Recently my daughter and I undertook to replace the bathroom creamic tilein her older home. The house was built in the 1920s. After removing the old tile (two layers), we found concrete poured over the floor joists about an inch thick. This layer of concrete has a few cracks but worst of all there doesn't appear to be any way to screw the next layer of cement board in constructing the new ceramic tile floor. My question is: Should we remove the concrete base and install plywood over the joists to give a solid base to screw the new cement board to?

TGSquare

That's what we had to do with a house that we renovated.

It is hard, heavy, sweaty work. I hope you can get some help. You will be amazed at how much will come out.

Emland, Thanks for your response. Fortunately, the bathroom is small, about 5'x5'. :)

Tgsquare,
I would place two layers of plywood if you are going to be going directly on top of the floor joist. The reason being is that you need a sub floor and a finish floor surface to prevent sagging in the future.

I would also place 30# felt paper between the two layers of flooring, and use screws.

my guess is that although it may seem there is no sub floor, there is, its just under the concrete,

what you should have is the joist, then they may have used oak flooring ran on a angle then a black felt plus wire lath, then the concrete, tile and so on.
you need to pull the concrete up, its not as bad as you may think, first you'll need a pry bar, and get a bigger one, they have some that's maybe a 1ft long or so , these will work but you need more leverage.
first take a slege hammer to make a hole to pry from.
and what you want to do is get up underneath the wire lath, from there just pry down, it will come up farly easily, don't be afraid of getting dirty, just get in there and knock it out.
5x5 should take 30min to and hour to do.(granted that's me and its what I do) do make sure you wear gloves will doing this the wire is really sharp.
after you get all the wire lath and concrete up, you'll have to measure how much you need to build up to start your finished floor, you maybe able to go back down with 3/4 first and then with a 5/8 or 11/16 then you finished floor,