospective teacher through questionnaire and interview, all five prospective teachers who participated in the study was lack of understanding the definition and could not correctly define the definition. Prospective teachers often failed to describe the definition of a "fundamental event" at all or wrote down the meaning by analogy. Regarding the "law of a large number", prospective teachers did not answer the definition correctly, such as writing completely irrelevant descriptions of the definition of 'a large number of laws', describing it in confusion with the central extreme theorem, or having a narrative error. And most of the prospective teachers defined 'normal distribution' in confusion with standard regular distribution. In addition, 'sample space' is defined as a set in high school textbooks, but prospective teachers often described it as a space extracted from the population. The definition of 'event' was defined by most prospective teachers as a usual meaning or rather vague expression, not the definition of the high school textbooks. In the case of 'independence', 'subordinate', 'expectation' and 'V(X)', there were cases in which the prospective teacher answered with routine definitions or described using definitions in middle school courses rather than high school courses.
As a result of analyzing the responses of each questionnaire, it was confirmed that the types of definitions that the prospective teachers are familiar with and the types of definitions that they think are appropriate to present to students do not match. When most prep teachers chose the definition to present to students, they recalled their previous easy-to-understand experience and chose the external definition that they thought would be acceptable to students, apparently because the prep teacher was heavily influenced by previous learning experiences. After summing up the responses of the prospective teachers, the distribution showed that each prospective teacher felt familiar with different types of definitions, and that the choice of types of definitions to be presented to students through the presented problem situation was also different for each prospective teacher. However, there were many prospective teachers who chose the definition that they thought would be easy for students to understand, unlike the definition in high school probability and statistics textbooks, despite the problem situations given when choosing the definition to present to students. Through this, it was confirmed that the prospective teachers lacked understanding of the definition and introduction method in high school probability and statistical textbooks.
The conclusions from this study are same as follows:
First, It is required to discuss about matching the definitions of each term and symbol in probability and statistical textbooks. According to this study, 12 out of 55 terms and symbols showed different definitions in different textbooks. Han Dae-hee (1998) referred that the mathematical term should be defined with consistency because the math curriculum is composed of a hierarchy and continues to affect subsequent learning. Thus, the mathematical terms of textbooks should be defined with consistency, given that teachers plan classes through textbooks and students understand terms through textbooks. In particular, as a result of analyzing high school probability and statistics textbooks based on the 2015 revised math and curriculum, two out of nine textbooks did not have a definition of 'fundamental event', so unlike other textbooks, the definition of 'mathematical probabilities' was defined without using 'fundamental event'. Shin Eun-joo (2002) said that students understand mathematical terms as definitions presented primarily in textbooks, and that the understanding of the terms is different according to the definition stipulated in the textbook. Therefore, terms defined differently in different textbooks can cause confusion to students, and this needs consideration.
Second, in-depth discussions are needed on the problems that various methods of definition presented in textbooks will cause. As a result of this study, there were 12 terms that matched the definition of each textbook, but were defined in several ways of definition or using different definitions of each textbook. Vinner (1991) said that if conceptual definitions are not formed by appropriate conceptual images by the presentation method of textbooks or by the teacher's explanation method, they may have misconceptions. Therefore, different definitions presented in textbooks can give different images of the definition method to teachers who plan and proceed with classes based on textbooks, which may also lead to different aspects of students learning the definition of terms. Therefore, discussions on this need to be made.
Third, it is necessary to supplement the teacher training process to ensure that prospective teachers correctly define the terms and symbols of probability and statistics and present appropriate definitions to students. In this study, the survey confirmed that the prospective teachers lacked the ability to describe the definitions of terms and symbols and that there were some errors. Since the mathematical knowledge of a prospective teacher can have a lot of influence on the student's understanding when they become a teacher in the future (Lee jean, 2015), it will be difficult for students to learn the definition of a textbook correctly if a prospective teacher who is not accurately aware of the mathematical definition does not educated enough in the training course and goes out to the field to guide students. Therefore, it is necessary for the teacher training course to provide opportunities for prospective teachers to explore various methods of definition in order to form a foundation for providing students with a correct understanding of mathematical terms.
Fourth, prospective teachers need to faithfully make efforts to study teaching materials and enhance their understanding of students. The results of the survey conducted in this study showed that the prospective teachers understood the students based on their existing learning experience and they chose the definitions that students could understand easily without understanding the intent of the questions presented in the questionnaire. This means that prospective teachers lack an understanding of how to introduce the definition of terms presented in textbooks in the current curriculum and an understanding of the various abilities of their abilities. Regardless of the student's difficulties, it is not desirable to teach using the knowledge or methods the teacher normally had, which may not be the best method (Ma, 1999). Therefore, prospective teachers should actively study textbooks in the teacher training process to improve their professionalism and strive to improve their understanding of students.
Based on the results of this study, a follow-up study is proposed as follows.
First, it is necessary to grasp the prospective teacher's perception of all terms and symbols in high school probability and statistics subjects. In this study, since only 10 terms and symbols were examined for recognition, it is necessary to check the remaining terms and symbols to confirm the overall perception of probability and statistical terms and symbols.
Second, we propose students' perception research on the definition of terms and symbols. By identifying which definition method students prefer and how familiar they are to probabilistic and statistical terms and symbols, it would help textbook authors produce textbooks to suit their level and interests, and it would provide implications for what teachers should be careful about when presenting definitions in the course of teaching and learning. Therefore, it is necessary to investigate students' perceptions.
Third, we suggest quantitative research on a number of prospective teachers. In this study, we conducted surveys and interviews with the few number of prospective teachers, so it is difficult to generalize. Therefore, it is necessary for subsequent studies to broadly identify the perception of prospective teachers through quantitative research.