I used to struggle with explaining the difference between these for a long time until I realized a very simple truth. Countable things are easy to count and clear to see; uncountable are difficult to count/separate into individual elements.
Next time you have class on this try bringing a glass of water and demonstrate how difficult it is to count each drop. Then show how easy it would be to count one glass of water and you have the essence of the shift between countable and uncountable nouns.
This exercise then asks students to decide if nouns are countable uncountable or both.
I recommend having them work with a partner and always give them time to come up with their own answers.