The brief enumeration of main causes of difficulties in mastering the multiplication tables described in my previous article needs to be detailed and motivated. Now I would like to dwell at some length upon the first three causes.
Cause #1. Pupils begin to learn the times tables while many of them have not mastered simple mental addition and subtraction totally.
Only the fact, that multiplication is repeated addition, defaults importance of the addition skills for understanding the multiplication idea. At the first stage of becoming accustomed to the times tables every pupil should obtain himself/herself each multiplication result from 2 x 2 = 2 + 2 = 4 to 9 x 9 = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 +9 = 81. Undoubtedly, only the pupils, which have mastered the mental addition skills within the limits of 100 totally, can perform such a work successfully. Only in this case the times tables will cease to be a meaningless set of numbers, and, furthermore, the results obtained by a pupil without assistance will become more accessible to be retained in the memory.
If a kid meets significant difficulties in addition, if he/she can not perform mentally, for instance, 7 + 7 + 7 quickly and effortlessly, then the exhausting calculating procedure will hide the importance of the desired goal. Hereupon the final result will lose meaning for the kid, and, as is generally known, it is not easy to memorize any meaningless information.
After the times tables have been shaped by pupils, the work upon learning them by heart begins. The process of the memorizing will require certain quantity of time and some, maybe, considerable efforts. We know that our memory has bad habit to let us down at the very unsuitable moment. So pupils are needed to become acquainted with some tips which can help them to recall the forgotten result. The first of the tips is already mentioned. If a pupil remembers that multiplication is repeated addition, then, when, for instance, the result of 6 x 4 has been forgotten, he/she can get it by adding 6 + 6 + 6 + 6. But there are other tips also. If you remember that 6 x 3 = 18, you can perform the next procedure 6 x 4 = 6 x 3 + 6 = 18 + 6 = 24. Or if you remember that 6 x 5 = 30, you can act so 6 x 4 = 6 x 5 - 6 = 30 - 6 = 24. To use these tips effectively, pupils have to master totally both the addition skills and the subtraction skills within the limits of 100.
Cause #2. Pupils learn the times tables without simultaneous learning the corresponding to them division facts.
If you need to memorize new information quickly and firmly, the best way is to form many connections between the new facts and any other facts in your brain. As for the multiplication tables, the first connection is the close relation between multiplication and addition. The second one is symmetrical multiplication results, and then the connections between the multiplication facts and division facts must be mentioned. My experience show that the simultaneous learning interconnected results (3 x 6 = 18, 6 x 3 = 18, 18 : 3 = 6, 18 : 6 = 3, for example) helps to memorize each of them better. Some my pupils had memorized well the division facts first, and only then they achieved success in the memorizing the multiplication results. Evidently, remembering that 18 : 6 = 3 helps to recall that 3 x 6 = 18.
Cause #3. To memorize the multiplication facts, many pupils use the learning by rote only, and the knowledge which has been crammed is forgotten very soon.
If a pupil is weak in mental addition and subtraction, he/she has the only way to recall a forgotten result – looking at the multiplication charts, and the only way to memorize the times tables – learning them by rote. Sometimes even teachers encourage children to do so by asking them to recite the times tables over and over again. As a result, many of pupils do not understand what have been crammed in their heads, and if they are asked for the answer to 7 x 8 they can give it only by reciting from 1 x 8.
It is doubtful whether we can manage without learning some math facts by rote at all, but it is evident that the learning by rote can not be the single or main way of learning math. In all events children must at first understand well the information which then they will try to memorize by heart, and, secondly, they must constantly use this information. We know well, that the phone number used often is memorized easier than the phone number which we use once in a while.
