Mathematically Yours

Following is the talk I delivered in a PACE Series. I intentionally picked the date to give tribute to my parents' wedding anniversary. It was the last of the series and I was glad a good number turned up for the talk.

12 November 2006

Dear Friends,

I do not indulge with mathematics because it is important. I study mathematics because I am fascinated with it and I am fascinated with it because it is motivating. It is motivating because it is attractive .

I am fortunate to have a job like mine. I have to admit that my stint in RECSAM is one of the most significant events in my life. I love what I do and I am challenged by the varieties the job affords me to do. I get to share what I do with other teachers and I learn how to integrate technology, humor, and history in teaching. I love meeting people and going places. Meeting people allows me to delve into their perspectives of mathematics and how mathematics influences their lives. Going places allows me to see beyond the familiar environment and be part of the strangeness that surrounds me.

I think that mathematics is about that, being strange. People perceive mathematics that way. Peculiar it may be to others but to me mathematics is one of the best things that man has engaged with. Its peculiarities made all the difference; it is more beautiful in its form and more appealing in its substance. The essence of mathematics lies in its form and substance. One just needs to experience them to fully appreciate what mathematics is.

I am a teacher. And that is the best gift this life has to offer me. I am a mathematics teacher. And that spells the distinction between being a good teacher and a mediocre teacher. But mathematics did not come to me on a silver platter. There were struggles and challenges; there were difficulties and frustrations. When I was a student, I was like everybody else; wishing that the teacher would not come or that he would be sick or that there would be a typhoon so that classes would be suspended or that there would be an urgent faculty meeting. I feared mathematics just like all the others in class. The mere sight of my mathematics teacher would give me goose bumps and make my heart beats so fast that it felt like it would leap out of my chest. I would tremble in every quiz and shiver with board work. Assignments and home works would render me sleepless nights.

But then I came to love mathematics. With a love like no other. Don’t ask why. I also do not know. I have been asking the same question myself and until now no answer would satisfy me.

I admire all my mathematics teachers. Each one of them had a unique way of delivering what mathematics is. Each one of them knew how to make students learn the best possible way. They all influenced the love I have for mathematics. When I was young, I told myself that if I would be given the opportunity to teach, I would emulate the good qualities of my teachers. Put together, I am the sum of how my teachers had been. In class, there were memorable moments and there were forgettable moments. In any way, these were mathematics moments. Vividly, I can still recall those moments and they come alive each time I am in class with my students.

I was amazed with the story of a young boy and how his teacher wanted to make the class busy because he would be marking some papers, and asked the students to get the sum of the first 100 natural numbers. Immediately, this particular boy came up to the teacher and gave him what he wanted. The teacher expected an incorrect response, of course. But he was stunned. In the paper was the genius of a young boy! The teacher might have spent the rest of the class time verifying the result. Do you think it is impossible? The young boy happened to be Friedrich Gauss, later to be regarded by mathematicians asPrince of Mathematics. How did he do it? Let’s take a look at it. The problem is as follows:

1+ 2 + 3 + 4 + 5 + . . . + 97 + 98 + 99 + 100 = X.

Find X.

Can you give it a try?

When I gave the same problem to my class and without the use of calculators, the students thought it difficult. Most of them struggled to add the numbers up one by one while others attempted to use some strategies.

Some started at the other end of the set of numbers.

100 + 99 + 98 + . . . + 3 + 2 + 1 =

Observe that there are 99 additions to be performed. But the problem remains the same. Yet students see that the order of numbers when adding does not really matter. This is associative law.

Some students worked on pairing the numbers.

1 + 2 + 3 + 4 + 5 + …+ 45 + 46 + 47 + 48 + 49 + 50

51 + 52 + 53 + 54 + 55 + …+ 95 + 96 + 97 + 98 + 99 + 100

Does this help? Not much.

How about this?

1 + 2 + 3 + 4 + 5 + …+ 45 + 46 + 47 + 48 + 49 + 50

100 + 99 + 98 + 97 + 96 + …+ 56 + 55 + 54 + 53 + 52 + 51

Do you see anything fascinating about this? Observe that the sum of each pair is the same and there are 50 of such pairs. That will give us 5050. This is what the young Gauss showed his teacher. Isn’t it beautiful?

It is all about recognizing patterns. To recognize patterns is one way to appreciate mathematics. It is like admiring a beautifully stacked pile of books and how the books spiral its way up. How could each book hold and not fall away? Wonderful!

A ketupat is a type of rice cake from Malaysia and is wrapped from woven coconut leaves. Weavingketupat is an art. The manner to weave it is mathematical in nature. Do you know how to weave aketupat? It is quite interesting and they say that it is a dying tradition. Start weaving ketupat, save the tradition and love mathematics!

Have you read Da Vinci Code? If you do, then you have come across the golden ratio. Golden ratio is a special number in mathematics. For a long time it baffles mathematicians and it becomes a source of fun and entertainment to most. How do we get a golden ratio? Take a line. Call it line A. Then have another line, call it line B. Divide line B into B’ and C so that the measure of segment B’ to line A is proportional to the ratio of segment B’ to C. This ratio is prevalent in nature. And that is the beauty of it. Look at our faces and try to verify the truthfulness of this claim. You can also check other parts of the body, things you see around and be amazed.

One of the places that caught my fascination in Penang is KOMTAR. Short for Kompleks Tun AbdulRazak (the 2 nd Prime Minister of Malaysia), Komtar was once the tallest in Asia. Located in the heart of Georgetown, it is a 65-storey complex and houses the government’s state offices. The structure of the building shows symmetry. Look at the way the vertical lines go and how each circular floor turns. Symmetry exists around us and it is a thing of beauty. Symmetry is a concept in mathematics that is common in nature. Look at yourself in a mirror; whatever you see on the right side can also be seen on the left side. That is, line symmetry. Symmetry can also be found among plants and animals. It is a good idea to introduce the concept of symmetry to students with something they experience and relate to.

I also love the games my teachers do in class. Well, they are not actually physical games but something that will incite our minds and excite our senses. Have you heard about the math claps? Math claps are hand claps following a rhythmic pattern. Try this clap patterns:

1-2-123-1-2-1234-1-2-123-1-2-1234-12

1-2-123-1-2-1234-1-2-123-1-2-1234-1-V-1

1-2-123-1-2-1234-1-2-123-1-2-1234-1-X-1

Another one that I fancy is the hand game called History Repeats Itself. It is about showing some numbers using your fingers on one hand and your partner has to name the number following the pattern as shown by the other. At the beginning, the person showing the finger numbers gives the names, the other person should discover the pattern in this. Then, it motivates me a lot since it made me think of how to figure out the names and discover the pattern. Though it seems meaningless, it is about developing thinking and enjoying it.

Paper folding and cutting is one mathematical activity that I enjoy most. It fascinates me to see a piece of paper turns into a fine work of art. It involves attention to details and consideration to forms. A mistake will make the entire thing wrong. This is a good exercise to teach students the value of patience, persistence and perseverance. Do you still remember how to make a circle or a square? It sounds easy, but can you do it without using a pair of scissors or a compass? Take a piece of paper. With it make a cone. Cut the cone horizontally. Open the paper. What do you have?

Who could ever forget the amusing questions in class? I am not referring to mathematical questions per se but questions that provoke thinking and require logic. How will you throw a ball and make it return to you without touching the ground? A ball fell into a vertical pipe with an opening only as wide as the ball, how will you get the ball? Suppose five days before the day after tomorrow was Wednesday, what day of the week was yesterday?

Students find lessons interesting when anchor on a discussion about relationship. I do this in class. Take the case of a right triangle. There are three sides in a triangle and each side defines a characteristic which is essential in a relationship. The base refers to a good foundation. In a relationship, anything can be fixed if it is grounded on a strong foundation. The height or the vertical line maintains the growth each of you should have while in the relationship. Both of you should grow and learn from it. The hypotenuse, the longest side, is the trust between each other. Trust is built upon sincere communication and this communication must be maintained. Students always find this lesson quite interesting and informative. During the lesson, they would be intently listening and asking questions.

There are different ways to learn mathematics. Some teachers are so strict that students find it stressful in class. There are good and bad things about being strict. The good thing is that students are forced to study. But students study not because they like it but because they are afraid of the teacher. The bad thing is that we can never guarantee if the learning is meaningful to the students.

Now that I am a teacher I emphatize with how students feel. I understand them. I believe that students can feel when you care for them and show them great concern.

It is a noble intention to touch students’ lives because it will make all the difference. I do not claim to be a good teacher but I strive to be one. As much as I could, I try to give them the best that I am and share with them the best that I am shared with.

Life is a journey. And this journey comes with a purpose. I am a teacher. Let me be. Give me that opportunity to make them appreciate the beauty that is mathematics. Give me that freedom to make them conquer new horizons and greater possibilities.

Mathematics is life. And life is devoid of mysteries without mathematics.

When we just open our eyes and see the world around us, then that is the only time we can believe that everything is mathematics.

Mathematically yours,

Jerome