The reason k, q or other array languages like J (which I use) help, not teach, people to solve novel problems is that they give you abstractions to quickly play with in order to tackle novel problems. My take on this whole thing is that these array languages (APL, k, J, q) are a perfect fit for things that are essentially arrays - big data, images, etc... And to those who shy away from the seemingly 'noisy' syntax, I point them to mathematics. Until you understand the sigma symbols and others, it is noise too. Why are all the new languages getting hot on 'vectorization' - Julia and company - because GPUs and data are all array-based. To have the array as your lingua franca puts you in a good position for all sorts of attacks on modern computing needs. And for those who cannot live with the syntax, and say how can you create a dsl - you can. Here is a J example of computing averages:

  +/%#

You can set this up as:

  avg=: +/%#

or

  tally =: #
  divide =: %
  sumall =: +/

How's that for a dsl? Which allows this now:

  avg=: sumall divide tally
  avg 2 4 6
  4

EDIT: I forgot to add that these operations, as others in J or APL, work on scalars, vectors and arrays without any special typing or handling. See this presentation for at least the first 5 minutes to see J in action with explanation: http://www.infoq.com/presentations/j-language The conciseness allows you to start seeing the small patterns in the small expressions just like mathematics, hence bringing clarity and speed of abstraction to your concept manipulations.