Andrea, There is a bit of over-kill sometimes in this hilling stuff. If you are able to plant the potatoes 4 inches or more deep, there isn't need for so much hilling. Just pull some dirt around the stems a couple times leaving those leaves alone. A small mound should be sufficient if the potatoes were planted deeply enough. Getting the dirt too high can impede rainfall soaking in deeply enough in hot dry weather.

I dug my trench about 8" and just filled it in about 1-2" at a time as the potatoes grew, covering the whole "sprout" (it didn't have any fully formed leaves, it was pretty much just a nub). Once it reached the top of the trench I stopped burying all together. We will see how it works out.

ZachS, That is good

Not everyone can go that deep in early spring...where rains are more frequent.

Or do they mean not to cover all of the leaves at the top of the plant?

Yes that is what they mean. Although even if you do - which many of us do - they grow right back up through it.

I disagree that hilling isn't important or isn't needed but there also isn't any need to make it over-complicated or get too detailed. The primary purpose is to keep the developing potatoes themselves (not the plant) protected from sun exposure.

Dave

I do like Zack.
Normally you plant more than a single potato. So you do it in rows. You dig a shallow tench/furrow lay down the seeds, cover (~3"). When they come up about 8". start filling the furrow back, little by little with some fluffed up soil. It is ok to bury part of the lower leaves. You can even pinch them off. It is just like pruning tomatoes lower leaves. Don't worry, they will keep growing more leave from the center.

As you you this filling thing (twice or thrice), eventually the furrow becomes a continuous hill. That is what hilling is about.
Your seed potato grows roots sideways and downward where you put it down. The buried stems grow stalons on which then grows the tubers . We hill to have a longer stem in the ground and also protect the tubers from light AND to keep them cooler.

I don't try for that last 3 inches of stem covering as I prefer to get fewer but larger potatoes rather than a slew of small ones.

"Not everyone can go that deep in early spring...where rains are more frequent."

That is true. In some (rare) cases, our lack of water can be useful.