| Teachers' knowledge is the main issue of mathematics education.Aside from the teaching knowledge theories of SMK,PCK,and others,Chevallard advocated the Anthropological Theory of Didactics(ATD)and Didactic transposition theory,which brought a new perspective to mathematics knowledge.According to the ATD,the source of what is taught at schools is not an invention of school but a creation outside of school,constructed and organized for practical sets of objects.The Didactic transposition starts from the scholarly knowledge transposed to the knowledge to be taught,taught knowledge,and to learned knowledge.In the traditional view of teaching in general,the focus is on the binary relationship between teachers and students,but in fact,mathematics knowledge is also one of the important factors in a mathematics classroom.Chevallard provides the ternary factors of teaching: teachers,students,and mathematics knowledge.Therefore,the didactic analysis is not only to focus on the classroom activities of teachers and students but also to the teaching methods of knowledge.According to ATD,knowledge is a result of human cognitive activities within a social system under the social and cultural background.Praxeology is the main concept of ADT,with two main components: praxis and logos.Generally,praxeology refers to human activities and behaviors,but in ATD,it also includes the intention and thinking of behaviors of members of the institution.The fraction division is one of the most difficult concepts to understand for both students and teachers.This paper observes primary school math classes in China and South Korea,analyzes the process of fraction division teaching knowledge generation in primary school classes and the social mathematics cultural norms generated in the interactions between teachers and students,and derives the significance of the mathematics knowledge formed in mathematics classes.The research questions are:1.What are the characteristics of didactic transformation in primary school mathematics classes of fraction division in China and South Korea?(1)What is the knowledge to be taught(KBT)presented in Chinese and Korean mathematics textbooks?(2)What is the taught knowledge(TK)in Chinese and Korean classrooms?(3)How does KBT change to TK in Chinese and Korean classrooms?The entire research process is divided into four stages: analyzing the structure of Chinese and Korean teaching materials and the content of the score division unit,pre-interview of teachers,video and voice recording of classrooms,and follow-up interviews of teachers.The study mainly analyzes teaching materials and classroom activities.The analysis framework is two main components of Proxeology(Chevallard & sensevy,2014,40): Praxis and logos.First of all,to analyze KBT lists in the content of Chinese and Korean mathematics textbooks,including text questions,models,and formulas.In addition,to analyze components of praxis,such as tasks,techniques,and logos: technology and theory.Thirdly,an analysis is made using lists of tasks,models,expressions,and language representations used by teachers and students in the class,four modules of PL are listed,and then the TK and transformed of KBT to TK.The results include the development and characteristics of fraction division in Chinese and Korean textbooks.It is found that there are different ways of interpreting the fraction division algorithm in the arrangement of Chinese and Korean textbooks.In Chinese textbooks,the concept of division as the inverse of an operator multiplication is used to explain the algorithm of reciprocal.In the development of textbooks,the arrangement position of reciprocal has also changed.The current textbooks of China describe putting the reciprocal first,and then exploring the calculation principle later.Korean textbooks represent different types of fraction division formulas,involving an algorithm of different denominator fractions division explore the transformation of reciprocal.However,the current Korean new textbook focuses on the different representations of fractional division,include the concept of determination of a unit rate and the double number line.Since the KBT in the textbook is different,the TK in the classroom is also different.In the mathematics class,both Chinese and Korean teachers follow the textbook activities.However,in the analysis of the class PL,it is found that the KBT in the teaching materials is different,and the TK techniques and technologies(mathematical terms,models,and formulas)in the class are also different.In Chinese classrooms,we found the hidden technology in the textbooks,and the teacher arranges the main technique of classroom activities.In Korean classrooms,the introduction of the real-life situation leads to the communication of finding out mathematics in life.The results show four-points of implications:(1)the arrangement of reciprocal in teaching materials;(2)the diversification of algorithm content of fraction division;(3)the use of line graph in teaching;(4)the reasons for the existence of teaching terms used in teaching(such as fraction rate,and multiple problems)(raison d'tre). |
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