The
Library for the Interactive Study of Mathematics is an Internet-based free library
of interactive mathematical and scientific explorations. It was funded from
1995-1997 by the National Science Foundation (NSF ILI-LLD DUE 955-1273). And
its plan is to offer a neutral resource on the World Wide Web that teachers
and students may use in a variety of ways for both classroom and private study.
The library gives free,
on-line access to explorations written by teachers (and sometimes by students)
that can support student-directed, individual and collaborative learning in
a wide range of topics encountered frequently in undergraduate mathematics instruction:
from college algebra and discrete mathematics, through single and multi-variable
calculus, linear algebra, and differential equations.
This Internet library
makes available to teachers all over the country a growing collection of interactive
explorations that they may freely download to their machines, use for classroom
demonstrations, and distribute to their students as laboratories or as directed
readings. This means that the explorations, together with the program (Mathwright
Library Player) that is needed to read them, are provided instantly, and free
of charge, over the Internet to any teachers or students who might desire to
have them.
In the first four years
of service to the academic community, we have learned much that is of value,
and the New Mathwright Library, both in structure and in function, reflects
what we have learned.
But there is still much
to do. We plan to modify our focus and to develop and provide many more secondary
school level explorations in both mathematics and science, that will serve the
growing population of pre-college students who use the world wide web for study
and research. The second (frankly more ambitious) goal is to offer resources
and infrastructure to teachers that will help them put web-based courses in
place in the coming years.
In order to help fix
ideas on the philosophy and goals of the Mathwright project, I will say a little
about the way I use the term: "interactive exploration." Interactive explorations
are lessons, laboratories, simulations and exercises that are designed to be
used by students (either individually or collaboratively) on a personal computer.
What distinguishes them from ordinary texts (or from "pushbutton-interactive"
materials developed in the style of CAI) is the support they give to participation
and visualization.
To me, Participation
means the active involvement of students in the construction of their knowledge.
The strategy often attempts to recapitulate the development of key mathematical
and scientific ideas by formulating problems in ordinary language, and encouraging
students to discover what is needed to refine that language and thereby to reconstruct
the formalism that it is intended to teach. This goal may be attained in a learning
environment that supports and encourages experimentation within the context
of simulations that give the answers to questions that students themselves formulate.
It works well, for example in an environment that supports collaboration and
discussion among students, such as a laboratory extension of a course.
Participation in this
sense puts students more in control of the pace and, to some extent, the direction
of their learning. It is an immediate consequence of Piagetian epistemology
that students are better prepared to understand the answers to the questions
they themselves ask, than they are to understand the answers to questions that
are asked for them. Interactive explorations, in the sense described here, encourage
students to experiment and to ask their own questions.
In order for this to
work, a delicate balance has to be struck. It is necessary to give students
the tools they need to formulate and ask their questions without "over structuring"
the activity so that their involvement becomes superfluous. Visualization refers
to the formation of stable and coherent mental pictures of abstract constructions
and processes, especially constructions encountered in mathematics and the sciences,
such as the notion of a Riemann integral, or the meaning of angular momentum.
But the constructions need not be mathematical; for example, biological processes
such as meiosis, or abstractions from chemistry such as covalent bonding, also
require visualization in this sense.
This ability to visualize
new constructions and processes requires practice and experience. The best students
enter their courses with refined and exercised habits of visualization, and
these students usually progress smoothly and quickly. But many students are
forced to spend much of their time "discovering" the right pictures. And with
the accelerated pace of many introductory courses, this often frustrates them
the first time through. Computers as heuristic tools can provide real support
both for visualization and participation. The role of interaction here to support
visualization is subtly different from that considered in the support of participation.
Here, it is a matter
of helping students who may not know what questions to ask form meaningful ideas
about key constructions and processes. This is often accomplished through a
rich variety of examples, simulations, and pictures at many different levels,
and by allowing the student to select and vary those examples as need be to
"see" the concepts they exemplify. The current Library holdings represent a
first approximation to this goal. The collection is not intended to be a heterogeneous
"database" of mathematical materials.
The collection intends
to represent and (if possible) to promote a point of view. The authors of the
books generally share this point of view (although, of course, not universally)
and you will find it reflected in many of the books in the Library. These books
are generally designed to support both participation and visualization on the
part of students. I have taken from my unique and continuous collaboration with
teachers over the years (as co-director of the MAA sponsored Interactive Mathematics
Text Project) several useful ideas. These I learned at the numerous workshops,
and in the review of over 200 curriculum and laboratory projects that teachers
(at the secondary and collegiate levels) have so far developed.
These ideas have been
incorporated, to the limits of my ability, in the present environment, Mathwright
2.1, which is the reader program for the Library. The Mathwright Authoring Program
itself has undergone many stages of refinement, in order to place into teachers'
hands a tool that they could use to translate their pedagogical ideas into genuinely
interactive teaching and learning environments that could support participation
and visualization in the sense described above. This meant that two conflicting
goals had to be satisfied.
First, it was necessary
that the authoring environment be "author friendly," so that teachers (who in
general are not programmers) can easily translate their ideas into interactive
explorations. This was done by creating an "object-oriented" interface that
allows teachers to design their books from the top down, page by page, object
by object, and script by script. It allows teachers to "cut and paste" pieces
of other explorations into their own, and in general, to build their books incrementally,
step by step, tending to details as that becomes necessary.
On the side of mathematical
functionality, there is an object-oriented Mathematics Scripting Language (a
graphics and computer algebra language) called MathScript that allows teachers
to create and manipulate mathematical objects (exact or decimal mathematical
expressions, functions, symbolic equations, matrices, dynamical systems, polyhedra,
surfaces, sprites for bitmap animation, and so on) in the same way. Thus object-oriented
support is provided both for the page design of interactive explorations (which
often contain videos, midi sound, hypertext navigation, and other "multimedia"
features) and for the mathematical functionality that makes the interactions
"smart enough" to answer a wide variety of student questions, even questions
not anticipated by the author. The second, and conflicting, constraint is that
these books must be easy to use by students; they must be "student friendly."
This means that students
should not be distracted from the pedagogical purpose of the text by strange
syntax or by clumsy interface. It is enough to place them in a mathematical
microworld where they are to think creatively about the mathematics. The objects
of that world should be convincing, and compelling (in the sense that they elicit
exploration). That cannot happen in general unless students can focus attention
on the content of the microworld. The Library thus represents, in its books,
the balance of a dynamic tension between conflicting goals. It is not (and will
never be) a finished product. No library is. We at the Library will continue
to depend on you, our readers, to help us refine and shape the collection in
accordance with the needs of learners. And we will welcome your comments and
suggestions, and especially your contributions to the Library in coming years.
James E.
White, Ph.D.
Mathwright Library Director
If you came here from the old building,
