The above map is an attempt at a curriculum development model. Traditional pedagogy stays at the top, as it is based on the belief that skills practice and teacher explanations are sufficient to build student understanding. Understanding acquired this way, plus the skills, allow the student to apply the ideas.
Would that it were that simple.
In my experience, concepts are best developed by spending a lot of time at the bottom of the map. The interaction of themes, representations, and tools provides a foundation on which concepts can develop. (In the map, I mention examples of themes, representations, and tools. Those lists are not meant to be exhaustive.)
The idea is not to rule out explanations and practice, but to give them a chance of being effective.
This is pretty complicated, you say? Where does one start? In thinking about curriculum one often starts with a concept one wants to teach. From there, one can look at the bottom of the map for ways to build up the necessary experiences for students.
But it's also possible to start with a theme. I developed a whole lot of curricular material, and wrote an article about it based on an exploration of how the theme of area interacts with various tools and representations, and leads to various concepts. Check out this out-of-control figure.
Or one could start with a tool, and ponder what uses it may be put to. We did just that at a recent meeting of Escape from the Textbook! I'll report on that in a future post. Meanwhile, read my article on a tool-rich pedagogy.