تحلیلی بر معرفی مفهوم مشتق در کتاب‌های درسی ریاضی دورة دوم متوسطه با تکیه بر نظریة نشانه‌شناسی پیرس در طی 45 سال

Abstract
The purpose of this study is to analyze the content of Iranian mathematics textbooks to introduce the concept of the derivative at a point during a period of 43 years (1980-2021). Textbooks are one of the main sources for teaching mathematics and are often used by students to learn mathematics. In Iran, the Ministry of Education is responsible for developing national textbooks. They form a team of senior mathematics lecturers, mathematics teachers, and mathematics educators to work closely together to develop such textbooks. Numerous studies focused on how teaching and learning derivatives could be improved due to their specific importance in mathematics and its applications in a wide range of disciplines. However, students in many countries still face various difficulties in understanding this concept. Part of these difficulties might be related to the approaches taken by textbooks for introducing this topic. Semiotics can be defined as knowledge or branch assessing signs and sign systems. Peirce defined a sign as a thing that is regarded as the symptom of something else for a person on a specific occasion and special topic. Signs play an important role in presenting and understanding mathematical concepts. Peirce provided different classifications based on sign users, among which the most important one, based on the relationship between sign and object as icons, index, and symbol, was used in this section. Icon: There is a similarity between sign and object for example a person's photo. Index: A sign which is related to an object. Sign and object should be related and a casual and natural relationship should be found between them such as the implication of smoke in a fire. The index itself consists of several icons. Arranging icons that are based on the hierarchy of Peirce's semantics is considered one of the important aspects of indexes. Symbol: A sign which needs an interpreter. Regarding symbols, the relationship between sign and object is contractual such as code. Pierce's theory can be used to show the importance of signs, how objects are represented by sign vehicles, how signs are intertwined and interpreted, and how they relate to and mean by signs. A mathematical concept, like a derivative, is not independent of its representations. Therefore, the construction of concepts is determined by the signs, so the signs play a primary role in presenting and correcting mathematical concepts. According to this semiotic view, we can consider that conceptualization and semantics are done by the connection between the three components of the sign. Like other sign vehicles, mathematical sign vehicles can only represent some aspects of a mathematical object, but all Its properties and properties do not show at the same time, they bring some to the background and keep some in the background. As a result, the process of conceptualizing and constructing the meaning of mathematical objects can be considered as a recursive process mediated by a variety of mathematical cues. This view led us to consider semiotics as a successful tool for analyzing the concept of derivation in mathematics textbooks. The math symbols used in the textbooks are mainly used as tools for coding and describing mathematical objects, for operations with these objects, as well as the communication of mathematical knowledge between teachers and students. As a result, the way mathematical symbols are presented and communicated in textbooks may activate or limit the process of derivation conceptualization. The research method is deductive qualitative content analysis. The unit of analysis is all the examples, activities, exercises, text, and pictures related to the textbook derivative chapter in both mathematics-physics and experimental sciences majors (Year 11 & Year 12). Because of applying the recent Calculus II textbook in the current educational system, it was used in the present study to assess introducing derivative concepts based on Peircean theory. This textbook studied the topics of functions, trigonometry, infinite limits, derivatives, and applications of derivatives in five chapters and 152 pages. Since authors mentioned in the introduction that this textbook emphasized understanding concepts, the present study highlighted the pattern of introducing a derivative concept which was presented in Chapter 4 named derivative and involved 40 pages. The first chapter of the textbook provided ideas and considerations about tangents. Additionally, intuitive concepts, definitions, some of the features, and techniques of calculating limits and properties of continuous functions are provided in the textbook. Lessons 1-3 regarding the derivative concepts were analyzed in the present study. The present exploratory and interpretive study aimed to analyze relevant components concerning a semiotic perspective to provide a derivative concept. This analysis aimed to assess how the mathematical signs existing in the Calculus II textbook were provided and connected. Some points about the possibility of activating or limiting students were used for conceptualizing derivatives in the present study. Further, the pattern of interpreting and understanding these signs during the use of textbooks was considered. The considerations of the present study on textbooks were classified into three classes and provided in the following parts. For this purpose, Pierce's semiotic framework is used, and in particular, his classification of sign vehicles is discussed. The sign vehicles associated with the concept of the derivative may be iconic, index, or symbolic. The authors have used a special classification of the icon (visual or graphical and metaphorical) and index (numerical, formula, and graphical) of Pierce’s framework. The validity of the measurement tool has been confirmed by three experts in mathematics education and its reliability has been done by reviewing two evaluators and 100% agreement. The results show that out of 14 textbooks reviewed, only 3 textbooks had both graphic or pictorial images and metaphors; Also, only 4 textbooks had all three numerical, formula, and graphical indexes. Calculus II and Mathematics III in Conceptualization of the derivative at a Point became more compatible with Pierce's theoretical framework of semiotics in presenting a derivative at a point than other textbooks. New textbooks on the subject of the derivative at a point have provided appropriate teaching and learning opportunities for students to learn meaningfully, With the help of the semiotics package.