There is nothing deep about the concepts of addition and subtraction. Tell a young kid you have two different piles of a number of objects. Combine them into a big pile and count how many are in it. Now they've mastered the concept of addition. Take a pile of a certain number of objects. Remove a certain number from the pile, how much do you have left? By gum, the concept of subtraction has been mastered. The CC processes are tricks to do the calculations more quickly. And since we have calculators that can do that anyways, who cares?
Things get more complicated with fractions. One part that trips people up is how dividing a number > 0 by a fraction > 0 and 1 leads to a number greater than what you started with? (Assuming positive numbers). Say you have a medicine of 8 oz and you must drink 1 oz each day, how many days does it take to finish it? 8 days from 8 divided by 1. Now take the same 8 ounces and you have to drink 1/8 of an ounce a day - how long? Now the correct answer matches your intuition and it makes sense that you'd come up with something larger. THAT is an example of concepts, not calculation tricks.
My favorite example of a mathematical concept, something to introduce to students after they know simple arithmetic, is the method that a young Gauss came up with to quickly add the integers from 1 to 100. It's easy to understand, clever, can be easy to show how to generalize up to any number, and it begins to show the difference between arithmetic and math.