It was an unusually pleasant Wednesday afternoon in July. A visitor, an experienced mathematics teacher from overseas, was eager to observe how a discussion in mathematics could be fostered in a mixed age classroom. The class was soon after the lunch break. So, to prepare the ground for discussion, an evocative story he narrated.
“It was an afternoon just like today, and an English teacher had been sent to substitute the Math teacher who was on leave. The students were a bunch of energetic seven-year-olds. In an attempt to occupy the class with what the English teacher thought was a painstaking exercise, he asked the students to add the numbers from 1 all the way up to 100. He then settled to attend to his own work. There was a prompt tug at the lapel of his coat within ten minutes by a lad who claimed he had 'finished'. Utterly disbelieving of the lad and yet in no position to actually check for himself the answer that the student quoted, the teacher asked the student for the one thing that he could: to justify his answer.
The student wrote:
1 + 2 + 3 + ...+ 98 + 99 + 100
(1 + 100) + ( 2 + 99) + (3 + 98 ) + ... and so on.
He reasoned that grouping the numbers in this manner should not alter the result. That would mean adding 50 pairs of 101. He had learnt that multiplication was repeated addition. So, the answer was 50 x 101. That could be further simplified as (50 x 100) + (50 x 1) which equalled 5000 + 50 = 5050.
The boy (who later came to be known as the famous mathematician Carl Friedrich Gauss) had found an elegant solution to a seemingly complex problem. “What can one say of the possibilities of the young mind!””
There was a palpably appreciative silence in the class. Now the challenge began. The teacher posed the question: “If you take all the numbers from 1 to 100, what would be more - the sum of odd numbers or the sum of even numbers? And by how much?”
Students were invited to make educated guesses. The majority felt that 'both the sums would be equal because there are an equal number of odd and even numbers up to 100'. A few said that 'even would be more because 100 was the last number and it was even'. The cautious remaining were looking for a basis to proceed.
I felt that students could now calculate using paper and pencil. The only condition I suggested was that every statement of an individual should be substantiated and justified.
Now energy had been unleashed. The following is a record of the conversations that took place. The technical complexity of the statements revealed the level of understanding of the student:
Student 1: Let me first try numbers from 1 to 10.
1 + 3 + 5 + 7 + 9 = 25.
2 + 4 + 6 + 8 + 10 = 30.
Even numbers add up to more, within 10. Therefore, up to 100 also it should be the same.
(Simplifying the problem and extending it inductively)
Student 2: How can you simply say that?
S1: Why? In the other numbers also, the trend is the same. Only tens and twenties are added.
S3: But that's no answer. In Maths you need to prove. You can't simply show 10 examples and say “it is so”.
S4: I have another logic. Look! Write the sum of odd numbers and even numbers one below the other:
1 + 3 + 5 + …… + 97 + 99
2 + 4 + 6 + ……. + 98 + 100
Each even number is 1 more than an odd number. So when you add them, the even numbers would be more.
(A completely lateral way of approaching)
S5: The explanation makes sense. But that still leaves the question: “how much more?”
S4: Since there are 50 even numbers, I feel the answer will be 50 more.
S3: But this is an estimation, at best an interesting one at that. That still doesn't prove the point.
S6: Since you are going on and on about proof, suppose you tell us what it means to PROVE!
S3: See, I too don't know how to prove this. But if some statement should be true, it should be true everywhere all the time. Not up to 10 or up to 100 or when you look at it one way and you look at it the other way. In fact that's understanding for yourself and not proving!
S7: Now let us try algebra. When you use 'x' and 'y' and a formula it means the statement is true for any random number and not necessarily particular numbers. That at least should be acceptable as proof.
(Moving from generic to specific; empirical to abstract)
Visitor: That appeals to me. Proof is the only thing that is specific to mathematics and we should not sacrifice that. In Physics you verify laws
learning initiatives with experiments. But in mathematics you prove statements starting from the beginning.
S7: Alright. Let us not go away in another direction. Hey, some seniors, tell about the formula that you learnt that day for adding numbers.
S8: Yes, I will write it on the board. In fact it is the same as that Class 2 kid mathematician's logic in the story we heard at the beginning. Only it has symbols instead of numbers.
(Moving on to formal proof)
S9: I know the rule. But I still can't apply it in this situation because you are adding odd and even numbers. There are gaps in them. The rule is for continuous numbers!
(Acknowledging assumptions and framework within which a precept is valid)
S3: A rule should be useful everywhere. Otherwise what's the point? Let's see if we can find the way to avoid the gaps…
S10: I've got it! I've got it! I've got it! See, we don't need to add numbers 2 times. We'll add numbers all the way up to 100. It is continuous. There are no gaps. We'll use the formula. Then we'll subtract the sum of even numbers. We'll get the sum of odd numbers. Then we can compare.
S11: Ha ha Einstein! How do you propose to add the even numbers? There are gaps. At least one of them with gaps you cannot avoid.
(Here the class was stuck. So the teacher offered a hint)
Teacher: You could think of 2 + 4 + 6 + …..+ 100 as two times 1 + 2 + 3 + …..50. Then there would be no gaps.
Class: Hey, that's cool.
Then students who knew algebra applied the formula and explained it to the class. The sum of odd numbers was calculated to be 2500 and the sum of even numbers to be 2550. So, the even numbers were more and by 50.
There was a student who had been very quiet and poring over numbers in a focussed manner. The teacher asked him to explain his attempt because he had been following his own reasoning. He expounded: In every set of tens, there is a group of 1 + 3 + 5 + 7 + 9 and 2 + 4 + 6 + 8 + 10. So, there are 10 groups of sums of 25 and 30. So that makes the odd number group 250 and even number group 300.
Then it was only a question of evenly distributing the tens in the other numbers. From 11 to 20, 5 tens go to the odd number group (11 + 13 + etc.) and 5 tens go to the evens group. Similarly, from 21 to 30, 5 twenties go the odds and 5 twenties to the evens.
Proceeding similarly, the student had calculated the sum of odd numbers to be 2500 and even numbers to be 2550 through elegant reasoning without knowing the formula.
The class applauded this painstaking effort. This concrete reasoning grounded all of the above discussion in a manner where each student still had a window of understanding into the solution. Suddenly the juniormost child in the class exclaimed:
S12: Hey! From 1 to 10 also, even numbers are more. From 1 to 100 also even numbers are more. So, in 1 to 1000 also even numbers will add more. In 1 to 10, 000 also even numbers will add to more.
(Not resting with one solution; extending the problem)
S7: Oh! True, come to think of it.
S13: Hey! Look at another pattern! From 1 to 10, the even numbers add up to 5 more than odd numbers. From, 1 to 100, even numbers add up to 50 more than odd numbers. In numbers 1 to 1000, will even numbers add up to 500 more than odd numbers?
S8: Yeah, let's find out!
S3: No, I don't want to say that it will always be even numbers which add up to more than odd numbers. It depends. If you take a string of continuous numbers from the middle with, of course, an equal number of odd and even numbers, if the first is an odd number and the last an even number, then even numbers will add to more. If the first is an even number and the last is an odd number, odd numbers will add to more.
S8: Can we prove it now? Please can we prove?
It was time for that class to end. It ended then, making way for another beginning the next time!