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THE MATHWRIGHT LIBRARY NEWSLETTER, September 2002,
VOL 4, #6
A publication of Bluejay Lispware
James E. White, Editor
The official publication of the New Mathwright Library and Café:
In this issue:
Featured Microworlds this Month!
A New Way to Find Your Books
1) Featured Microworlds this Month!
Experience...The
Magical Gravity Tour
For
the Teacher or Parent...
The
Magical Gravity
Tour is an
Interactive Mathematics/Science Game that is designed to be used by
students ages 14 and older for private, self-directed study and recreation.
This 55-page, 3-Dimensional PlayBook can easily give the player weeks (or
even months) of recreational exploration and learning pleasure as she tries
to collect her trophies. Gravity aims to explore some of the promising new
capabilities of interactive media. In this way, it differs from traditional
texts. For example, it is not structured on a linear, sequential plan, but
places the learner in an environment where (with hints from Salviati, the
Instructor) she chooses which way to go.
It
can, however, be used as a resource to enrich mathematics and physics courses
at the Secondary and College levels, can support group projects and laboratories,
and can be used by an instructor to illustrate (with simulations) a number
of key ideas in the development of Calculus and of Classical Mechanics. Since
one of the aims of the book is to stimulate questions and generate discussion,
it seems that it would lend itself to a number of collaborative teaching strategies.
The
Microworld has a number of game-like explorations (such as a Lunar Lander
or a Space Shuttle Docking) that also teach. Further, students have the opportunity
to experiment with each of Kepler's 3 laws. If they choose to do so, they
may read a formal derivation of those laws from Newton's Principle of Universal
Gravitation, using Calculus, but the Microworld requires no Calculus to play.
In fact, it is intended to provide motivation for later study of Calculus.
See
more focused comments for teachers in our Cafe
article: The Magical Gravity Tour
For
the Player...
Imagine
that you wake up one day in 17th century Italy, and meet a teddy bear named
Salviati. Salviati is your guide to a great adventure in ideas. He explains
that he was a student of the Master, Signore Galileo Galilei, and was right
there when it all came together. You are in a 3D virtual world where you may
follow some of the footsteps of the Giants. He shows you the libraries where
you can read, as you like, the words of Kepler, Galileo, and of Isaac Newton.
And he shows you the
‘Questions’ that guided these men eventually to an understanding of the riddle
of Gravity. The questions were simple questions, and you are encouraged yourself
to find the answers. To answer these questions, you must do ‘experiments’
and these experiments will lead you to a deeper and richer appreciation of
ratios, algebra, and geometry in the service of scientific inquiry and measurement.
And the experiments are fun to do! You learn about inclined planes and pendula,
about ballistic trajectories and orbits of solar probes by setting them into
motion yourself, and measuring the results of various trials.
As you answer more
and more questions correctly, you gain access to more interesting experiments,
and you gain “trophies.” You may watch the planets move about the Sun in the
starry sky, or dock the space shuttle with a satellite...or land a rocket
on the moon. You may save your work and leave at any time, and when you ‘sign
in’ later, the program remembers where you left off and restores your trophies.
All of these experiments have to do with the unfolding puzzle of Gravity.
And the aim is to bring you to the place where you understand Newton’s derivation
of Kepler’s laws of planetary motion. You may understand this either experimentally
or, if you like, theoretically, using Calculus.
Whether you pursue
the Calculus or not, you will learn about the geometry and algebra of conic
sections (of which the elliptical orbits of planets are an example) the use
of Rectangular or Polar coordinates to represent motion, about conservation
of energy, escape velocity, and a great number of other things. But the path
you take through this wondrous place is your own to choose.
If you remember the
game Myst™, you know that half the fun is guessing which way to turn next.
Salviati explains that it was exactly that way for his Master. So no one will
tell you what to do. To arrive at the end, and to guide a rocket to the surface
of the Moon in 3-Dimensional simulation, you must ‘earn your wings’. That
is the game. This game is a 3-Dimensional Mathwright Interactive Web PlayBook
that is 55 pages in length, and to which the reader may return again and again
until she also walks on the Moon.
Salviati,
who is now a member of the Mathwright Library Staff, has agreed to answer
questions from Library Members about any aspect of the PlayBook (Science,
Mathematics, or Instructional Technology). Send him those questions if you
get stuck at the Talk to a Librarian
page. You can also get to it when you press Start
Here on the home page. You'll see it.
This
38-page Microworld is about the reflection and refraction of light rays. It
is suitable for both exploration and instruction. The material can be appreciated
by a wide range of students, from high school through college. There is material
on conic sections, trigonometry, visualizing "virtual" images, optimization
done both experimentally and with calculus, and the slope field of a differential
equation. One can use this book to motivate and investigate student research
problems. Step through these pages for hours of guided experimentation and
fun with the physics of Light.
This
Microworld is
a tool that you may use to row reduce matrices. In it, you may create a new
zero-initialized or randomly initialized matrix of any dimension. You may
also enter your own matrix, or modify the most recent displayed matrix by
editing it onscreen. Finally, you may augment the current matrix with the
appropriate identity matrix, so that you can generate the inverse matrix onscreen
as you row reduce.
The
row operations are:
- Exchange two rows
- Multiply a row by
a nonzero constant
- Add a constant times a row to another row
There
are two pages. On the first page, all operations are exact rational operations,
unless the reader enters decimals in the matrices. And on the second page,
all operations are decimal to the decimal precision that you choose.
There
is a "Help" page that explains how to enter your own matrices. On the main
page, the reader has an array of standard tools to implement the row operation
on the current matrix, or to start from scratch with a new matrix.
In this nine-page Microworld, you
will have the opportunity to explore and experiment with four topics from
Calculus:
- Defining and graphing functions and their iterated derivatives.
- Calculating and graphing Rational Taylor Polynomial approximations to arbitrary order.
- Creating series approximations for any transcendental functions that you define by specifying recursive relations on the derivative and the definition at a single evaluation point.
- Experimenting algebraically and graphically with these rational polynomial forms.
2) A New Way to Find Your Books!
As
our collection grows (it has now grown to 207 Microworlds and WorkBooks),
it becomes increasingly difficult to find just what you want in a hurry. That
is probably a good thing for those who have the time and patience to look
around, but it can be an annoyance if you do not. We have provided several
ways to search for your books. Among those are our 75 rooms, our listings
by title, by purpose and topic in the Stacks. And recently, we installed a
"Card Catalog" for more comprehensive searches, including text search.
Well,
now there is a new (perhaps quicker) way to find your books. If you click
the Start Here button on the Home page, you will be taken to a listing
of 28 new "classrooms" in the Stacks. Each book in the Library belongs
to just one such classroom. The classrooms are named by topic: