‘Multiplication is a bummer;
the division is just as bad;
the rule of three puzzles me,
and practice drives me crazy. “
John Napier, 1570.
The verses that head this article correspond to a children’s song, Multiplication is a nuisance, which dates back to an Elizabethan document from 1570 entitled Description of the admirable table of logarithms, written by the Scottish mathematician John Napier (1550-1617) and printed for Simon Waterson in 1618.
John Napier was recognized for being the first to define logarithms. In fact, from its Latin name, John Neper, comes the one of the Naperian logarithms.
Napier was also the inventor of an abacus, the description of which was published in his work Rhabdologia, printed in Edinburgh at the end of 1617. That abacus is known in English by the curious name of ‘napier bones‘, a first mechanical device for calculating multiplication and division.
Napier was the son of famous people: his father was sir Archibald Napier, Merchiston landowner. Naper was born in Merchiston Castle, and was nicknamed for this as’the wonderful Merchiston‘.
The rule of three
Let’s go back to the rule of three that ‘puzzled’ the mathematician. Much later in time Abraham LincolnIn a short biography provided to friends who supported his candidacy in 1860, he wrote: ‘He could read, write and calculate with the rule of three; but that was all. It seems that the rule of three had a value in those days.
We know that the rule of three is a way of solving proportions, which are solved with cross multiplication in which the problem is posed in such a way that the unknown quantity is the last extreme of a series of numbers that have a proportional relationship.
We know a, b and c, and we compute x. And that as for the simple or direct rule of three, which we already know that we can complicate it more with the inverse and the compound rule of three.
In my school days I had to solve many arithmetic problems with the rule of three, which became the universal panacea. This is probably what Lincoln had to do in his days as a young grocer in New Salem (although he studied a lot of other math stuff, like Euclid’s elements). Surely this apprenticeship with numbers helped him in his later duties as President of the United States.
The rule of three was known to the Arabs, as al-Jwarizmi in your Algebra, and al-Biruni (973-1050), who dedicates a complete work to this subject, On the rules of three of India. Aryabhatiya described it in these poetic terms: “In the rule of three multiply the fruit by the desire and divide by the measure; the result will be the fruit of desire.
The rule of three has been collected in many texts. For example, in the mad gardener song, Lewis Carroll includes the lines:
«He thought he saw a garden gate
that was opened with a key:
he looked again, and found that it was
a double rule of three.
And also Rudyard Kipling He mentions her in The Jungle Book:
«You can solve it by fractions or by simple rule of three,
But the Tweedle-dum way is not the Tweedle-dee way.
You can twist it, you can twist it, you can braid it until it falls off,
But the way of Pilly Winky is not the way of Winkie Pop.
The ‘golden rule’ in France
In France the rule of three is used at least from 1520, although everything indicates that it was already used some centuries before. In L’arithmétique nouvellement composée, Estienne de La Roche dedicates an entire chapter to her, and considers her the most beautiful ruler of all.
The recipe became popular at the beginning of the 18th century thanks to the numerous editions of the book of Francois Barrême, ‘The Arithmetic of Sieur Barrême, or the easy book to learn arithmetic on your own and without a master’. Este es autor de obras de cálculos prácticos y tablas de correspondencia that han pasado has the posteridad con el number of baremos.
Barrême is no longer concerned with proportionality, but in the article of the Encyclopedia of Diderot and d’Alembert yes there is this concern. The two encyclopedists call it the ‘golden rule’. And that presentation as the rule itself or as a result of proportions continues in later decades.
As an example, when you enter 1960 and 1970 the wrong calls are introduced ‘modern mathematics‘, the interpretation behind the rule of three is sought, highlighting the mathematical concept that supports it, proportionality.
In 1963, Gilbert Walusinski, a member of the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP), wrote an article entitled The rule of three will not take place, paraphrasing the theatrical work of Jean Giraudoux, criticizing the automatism of the rule of three and proposing problems in situations that will mobilize the critical spirit of the students.
A few simple examples are enough to realize the insubstantiality of the rule of three:
If a circle of radius 2 meters has an area of 4 π, then one of radius 4 meters would have, if we apply the rule of three, 8 π, when the correct answer is 16 π. Because the relationship between the area of the circle and its radius is not linear, it is quadratic.
If Juanito has a height of 1.25 meters at 5 years old, when he is 10 years old, he would measure 2.50 meters, a future NBA player.
So we could go on indefinitely.
Reflection and analysis against any rule
The teaching of mathematics in Spain does not differ much from what happens in France (and, in fact, in any other country because the problems are similar in almost all of them).
I do not advocate a return to those ‘modern mathematics’, although I do not repudiate them, because the grupo Bourbaki I was pursuing a better foundation for mathematics and it made an impact that hasn’t stopped (that’s what happens when you put great minds together).
Questioning what is done at all times means always reflecting on what is best, and that can lead to substantial changes. But I do see that repeating exercises over and over again without knowing what is actually being done is not going to mean that the mathematical level of our students will improve.
It is much more useful for their minds to know how some quantities are related to others, than to apply rules of thumb without an analysis of their applicability to the case at hand.
Manuel de León Rodríguez is a research professor at the CSIC, Royal Academy of Sciences, Institute of Mathematical Sciences (ICMAT-CSIC)