|DOING INVESTIGATIONS: A RESOURCE BOOK FOR GET & FET MATHEMATICS & SCIENCE EDUCATORS|
Introduction to the book and its theme: investigations
Introduction to the book and its theme: investigationsEdit
The theme of this resource book is investigations. Investigation is important to science, mathematics, engineering and technology educators because investigation lies at the heart of those disciplines. In the natural sciences, we investigate phenomena in our physical and biological environments, both the seen and unseen and both the very large and very small. In mathematics we investigate abstract ideas that arise from our chosen axioms. We look for mathematical relationships between variables that we study and we seek mathematical models in the quest to find the "best fit" between physical reality or human activities and the data we collect about them. In technology we investigate problems arising out of human "needs and wants" and we seek the best or optimal solutions to those problems. We are homo sapiens not so much because we are wise but because we are forever trying to become wiser and to know and understand more and better than we did before.
How do we know this? Well, just start by looking at children and the way young, pre- school children will probe, examine, try out, explore and inquire if they are left to themselves in an unfamiliar but friendly environment. What children do is natural to the type of being we are. Unfortunately the way we teach children - more than the actual content of our teaching - often removes the natural spirit of inquiry more than encourages it.
We can also look at the lives of any of the great scientists, mathematicians, engineers and technologists who could also be regarded as inventors. Some of them may have been motivated by their pride, their competitive nature, a desire to "be the first" or the desire to claim authorship of an idea and be recognized by their peers and by history. Others may have been driven by commercial motives and what they stood to make from their discoveries and inventions. But the deepest and most common motivation of all scientists, mathematicians, engineers and technologists finds expression in three, simple, human needs:
1. the need to know more;
2. the need to understand better; and
3. the need to find solutions to problems (and the need to make a device work well).
Six great mathematical and scientific investigatorsEdit
We can speculate about what drives someone to wrestle doggedly with a problem for years and years, even when no obvious solution is in sight. Take someone like Andrew Wiles, the young mathematician who found the solution to perhaps the most famous mathematical problem of the past thousand years: Fermat's Last Theorem. Despite having got it wrong when he first announced that he had solved the problem (and despite the scorn of some other mathematicians at the time) he returned to the problem (with encouragement from his friends, it must be said) and found the correct solution.
Why did he do it? Why did he sit for years in a small study and forego beautiful spring and summer days, the company of friends, the laughter of children and the excitement of great world events? What drives humankind to such extraordinary and single-minded feats of the intellect? [A very readable account of Fermat's Last Theorem and the quest to prove it can be found at [www.prometheus.demon.co.uk/01/01/fermat.htm] . An interview with Wiles is reproduced at [www.pbs.org/wgbh/nova/proof/wiles.html].
Johannes Kepler studied data about the planets for over 20 years, (1596 to 1619) before he was able to formulate his three laws of planetary motion. What Kepler found was a set mathematical formulae or a mathematical model that could fit the available data. However, he only explained that planets moved on the orbits they did and not why. It took one of the greatest scientific investigators of all time, Sir Isaac Newton, to give an explanation for the planets' elliptical orbits through his invention of the calculus as a mathematical tool to help him solve the problem.
Newton did not think to publish his results until he was urged to do so by the astronomer Edmund Halley, after whom the comet was named. Why? Probably, for Newton, his reward was the thrill of discovery after years of investigation and the simple satisfaction of knowing that he knew. Newton investigated many things as he fed his insatiable curiosity about the phenomena in his universe. He is rumoured, during his investigations of light and sight, to have inserted a probe into his eye socket to discover how his eye was fixed in it. His eye took several weeks to recover and one hopes that satisfying his curiosity compensated for his discomfort! He also tasted enough heavy metal compounds to give himself mercury poisoning, a fact which probably explains his legendary bad temper, paranoia and erratic behaviour at certain times of his life.
Michael Faraday was a man of little formal schooling and was self-taught in the ways of scientific investigation. Yet he discovered how to generate electricity using a moving magnet inside a coil of wire. Magnetic induction, as the effect is known, is still one of the most important, practical discoveries ever made in physics. "But what's the use of it?" asked Mr. Gladstone, then Chancellor of the Exchequer (Finance Minister) and later Prime Minister of England. "Why, sir," replied Faraday in a flash, "there's every probability that you'll soon be able to tax it." But taxes and commercial interests were furthest from Faraday's mind during his years of investigation which led to his discovery. He consistently refused the honours and riches that would have been lavished on him had he chosen to accept them. In 1875, six years after Faraday's death, the German scientist A.W. Hoffman, wrote
"...Faraday belonged, from the universality of the benefits conferred by his genius on the human race, not merely to the island of his birth, but to all the civilized countries of the globe ..."
Perhaps the greatest investigator of the modern, western world in the past millennium was an Italian painter, sculptor, anatomist, engineer and inventor called Leonardo da Vinci. Some have called him "the Renaissance Man". (The word "renaissance" means "rebirth".) For many centuries in Europe a tight lid had been kept on the spirit of investigation. It was as though the authorities, both religious and secular, believed that if people were allowed to investigate and question freely it would result in chaos and anarchy. The same authorities brutally discouraged people from asking awkward questions or from investigating the unknown. What was known was known, they seemed to say, and everyone should be satisfied with that state of affairs. But someone forgot to tell da Vinci and he studied and investigated everything that caught his interest. And that was just about everything. He dissected dead animals. He dissected dead humans. He made such accurate and detailed sketches of the muscles in these bodies that they have never really been improved on. He drew plans for a winged flying machine after studying birds in flight, parachutes after watching falling leaves and a type of helicopter after studying seeds falling from pods. He never built them because the light, strong materials he would have needed did not exist as they do today. But he imagined and designed them nonetheless. And Leonardo da Vinci lived from 1452 to 1519!
Albert Einstein's particular genius was to investigate crazy questions in his mind. "What," asked Einstein, "would I see if I traveled on a beam of light?" By using his imagination, he found an answer to his question, which surprised even him. But his belief in his conclusion, even though it contradicted what every other theoretical physicist believed at the time, led him to state "imagination is more important than knowledge." Another great scientist of his day, Max Planck, explained his observations of heat radiating from an object by cautiously suggesting that energy might occur as tiny, separate "packets" rather than a steady, unbroken stream. He was so disturbed by his own conclusion that he immediately began looking for ways to disprove it. Again, Einstein used his amazing imagination to prove Planck's theory.
Investigations in the curriculumEdit
So how do we justify investigation in the curriculum? Easy. Investigations develop important skills e.g. identifying interesting questions; systematically seeking answers to puzzling questions; and solving intellectual and practical problems. Too many of us spend our lives seeing our environments fleetingly through unquestioning eyes. We see, but we don't really look. Educators have the responsibility to develop learners' spirit of open-minded and active looking instead of just passively seeing. Today's learner, who is tomorrow's citizen, must learn to look at and investigate the world, not as a school activity, but because it is a life habit. "LOOK at that!" "Wow, that's interesting!" "I wonder why it does that? I would have expected it to ... ." "Isn't that pattern amazing? Is it perfectly symmetrical?" "Look how this has changed." "I wonder when ...?" and "what if ...?" and "how would ...?" and "why does ...?" and "why can ...?" and "why should ...?" And ultimately: "Aha! I have found it!"
In our science, mathematics and technology classes we cannot afford seeing without looking. We must nurture curiosity and we must turn curiosity into investigate-able questions. Learners must be taught how to ask meaningful questions and educators must make opportunities for their learners to answer them. Above all, we must encourage learners to dream. When you make the investigation of meaningful things a part of your curriculum, you will find it unlocking something powerful in your learners: a genuine enthusiasm for learning.