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How even a child can make sense of Algebra- Lesson 1
Using imagination to deal with algebra
When I was first introduced to algebra as a boy I was very confused. The teacher had gone from talking about numbers to now doing mathematics with the letters of the alphabet. At the time the leap was too great and I found myself like many other boys in my class just plain ... lost. I am not sure why or how it happened but I began to create cartoons in my mind to make sense of this “new” mathematics. I created a vibrant cartoon world in my mind for the lifeless algebra being slung on to the board by a teacher that just droned away for the whole lesson.
My first discoveries about algebraic equations
You can only add like terms:
The equation X + X + X = ?, only made sense to me when I saw each X as a dog.
So the equation became Dog + Dog + Dog and the answer is 3 Dogs.
This then led me to correctly answer the original question X + X + X = 3X
You can’t add unlike terms:
X + 2Y + 3X + Y = ? Became in my mind
1Dod + 2Cats + 3Dogs + 1Cat
I knew that I couldn’t add a dog and a cat together because they were different animals so I just added the dogs to the dogs and the cats to the cats to get
4 Dogs and 3 Cats.
X + 2Y + 3X + Y = 4X + 3Y
Recognising like terms:
When I then came to more complicated looking expressions I could still apply the same basic rule for adding like terms.
X Y² + 3 X² Y + 5 Y² X + 2 X² Y + Y X²
The first step was to identify the different animals
X² Y is a chameleon ( remember what a chameleon can do?)
Y X² is also a chameleon
X Y² is a rabbit
Y²X is also a rabbit because it doesn’t matter the order in which you multiply two numbers.
X Y² + 3 X² Y + 5 Y² X + 2 X² Y + Y X² becomes
1 Rabbit + 3 Chameleons + 5 Rabbits + 2 Chameleons + 1 Chameleon
And that altogether is 6 Rabbits and 6 Chameleons.
X Y² + 3 X² Y + 5 Y² X + 2 X² Y + Y X² = 6 X Y² + 6X²Y
The Human Value in learning Algebra
This lesson in algebra taught me something about myself too. I realised that I had a great imagination and by using my imagination I could overcome any challenge that life placed before me. My imagination was like a stage upon which I could rehearse what I was going to do. It was safe to make mistakes on the stage of my imagination and it was fun. The lifeless numbers came to life and algebra started to become a dance with each number and letter having character and personality. The rules of algebra made sense on the stage I had created and my confidence increased.
Though we are all human and come from the same ocean of life we are each a unique wave. We rise into the air for a moment but always merge back with the sea when our energy is expended. Our personalities can be likened to the shape of the wave and each wave is unique. No two waves are alike. The wave that you are has never been seen before nor will it ever be seen again. Use the energy of your form to express you as you are and celebrate your wonderful existence.
We sometimes imagine that we are separate from one another in some way as we experience our lives as unique waves. Just as all the combinations of Xs and Ys in algebra are part of the alphabet. We all come from the ocean of life and we will all return to that ocean someday.
I went on to study Physics and Mathematics in University and also qualified as a high school teacher. I taught in schools from Ireland to Japan and though I retired from full-time teacher and now run my own private hypnotherapy practice here in the Isle of Man I still teach for a couple of hours at the local high school and tutor children who ask for my help. Every child has a wonderful imagaination and by learning how to use the imagination every child can succeed.
Before I left my home in Ireland to start my first teaching job in Yokohama Japan my aunt gave me a present. It was a sheet of paper with the following words on it by an anonymous author:
"Tell a child what to think and you make him a slave to your knowledge. Tell him how to think and you make all knowledge his slave".
These words have been my source of inspiration thoughout my career and continue to be my guiding light to this day.
Personal Math Tutor
Some years ago I created a program to teach basic arithmetic to children and I called it Math Tutor. I designed it to make children think for themselves and notice patterns that they may have missed in their normal lessons. Every child needs to also remember some basic information upon which to build and that includes learning their tables. The program includes this very importand element often neglected in the present day system of teaching Math. The beauty of the program I designed is that it is self correcting so the child cannot move forward without mastering what has gone before. This takes the pressure off parents to sit and monitor their children in a subject that may have been challenging for them. Each of my own children have used Math Tutor with excellent results.
To order Maths Tutor simply send me an e-mail.
I have also been very interested in how mathematics is treated in the Vedas of ancient India. The Vedas are the oldest sacred texts considered to be the storehouse of all knowledge written over 7000 years ago. I have studied Vedic Mathematics and found it brought meaning to the subject in a way I find hard to explain. Mathematics becomes part of the human experience and not something separate and only for those born with some kind of special gift. The techniques taught in Vedic Mathematics cultivate an intuitive appreciation for the subject and involves seeing the simplicity of even the most complex calculations allowing them to be done mentally. Calculations that would normally have taken a calculator or many lines of mathematics can be done simply in one line. The principles involved are easy to learn and encourage creativity so the subject is fun and exciting for the learner. I highly recommend Vedic Mathematics for anyone child or adult who wishes a completely fresh approach that ultimately leads to mastery.
It is high time that we as parents and teachers recognise how special our children are and how, rather than forcing them to conform, our duty is to encourage them to be creative. Education is not just for making a living, it is for life!
Links to Connected Articles by Spirit Whisperer on Mathematics
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- Solving Algebraic Equations Part 1
In order to simplify what I was being taught in Algebra my mind came up with cartoon like descriptions for the equations I was expected to solve. I found that there were only two kinds of algebraic equations....