What your maths teacher didn't teach you at school

It ain’t what you don’t know that counts. It’s what you know that ain’t so – Will Rogers On Teachers Day, it may also be a good time to bring up things they don’t teach too well. Maths, for example. The year was 1986. I was in the fourth standard. My maths teacher, Mrs Leila Abraham (popularly known as Mrs Cherian because her husband’s name was Cherian Abraham) had just asked us to get an Amul or Cadbury’s chocolate for the next day’s class. She wanted to teach fractions through a bar of chocolate. The idea was exciting enough to motivate a few students to blackmail their parents into giving them what she had asked for. Over the next few days she taught fractions to the class by breaking the bar into half, three-fourths, one-fourths and so on. Even the dullest students picked up the concept very quickly. As my interest in the subject grew, the quality of teachers who taught me went rapidly downhill. The ordeal ended when I graduated with a BSc in Mathematics from St Xavier’s College, Ranchi. The quality of teaching was so bad that before the last class in the third year started I wrote this on the blackboard: Mentally (M) Agitated (A) Teachers (T) Harassing (H) Students (S). An English professor in the college, who was also the best quiz master going around in Ranchi, had come up with this expansion for M.A.T.H.S. Prof Pankaj Chattoraj, who taught us coordinate geometry, among other things, was supposed to take the last class. He was the best of the six professors who taught us. So the joke wasn’t really on him. He took it in a good spirit made a few more jokes, taught what he had to and left. I have no numbers or research to back this but I feel that Maths often ends up being taught by the worst teachers. The impact of bad teaching in mathematics is clearly seen when people have to apply it. Let me share a few examples which I have come across over the last few years. Justice Markandey Katju, in a recent column in The Hindu titled Professor, teach thyself, wrote: “When I was a judge of Allahabad High Court I had a case relating to a service matter of a mathematics lecturer in a university in Uttar Pradesh. Since the teacher was present in court I asked him how much one divided by zero is equal to. He replied, “Infinity.” I told him that his answer was incorrect, and it was evident that he was not even fit to be a teacher in an intermediate college. I wondered how had he become a university lecturer. (In mathematics it is impermissible to divide by zero. Hence anything divided by zero is known as an indeterminate number, not infinity).” Rather ironically, the teacher Katju castigated was right. Any non-zero number divided by zero is infinity. But when zero is divided you get what is known as an indeterminate. The following example should explain things a little better: A2 = A2 A2- A2= A2- A2 A(A-A)= (A-A)(A+A) [A(A-A)/(A-A)] = (A+A) A=A+A A=2A 1=2 In the fourth step of the equation we are dividing (A-A) by (A-A)  and that allows us to come to the  fifth step, i.e. A=A+A and which finally leads to 1=2. Now it need not be said that one cannot be equal to two. When we divide zero by zero we can prove anything. Hence dividing 0 by 0 (which is what A-A is) is not allowed in mathematics. [caption id=“attachment_445033” align=“alignleft” width=“380”] Pic courtesy IBNLive (from the movie 3 Idiots).[/caption] So I guess Justice Katju’s maths teachers either did not manage to get some things through or the good judge was AWOL when the subject was being taught. Justice Katju’s being wrong did not harm anyone and was more confined to the realms of what we can call an esoteric argument. But there are occasions when a lack of basic understanding of maths can lead to totally wrong interpretations. Recently ABP News (formerly Star News) ran a report with a headline “Mahangai ghati kya aapko pata chala kya?”. This was in response to the consumer price inflation falling to 9.86 percent in July against 9.93 percent in June. The report went on to show that how the prices of vegetables and a lot of other goods had actually gone up. So it then questioned: how was the government claiming that prices are down? This again shows the lack of basic understanding of maths. When inflation comes down no government can claim that prices are coming down. What they can only claim is that the rate of increase in prices is coming down. Let me explain this through an example. If the price of a product increases from Rs 10 to Rs 12, we say inflation is 20 percent ((Rs 2/Rs 10) x 100%). Let us say the next month the cost of the product goes up to Rs 13. What is the month-on-month inflation now? The inflation is 8.33 percent ((Re 1/ Rs 12) x 100%). Now the inflation has fallen from 20 percent to around 8.33 percent. But does that mean that price has fallen? No, it hasn’t. What has fallen is the rate of increase in price, not the price itself. This is something very basic which a lot of people don’t seem to understand. On more than one occasion in the past I have been asked by fairly senior colleagues in the media “But why aren’t prices falling, if inflation is falling?". Another common mistake that people make is that they add or subtract percentages.  Take the case of what Jerry Rao (an alumnus of IIM, Ahmedabad, founder of the IT company mPhasis Corporation, and the former head of consumer banking of Citibank in India) wrote in a column in The Indian Express on 6 October 2008. “If stock market wealth drops by 50 per cent in six months, we get concerned. We conveniently forget that it went up by 200 per cent over the previous two years. At the end of 30 months we are still 150 per cent ahead.” (Read the full article here ) At the end of 30 months we are not 150 percent ahead but 50 percent ahead. Let us say an individual invests Rs 100. A 200 percent gain on this would mean that Rs 100 invested initially has grown to Rs 300 ( Rs 100 + 200% of Rs 100). A 50 percent fall would mean Rs 300 has fallen to Rs 150 (Rs 300 - 50% of Rs 300). This in turn means that we are 50 percent (((Rs 150 - Rs 100)/Rs 100) x 100%) ahead and not 150 percent ahead, as was written. So what this means in simple English is that a 50 percent loss can wipe off a previous 100 percent gain. Let us say an investor buys a stock at Rs 50. The stock does well and runs up to a price of Rs 100. What was the gain? The gain was Rs 50 (Rs 100- Rs 50). What was the gain in percentage terms? 100 percent. ((Rs 50/Rs 50) x 100%). After achieving its peak, the stock started to fall and is back at Rs 50. What is the loss from the peak? Of course Rs 50 (Rs 100- Rs 50). But what is the loss in percentage terms? 50 percent ((Rs 50/Rs 100) x 100%). The point is that a 50 percent loss can wipe off a 100 percent gain. Or to flip it around, a 100 percent gain would be needed to wipe off a 50 percent loss. But the example that clearly takes the cake was when a former colleague remarked that the sales of a company she was tracking had fallen by 110 percent. Anyone who understands percentages wouldn’t make a remark like that. Anything cannot fall more than 100 percent (Unless we are talking about things like temperature, which can become negative. Then the concept of percentage becomes meaningless). Let me elaborate. Let us say a product sells 700 units in a month. In the next month no units are sold. What does this mean? It means sales are down by 700 units or 100 percent. On the flip side, when it comes to gain, gains can be unlimited. A product sells one unit in a month and in the next month it sells 71 units, or 70 units more than the previous month. Or a gain of 7,000 percent. Now, theoretically, there is no upper limit to the number of units that the product can sell. And so there is no upper limit to the gains can that can be expressed in percentages. These are a few examples of lack of basic understanding of maths that came to my mind on this Teachers’ Day. The bigger question is why is there such a lack of understanding of basic mathematics? My theory on this is that it all boils down to the way teachers teach mathematics in schools. The entire emphasis is on solving a problem, rather than trying to explain to students why we are trying to solve a problem, and then getting into the nitty gritty. In colleges, it gets even worse. Vivek Kaul is a writer and can be reached at vivek.kaul@gmail.com . After 11 years in school and eight years in college, from all that he was taught the only thing he partly remembers is some elementary mathematics