A method and an apparatus are disclosed for separating or slitting a rigid material, in particular a piece of wood, having a modulus of elasticity of between 50,000 and 400,000 kg/cm.sup.2. A narrow tool having chip-removing elements, in particular a circular saw blade, produces a slot or kerf of finite width in the material at a cutting speed of more than 40 m/s. After the chip-removing elements having passed, a separator element bends a side piece out of the slot plane so that after the chip-removing elements have passed, the side piece is always at a distance from the tool. When viewed along the feed direction of the material, the separator element is provided with a hump at a distance from its periphery adjoining the tool and being first to come into contact with the material when the material is fed in. An imaginary connection line extending between the hump and the chip-removing elements is at a distance above a first portion of the separator element lying between the hump and the periphery.