The phrase is nine hundred years older than the notation it now describes, and it once meant exactly what it says.
Around the year 820, in Baghdad, a scholar wrote a short book on solving problems about unknown quantities. His name points to Khwarazm, the region south of the Aral Sea, and it has come down to us twice: the Latin form of his name gave us the word algorithm, and a word from the title of the book gave us algebra.
He had no symbols. There was no letter standing for the unknown, no equals sign, not even a plus. Every problem was set out in words, and every solution was justified by a drawing.
Here is one of his, in his own shape. A square field of unknown side, together with two strips five paces wide laid along two of its edges, covers thirty-nine square paces in all. Draw it: the square, a strip down one side, a strip along the bottom. One corner of the picture is empty — a small square, five paces by five paces, missing. Complete it. Adding that corner adds twenty-five square paces, so the finished figure covers sixty-four. A square of sixty-four square paces has a side of eight paces. Take away the five-pace strip, and the field's side is three.
Nothing about that argument needs a symbol, and nothing about it has changed. When a modern pupil adds a number to both sides of an equation to make a perfect square appear, they are drawing his corner — in notation he never had, for a reason he would recognise at once.