Sam H. Jones
The development of mathematics, from its very beginnings, has been about problem solving. Whether it has been commerce, trade or navigation we have always used early mathematical explorations of our world for very practical purposes.
Although the title suggests that Einstein’s theory of relativity may be incorporated into the unit, it will only be mentioned in passing. One of the main themes of the unit will be the role of mathematical modeling in describing our world. Despite more advanced theories, Sir Isaac Newton’s model is still widely used to this day.
The study of standard gravity, planetary motion, and the motion of tides, in addition to the quest for knowledge about the world around us, was required in order for human beings to be able to better navigate, and map, the vast world around them. It is in this context, a practical applications approach, in which I would like to develop this unit.
Metropolitan Business Academy is an inter-district magnet high school in New Haven. The students have primarily chosen to pursue careers in business. Historically these students have done better on the verbal portion of standardized tests such as the CAPT. It is the intent of this unit to draw upon these superior verbal skills to teach about the Fundamental Theorem of Calculus and periodic functions.
Rather than focus on the abstract and procedural nature of the subject, the unit will attempt to put a human face on what is a very human endeavor. The unit will use the story of Isaac Newton, and his discoveries, to illustrate the scientific process and the role of mathematics in that process. Additionally, the unit will emphasize the problem solving and practical aspects of the whole enterprise. By having students “discover” ways to solve these problems in a historical context we hope to facilitate a deeper and better understanding.
The seminar will allow me access to source materials which will be invaluable in the preparation of this curriculum unit. The materials will be used to develop a curriculum which is relevant to students of business in the practical application of solving real world problems.
The New Haven calculus curriculum covers the Fundamental Theorem of Calculus and the Precalculus curriculum covers periodic functions. The unit specifically directs students to solve real world problems. As with many - if not most - mathematical discoveries, Newton’s work was developed in conjunction with solving real world problems. In this case it is the study of the universal force of gravity and the effect upon planetary motion and tides on Earth. The unit will be used to teach students in this problem solving, real world, context.
The running theme through the unit will be the contribution of Newton’s discoveries of universal gravitation as it specifically relates to standard gravity, planetary motion and the role of gravity’s influence upon the ocean tides. Although there are more modern theories of gravity, calculus was developed concurrent with Newton’s theories of motion and gravity. As such it is more appropriate for study in an introductory calculus course.
Newton’s work in explaining and predicting ocean tides will be used in a Precalculus course demonstrating the practical application of periodic functions. By making and recording observations students will learn something of the scientific method. Students will also come to better understand the role of mathematics in that process. The power of mathematical models will be demonstrated.
When Newton’s
Principia
was published, it was thought by many to be impenetrable. Others thought it was a step backward by invoking some sort of demon or supernatural force. Scientific thinking (particularly Descartes) had quelled some of the importance people attributed to invisible spirits, but Newton invoked an invisible force called gravity, a force which ruled the apple, the Moon, and the Earth, and which caused the tides. Newton could only maintain that it was not occult. The essence of gravity was not something that he or anyone else understood, but it was demonstrated by mathematics. “It is enough,” he wrote, “that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.” For Newton, the mathematical model was sufficient.
The physicist Richard Feynman liked to tell a story about how when he was a little kid, he asked his father, “Why do things fall?” As an adult, he praised his father for answering, “Nobody knows why things fall. It’s a deep mystery, and the smartest people in the world don’t know the basic reason for it.” Feynman liked his father’s answer, because his father realized that simply giving a name to something didn’t mean that you understood it. The radical thing about Galileo’s and Newton’s approach to science was that they concentrated first on describing mathematically what really did happen, rather than spending a lot of time on untestable speculation such as Aristotle’s statement that “Things fall because they are trying to reach their natural place in contact with the earth.”
The fundamental theorem of calculus, in physical terms, states the relationship between acceleration, velocity, and position (or displacement). Given the function of one it is possible to calculate the others. The unit will use falling objects and other motions to demonstrate the principle.
Moon phases and the tides may be modeled with periodic functions. Students will collect and use data to derive periodic functions to model this behavior.
