Instead of having our children to memorize the product of numbers, what is a more rational approach to make them learn to determine them instantly. We have to raise them to be rational and educated humans and I feel that memorizing product of numbers is not a very good approach to learn arithmetic and multiplication.
Teach the child that at the most basic level, multiplication is a shortcut for adding numbers many time. If the child understands addition, they can always derive the corresponding product by repeatedly adding numbers. For example:
2 * 3 = 2 + 2 + 2 = (2 + 2) + 2 = 4 + 2 = 6
Multiplication is merely a trick for quick addition, at least it can be understood so fairly intuitively when multiplying by natural numbers. In our practice it helps playing around with multiplication until it becomes familiar. Games were very handy. We had success with this game, for example, although there are probably other similar ones:
MultiBloom, by The Brainy Band: https://www.amazon.com/The-brainy-Band-Multibloom/dp/B01N91Y2V6
Multiplication is successive addition and therefore it is important for a child to learn conventional multiplication. But relying only on memory and perceiving facts as the way they are must not be the emphasis for a good parent or teacher. And if we teach them to rely on their memories and accepting facts because they actually always exist then we are simply deterring their potential to become rational individuals and learners.
Here is a Japanese way of multiplying numbers. By using this methods along with the conventional one, we can make the children realize that there can be many ways to achieve a solution as well.
The fundament of mathEmatical operations is handling and acting.
The concrete operation of multiplication are the square areas. Let the children play and help working in the garden, doing the table and the dishes cleaning the plates of the bathroom walls and floor with rows and columns. Make them reflect what they are doing: how many rows, how many plates?
Play board games with more several dices where the eyes have to be multiplied and the piece can advance the number of steps of the product of the dices.
Let them draw binomials (a+b) in square and let them invent their own tasks.