Is this really true? Yes. Look, I can prove it. Here is a graph of the trajectories of the Earth and the Moon as they orbit the Sun over the course of half a month. The blue curve is the Earth's path from left to right, and the red represents the Moon.
Graph: Rat Allen
Here, position is measured in astronomical units (AU), which were originally defined as the Earth's average distance from the Sun. The trajectories look flat because I had to use different scales on the two axes, for reasons we've already talked about. Trust me, it's a circular orbit.
Let us now look at the path of the Moon. First, notice that during this half month the Moon starts on the far side of the Earth and ends closest to the Sun (that's half an orbit around the Earth). Second, at no point does the Moon's path turn away from the Sun. In fact, it's always accelerating towards Sun. And that's because the Sun's gravitational force is stronger than Earth's.
So, again: Does the Moon revolve around the Sun or the Earth? The answer is yes—it orbits both. While the Moon revolves around the Sun, it also revolves around the Earth. And it revolves around the Earth without turning from the Sun! It's hard to picture it in your mind, and the bad news is that we can't draw an exact diagram as an aid to understanding.
The whole problem is that, on the scale of the Solar System, the Earth and the Moon are so close together that you can't see their relative motion. But I'm sure textbook writers will keep trying.
Finally, for those who still think the Earth and humanity are the center of the universe, consider this: If our planet suddenly disappeared from existence, the Moon would continue to orbit the Sun once a year. He won't even remember us.