Think by Analyzing Differences before Generalizing Similarities

Politicians, business leaders, consultants, and motivational speakers like to quote the following sentence from a speech that John F. Kennedy gave in 1959: “When written in Chinese, the word ‘crisis’ is composed of two characters—one represents danger and one represents opportunity.”[i]

The idea that a crisis can create opportunities is a great meme for some people. Yet, others mock it as either political spin or wishful thinking. These opposing views reflect two distinct cognitive attributes, which William James—the first educator to offer a psychology course in the United States—described as (1) “reasoning” and (2) “associative wisdom”:

Reasoning helps us out of unprecedented situations—situations for which all our common associative wisdom, all the “education” which we share in common with the beasts, leaves us without resource.[ii]

A simple example can illustrate these two attributes. A chef in a restaurant receives a meal order from a couple of regular customers. The order is for their standard parmesan-crusted salmon with asparagus, but it sends danger signals through the chef’s brain. He used all asparagus the day before and forgot about it. In the absence of any reasoning, his culinary education cannot help him. The obvious option is to apologize and to ask the customers to order something else. After some thinking, however, the chef decides to offer a new version of the dish, which would be parmesan-crusted salmon with mushrooms and leeks.?

In principle, no two cuts of salmon are the same and no two plates of a salmon dish can be identical, but we ignore these differences as irrelevant. We generalize similarities to say that the salmon cuts are the same. However, we wouldn’t say that salmon with mushrooms is the same dish as salmon with asparagus. This difference matters.

With this difference, the chef is taking a risk. The proposed version of the dish is not on the menu and he hasn’t cooked it for a while. Even if he manages to do it well, the customers may not like it. But it also opens the opportunity to delight the customers with the suggested change. If this happens, it would be an example of what Harry Potter author J.K. Rowling has said: Every success begins with a failure.

Chapter 5 of Trial, Error, and Success is about thinking that can help us out of unprecedented situations, can help us solve problems, adapt to change, innovate, and see opportunities in a crisis. This kind of thinking makes the difference between intelligence and dumb reliance on existing knowledge to act in an unknown situation. The following text is an excerpt from that chapter. You can subscribe through Substack if you wish to receive the whole chapter as a PDF file (for free), or read first the following excerpt from that chapter.

To trigger thinking, spot and analyze differences.

Thinking is disheartening when we don’t know where to start this process. A dumb use of what we know rather well is the easy way out.? The famous Monty Hall problem is an excellent example to illustrate this point.

Inspired by the television show Let’s Make a Deal, hosted by Monty Hall, a professor from the University of California, Berkley, formulated this problem in a Letter to the Editor of the American Statistician in 1975.[iii] The solution of the problem sparked a passionate debate fifteen years later when a reader of Marilyn vos Savant's column in Parade?magazine rephrased the original problem to ask the following succinct question:

Dear Marilyn:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what’s behind the doors, opens another door, say number 3, which has a goat. He says to you, “Do you want to pick door number 2?” Is it to your advantage to switch your choice of doors.[iv]

Consistent with the solution in the original letter, Marilyn answered that the chance of winning is two in three by switching to door number 2, which is higher than the chance of one in three for door number 1. This answer flies in the face of the wisdom associating doors 1 and 2 with the equal head and tail choices in the analogous coin-flipping scenario. To use James’s terminology, this is an example where the associative wisdom is faster but inferior to reasoning.

After posting her answer, and even after offering a follow-up explanation, Marilyn received thousands of patronizing letters from prominent experts. Here is one from a Georgetown University professor, Dr. E. Ray Bobo:

You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. How many irate mathematicians are needed to get you to change your mind?[v]

The chance that Marilyn made the right choice by picking door 1 is one in three and, in that case, she would lose by her decision to switch. On the other hand, the chance that the car is not behind door 1 is two in three, which becomes the chance to win the car by switching from the initial choice of door 1. Ignoring the fact that Monty Hall eliminated the third choice by opening door 3, the choice between doors 1 and 2 appears analogous to the equal coin-flipping probabilities. However, the ignored fact makes a difference because Monty would have opened door 2 if the car were behind door 3, in which case the decision to switch to the then available door 3 would be also a win.

This thinking is from the point of view of a contestant who has one opportunity to play the game, without the possibility to run repeated experiments and to calculate probabilities from their outcomes. However, it’s easy to perform repeated analogous experiments, and that is what Marilyn asked math classes across the country to do. Following her call, many excited teachers wrote about their experience, all confirming that switching resulted in wins in about 67 percent of trials.

Presented with two options, door 1 or door 2, most people tend to ignore the detail that Monty opened door 3. It would be the first step of James’s reasoning to analyze what difference the opening of door 3 makes, but that would require time, effort, and some healthy doubt in math expertise. It would be the first step of slow thinking in the terminology of Daniel Kahneman, a psychologist who won the 2002 Nobel Prize in Economic Sciences, and the author of Thinking, Fast and Slow.[vi] This process would have to evolve slowly through back-and-forth neuron firing in the brain’s grey matter. Richard Thaler and Cass Sunstein, the authors of Nudge,[vii] assign this process to what they call “reflective system” of our brain.

In contrast to thinking about the difference that opening of door 3 could make, the strength of the wisdom associating doors 1 and 2 with the familiar head and tail choices leads to what Kahneman calls “fast thinking.” In this case, our “automatic system”—in Thaler and Sunstein’s terminology—executes the fast, effortless, and wrong decision that the probabilities are equal.?

The strength of knowledge, residing in our automatic system, makes it easy to ignore relevant differences and, consequently, makes it harder to both think and learn. It can make learning so unlikely that even pigeons learn faster when presented with Monty-Hall type of choices; and they do it not because they are smarter than humans but because their small brains are less polluted with strong knowledge.

In 2010, psychologists Walter T. Herbranson and Julia Schroeder published a paper entitled “Are Birds Smarter than Mathematicians?”[viii], which investigated whether pigeons, like most humans, would fail to maximize their expected winnings in a version of Monty-Hall choices. They used three pecking keys and a feeder through which birds could gain access to mixed grain. At the beginning of a trial, a computer randomly linked one of the three keys to the feeder and illuminated all three keys with white light. After the bird pecked any one of the lit keys, the computer followed Monty-Hall’s scheme to deactivate and darken one of the keys, in analogy with the door opening in the game. The key that the bird pecked and the additional active key were then illuminated with green light. If a bird then pecked the key that the computer selected initially as the prize key, it was given a short access to the mixed grain before starting a new trial.?

For a comparison, Herbranson and Schroeder performed analogous experiments with undergraduates at Whitman College, where the two were on the faculty. The students were not informed about the Monty Hall problem.

The results are unambiguous. At the beginning, the pigeons switched from the initially pecked key about a third of the time. As the trials proceeded, the pigeons learned that switching from the original choice resulted in a better chance to access the grain.

Not knowing anything about the Monty Hall problem, the students’ initial rate of switching was close to the coin-flipping probability of 50 percent. The knowledge that the choice between two illuminated keys is analogous to coin flipping seemed to inhibit students’ learning. Unlike the pigeons, whose switching rate was nearly 100 percent, the students only improved slightly, reaching just 66 percent.

Like most other people, the students kept ignoring the relevant difference between coin-flipping probabilities and Monty-Hall problem. Yet some differences matter and the first step in a thinking process is to identify them. After that, the process is to decide how to deal with them.

The decision what to do with a relevant difference—once it’s identified—is often obvious. The following hypothetical example illustrates again that the most important thinking trick is not to ignore a relevant difference in the first place. In the example, we have a restaurant manager who wants to quantify the demand for three different types of salad dressings offered by the restaurant. Recorded choices of recent customers show that a hundred people selected option A, seventy picked option B, and thirty went for option C. A side note stated that two customers asked for a reduced-salt dressing, even though this possibility was not on the list. These two unexpected responses correspond to one percent of the two hundred recorded selections. However, this mathematically correct calculation does not mean that only one percent of customers would order a reduced-salt dressing if offered in the future. A proper analysis of the difference that reduced-salt options could make is likely to show that offering these options would be attractive to a significant number of customers. The manager would be ignoring a signal for business opportunity by interpreting the two unexpected responses as only one percent of customers’ choices.

Subscribe through Substack if you wish to receive the whole chapter as a PDF file (for free).

Here are the titles of the remaining sections in Chapter 5 of Trial, Error, and Success:?

·?????? Do not ignore differences that can make a big difference.

·?????? Distinguish generalizations with and without specific meaning.

·?????? Utilize feedback to improve future trials.

·?????? Think positively about negative feedback.

[i] John F. Kennedy Speeches, Indianapolis, Indiana, April, 12, 1959, in John F. Kennedy Presidential Library and Museum; https://www.jfklibrary.org/Research/Research-Aids/JFK-Speeches/Indianapolis-IN_19590412.aspx

[ii] W. James, The Principles of Psychology, vol. 2 (New York: Dover Publications, Republished Edition, 1950), p. 330.

[iii] S. Selvin, “A problem in probability,” The American Statistician, vol. 29, p. 67, 1975.

[iv] M. vos Savant, The Power of Logical Thinking: Easy Lessons in the Art of Reasoning… and Hard Facts About Its Absence in Our Lives (New York: St Martin’s Griffin, 1997), p. 6.

[v] M. vos Savant, “Game show problem”; https://marilynvossavant.com/game-show-problem/

[vi] D. Kahneman, Thinking, Fast and Slow (New York: Farrar, Straus and Giroux, 2011).

[vii] R.H. Thaler and C.R. Sunstein, Nudge: Improving Decisions about Health, Wealth, and Happiness (New York: Penguin, 2009).

[viii] W.T. Herbranson and J. Schroeder, “Are birds smarter than mathematicians? Pigeons (Columba livia) perform optimally on a version of the Monty Hall dilemma,” Journal of Comparative Psychology, vol. 124, pp. 1–13, 2010.

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