The purpose of this paper is to clarify the effect of differences in the structure of natural language and mathematical language at the stage of “mathematisation” where the translation from the language in the real world to the language in the mathematical world.  To achieve the purpose, three sub-tasks were set in this paper; (1) How have language problems been dealt with in mathematics education and their research? (2) What kind of suggestions can be obtained from linguistics in order to qualitatively realize the structure and cognitive process of Japanese and mathematical languages? (3) From the perspective of linguistics, how does the differences in structure between Japanese and mathematical languages affect learners’ thinking?  

　　The results corresponding to the sub-tasks are as follows; (1) It is organized from an international perspective and a Japanese perspective.  In particular, while language problems have been attracting attention internationally in mathematical modeling research in recent years, the reason why it was difficult to study language problems in Japan was clarified.  (2) We pointed out that the field of linguistics, which has a high affinity with mathematics education and its research, is cognitive linguistics, and organized basic ideas of cognitive linguistics.  Furthermore, based on cognitive linguistics, we organized two relativist perspectives: “relativism of interpretation between individuals (conceptual relativism)” and “relativism of languages  (linguistic relativism)”. And we also theoretically argued that these relativism provide a new perspective that are different from the existing framework of mathematical modelling research.  (3) We conducted a survey and empirically verified that the perspectives of “conceptual relativism” and “linguistic relativism” are effective frameworks for capturing the influence of language on mathematisation.

　　The content of this paper may also be useful as a basic study that examining the possibility of using cognitive linguistics in mathematics education and research.  This is the significance of this paper.

　　This study interprets the change of the ontological status of “subjects,” which have a tripled structure consisting of a semiotic object, interpretant, and media.  The study was carried out during a series of secondary school mathematics lessons from the embodiment perspective including embodied acts and language use.  The authors presented the dynamic “zig-zag” phenomena of the semiotic classes, analyzed from the semiotic perspective, and assumed that perceptual media affected individual and collective activity in the classroom. Hence, we adopted the embodiment theory to make sense of the phenomena because it might prepare us for useful, detailed interpretations.  

　　Embodiment theory has at least two theoretical assumptions concerning complementary analytical methodology: (1) “subjects” could exist through embodied actions with perceptual media, and (2) the way to deal with “subjects” is language use.  These metaphorical expressions are based on some prior discussions about possible roots of embodied mathematics.  In recent mathematics education research, the metaphorical approach is one of the most familiar and strongest methods to analyze the multimodal nature of human mathematics. Gesture appears to be an important factor to form mathematical meaning during mathematical communication. Thus, we developed the analytical framework under the above two assumptions.

　　Through embodiment analysis, we could identify several typical metaphors, such as “a magnitude relation between numbers is a segment.” In addition, we could establish that the segment metaphor might create a foundation for students to understand an approximation value of a square root.  Finally, we identified whether students could recognize isomorphism among various media or not as the critical factor under the special “zigzag” phenomena.

　　The purpose of this paper is to problematize the characterization of indirect proof in previous research and to propose a cluster model for indirect proof in order to capture a student’s overall process of developing the concept of indirect proof. We review and analyze polysemous characterizations of “indirect proof” in research and practice from a theoretical perspective of “cluster models.” As a result, we find that we should consider the concept of indirect proof as essentially polysemous. It has dual aspects of a method and its product, and has at least the following five meanings: A) proof based on logical laws such as law of excluded middle and double negation elimination; B) an approach to prove an equivalent different statement, especially with a meta-theorem in logic; C) a heuristic device through the introduction of an assumption “if …”; D) any method for proving in which (1) we prove an equivalent statement with the given statement or (2) we disprove a contradictory of the given statement; and E) a collective term for proof by contradiction, by contraposition, and so on. In cognitive linguistic term, the concept of indirect proof can be understood as a cluster concept of the above four meanings from A) to D).

　　Based on this result, we draw an implication for more constructive research on indirect proof. We need to flexibly design a research plan and to explore the relationships between different meanings of indirect proof. In addition, we also need to explore new potential meanings of indirect proof based on the meaning E) as an extensional definition and to continue to update the cluster model of indirect proof.