Subitizing: The ability to recognize(or guess) the number of a small group of objects without counting.

The name subitizing comes from the Latin word “subitus” which means “sudden”.

Subitizing can be seen in many every-day activities. One of them is a six-pack soda. No matter how they are lined up, we know that the number of soda bottles is 6. We inherit this knowledge without counting the bottles. And if we decide to drink one of them, we automatically know(without having to count them) that the number of soda bottles left is 5.

You don’t have to count the dots on the surfaces. You just know that it is 5.

Another example of subitizing can be given from the game backgammon. Assume that two dices are rolled and you identify them as 2 and 5. The process of identifying the dices can be measured in milliseconds. This can be even shortened as you spend more time playing the game. In short; subitizing is a skill that can be developed if one spends time and work on it.

Research studies showed that 6-month olds can differentiate, visually (a top bounces 3 times) and from sounds (clapping hands 3 times), between 1, 2, and even 3. In other words; humans start developing the number concept when they are just infants.

Kebab Truck & Subitizing Subitizing is hidden behind the number of customer groups in the game of Kebab Truck. As the game is played, scores become higher and higher. The reason behind this is that players’ subitizing skills are improving.

Let’s check this scene from Kebab Truck:

In the beginning, you will be making certain moves during the game. Nevertheless, in time, your moves will differ substantially. The biggest reason behind this is that your subitizing skills were improved while you were playing the game.

Kebab Truck also helps the players to develop their basic arithmetic skills. These improvements are not limited to adding and subtracting the number of customers. Once you understand how the scoring system formulated, you will realize that (to maximize your scoring) multiplication is an important part of this game as well.

A very long time ago in Mesopotamia, a few hundreds of people lived together in the village of Badaks. Badaks were very hard-working people, and they were among the first farmer communities. Their lives depended on two things more than anything: Their farms and sheep.

Monday syndrome in Badaks village…

There was a lot of sheep in the village of Badaks. Thanks to them, people of Badaks were able to protect themselves from cold weather. Their milk and meat were also important to Badaks as food sources. Because of their importance, the person in charge of the sheep had to be wise and trustworthy.

Zaylin a.k.a. the protector of sheep!

Zaylin, head of Badaks, was in charge of this crucial duty.

Every day, with the first sunlight, Zaylin took the sheep out of their pens for them to explore the hills and graze the green grass of the village of Badaks. Before the sun is gone, Zaylin had to gather the sheep and be sure that every sheep returned to the pens.

Omg! Where are the rest of you?!

Even though Badaks were one of the most progressive communities of their time, they didn’t know the use of numbers like the rest of humanity.

At this point, Zaylin had a bit of a problem: How did he know that he returned with the same number of sheep as left in the morning? Don’t get me wrong; Zaylin was an intelligent person for his time. But like everybody else, he didn’t know how to count.

One wonders…

Put yourself in Zaylin’s shoes: Is it possible to detect if you lost any sheep or not when you finish a day without counting or any use of numbers?

A game for kids who would like to get better at arithmetic operations, decimal system and numbers in overall:

European Championship

Only materials needed for this game are a twelve-faced dice, pen/pencil and a piece of paper.

Game consists of encounters between two players.

In each encounter players roll the dice four times in order.

Outcome of a rolled twelve-faced dice is like the following:

For every player only ambition is to write the biggest possible four-digit number.

Difference of players’ four-digit numbers decides the winner.

Scoring of the game

Player with the bigger number would get:

4 points if the difference is a four-digit number.

3 points if the difference is a three-digit number.

2 points if the difference is a two-digit number.

1 point if the difference is a single-digit number.

If the difference is zero; meaning that the numbers are equal to one another, then both players get no points.

Every encounter finishes when one of the players gets to 7 points.

League

In case there is enough number of students, it is possible to construct a league version of the game that finishes after playoffs. For instance if there were 20 students we could divide them into 4 groups with 5 teams. In each group every player would play 4 games and after the group stage group leaders would go onto the playoffs where the champion can be decided after semi-final and final games.

World Cup

In this version of the game players would roll the dice three times and write the biggest possible three-digit number. Although this time winner gets to be decided like following:

If the difference is odd, biggest number wins.

If the difference is even, smallest number wins.

Winner gets 3 points as loser gets nothing. Differences are kept as averages.

If numbers are the same, players get 1 point each.

The word fraction has a Latin root “fractio” which means “to break”. Let’s take a slightly different approach for fractions. Instead of breaking, we will use “folding”.

Fractions with Papers

In order to understand the four arithmetical operations in fractions, we can use a standard A4 paper. First we will fold the paper into two halves. Every half represents the fraction ½. We can continue folding one of the pieces in two halves. In the end we will have the following papers which we can use for four mathematical operations:

a. 1b. 1/2+1/2c. 1/2+1/4+1/4d. 1/2+1/4+1/8+1/8e. 1/2+1/4+1/8+1/16+1/16

Addition

We are aware of the fact that all five of the versions are equal to one another and they each add up to 1. If we take a look at versions b and c we can conclude that ½ is equal to ¼ + ¼. If we substitute 1/2s in the version b we can find the following result:

1 = ½ + ½ = ¼ + ¼ + ¼ + ¼.

Subtraction

Let’s continue from the previous. If we subtract ¼ from both sides of ½ = ¼ + ¼, then we get:

½ – ¼ = ¼ + ¼ – ¼

½ – ¼ = ¼

Multiplication

Let’s start with another A4 paper and take its whole as 1. This time we will fold the paper and use the marks on it.

Assume that we are trying to find ½ * ¾.

Check the first fraction and fold the paper into two halves since it is ½.

Then check the second fraction and fold the paper into four equal parts.

Since second fraction is ¾, mark 3 parts of the paper.

Now unfold the A4 paper completely. It shows that there are 8 equal parts and 3 of them are marked. Hence solution is 3/8.

One wonders…

Using the paper method show the connection(s) between 1/8 and 1/32.

Find 1/8 – 1/32 with the paper method.

Game

Imagine a gambling game in which you don’t have to gamble your money. In the worst case scenario, you will win nothing. A dream for gamblers, isn’t it?

Let’s say you get 128.000 dollars and 6 red and black cards (3 for each). Here is how the game goes: In every hand you must bet half of your total money. When you pick a red card, you will get twice what you played. When you pick a black card, you will lose all the money you bet in that hand.

When the game is finished (after all 6 cards are played) if you end up with more than 128.000 dollars you will get that surplus.

What is your strategy to win? Explain your answer.

In schools we start learning mathematics with learning what numbers are. Unfortunately numbers are taken for granted and being overlooked just because it starts in the elementary school. The truth is this part of mathematics is a joint work of countless civilizations that lasted thousands of years. Although, categorizing and defining all those information were done only in the near past. This means that things we learn in the first few years of school have so much more depth than we think they have.

Especially fractions (or rational numbers) weren’t used in Europe in the sense we understand them today until the 17^{th} century. In fact for a long time people thought of fractions not as numbers but as two numbers being divided to one another.

Rhind

Ancient Egyptians were one of the first known civilizations that used fractions. They created one of the most important and oldest documents in the history of civilizations using papyrus trees. Around 4000 years ago they started writing valuable information on papyrus leafs. Rhind papyrus is one of those documents. It is believed to be written around 1800 BC. Thanks to Rhind, we can understand how ancient Egyptians used fractions.

It is uncanny how commonly they used fractions in Rhind. Although they were obsessed with unit fractions as they found ways to describe every fraction with them.

Unit Fraction: Fractions that have 1 on their numerators.

In the ancient Egypt they used a shaped that looks like an open mouth (or an eye). This shape was the notation for the unit fraction. Denominator of the fraction would be placed under the mouth.

Table of 2/n

Inside Rhind there is a method for describing fractions in the form of 2/n (when n is odd) with two unit fractions. Table starts with 2/3 and ends with 2/101. In the papyrus it says that 2/3 is equal to ½ + 1/6. For the rest of the papyrus a formula was given in order to describe fractions in the form of 2/3k: It is 1/2k + 1/6k.

Let’s try it for 2/9. 9 is 3k, hence k=3. This gives 2/9 = 1/6 + 1/18. Ingenious, isn’t it?

Next number on the table is 2/5 = 1/3 + 1/15. This is also a general formula just like the previous one. Any fraction in the form of 2/5k can be shown as 1/3k + 1/15k.

Fraction Line

Besides ancient Egyptians, Babylonians used fractions too. But their choice of symbols was so confusing, it is impossible to understand which number is written. You could only check the rest of the calculation (if there is any) and guess which number is being used.

In the Babylon civilization number system has base 60. The one on the left is 12, other one is 15. But they can also mean 12+(15/60) too. Lack of symbol for fractions caused a lot of problems in this civilization.

Around 1500 years ago Indian mathematicians were shining. They found the number system we use today and even the number zero was invented (or discovered). Their brilliance was key for fractions too as they showed fractions one under the other. Muslims were the ones who thought of putting the fraction line between numbers.

In the end we owe our modern notation to Indian and Muslim mathematicians.

The way 7/15 was shown in the old Indian symbols.

One wonders…

Try to find answers to following questions about the Rhind:

Why did they only consider odd numbers in the denominator of the fractions?

Find 2/7 and 2/11 using their methods.

What happens after 11?

Try to come up with a general formula for 3, 5, 7 and 11.

There are such questions, even though they seem useless (or unnecessary) to others they can help one to get much better at the number theory. One of the main reasons why some people label those mathematics questions as “useless” is that they are actually scared of the question (or of mathematics).

What those people really feel is like the feeling you get whenever you are walking down on an unknown street, city or country. Those people are away from their comfort zone and they would never feel as relaxed as they feel at home as long as they don’t “try”. Defining mathematics questions as unnecessary is in fact a way of expressing the fear of mathematics.

A different kind of magician.

In that case trying and/or striving are essential if you’d like to get better at mathematics. This is how you could find your own methods and accomplish things that will shock others. There is another matter I’d like to point out: If the trick is obvious no one would watch your show second time. Magic is beautiful when people don’t understand what you are doing.

Equal Sums

Assume that we have ten different numbers between 1 and 50. Our goal is to divide those ten numbers into two groups of five such that their summations will be equal to each other.

Example 1: My random ten numbers are: 2, 12, 23, 24, 30, 33, 39, 41, 44 and 48.

Goal is to divide these numbers into two groups of five such that their summations are the same.

I managed to do so after a short amount of time:

48+41+33+24+2 = 148 = 44+39+30+23+12

Maybe you think that I choose those numbers on purpose. This is why I asked some of my friends to send me random ten numbers from 1 to 50.

Example 2: Random numbers: 34, 21, 7, 42, 22, 33, 13, 27, 20 and 19.

After a minute or so I found the following:

34+33+13+20+19 = 119 = 21+22+27+42+7

Q: How do I do it? Can you speculate about what kind of method I might be using?

Example 3: I got these numbers from another friend: 3, 9, 13, 19, 21, 27, 36, 33, 39 and 45.

Example 4: And these are the numbers I received from one last friend: 7, 10, 11, 14, 21, 23, 30, 33, 43 and 49.

For the examples 3 and 4, I found out that there can’t be such groups and I came to this conclusion in a matter of second.

One wonders…

How did I decide so quickly?

Hint: Take a look at how many of the numbers are odd or even.

No matter how hard you try to avoid them; one day you may need negative numbers. There are a respectable number of people from Britain who would agree with me on this matter. Back in 2007, a British lottery company released a new lottery game named Cool Cash. On every Cool Cash card, there were five boxes that you had to scratch. The box which was on the left bottom corner was called the “temperature of the day”.

Other four boxes were assigned to a prize. In order to win at Cool Cash, your box or boxes should have had lower temperature than the temperature of the day.

In this example our temperature of the day is given as -8 Celsius degrees and our four temperatures are -4, -6, -7 and -7 Celsius degrees. All four degrees are higher than the temperature of the day which meant that this Cool Cash card gives away no prize whatsoever.

Unfortunately this game caused huge problems for the lottery company. It turned out that British people had a poor understanding on negative number notion. Majority of the people called the company and claimed that they won even though they really did not. Ironically, some people thought they lost and threw away their cards which in fact had won a prize. After increasing number of complaints executives of the lottery company realized that they had two choices: Teach mathematics or stop producing Cool Cash cards. They had chosen the second option which was clearly the easiest of the options.

Smallest Number

Negative numbers had challenged mathematicians throughout the history. That is why it is not surprising to see kids having trouble when they try to make calculations with them. Although in most mathematics curricula negative numbers are ought to be thought for 11-12 years old students, some high school students have problems with negative number notion. Even in the 17^{th} century Europe where modern science flourished, many scientists claimed that there can’t be a number that is less than zero.

Chinese rod numerals.

When one searches for a counter example, it is possible to see the negative number concept in the early civilizations. For instance in ancient China there was a system for distinguishing positive and negative numbers. In ancient China, numbers were represented by rods in two different colors: Red rods were used for positive numbers as black rods for negative numbers. (Today we use red and black to show if an account is in plus or in minus.) Nevertheless Chinese mathematicians believed that a problem or an equation can never have a negative answer.

Adding Words

Ancient China and Egypt was using this technique quite often: Addition to the statement. People were adding extra words in their statements such that these words would change the whole set of the problem. In the end these extra words prevented people from the need of negative numbers. Let me explain this with an example:

Example 1-a: If one has 50 euro in his/her account and shops for 70 euro, that person should end up with -20 euro in total.

Example 1-b: If one has 50 euro in his/her account and shops for 70 euro, that person should end up with 20 euro debt in total.

Addition of the word “debt” changes a negative number into a positive one. This technique is still used today in our daily lives.

Opposition

People also developed an opposition technique in their language to prevent the need of negative numbers: Choose the opposite of the term.

Example 2-a: A submarine is compared to sea level at -2400 meters height.

Example 2-b: A submarine is compared to sea level at 2400 meters depth.

Oppositions such as up-down, forward-backward, profit-loss, more-loss … are useful for turning a number from negative to positive.

Real Issue

The real issue for understanding negative number notion is that there are no negative numbered things in nature. This is really crucial as we don’t let students think over number concept. Actually whole number notion is not natural; we create numbers in our minds. You can’t go out to nature and run into something (living or not) that is in the shape of number 3. The number 3 makes sense to us when we see 3 cows in a cattle farm.

Brahmagupta (598-668)

3 apples, 3 trees, 3 cats… Numbers make sense to us whenever you bring them with another concept. However, if someone asks you “what makes 2+1?” you would answer “3”. Here, “3” has no concept attached to itself, yet you used it perfectly.

Almost 1400 years ago, a brilliant Indian scientist named Brahmagupta used negative numbers as we use it today. He explained the negative number notion with another concept attached to it: “Negative numbers stay in the debt side of the number zero.” Brahmagupta also gave rules to negative numbers in calculations. He is the first known person who claimed that a negative multiplying another negative makes a positive.

One wonders…

This is the picture of +2 apples. How can you photograph -2 apples?

First thing first; mathematics and philosophy can’t be separated. I can back this idea with a historical fact: Compared to history of science, “scientist” is a relatively new title. For instance, Isaac Newton who is regarded as the father of modern physics was known as a “natural philosopher”. Use of the word scientist had made us forget the fact that a scientist is actually a person who does deep thought sessions in majority of his/her work.

Mathematics Education – “This is that, and hence the answer should be that.”

Unfortunately, in traditional mathematics education there is almost no time for one to think. I think I should explain what I mean with the term “traditional mathematics education”: Imagine that numbers is the subject of a specific mathematics class. Teacher follows the traditional way and gives definitions to number types.

Counting Numbers: These types are also known as whole numbers. They start from 1 and go to infinity one by one:

1, 2, 3, 4 …

Natural Numbers: They start from 0 and continue to the infinity one by one. Their only difference with counting numbers is the mighty 0 (zero).

0, 1, 2, 3, 4 …

Odd Natural Numbers: They are the numbers which has remainder 1 when divided to 2. First odd number is 1. Odd numbers increase by 2. So their sequence is:

1, 3, 5, 7, 9 …

Even Natural Numbers: They are the numbers which can be divided by 2 with remainder 0 (zero). First even number is 0. Their sequence is:

0, 2, 4, 6, 8 …

Integers: They are the sum of natural numbers and negative of counting numbers. They can be shown like an infinite line where left goes to negative infinity as right goes to positive infinity. Their sequence is:

… -3, -2, -1, 0, 1, 2, 3 …

Prime Numbers: They are greater than 1 and they can only be divided by themselves and 1 to give and integer answer. They are:

2, 3, 5, 7, 11, 13, 17 …

“What is 7 times 8? Bravo! You are so fast!”

We need to define these, so there is no problem up to this point. What is done wrong with traditional approach is that it rapidly focuses on problems. There is nothing wrong with asking questions to kids, although with the traditional approach important thing isn’t the question: It is the answer! After a while kids focus on the answer rather than the question. That is one of the reasons why when asked (I am not talking about mathematics per se) children try to answer even without thinking. Sometimes even families contribute that with awarding how fast the answer is given.

Even before they start high school most of the kids lose the ability to think in mathematics classes.

What needs to be done?

If you are good at observation, you probably realized that those definitions above include notions such as negative, zero and infinity. These three notions are relatively new to humans and they all possess deep meanings underneath them. For instance only 1350 years ago humans first explained the number zero mathematically. Actually first use of the number zero in Europe was around 12^{th} century which was almost 500 years after its first discovery in India! (I will get back to this subject in another article.)

Nowadays we define negative numbers and we expect kids to do calculations with it immediately. Actually negative numbers were known to civilizations way before the number zero. Although in the 16^{th} century important European mathematicians were referring to negative numbers as “wrong numbers”. It has been only 200 years or so since humans made peace with the negative number notion. Maybe it is clearer now to you how wrong it is to expect kids understand negative numbers instantly.

Infinity might seem the hardest of these three notions. There is a popular definition for it: Something that never ends, keeps going until the end of time, everlasting. However there are much more profound meanings for infinity in mathematics. For example, some infinities are countable and some are bigger than others.

Hilbert’s Infinity Hotel

This paradox was thrown out for consideration by David Hilbert, one of the leading mathematicians of 20^{th} century. In Hilbert’s hotel there is infinite number of rooms. Imagine that you are the manager of this hotel and you work on commission. One day business was ticking and every room was filled with customers.

Would you turn down if a new customer arrives? If you decide to give this customer a room, which room will it be? How can you decide that room number?

A couple of hours later you see infinite number of customers in front of the hotel. How can you arrange rooms for this many people?

Just when you thought you could relax for a bit, you hear honking coming outside of the hotel. You go outside and see infinite number of buses in the parking lot. Not only that, each bus has infinite number of tourists. WHAT WILL YOU DO?!

Answers to these three questions will help students comprehend different number notions. You should stay away from internet in case you have never heard of this paradox. Give yourself time, even if it is limited, at least 1-2 hours. Just think on these questions. Try to remember this: It is not about “when”, it is about “how”.

Negative integers are a “game changer” when they sneak into our lives. Everything we learn about calculation gets a little bit more complex with the introduction of negative integers. Now we have rules that say crazy things such as “multiplication of two negative integers gives you a positive one”. How come two negatives make a positive? Is there a sensible explanation?

Q: (-2).(-3) = ?

Algebraic Method: It is being taught that when two negative integers are being multiplied, absolute values 2 and 3 are being done and a plus sign comes to the top of the calculation. This gives +6 or 6 as a result. Almost all of the students learn how to do this calculation but most of them have no idea why it is being done like this.

There are numerous examples to explain the reason. I prefer giving a specific geometric explanation as an example.

Geometric Method: We teach that integers can be shown on an infinite straight line what we call “number line”. Middle of this line is assigned to the number zero. Left of zero is for negative integers as right side is for positives.

While teaching multiplication with negative integers, you could imagine a line that is perpendicular to the original number line. Upwards would be assigned to the positive integers as negative integers go to downwards.

Rules

Assume that we have to multiply two numbers.

Sign of the first number tells us which way we are facing: Upwards or downwards. Positive would mean “look up” as negative means “look down”.

Sign of the second number tells us the way we are taking our steps: We could either go forward (which means the number is positive) or go backwards (meaning that the number is negative). You could imagine that backward steps are like how Michael Jackson was “moonwalking”.

Value of the first number tells us how many steps are taken in each time.

Value of the second number tells us how many times those steps are going to be taken.

Let’s solve our original problem with the geometric method: What is (-2).(-3) ?

First number is negative: We are facing downwards.

Second number is negative: We are taking our steps backwards.

Value of first number is 2: We are taking 2 steps in each time.

Value of the second number is 3: We are taking those steps 3 times.

In the end, we are facing down, moving backwards (like Michael Jackson’ moonwalk), 2 steps at a time and 3 times. Our arrival would be +6 as shown on the graphic.

One wonders…

Try to prove that the algebraic method is true. (Hint: Start with assuming that the multiplying two negatives won’t make a positive.)

Try to find another example for geometric method from real life.

Almost everyone knows how to play tic-tac-toe. Dotted tic-tac-toe is a multiplayer game to teach kids about number systems.

Rules:

Players throw a dice in turns.

Each square in the board has the capacity of 9 dots.

Players should put dots into one of the boxes as many as their dice shows.

While putting down dots, players must be careful not to exceed the capacity of the box. In such situations they have to put dots to another box that has capacity.

Whoever gets right to left, upside-down or diagonal three boxes will be the winner.

Beauty of this game is although it is competitive, it requires help of your opponent.

An example:

In the first three rounds players get 5, 6 and 3 from the dice:

Fourth dice is 1 which gives second player a chance to complete a box.

In the next four rounds players get 5, 5, 6 and 4 which gives again second player to complete another box.

Game resolves in the next rounds as shown in the following:

Binary Tic-Tac-Toe

In this version of the game, use a coin instead of a dice.

Heads: 1

Tails: 0

Consider that a box can have either of the following: A head & tail (tail & head is the same) or a tail & tail.

A square can never had head & head, since 1+1=2 which is overload for the binary number system.

This means that every box is either 1+0=0+1=1 or 0+0=0.

So in this version of the game, x-o-x turns into 1-0-1. Right to left, upside-down or diagonal 1s or 0s wins the game.

Once again, although the game is competitive, you’ll need your opponent’s help.