What is the first step of the man's part in the box step? My PE teacher told me it's moving the right foot back. But I think it's moving the left foot forward. Can anyone answer this please? And is the first step of the man's part in the underarm turn moving the left foot forward? thank you.

The pattern has forward and backward halves, and can be done to either the right or left.

There is a bit of a tradition in recent decades of starting a dance with the version where the man moves forward onto his left foot. Or, if starting with a figure that begins with a step forward onto the right foot, of preceding it with a step onto the left occuring at the end of the previous measure of music.

In class you might start directly into some later portion of the dance being presented, rather than always from the first step of the first figure.

In the grand scheme of things most attention will go into quality of movement - the steps are of only temporary concern (though obviously a lot of it when everything is new!)

In American style Waltz, normally the man's first step of the basic (left) box is moving his LEFT (sorry, I wrote "right" previously and missed it when I proofread what I wrote) foot forward. The man continues dancing the box during a Natural Underarm Turn; he raises his left arm to lead the turn on the "back half" of the box, which begins when he steps back on his right foot. (This isn't just arbitrary; there is a good reason for it.)

As another poster has pointed out, the steps are only a small part of dancing--though that doesn't mean that the foot positions are unimportant. You need to learn them, but you will be better off not concentrating too hard on them. Practice enough so you don't have to think hard about which foot moves next and where it goes. You don't do it all with your feet. You really want to think of moving your "center." When you walk you don't stick out your foot and fall on it (unless your name is Charlie Chaplin). When you dance, you don't do that, either.

A former (because he retired) instructor put it very well: "The body places the feet. Your feet just keep the rhythm." That will mean more to you as you progress. I hope you continue to enjoy dancing.