What is happening here?
In grade 2, multiplying means first of all: putting equal groups together. 4 · 6 is four rows of six dots each. Memorizing only pays off once that picture is there.
That is why the book goes in this order: the dot array first, then the tables, then the key facts. A child who can only recite a table does not yet have the times tables.
What helps at home
Key facts first: the 1s, 2s, 5s and 10s and the squares (3 · 3, 4 · 4 …) are the anchors. Everything else can be derived from them: 6 · 7 is 5 · 7 and one more 7. Swapping halves the workload: a child who knows 3 · 8 also knows 8 · 3.
Quiz in short, mixed-up bursts. Five facts from across the tables are worth more than one table recited in order, because when you calculate, a fact never comes in its table's order.
What can go wrong
If your child recites the whole 8s table to get to 7 · 8, the table has been learned but the fact has not. Then a detour via a key fact (5 · 8 = 40, plus two more 8s makes 56) helps more than further reciting.
The other way round, knowing facts by heart without the picture is just as fragile. So now and then ask "Show me 4 · 6 with building blocks". A child who can lay it out really understands it.