What happens when Mentally Agitated Teachers Harass Students (M.A.T.H.S)

Vivek Kaul

It ain’t what you don’t know that counts. It’s what you know that ain’t so – Will Rogers
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 a Cadbury 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 to get 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.
Professor Pankaj Chattoraj who taught us co-ordinate 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 ends up being taught by the worst teachers. The impact of bad teaching of Mathematics is clearly seen when people have to apply Maths.
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)
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.
So I guess Justice Katju’s Maths teachers did not teach him the right thing here. 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% in July against 9.93% in June. The report went onto show that how the prices of vegetables and a lot of other goods had gone up. So it then questioned that 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 a product increases from Rs 10 to Rs 12, we say inflation is 20% ((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% ((Re 1/ Rs 12) x 100%). Now the inflation has fallen from 20% to around 8.33%. Does that mean that price has fallen? No it hasn’t. What has fallen is the rate of increase in price, not the price.
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 October 6,2008.
“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% ahead but 50% ahead. Let us say an individual invests Rs 100. A 200% gain on this would mean that Rs 100 invested initially has grown to Rs 300 ( Rs 100 + 200% of Rs 100). A 50% fall would mean Rs 300 has fallen to Rs 150 (Rs 300 – 50% of Rs 300). This in turn means that we are 50% (((Rs 150 – Rs 100)/Rs 100) x 100%) ahead and not 150% ahead, as was written.
So what this means in simple English is that a 50% loss can wipe off a 100% 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%. ((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% ((Rs 50/Rs 100) x 100%).
The point I was trying to make was that a 50% loss can wipe off a 100% gain. Or to flip it around, a 100% gain would be needed to wipe off a 50% loss.
But the example that clearly takes the cake was when a former colleague remarked that sales of a company that she was tracking had fallen by 110%. Anyone who understands percentages wouldn’t make a remark like that. Anything cannot fall more than 100% (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%.
On the flip side when it comes to gains, they 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 7000%.
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 lack 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.
So it’s time we stopped respecting our Maths teachers and re-title them as Mentally Agitated Teachers Harassing Students.
(The article originally appeared on www.firstpost.com on September 5,2012, with a different headline. http://www.firstpost.com/living/what-your-maths-teacher-didnt-teach-you-at-school-444727.html)
(Vivek Kaul is a writer and can be reached at [email protected]. After eleven years in school and eight years in college, from all that he was taught the only thing he partly remembers is some elementary mathematics)

People prefer 50% free to 33% lower price…

Here is a question. You invested your hard earned money in stocks. Your investments rose in value by 100%. Then things turned around and stock prices fell by 50%. How much money did you gain in the end? Before you jump up and say 50%, just hold on. You have ended up where you started. Let’s say you invested Rs 1 lakh to start with. A 100% gain on that meant that now your investment is worth Rs 2 lakh. A 50% loss on this meant that you were back at Rs 1 lakh. A 50% loss had wiped out a 100% gain. “People are not very good at performing arithmetic with complex forms such as logarithms, fractions, probability and percentages, because, for evolutionary reasons, the human brain has not evolved to perform these functions,” says Akshay R. Rao who holds the General Mills Chair in Marketing at the Carlson School of Management, University of Minnesota. In recent research which has received widespread international attention, Rao and his colleagues found that shoppers prefer getting something extra free to getting something cheaper. This happens because of their inability at handling percentages. Rao talks about this and more in an interview with Vivek Kaul.
In recent research you found that “Shoppers prefer getting something extra free to getting something cheaper.” How did you come to that conclusion?
Our research examined the phenomenon of bonus packs in which the consumer gets a larger quantity for the same price. We contrast this offer with a standard price discount, where the consumer gets the same quantity for a lower price. Imagine that I am selling coffee beans, and I offer you 100 beans for Rs. 100 on a normal day. Then, one day, I offer you a 33% discount, so you receive 100 beans for Rs. 67. On another day, I offer you 50% extra (or free). You now get 150 beans for Rs. 100. But, I impose no limit on how many or how few coffee beans you can buy, on either day. So, on the day in which I offer 50% extra, you could quite easily have bought 100 beans for Rs. 67! Yet, most people prefer 50% more to a 33% lower price, even though the two options are economically equivalent! In fact, we find that when we offer 33% more and a 33% price discount (which is economically superior), people are indifferent.
Can you give us an example?
In India, particularly for products that are sold in bulk (such as dal, rice, cooking oil etc.) in the unorganized retail market, this tendency on the part of consumers to prefer free products is likely to be successfully employed by the retailer. In our research, we were able to increase sales of an inexpensive consumer packaged good by over 70% in a retail store, when we employed the extra/free bonus pack format relative to the price reduction format.
What sort of experiment did you carry out?
As I mentioned above, we conducted one experiment in a retail store, in which we varied the promotion format for one product (hand lotion) each week. All the other products in the store were not promoted, providing us a control for comparison. We measured sales volume during each of the weeks to compare consumer response to format variation, and found, as predicted, that offering a quantity increment yielded substantially higher sales than offering a price discount that was economically equivalent. We followed up this naturalistic study with a survey of adult consumers in a shopping mall, asking them to express their preference for options that were either reduced in price or featured an increase in quantity. We also asked our respondents to respond to some simple maths questions, to assess their computational skills. Again, as we expected, we found that those with better maths skills did not display this error, while those with poor maths skills displayed the erroneous preference for quantity increments.
Finally, we also showed that the error occurs for harmful as well as beneficial changes — people prefer a quantity decrease of 33% relative to a price increase of 50%, though both are economically equivalent.
What do people behave in this way?
Essentially, we demonstrate that this occurs because of “base value neglect” when dealing with percentages, a phenomenon akin to “denominator neglect”, a term coined by the illustrious psychologist Paul Slovic. According to this human tendency, people are not very good at performing arithmetic with complex forms such as logarithms, fractions, probability and percentages, because, for evolutionary reasons, the human brain has not evolved to perform these functions. For existence and survival, to find food and avoid becoming prey, we are quite successful as a species if we operate as “frequentists”, that is number counters. Hence, people treat percentages as whole numbers and make predictable errors in computation.
Do companies already realize that shoppers prefer something extra free rather than getting something cheaper?
Companies intuitively use some of this logic, but I am not sure they have thought this through the way we have. (If they had, our paper would not have been novel and would probably not have been published!). However, now that our paper has been published and has received widespread attention in the business press, I expect that companies will start experimenting with our results to assess whether and when they can profitably employ our theory and findings.
You have in the past said “errors in peoples’ judgments of the net effect of multiple price discounts on the same product or on different products in a bundle have implications for a variety of marketing settings including advertising, promotion, pricing and public policy”. Can you explain this in detail to our readers?
A classic problem in numerical competence with regard to the processing of percentage information is how people process multiple percentages. Think of the following example which first appeared in The New York Times and was quoted in the bestselling book How To Lie With Statistics (Huff 1954, 111):
“The depression took a stiff wallop on the chin here today. Plumbers, plasterers, carpenters, painters and others affiliated with the Indianapolis Building Trades Unions were given a 5 percent increase in wages. That gave back to the men one-fourth of the 20 percent cut they took last winter.”
A little thought will show that the maths is wrong here. If the workers were making $100 at the beginning, and experienced a 20% cut, their wages had dropped to $80. A subsequent 5% increase constitutes $4, which is one-fifth, not one-fourth of the original wage cut! Even the venerable New York Times makes maths errors!
That was an interesting example!
Now, take this example to the marketplace. Imagine that a store offers a 20% off Diwali sale, and offers an additional 25% off on Diwali sweets. What is the total discount? It is not the sum of the two percentages (45%), it is actually, only 40%! But, people systematically ignore the base value and add up percentages as if they are whole numbers. The problem becomes even more interesting when there are gains and losses. Imagine if your stock portfolio goes up by 40% and then declines by 30%. You might think you are still better off from where you started, by 10%. But, you would be wrong — you are actually worse off by 2%!
So what are the practical applications of this?
The application of these errors in advertising, promotion and pricing should be obvious. Consumers can be tricked by stores into thinking an offer is better than it actually is. From a consumer welfare standpoint, this is obviously not a good thing. So, we suggest that, the scientific insight we offer can be used to improve consumer welfare, by the introduction of regulations to require purveyors of numerical information to present absolute as well as percentage information. Particularly with regard to consumer finance (credit card interest rates) or the petrol consumption improvement of a car, it is possible for consumers to be fooled by multiple percentage changes that appear beneficial, when they are in fact harmful.
In one of your articles you discuss a study which points out: a) price may exert a non-conscious influence on expectancies about product quality b) such expectancies may have an impact on actual product performance c) such expectancies can also be induced through non-price information such as advertising claims about product quality. Can you explain this in detail?
This was a fascinating study authored by Baba Shiv of the University of Iowa (now at Stanford) and his colleagues Dan Ariely and Ziv Carmon, about which I was invited to write a commentary. The essence of their research was as follows. Some people were given a drink that was claimed to enhance mental acuity. Half the people were told that the product was purchased at $1.89 while the other half were told that the product, priced at $1.89 was purchased at a discounted price of $.89. Then, after they had drunk the liquid, they were given puzzles to solve. The group that drank the full-priced product solved significantly more puzzles (9.1) than the group that drank the discounted product (7.7)! Seemingly, people who drank the discounted drink expected that it would be less efficacious at increasing their mental acuity, and performed relatively poorly on the puzzle solving task! Their expectations sub-consciously influenced actual product performance. This is a powerful demonstration of the price-quality effect, a phenomenon that I have studied extensively over the last 25 years.
What is the learning for marketers here?
The remarkable thing about this research is the power of the placebo (as the authors term it). This placebo effect suggests that the human brain can be fooled into performing because it expects to perform. The implicit argument is that in many instances, psychology may be more important than engineering, in product design. The obvious learning for marketers is that consumers are subject to many subtle influences that have an impact on their subjective experiences. They may not be entirely rational and the creative and astute marketer is able to identify and exploit opportunities to influence consumer behavior through clever marketing.
Car dealers frequently draw customers into their establishment with the promise of an attractive advertised deal. However, upon arrival, the car buyer discovers that the deal does not apply to the model he wishes to buy. Nonetheless, after a few minutes of consultation with a ‘‘sales manager,’’ the salesperson returns with the news that an exception has been made and the deal has been approved. The buyer is relieved. Why does the dealership not simply offer the deal on the buyer’s preferred model in the first place?
The answer to this question lies in the psychological impact of the consumer experiencing relief. It is rooted in the “Good news, bad news” sequence phenomenon. People generally prefer to end on a high note (there is a good reason that dessert comes at the end of a meal!), and prefer to get bad news first and good news later, rather than the other way around. So, imagine you are boarding an international flight and reach for your passport in your pocket. It’s not there. Panic ensues. You search frantically, and find it in your briefcase. You experience an immense sense of relief because of good news following bad; this sense of relief would not have been experienced had you found your passport when you first reached into your pocket. It turns out that, when you experience this relief, you are psychologically vulnerable. Your cognitive resources to process bad news has been depleted, so your tendency to be loss averse increases. So, when a car dealer tells you that your preferred deal is not available, and then subsequently tells you that it may still be available, you experience relief and your desire to accept the deal increases, because you don’t want to lose the car. You negotiate with less vigor and wind up accepting a deal that is less advantageous to you than if you had not been taken through the bad news, good news roller coaster in the first place.
Companies often indulge in price wars when more often than not they turn out to be a race to the bottom. So why do they do it in the first place?
The impulse to fight on price is driven by many factors, not least of which is that the price variable is easy to change. Remember how Usha used to advertise in the 1980s – their slogan was “Massive Price Cut!” Dropping price in response to a competitor’s price cut is easier, particularly in the short-run, than engaging in creative actions such as promoting benefits, emphasizing your brand’s trustworthiness, alerting consumers to the risk of purchasing low-priced options that may perform poorly, and so forth. In my research and consulting work with companies in the U. S., Europe and Asia, I have found that most managers have been taught and instinctively feel that price is the most potent weapon in the marketplace. Therefore, it becomes the weapon of first resort — it is easy, available and often measurable (through changes in market share).
One sector that has been completely destroyed over the years because of price wars is the airline sector. But they still seem to have not got the point. How do you explain that?
In the year 1992, following a recession, there was a mad scramble in the airline industry in the U. S. to acquire market share. The easiest way to do so was to cut prices. And, it worked. Leisure travel increased that summer, revenues went up, but profits declined. Many experts have analyzed this continuous emphasis on price in the airline industry. At the present time, thanks to the internet and travel sites that routinely search for low-priced options, consumers have been taught by the industry to engage in price-comparison shopping. This is a phenomenon that is here to stay. Airlines have employed other means to enhance loyalty and revenue without suffering on price, through frequent flyer programs, through additional fees (such as baggage fees), but the fact remains that coach/economy travel is much less profitable than business and first class travel, particularly on international routes. Now, the managerial mindset is that price is the principal means of attracting and retaining customers.
You also write that “Managers can localize a price war to a limited theater of operation – and cut down the opportunities for the war to spill into other markets”. How do they go about doing that?
Price wars need not be global. They can be limited geographically, or to certain segments, or to certain product categories, at certain times, and so forth. Smart companies realize this. Returning to the airline industry, when a low price competitor enters a market, it offers inexpensive flights to a limited set of locations, and at certain times. The incumbent would be smart to cut prices on flights that cater to those destinations and at those times, at which the low-price competitor operates. It is not necessary to cut prices across the board. This is what Northwest Airlines did when a small rival — Sun Country — entered the market. To counter Sun Country, Northwest dropped its fares on the Minneapolis-Boston route, on flights that operated between 7:10 am and 11:10 am. That was the time at which Sun Country operated its Minneapolis – Boston flight.
Any other example?
Or, consider another creative strategy that is based on an intimate understanding of consumer price sensitivity. 3M used to make diskettes. It faced a new low-priced competitor — a Korean brand called Kao — and was faced with the prospect of a price war. Instead of dropping the price on 3M diskettes, it launched a “flanking brand” called Highland, at a low price to match Kao. Now, price sensitive customers had the choice of Highland and Kao, while quality sensitive customers stayed with 3M. This was a more profitable strategy than simply dropping the price for 3M diskettes, even though the Highland diskettes came off the same shop floor! Simultaneously, 3M signaled to Kao its intent to protect its market, and Kao eventually withdrew.
The interview appeared in the Daily News and Analysis on July 16,2012.
(Interviewer Kaul is a writer and can be reached at [email protected])