You’ll have to decide for yourself what is best in this case.
The issue is one of form drag or body position and which you think is more important. The coaches background will have a lot to do with what he or she believes.
Your coach is thinking form drag is very important and as such you need to have a streamlined shape hitting the water. On the other hand I’ll go ahead and make the case that body position is more important and that the shape your head presents to the water is less important.
Her case, basically you need to present the streamlined shape to the water. Some folks believe (although she may not) that the water at the eyebrow position makes you ride higher in the water because fo the sloped surface your face will make.
My case, your body is somewhat like an iceberg it will naturally have about 10% above the water and 90% below (not the actual numbers). You can fight it into other positions but it is a costly proposition energywise.
Where your head is positioned will determine to a large degree where the 10% comes from. If you look forward most of the back of your head will be out of the water and that’s pretty much all of your 10% right there. On the other hand if you are looking down then a small sliver of your head is out of the water and the 10% is evenly distributed across the body so a little of your head a little of your shoulders and little of your hips will be out o the water. That’s the key to the look down position. The hips ride high in the water.
The high hips ends the problem of your legs feeling like they are sinking as you swim.
The high hips is what you’ll sacrifice if you look forward. Look forward and you’ll get more of the legs sinking feeling.
Now here is where the background of the teacher comes is. First, how they were taught is probably going to be very similar to how you are taught. We all try to be objective but the methods used to teach us will still have a strong pull. Also the sport background matters if your teacher was a swimmer then the answer is easy. Swimmers trained ever since they were kids mostly for distances lasting 200 yards or less. After they had been swimming for a couple of years and developed the aerobic capacity then they moved up to 1000 and 1500 meters. Why is this important? Because over the course of 400 yards with appropriate training you can kick your legs up. If well trained you can even do it over the course of 1500 meters.
Since you’re going to lessons it seems you don’t have the background that the age group swimmers do and might struggle with keeping your legs up by kicking. But only you will be able to tell. I suspect your coach probably has a big swimming background even if she currently might be a triathlete or somethign else.
On the other hand i came from the open water world where it’s much harder to kick the legs up over the course of 3 to 7 miles. In that light somewhat predictably my bias lies toward keeping the head looking down.
Please note this doesn’t solve your problem; there are plenty of folks who take the looking down part too far and in fact are plowing their shoulders through the water. If that’s what you were doing then looking ahead may be a great improvement for you.
Chris i don’t know what will be right for you, but hopefully this will help you figure out what you will gain and what you will lose by trying either technique.