A Brief
History of the Library:
Many of you know that
the Mathwright Library has been at home on the web since 1994. Initially funded
by the National Science Foundation, its mission has been, and continues to be
this: to provide a free and neutral resource for teachers and students of mathematics
on the world wide web.
This resource consists
of our growing collection of Interactive Mathematics WorkBooks, all written
by teachers (or by students) and all designed with a single, simple purpose.
That purpose is to bring mathematical ideas to life, by allowing readers to
experiment and to play with the topics the WorkBooks develop.
At the heart of this
strategy is the basic idea that students understand the answers to their own
questions more readily than they understand the answers to questions that are
asked for them.
What are Mathwright
WorkBooks? The WorkBooks in the Library are often more ambitious than what is
available on the web in the form of "electronic textbooks" or of "pushbutton
interactive" modules. Our books attempt to be open-ended, allowing the reader
to pose the question that makes sense to her, and then to see the answer.
An Example: To
illustrate this, I'll discuss a 2-page book called "Slider," in the Calculus
2 Collection, that asks a simple question (one that John Bernoulli asked 300
years ago). Given two pegs on a wall, say, one slightly higher than the other,
and to its left, there are many ways to connect them by a wire. Given any such
wire connecting them, let a bead slide from the higher to the lower point. It
will take a certain amount of time. What shape for the wire will require the
shortest time of descent?
This problem begs for
experimentation. But how do you let a student try a wide variety of shapes for
wires? Slider allows the student to click on the screen, and determine thereby
a polynomial (of degree 1 through 10, depending on how many times clicked) that
connects the points and passes through the integer approximations to the clicked
points. If she opts to click no times at all, she gets the straight line connecting
the points. One click gives a quadratic function passing through them. Two clicks
give a cubic, and so on.
For each trial, Slider
prints the exact algebraic form that defines the function, and draws the curve.
Then it "slides" the bead from start to end, running a clock that shows how
long it took. It also keeps track of the bead's velocity allowing some students
to observe the conservation of energy (it always finishes with the same speed).
Students get to do the
experiment in this way, with increasingly complex curves. And Slider supplies
two special curves: the arc of a circle, and the "Brachistochrone" for comparison.
The Brachistochrone of course gives the shortest time and the straight line
connection fails miserably. So they ask (usually) what is a Brachistochrone?
And they are taken to
a page where the curve is constructed as hyperboloid by rolling a circle without
sliding on a line. From this, they may deduce the equation (in polar form) of
the Brachistochrone. And then they may vary parameters in the "rolling circle"
construction to see whole new families of curves. For these, they may "guess"
the equations, and their guess is drawn over the actual curve to give them feedback
on the quality of their guess.
The point of the long
description was to show that even in this simple problem, there are many learning
opportunities that might start with a student's question, or might be "seeded"
by a teacher's question. The visual (almost tactile) feedback can capture the
student's imagination. This virtual environment brings the geometry, the algebra,
and the dynamics to life in the student's eyes.
This would be a very
long newsletter indeed if I attempted to describe the strategies that our other
Library books use to achieve similar goals. And, it must be admitted, different
Mathwright authors use these strategies to different degrees, and in different
ways. At the Library, we believe that the potential in the hardware and software
for creating genuinely new and powerful Interactive learning environments has
barely been explored.
In the third article
in this newsletter, we'll talk more about that. We encourage our authors to
"test the envelope" and we hope that teachers who browse the Library for materials
for their students will judge the books with the same critical eye. A good question
is worth a thousand answers.
Discussion
Forum:
Another new feature
of the Library is the Discussion Forum. This is a threaded discussion in which
we hope visitors to the Library will participate. There are six main discussion
topics at present, and you may start a new thread in any topic, or respond to
an ongoing thread. You may get there easily from the Home Page by selecting:
"The Forum." The topics are:
1) Mathwright Library Books
2) 3D Graphics and Visualization
3) Simulations and Laboratories
4) Web Course Support in Mathematics and Science
5) Computer Pedagogy and Heuristics
6) Newsletters
The
notion of a discussion forum may be unfamiliar. It works this way. If you would
like to comment on what has been said, or to start a new thread, just do it
from your machine. The result will appear automatically in the forum once you
submit. It's as simple as that.
Web Course
Support:
Another new component
in the Library is the growing collection of Web Courses that we host. We will
talk more about this in article 3, but will say here that if you are interested
in teaching over the web, the Library may simplify the logistical questions
of how students access your Interactive course materials.
The Library Player is
free, and so all your students have to do is register in order to locate and
download the materials you park here. Of course, we do exercise editorial discretion
in what we publish at the website, because we would expect that others might
also access and download those materials.
Write us for our guidelines
at webcourses@mathwright.com
Library
Protocol: Why not Java?
Some visitors to the
Library may wonder why the WorkBooks are not "web interactive" meaning, I suppose,
that they should run "in the browser." The short answer to this question is
that browsers are not yet "smart enough" to answer the sorts of questions students
may ask. Java applets may do a few routine things, such as drawing graphs, but
much more is required. To use "Slider" as an example, let's consider how it
responds when the student clicks on the screen to draw an interpolating polynomial.
That is not diffcult, and an applet could feasibly calculate the interpolating
polynomial in short order. But then it would have to write the EXACT algebraic
function whose graph the polynomial was.
That is a different
matter, because now the applet must do exact symbolic algebra rather than floating
point arithmetic. And after that, it would have to print the exact rational
representation for the polynomial. For polynomials, it's not too dfficult, but
you can imagine that other expressions (say rational functions, or various compositions)
would be extremely difficult to print. This is why browsers do not even try
(yet) to pretty print algebraic expressions.
MathML won't solve the
problem by itself, because it will only enable humans to input mathematical
expressions into browsers. Eventually some program (client-side or server-side?)
will have to generate the MathML for the browser. But there is a bigger problem
than that. Once the curve is drawn, Slider simulates the sliding motion of the
bead, timing it with an onscreen clock. Now this means that even if an applet
could create the polynomial, and another applet would print it, then a third
applet would have to accept the algebraic expression for the curve, and solve
the second-order differential equation of motion while drawing (in real-time)
a sliding bead that follows that equation.
The problem I'm highlighting
here is one of intercommunicability. The various mathematical objects: exact
polynomial, function and graph, differential equation are related semantically,
and a system of Java programs that would use them would have to "talk" to one
another. But this means that a "language" has to be available for them all to
speak. That language might eventually be a Java Application, running on your
machine.
This is precisely why
we provide the Mathwright Library Player when you register in the Library. The
player contains the Mathematics Scripting Language: MathScript, that can regulate
all of the intercommunication that is necessary for the various sorts of mathematical
objects that are created in an interaction to intercommunicate. Sometimes data
needs to be printed in exact algebraic form, sometimes it needs to be printed
to a spreadsheet table, sometimes in needs to be graphed, and sometimes it simply
needs to move a sprite around. All of these things can happen within the context
of a language. And for the forseeable future (~ 10 years) the language will
have to be on your machine, because to place it on the server would spoil the
realism of the interaction. It would "break the spell" that might capture the
learner's imagination.
And finally, of course,
once a student downloads a book it is hers for as long as she wants it. A student
can return to it again and again as needs be to gain new insight, or to ask
sharper questions. This is the model of the Library. Using it, we have provided
some rather large and versatile learning environments. Once again, we welcome
you to the Library, and we hope that you will consider contributing to this
ongoing effort, this exploration of the possible in computer-based pedagogy.
You no longer have to settle for electronic text books. - J. White
2] A Library for Students
One
of the principal ways in which the New Mathwright Library extends the original
one is that we want to invite students to be a part of the project on every
level. The Discussion Forum is not only for teachers. Students are invited to
participate too. The first discussion Topic: "Mathwright Library Books" is a
good place to give us feedback on the effectiveness (or failure) of specific
books in the Library to meet your needs. It is also a good place to suggest
new material or explorations.
The "Math Playground"
is a shelf of books that are written for students in grades 3-5. And most of
those WorkBooks were written BY students. This is an exciting new possibility.
Those of you who are students, or who know students who might like to create
interactive books, please get in touch with me, James White, Library Director,
via email at students@mathwright.com
If some of you have
an interest in seeing a book that covers a specific topic that is not here,
do the same. I cannot promise that I will respond to all of your requests, but
I (and the other Library authors) are teachers. And you can be certain that
we will do what we can.
The New Library also
hosts a 'Puzzler' page with mathematical problems at a variety of levels. Anyone
is welcome to try them, but they are obviously designed for students. More often
than not, these Puzzlers will refer the problem-solver to a WorkBook that contains
a hint on its solution. We feel that these Puzzlers may invite the problem solvers
to look at some of the books that are perhaps not directly related to their
courses. And that's the point. Students may use the Library as they choose,
but it is our hope that some will come to browse, just to play and so to learn.
The shelves: "Math Exploratorium",
and "Games and Diversions" are especially designed for gratuitous play. They
hold books that are not designed principally to accompany courses. Students
will have the opportunity there to attempt to Launch a Space Shuttle, land a
rocket on the moon, play Mastermind against the computer (The computer can guess
codes!), or explore the Mandelbrot set, among other things. So, for the students
out there. This Library is for you. - J. White
3] Web Mathematics Courses: Where have all the teachers gone?
A friend who is teaching a calculus course over the web complained to me the
other day that he feels more like an "information manager" than a teacher. His
students essentially read their Calculus book at home on a CD and correspond
with him by email. The CD is little more than an electronic version of the text,
offering only in addition to the text, hypertext search and a few highly restricted
programs that illustrate the skills they are to acquire. They have no real opportunity
to ask questions in the "drill and practice" environment, because the programs
simply ask questions, and let the students know if their answers are right.
So of course the students
ask many questions via email, and the entire enterprise becomes so taxing and
time-consuming that this teacher wonders whether he is teaching at all, or is
inculcating "behaviors" that will elicit the correct response from the electronic
text. In this article, we take up a sensitive question, but one that asserts
itself with increasing urgency as web technology develops.
The Internet has become
(in a few short years) the most extensive and accessible source of "information"
in the entire history of our species. It is also rapidly becoming a principal
medium for intercommunication and transaction, that binds us together socially,
economically, and in many other important ways. It is inevitable to ask what
roles this medium may play for education. Teachers have of course been asking
this question, and have invented ingenious and powerful extensions of the classroom
on the web. These are places where students may do research, may "visit" libraries
and museums, and may in general avail themselves of the vast repositories of
information (in textual, pictorial, musical, and other forms) that exist out
there.
But what about teaching?
While there is much to be said for making factual information available to students,
we cannot neglect the importance of the guidance teachers have always provided
in the organization of facts, and in the interpretation of them. This is especially
true in mathematics, where knowledge is not facts alone, but is knitted into
patterns and habits of thought. Here, we require especially that learners actively
coordinate and assimilate their understandings into increasingly stable and
adequate conceptual structures.
Activity is the key
word. A student passes from the understanding of whole number in counting activity,
to the understanding of fraction (or eventually ratio) in measurement activity
only after a very long process of experimentation. To pass from that understanding
to the abstract notion of number as variable in algebraic and geometric activities
requires a further development and activity with representation of problems
with equations, or with geometric pictures that lead to equations. And to pass
from this idea to the representations of the properties of abstract functions
as representing relations between covarying quantities in the differential and
integral calculus requires a further conceptual development.
All of the steps in
this epigenesis of mathematical knowledge require, as we know, the active critical
exploration of the adequacy of current schemes of interpretation to new categories
of analysis. The steps must be taken more or less sequentially as Piaget points
out. Success with each new interpretive activity opens the door to new questions,
and to more sophisticated understandings.
The view that the learning
environment is simply a substitute for the textbook is short sighted and naive.
It is based on the idea that what students need is information, or the answers
to questions. But the role of the teacher must certainly be to generate questions,
not answers. This is because questions are much more important than answers.
In the first article,
we discussed the notion of "open-ended" explorations. These place students into
a context where they are presented with ordinary-language problems or situations
to analyze, and may experiment in the "what-if" style to discover the implications
of their own beliefs and hypotheses. For that, it is important that the environment
be expressive and flexible enough to allow them to experiment in the first place,
and to provide some answers to reasonable and relevant questions. Environments
of this sort can add a dynamic dimension to learning that is simply not present
in textbooks.
They can supplement
(not replace) texts with "experiences" rather than information. These "experiences"
evoke questions which lead to new experiences, and new questions, and so on.
It seems that one of the genuine promises of computer-based pedagogy is to make
such experiences available to students in a variety of settings. The web is
one such setting. Collaborative learning groups in laboratory classes are another.
For distance learning,
the collation of a good text with pregnant experiences, strategically placed,
is a possibility teachers will explore. To return to the original question of
the role of the teacher in web pedagogy, I suggest that teachers may design
their courses in such a way that the computer interface becomes principally
the place where students actively explore new ideas (individually, or in groups).
This means that the teacher, who understands the flow of ideas developed in
the text would actually create, at various points, interactive learning opportunities
for her students, in order to elicit certain questions from them. These might
be experiments, simulations, puzzles, and, yes, "drill and practice" exercises,
too. But the result would be an enrichment of the textbook (which the computer
environment might also deliver in hypertext and multimedia form, if that is
felt to be useful). The
Mathwright Library contains many examples of this strategy already. Teachers
might simply direct their students to these WorkBooks as a first step. But we
encourage teachers to become actively involved in the creation of new learning
experiences of this type. It may be easier than you think. The books in this
Library were, after all, written by teachers and students, not by programmers.
They use the authoring program: Mathwright Author 2.1, which teaches in a detailed
175-page tutorial everything a teacher needs to know to build such interactive
books. So, look at the books on the stacks. If you like what you see, think
about writing your own. Get in touch with us at info@mathwright.com J. White
If you came here from the old building,
