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Webquest
The
Historical Roots of the Calculus
"As its campfires glow against the dark, every
culture tells stories to itself about how the gods lit up the morning
sky and set the wheel of being into motion. The great scientific
culture of the West--our culture--is no exception. The calculus is
the story this world first told itself as it became the modern
world."
From: A TOUR OF THE CALCULUS by David Berlinski
Introduction
Calculus is a subject many students dread. It is
the way mathematicians study moving objects, even fluids like the
movement of air across an aircraft wing that gives it lift making
flight possible. It was calculus that first allowed Newton to
describe the motion of planets. Calculus made modern science possible
and no physical theory has ever broken the link to the calculus.
Length, Volume, Area, these are static. If you want to study motion,
you must study the calculus.
"The overall structure of the calculus is simple.
The subject is defined by a fantastic leading idea, one basic axiom,
a calm and profound intellectual invention, a deep property, two
crucial definitions, one ancillary definition, one major theorem, and
the fundamental theorem of the calculus.
- The fantastic leading idea: the real
world may be understood in terms of the real numbers.
- The basic axiom: brings the real
numbers into existence.
- The calm and profound invention: the
mathematical function.
- The deep property:
continuity.
- The crucial definitions: instantaneous
speed and the area underneath a curve.
- The ancillary definition:
a
limit
- The major theorem: the mean value
theorem.
- The fundamental theorem of the calculus
is the fundamental theorem of the calculus.
- These are the massive load-bearing walls and
buttresses of the subject."
From: A TOUR OF
THE CALCULUS by David Berlinski
This exercise will briefly explore the historical
development of calculus from Zeno of Elea (about 450 BC) to Cauchy in
the 19th Century. Along the way you should gain an appreciation of
inquiry and discovery and the individuals and their world views that
brought about modern science.
The
Task
Review the list of activities below and the list
of resources provided. Write down any ideas or questions that you may
want to research.
You will be working in groups of four. Each member
of the group will read three of the twelve biographies in the
resource list and write a paper on each one answering these
questions.
- What mathematical principles did this person
discover or refine (include any examples in graphical
form)?
- How were these principles related to the
historical development of calculus?
- How did this work depend on the work of
others?
- Why was this inquiry and discovery in
mathematics attempted at that time?
- How long did it take for the discoveries to be
applied or to be validated in other ways?
- Stress their motives, successes, and failures
in the study of mathematics.
With your group develop a time line showing the
mathematicians and their contributions to the development or
refinement of the calculus.
Return to the eight foundations of calculus
mentioned in the introduction and place these on the time line
as accurately as you can.
After you have completed your time line , read
"An Overview of the History of
Mathematics" and read one of the following:
- "Mathematical discovery of the
planets"
- "Orbits and
Gravitation"
- "A Brief History of
Cosmology"
Participate in discussion at the conclusion of
this project about what you learned and be prepared to share your
understanding of how scientific and mathematical discoveries are
made. Have an answer ready to the Challenge Question at the end of
the article: "An Overview of the History of Mathematics".
Write a one page reflection of what you learned.
Include comments on the last two readings.
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an example of an integration problem and a differentiation problem
for each of the pictures below. |
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Water
transportation and service |
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A
construction crane |
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Microwave
communications |
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Population, traffic,
and pollution |
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A model of the
International Space Station. Az. Science Center |
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The
Process
and Evaluation
Three class periods will be allotted to do your
research, but after-class time will be necessary to complete your
writing assignments.
One class period will be allotted for putting
together your time line and comparing it to the others in the class.
They will be displayed.
One class period will be allotted to a discussion
of the development of science, the calculus and some of the
personalities that helped change our view of the Universe. Your
written Reflection paper will help you prepare for the
discussion.
Your grade will be based on the
following:
60% ----------Your three papers answering the 6
questions in the Task
15% ----------Group grade for the time
line
15% ----------Your Reflection paper
10% ----------Participation
Grading Scale:
90 - 100%-----A
80 - 89%------B
70 - 79%------C
60 - 69%------D
below 60%----F
Conclusion
"Space and time are the great imponderables of
human experience, the continuum within which every life is lived and
every river flows. In its largest, its most architectural aspect, the
calculus is a great, even spectacular theory of space and time, a
demonstration that in the real numbers there is an instrument
adequate to their representation. If science begins in awe as the eye
extends itself throughout the cold of space, past the girdle of Orion
and past the galaxies pinwheeling on their axes, then in the calculus
mankind has created an instrument commensurate with its capacity to
wonder."
From: A TOUR OF THE CALCULUS by David
Berlinski
The calculus is not a subject to be avoided. It
has helped to give us a way of life undreamed of by past generations.
It is not the end of mathematics, but for many it has opened up new
worlds of thought and possibility. I hope this exercise has given you
an appreciation for the work and flashes of brilliance on the long
road to the calculus. You will be studying something that has a sort
of beauty in the realm of the mind. If this WebQuest has given you an
interest in the history of science and mathematics, you may return to
it at any time and submit work for extra credit with the approval of
the teacher.
Resources
for the
Webquest
Mathematicians
Other Resources
Notes
to the
Teacher
This WebQuest is for pre-calculus and calculus
students.
My goals were to:
- Have students demonstrate an understanding of
the historical development of calculus.
- Have students gain an appreciation of the
struggle of mathematical and scientific discoveries.
- Have students connect the abstractions of
calculus with the real world.
- Motivate students to study
calculus.
Links
Yahoo
Science and Mathematics
MacTutor
History of Mathematics Archive
Math
Forum Internet Mathematics Library
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