Affiliation: Association for International Mathematics Education
and Research – India
When teaching mathematics usually we do not think of a story to tell in the class. On the other hand people think that it is a remote thought to present a topic in mathematics through a story. Not only that. A story is one which is told elaborately and finally deriving a crux of the whole idea. So to say, a small idea or a fact is amplified, connected to all imaginative things and finally a story is formed. But, mathematics usually demands to tell the crux of a whole idea. Both are contradictory. How can one think of combining two contradictory ones and develop pedagogy? Let us see some recent developments happened in the mathematics education regarding this aspect.
Kieran Egan says that the great power of stories is in their dual mission, they communicate information in a memorable form and they shape the feeling of the hearer about the information being communicated. September 25 is observed as the “Math story Telling Day” Maria DroujKova of National math.com created Maths story telling day in 2009 after reading a blog posted by Seth Godin entitled “What should I do on your birthday? Maria took this idea and declared that her daughter’s birthday September 25 as the Math storytelling Day and stared a math story with her friends and relatives.
August 15, 2017 is called Pythagoras Theorem Day. Why?
But, telling a story in a mathematics classroom seems to be an off-task activity. When the topic of storytelling in mathematics class room was discussed with some teachers, several questions arose.
Teacher 1: Stories means what stories? About mathematicians?
Teacher 2: If I want to tell a story about the mathematician who designed multiplication, first of all I must know who developed multiplication. So, it is highly unsuitable in lower classes because there is no clear history of the 4 fundamental operations.
Teacher 3: If you say that it need not be on mathematician, what story one can I till about a long division problem?
Teacher 4: If I tell in simple terms the process of division, my student understands. Then why to waste time on creating and telling a story on this?
All of them in a way are genuine questions every teacher faces.
But one great thing about story telling in mathematics classroom is humanizing mathematics. Mathematics was developed and taught in every ancient culture. How did they do? Let us take the old civilizations and study the methods adopted in teaching mathematics.
Ancient Indian Mathematics
If we study the Plimpton 322 clay tablet, we do not find much word problems in Babylonian mathematics. In Egyptian arithmetic, a little algebra and a very little geometry, word problems are not available. Greek mathematics was the first one which introduced deductive logic in geometry by Euclid, which did not concentrate on word problems. In a similar way Chinese and Japanese mathematics also, no word problems are available. But we find lot of word problems in Ancient Indian Mathematics in the treatise called “Lilavati”, Bhaskaracharya, wrote in his book, mathematical problems in a story telling way. Here given an example.
“There was a snake barrow at the foot of a pole, on the top of which Seated a domesticated peacock. The pole was 9 cubits high. The peacock saw a snake crawling towards the post at a distance of 27 cubits from the post. The peacock pounces on the snake at the same speed as the snakes crawl and caught it at a certain distance from the pole. Find this distance quickly”.
What are the types of storytelling associated with teaching mathematics? The following are worth probing.
There is a big compound and inside the compound there are two streets. In each street there are three houses, and in each house there are four rooms. In each room there are five cupboards and in each cupboard there are six boxes and in each box there are seven toys. How many toys totally we have?
The number of toys - 1234567=7! (The factorial)
Stories that demand an answer
Sita has to go to school on a holiday for her dance practice. She has four different tops and three different skirts (bottoms). The tops are of different colours (Red, Blue, Green and Pink) and the skirts are of different colours (Purple, Violet and Orange). She can wear any top for any skirt. How many ways she can choose for any outfit?
Stories that initiate a research attitude
Camel stories: A man had 17 camels and three sons. He wrote a will regarding the sharing of the camels among his sons after his death. As per the will of the total number of camels must go to the eldest son, of the total to the second and of the total to the youngest. After his death, the sons had a problem, since of 17 is not a whole number. They do not want to cut camels. They approached a wise man in the village and told about the will. The wise man told them not to worry and came with his only one camel to them. Then he made the division. There are 18 camels of it is 9, of it is 6 and of it is 2. Total is 9+6+2 = 17. He divided them and took back his camel.
Now there are two questions.
Is this a fair division?
Can such a problem be posed with different number of camels?
Let us dismiss the first question by the obvious reason that it is a story to illustrate a mathematical technique.
The second question was given to a set of teachers and students. The outcomes were as follows.
1) Some told that the total number is 35. Then the divisions are 18, 12 and 4 and the total is 34. All 35 camels are not distributed. What to do?
2) Let us consider that the number of Camels be 53. If the wiseman joins his camel to make it 54, then the divisions are 27, 18 and 6. But the total is 51. We get 2 camels extra. What to do?
They realized that as the number goes up, the number of undistributed camels increases.
Since the purpose of the workshop is not pure mathematics but storytelling, they were told to construct a story fitting to their numbers.
One teacher gave an interesting story. A father had 35 camels to be distributed as ,, to his three sons, They approached a wise man. The wiseman told, that he will do that but what benefit he gets. The three sons told that they will give away one camel as a gift. The wiseman got back his camel and got one more camel as a gift.
Then it is pointed out, that in a different disguise the same story appeared in the book titled, “The man who counted” by Malba Tahan.
A story which involves technology to solve
Mrs Euclidea has two daughters, Archimedia and Apollonia. Archimedia knows a good amount of mathematics and the other is not familiar with mathematics. One day there has to be a together in their house. Euclidea wants to put up a welcome flower bouquet in front of the gate. There are several bamboo sticks of different integer lengths. She gave instructions to her daughters to make a flower welcome bouquet as shown in below, 6ft above the ground.
The two daughters did that. When Euclidea saw this she found that the flower arrangement is at a height of 6 feet from the ground. She told her daughters to rearrange it so that tall visitors will not hit on it. Apollonia started re-arranging the sticks to and fro, but Archimedia told this will not at all help to raise the flower arrangement. How did she come to such a conclusion?
The geogebra software is used to this situation and when the trace of the point of intersection is investigated we find that it is a straight line parallel to the ground.
This is the semi-harmonic mean of the lengths of the poles.
Clearly, the distance between the poles is not coming in this expression This implies that the height ‘h’ only depends on a, b. So, when the relative position of the poles is changed, h will not alter. That is how, Archimedia told that if the relative position of the poles is changed, it will not improve the situation.
Researching on the problem:
Now, first of all we want the height of the flower arrangement to be 6 feet. What are the heights of the poles to be erected? It leads to the problem to find natural numbers a, b such that .
When a = b we have or a=12
When two poles of 12 feet each are arranged then h=6 feet
But for what values of a (not equal to b) and b we have
This leads to the Diophantine equation 6a+6b = ab, where the natural number roots are to be found. When the problem was given to a set of students of age group 14–15; some of them framed rightly the equation and some got
Eventually, both are in accordance with the right direction of getting a solution.
Many of them made the correct obvious observation that .
Let us concentrate on the group who got
They started a trial and error method, which is a part of natural thinking.
When b = 12,
They got the known result that the poles have to be equal in length.
Some of them tried b= 18. Then
They stumbled upon a=9, b=18 which is a solution.
Next trial is b=24 which gives
a=8 and b=24 is another solution.
A natural thinking is why not try for a=7. They worked in the reverse way
when a=7 gives
, and h
They found 4 solutions (a, b) = (7, 42), (8, 24) and (9, 18)
Some started trying a=10. Which yielded b=15 (10, 15) is another solution.
a = 11 did not work. a=12 yielded b=12 a known solution.
a=13 did not work. a=14 gave b not a natural number. Because of the symmetry of the equation a=15 is the same as b=15.
They concluded that there are only three possibilities of non-equal poles to get the flowerpot at 6 feet from the ground.
They are (a, b) (7, 42), (8, 24) and (10, 15)
This problem gives rise to several new problems because ‘h’ can be changed to any natural number.
With the help of technology we understand the situation visually. By changing the values of a,b (where a,b are unequal natural numbers) using Geogebra Software, one can find the situation when the height of the flower boque is 7 feet (or 8, 9. . . .feet). Correspondingly working with known mathematical techniques, we can find all the solutions of such problems.
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