The next two causes of difficulties in mastering the multiplication tables are considered in this article: criteria of mastering and the lack of drill.
Cause #4. Criteria of mastering the multiplication tables are ill-defined. Teachers and parents have no exact method to determine whether a kid has mastered the skills totally or not.
Most teachers and educators acknowledge importance of mastering the times tables for further math learning. The simple multiplication facts are considered as critical because they form the foundation for virtually all future learning with numbers. They allow students to succeed at higher math skills and to gain higher math concepts.
According to most school curriculums, students should know their multiplication facts by heart by the end of the fourth grade. The skills should be carried to automatic recall of multiplication facts up to 10 x 10. It is noticed very often that a certain amount of math simply needs to become automatic. But how can we determine whether the skills have become automatic or not?
If we try to clear up the meaning of the term “automatic”, we will find only approximate descriptions. There are metaphors like the followings: “stone cold in the bones”, “could recite them if awakened at two in the morning”, “do it in their sleep”. We can see hyperboles also: “that the answers just pop out without trying”, “should know without having to think”.
And, finally, there are a bit more certain descriptions. For instance, “ability to quickly, correctly and consistently answer questions across all tables sets, selected entirely at random” or “Mastering the times tables does NOT simply mean being able to recite them off in sequence like a parrot. It means being able to quickly, correctly and repeatedly answer questions from any tables group in any sequence at any time”.
In this case the only unambiguous demand is that a sequence of questions should be taken from the times tables at random. But how can we interpret the terms “quickly” and “correctly”? I imply we need more precise definitions explaining how quickly? and insofar correctly?
Let us imagine that a pupil needs to answer 30 questions from the times tables in writing. How much time he/she must spend on the answers to allow the skills will be judged as automatic? 30 seconds, 60 seconds or, maybe, 120 seconds? And how many errors can be admitted to occur herewith?
I hope, you do not think that any sequence of answers must be error free completely. An error may be caused not only by lack of the skills. There are many other outside causes - a bad condition of a pupil, a brief distraction of attention, and so on. All we are humans, and to err is human.
So I consider that the criteria of mastering the multiplication tables are ill-defined. To improve the situation, we need to know the largest possible average running time of one operation and the largest possible relative frequency of occurrence of errors (the last criterion must be sufficiently low, but, of course, not equal to zero). Only in this case we can calculate the time required for any sequence of operations and the largest possible number of errors, which may be permitted herewith.
Cause #5. The memorized results of the times tables are not used in ordinary calculations on a regular basis. Because of that they are forgotten gradually.
Even though pupils have memorized the results of the times tables by heart, we can not consider that the work upon mastering the times tables is over. The pupils should learn to use the memorized multiplication facts in practice. If a kid finds it difficult to carry out any computations that include multiplication facts, then he/she has not mastered the times tables totally.
What is more, pupils must over and over again use the skills in practice to make stronger and expand them. The best tool for such practice is multiplication and division of numbers expressed by several digits. At the same time, according to many school curriculums, “there should be decreased emphasis on such activities as: isolated drill with numbers apart from problem contexts; performing paper-and-pencil calculations with numbers of more than two digits”. Very often we can read instructions in covering notes to math curriculums like the following: “The extensive use of drill in multi-digit operations, necessary in the past to enable people to perform calculations rapidly and automatically, is no longer necessary and should play a much smaller role in today's curriculum.”
When we abandon to wide use of multi-digit operations while teaching math, we lose the opportunity for regular development of the times-tables skills. Because of that they gradually grow worse and, in some cases, are forgotten at all. As a result, we turn out to be in a vicious circle. We allow the multiplications facts be forgotten and, hereunder, tear down the "crucial milestone in children's elementary math development". In consequence of which our pupils turn out to be unable to master well all following computation skills (fractions and negative numbers, particularly), many of them can not perform the calculations on their own, and, for want of drill, the times-tables skills continue to go from bad to worse.