The Pythagorean Theorem

In the case of a right triangle, the square of the hypotenuse (the side opposite of the right angle) is equal to the sum of the squares of the other two sides.

The ability to access and add to collective knowledge is one of the fundamental qualities that differentiates human beings from all other species on Earth. Knowledge that is gained by one generation can be passed on to the next through teaching. It is not incumbent upon each person to rediscover all that those before them have come to understand. As time rolls forward, the offspring of the people of today can learn and then stand upon the shoulders of knowledge previously gained by others as they reach for and discover new things. In this scenario each person will at some point in life find themselves in the role of student and teacher. One of the defining measurements of a successful life is the longevity of a person’s ability to embrace the role of either teacher or student as each unique situation dictates.

I was about eighteen years old and working as an apprentice lather on the construction of a high-rise apartment building. My father, who was the foreman on the project had teamed me up with a man named George Barker who was an extremely gifted journeyman in our trade. In hindsight I can now see that my father knew that George would teach and he hoped that I would learn. One day George and I were trying to get the layout of a hallway off a parallel line running across the floor. George told me that we were going to use a little trick called the “three four five”. I smiled because I knew that, even without formal understanding, he was going to apply Pythagorean Theorem to solve the problem of the hallway. We used his trick and sure enough we were able to get the two sides of the hallway laid out perfectly parallel to each other. I then told George that his trick was not limited to using the specific measurements of three, four, and five feet. He looked at me quizzically as I began to demonstrate my knowledge. Using a piece of sheetrock like a blackboard I illustrated and explained the Pythagorean Theorem. Fifteen minutes later and very satisfied with my explanation I turned to George and said, “So do you understand?” He just laughed and replied, “Yea, three four five” and walked away. My explanation might as well have been in a foreign language. I did not teach George anything that day because I did not find the path to communicate my knowledge to him. George was a master builder who understood almost everything about construction including the practical application of the Pythagorean Theorem and yet I could not add to his knowledge. Although I could have been, I was not a teacher that day.

In order for one person to teach another, a form of communication must be established based on the common knowledge and experience of both parties. Difficulties arise out of the uniqueness of each person’s intellect and individual predisposition for acquiring knowledge. Teaching becomes the process of building a bridge to transport that knowledge from teacher to student. An effort must be made on the part of the teacher to discover already existing knowledge possessed by the pupil and through specific dialogue work back toward the new knowledge that they are trying to teach. By accessing knowledge the pupil already has acquired the teacher will create interest and confidence in the person they are trying to teach. Teachers that drone on in a specific subject-driven lecture have failed to find a common language for learning. While it is true that there will always be some who are of like-mind that will understand the lesson being taught, many will not gain understanding. This lack of understanding is not an indication of the student’s inability to learn. It is a failure of the teacher to find a common language to impart understanding. In its most elementary form, teaching is really the art of helping a person find a way to use what they already know to discover new things.

When I was in my forties I was given the opportunity to coach soccer to a group of ten year old children. Most of the kids were not very athletic but all were academically gifted. They all attended an extremely rigorous, bilingual school that taught all subjects at a high pace and level. When I would listen to them talk before and after practice I was astonished at the subject matter that they were already studying in school. Over the first year, I tried to teach them about the game and we all struggled with our progress. One day I decided that it was time for me to show them how to kick a ball for distance and accuracy. Kicking a ball was easy for me so I was sure I could just tell them how to do it and they would learn. I lined them up, gave some brief instruction, and let them start kicking. For ten minutes I watched as they kicked the ball everywhere except where they intended it to go. I gathered them around me and looked down at their frustrated, defeated little faces. In those few seconds that we stared at each other I had a moment of divine inspiration. It occurred to me that perhaps I could teach them about kicking by using their academic prowess as the bridge. So I asked them if they had studied the Pythagorean Theorem in school. Their faces looked up at me with a combination of confidence and childhood arrogance. Hearing their chorus of almost dismissive affirmation I simply explained that kicking a soccer ball was a real-life application of the Pythagorean Theorem. I now had their complete attention but more importantly I had turned on their intellectual confidence. We laid out cones in a right triangle and I explained that they should start at one end of the hypotenuse and approach the ball along its length striking it at the opposite angle. We began again but this time they all felt empowered by the knowledge they had already acquired in geometry class. Some had instant success and began to encourage the others. I just stood there and watched exceptional geometry students teach each other how to kick a soccer ball. For all of us the transformation from disappointment of failure to the excitement of success was nothing short of amazing. They had learned how to kick a soccer ball and I had learned a little about teaching.

Teaching is not an opportunity for the teacher to celebrate or demonstrate their own level of knowledge and understanding of a subject. Teaching is a challenge for the teacher to find the language to give knowledge and understanding to the student.