Let's think about 54x15. Front-door maths is about recognising the obvious in a problem and asking some straightforward lower-order questions. For example: "What is the place and face value of each digit?", "True or false, 15 is less than a score?", "Is the product divisible by 3?"
Back-door maths is about knocking on the minds of learners, using middle-order and higher-order probes to squeeze more maths out of a situation. For example: "Describe how you would check your answer using two other methods," "Make up a number story and tell it to a friend," "What could you do to make the sum easier?"
Skylight maths is what comes from children - they pose their own questions.
They might ask: "I wonder if the product is a square number?", "What is the digital root of the product?"
Entry and exit maths is about exploiting the potential in a moment so that there are no missed opportunities, and deepening the maths experience for children. Milk the maths whenever you can
John Dabell is a numeracy consultant and teacher trainer