(This essay is inspired by Peter Watts "Blindsight" which deals with the theme of alien encounters in a "novel" way. Readers may wish to contrast it with an earlier essay.)
To paraphrase my friend Paritosh Pandya, one may posit that intelligence is the ability to find solutions to problems that have not been seen before. This is to be contrasted with "just running through the playbook".
In most discussions regarding artificial intelligence, the opponents of "strong AI" have argued that sentience (which is not purely mechanical) is what gives us this ability, while the proponents have argued that sentience is the result of a calculating machine "going critical".
In contrast, Peter Watts argues that sentience is often at odds with intelligence.
Clearly, (non-routine) problem solving is a consequence of the ability to "abstract out" some component of past experiences to apply them to new contexts. Given our experience with statistical machine learning (such as spam filters), we can imagine this happening without the formulation of "universal solution pragmas".
Recently, I argued that mathematics is "conscious" abstraction---the formulation of rules and recipes. One could argue that the primary purpose of such activity is to pass on those rules to others or to write computer programs that can carry out recipes.
If one can solve "ten dimensional integrals in one's head while simultaneously participating in dogfight in space (he just did it; he didn't know how he did it!)" (Lensman series by E.E. "Doc" Smith), then does one really need to learn or do mathematics?
Going even further, one might argue that conscious abstraction may hinder one's ability to react quickly and instinctively. Indeed, physicists and other scientists often complain that mathematics blocks (or at least hinders) intuitive thinking!
To put it differently: Practice makes perfect. It may not be possible to teach such perfection without putting the trainee through the (seemingly endless) hours of practice. Moreover, reading the "how-to" out of a book may actually impede such training.