I think there is still some algorithm-based teaching, or at least the students I taught had been taught via algorithms. The simple O(n^2) multiplication one, or long division, or adding/subtracting with carrying/borrowing. I know a lot of the perceived complaints about the 'common core' change were a push away from this, though I never had any of those students (too young, and I left teaching two years ago, at least for now). That said, I think the common core way was a better way to go about it to help kids get a number sense.
I see so many kids who don't know that, for, say, 28 + 35 you can simply rearrange this to be 30+33. They have no concept of what addition means, or how you can regroup parts. Same with breaking down multiplication and how you can do it by parts and then sum it. They have no concept of how the numbers work, basically, and I don't think algorithms would give them that sense, it'd just give them steps to compute stuff.
I see so many kids who don't know that, for, say, 28 + 35 you can simply rearrange this to be 30+33. They have no concept of what addition means, or how you can regroup parts. Same with breaking down multiplication and how you can do it by parts and then sum it. They have no concept of how the numbers work, basically, and I don't think algorithms would give them that sense, it'd just give them steps to compute stuff.