Prospective secondary mathematics teachers read Clairaut: professional knowledge and original sources

Abstract

The use of original sources is a useful resource not only to be used with secondary school students but also with prospective mathematics teachers. In this work, we designed a series of tasks based on a fragment excerpted from Clairaut’s Éléments de Géométrie to be carried out with 24 participants enrolled on a Masters’ Degree in Secondary School Mathematics Teaching. This fragment was chosen both due to its content and to its narrative structure and our main goal was to determine which elements of professional knowledge were used by prospective secondary mathematics teachers when reading this fragment. In order to do so, we used the MKT model as an analytical tool and we also assessed some aspects related to literacy skills. The prospective teachers were able to recognize mathematical and pedagogical components within the source that relate to their future practice. In addition, the participant’s literacy skills seem to play a role in the richness of their reading.

APPENDIX

In this appendix, we provide the French original as well as an English translation of Clairaut’s fragment. The original text (Fig. 9) corresponds to the Third Part, Section XVIII of the Éléments de géométrie (Clairaut, 1741). The English translation that we provide here has been slightly adapted from Chorlay (2015), p. 490):

Fig. 9

Original French text (Clairaut, 1741, pp. 125–128)

“Since we saw that the angles on the perimeter AEB, AFB, AHB are all equal, one wonders what becomes of angle AQB as its vertex Q coincides with point B, the extremity of its base. Would this angle then vanish? It does not seem possible that it suddenly vanishes without gradually decreasing. Also, one cannot see after which point this angle would cease to exist; how, then, could we measure this angle? The only way out of this conundrum is to resort to the geometry of the infinite; a geometry of which all men have some (maybe imperfect) grasp, and which we aim at improving.

Let us first observe that, as point E approaches point B, thus becoming F, H, Q etc., line EB gradually decreases, as the angle EBA which it makes with line AB increases ever more. But, however short line QB may become, the angle QBA will not cease to be an angle, since, to make it perceptible, we only need to extend line QB to point R. Will the same hold for line QB once it has decreased to the point of vanishing? What has then become of its position? What about its extension QR? It is obvious that it becomes no other than the line BS which touches the circle only at B, without cutting it at any other points; for this reason, this line is called the tangent.

Moreover, it is clear that as line EB continuously decreases and eventually vanishes, the line AE, which successively becomes AF, AH and AQ etc., comes ever closer to AB, and eventually coincides with it: hence the angle AEB subtended at the perimeter, after becoming AFB, AHB and AQB, eventually becomes the angle ABS between chord AB and tangent BS; and this angle, which is called the alternate-segment angle, must retain the property of being half of the measure of arc AGB.

In spite of the fact that this proof may be a little abstract for the beginner, I thought fit to include it, since it will be very useful for those who will further their study into the geometry of the infinite to have been accustomed to these considerations fairly early on. However, if beginners find it too difficult, they can be led to the discovery of another one explaining them the main property of tangents.”