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MegaMath: Expanding and Connecting the Mathematics Community

MegaMath: Expanding and Connecting the Mathematics Community

April 28, 1995

Nancy Casey


Abstract


Mathematics is a live science with new discoveries being made every day. The frontier of mathematics is an exciting place, where mathematicians experiment and play with creative and imaginative ideas. Many of these ideas are accessible to young children, yet unknown to teachers and absent from the school curriculum.. Others (infinity is a good example) are ideas that have already piqued many children's curiosity, but their profound mathematical importance is not widely known or understood.

The MegaMath project is intended to bring unusual and important mathematical ideas to elementary school classrooms so that young people and their teachers can think about them together. Through this process, the live discipline of mathematics can be explored and teachers can become central figures in mathematics education reform.

See below for more information about this document and how to use it.


Contents

1 The Genesis of the MegaMath Project

2 The Central Question

3 Mathematics: Alive or Dead?

A Community of Mathematics Learners

Value of the Internet for Emerging Mathematics Communities

Using MegaMath

References

Author Information


1 The Genesis of the MegaMath Project


The MegaMath project was founded by two individuals: Mike Fellows, a theoretical computer scientist and mathematician, then at the University of Idaho, and myself, a former elementary school literacy specialist curious about mathematics. We met because our children were the same ages and in the same school.

Mike had been inventing games and puzzles for elementary school children that exposed them to topics of current research interest in mathematics and computer science. He wanted to help them experience the excitement and frustration of grappling with the unknown and to portray mathematics as the lively endeavor that it is [1]. I had been studying how young children's reading, writing, and speaking skills develop within the context of meaningful communication, and conducting observations in classrooms where teachers manage assessment, skill development, evaluation, and lesson planning in environments that are language-rich and student-centered.[2],[3],[4]

On the whole, elementary school teachers seemed to be far more comfortable with the language side of the curriculum than with mathematics. They were able to draw upon their vast experiences with language and literature to integrate reading, writing, social studies, science and art into a coherent and fluid classroom experience. In mathematics, on the other hand, they had only their (often dismal) school mathematics experience and one or two university courses of sometimes dubious quality as their foundation for lesson design. They could help students articulate their ideas, think critically, and engage in problem solving under the guise of ``language arts'', but lacking both mathematical background and confidence in their ability to learn and do mathematics, their mathematics lessons did not stray far from the workbook.

Many of the most creative and successful language arts teachers sensed that mathematics in their classrooms ought to be different, but they had no idea what that difference should be or how to bring it about. These are the teachers that we sought out as collaborators at the beginning of the project. Our goal was to dissolve the barrier that separates language arts and mathematics and explore a truly integrated curriculum.

2 The Central Question

The central question of our collaboration remains: What is mathematical knowledge and how can one establish a mathematically-rich environment in which it can develop?

Mathematicians describe and recognize patterns and make abstractions about them. They may use what they learn to draw conclusions about the natural world, or they may manipulate pure abstractions out of sheer joy. Drawing upon a common vocabulary, notation, traditions, and body of knowledge, mathematicians communicate with one another about their perceptions and insights. The language they use is mathematics. [5]
Using the above as a working definition of mathematics, it became useful to view the mathematics community as a cultural group complete with its own specialized language, modes of discourse, behaviors and power structure. From this perspective, mathematics education emerges as an acculturation process fraught with pitfalls of communication, access and power. Fortunately, however, these issues are not new to educators in multicultural and multilingual settings, so mathematics educators can benefit from their experiences. [6] I became particularly interested in questions such as: How do I connectthe mathematics that I wanted to ``teach'' to the ways that individual children already collect, organize, analyze and ask questions about information? Who controls the content of the school mathematics curriculum? What is the basis of elementary school teachers' alienation from and fears about mathematics? [7]

3 Mathematics: Alive or Dead?

In the United States and in all other nations who take their cues about trends in education from the US, the Curriculum and Evaluation Standards of the National Council of Teachers of Mathematics [8] is the primary reference for for forming and reforming the structure and content of mathematics education. Clearly the work of a committee, this work is a largely compilation of lists intended to provide a framework for designing school mathematics programs but not a roadmap for doing so. For every grade level, a list of about a dozen standards is set forth. Interestingly, at every grade level, the same four items appear:

These four standards get to the heart of what it means to do mathematics. At each level, the remaining standards are about specific content items that one is to do mathematics with---number operations, fractions, measurement, probability, variables. No doubt these are valuable, they have been a part of (western) mathematics for hundreds of years. If these topics and these alone become frozen as the exhaustive list of what mathematics is about, then mathematics is indeed a dead discipline.

MegaMath seeks to avert this unnecessary mummification of mathematics by making available to elementary school students and their teachers ideas in mathematics which are not on the traditional lists, which are of current interest in the mathematics community, and whose critical prerequisites are curiosity and a desire to experiment with something new. MegaMath makes no attempt to articulate what items do belong on any curriculum list, only to push the boundaries that traditionally limit what children and their teachers do in the name of mathematics.

Some of these topics in MegaMath include infinity, the mathematical theory of knots, coloring problems in mathematics, and ideas from computer science. Practically everything is from 20th century western mathematics , so topically, MegaMath has much growing to do. The approach, however, is one that lends itself to students and teachers doing mathematics together and on their own terms. It creates opportunities for analyzing, organizing, looking for patterns, making abstractions, asking questions, and communicating ideas. There is an abundance pictures and relative scarcity of numbers. Art projects, storytelling, drama, and activities for the playground form the core of what students do, and there are many suggesions and ideas for thinking about, discussing, and extending what one has done. Particular care has been taken to write for an audience that is has little or no experience with mathematics.

4 A Community of Mathematics Learners

An important aspect of mathematics as a live science is that mathematicians collaborate. Research mathematicians function within a large comunity where they exchange and discuss ideas, share their triumphs and frustrations, and encourage one another. As recognizers of patterns and manipulators of abstract concepts, we are all mathematicians. We differ in our levels of skill and passion, but we all can benefit by sharing ideas, questions, problems and solutions with one another. The unskilled can learn so much from contact with the skilled, and the skilled should not consider themselves above learning from the unskilled.

For years, language arts teachers have been creating communities in their classrooms where children share and articulate ideas through reading and writing. These teachers have access to resources and people which allow them to extend these classroom communities to ever larger circles, whose members continue to inform and inspire one another. Unfortunately, mathematics communities which parallel or overlap these literary communities and within which uninitiated mathematicians (whether elementary school teachers or children) can actively and confidently participate do not seem to exist. Without them, school children are not participating in mathematics as a live discipline. Teachers, who so desperately need to access and develop their own mathematical talents cannot hope to do so in such a vacuum. They cannot initiate their students into a community to which they do not belong.

5 Value of the Internet for Emerging Mathematics Communities

In the beginning, ideas from MegaMath were disseminated through personal exchanges. Progress was slow. Teachers with the inclination to explore the ideas and experiment did not always have very much time to devote to them, and even if they did, they were acting alone on ideas of their own, without much support from their colleagues.

Over time, a body of stories from the classroom were developed. Anecdotes about students' enthusiasm, their responses, their thinking encouraged teachers to use more of these kinds of activities and topics in the classroom, and gave them ideas for what to look for and how to assess and evaluate the work that their students were doing. Teachers who insisted, "I am not a math person" began to see how they regularly applied "mathematical" skills in other contexts that called for creativity and critical thinking, and that their experiences with mathematics had been so narrow (and demeaning, and frightening) that they never really understood what the discipline was about. Some of these stories were written down and distributed via email and snail mail to people who heard about them and became interested. Eventually a book of classroom activities with background material was published. [9]

Still progress was slow. Interested teachers were far-flung and their time was at a premium. We ran out of activity books.

Finally, all of the MegaMath material was rewritten for the World Wide Web. Establishing a Web site has been the first step in establishing the mathematics community that emerging young mathematicians and their teachers need. Within MegaMath, teachers can find other teachers interested in the same topics and approaches and exchange ideas about helping students, planning, and integration with other lessons. Students can explore MegaMath on their own without waiting for their teachers to decide that the whole class is ready for a certain activity. The hypertext presentation lends itself to the needs of users from a wide age and skill range, since background information, definitions, and related topics can be clicked on and accessed as desired.

MegaMath is also affilliated with other projects sharing similar goals:

6 Using MegaMath

The MegaMath home page is a large (100K+) picture with multiple hot spots that the user can click to navigate the material.

Users who do not have sufficient bandwidth to download the home page image (100K+) may prefer to navigate MegaMath through a text menu.

Since MegaMath is intended to be forum that links elementary school teachers, their students, and professional mathematicians, people from each of these groups will approach the material with different interests and points of view.

On every page there is a mailbox icon which users may click to ask questions, make comments, or ask to be put in touch with other users.

There is also more extensive information about the sponsorship of the MegaMath project and ideas about it's direction and potential see:

Essays about mathematics education and stories about classroom experiences written by MegaMath founders and collaborators are also provided.

Users who do not have access to the World Wide Web can obtain copies of much of the MegaMath materials via anonymous ftp:

 ftp ftp.cs.uidaho.edu 
 login: anonymous 
 password: (your email address)
 directory:  pub/mega-math/ 
 file: readme

Users who cannot access MegaMath through any of the above means can obtain materials and information by contacting the author.


References

[1]
Fellows, Michael, ``Computer Science and Mathematics in the Elementary School," CBMS Issues in Mathematics Education. American Mathematical Society, 1993. (Postscript version of this paper.)

[2]
Goodman, Kenneth. What's Whole in Whole Language? Portsmouth, NH: Heinemann. 1993

[3]
Calkins, Lucy. The Art of Teaching Writing. Porstmouth, NH: Heinemann. 1990

[4]
Casey, Nancy, ``Enhanced Lives: Classroom Structures for Human Development'', unpublished ethnographic report, 1992. (Postscript or ascii version.)

[5]
Casey, Nancy, ``Literacy Lessons and Mathematics Learning'', CBMS Issues in Mathematics Education. American Mathematical Society, 1993.(Postscript version of this paper.)

[6]
Horton, Myles and Paulo Friere. We Make the Road by Walking: Conversations on Education and Social Change. Philadelphia: Temple University Press.1990

[7]
Casey, Nancy. ``Mathmatics, Metaphor and Method'', unpublished essay, 1993. (ASCII version. Postscript version.)

[8]
Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. 1989

[9]
Casey, Nancy and Michael Fellows. This is MegaMathematics: Stories and Activities for Mathematical Thinking and Problem Solving. Los Alamos, NM: Los Alamos National Laboratory. (Postscript version, approx. 1MB)


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