In his book "Surely you're joking Mr. Feynman", Richard Feynman relates the following story. As he was supervising a group of calculators for Manhattan project, he at some point gave them a lecture on what they were actually calculating. He claimed that this improved their performance - even though their job was, essentially, that of arithmetic gates in a computational scheme, so the lecture wasn't directly useful in their work.
Is there evidence for a similar effect in teaching mathematics?
For example, suppose I want to teach Gauss elimination. Would students learn better if I chose examples from some actual real-world problems? Or, when teaching determinants, many students are happy when presented with the "volume" interpretation, but does it actually enhance learning of the usual algebraic/computational aspects? When teaching integration or ODEs, is it beneficial to choose examples relevant to physics/biology/...?
Generally, is providing a context/big picture helpful or a waste of time when teaching "hard skills"? Disregarding the "added value" of the students knowing the big picture, being able to apply the skills, etc., does the learning of skills per se improve?
I'm mostly interested in rigorous research answering this, but anecdotal evidence is also welcome.