If I read that article right, the fall of K being studied was fall 1998, so indeed, it'll be interesting to see how things have changed.
I've talked before about the surprise I felt the first time I found that DS-then-just-turned-3 could indeed answer "which number is less?" questions not, as I'd expected, for all pairs of the numbers he apparently understood, but only for some of them, just as the developmental list I'd been reading predicted (albeit for kids several years older!). So it could be that teachers observed "teaching counting" were actually doing something much more sophisticated and appropriate... no, who am I kidding?
But seriously, part of it may be a mismatch between what mathematically aware people intend and what actually happens in the classroom. For example, identifying shapes is a grand thing to do if the follow-up question is "How do you know?". Children may start school being able to identify circles, but they don't, typically, start school knowing the definition of a circle. I would think there is lots of good maths to be learned by a Socratic dialogue along the lines of "What is it?"... "How do you know? Could you explain what a circle is to a Martian who doesn't know the word circle?"... "So is this a circle, too?"... "Why not?"... "Ah, now I understand. So this is one, and this isn't?" etc.
I'll have what she's having. This is one of the ONLY types of instruction that truly work with my DD. She loves to think and talk about her thinking and learning. But she does NOT love to do things that are seemingly pointless just because that's what the directions said to do. (ugh)
