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 launch a space shuttle into circular orbit...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 watch Neil Armstrong take his first
“small step” to the surface of the Moon, you must ‘earn your wings’ That is
the game. This game is a 3-Dimensional Mathwright WorkBook that is 55 pages
in length, and to which the reader may return again and again until she also
walks on the Moon. It is much too large to include in the Library. But it is
available on CD, and runs in the same Library Player with which you read our
WorkBooks now. To learn how you can get it, along with the entire Mathwright
Library on CD, check out the page: http://www.mathwright.com/cdrequest.htm James
E. White, Editor
2] Computer as Heuristic Machine: The Aims of the Gravitation CD
Gravitation
is an Interactive Mathematics/Science book that is designed to be used by students
ages 14 and older for private, self-directed study and recreation. It breaks
so many of the rules of textbook design, that it will almost certainly be unusable
for traditional classroom textbook use. 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 term
"interactive" has been used to describe such a wide variety of electronic appliances,
from ATM machines to Web pages to Video games, that it has come to be almost
devoid of meaning.
In a sense, the electronic
media are defining for each context, an evolving meaning that will be appropriate
for that context. This depends, as always, on two things: the competence of
the hardware, and the virtuosity of the software. Much thought has been given
to the role that electronic media can have in educational computing. On the
one hand, those media are defining new ways to think about information sharing
and retrieval, and the blossoming of the World Wide Web illustrates with dazzling
clarity what effect a sudden shift of paradigm can have on our society. No one
can forecast how that will go, even into the next decade. On
the other hand, both hardware and software have arrived (via the new object-oriented
paradigms) at a place where they present genuinely intuitive and powerful interfaces
that make it possible for much wider segments of the population to have productive
interactions with the media.
But there has been a
certain ambivalence (perhaps even naive optimism) about the role of these media
in teaching and learning mathematics and science. Certainly, with the vast computational
power at the disposal of personal workstations and networks, one temptation
has been to focus the software on computational depth and perspicuity. This
is, for some purposes appropriate (for example the support of student research).
Another temptation, that so far has borne little fruit, was to build a "model"
of the learner's understanding based on performance, and to "tutor" the learner.
This latter is very
ambitious, given our lack of understanding of the semantics of problem-solving.
At this stage, we know so little even about our use of natural language, that
such roles for computers probably belong more to science fiction than to science.
But the short history of computational media can already teach us what realistic
goals for education lie on the immediate horizon. These media are fundamentally
dynamic and they can provide unique experiences for students that can support
the constructive elements of learning, and that can aid visualization. They
can do this by creating microworlds that bring the learner under their spell,
and encourage the learner to accept their rules and to adopt their reality.
When the rules are the mathematical and scientific conventions that we would
like to teach, then these microworlds can have great pedagogic power and heuristic
value.
It has been observed
by Jean Piaget, Henri Poincare, Rene Thom, Hans Freudenthal, and others that
children construct their knowledge of the world in a process that recapitulates
in many ways (as ontogeny recapitulates phylogeny) the history of key ideas
in their culture. This epigenesis is driven, to a large extent, by internal
(innate) factors, such as curiosity, the need to balance and disambiguate conceptualizations,
the use of language and logic to form ideas about causality, conservation of
substance, continued existence, and so on. These innate factors often get lumped
together under the vague rubric of "instinct" or "common sense".
And there are also external
factors in this epigenesis. Initially, for the formation of the "mother concepts"
of space, time and substance, the teacher is the natural environment (and our
senses), and we learn through continual experimentation and constructive synthesis
(what Piaget called "assimilation and accomodation"). Eventually, the teacher
becomes the social environment as manifest in language and culture, and then
the role of education is to transmit the cultural inheritance, and thus to preserve
the culture.
What does this have
to do with electronic media? Anyone who asks the question: "How do children
learn mathematics?" will come to the conclusion that, as important as the influence
of external factors such as education are, the internal factors in cognitive
development, that lead the child to a prelinguistic understanding of substance,
spatial relation, causality, correspondence, and so on, are more important.
These things determine the later articulation of those understandings in the
conventional and linguistic structures that we aim to teach in our schools (ratio
and measurement, covariation, graphical representations, and so on).
These internal factors
acquire meaning in the developing child's mind quite independently of formal
education, as Piaget points out. They are focused by the spontaneous playful
and experimental activity of the child, beginning in sensorimotor coordinations,
and continuing through constructive synthesis to the prelinguistic understandings
mentioned above that give meaning to measurement, spatial reasoning, and counting,
and eventually to a refined notion of number and geometry. What is true for
the developing child is also true, to a large extent, for any learner who embarks
on performances in unfamiliar territory (whether it be driving a car, playing
a piano, or solving algebraic equations).
In the beginning, there
is always a period when strategies are fluid and somewhat amorphous, when they
have not yet been focused by experience. And there is the attendant "anxiety."
One of the strong contributions that educational media can make in this regard
(Think of "flight simulation.") is to provide a variety of rich and dynamic
experiences that can, if they represent the content area well, help the learner
explore that content, to ask questions, and to obtain answers that can refine
her understanding.
This happens in two
ways. First, there is the support these environments can give to the internal
process, the spontaneous questioning (what if ?), and second, there is the reinforcement
they can give to the external, more formal, process of teaching conventions
and notations for articulating and communicating understandings to others. The
environment must be "responsive" enough to give answers to the learner's meaningful
questions, but it must also be "open" enough to encourage those questions in
the first place. It is, I believe, in these mutual capacities: openness and
responsivity that the meaning of "interactivity" for a learning environment
is seated.
In this view, the learner,
as the center of the process, actively constructs her deepening understanding
of the mathematical and scientific formalisms that we intend to teach by asking
at each stage just those questions whose answers can be meaningful for her.
This is the place, in their openness and responsivity, where the electronic
media can supplement and extend the more traditional educational media.
The aim of Gravitation
is to guide the student, but not to control the process of inquiry that can
lead the student to deeper and more balanced understandings. It creates, as
Seymour Papert described it, a "math-rich" environment (or "microworld") that,
in the best circumstances, can cultivate the student's own curiousity, and can
support her intuition. It aims to do this in a number of ways, all fundamentally
dependent (in a new way) on the use of the computer as heuristic tool. In particular,
it does not "package" its ideas in a sequential way, nor does it establish,
either implicitly or explicitly, specific behavioral goals for the student.
It rather attempts to create an environment that will stimulate questions, generate
student discussion, and will, to a large extent, provide an experimental context
in which students can give answers both to their own questions, and to the "seed"
questions that we pose.
In order to guide the
student in this process, Gravitation attempts to develop certain themes that
play a central role in the study of classical mathematics and physics. It does
this by providing readings from original sources, such as Galileo's Two New
Sciences, and Newton's Principia, that place students in the exciting historical
context in which the basic scientific problems were formulated. Further, it
poses a number of problems, presented in ordinary language, and refers students
to the readings, and to interactive laboratories where they can experiment with
the concepts and constructions. In this way, the student is in control of the
process
The topic we chose,
is of course a venerable one, and marks a watershed in Western thought that
led to major developments both in the history of mathematics, and in the history
of science. On the belief that, in some things, ontogeny really does recapitulate
phylogeny, we thought it worthwhile to allow the student to hear the voices
of the discoverers of these monumental ideas. The problems of gravity (and of
measurement in general), as formulated for example by Galileo, and in a new
and different way by Newton, have a marvelous simplicity and freshness that
modern formulations, laden as they are with our powerful notations and conventions,
often miss.
Students of mathematics
often have their first difficulties with new constructions and relationships,
not in concepts, where we expect them to be, but in the conventions and notations
that are necessary even to describe those constructions and relationships. So
we use the microworld as heuristic tool to address that problem obliquely. It
simply presents the ideas to be investigated in the language of those conventions
and notations. To the extent that a student has a grasp of the idea, this presentation
will reinforce intuition, and teach convention.
There are in fact over
twenty-five experiments to which the students have immediate access. All of
the experiments are accompanied by discussions that explain the questions those
experiments ask, and how the answers are interpreted conventionally. And these
experiments exercise and illustrate certain techniques of visualization (for
example, addition of vectors, various descriptions of conic sections, graphs
of linear and quadratic functions, Polar and Cartesian representations, and
so on). The experiments are organized in such a way that a student, depending
on her level, can arrive at either an empirical or a theoretical understanding
of Newton's deduction of Kepler's three laws of planetary motion. But the questions
we present, and the experiments that illustrate them needn't be pursued in any
particular order.
The student may choose
where to start, and what to read as she likes. As she answers questions successfully,
the student gains access to a number of additional microworlds which, while
like the experiments, have a more playful aspect, and which actually reinforce
and extend the knowledge gained in the experiments. Those exercises can, in
fact, take on the appearance of games of skill, but they are also designed to
give a realistic representation of the major themes of the book. So, for example,
in order successfully to launch a space shuttle into nearly circular orbit,
the student must come to terms with the idea of the balance of centrifugal and
gravitational acceleration. The Gravitation book attempts to create a spell,
and to bring the learner under that spell. It cannot completely succeed, of
course. But if it brings a few students to ask a few new questions, then that
will be success enough to justify the effort. James E. White, Editor
If you came here from the old building,
