Two points come to mind that you might find interesting.
First, Myers-Briggs uses four dimensions to describe personality, based loosely on Jung's system. One dimension in particular, namely Sensing versus iNtuiting, describes (among other things) the inclination to prefer the general rule first (as iNtuitives do) or specific examples first (as Sensates do). Intuitives tends to baffle Sensates by describing high-level abstractions and often forgetting to ground them in examples at all, which can make Sensates look like simpletons to the iNtuitives in question.
If that sounds like it might fit, you could try making a point of starting with examples when talking to this particular lady. I usually find that Sensates follow a lot more easily if I start with examples and then label the appropriate parts of the example as needed.
(If that's not quite what you're talking about, then my apologies for taking this on an irrelevant tangent!)
Second, this point:
Someone fully understanding a concept ought to be able to use that understanding as a guide to understand analogous unfamiliar topics.
...refers to something that in education research (and I think in psychology?) is known as transfer. I'm not deeply familiar with the transfer literature, but I do know enough to say that "fully understanding a concept" ends up functioning a little like a "no true Scotsman" argument in this context. Analogies aren't in the environment; they're part of how we as intelligent beings perceive similarities in our environment. The fact is that someone can understand a concept perfectly well but utterly fail to notice its analogous application in instances that might be downright obvious to someone else. This is a huge problem in education because we often don't notice when we're assuming that students will "obviously get" some critical part of the lesson; the only reason they'd "obviously get" it is that it's obvious to us.
To learn more about the history of transfer research and/or some modern approaches to this, I'd suggest looking up some of Dr. Joanne Lobato's work on "actor-oriented transfer". (That's just a bit outside my specialty, so I'm not sure which specific papers to recommend. Sorry!)
"fully understanding a concept" ends up functioning a little like a "no true Scotsman" argument in this context.
I'm partial to the regeneration theory of knowledge: you really know something when it's interwoven with the rest of your knowledge such that upon forgetting it, you would readily rediscover it again.
I think you ought to use a straightforward application of the concept as a test of understanding (e.g., if teaching how to use a hammer, have them put in some nails). Only after the concept has been verified as solidly understood should you try to make use of it as a stepping-stone to guide their understanding of a different unfamiliar topic.
Someone fully understanding a concept ought to be able to use that understanding as a guide to understand analogous unfamiliar topics.
I agree about this normatively – that we ideally should be able to apply the same principle across multiple contexts – but you shouldn't expect it. Whether or not you fully buy the "massive modularity" hypothesis, cognition is very context-dependent. As Mercurial notes, cross-contextual learning is the big thing in education and goes by the name of transfer. They've been working on it for a long time and I think is one of the central issues in practical rationality. If you are primed to think a certain bit of knowledge is cross-applicable, it's not that hard. The real difficulty is spontaneously realizing what is going on, along the lines of the 5-second skills Eliezer recently touched on.
I told someone that I learn best by first hearing a general principle and only afterward being given examples and analogies. She replied that my explanations are hard to follow when my analogies are not from subjects already familiar to and well understood by her. She went further and said that sometimes she understood novel things I was trying to explain, only to be confounded by my subsequent analogies. I immediately replied that in my opinion, analogies to familiar topics are of course much better teaching tools than those to unfamiliar ones, but obscure analogies primarily function as tests to ensure understanding rather than tools to convey it. Someone fully understanding a concept ought to be able to use that understanding as a guide to understand analogous unfamiliar topics.
I am very interested in what others have to say about my last point in particular and would appreciate comments.