The Chinese algebraic method, the tian yuan shu, was developed during Song period (960-1279), of which Li Ye's works contain the earliest testimony. Two 18th century editors commentated on his works: the editor of the Siku quanshu and Li Rui, the latter responding to the former. Korean scholar Nam Byeong-gil added another response in 1855. Differences can be found in the way these commentators considered mathematical objects and procedures. The conflicting nature of these commentaries shows that the same object, the quadratic equation, can beget different interpretations, either a procedure or an assertion of equality. Textual elements in this paper help modern readers reconstruct different authors' understandings and reconsider the evolution of the definition of the object we now call 'equation'.

In this study we reveal that there were positive changes after the practicums of pre-service math teachers, in that, they changed contents of class and the way of explanation in order to enhance students' understanding, and provided opportunities for students to participate in the class. In addition, they made more use of textbooks in their class instead of depending on their experience acquired before the practicums. However, no difference was found before and after their practicums, neither in their review of lessons nor in their use of simple 'question and answer' strategies. In this regard, we suggest continuous researches be conducted in order to enhance pre-service mathematics teachers' teaching ability, especially those which is hard to be acquired in a short period of time. And, reflecting the results of the research, systematic development and operation of the curriculum are required in the teacher training institutes.

In tropical geometry, the sum of two numbers is defined as the minimum, and the multiplication as the sum. As a way to build tropical plane curves, we could use Newton polygons or amoebas. We study one method to convert the representation of an algebraic variety from an image of a rational map to the zero set of some multivariate polynomials. Mikhalkin proved that complex curves can be replaced by tropical curves, and induced a combination formula which counts the number of tropical curves in complex projective plane. In this paper, we present close examinations of this particular combination formula.