The next three causes of difficulties in mastering the multiplication tables are considered in this article: the times tables from 11 to 20, poor memory of pupils, and the use of calculators.
Cause #6. Some educators suppose that pupils must learn the times tables up to 20 by 20. But several results (16 by 17, 19 by 18, etc.) are too difficult for kids to remember for a long time.
A few months ago I had written that I could not even imagine some explanation to justify this immoderate demand. Recently I have found one reason - the 12 times and 16 times tables are needed to help with inches and ounces. If so, I can not be a referee, because I have never dealt with inches and ounces at school.
If a math curriculum provides for wide use of inches and ounces, then, maybe, the demand for knowing by heart the 12 times and 16 times tables is justified. But what about the13 times tables or the 19 times tables, etc.? Even if a pupil has learnt them by heart, he/she will forget them soon because of the lack of use, and the time spent on the learning will turn out to be wasted in vain.
Cause #7. Many pupils have poor memory in view of the fact that the methods of teaching mathematics which are used at school do not encourage the development of their memorizing abilities.
Teachers and parents notice often that many children have weak memory now. They can not memorize simple math facts totally; they do not remember the rules which have been learnt a month or even a week ago; they can not recall the methods of solving tasks used recently, and so on. I suppose, the cause is connected with our methods of teaching math, and not only math, but other school subjects too. Now there is decreased emphasis on memorizing at school. We do not develop pupils' memory duly; we do not use proper exercises to drill their memorizing abilities. At the same time, most researchers agree that memorizing abilities play a significant role when kids think with numbers.
Learning multiplication affects different parts of the brain. Drilling the times tables uses one area, while learning multiplying methods uses others. According to Brian Butterworth, Professor of Cognitive Neuropsychology at University College London, anything that is memorized is consigned to the back (memory) part of the brain, freeing the front of the brain (the learning area) for more learning. So the practice of learning the times tables by rote (which is named unfashionable very often) turns out to be not as ineffective as it is considered. It looks like kids are "born with a start-up kit for numbers," Butterworth said, and have to practice, just as musicians do. “Processing is done at the rear as you become more experienced,” he explains. In other words, if pupils have memorized the times tables by heart (the back part of the brain), they become more experienced, and the front parts of the brains become more receptive for advanced learning.
I think that multi-digit operations, for instance, are important not only as a drill in calculations. During implementation of such operations pupil's memory is continuously working. The pupil recalls and uses simple math facts (the results of the times tables, for example), memorizes the intermediate results, performs simple mental computations, and, hereupon, his/her memory gets a good training.
Cause #8. Pupils are allowed to use calculators too early, and, instead of development of their independent computational abilities, their brains get a cripple wheel-chair for moving in the world of numbers.
The debates on the use of calculators at school have been lasting thirty years or even more. The main claim of the proponents is that calculators allow students to spend less time on calculations and more time on understanding and solving problems. They believe that the use of calculators helps students develop better number sense. Moreover, some of modern curriculums can tell you that basic arithmetical skills are not necessary now, because pupils can use calculators.
The opponents say that calculators produce students who can't perform basic tasks without calculator. One of their mantras is "Calculators were invented by vampires to suck your brains out."
I am not an opponent or a proponent of calculators. My opinion is closer to standpoint of those educators who consider calculators as useful devices only, but not as a mean of teaching math. Calculators can not be used to excuse pupils from learning the fundamentals. If pupils do not master mental arithmetic, they will also be poor users of calculators. To use a calculator effectively, a kid should learn to monitor and control the obtained results. This requires good developed skills in mental calculations.
I agree, there are many problems which are too difficult to be solved without the use of calculators, but we can postpone these problems until our pupils master mental computational skills totally, and only then the use of calculators may be begun. In the hands of an inexperienced kid a calculator ceases to be a useful device – it becomes a useless toy. Regrettably, we can often observe pupils who implement the same calculations with the use of calculator several times and, herewith, each time they receive different results. It seems that only one part of their brains is acting at that moment - the part which has charge of moving hands, and, if they need to add 15 and 10, their fingers stretch to the keys again, press them mechanically, and, without slightest delay, write in the copy-books a ridiculous result: 150 (or, maybe, 5 or 1.5).