I definitely think analogies are a great starting point. Another useful strategy that is as ancient as analogies and parables is refutation of misconceptions. Learning the misconceptions and having them refuted will further understanding beyond painting with your hands. It’s a lot like taking useful critique on your technique. It’s not like DeviantART critique, which often sounds like “This isn’t in the style I usually like, and here’s what that entails”. It’s useful critique, like “instead of building your skeleton this way, which is hard to get right on the first try, build it this way which is more accurate every time, and easier to adjust in the later stages of development”.

You probably won’t hear a lot of mathematical misconceptions around the corner, but they do exist if you look for them. They happen a lot more in the less rigorous sciences, ones that are several steps removed from math, but can still be observed in controlled situations. Still, something I would keep in mind!