Yep, I have a monthly one on my Oyster, as well as some PAYG credit for when I step outside zones 1-3.

And a travelcard should definitely be cheaper than PAYG. What TFL don't tell you (and I found out to my cost recently) is that the PAYG fares they quote on the TFL website are for travelling on the Overground and Tube. If you take a Southern train to London Bridge as part of the journey, it's rather more expensive. I thus ended up paying more for four days of travel than it would have cost me to buy a weekly travelcard (which is how I found out this well-concealed bit of info).

Of course, as your daughter is much younger than me it might be different with young people's fares - but I suspect not.