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THE MATHWRIGHT LIBRARY NEWSLETTER, Nov 2003, VOL 5,
#7
A publication of Bluejay Lispware
James E. White, Editor
The official publication of the New Mathwright Library and Café:
In this issue:
Visit the new Mathwright Theater
What
are Problem Objects ?
1) Visit the new Mathwright Theater
Mathwright
Microworlds are a little different from what you may be used to. They are
different from textbooks, because you can ask them questions, and they will
often answer you. They are different from various forms of Applets because
they tell coherent stories over
many pages, and they remember what you have done as you move from page to
page. You may also read them either online or offline, as you wish. But until
recently, visitors to the Library could have only static, text-based descriptions
of our Microworlds and WorkBooks. That has changed.
Thanks
to streaming Flash™ video, you may now visit our Mathwright
Theater (or from the home page) and see in your browser what our Microworlds
look like and -- what is more important -- how they behave. Of course, videos
also lack the key element of interactivity, but they will communicate what
words cannot do easily. And we hope that they will encourage you to take the
next step and peruse a few free Microworlds at the MATH Cafe.
In
fact, we would like to accomplish two things with these videos. First, we
would like to give you the experience of being in a Microworld, and of seeing
how, for example, our 3D Graphics objects can support intuition, and at the
same time can help you visualize a complex situation. For that, and always
faithful to our belief in the value of play, we show you a valiant (but unsuccessful)
effort by yours truly to defeat Mathwright in a game of 3D Tic-Tac-Toe. Why
Tic-Tac-Toe? Because anyone who has been there knows that it is more difficult
(and more interesting) to make an attractive and intuitive game board than
it is to teach the computer to play.
We use
OpenGL 3D Graphics for this, and we hope that you will guess
that those same 3D graphics provide excellent service to visualization in
many of our more "mathematical" Microworlds, in topics ranging from
Special Relativity and Hyperbolic 2+1 spacetime geometry, to Spherical Logo
(and Riemannian Geometry), to the physics of docking with a satellite. Visit
the 3D Gallery for some more examples.
Another
Microworld that we visit in a video at the theater has the innocuous title:
How to Draw a Star. In this Microworld
by Dan Kalman and Angela Hare, you will see how the visual and dynamic environment
of this 5-page Microworld can elicit (and answer) a range of questions from
elementary geometry to modular arithmetic. The powerful idea in this Microworld
is that a reader can move back and forth among the pages, bringing new knowledge
to old questions, and that this reader can engage the Microworld with questions
and constructions that make sense to her. We hope you will enjoy the movie,
and then visit the Microworld yourself.
The
second thing we hope to accomplish with these videos (and with more to come)
is give a guided tour of the Library itself. Our Welcome to Mathwright
video gives a synoptic view of how the Library is organized, and how to get
started using it. Later videos explain in more detail the various options
readers have: to read Microworlds in their browsers, or to build their private
collections on their own computers to read offline, to search for their books
in various ways, and so on. We hope that you will find that the Mathwright
Theater is both informative and entertaining.
2) What are Problem Objects ?
The
October, 2003 issue of the Maths CAA Series in
the English journal "LTSN Maths Stats and OR
Network" features a Mathwright Library article entitled The
Anatomy of a Problem Object. It begins with a discussion of our Microworld
SAT
Math Practice and Tutorial 1
and goes on to discuss Mathwright "Problem Objects," which are a
natural extension of the objects we built for the SAT Microworld.
That
Microworld uses 25 "problem objects" and it generates an essentially
unlimited number of multiple choice tests that students may use to practice
for the Mathematics section of the SAT. Students may read the microworld online
in their browsers in "test mode," in which case they are given 30
minutes to complete a 25 question test. At the end of that time, or whenever
they complete the test (if sooner), they are taken to a "review page"
that displays their overall score, the time used, and that reports which questions
they answered correctly, and which they did not. At this point, the student
may do one of two things. She may choose to enter "answer" mode,
in which she may visit any problems to see the correct answers to each one.
The
real pedagogic promise for problem objects is the possibility of creating
tree-like taxonomies of problem types that can guide a student through a series
of increasingly sophisticated problems. There are many examples of strategies
that require prior understandings and skills in order to be properly assimilated.
These strategies can be organized and illustrated in a tree of problem objects
that are chosen in advance.
Approaches
such as this have of course been in use for decades. They are a common feature
of CAI methods. Problem objects can, however, offer something new. It is possible
to equip problem objects with the capability of providing step-by-step explanations
of how each solution was obtained. Of course one may always give abstract
explanations, but it is also feasible to provide such explanations using the
generated parameters of each activation of a problem object. In addition to
this, problem objects may allow the student to vary parameters herself and
see the step-by-step solution to her variation of the problem.
This
idea, which has guided the overall development of Mathwright, has already
been implemented in several Microworlds by Mathwright authors. It is inspired
by Jean Piaget's observation that the learner only ever acquires new understanding
by asking (and answering) her own questions. As long as computational environments
ask the questions for the learner, then tell her whether her answer was correct
or not, they cannot teach. A teacher must listen for a student's question,
and, at that moment, give a meaningful response.
Please
visit the article online: The
Anatomy of a Problem Object if you would like a glimpse of where
we are going at the Library.