How you go about teaching your child in the pool can make or break them. If you force it too much, then you'll leave a mental scar. Then it will make it a lot harder to teach them down the line, either yourself, or even for an instructor.
You can't force it.
Think about the water itself, you can't force it to do what you want to do, instead you have to go with the current. Let it flow.
Why + Philosophy:
Swimming is as simple as 1, 2, 3, hence my 3 steps to swimming. However LEARNING those steps is a whole other ball game.
When you get into the water your whole mind changes, and thus your actions will as well. You have to realize that you yourself are in a different environment. Let alone the kid you're trying to teach.
My take is that I teach first to get comfortable in the water. Usually by sinking a LOT, and while that may seem counter intuitive. It often leads to 10x more comfort, a lot stronger lungs, and more. Now more lung capacity equals more time under water, and more time under water equals more time to literally practice the act of swimming.
How + Physics:
Once you've garnered enough comfort training, you can even skip to part of this towards the beginning. Then you need to start working on the 3 steps.
I made them simple so that anyone can understand them, let alone anyone can teach/learn them.
Kicking comes first. Have them hold onto your arm, or hands, and then focus on kicking straight.
After doing that for a while. Combine the under water training and the kicking, by having them hold their breath while kicking. Perhaps at the end blowing some bubbles out of their nose too.
Finally, you can eventually get them to move their arms, and this is something I'd wait as long as you can for. So they have time to master the other parts first. Then it all comes together!
What + Psychology:
Your kid, regardless of age, is going to see you as an authority figure. Perhaps not as a teacher, so the mindset is going to be hard to overcome. I even had a hard time teaching my sister for this exact reason.
I eventually was able to, but not until she was a bit older.
Now regardless you have to think about it in the sense that you are having an exciting time. I teach swimming, even on the hard days, by making it fun but also a challenge.
Teaching isn't just information sharing, which is where a lot of parents mess up, as you have to let them try it themselves too. You gotta let them fail.
Conclusion
Notice how I take a very relaxed approach to how they learn. It is sequential, it is calm, and it is focused.
Yet if you were to see me actually teach, you might hear KICK KICK KICK, FACE DOWN, or me yelling about something. Mostly because they are under water, and can barely hear me.
However I'm a tough instructor, but I know how to tone it down when I need to.
Oftentimes parents default to either too sweet or too tough when you first start, which then sets a precedent in the kid. I've seen this first hand as a lifeguard.