Real Mathematics – Numbers #2

Lottery Bust

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.

card

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 17th century Europe where modern science flourished, many scientists claimed that there can’t be a number that is less than zero.

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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.

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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…

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

M. Serkan Kalaycıoğlu

Real Mathematics – Numbers #1

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 12th 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 16th 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 20th 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.

  1. 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?
  2. 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?
  3. 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?!

hilbert

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”.

M. Serkan Kalaycıoğlu

Real Mathematics: Numbers #3

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.

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Rules

Assume that we have to multiply two numbers.

  1. 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”.
  2. 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”.
  3. Value of the first number tells us how many steps are taken in each time.
  4. 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.

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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…

  1. Try to prove that the algebraic method is true. (Hint: Start with assuming that the multiplying two negatives won’t make a positive.)
  2. Try to find another example for geometric method from real life.

M. Serkan Kalaycıoğlu