An ordinary person has to make up to 35,000 trivial decisions every day. Two hundred twenty-seven of them are about food alone. If you make the wrong choice, it is not a significant event in your life if it comes to your coffee; you make a face and move on. What about your decision to choose your soul mate, health, or work? Errors in decision-making lead you in the wrong directions. Your choices can irreversibly impact not only that day but also the years after.

Some choices can be easier than others. Let’s say there is a bicycle behind one door and an expensive car behind the other. Before asking you which door you would choose, you also know which door the bicycle is behind and which door the car is behind. There is a bicycle at door A and the car door is B. Now let’s ask our question: Which door would you choose? A or B? The answer is pretty obvious: of course, you would choose door B. 

Let’s make the decision a little harder: things might not be as they seem. We have doors A and B again. There is a possibility that there is a bicycle behind door A and a car behind door B. There may or may not be a bicycle and a car behind doors. I don’t think your choice will change. In the end, the chance of them existing or not existing is the same, but because the car’s worth outweighs the bicycle’s worth, it is the most sensible decision.

We can prove what we said by calculating the expected value in probability. First considering door A: there is either a bicycle behind it or there is nothing. The chance of getting a bicycle is ½, and let’s say the bicycle is worth $100; the profit will be $100. If there is no bike behind door A, the probability of this happening is ½ again, but there is no profit this time. In statistics and probability analysis, the expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values. So, expected value in this scenario is calculated as:

E(A) = $100 x (1/2) + $0 x (1/2) = $50.

The expected value of door A is $50. The expected value for door B is also considered and calculated in the same way. Let’s say the price of the car here is $100,000. From the same formula:

E(B) = $100,000 x (1/2) + $0 x (1/2) = $50,000.

The expected value of door B is a thousand times that of door A. If a choice is to be made between two doors, the wise choice will undoubtedly be door B.

Pascal’s stake is very similar to this game of chance. Blaise Pascal was a scientist and mathematician who lived in the 17^th century and developed his own theory of probability. His theory was quite successful. Using his own approach, he showed the existence of God and the necessity of believing in God. Instead of the door metaphor we used, he used two people, one who believed in God and one that didn’t believe in God. We assume that there is a fifty percent chance of God’s existence—that God either exists or he doesn’t. People that believe in God would pass through door B, similar to the example above. Under these conditions, if a person believes in God and God doesn’t exist, they lose or gain nothing. If God does exist, the believers go to Heaven and achieve infinite happiness. When we formulate this situation, we find out that the expected value of believing in God is infinite. We consider the probability of not believing in God is fifty percent, and there is no gain if it is true. On the other side, the existence of God is also fifty percent, and one gains infinity of happiness. Therefore, the expected value of believing in God, E(Believing), is calculated as above in the car/bicycle example:

E(Believing)= 0 x (1/2) + ∞ . (1/2) = ∞

Remember that half of infinity is also infinite.

So, even if God doesn’t exist, the faith that the believer has is priceless.

For those people who choose door A (not believing in God), there is nothing so beneficial for them. Even if they are correct, there is no gain for them in the end. If there is no God, then there is no afterlife. They receive finite pleasure from this life: choosing the bicycle still has some value but choosing the car is way more sensible. What if the other possibility happens? If a person does not believe in God and God exists, they will be deprived of all believers will get. When we think of the second option, that is door A, we formulate the expected value of not believing as:

E(not Believing)= 0 x (1/2) + (-∞) x (1/2) = -∞

Not believing in God also has infinite value but in the opposite way. You are in debt, bankrupt.

One of our choices is God existing, which is fifty percent. Someone may say that it is a high probability. So, despite all the evidence to believe in God, what if we reduce the probability of His existence to one in a thousand? Then the chance that God does not exist is 999/1000. In this case:

Probability of God not existing: 999/1000 and no gain.

Probability of God exists: 1/1000 and the gain is infinite.

E(Believing)= 0 x (999/1000) + (∞) x (1/1000) = ∞

As can be seen, the expected values do not change. Likewise, an atheist will get the same negative infinity. According to possibility calculations, belief in God means an infinite gain in any case. According to mathematics, believing in God, even if the likelihood is one in a googolplex, is still profitable. This topic leaves no room for choice. The chance to gain infinite value is infinite, and the value you lose is finite or, more likely, non-existent. There is no doubt that you have to give everything in this game of life that you are obliged to play. There is no other way to end it than to risk your finite life to win eternal life. Preferring the other option is like resigning from one’s mind.

On the other hand, you may say that the profit is uncertain, and everything is left to luck. But when considering the infinite distance between the certainty of the misery endured and the uncertainty of gain, it makes no sense to argue that the finite life we have should be used to endanger the infinite gain. Everyone who enters a game of chance tries their luck by putting in something sure, but it is always doubtful whether they will win. He puts something limited and negligible in danger without offending the mind. As Pascal states, “Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.”

The following story is related to Pascal’s argument. Someone came to Ali, the Caliph. The man denied resurrection, reckoning in the hereafter, heaven, and hell. He asked Ali:

“Ali, you believe in hereafter; we don’t. You worship, spend a lot of money, and go through the trouble to be saved from hell and enter heaven. Is it worth it? And how do you know that there will be a resurrection after death?”

Ali (R.A.) listened to the man calmly, then gave him the following answer:

“Let’s first assume that what you say is true. If there is no afterlife, we are in the same situation as you. We are even, and there is no gain for you or us. In the meantime, our prayers, good deeds, good morals, and charity we give for the sake of God will not harm us. On the contrary, it will be beneficial for improving the community. But what if there is an afterlife, and what we say turns out to be true? What will you do?”

As an instructor of statistics, I sometimes cannot help but explain the phenomena of the social sciences through mathematical expressions. This is especially true when there is almost complete correspondence between the two. One such topic is the “powers” of ignorance. When I say “powers,” I obviously do not mean how “mighty” ignorance makes a person. I mean the exponential expressions of ignorance:

Ignorance to the power of one (ignorance¹);

Ignorance to the power of two (ignorance²);

Ignorance to the power of three (ignorance³).

Powers of ignorance

Let us start with the simple definitions of these three powers:

Ignorance¹: Does not know;

Ignorance²: Does not know AND does not know that s/he does not know;

Ignorance³: Does not know AND does not know that s/he does not know AND insists that s/he does know.

Among the three, the first power is the most powerless in terms of leading the person to mistakes. The person is aware that s/he does not know. The solution to this case is simple: the person would go after the needed knowledge and acquire it, remedying this form of ignorance.

The second power is not that bad, either. The person does not know AND is not aware of the knowledge s/he lacks, either. Therefore, this person, when alerted to the necessity and existence of the knowledge in question, would acquire it, successfully ending the ignorance. In this sense, this power is also not that powerful in terms of leading the person to mistakes.

The third power is the most powerful at fooling its owner. The person does not know AND does not realize that s/he lacks certain knowledge. What is more, when alerted to the necessity and existence of the knowledge in question, s/he insists that s/he has nothing to learn. As you can tell, the ignorance cannot be eliminated easily—if at all—in this case. Such ignorance cannot be overcome by alerting the person to the knowledge in question and/or providing it. This is an attitudinal problem that “guards” the person against learning.­ ­

Now, let us return briefly to the math. Just as the magnitude of the difference between the consecutive powers of a mathematical entity increases as the powers increase, the differences between consecutive powers of ignorance are also different. For example, while the difference between the first power of two (2¹) and its second power (2²) is two (four minus two), the difference between the second power (2²) and the third (2³) is four (eight minus four). Likewise, there is a bigger leap in the severity of the problem from the second to the third power of ignorance, compared to the difference between the first and the second.

Known unknowns vs. unknown unknowns

I believe this hierarchical definition of ignorance might provide some insight into the relationship between the teacher and the learner. From the perspective of the constructivist theory of learning, the best scenario for a learner is to be the architect of his/her own knowledge palace. In this approach, the teacher is a facilitator of the learning process while the learner builds his/her knowledge palace piece by piece. For a learner who smoothly goes through this process, the first power of ignorance can easily be eliminated without much assistance, especially in this information age. In other words, in this kind of ignorance, there are “known unknowns” that can be learned.

Some initial guidance might be needed to eliminate the second power of ignorance. The learner might successfully decorate the existing rooms of her/his palace by herself/himself, yet might be unaware that there might be brand new rooms that can be added. At this point, the teacher might provide guidance and vision to the learner, so that s/he might start sailing in these uncharted territories. Once new land is discovered, the teacher might return to the role of facilitator. While there are “unknown unknowns” in this case, there is no resistance to the realization and acknowledgement of this situation by the learner.

The biggest challenge we face nowadays is undoubtedly around the third power. The teacher facing the third power of ignorance is helpless in helping the learner. In addition to “unknown unknowns”, there is this denial of the “unknowing”. In old times, the difficulty in overcoming ignorance was mostly about accessing knowledge. People who appreciated knowledge used to get very happy when they attained it. They also mostly exhibited the attitude that the more they learned, the more they realized how ignorant they were. In this sense, it is very odd that we can now access knowledge in seconds, yet we do not appreciate it. Nor do we have the wisdom that the more we learn, the humbler we should become about our knowledge.

Such an attitude also has a high potential of hurting the teacher-learner relationship, since the learner denies the fact that s/he sometimes needs guidance. Such learners only seek “depositories” of information, with the thinking that they, by themselves, can correctly and sufficiently process and digest all the information taken in. In this sense, they are closed to any advice, guidance, perspective, or vision provided by the teacher. Their approach implicitly says, “Just give me the information I am asking for and shut up!” This is where wisdom is lost between generations.

The missing link: wisdom

This might remind some readers of movies or TV series that depict the wisdom exhibited by a Far Eastern master of martial arts. The master would accept a Western person as his apprentice. Yet, he asks the apprentice to do extremely difficult and exhausting chores that are seemingly unrelated to the martial arts. Moreover, these chores appear as if they are geared towards the comfort of the master. Yet, as the apprentice proceeds in her/his training, s/he realizes how vital those chores have been in acquiring the necessary skills as well as attitudes, such as stamina, perseverance, and patience. Only then does s/he realize that it should be the master, not the apprentice, who defines the format of the master-apprentice relationship.

The more learners see knowledgeable people as only “depositories of information”, the more damage is done to a productive and meaningful teacher-learner relationship. Learners start to think that the second they acquire the information they lack, they become “even” with their teachers. And this exactly is the way of thinking that is a major source of the third power of ignorance.

Let us give a simple and practical example to illustrate this phenomenon. Think of a learner who wants to cook a dish. In response, the teacher provides the learner with the full list of ingredients for the dish. The learner, in turn, adds this to her/his library of information. As a second step, the teacher attempts to explain how and when each ingredient should be added to the pot, at what heat the dish should be cooked, and suggestions to add more flavor, etc. However, the learner declines this offer, claiming that s/he can handle these by herself/himself. Just like in this example, obtaining information about a topic is just the beginning. Applying this information to various situations, analyzing or synthesizing it, and comparing it with similar information are examples of tasks that require experience and are more important than simply depositing unprocessed information into one’s library. What is needed is to continue the learning process during these stages as well. This is only possible if the learner exhibits humility through awareness that s/he has yet learned little and keeps the channels of learning open.

Two disclaimers are in order here. First, this article is not to discredit learner-centered learning; rather, it seeks to highlight some points during the process where the teacher’s guidance and vision should play a critical role in the learner’s healthy progress in acquiring, digesting, and internalizing information. Second, it is not the intent here to imply that it should be the teacher alone who shapes the format of the learning-teaching process, and that the student should only be the object of it, as has been the case in many traditional educational systems. There is no doubt that a successful teacher will continually gauge the learner’s progress and gather constant feedback about the strengths and weaknesses of the learner and use these to provide proper guidance and vision.

It is crucial to help learners acquire the right attitude towards learning and knowledge. In an era where individuals are flooded with information and asked to mostly make pragmatic use of it, it is of the utmost importance to caution learners against the pitfalls of this approach and guide them towards successfully turning it into internalized knowledge. That would also bring about the wisdom and humility necessary for learners to continue to build their wonderful palaces of knowledge.