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In order to participate proficiently in mathematical activity, a student needs to become fluent in both mathematical sign activity and speaking about the activity with signs. From the theoretical viewpoint of the philosophers Wittgenstein and Peirce, on the one hand, mathematics is seen as a sign game, while on the other hand experiments with signs allow for observable, communicable and describable reasoning. In this article, the focus is on the student’s sign activity and his or her speaking about it and the question of how they are intertwined. To that end, a method is presented to display the interplay of student’s activity with signs and speaking about it, and in particular, to reconstruct the student’s line of argumentation. This is applied to a case study where imaginary dialogues written by a class of grade 11 students on the topic “derivative” were analyzed.

Activity with Signs and Speaking About It: Exploring Students’ Mathematical Lines of Thought Regarding the Derivative