Throughout its history, humanity has developed numerous methods of multiplication or repeated addition. These can be more or less time-consuming and also vary in effectiveness: while some are better for education, others pursue purely practical purposes. Three such methods are traditional, Ancient Egyptian, and the method of cups and counters. Each of them has its pros, cons, and application: the first is better for comprehending the principle of multiplication in the decimal system, the second – for practical purposes, and the third – for education.
The traditional or long multiplication method is one of the most famous and widely accepted in the world. This method goes back to medieval Italian mathematics and relies on the decimal system as its main underlying concept. This algorithm amounts to multiplying ones, tens, hundreds, etc. of one factor by the ones, tens, hundreds, etc. of another factor, thus reducing any problem to several one-digit multiplications.
The method is similar to cups and counters in the sense that it also relies on the decimal system, but different from the Ancient Egyptian one, which relies on binaries. The pros of this method are its easy operability once mastered and the availability of different strategies to simplify it while learning, such as writing the factors on different paper pieces for convenience (Anu & Vinod, 2016). On the negative side, it requires an understanding of the decimal system and knowledge of multiplication tables before one can proceed. Its best use is for multiplying large numbers in writing.
Ancient Egyptian is another method quite different from long multiplication. As the name suggests, it comes from Ancient Egypt historically, and its central underlying concept is binary arithmetic. Ancient Egyptian multiplication relies on extensive use of multiplication tables for different numbers structured by the binary principle. Since Ancient Egyptian hieroglyphic writing was ill-suited for written calculation, the primary purpose of this method was to reduce the calculation to a sequence of steps that could be carried out mentally (Imhausen, 2018).
Thus, it is similar to both other methods in its intent to reduce multiplication to several simpler steps, but its devotion to binary arithmetic sets it apart. Its pros are convenience – it is easy to operate with multiplication tables at hand – and practical orientation, since it does not require much theoretical understanding of mathematics. Yet it is virtually impossible to operate without physical multiplication tables, and memorizing those would be hard, as they are not limited to one-digit multiplication in a decimal system. Thus, the best use for this method is limited practical purposes, but it is of questionable value in terms of education and theoretical mathematics alike.
Finally, the method of cups and counters is yet another way to approach multiplication. It is hard to pinpoint it historically since it relies on the concept of groups inherent in any multiplication operations. This approach uses countable physical objects of the same type and separates them into groups according to the number system used (“Number Facts,” n.d.). For better visualization, each group is represented by a container with the same number of objects, such as a cup – hence, cups and counters.
In its contemporary form, this method uses the decimal system, which makes it similar to long multiplication, but sets it apart from the Ancient Egyptian way. Its pros, such as easy visualization and the understanding it promotes, make it suitable for the use in education for the first acquaintance with the concept of multiplication. Yet its cons are quite obvious: one cannot rely on cups and counters in multiplying large numbers, and the method does little except for explaining the concept of groups.
As one can see, different multiplication methods have their pros and cons alike. Long multiplication, with its reliance on the decimal system, is easy once comprehended but requires more pre-existing knowledge than others. Ancient Egyptian method with its extensive multiplication tables is suitable for practice since it reduces actual calculations to a minimum, but is impossible without the said tables at hand. Cups and counters method, while good for early education, is limited to small numbers.
References
Anu, S.R., & Vinod, K. A. (2016). Impact of new multiplication method (paper strip multiplication) to enhance mathematical achievement among primary school students. Academicia: An International Multidisciplinary Research Journal, 6(11), 52-61.
Imhausen, A. (2018). Mathematics in Egypt. In P. T. Keyser & J. Scarborough (Eds.), The Oxford handbook of science and medicine in the classical world (pp. 49-60). Oxford UP.
Number facts: Multiplication and division. (n.d.). Queensland Government Education. Web.