Since I was a professor of maths education for most of my working life my brain is pretty much dominated by the idea of number so when I retired all sorts of numerical thoughts began to spring into my mind. Most of them were things that had never occurred to me before or things that I may have taken for granted. For example, I no longer meet upwards of 80 or so new people, students, in my life each year. There were usually around this many students in the courses I taught each year. And then when I supervised student teachers in public school classrooms I would frequently meet another 100 or so new children each year. Only now that I am retired have I become acutely aware of just how much I learned from all these new people I met each year; the interactions, the term papers I read, the classroom events and experiences all conspired to enrich my life each year, I miss these interactions so much and frequently wonder if I did take them for granted. I don't think I did but it all looks so different now.
So I now have time to watch and interact with my two and a half year old grandson as he comes to terms with the wonderful world of maths. He has already learned to say "Sierpinski triangle" and can pick one out in a whole bunch of different triangles. There is nothing quite like hearing him say those two words and pointing at an example of one, Numerically, he is going through the process of learning the number names and, dare I say it, has already the beginnings of a sense of cardinality at least with two and maybe three objects. His mom, my daughter Marie, took my maths ed. grad course several years ago and so is really in tune with the growth and development of a child's counting skills. She demonstrates so wonderfully the two most importance things in teaching maths. First the importance of observing the student, her son, and second, just how much a full understanding of the most basic mathematical ideas is to the teaching process. She doesn't push maths on him at all but just makes him mathematically aware of the world in which he lives.
Even something as seemingly simple as helping a student count requires a deep understanding of the ordinal, nominal, and cardinal use of number. Lachlan, my grandson, is currently learning the sequence of the number names. He can number name more or less up to twelve but hasn't yet quite got the teen sequence. I say number naming because he really is not counting yet in the true sense of the word apart from, perhaps, "twoness" and "threeness". When he number names he is just learning the order in which the number names occur. He has, yet, no sense of "fiveness", for example. More next time on the nitty gritty of learning to count.