A pedagogical potential of one mathematical inaccuracy Alik Palatnik 1 # Springer Nature B.V. 2020 Dear editor of Educational Studies in Mathematics, This letter follows up the article by Fan, Qi, Liu, Wang, and Lin (2017) “Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems? An intervention-based study with two Chinese classrooms,” which appeared in ESM 96(2). It appears that the article contains a task with a mathematical inaccuracy (Fig. 1). In what follows, I first describe the inaccuracy and then outline the pedagogical opportunity that it provides. MP = NQ does not imply MP⊥NQ (see Fig. 2). In Fig. 2a, NQ = MP = MP 1 , MP is perpendicular to NQ, but MP 1 is not perpendicular to NQ. Moreover, for some choices of M, N, and Q, it is impossible to obtain a right angle between MP and NQ at all (e.g., Fig. 2b). I wrote a letter to the editor, driven by a belief that this task, like many others, published in ESM may become a source of reflection for researchers, as well as for the future activities of teachers and student learning. As a researcher interested in the topic of introducing auxiliary lines in situations of proof and problem solving (Palatnik & Dreyfus, 2018; Palatnik & Sigler, 2018), I found it necessary to refer to the study by Fan et al. (2017). As a practitioner, I liked the task in question, which the authors used for the intervention and decided to use it with my undergraduate students. I noticed the counterexamples mentioned above using the dynamic sketch that I prepared for a lesson. Thus, a sequence of an original paper and pencil task and a DGE task (Fig. 3) provides an opportunity for students’ exploration activity with a surprising twist. For instance, first, the original statement is proved, and then its shortcomings are discovered. 1 Among the possible conjectures to discover and questions to explore in a due course of this activity are Educational Studies in Mathematics https://doi.org/10.1007/s10649-019-09922-6 1 The readers are invited to use applets, which were prepared for the lessons, or to create similar ones: https://www.geogebra.org/classic/fpuaceds; https://www.geogebra.org/classic/huvywfzq; https://www.geogebra. org/m/arq9frkj; https://www.geogebra.org/classic/gp4bpk2q. * Alik Palatnik alik.palatnik@mail.huji.ac.il 1 The Hebrew University of Jerusalem, Mt. Scopus, 9190501 Jerusalem, Israel