Here is the secomd part of my rephrase. I urgently need to rephrase point 5 (symplifying it).
1) He addresses one of the pupils, identifying her by her number in the class, not by her name. She introduces herself as Sissy Jupe.
2) He takes offence at this version of her given name and insists that she is properly called Cecilia. When she points out that her father calls her Sissy, he takes to opportunity of questioning about her father. She replies that he belongs to the horse-riding.
3) Mr Gradgrind frowns and says that she must describe her father as a horsebreaker (in a circus??), who doctors sick horses. Or better as a veterinary surgeon, a farrier and a horsebreaker.
4) Gradgrind then insists on hearing a definition of a horse, and when Sissy fails to produce it, he turns his attention to another pupil called Bitzer who promptly delivers a dictionary definition of a horse.
5) "The square finger, moving here and there, lighted suddenly on Bitzer, perhaps because he chanced to sit in the same ray of sunlight which, darting in at one of the bare windows of the intensely white-washed room, irradiated Sissy. For, the boys and girls sat on the face of the inclined plane in two compact bodies, divided up the centre by a narrow interval; and Sissy, being at the corner of a row on the sunny side, came in for the beginning of a sunbeam, of which Bitzer, being at the comer of a row on the other side, a few rows in advance, caught the end. "