Creating your own mathematics

Wanting to do something different for a change a few days ago, I amused myself with a little bit of mathematics. In our work, we are always trying to make the life of the teachers and children much better in the primary school classroom. Any mathematical activity which we undertake is with this objective.

I was playing around with a seemingly innocent problem of subtracting 169 from 637. However, I did it “my” way which was different from the “normal” and “accepted” method taught in school. Here goes:

100 + 100 + 100 + 100 + 31 + 37 = 468 (answer)

169 --- 269 --- 369 --- 469 --- 569 --- 600 --- 637

As you can see, we start with 169 and then progressively keep adding hundred. Why add hundred? Because it is easier to add hundred at a time. You are correcting an annual examination paper, and one of your children has come out with a working process with the answer as written above – what would your reaction be? Without being harsh on the teacher community, I would be tempted to say that this would invoke the much dreaded red cross against (and across) the answer. Not only that, the child would be pulled up with “This kind of an answer will not get you anywhere. No marks for this! Where is your working?” This would be followed by a “recapitulation” (torture) of the method:

637
- 169
-------------
468
-------------

Of course, you’ll have to remember how to “carry one”, “borrow one”, and “pay it back”.

Have you noticed how bus conductors deal with cash and return change? Suppose you have bought a ticket for Rs.5/-, and you hand over a Rs.50/- note. Many conductors, who have the habit being vocal about how they return the change, will be heard to say, “Ten, twenty, thirty, forty, fifty”. When “ten” is being said, the conductor actually gives you Rs.5/-, and then successively gives ten rupee notes till he comes to 50. In the process, you’ll have been given

5 + 10 + 10 + 10 + 10 = 45

Sabjiwalas use this method every minute. Where do the conductors and sabjiwalas have the time to “carry, borrow and pay it back?” Some of the sabjiwalas may not be even “numerate” in our definition. How are they able to manage?

Inspired with this discovery, I continued playing around with more problems. Consider the product: 17 X 14. This is how we normally do it in school:

17 X 14
----------
68
170
----------
238

There are other ways of doing this problem. One way would be:

(10 + 7) X 10 + 4)
= (10 X 10) + (10 X 4) + (7 X 10) + (7 X 4)
= 100 + 40 + 0 + 28
= 238

Can you think of other ways? How many of us wonder what takes place in the mind of the child when given a problem like

56 + 27. Answer?

There is something about the ease with which we use a ready made algorithm (an algorithm is a set of rules or procedures) whose logic we may not even understand, which stops us thinking about how we would approach a problem like 56 + 27. Mechanically, we follow the rule of “carrying over”. As teachers, we do not take the trouble to find out what happens in the mind of a child when it sees the problem. It may be more natural for many children to add 50 and 20 to get 70, and then add 7 and 6 to obtain 13. The final answer can then be got by adding 70 and 13. No wonder, we then let the children fall into a set of habits which, in the long run, close their minds to other possibilities. Our associations with particular procedures and our rigidity with particular symbols are so often tight that a child of eight may not know the answer to 7 multiplied by 5 but know straight away what 7 times 5 gives!

To subtract 169 from 639, it is surprising how many personal procedures (or algorithms) there are, and yet, we are often stuck with the method we have learned in school. Algorithms may help us to ease the problem of writing procedures in a symbolic way, but that is not the end in itself. Consider another personal procedure for the same problem (i.e., 637 – 169):

169:31
31 and 37 is 68
and there are 6-less-2 hundreds
468 (answer)

Here is a final example:
Since 637 = 100 + 100 +100 +100 + 100 +100 + 30 + 7, and
169 = 100 + 60 + 9, it follows that 637 – 169 can be written as

100 + 100 + 100 + 100 + 100 + 100 + 30 + 7
-- 100 + 60 + 9)
------------------------------------------------------------
0 + 40 + 91 + 100 + 100 + 100 + 30 + 7
------------------------------------------------------------
The reader is left to obtain the answer in whichever way is convenient. Notice how the above approach also helps in understanding the idea of the expanded notation. Notice also, that, in this case, “borrowing” as we would do it normally in a subtraction problem, is not required at all! After enough practice has been provided in the use of the expanded notation, the same problem could be now written as:

600 + 30 + 7
-- (100 + 60 + 9)
---------------------
468 + 0 + 0
---------------------

The process would be: Nine is greater than 7. Therefore we would need to borrow at least 2 from 30, and add it to 7. 9 – 9 is now equal to 0. By taking 2 from 30, we have made it 28. To subtract 60, we would need to take at least 32 from 600. Similarly, 60 – 60 is 0. Since we have removed 32 from 600, we are left with 568. 568 – 100 is therefore 468, the answer.

Another typical way of solving this problem would be – starting from the left, i.e., from 600, we could begin by removing 100. We are now left with 500. But we see that 30 is lesser than 60, and 7 is lesser than 9. Therefore, how much would we need to borrow from 500 to subtract 60 and 9? First, take away 30, and add this to the 30 we already have. 60 – 60 is zero, and 500 – 30 is 470. We need to add 2 to 7, so that 9 is also cancelled in the same manner. Finally, we are left with 470 – 2, which is 468.

Notice how the expanded notation is used, and how this gradually gives way to the illustration and use of the place value. The procedure of “borrowing” is very clearly shown. You don’t always have to borrow 10 or 100. On the other hand, the amount that you need to borrow is flexible, and depends on what is required to be borrowed. Borrowing can also be done in many different ways. By admitting this idea, we are allowing enough scope for the child to think and explore to find out how to go about a particular problem. And each problem brings with it a new experience and challenge. Often, the tendency is to teach expanded notation, place value and operations on numbers separately. This piece meal approach prevents one from seeing the connections.

The methods explained so far do not destroy for me the other ways of subtracting that I know. Very often, the procedures we follow in our minds when doing a problem cannot be put on paper without making them to appear clumsy and chaotic to the reader. The above examples are sufficient to illustrate this. This does not mean that these methods are not correct, are ‘slow’, and therefore should not be followed. The only advantage of following the method learned in school is that it can be put down on paper without the need for elaboration. Secondly, these methods help us to compute quickly. This brings us to the next question…

“What is the best method?” I do not want to ask this question without counter-demanding, “For what purpose?” There is nothing sacred about a particular method. In fact, the popular perception which tremendously influences our attitudes as teachers and parents towards children is that:

Speed = Brilliance,
Slowness = Dullness

There are certain misplaced notions about what about the ‘qualities’ of a ‘good’ student of mathematics - the ability to compute fast, and the ability to handle big numbers. Shakuntala Devi is often referred to as a great mathematician (which she’s not!), because she can multiply two twelve digit numbers with ease, or obtain the square root of a ten digit number faster than the computer. Often, parents and teachers take pride in such skills that their children may have developed. Pray, what purpose will this serve to a child in an ordinary school classroom and later on in life? As adults, we can only pretend to understand the value of, say, 1 light year (the distance covered by light in one year, at the speed of 3,00,000 km/sec) which is 9460800000000 Km. Why should we torture our children then?

Mathematics is not just about how fast you can calculate, or your ability to play around with big numbers which may mean little to you in everyday life. It is not limited to the application of ready made, uniform procedures to the solution of problems. It is about cultivating the ability to create and explore paths which we can identify with. It is often said that in order to learn mathematics, one needs to create (re-create) it for oneself. The examples discussed so far clearly illustrate this. What we consider to be the “fundamental” or “basic” principles of mathematics at the school level have taken thousands of years to develop. It necessarily follows that we cannot force the learning pace with children. Yet, how easily frustrated we become when we see a “wrong” answer! The truth may be that this wrong answer represents a genuine exploration on the part of the child, a struggle to comprehend.

Most often, we do not let out children explore different ways to arrive at an answer with the argument that forming habits (in my words, the ability to mindlessly repeat) are a protection against the confusion that could take over if the mind began to charge off in too many directions. This uncertainty of not knowing what will happen makes us hold our cards close to our chests, and “protect” the interests of the child.

The truth is that, as parents and teachers, we would like our children to cultivate and perfect these skills and habits so that they can ‘do well’ in the examinations and score high marks. Remember, the competition is tough out there! But, in the name of this competition, are we not inhibiting the natural ways of learning in our children? You decide…

How can we have an environment where both experiences, i.e., formalized procedures and treatment of topics, are reconciled with exploration, imagination and the 'freeness' to think? While it is possible to go in all kinds of directions without necessarily having the ability to be able to compute fast, or be precise, this imagination would be useless without care in developing appropriate skills. On the other hand, these skills (of calculation, of being able to apply procedures, etc.) cannot be developed in isolation of the ability to be able to explore, imagine and think freely.

New Delhi
26th April
1998