Start with “how do you think the king moves?”.
Put a king on a central square of an empty board. Ask the student to point out each of the squares the king can move to. Mark the squares with pawns of one color. After eight squares are marked, take the king away. Ask what kind of geometric pattern has been drawn. It’s a square or a box, yay. How far away are the points on the square from the center? One square. Kings move one step. To which direction? Any direction. Take a straightedge (ruler, postcard, whatever). Place the straightedge between the center of that square — that is, where the king started — and each point on the outer square — the points to which the king may move. Does the king move in a straight line?
Everyone can agree that the king moves in a straight line, one square away, in any direction.
Now draw a different square. Put the king back in the center of that square, remove the pawns. Now ask to mark points on a square that is two squares away from the king.
When the student names or marks a square that is the same color square as the center point (the king’s square), mark it with a pawn that’s the same color piece as the king. When the student marks an opposite-color square, mark it with an opposite-colored pawn.
When 16 squares are marked, take the king away from the center, replace it with a knight. Then remove the same-colored pawns. What’s left behind is the typical star pattern that beginning chess books use to demonstrate a knight move.
Take the straight edge. Place the straight edge between the knight and the points on the square that were two squares away. Ask the same questions.
Do knights move in a straight line? Yes.
How far do knights move? Two squares away.
To which color square? The opposite color.
Draw a giant L in space. Ask the student(s) how many penstrokes it takes to draw an L. That takes two. People who learn that “knights move in an L shape” learn knight moves in two steps, one long penstroke, then a shorter one. After they relearn the right way to move a knight, they save half the time forevermore. Same with “forward one, diagonally forward another”. Those are two-step thoughts; knights move in one step just like every other unit.
Eight years ago, I couldn’t teach that as well as I do now. But I wrote a four-page magazine article How to Move a Knight on Page 20 of the March/April 2002 issue of the California Chess Journal. I’ll have to write that piece again, incorporating the good bit about extending the king’s one-step square to two squares, then replacing the king.