^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Thoughts on a Geometry Center Materials Development Program ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The following is a position paper on what I think the Geometry Center Materials Development Program might be. The views expressed herein are solely my own, and are being circulated with the hopes that your comments will help improve them as we attempt to construct a proposal to be submitted to the NSF. The paper is longer than I would like, but this promises to be a complicated undertaking.
Do note that some of the activities which might interest us may not be supportable under the Materials Development program of the NSF. Certain very worthwhile activities--perhaps some aspects of educational research and teacher training--may need to be supported elsewhere. Nonetheless, for the moment we should be concerned with constructing the best possible program, and be prepared to make hard choices and further efforts when the situation demands.
Gene Klotz 7/28/93 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Table of Contents PURPOSE STRUCTURE FOCUS THE MEDIA HOW? who? our role WHAT? ROLES how we'll interact with some associated groups (1) researchers in mathematics education (2) other materials developers (3) publishers roles of some special needs in our program (1) gender, race, the handicapped (2) assessment (3) testing TECHNOLOGY computer programs computer platforms video manipulatives telecommunications PERSONNEL (1) needed (2) interested parties TIMELINE APPENDIX: AN EXAMPLE ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ PURPOSE The proposed Geometry Center Materials Development Program will provide a natural and convenient means to harness the geometric visualization power of the Center for the purposes of education, and in particular to provide an interface to the education community whereby the intellectual power of research mathematicians can be well utilized.
STRUCTURE We have in mind an atelier for materials development, patterned loosely on the existing Center, with programmers, software developers, mathematicians, classroom teachers, appropriate visitors (sabbatical, apprentice, short-term, etc.), all toiling together within a context which allows for both free interaction and focused work.
Of course, there are many differences between software development for schools and for research mathematics. For the former much more time is necessary for re-design to meet user needs, and to obtain a good user interface. Debugging must be more thorough, and documentation and ancillary support materials carefully prepared. To the mix of participants it is necessary to add persons from mathematics education, testing and assessment, publishers, teachers, and students--a computer classroom will be necessary.
FOCUS The program will focus on three-dimensional geometry. As several conference attendees independently pointed out, kids live in a three-dimensional world, and they often appear more interested in spatial than planar geometry. Intuition and judgments about three-dimensional objects--and about these objects depicted on a two-dimensional screen--is called for on computer monitors in industry, in scientific visualization, and on video displays in many aspects of everyday life.
Unfortunately, three-dimensional geometry has all but disappeared from both the school and college curriculum. Fortunately, most of the present large curriculum development projects deal directly with three-dimensional geometry, and it was the subject most frequently mentioned by the project representatives at our conference in which they would welcome assistance.
THE MEDIA We see three separate media which have great potential to help students and teachers learn about three-dimensional geometry: computer programs, videos, and "manipulatives"--and these hands-on models have an especially important role to play in understanding how to interpret two-dimensional images as three-dimensional objects (perhaps when used in conjunction with practice in drawing). We will discuss this further under "Technology" below.
HOW? How can we be sure of producing materials that will be widely used? We will work with interested groups who are in the process of developing curricula and are interested in the educational possibilities afforded by new technology, and who see the need to train students for a world permeated by technology.
who? In particular, we are following up on the interest expressed by representatives from a number of the large NSF sponsored curriculum development groups: Mike Battista of the TERC project, Phil Wagreich of TIMS, Jim Middleton of NCRMSE, Johnny Lott of SIMMS, and Bill Berlinghoff of the Hartford Alliance. We will explore possible collaboration with the other NSF groups, as well; members of the planning team will hold discussions with all of the other projects interested in possible collaboration.
We hope to work with other groups involved in curriculum development relating to three-dimensional geometry. In particular, the Connected Geometry Project at EDC has expressed an interest in collaborating with us. We would also welcome possible involvement with others to develop three-dimensional geometry for a community college course, and with authors of college geometry texts.
In a different direction, geometric visualization has become an indispensable tool of research scientists, and their programs are filtering down to the college level, and also to schools (for example, there are several projects devoted to this being supported be NSF's Applications of Advanced Technology program). Science education frequently needs mathematics beyond that formally studied by the students, and this will be exacerbated in computer visualization programs, which demand mathematics such as multivariate statistics, partial differential equations, and the like. Following Jeff Weeks' rallying cry of "intuition before theory!", we would like to explore working with scientists to develop videotapes which could give students a sense of what is going on mathematically in scientific visualization, and which mathematics courses study these topics. If mathematicians fail to meet the new needs of the scientific community, we run the risk of scientists attempting to meet these needs and succeeding indifferently.
our role In general, our plan is to share ideas and to develop technology in close collaboration with curriculum development groups, but to have these groups develop the curriculum. We do not plan to undertake curriculum development on our own, since we have no experience and lack appropriate personnel. Moreover, the standard curriculum lacks the space for the course we would be most in a position to develop, one on three dimensional geometry (there are battles already being waged to introduce statistics, discrete math, and business math into the curriculum). Also, successful new curricula may obviate the need for such a course.
While we are reluctant to develop curricula, we think it important that we not only work with teachers to familiarize them with the new technology and how to use it, but that we help provide written materials and videos to assist in this endeavor. We will collaborate with teacher groups to develop such materials for teachers, and to develop technology-oriented classroom materials for students to augment what will be done by the curriculum development programs.
To this end, we are building ties with several teacher groups. Jim King has an organization spawned by a Regional Geometry Institute. Herb Clemens is involved with a cross-section of elementary schools in Utah whose teachers and students could examine, try, and react realistically to our materials. Martha Wallace has a number of geometry teachers trained in the St. Olaf program. Naturally, we also hope to use our excellent contacts to develop a local Minneapolis teacher group so we will have teachers and their students available for immediate testing and feedback. The Center is already beginning to work with the local base being established in Minneapolis for the Interactive Mathematics Project (which is attempting to provide meaningful contexts for students to use computers as part of their problem-solving strategies).
Jim King suggests that we subcontract out to some teacher groups to write materials and train other teachers. We invite interested parties to submit rough ideas and the financial implications by early September, so we can include this in the preliminary proposal. This could be a very helpful approach to disseminating our materials.
WHAT? There are some ideas on what three-dimensional geometry to teach in the NCTM Standards (even down through the elementary grades) and members of the curriculum development projects have thought hard about these issues. However, with new expressions of interest from those involved in research in mathematics education, with the Center's experience in technology and contact with research mathematicians concerned about mathematics education, with the advent of entirely new types of software since the Standards were written, and with the need to develop software and other materials of broad applicability, we think it necessary to further examine this topic. Consequently, we propose holding a small, focused, reflective conference on this subject late next summer. Groups and foci would include o mathematicians: what three-dimensional geometry should be taught? o members of curriculum development groups: what are they planning to teach? o researchers in mathematics education: what is known about learning three-dimensional geometry? what would we like to know? what do they hypothesize? o teachers: what are their reactions to the discussion? what do they think? [SUGGESTIONS NEEDED FOR CONFEREES!]
We would also like to hold a conference Fall, 1994, whose main guests would be scientists willing to speculate on: what three-dimensional geometry is needed for science? when? what for geometric visualization? Some persons representing the groups at the previous conference would also be invited. [SUGGESTIONS NEEDED FOR CONFEREES, EVEN MORE!]
ROLES how we'll interact with some associated groups (1) researchers in mathematics education In his paper, Alan Hoffer mentions the scarcity of investigations that deal with space geometry, as compared with plane geometry. He calls for a research agenda for space geometry, and cites both new research methods and powerful computer-related tools as contributing to the timeliness of the endeavor. Rich Lehrer has mentioned that the challenges presented are new and provocative, and several other persons from the discipline have expressed potential interest.
It is our hope to make use of the special possibilities afforded by the proposed program to work closely with researchers in mathematics education. Their involvement can strengthen what we do and at the same time contribute to the progress of their discipline. We encourage interested parties in mathematics education to suggest specific programs for fruitful interaction. Time is of the essence, since we must get a preliminary proposal in by mid-September.
(2) other materials developers There are others who have already made important contributions to technology for teaching three-dimensional geometry. Alan Hoffer has some pioneering and still very relevant software. Key Curriculum Press has some videos (with associated workbooks and manipulatives). Jean-Marie Laborde has announced that a three-dimensional version of Cabri will soon be out. There is also some usage of CAD programs, and of programs such as Maple, Mathematica, and Theorist, even though they are not aimed at the pre-college market.
We wish to foster cooperation with other materials developers, and plan to make available to them information gleaned from our work with the curriculum development groups and our other resources. Those who are willing to give as well as to take will be welcome to visit the Center and mutually explore technical and other problems. There is important and difficult work to be done, and it needs to be done in a spirit of openness.
(3) publishers Publishers have important roles to play in software support, materials dissemination, and market assessment. Moreover, the publishing industry seems on the verge of a major revolution in which electronic multimedia and hypermedia will partially replace paper books. This offers great promise for the dissemination of new technology (and will possibly allow different versions of the same material, aimed at usage with different resources, such as various kinds of computers). It has even been suggested that we foster multimedia development so that text will routinely be integrated with the means to visualize it. In any event, we plan to keep in contact with several forward-looking publishers and to involve them early on and whenever appropriate.
roles of some special needs in our program (1) gender, race, the handicapped It is necessary that we produce materials which are supportive of women, minorities, and the handicapped. It seems possible that among our personnel we will lack specialists in at least some of these areas. Perhaps the best way to meet our goals is to have appropriate consultants. Consultants with much experience in technology and three-dimensional geometry may be hard to come by. Your suggestions will be most welcome!
(2) assessment If our materials are to be broadly used by busy teachers we must provide assessment tools which will be easy to use with our new technology. We might even try to improve upon the current situation--Dick Lesh asked "how do we keep the tests from screwing up what we are teaching?" He's potentially interested in working with us to help answer the question. His colleague Mark Hoover has also expressed some interest, and we would like to hear from others.
(3) testing We need to be able to undertake frequent simple testing to see how users react to options and interface, and to get some sense as to whether our materials achieve teaching goals (a topic which we expect to crop up more elaborately in the work of the mathematics education researchers).
TECHNOLOGY computer programs For a foundation, we plan to construct an engine which is modeled after the relevant features of Geomview, the Center's very successful visualization program. Many of our software activities undertaken with curriculum development groups would then involve writing an appropriate front end for this engine. This approach would allow us to achieve highly vertical software--that is, users could work with different versions of the same underlying program in different grade levels.
Moreover, this approach might solve the problem of what should be published and what should be free: the engine could be given away to be used by other developers (and by experienced teachers), while the programs with their special user interface would be sold by publishers (who would thus be reimbursed for the necessary user support and dissemination efforts they would make).
computer platforms In the rapidly changing world of computer technology, which platform(s) shall we focus upon? Shall we go for the large installed base of Macintoshes, or the Power PC, the RISC version of the Macintosh which will soon be out? Perhaps PCs running the highly regarded NeXTSTEPS operating environment? Maybe we should work with one of the long-awaited cross-development systems for the Macintosh and PC together? There may be a version of Macintosh's hand-held Newton or equivalents which would be the way to go. Or how about Nintendo, which combines very good graphics, low cost, a huge user base in student homes, but none in the schools? Or their chief rival Sega, which has a smaller user base, but is soon to come out with impressive sounding virtual reality?
My inclination is to hold off on the final decisions as long as possible because of the volatility of the situation. Whatever we choose, if we take care (and are lucky), porting our programs to different platforms should involve little more than porting our engine (which will of course be designed with portability in mind).
video Whatever platforms we choose, not every schoolroom will have them. In many instances we may be able to supply useful (albeit non-interactive) visual experience with video. As we develop a new computer application, we will become highly experienced in placing and moving important examples of the objects being studied in the environment. The Geomview program is convenient for real-time animation on the sophisticated hardware at the Center, and since our engine will be patterned after this program, we will be able to make relevant videos almost as a byproduct of our software development.
Groups have already expressed interest in using the Center's resources to make their own videos on both school and college levels. Such activity could fit in well with our intended program of working with visitors. More "professional" computer animated videos might be undertaken when warranted, both for the educational value and to strengthen the available video resources.
manipulatives As mentioned, we see these as imperative. Many of the curriculum development projects will have their own recommended favorites, and we will work with them to help illustrate and make tactile the ideas we are trying to convey. In addition, we will make sure that our programs have the capability of printing paper nets for objects which can be constructed therefrom, and at appropriate scales to match objects such as spheres for which nets are impossible but which are readily available in a variety of sizes (pingpong balls, basketballs, etc.)
telecommunications We will make strong use of email and Internet-accessible newsgroups throughout the project. This position paper is being posted on the Geometry Forum newsgroup in an attempt to include a broad base in our planning, and we will continue to be open in our discussions throughout the project. We plan to more closely link the wide variety of persons involved through this medium, to encourage broader participation in the program, to allow non-attendees to be included in conferences, and "outsiders" to participate in open discussions. We will use telecommunications to offer better support for teachers (and to receive better feedback).
PERSONNEL (1) needed We anticipate that the program will require at least four high level programmers. At least one should be a specialist in user interface design, and at least one should focus on the design and execution of the basic engine. We will also need a separate system administrator (who would also be in charge of the computer classroom), and quite likely a video person (although final postproduction on videos might be contracted out). We may well need a writer and a person skilled in desktop publishing. A multimedia specialist may even be necessary.
Mathematicians, teachers, and those in mathematics education are likely to serve as intellectual parents to the various projects undertaken, but they are unlikely to be working on the project full time.
To administer the software development, the interaction with the curriculum development groups, teachers and teacher groups, publishers, assessment and testing, etc., the project needs a director and an associate director. Appropriate support personnel are of course needed.
Not all of the above need to be full-time, nor do all need to be continually in residence at the Center--although there should always be enough to maintain the feeling of an atelier in which the whole is greater than the sum of its parts. Some of the personnel might be shared with the existing Geometry Center, providing some relief to a hard working and over-extended staff.
Careful thought will have to be given to an appropriate advisory board, since this is a project which is very broad in scope. There may have to be sub-boards specializing in some of the aspects, or a special representative local group when fast action in a person-to-person meeting is called for.
(2) interested parties At the moment we have expressions of interest from Bjorn Felsager, Nick Jackiw, and Jeff Weeks to serve as experienced programmers/software developers. We'd love to hear from others, and we also need to cultivate a list of potential junior programmers (we already have several possibilities).
Both Arnie Cutler and I (GK) are interested in serving in some administrative capacities.
Please don't think that anything is filled, just because a name is there. We need as many expressions of interest and nominations as we can find!
TIMELINE Summer 1994. Program funded and preparation begins. Conference on what three-dimensional geometry to teach. Work begins on design of the basic software engine as soon as possible.
Fall, 1994. Conference on what to teach for scientists and scientific visualization. Lots of interaction with curriculum development groups (who seem too busy for this in the summer). Work begins on several software projects as soon as possible.
Spring, 1995. Jim King and Herb Clemens spend some time at the Center helping to work on issues of teacher training, support, and dissemination. Tom Banchoff may be there for a portion of this year, as well. (Other people, too, let us hear from you).
Summer, 1995. Carefully work with some teacher groups with the modest amount of software and videos we've produced.
The future is shrouded in the mists of time, but hey, this is only a position paper.
APPENDIX: EXAMPLES There follow a number of sketches (most very brief) of ideas from various curriculum development groups. They will be filled out in discussion with the groups.
Mike Battista has submitted the following idea for a simple program to be used in elementary schools. I particularly like its potential for helping students think abstractly and to envision the depiction of three-dimensional objects on a two-dimensional surface.
Initial Specifications for Cube Constructions and Perspectives Program
Linked to several 3d geometry units of the NSF funded Investigations in Number, Data, and Space curriculum.
Goals: o Help students visualize and "mentally construct" 3d cube configurations o Engender mental operations required in the mental construction and enumerations of 3d rectangular arrays of cubes - coordination of different orthogonal views of faces so that one coherent mental model of the rectangular prism of cubes can be constructed - construction of composite units of various ranks, units that can be used in both spatial and numerical operations
Features: o There is a "manipulation window" in which isometric drawings of unit cubes can be moved around with a mouse; cubes are created by clicking on a palette icon. o The current front, top, and side views of cube configurations in the manipulation window appear in real-time in three separate windows. (Objects in the four windows are the same size.) The user should be able to hide these orthogonal-view windows. o There is a capability to group any set of cubes so that they can be manipulated as a unit. (That is, composite units of cubes, and even composites of composites can be formed.) o Any composite unit of cubes can be replicated by selecting it and clicking on a button. o Any composite unit can be translated by moving it with the mouse. Such a unit can also be rotated 90d about an x, y, or z axis through its center. o Capability to put a wire frame rectangular prism in the manipulation window so that it can be filled with cubes. The user inputs the dimensions of the prism. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Phil Wagreich has different interests for an elementary school program. He would like to be able to work with "CAD for kids"--with as many features as possible available from the type of program that Johnny Lott expressed interest in below. It would appear that he would prefer a wide variety of tools available to call upon, rather than a more sharply focused instrument.
[A middle school program will be put here, perhaps some of the ideas that Jim Middleton began to sketch, such as a 3D wire frame program.]
[Perhaps it would be better to have a single, more detailed high school program; I'll speak of two possible directions]
Bjorn Felsager discussed the problems of approaching a three-dimensional geometry program with equations--it is difficult to develop a user-friendly interface, and there are many possibilities: implicit equations, parametric equations, lists of vertices for polyhedral objects, etc. (Moreover, it's not clear that many users will have much need for formulas!). Instead, he recommends using the geometric characteristics of the object.
One would have a palette of standard objects--sphere, cone, cylinder, torus, box, pyramid, regular polyhedra, etc., which are in standard positions. The prototypes could then be manipulated by changing their geometric characteristics by means of translations, rotations, dilatations, reflections. One would want to be able to use the boolean operations of union, intersection, and symmetric difference on these objects to create new objects.
In addition Felsager would like to be able to handle metamorphoses from one object to another by means of interpolation (for parametric equations this could change a rectangle into a moebius strip; for lists of vertices this could transform one polyhedron into another). Taking into account the multiplicity of vertices, this metamorphosis should be able to transform a polyhedron into its dual.
Johnny Lott had a somewhat different list of desiderata, which we hope could be accommodated by a different program built on the same engine. (Rather than a different program, we also wish to consider having the same program be able to call separate tools according to user need--a type of horizontality we would like to achieve with some of our programs).
Lott was interested in coordinates, on having convenient vector representations, and on a CAD program which was easy to use and cheap. Several others also mentioned the need for CAD-type programs.