For me it's part of a set of ways of thinking that make up an active attitude.
Once, taking my class out on a museum visit, I was struck by how the students moved around town, how they crossed the road. They seemed not to be actively looking around much at all, not to be be checking the road was safe. It was almost a kind of trance, maybe talking as they walked, but relying on the teacher to make sure they were safe. It occurred to me that this was a metaphor for a lot of learning in school, a kind of passive reliance on the teacher to do the work. Mirrored by the teacher's expectation of being the authority.
I've just read Danny Brown's great post The student’s passive attitude towards Mathematics and his [sic] activities which is all about this kind of passivity.
So, inquiry for me is part of an active state of mind. Think Sherlock Holmes rather than Inspector Lestrade ("wholly conventional, lacking in imagination, and normally out of his depth"). What does Holmes do? He notices things. He wonders things. He, perhaps implicitly, asks himself questions. He goes out, looking for further clues. There is something of the hunter noticing the traces of his prey in him, perhaps going back to the roots of detective fiction in stories like The Three Princes of Serendip. The alertness of these detectives... the opposite of trance. The present case is not routine, there is not a procedure to follow that will bring sure results. It must be interrogated, understood. There may need to be experiment.
So how do we help our students to this kind of attitude?
That must be the subject of a whole book, and I've posted thoughts around this question already. But here are today's thoughts.
I came across the booklet Cognitive Activation in Maths and it seemed to be getting to the same thing (Cognitive activation = "starting to think actively"?). Here are some thoughts:
Another viewpoint: the idea of "open". An openended problem is one where pupils can follow something up in different ways. Dan Meyer, among much else, talks about openmiddle tasks. And then in a unit of inquiry in the PYP there's meant to be space for openbeginning inquiry. What does open mean? To me, it means that students can do some choosing:
Openended

Where could I explore in this place the
teacher has shown us?

Openmiddle

How can I get to the place the teacher
has shown us?

Openbeginning

Where would I like to go?

That last one is a hard one to ask students to do, and for the teacher to manage. It's one I'd like to find ways into.
All of these seem ways of encouraging students to be active, not passive, in their learning. Alongside this there are all sorts of number talk, estimation and quick image activities that ask students how they see things, how would they explain things. They are asked to give not the answer, but their answer.
I like this table in Making the PYP Happen (though I'm not sure whether this is how mathematical practices are really changing  I'd like it to be):
"Instruction built on what students know, what they want to know, and how they might best find out".
It would connect the motivation to play, explore and create that children are allowed to exercise in Foundation Stage (K) to the wonderful explorations and results of mathematics that have come to us across the centuries.

I'll be reflecting on this idea of being active next term, how best to encourage students to follow up their spoken or unspoken questions.
This post has been hopelessly general, but I'd be interested in your ideas about how to allow and encourage an active and inquiring attitude.
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