An ice cube was put on a steel plate and an identical one on a wooden board

Why

Kamla's observation is completely correct — steel does melt the ice faster, and steel truly does feel colder to the hand, so her reason hangs together. Here is the catch she could not have guessed: both objects spent the night in the same room, so both are at exactly the same temperature; a thermometer on each would agree. What differs is how fast each moves heat. Steel carries heat quickly, so it pours warmth into the ice from the whole plate and from the room beyond it. Wood passes heat slowly, so the patch under the ice chills and then just sits there. The same property fools your hand: steel pulls heat out of your fingers fast, and 'cold' is what fast heat loss feels like. Her answer about which melts faster is right; her reason is not — steel is not colder, it is a better conductor.