Not so many years ago, I was one of the people in the education system, spending much of the working day being lectured to. Being a geek, most of my subjects involved mathematical, scientific or computing related material, which is often complex and requires (from me at least) intense objective mental effort to understand them, and just like others around me, I would desperately try to wrap my brain around the convoluted concepts my teachers were trying to crowbar into my cranium. For educators, introducing people to these sorts of concepts is a hard problem. After all, how do you introduce someone to ideas like quantum mechanics, or biological evolution, or the theory of complexity. Start with a definition? Maybe lead with some examples? I know of no "right" answer.
These days, I am on the other side; I am an educator. It is part of my job to teach people, and my subject matter involves heavy technical detail. Because of this, I try desperately to cling onto my memories of what it was like to be in the audience of the lecture hall, rather than at the podium. I want to remember what I thought as the lecturer droned on at me, what I wanted him/her to actually say in order to make sense. I still have these memories, but some day they will slip away, and I'll be doomed to delivering my obtuse explanations, all the while berating the students, asking myself "What's wrong with them, why don't they get what I'm saying?"
But, until that day comes, I am determined to remain empathetic and keep in mind how I favoured being introduced to new complex new ideas. My favoured way was by examples, and especially by visualisations although these were exceedingly rare. I do not think it is an accident that one of the degree courses I can remember most clearly, Program Slicing with Professor Keith Gallagher, began by the professor refusing to give any definitions until he had thrown wads of illustrative examples at us. As another example, over the pond, Professor Walter Lewin won great acclaim for his very physical demonstrations of physical principles at MIT.
Now, why should I bias my approach towards the way I favoured learning? A fair point, but I think I can justify it by arguing that my way generalises to the great majority of humans. (And remember that I'm only talking about introducing someone to a topic -- once someone is knowledgeable, they should be expected to deal in the equations, models or code.) Humans cope best with visual data -- a huge proportion of the brain is devoted to processing images -- and humans are also intuitive creatures, whose grasp of something (be it correct or not) is improved by subjectively getting a "feel" for it.
Let's take an example from my own domain, computer science, a particularly challenging subject in which to visualise concepts, so maddeningly abstract are they. If you enter computing, you have to know about algorithms, which are simply lists of instructions used for solving a problem. You have to know things like: what they do, how they work, how efficient they are. One particular algorithm, called merge sort, I could introduce with a definition (from U.S. National Institute of Standards and Technology):
A sort algorithm that splits the items to be sorted into two groups, recursively sorts each group, and merges them into a final, sorted sequence.
Or I could show you the code (taken from Wikipedia):
function merge_sort(m) if length(m) ≤ 1 return m var list left, right, result var integer middle = length(m) / 2 for each x in m up to middle add x to left for each x in m after middle add x to right left = merge_sort(left) right = merge_sort(right) result = merge(left, right) return result function merge(left,right) var list result while length(left) > 0 or length(right) > 0 if length(left) > 0 and length(right) > 0 if first(left) ≤ first(right) append first(left) to result left = rest(left) else append first(right) to result right = rest(right) else if length(left) > 0 append first(left) to result left = rest(left) else if length(right) > 0 append first(right) to result right = rest(right) end while return result
And that's everything you need to know to learn merge sort. Does that help? Do you get a feel for it? Can you run through an example in your head? Eventually, you could gain an understanding of the way it works, but if you're being introduced to sorting algorithms, what use is it really? What's more, you would you need to be able to compare merge sort to other sorting algorithms. Would you have an intuitive understanding of that?
I use the example of sorting algorithms, not only because it is relevant to computer scientists and time-consuming to gain a feel for, but also because I came across this attempt to give sounds to sorting algorithms. This, I think, is a great way to provide intuitive understanding to something as abstract and difficult as an algorithm in process.
So much of our time learning is spent providing ourselves with an intuitive grasp of things, the ideal being that we can "see" them in action within our minds. Those with extraordinary talents, from mathematicians to musicians, often tell us mere mortals that they can actually see the numbers or the notes in front of them. The memory masters, reciting π to a million decimal places, only enable their great talent by using images to power their feats of recollection.
We can't all be extraordinary, but we can help others to achieve a correct, intuitive and "visual" understanding of things to cement them more firmly in our minds.