A recent UK study “Testing times: which times tables do kids find the hardest?” showed that “The hardest multiplication was six times eight, which students got wrong 63% of the time. This was closely followed by 8×6, then 11×12, 12×8 and 8×12”. It is interesting to note that the 8x and the 12x feature prominently in this list.  The grid below represents the relative difficulty of multiplication facts in a visual form. The blue are the correctly answered questions and the red the most incorrectly answered:

difficult times tables
This graphic helps show why it is important that students have a strategies to help memorize answers rather than just learning by “parroting” rote methods. Note that the easily remembered facts are the 2x , 10x and 5x, the most difficult 12x and 8x. Using a strategy approach to memorizing number facts helps students to see the relationship between these easy facts and the difficult ones.

For example: 12x is just 10x + 2x the number. It is so easy to work out when 12x is thought of like that, but many students never make the connection. No-one has ever drawn their attention to this.

Another example: 8x is “double double double” – a little more tricky but not really that hard if practised regularly. The 6x facts are the 5x facts add one more set, so 6×7 is 5×7 + one more 7. Strategies give students a way of working out the answer when they have forgotten it. Rote leaves them with nothing but a guess.