Abacus Vs Algorithm

Procedures in Arithmetic operations

Indians invented the place value system and the number zero. Based on these ideas, they developed easy procedures for performing all the 4 operations on numbers. These reached Europe through Arab scholars.

Al-Khwārizmī (780- 850 CE) was a scholar in the House of Wisdom (Bayt al-Hikmah) in Baghdad. At this intellectual hub, scholars were translating knowledge from around the world into Arabic, synthesizing it to make meaningful progress in a range of disciplines.

He wrote 2 books; one on Algebra and the other on Arithmetic.

The 1st book "On Calculation With Hindu Numerals" was a popular math book written at the behest of al-Rashid’s son and successor al-Ma’mun. It described the Indian Positional Number system using 0 and also the various algorithms for the 4 operations. Its Latin translation awoke Mediaeval Europe to an amazing new way to do arithmetic using just ten number symbols. In Latin, his name became ‘Algoritmi’ and his systematic calculating procedures were called algorisms — later, algorithms.

His 2nd book was "al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala" — The Compendious Book on Calculation by Completion and Balancing. Al-jabr, Latinised to ‘algebra’, became a word in its own right. Al-Khwarizmi didn’t use symbols, but he focused on how to manipulate algebraic expressions in their own right, rather than thinking of the numbers they represent.

Before even the Arabs, in a fragment, dated 662, of a work by Severus Sebokht , the learned bishop of the monastery of Quinnasrin in Syria, expressed his admiration for the Indians because of their valuable method of computation 'done by means of nine signs.' Severus had probably learned about the system from Eastern merchants active in Syria.

Before the arrival of these ideas, Europe was using Roman numerals. Since it was very difficult to compute with Roman numerals, the abacus was used to compute numbers written the Roman system. So "Abacists" or arithmetic experts emerged, who could perform these operations & keep accounts using the abacus.

The Hindu methods were found very easy to use and were adopted by merchants in Venice who were trading with many countries and welcomed a number system which made it easier for them to keep track of their trade and goods. "Algorists" who had mastered the algorithms for computing emerged, using the Hindu Arabic place value system.

It took several centuries after the introduction of these procedures for a common man on the street to do the 4 operations without depending upon a "Algorist".

Today the procedures for operating on numbers using the 4 basic operations have become simple & standardized. They are simple enough to be taught in schools as mechanical step-by-step procedures. These algorithms also enabled arithmetic operations to be mechanized using machines and then to be coded into computer software.

We see the effect of traders & accountants having used these procedures extensively in phrases like "carry over" addition, "borrow" subtraction and "use of Xs to fill up blank spaces during multiplication" which are still used in our classrooms. The publication of "multiplication tables" and their being used very much like the "Panchang" is also a carryover from those days.

Algorithms & Math Education

While algorithms help in mechanizing calculations, they are not very effective tools for learning math in schools. They hide the math concepts underlying the procedures. Hence they could be taught to "weaker students" who find it difficult to understand the concepts.

An interesting way to teach would be to investigate the concepts underlying why these algorithms work. E.g what is the meaning of the “carry-over” algorithm in addition?

Use Algorithms with Caution

Students mostly "memorise" these procedures as well as the "computational facts of adding, subtracting, multiplying and dividing with 2 numbers" and apply them. Since students apply these procedures like "magic formulae", many a time they make mistakes. An overuse of algorithms may result in a reduction in number sense skills. The only way most teachers try to rectify these mistakes is to "drill the students". Invariably, drilling produces boredom as well as a fear and hatred for mathematics.

With a little thinking, the various algorithms for any operation can be arranged in a sequence starting with "most easy" to he "least easy" in terms of understanding what is happening in that operation. The least easy to understand, will also be the "standard algorithm", which has been honed and practiced over so many years that it has become most efficient but with loss of ease of understanding.

The algorithm which is easiest to understand will typically, will use "number sense ideas" and take more time also. All children should start with the "easiest to understand". As their understanding becomes better, they can move to the others and ultimately end at the standard algorithm.

Primary school students can be allowed to spend as much time as required at each stage, until they are ready to move up. Teaching ONLY the "least understandable" standard algorithm, for ALL is a sure recipe for "the math anxiety bug" to infect students.

Primary school students should also not be subjected to the tyranny of the "timed tests". The focus in primary school should be to enable all students to understand the curriculum and not in making them "fast" in calculations.

In dealing with each of the 4 operations, in the next few chapters, we will present several ways in which an operation works. These could be seen as varying in "understandability" as well as "speed of doing"

We will not, however, be explaining different techniques used in computations in detail, except to clarify the "concepts" underlying them.

Algorithms & Computers

In the early 20th century, the word algorithm came into its current definition and usage: “a procedure for solving a mathematical problem in a finite number of steps; a step-by-step procedure for solving a problem”. Algorithms have got very closely associated with computer software.

Hence we can think of Al Khwarizmi as the grandfather of computer coding.