In elementary school, we had to memorize the multiplication tables (1-10 times 1-10).
We were then quizzed on this.
Being able to answer "5x5 = 25" meant we had memorized the tables
The teacher then asked what "5 x 12" was, which checked to make sure we actually understood multiplication.
Hopefully that clarifies the difference :)
I understand the difference. What I don't understand is the odd conjunction.
so concerned with getting the right arithmetic answers that they weren't thinking about arithmetic.
I parse this as saying: because they were so (strongly) concerned with getting the right answers, they weren't thinking. If they had been less concerned with getting the right answers, they would have been thinking more.
What is the idea here? That if they didn't spend so much time, effort, or worry on memorizing, then they would have had more time to think?
As a teacher, I wonder if it is possible to instill this skill into students the skills of rationality and critical thinking. I teach the third grade, and it is not immediately apparent how to apply this with my own class.
The problems I foresee are as follows:
In the sequences, it is suggested teachers should drill into students words don't count, only anticipation-controllers. How practical is this for an elementary school level? Also appreciated would be any ideas or experiences on how to do this, or how to combat the above problems. Hearing from other teachers would be excellent especially.