Mathematics appears to be almost entirely neglected in early childhood learning. As both a mathematics teacher and father home-schooling his four-year-old daughter, I am particularly focused on teaching mathematical reasoning.
Consistent with the conclusions of Ginsburg et al, my daughter had a very natural grasp of number, operations, shape, space, measurement, and pattern from the time we began (age two). Imitation and rote were clearly the more natural mechanisms of learning at that age, as they remain now. And, as was discussed in another post, mathematical logic did not appear until she was approaching four years old. Now, her logic is limited more by her ability (or otherwise) to focus, and to organise her thoughts.
A lot of data exist relating age with typical cognitive abilities, but these are necessarily influenced by our collective culture of education. Has any rigorous research been done to determine when particular cognitive abilities can be developed, for example deductive as opposed to inductive reasoning? For early childhood development, can educators know what is possible or (generally/typically) impossible to teach at a certain age, or what is appropriate or inappropriate?