Using pivot signs to reach an inclusive definition of squares and   rectangles
  We present some fragments of a teaching experiment realized in a first grade classroom, to sow the seeds for a mathematical definition of rectangles that includes squares. Within the paradigm of semiotic mediation, we studied the emergence of pivot signs, which were exploited by the teacher to pave the way towards an inclusive definition of rectangles and squares. This was done to favor overcoming children's spontaneous distinction of these figures into distinct categories, reinforced by everyday language. The experiment is an example of an approach towards the theoretical dimension of mathematics in early childhood. 
Maria Giuseppina Bartolini Bussi, Anna Baccaglini-Frank
math.HO