Many teachers regard the responsibility of learning the times table as something the student simply has to do, normally in their own time e.g. as a homework assignment. Then the teacher will use teaching time to test their students; to make sure they've completed their assignment. This is not teaching, but testing; and it is often embarrassing or even humiliating for students who have difficulty memorizing those lists of times table answers.
A far better approach, more in line with the expected role of a teacher, would be to actually teach students mental multiplication skill, which they could of course practice as a homework assignment. In the process of helping students develop such skills, the teacher would test as part of coaching that skill. This approach is far less threatening to students.
In recent years, many people have challenged how appropriate it is to attempt to learn the times table up to 12x12. Some prefer to limit the learning objective to 10x10 often citing decimalization as the reason for this, without speaking out the truth which underlies their motivation, being a preference to dumb-down the learning objective as it is then easier to accomplish.
On the other hand, those who value the ability to multiply (typically those who are comfortable themselves and often fail to see the problem through other people's eyes) suggest raising the bar to empower students with skills based on enlarging the times table to cover up to 20x20 or even 100x100. This kind of goal cries out for a skill-based approach, rather than memorization of number facts.
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