Math too has a history!

S. Sundaram

It is safe to say that the two school subjects which evoke extreme reactions from students are mathematics and history, for entirely different reasons.

History is disliked because it has plenty of unconnected information about kings and their victories to be memorized. But students never learn that “everything” in this universe has a history!

Mathematics, on the other hand, is disliked by students for several reasons. It is abstract, has too many rules, looks like a patchwork of many disjointed topics and most importantly many teachers and parents use students’ performance in mathematics as a yardstick of intelligence.

Many of the solutions to these issues in mathematics are long-term in nature and need the sustained cooperation of schools, parents, and examination boards.

In this article I will suggest an easier but radical alternative – Use anecdotes from the history of mathematics to enrich the understanding of students and make mathematics meaningful to them.

I will provide a few examples of this approach illustrated with some anecdotes from the history of mathematics.

Children like stories and remember the concepts embedded in them easily
The first advantage of this approach is that children like to listen to stories. They remember them easily, unlike definitions in mathematics.

How did humans keep count of sheep before knowing how to count?
It is likely that they used a set of pebbles to keep track of their flock. They probably had two bags, one filled with pebbles and the other empty. As the sheep go out of the paddock gate, one pebble could be transferred from the “full” bag to the “empty” bag. When the sheep return, the whole process is repeated but in the reverse direction.

This came to be known as the principle of “one-to-one correspondence.”
Taxi Cab Number – the mind of a mathematics lover
Most students know that 1729 is the “Taxi Cab” or the Ramanujan number. But they may not know its significance.
While Prof Hardy thought that 1729 was an uninteresting number, Ramanujan immediately recognized its significance.

This incident shows how the mind of a mathematician like Ramanujan was constantly playing around with numbers, their properties, and relations. He did not have to memorize them. Today such a skill is called Number Sense.

Gauss’s method of adding numbers from 1 to 100
The mathematician Carl Friedrich Gauss was a very precocious child in school. One day, to keep him occupied, the teacher gave Gauss the task of adding all numbers from 1 to 100. While the teacher expected Gauss to take a long time, Gauss gave the answer in a few minutes.

The simple method that he used can easily be understood even by a grade 3 student. The same technique is used today in summing many numerical and algebraic series.

The chessboard and rice puzzle
This is a classic story from ancient India of a king trying to fill the 64 squares of the chess board with rice in such a way that the subsequent square should have twice the number in the previous square.

This puzzle is a simple introduction to the idea of exponential growth and its counter-intuitive results.

History reveals the simplicity underlying complexity
Students have a wrong notion that complex mathematical ideas which they encounter in their textbooks were “discovered” in their final form. They need to know that their notion is not true. Mathematical ideas started in a simple way from humans observing the patterns and relations of objects and events around them. Over a period of time, these ideas acquired layers of abstraction until they reached the form in which we meet them in our textbooks.

History can reveal the gradual development of mathematical ideas from the concrete to the abstract. This makes it easier for students to understand these mathematical ideas.

The decimal place value system
The decimal place value system is one of the most basic concepts in mathematics, necessary both for understanding different types of numbers as well as operations with them.

Most cultures invented number systems based on ten and a rudimentary “place value” system, because most of us are born with ten fingers. But they lacked a way to express this idea in a clear manner.

Today we write a number like “four hundred four” as 404. It is obvious that both the 4s indicate different quantities; one denotes four hundreds and the other denotes just four. 0 indicates that there are no tens.

But early cultures like Sumerians and Egyptians wrote “four hundred four” as 4 4, with a “blank” space between the two 4s. They could not think of a “blank space” as a number.

This system “of just leaving a blank space” was confusing since “4 4” can many times be read as “44”. The value of the number had to be figured out from the context and this led to mistakes on several occasions.

This system of writing continued for more than 2000 years, until Indians “treated zero as a number and invented a numeral (0) for zero” on par with numbers 1 to 9. The symbol they used for zero started as a dot and ultimately became “0”. This notation also made computations very easy.

This idea was so easy as well as powerful that it reached Europe through Arab traders and completely replaced the abacus-aided calculations based on the Roman number system.

Today students are familiar with these methods even in lower primary school. We are so used to this idea that we do not realize how revolutionary this idea was!

Why do we have a 24-hour day?
The natural measure of an angle was the “complete angle.” This was similar to the annual revolution of the earth around the sun. Since the year was initially estimated as 360 days, the complete angle was divided into 360 degrees.

Dividing the complete angle into four quarters gave the measure of a right angle as 90. The equilateral triangle with all sides equal gave an angle with a measure of 60. The difference between the right angle and 60 gave an angle with a measure of 30. Bisecting an angle of 30 gave an angle of 15.

Mathematicians could not get an angle smaller than 15, without getting into fractions, which they wanted to avoid at any cost.

Hence they notionally divided the complete angle (360) into units of 15 which yielded 24! Hence the day was divided into 24 hours.

Interestingly, Hindu astrology divides the day into 60 units of 24 minutes! Each of these 24-minute periods is called a “ghati.”

Most mathematics was created by humans like us and our ancestors
Mathematical ideas were not discovered only by superhumans and world-famous mathematicians.

Starting ideas in mathematics were simple and were invented by humans like you and me as solutions to day-to-day problems encountered by them daily. Several mathematical ideas were invented simultaneously by several cultures which had no contact between them.

There are two underlying messages in this for students.
One is that they have a heritage of mathematics which they can be proud of.
Second is that “if we put in the necessary effort, we also can understand the mathematics involved.” This is called a “Growth Mindset.”

Number Systems
When humans turned to agriculture from hunting and gathering, they started accumulating possessions in terms of cattle, sheep, grains, land, etc.

Initially they were able to use the idea of one-to-one correspondence to keep track of their possessions. But in the long run they were forced to invent counting and number systems to keep track of large magnitudes.

Operations like addition, subtraction
When humans started breeding livestock, their numbers increased and decreased depending on births and deaths. Addition and subtraction operations were invented as ways to keep track of the changing quantity of possessions.

Fractions and Decimals
In families and societies “sharing” food and possessions is a daily occurrence. Humans invented fractions and division to keep track of such sharing.

When fractions were invented, the rules of operations were complicated to remember and master. Even today, all over the world, fractions are considered the most difficult topic in schools. Hence there was pressure to invent a simpler system of representing fractions.

Decimal numbers were the result. Decimal numbers are nothing but an attempt to represent fractions (quantities less than a whole) using the decimal place value system.

Mathematics was also a product of cultural, social, and commercial interactions
Hindu invention of zero
Scholars agree that Hindus could imagine a number like “zero” which was called “shunya”, which signified both “fullness” and “emptiness.” This was because these ideas were part of their philosophy!

The absence of such a philosophy, possibly prevented Egyptians, Sumerians, and Greeks from thinking of “zero or nothing” as a number.

Π was discovered by many cultures independently
Most cultures developed into agricultural societies from hunting and gathering. Agriculture needed plots of land, water from wells and canals to distribute them.

Circle was one of the easiest shapes that could be drawn for digging wells. So, there was a universal interest in studying circles and their properties.

Discovery of π, which is the relationship between the diameter and circumference, was one of the earliest discoveries in all these societies.

The Right Angle
Similarly in geometries of all cultures, the Right Angle became very important, because it was the “right” angle at which humans had to stand on the ground or a wall had to be built!

It was also the easiest angle to draw anywhere in the world with minimum equipment. A simple plumbline makes a right angle with the ground and a piece of parchment or leaf folded twice can make a right angle.

The Pythagoras Theorem
The right-angled triangle was a much-studied shape. This led to the discovery of the Pythagorean Theorem in many cultures, centuries before the theorem got associated with Pythagoras.

Mathematics also emerged from trivial pursuits
A lot of mathematics ideas emerged from simple games and puzzles invented by people for spending their leisure time. Mathematics provides a lot of fun along with intellectual rigor. Sometimes day-to-day problems faced by people give rise to topics in mathematics.

Bridges of Konigsburg
The city of Konigsberg had a river flowing through it and seven bridges which crossed it. A common puzzle for its citizens was whether all the seven bridges can be crossed once, without any bridge being crossed twice.

When this problem reached Leonard Euler, he not only solved the problem but also sowed the seeds for a new branch of mathematics called Graph Theory.

He also gave a simple visual explanation that the problem of crossing all the bridges just once was not possible.

Games and puzzles
Games and puzzles based on mathematics have been part of folk-mathematics and leisure-time activities.

Some examples are constructing magic squares, games like snakes and ladders, ludo, NIM, etc. There also have been puzzles involving weighing or measuring out certain quantities and altering geometrical shapes made with matchsticks.

Mathematics also has “difficult problems” which can be understood even by primary school students
Many problems in mathematics are simple enough for primary school students to understand them. What will be surprising to them is that many of them have not been “solved” or “fully understood” till date.

Goldbach’s Conjecture or Collatz’s Conjecture or Fermat’s Last Theorem are examples.

Fermat’s Last Theorem was “solved” after almost 250 years of work by many mathematicians. The other two have not been solved till date.

History provides a “bird’s eye-view” showing connections across regions and time scales
It is similar to having a map of a city rather than verbal instructions to travel around the city. The map reveals many paths and connections all of which cannot be understood with just a verbal explanation.

The Konigsburg bridges are an example of the connection between a puzzle faced in daily-life and a sophisticated topic of Graph Theory in mathematics.

The simple art of counting is connected to the topic of Permutations and Computations and Probability.

The principle of one-to-one-correspondence was used by Georg Cantor to prove that the infinity of irrational numbers was more than the infinity of rational numbers.

Conclusion
Due to lack of space, I have provided only a few examples and that too in summary form. Many more such examples are available on the web and several books.

I hope this short article will kindle interest in mathematics and history teachers to collaborate and find many such historical anecdotes, which are not only interesting in themselves but also make mathematics an interesting subject.

The author has worked as a principal, teacher trainer and educational consultant in several schools in the country. He retired as the principal of Reliance Foundation School in Jamnagar. His areas of interest are primary mathematics, school leadership, quality in education and technology in education. His book “Understanding Primary Math” can be accessed at https://primarymath.miraheze.org/wiki/. He also has a FB page “Primary Math Is Easy by S Sundaram”. He can be reached at sundaram021148@gmail.com.