The effort-praise finding is real, smaller, and often misquoted
Few findings have travelled further from their evidence than this one. The popular version says telling a child they are clever damages them and telling them they worked hard fixes it. The literature is more modest.
What was found
Children praised for ability after success tend to choose easier tasks afterwards and to give up sooner when a task gets hard; children praised for effort or strategy tend to persist somewhat longer. The effect is consistent in direction and small to moderate in size, and it is measured in the minutes after the praise, not across a childhood.
What was not found
That a single sentence permanently alters a child's beliefs about their own capacity. That praising ability is harmful in the ordinary sense of harm. That the effect holds equally across ages, cultures and task types — replication has been uneven, and several large attempts found smaller effects than the early work.
A reliable small effect is worth acting on. It is not worth anxiety about a sentence you said at breakfast.
What is usable
Praise that names something specific — the strategy, the fix, the part that was hard — gives the child information they can reuse. "You checked it twice" tells them what to do again. "You're so clever" does not, which is the mechanism, and it is a good enough reason to prefer the first without treating the second as damage.