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?!


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


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