Hi all,
I think trying to tease out problems from solutions goes against what design is all about - they co-evolve as a conversation.
Here are 2 references I use that speak to characteristics of design problems. The first you probably know (Goel and Pirolli), the second is by David Jonassen who speaks from instructional design. His taxonomy of problem types uses problem solving as the big umbrella of a whole class of different problems...of which one of the most "complex" forms is design. I like both of these because they link the nature of the activity to the kinds of thinking that are engaged (e.g., creativity, routine skill, collaboration...).
Goel, V., & Pirolli, P. (1992). The structure of design problem spaces. Cognitive Science, 16, 395-429.
Jonassen, D. H. (2000). Toward a design theory of problem solving. Educational Technology: Research & Development, 48(4), 63-85.
More recently, Johannes Strobel has been exploring the idea of "compound problems" - that very complex problems are made up of a collection of different problem types.
-Robin
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