Adding the Dunning-Kruger variable in the mathematics classroom

In school we have been taught that 1+2 = 3 and that ¾= 0.75. Ok, so far everything seems more or less fine. That is, if in a regular classroom the teacher is explaining the subject of real numbers and asks a couple of questions about addition and multiplication, the students, for their part, raise their hands quickly trying to beat the classmate sitting next to them and answer without waiting for the teacher to give them the opportunity to intervene. Of course, in this regular class, whoever answers the fastest and has the greatest number of interventions is generally evaluated as something like a wiseguy, like a math genius.

But does the same thing happen when we get into a rougher topic, say square roots? It seems that the average student has forgotten at this point how to obtain the square root of, for example, 26, and prefers to remain silent before stopping to mentally calculate the result, he would even prefer to have his cell phone at hand to somehow obtain the answer.

Why does the student prefer to answer more or less obvious questions with exquisite speed instead of taking the time to calculate a slightly longer problem? The answer is not really in the field of mathematics, it lies in the realm of the unknown.

Some devastating phrases frequently found on the internet are “According to a study rude people are more intelligent and direct”, “Eating certain foods favors a flat stomach”, “The Earth is flat”, “Millionaire gives away his entire fortune, click here” , “Science is incapable of solving this mystery of nature”¸ most of them imply that the reader, or the student, does not make a greater effort of analysis and that information is taken as true.

Leaving aside certain sentimental blackmail in these phrases, we must recognize their impact on our existence. Some of us might wonder how we are able to believe such phrases, but we must also accept that almost all of us are susceptible to not performing an analysis. So not only in the classroom we are attracted to the idea of not thinking and calculating, but we extend it to daily life, so we validate stories because so-and-so said them or because he saw them on a TV show.

Let’s return to the classroom with the dynamics of whoever answers faster and more often “earns more points”, which makes us feel happy, fulfilled, stepping on known ground. This dynamic leads us to overestimate our abilities, generating an effect in which we believe we have sufficient knowledge on various topics. This effect was analyzed by the psychologist Dunning-Kruger , who shows that the less knowledge we have in a field, the more we tend to believe that we know a lot.

Dunning–Kruger effect . Taken from https://citadinamx.com/efecto-dunning-kruger/

People who have more knowledge are more aware of what they ignore, in this sense Dunning-Kruger ‘s statement recovers Socrates’ phrase “The only true wisdom is in knowing you know nothing.” But why does this phenomenon of rapid response to the simplest thing occur? The rapid response phenomenon in the classroom is due to the survival trend of the species itself, based on the available information, which does not imply a sensible analysis.

Recently a group of students approached me asking for help with a somewhat simple mathematical problem (in terms of writing), but it had been a week since they had found the solution: If a zombie were released in any city on the first day and this bites three people the next day, turning into zombies, and thus, how much time is required to convert a city like Queretaro, then the time needed for the entire country, and already in the midst of the apocalypse, the entire world.

Apparently said teacher did not explain the subject of exponents before, so the students not only fought against time but their entire group in their desperation, looked for the most logical way out but no function known to them gave them the solution.

I decided to intervene and support them by explaining that the function is exponential and we proposed it like this

x=z(z+3)^t

We use z to indicate the initial condition of the function, in this case 1 zombie , multiplied by the transformation variable in which each zombie bites three inhabitants, all raised to time as an exponent, that is, the Monod equation .

Monod equation . Wikipedia

Interesting, so far the boys found a small treasure that brought back their joy and goodbye to the tension. Approximately 10 days and 34 minutes are required to turn the population of Querétaro into zombies ; and thirteen days to consolidate the conquest of Mexico.

When they presented the answer to the Mathematics teacher, she replied that it was more or less fine, but that they should simplify the function, in this way:

=(4)^t

Completely ignoring that the standard function is not this, and the initial condition indicated by z should not be simplified, is a gross error. Again I had to explain to the students the importance of the initial condition in z, what would happen if we had two or three original zombies ?

Far from the exponential function, in this story the phenomenon of cognitive bias in mathematics begins to appear. On the one hand, the teacher who has already generated a certain profile of rejection and acceptance at her students’ discretion, on the other, the profile of the group inversely proportional to what the teacher knows about the subject. Apparently we have two different circumstances, but they can be simplified into a single variable, Cognitive bias.

We all like to be in scenarios where thinking fast and responding even faster is the best response to survive. Students can quickly answer a basic question, but when pushed, they tend to neglect mathematical analysis. On the other hand, said professor also acted quickly in her response by simplifying the initial condition z. Psychologist Daniel Kahneman explains it thoroughly in his study “Thinking Fast, Thinking Slow.” In this study, Kahneman explains that when solving problems or making decisions, two systems operate in our minds, System 1 and System 2.

System 1 and System 2, Daniel Kahnemann .

System 1 has predefined response systems, acts very quickly, is guided by intuition, working most of the time. System 2 analyzes things carefully but requires more time, and therefore more energy. Both systems are just two ways to deal with a particular situation.

When we answer the math teacher the equation 2 * 4=8 we have System 1 in action. But when we are asked for the square root of 26 then we have System 2 working.

Of course, like most people, the square root question takes longer, you even made the important decision to answer it or not, and you probably take the last option.

System 1 is very fast and works most of the time, it is based on intuition and the possibility of surviving the problem. It is much more convenient to react quickly than to be right. Its origin lies in prehistoric times when the best defense was to immediately seek protection at the slightest sign of danger, protecting the species from extinction. System 2 asks us to stop to think and analyze the circumstances but we may excuse ourselves from doing so. In this sense, Cervantes de Saavedra himself shows us how both systems work in unison, while Sancho is System 1 and Don Quixote represents System 2, when facing the situation of the frightening noise of the fulling mills, in chapter XX:

“ And there were -if you haven’t, oh reader!, out of sorrow and anger- six fulling mallets, which with their alternate blows formed that din.”

Don Quixote and our good Sancho have spent a whole night without sleeping due to the frightful noise that filled the dark night, making them imagine all kinds of monsters and beasts.

In short, our irrational fear of facing the unknown, the engine of an expeditious response with the little knowledge we have, is the result of evolution itself.

How far can System 1 take us? Solving everyday problems with this system we find ourselves in a field where mistakes are continuous and more or less accepted.

Now, how is this phenomenon integrated in the field of mathematics? Let’s meet Antonio, he is 22 years old, studious, hard-working, introverted, dreamy, imaginative, likes to read, likes to draw, and is orderly.

What is more likely? Option A: Antonio is a bank teller; Option B: Antonio is a bank teller and he loves Star Wars .

Which option have you chosen? Option B? Congratulations, you have put to work System 1, rapid response; if it was Option A, mostly congratulations because you stopped to think using System 2. This is a mathematical truth.

intersection of two sets. Wikipedia

The intersection of these two sets represented by Antonio who is a fan of Star Wars (set A) and Antonio who has no fans (set B) is less than or equal to either of the two sets A and B.

This type of thinking in which the rapid response System 1 is present is called Cognitive Bias. And there are a huge number of examples in daily life:

a) Halo effect. It happens when we judge the totality of a person, based on the first characteristics that we know about him. When we meet an attractive and neat person we think that he is also kind and trustworthy. In the case of the classroom, if a student gets poor grades at the beginning of the course, it is very likely that the teacher will continue with a very harsh evaluation even if the quality of their work improves. If we like or dislike a politician from the beginning, it is very difficult for his future actions to change our point of view.

There are even studies indicating that this cognitive bias leads us to evaluate as True through system 1 the quality of written texts if they are made with a certain type of clear, large and formal letter, comparing it with statements written with small letter not formal.

Which of these two statements is true?

- World War II ended in 1943

- World War II ended in 1946

Answer: neither. World War II ended in 1945.

b) confirmation bias. It occurs when we look for information that reinforces our mental schemes and ignore that which contradicts them. In the classic example of a certain social group prone to violence, we will more easily remember the news where that group was violent and we will ignore those in which they were generous or responsible.

Currently the search algorithms of Youtube, Facebook, among others, occupy System 1, so they keep us sitting in this comfort zone, presenting information that more or less fit our search profiles.

c) Drag effect. It occurs when we adopt an opinion only because many people do. Kahnemann states that “for some major beliefs we have not the slightest evidence other than that the people we believe and trust hold those beliefs.” People can maintain unshakable faith in a claim, no matter how absurd, if they feel supported by a community that supports them in that regard.

d) Loss aversion. We are very reluctant to lose anything we have or have invested in. Would you be willing to enter a business where you are “offered” 90% of losing $50? Or would you accept the opportunity to enter a business where you buy a ticket for $50, but there are only 10 numbers and whoever wins gets $1000?

Obviously the second option, but have you noticed that it is exactly the same, except for the wording? The change in conceptualization will help change our attitude to loss, calling it investment, support, reward.

Affirmation of the consequent. Error present in the field of logical fallacies, for example the affirmation of the consequent, in which it starts from a true premise, for example all fish have gills, the sole is a fish, therefore the sole has gills. So far everything is correct; but what if we apply the same premise to octopuses, fish have gills, octopuses have gills, therefore octopuses are fish, wrong conclusion.

False Causality. When we think that two events happen one after the other, then the one that happened first must be the cause and the one that happened second after the consequence. Most of the time our minds work by concatenating events of this type, so we fall into this fallacy.

The spurious correlation appears when we relate a certain object or action with another, for example, we think that wearing a certain jersey of our favorite team brought him victory that day; or the relationship of a certain medicine to counteract infection or disease at a given time, without having been prescribed by the doctor.

To validate the studies of the various relationships between events, scientists rely on statistics, comparing groups and controlling the factors that may or may not influence the effects obtained.

Finally, one of the most dangerous features is the false dilemma. The mind works in a binary sense, if you are not with me you are against me. It probably works in certain situations, but not in all. The truth is that it is one of the most dangerous statements. “If you are not with us, you are in favor of those who attack us”, “Do you join the game or not?”, the only option that saves us from this dangerous situation is to propose other options.

Finally, the various strategies that we have decided to survive in the classroom have formed our student profile, we have decided to imitate those who only responded easily to simple problems. Or we have taken the path of questioning, proving again and again the various conclusions of mathematical hypotheses and problems. Definitely the second option is the least taken up today.

In summary, the next time you are faced with a mathematical problem, try to take a few minutes to analyze it, try the solution several times and find those numbers for which the equation is valid and those that it is not, and then you will know that you have started System 2.

Bibliographic references

(Cervantes Saavedra, Biblioteca Virtual Miguel de Cervantes, 2001)

Cervantes Saavedra, Miguel de. Don Quijote of La Mancha. Chapter XX. Edition by Florencio Sevilla Arroyo. Alicante, Spain. Miguel de Cervantes Virtual Library, 2001. Retrieved from https://www.cervantesvirtual.com/obra-visor/el-ingenioso-hidalgo-don-quijote-de-la-mancha--1/html/ff31184c-82b1-11df -acc7–002185ce6064_222.html

Intersection of sets. Wikipedia. April 13, 2022. Retrieved from https://es.wikipedia.org/wiki/Intersecci%C3%B3n_de_juntos#:~:text=En%20teor%C3%ADa%20de%20juntos%2C%20la,a%20los %20sets%20of%20game.

Kahnemmann , Daniel. Wikipedia. April 13, 2022. Retrieved from https://es.wikipedia.org/wiki/Daniel_Kahneman

Dunning–Kruger effect . Edgar Larios at Citadinamx . April 13, 2022 Retrieved from https://citadinamx.com/efecto-dunning-kruger/