(What’s The Story) Triangle Glory?

Triangles. Lovely shapes whose angles measure a total of 180º degrees. They appear in seemingly tedious admission tests, Math books, religious icons, fashion, extravagant dishes, outlandish furniture, cars, lamp posts, stairs, wings, windows... In other words, they’re everywhere. Why are they so important? How come there is even a whole branch of Mathematics devoted to a special category of them (Trigonometry)? Why do we need to analyze them?
The answer is simple. Again, they’re everywhere. Since they are geometric shapes based on a particular angular arrangement, it would be quite impossible to imagine some of our daily actions without them. The movement of our fingers, the way we walk, the incidence of sun rays and the way Earth receives them to provide life; are all events related to triangles. Now, thanks to computers, triangles are the foundation of some of the coolest special effects or videogames. The word polygon comes to mind. A polygon is any geometrical shape with more than one angle. Get your Greek together: poly= many gon= angle. A collection of polygons may be calculated inside a computer to produce masterpieces such as… Toy Story, Avatar, The Matrix. Or whole scenes such as the stampede from the Lion King, the flying scenes of How To Train your Dragon or the expressions on Boo’s face. In terms of videogames, making a ball bounce, performing a shot, making a wall break, or moving a character is based on directions and angles. On the other hand, triangles have been closely related to concepts such as the feminine and the divine. The shape of a triangle may refer to a waistline, fertility, fragility or sensuality. In any fashion show or marketing campaign related to beauty, you will surely find the reference to an angle. Divinely, triangles represent Providence, the Holy Trinity or the Star of David (Israel’s symbol made of two intersected and opposite triangles). There is also a very interesting fractal known as Serpinsky’s Triangle.
So… do not hate triangles just because you need to study them to get a grade. Take a look at the world around you and see how many triangles you may find. It is just like finding Hidden Mickeys in a Disney Theme Park.

Triangles Everywhere:

Lost Generation: Hope

An excellent video recommended by a special friend of mine (she actually won a tie and a fancy watch for following me ;) ):

Purpose. It Is All About Purpose.

Today I was struck by the sudden question of a Math II (Algebra) student--Why do we need to study the properties of every single mathematical concept?. I did not have time to answer the question thoroughly, or as thoroughly as I wanted to. It is a natural question, although it may really shake the ground of any educator. It is one of those questions that truly makes us think about the purpose of our craft. After all, it is one of those questions we expect from students. The answer to it is that properties let us comprehend in a deeper manner any phenomenom or event that happens in nature. It allows us to appreciate its PURPOSE. Features and properties are what attract us about... ANYTHING. Whenever we like a film, or a piece of music, we may understand each one of them better if we dig into the production notes of that particular movie, or the story of the song's composer. Learning about the properties of equalities (in today's topic), broadens our vision about their use and application. It may seem dry and slow and times, and some properties may feel stupid, boring or even pointless. However, it is a Math Educator's responsibility to build upon those concepts to show students how such concepts fit in a wider picture. A piece of advice... try allowing students to formulate by themselves any mathematical property.

Mathematics is a huge Lego wall. It is made upon tiny conceptual blocks that are linked together in order to allow us to... move the world.

Today's Recommended Mathematical Concept: Fractal

Soundtrack
The Verve--Lucky Man
Collective Soul--December
Moby--Bodyrock
Moby--Flower

Pic: http://www.euphoria-magazine.com/images/stories/photography/100-wonderful-fractal-images/fractal-images12.jpg

Beginning the End or Ending the Beginning

Back on Friday I had the opportunity to guide the introductory workshop for prospective students who are about to take Tec's admission test. I clearly remember how I felt during those courses 10 years ago--nervous and overwhelmed. The main difference is that today High School students are separated from those of College. In my opinion, it provides a better platform since High School students learn and understand in a different manner from the more adult ones. Moreover, the learning environment tends to be more relaxed than it used to be since we are talking about teenagers in their mid 10s trying to overcome an intellectual challenge--by themselves.
A method that truly worked out was not only the obvious approach of giving away hints and tricks to answer a SAT-ish/GRE-ish admission test, but motivating. Talking about life, and the tests we endure in order to forge our character rings a bell within the students' minds. Not only do they think about the challenge, but they start to FEEL the responsibility and the possible option of TRULY succeeding.

Introductory workshops, especially the Math/analytical reasoning ones, tend to be solely focused on their subject. However, having a guide who understands what the students are going through generates empathy. Let me give you an example. One of the attendees ensured that he does not even have a slight chance of passing the exam. We listened to him, supported him, and during the second session he started to seem more calm towards solving an exercise. He even provided innovative points of view about how to solve an exercise! Yesterday he contacted me via Facebook telling me that he thought the test was easy :D

Another characteristic that an introductory course guide should take into account is the fact that, in terms of the students I had, they are about to experience a life changing event. Starting High School, and opening their eyes to the world may seem daunting to them. Thus, it is a good idea to talk to them about what awaits them.

I ended the workshop by showing them the EXCELLENT Steve Jobs commencement address. They loved it :D (I'll post it later ;) )

So, this weekend's lesson--practice and motivate. Practice and Motivate. PRACTICE AND MOTIVATE. If Math is the result of a long process of wonder and discovery by motivated individuals... then the way we approach it should pay homage to that fact. Let's not forget that, after all, Math is a poetic way of understanding the universe.

More of this tomorrow...

Ken Robinson--Schools Kill Creativity

A fascinating TED talk about why contemporary schooling rewards expected results, instead of fostering innovation. Worth seeing!!!

Simplifying, Complementing and Swans

Another day, another post. I reviewed operations with rational expressions today in the Math II courses. A concept truly mesmerized me for the rest of the day--the power of the number and the variable. Watching the whole blackboard being filled with hundreds of signs which provide meaning and order feels like composing. It feels like giving harmony and beauty to chaos. Moreover, it provides a framework that compels , seduces, and challenges the mind. Some of the most active students suddenly became quiet throughout the simplification of a long expression. This is an interesting lesson and aid when dealing with discipline and disorder. NEVER yell at the students under your wing. Only start writing procedures that deal with long and continuos simplification processes. They will definitely pay attention to the board. Such procedures also deal with passion and perseverance, something that new generations are somewhat lacking since everything nowadays seems immediately obtained; even success.

As for the Trigonometry courses, I decided to move a quiz that was supposed to be answered today for next Friday. This is also an interesting method--never force a quiz unless there is a huge need for it. Try teaching concepts patiently. Let the students grasp their meaning. Let them wonder, and allow them to find their own conclusions. Then, put them to the test. They will show a more positive attitude than when threatened by the "immediate quiz".

I also went to the movies today, and watched the critically acclaimed Black Swan. It is truly a coda of close ups. Although I would not consider it as Aronofsky's finest, it is a peculiar take on the Black Swan tale. Questions such as Is being good the answer for happiness? Why is evil so seductive? Is perfection absolute control or absolute deliverance? come to mind. The music; well... Tchaikovsky at its best.

Today's soundtrack:

Oasis--Stop Crying Your Heart Out

Augustana--Boston

Jimmy Eat World--The Middle

Of Details and Flowers

Ok, here it goes. The first sessions I'm blogging about happened during Valentine's Day :) During the Math II courses (Algebra), I reviewed how to factorize in order to perform operations of rational expressions. The first class went smoothly. I asked students about the purpose of identifying how to factorize a expression. During my High School and College years there was not a single teacher who told me about the cognitive processes involved while factoring. Well, here is my take. While we try to factorize an expression in order to simplify it, we are developing an EXTREMELY IMPORTANT technique. Its name??? ATTENTION TO DETAIL. Why is it so important? Well, it is not only important. It is actually EVERYTHING. Think about your favorite movie. Your favorite song. Your favorite book. Think about the experience of using an iPod Touch, a Blackberry, or updating your Facebook status. Think about the clothes designed by your favorite fashion designer. They are all made up of tiny details that provide a wonderful experience. For example, you have a date, and you try to impress that girl or that guy... most surely you will use a strong perfume/cologne. Attention to detail is one of those concepts closely related to beauty, happiness and perfection. Knowing how to factorize is all about details... (is it a sum of cubes? a square trinomial?), and details mean everything.

The Trigonometry courses today were quite hectic since I had to review logarithmic and exponential equations, and explain concepts related to angles. An important fact that I like to stress in angles, and which makes students think is the fact that "the concept of a straight line is the proof of mankind's dominion upon nature". Think about it. Do you see any straight lines in nature? Is a tree trunk perfectly straight? The flow of a river? The sun? Grass? A flower? Animal fur? There are no straight lines in nature. Thus, mankind develops the straight line concept to build and master nature.

The songs I played today during each course (quite Valentine-ish):

Nickelback--Photograph

The Calling--Wherever you will go

Goo Goo Dolls--Iris

Savage Garden--Crash and Burn

Radiohead--Fake Plastic Trees

Oasis--Sunday Morning Call

Coldplay--Green Eyes

U2 feat. Mary J. Blige--One

Aerosmith--I Don't Want To Miss a Thing

Snow Patrol--Chasing Cars

A journey begins

I've always been fascinated by the idea of a Journey. Every step I take. Every sound I listen to. Every word I say. Every glance I take. Every letter I write; feels like a journey. I recently started to feel a mixture of what I like to call the "Forrest Gump Syndrome" and the "Rocky Training Process". The Forrest Gump Syndrome refers to the act of just running. For no particular reason. Just running because you like doing it. Personally, I feel like writing. I've experienced a bunch of events in my life that prompt me to write. On the other hand, the Rocky Training Process refers to putting oneself into a path of continuous discipline for a very particular objective. My personal objective is to bring the experience of teaching Math and Science into the web. I started this blog as a companion to fellow educators, students, and people who are in love with the art of living. No matter where you are, this blog is for you. I hope that it gives the reader several topics and ideas to think about for a couple of minutes a day. Topics and ideas to share, to inspire... Welcome to my blog.