How do you visualize inquiry and learning?

I am trying to hash out a couple of ideas in my head right now, so bear with me.

The book I am reading has me in a contemplative mood.  My question is; how do you visualize inquiry?  When you think of inquiry, what image comes to mind?  What is the organization of that image?  What are the implications of that image on teaching, learning, and knowing?  Here are three possible images (among a sea of endless ones):

Image #1 – The linear graph

This image brings to mind an input/output system.  The teacher inputs the information into the student and the student outputs it back in the form of knowing.  We then increase along the line, building knowledge in a straight linear fashion.  It assumes that the previous knowledge has been acquired, and it therefore is known.  Think of how prevalent this is in school; linear curriculum, direct instruction, and top-down management (not to mention schedule blocks, standing in line, desks in rows; if you really wanted to tease apart all of the linear metaphors we use in education, it would take forever).  This type of model also assumes that there is a desired destination at the end of the journey.  Perhaps the line can continue to infinity, but it always going along a straight trajectory.  This type of view is problematic.  Think of a group of students, and each student is expected to travel on the line, and learn the same things in the same way.  Or, perhaps each student has their own line, and each student builds their own knowledge upwards, one piece of information at a time.  My biggest problem with a model like this (and I have many) is that it assumes there is only way path to follow, and one way to know.  That is, the way of the line.  That is the only possibility.

Image #2 – The spiral

This image is a variation on the line.  It curls back over itself (review and relearning and rediscovering) and gradually gets bigger and wider.  Near the bottom of the spiral the partial circle is small, but as the learner grows the circle and the space in-between the edges of the boundary grow, suggesting their is more room to build.  Again, this line could theoretically continue off to infinity, with the space in between ever widening, and the students perspective of the world ever increasing.  Many curriculums are built on this model, as each year there is a gradual increase in the level of difficulty after a brief review of what was studied last year.  Assessment takes on this form as we build formative tasks that finish off in a culminating activity that is designed to widen the circle and encompass all that came before it.

Yet, it is still a line.  It still suggests growth towards a known goal (knowledge?).  It makes the assumption that there is only one possibility, and that possibility moves in this shape.  It is still, at its core, a simple model and simple view of how we know.

Image #3 – The fractal tree

This image is taken from fractal geometry.  It starts with a simple seed, in this case a Y.  There is only one rule to build this image; at the end of every branch of the Y, build another Y.  If the rule is allowed to iterate, we get a picture that looks very much like a tree (using this seed, other seeds may cause other images).  What is the seed?  The person, the idea, the curriculum, the content?  How this relates to inquiry is obvious; as you travel along the branches of inquiry, you are faced with an extraordinary amount of choice. Do I go this way, or that way?  The branches you travel along will bring you to different points, with different perspectives.  Students, and teachers, are free to choose a path that is interesting to them, based on their own perspective of what they are learning and how they are knowing.  This image too can continue into infinity.  Also, in the other two diagrams, what happens if you were to go backwards on the line or the spiral?  Well, it would result you going back on your learning, or your circle getting smaller.  In this case, going backwards should be encouraged, because it opens up more possibilities, and presents different paths and ways of knowing.  Then, if we accept this image as a metaphor for inquiry and learning; what is the role of the teacher?  To help their students see as much of the tree as possible?  To orient students to a particular branch?  To create an environment where there are endless possibilities?

Any other images or interpretations come to your mind?