1. numbers (a need to move forward)

1-3. We learn numbers so early that we might not be able to recall it, so we take the ability to work with numbers for granted. We also live in a world which has been quantified and digitized so we might find it difficult to imagine a world without number. However, there are still people who haven’t developed the concept of number and they hardly use it. This video shows how a member of Walpiri tribe expressing 4.

This behavior has been observed in many primitive tribes whose members were able to count up to a certain level, let’s say 2, 3 or 6; and beyond that they use the word “many”. Dobrizhoffer observed that counting bores primitive people.

“Primitive man not only does not know arithmetic, he avoids it. His memory proves to be a major handicap; counting bores him, so he is unwilling to use it. “When members of a primitive tribe return from an expedition hunting from wild horses, nobody asks how many wild horses they have brought back. The question asked is, “How large an area would the herd you brought back occupy?”

How can inability to understand, work with, or express the large numbers, be justified? Since most of the 6 year old kids are able to count up to 100, can we conclude that there is a faculty in our brain, which enables us to understand concept of numbers, therefore primitive people lack it?

In order to answer the question, we should look through the process of creating the concept of number step by step. At the first step, we perceive and conceive the difference between two different amounts. To some extent animals can compare two quantities and select one of them. This video shows the ability of pigeon to distinguish the quantities and select the smaller value (in order to get rewarded).

Is pigeon able to compare every two values? We have to consider that comparing 1 object with 2 objects is easier than comparing 8 objects with 9 objects. Through a research Gatton found that if he took 1 of the 4 eggs out of a nest, “the bird didn’t seem to mind it, but if he took two eggs, then the bird destroyed its nest”. You might find it difficult to compare 10 flowers to 11 flowers, eventually you would need to count each group in order to compare their quantities.

Primitive people might not be able to recognize the difference between two close quantities, so they don’t use different words for them, or the difference between the quantities doesn't affect their living therefore, they haven’t felt the need to come up with proper names for each number.

Evolution of numbers was propelled by individual and social needs. Even in ancient Greece when geometry was flourishing, mathematicians almost disregarded arithmetic since they thought recording and working with numbers is a slavish job (their slaves were in charge of reckoning). Development of numbers can be directly related to the civilizations in which people needed to deal with the surplus of their grains, to share something evenly with a group of people, to deal with the inheritance, to measure the quantities such as time, length and mass, and many other needs. If you ate whatever you hunted or obtained, you wouldn't have any belonging to be counted; besides, if you dwelt in a barren land you wouldn't be able to see the periodic growth and decay of the plants, therefore you wouldn't come up with the idea of agriculture.

Jean Piaget, whose name was mentioned earlier, conducted several experiments on many kids to monitor the cognitive stages of development. These two videos demonstrate his experiments.

We can see that the concept of quantity and specifically the number (in the coin experiment) needs to be refined in the kids' brains, since they can't still separate it from the size or the change of shape. It shows that creating the concept of numbers and quantity needs a huge development in our brains (what we take for granted) that if it didn't take place we couldn't understand the numbers, therefore there couldn't be almost any developments in our societies. 

In conclusion, we can say that the ability to understand the numbers is not built in; even though our brains have been evolved through centuries so we can understand them faster than our primitive ancestors, it has been acquired when we were small kids and it can be reinforced and developed when we need the numbers for counting or calculations.

1. numbers (mathematical number vs. physical number)

1-2. Story of numbers started when our ancestors found out there was a difference between one apple and two apples. In order to communicate clearly, they needed to come up with two different names for them. We don’t know what they called them, but we are sure that soon after they faced another problem which was a new word for three apples. It took a while that they found out the similarity between them, so they decided to disintegrate the name into two parts: one part to indicate the difference and the other to reflect the similarity. Then they said one apple, two apples, and three apples.

However, there was another issue that they used different adjectives for different objects; for example the adjectives that they used for apples were different from swords or cats. Another big leap was needed that they conceived the similarity between two men and two apples.

To understand the importance of this abstraction, we should know that our brain is removing all components of apples and men to find one similarity between them which is their quantity, apart from that they are completely different. We can’t even imagine it because we’re removing their shape, size and every feature; we only understand it. So they invented one set of adjectives for all different quantities which was a huge development.

Here one is not a number. It’s merely an adjective which is serving a noun. When one as a number was born that our ancestors were able to separate it from the nouns which they had counted. Physical one is different from mathematical one, physical one is meaningless without a noun or unit; if you told somebody that I'd been waiting for you for one, how could they know if you meant one second, minute, hour or even year, however in math nobody needs to know what one is referring to. That’s why they invented a different symbol for it: 1. 

This segregation enabled humankind to develop a different system for numbers which is the foundation of mathematics. This was the moment that the plane took off.

1. numbers (pre-requisite: set)

1-1. We discussed how a concept is created by our mind. It abstracts the similarity from several things in order to group them. In mathematics we call these concepts SET. Dog is a concept for the animals with particular similarities, in mathematics dog is defined as a set of animals which have those common properties.

Set is one of the basic concepts which cannot be defined; however almost everybody can understand it. Set of mathematicians, set of bicycles, set of psychological disorders, and set of smiles.

Members or elements are things that belong to a set. Gauss is a member of mathematicians, but Lionel Messi is not one of its elements. Since we use this format frequently that a particular member is an element of a particular set, we replace “is an Element of” with this symbol ∈ (stretched E). So translated into math language, we write Gauss ∈ mathematicians. We could even simplify it more, if we used Gauss and mathematicians frequently. Therefore, we could write that G ∈ m (In the context G and m must be defined). “Messi is not an element of mathematicians” can be simplified as M ∉ m. You can see how a sentence can be written in mathematical language more simply and easily. 

Like the words in English that we memorize for communication,  we should keep in mind the symbols in order to read them correctly, otherwise they look like some weird, mysterious or even terrifying codes.

  

If we change the arrangement such that we say "mathematicians" consists the element named Gauss, we write it m ∋ G, or m ∌ M (for mathematicians which doesn't have an element called Messi). Most of the times we use the first arrangement " the member is (or is not) an element of the set".

0. Introduction (what we talk when we talk about math)

0 – 5: Water is a chemical compound with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms that are connected by covalent bonds. Water is a liquid at standard ambient temperature and pressure, but it often co-exists on Earth with its solid state, ice, and gaseous state, steam (water vapor).

That is a definition for water. Does it motivate anybody to learn swimming? Definitely no. The best way to define swimming is jumping in water. You can find many definitions for math throughout Internet, but they are not as attractive and mesmerizing as solving a math problem. So instead of defining math, I’ll try to portray it.

Mathematics is a language in which we speak of quantifiable problems. It has its own words, sentences, grammar and punctuation. The big difference between English and math is if you made mistake in an essay, it would probably result in a weak essay, not a wrong one; however in math, your solution would be certainly wrong. We can analogize communicating in English and math to driving a car and an airplane respectively; if airplane pilot made a minor mistake, it would end up in a deadly accident. That’s why math is the most challenging subject for majority of the students.

The second aspect of this portrait is the elegance and efficiency of math in order to solve problems. If solving a problem can be analogized to a journey between two places, math is the aerial travel. In this analogy, land is the reality or the physical approach, and sky is the realm of mind.

Imagine that you’re asked to count the number of planted trees in a land. You could count them one by one and if there were 50000 trees, it would take a long time (if you could count one tree per second, it would take 15 hours approximately). However, if you could find a pattern (such as grid plantation), you would need to count the rows and columns (say 500 by 100) so instead of tallying 50000 tress you’d need to count 600 trees, then your brain could process the numbers (in the sky!) to find the answer (it would roughly take 10 minutes)

Without science and math as its language, who would trust to get on an airplane? If an engineer announced that this plane had been designed and manufactured after 1000 experiments (without applying mathematical approaches), would you trust them?

So, wherever we talk about a quantity, we can see math, coming along.

0. Introduction (from imitation to understanding)

0 - 4: So far, we’ve learnt what subject and object are and how brain can create a concept based on abstraction. Through the last post the process of creating concept of red in a kid’s mind was explained. It was mentioned that we can’t transfer a concept; like agriculture, we cultivate it; and then we hope that the kid would understand it. If not, we need to change our method, changing the examples probably, until the kid could conceive it.

However, when we teach a concept to somebody, especially kids, before they understand it, they imitate how to apply it, and they might be trained to use the concept just by imitating.

Let’s study a case: you have a bunch of keys and locks. You teach a kid that key A can unlock lock 1, key B for lock 2, etc. Then the kid learns how to use the key to unlock the locks. Who can say if the kid was imitating because he memorized the correspondence between keys and locks or he understood the concept of specific (sometimes unique) key for specific lock? Nobody could answer the question, sometimes understanding a concept is not apart from its application. However, we can test the kid by giving them a new bunch of keys and locks and ask the kid to unlock them. If the kid tries driving every key through every keyhole one by one, we can say that they couldn’t understand the concept which is the key must fit into the keyhole, so their shape and size must look alike. If the kid just tries the key and keyhole with the same size, they are solving the problem intelligently, it means that they’ve understood the concept of key and lock. So we can conclude that unfamiliar problems can show how deeply people have understood a concept.

I don’t believe in categorizing people. I believe in a continuum such as a spectrum to show the diversity of thoughts and actions. To show people’s capability to solve a problem, I use the gradient of blue.

For the end points of this bar, I use operator and engineer. I redefine these terms.

Here, operator means somebody who solves problems exactly by imitating. Operator follows the instruction given and doesn't think about it personally. They don’t have the slightest idea what they are doing. On contrary, engineer means someone who can solve problems using their own intelligence solely. They can design or fix a system because they have understood how it works. They are fully aware of what they are doing step by step.

In real life, nobody is an absolute operator or an engineer; however this gradient can help us to determine how far we are from each endpoint, to see whether we’re closer to either engineers or operators. Normal people use combination of others ideas and their own intelligence to solve problems. Relying on your intelligence exclusively is as foolish as implementing what other people have prescribed without a single thought.

Jean Piaget, the Swiss psychologist and philosopher, stated that intelligence develops to adapt the humankind to the world. Two complementary agents of this process are: assimilation and accommodation.

When our mind processes a new object, it uses the information stored in memory to analyze it: at the first place to find the similarities and differences. Assimilation tends to fit the object into the subjects that our brain has created, so it tries to find the similarities. Imagine that you had seen a horse, then you saw a donkey for the first time, based on the concept of horse created in your mind, you might compare the horse and the donkey, and might find similarities, here assimilation tries to fit the donkey into the image of the horse, so you might conclude that this animal is a horse. The reason that kids confuse naming the things around can be assimilating it with something in their mind.

On the other side, accommodation tends to change the block of information that our brain has created to fit them into what we perceive. If we examined the donkey more closely, we could find the differences between donkey and horse, so donkey doesn’t completely fit into the concept of horse, therefore our mind creates the concept of donkey. That’s how it fits the subjects into objects.

Here assimilation looks clumsy. Obviously donkey is different from horse. Now let’s study another case.

If there were no shoes, accommodation would tell us to walk on the smooth paths without sharp or spiky stones, we would adapt ourselves to the ground we would walk on. If we hadn’t had assimilation, nobody could have invented shoes. By inventing shoes, humankind changed the world around to fit it into their image (previously the concept of a cover to protect the feet must have been created in the mind, then humankind tried to make it come true).

Now it seems that assimilation is more attractive. It is the power that scientists, philosophers, inventors, and leaders use to change the world based on what they have had in their mind. However, assimilation without accommodation can be dangerous and destructive. Don Quijote is the most dramatic example of a person who just assimilated. He fit the reality into his imaginary world. That’s why he fought a windmill which he imagined to be a giant. In English there’s an adjective “procrustean” derived from Procrustes (Greek Mythology) who stretched the victims or cut their hands or legs to fit them into his bed, so this adjective is used for person, attitude or solution which uses other ideas but fit them into their own philosophy. The dictators are other examples of uncontrolled assimilation; and religious people who try to interpret everything, especially new discoveries, to fit them into their beliefs. So you see that the only difference between the insanity and innovation is the way we use accommodation. If Einstein’s theory hadn’t been verified by experimental findings (reality), everybody could have assumed him as an insane for the relativity theorem. 

Piaget mentioned that intelligence develops if assimilation and accommodation complement each other in a balanced way. If assimilation dominates, the brain tends to the game, if accommodation dominates, the brain tends to the imitation.

In my opinion, dominance of either assimilation or accommodation takes place while the other agent can’t develop accordingly. Brain evaluates itself statistically. If  kids are forced to do what they ought to, or their thoughts or opinions are belittled or devalued, in their mind assimilation doesn’t develop as much as accommodation, so they fall in imitation, consequently being a good follower. When we don’t tell our kids the reason of the expectations or duties, we don’t give them the material to think of, therefore both assimilation and accommodation can’t grow.

When we teach numbers to 5 year old kids, we can’t explain what the number is. We just teach them how to use them to count, then step by step other arithmetic operations come after. Like teaching words, we don’t tell them that orange is a noun and eat is a verb, we just teach them how to use the words appropriately. Kids ought to follow what teachers or parents ask them to do. This teaching method is supposed to transform to a conceptual method, and if it doesn’t, kids will fall in imitation. In mathematics if they’re just required to solve problems correctly, regardless of understanding it, kids don’t learn to think independently and critically. That’s why the problem arises in math when they go to higher classes where they are expected to use the concepts that they were supposed to create previously.

Based on my experience most of the kids who had problems with math told me that they loved it in elementary school or in the first or second year. Why? Because it was easy for them at that time like the key and lock problem, and their brain had developed enough to cope with addition or subtraction. But they just learnt how to imitate the approaches teacher applied to solve the problem without understanding it. Now the solutions can’t be applied to solve unfamiliar problems.

In conclusion, imitation is the first step of learning, but we are supposed to leap from imitation to understanding which can’t be achieved in one step. Concepts can grow, or they can become more profound when we move forward. When you walk through a city, you just see the houses and shops and streets. If you climb a mountain next to the city, you can see how the houses and shops and streets are related to each other. So in your mind a bigger integrated image is created (what maps can do for us). When you climb higher, you can see more of the city; likewise, the concepts can grow in our mind when we find the relation or the interaction between concepts. Therefore, we should equip ourselves for this journey.