2021 Volume 27 Issue 1 Pages 15-32
In this research, we consider the “subjects”─the targets of consideration of mathematically active persons ─as “SIGNs”─in Peirce’s semiotics─consisting of representamen, object, and interpretant, and explore the ontological status of this topic’s evolutionary development. The purpose of this paper is to clarify the status of the evolutionary development of the subjects in a lesson on the magnitude relationship between square roots for 9th graders through an analysis using semiotics and comparing the lesson with a lesson introducing square roots analyzed in previous research.
In the beginning of the lesson on the magnitude relationship between square roots, the subject was “a sign that looks like a number,” but with the introduction of a diagram, this was changed to “the length of a side of a square.” During the exploration of the successive approximation method, the initial “ squared to something close to 2” continued to exist. In this process, the possible range of existence and the method of diminishing intervals were revealed, so by “matome (summing up)” along the way, the subject became “something in between other numbers.” Finally, the existence of “something that is approximated by the successive approximation method” involved the process and product of the final matome.
The following hypothesis was thus obtained: In a classroom situation where the developmental process of accidental evolution is shown, a subject that has a “legisign (sign as law)” as its representamen can be changed to a “sinsign (individual sign).” It is suggested that in order for a new subject to become a being for many participants, it needs to become at least a productive object of existence, and it thus changes into a being as a class containing a sinsign within Peirce’s “ten classes of signs.” If the same situation had not changed to a subject containing a sinsign, it would be difficult for the participants to interpret the rules for application, etc., of the subjects, so another subject could emerge that transformed in to a different mode of representamen or the representamen might not change but the object or interpretant could change. We also obtained some suggestions about matome, which has a condensing and integrating function.
