I would like to relate a little story about ski terrain and my hometown mountain, the Arizona Snowbowl. I was skiing there with my son, who had recently progressed to skiing off a leash, making turns and stopping all on his own, and he wanted to ski a run called Ridge to celebrate his new-found freedom. Most of the Snowbowl's runs are intermediate, and easy intermediate at that, so I saw no reason to say no, and off we went.
Now, what I noticed right away skiing with a four-year-old was that 'easy intermediate' was a LOT steeper than I thought. According to my memory, the entire run was gentle, the kind of thing you schuss down in one breezy go. But crawling down, turn by turn, with my son, I realized the pitch was significant, and the run was narrow. Worse, we soon arrived at the top of a short section called Waterfall, which I had forgotten about entirely.
What was to me an insignificant dropoff, a place where I typically made two or three blazing GS turns, was in fact a steep headwall probably nearing 300 vertical feet high. Waterfall, I saw at once, was utterly beyond my son's ability. How could I have forgotten it? Since then this drama has repeated itself, in one way or another, over and over, and I've come to realize something important: I see mountains with an expert skier's eyes.
Oddly enough, this creates a sort of blind spot. Ask me how difficult a ski run is, and I'll naturally frame it relative to my own experience, which may be very different from yours, and which is certainly very different from a child's. Seeing terrain through a child's eyes turns out to be challenging.
How can you learn to see terrain from a beginner's perspective? Try this experiment: visualize yourself in a full-body cast, lying on a stretcher. Now imagine some madman has attached skis to the bottom of the stretcher, and is preparing to release you atop a ski run. If that scenario induces any anxiety, you've probably found a pitch that's not appropriate—or at least less than ideal—for a true beginner.
Put another way, the ideal starter run for a kid's first day on skis is one where you can let them go and know with absolute certainty they're not going to accelerate wildly until they blow up. Find a run where their speed tops out right around a brisk walking pace, and where they eventually slide to a stop with no help from you, and you've found your perfect training ground.
Runs like these, I'm sorry to say, aren't that common—even at kid-dedicated zones like places with magic carpets. You'll need to search around, remembering always that what looks completely flat to you probably isn't flat enough. When your child does start developing the skills to stop and turn without help, you'll naturally want to up the difficulty level. Manage that impulse carefully!
It is an axiom among professional instructors that parents habitually put their kids on inappropriately difficult terrain, and that this hinders rather than helps development.
To make matters more challenging, what is appropriate one day may well be inappropriate the next. Changing conditions dramatically alter a run's difficulty, including not just snow quality but also crowds and weather. And just as you and I have bad days, so too does your child. Some days, things just look scarier. Some days, solid and stable turns just don't come as easily.
I suppose the simplest solution is to always let your child pick the run. Let them decide what they want to ski. When they want to challenge themselves, use your judgment to gauge whether the level of challenge is appropriate at that given moment. And when they want to stay on the flats—let them. I can tell you honestly, if you're like me you'll find it difficult to back off. We want to push them—we really do. But I say, as best you can, try to err in the opposite direction. Let them push themselves.