Does the value of pi or the calculation of square roots change in some way in going from one f-stop to another? If you were talking about focal distance, I can more easily see your point, as a lens' focal length is only "correct" at a single focal distance and changing that focal distance can change the projected angle of view slightly (which is why a 400mm lens may only be 389mm at infinity but 400mm at MFD).
In the days of 8mm film projectors, when the movie ended and white light only was being projected, everyone would make rabbit head shadows on the screen with their fingers. Turns out so many years later that this bit of fun with two hands at the same time held equal distance (focal length) from their eyes becomes the perfect example for teaching this basic concept. 75% of students immediately have this moment of understanding (I love teaching when that happens), and then immediately start sharing it with those who don't. Think about it and you may be able to figure out the fun.
Angle of view also changes with aspect ratio. A 50mm f1.8 lens is always a 50mm f1.8 lens regardless of the size of sensor behind the lens or the aspect ratio.
What's more, when we're studying Ansel Adams' prints or maybe even using a large-format view camera, we wouldn't have to guess what focal length was used if we were told the AOV. It sets up understanding of perspective in the results. In other words, AOV would make us as independent of actual focal length (which occurs inside where we can't see it) as we are from the physical size of the aperture by using the single f-stop number, instead of two factors (aperture physical size and focal length).
Why originators of the units in the 19th century simplified one factor and not the other has always baffled me.
Because film sizes and aspect ratios weren't standardized back in the 19th century either.
Trying to teach students the effects of crop factor and what lens to buy based on their sensor size can so puzzle them that they get scared of the technology and learn less. Of course, once you "get it" it's obvious, but not as easy and long-lasting as it could be.
So I applaud opening this subject.
If you really want to scare people, start talking about angles of view. Geometry scares the crap out of most people.
When people are not willing to work with basic concepts to make their work or hobbies or passions or life more enjoyable, that scares the crap out of most teachers. And angle of view is REALLY a basic concept that doesn't even require saying the word geometry. It usually takes just a little bit of knowledge to make life a lot less scary.
Angle of view is really only a basic concept when discussing lenses from fisheye to very short telephoto. Once you get above 200mm on full frame, your angle of view is down in the single digits and is really difficult to visualize. Actually, anything other than 45 or 90 degrees is difficult for a large number of people to visualize. Unless of course if they took geometry.
Not necessarily. Leica told us a century ago that a 46° angle of view (the first lens designed for 35mm film had a 50mm focal length) was very easy for people to discern, as that was perceived to be the angle we are paying general attention to. We also know that it is fairly easy for us to pay attention to more detail by progressively cutting that angle in half. We are using about a 23° (~100mm focal length on 35mm film) angle of view when looking at a person from a short distance and 12° (~200mm) angle when having a conversation. The concept of continuing to cut the angle paid attention to in half, 6° at ~400mm, 3° at ~800mm becomes easy to perceive visually without any need for even simple arithmetic, no geometry required. Even with all the science, photography is a VISUAL art of perceiving light and space. Angles of view become easy.
What will be interesting to see is how the Smartphone camera case evolves. People are accustomed to having their ~4mm lenses on their 1/2.5 or 1/2.3 sensors, though they probably don't know any of those specifications. If they want a different angle of view, they use an app. But more and more auxiliary lens solutions are appearing. Will the OEMs start telling them the focal lengths of these auxiliary lenses? Tell them the nebulous 4X, 8X, etc. factor? Or just let them realize they are now getting a different angle of view.
