A Process Approach to Mathematics: Mathematics as Communication

Rebecca B. Corwin

The push for change in the school mathematics curriculum is growing. As the mandates become stronger to produce students able to think critically and creatively about mathematics, many of us wonder whether these changes can reasonably be achieved in the next few years. The mathematics education community can learn much, I believe, from looking at the successful revision of the writing curriculum during the past decade. We can find ways to apply this "process approach" to the teaching of mathematics.

The Process Approach to Writing

About ten years ago, teachers began to be stirred by research results which indicated that students learned to write best when their writing was treated as communication, rather than being considered an object to be evaluated. Donald Graves (1983) was one of the wave of teacher educators who caught the fancy of the K-12 language arts teachers. Workshops and networks of interested teachers created strong, broadbased support for this way of teaching children to write which has become known as "the process approach to writing."

Most descriptions of process approaches to writing include four phases. First, there is a planning phase (often called prewriting). In this phase, students use various brainstorming and discussion techniques to explore possible topics which come from their own concerns and experiences. Often children and teachers take notes on possibilities, filing those ideas for future development.

The writing phase then follows the planning -- in this, the writer actually does the writing in what Murray (1984) calls the "discovery draft." This draft is treated as a piece of raw material which will be shaped and modified to make sure the author's ideas come alive.

In the next phase, revision, the work is shared with colleagues -- perhaps read aloud to a small group of peers or read quietly by one or two "editors." Sometimes a student reads her work to the whole class for comments. Listeners or readers respond with questions, editorial comments, thoughtful reflection on the work at hand. Even young students will become involved in the revision process which may occur over and over as the writer shapes the raw material more and more.

Finally, in the publication phase, the work takes a "final form." Perhaps it is published on the classroom computer; perhaps it is copied onto "best" paper; perhaps it is literally published in a class newspaper, using desktop publishing software.

Graves (1983) likens the process of writing to a studio craft in which the artist completes some pieces, yet leaves others unfinished. This notion of the malleability of writing is central to the process approach to writing instruction, and it is here that I believe the link to mathematics becomes most persuasive.

Mathematics as Communication

Mathematics, as a very specialized form of writing, becomes a tool for discovery and exploration of ideas. Adult mathematicians, like adult writers, write, throw out ideas, rewrite, reshape, communicate, take feedback, and rewrite. They use their language to express their ideas.

Why does this seem so foreign to most people's experience of mathematics in school? School math focuses, like writing instruction used to, on correctness rather than communication. Writing instruction used to be (and sometimes still is) an excesses in writing what the teacher wanted. Students' work was done for the teacher and was never used to communicate ideas. Corrections focused more on the form of the work than on the structure of an argument.

In mathematics, similarly, the focus has been on the form of doing computation, not the reasons for using numbers. It is not surprising that studies of mathematical achievement in our country show that our students can compute but cannot apply their knowledge in real life problems -- they learn quite early that school mathematics and informal mathematics are quite separate (Ginsburg, 1986).

But in the real world, mathematical work is quite different. For the past three years I have been working on the Used Numbers project at TERC developing statistics and data analysis curriculum modules for grades K-6. Students have been asked to participate in collecting, displaying, and analyzing real data. They have had to talk about the mathematics that they are doing, to define their terms, to debate and discuss their results and their theories. They are functioning like adult mathematicians -- and I believe they are using a process similar to the writing process.

In brainstorming and planning, students define their terms and discuss the question to be explored. What question will we ask? How will we get data? How might we record it?

After data collection, students look at compiled data and compose "sketch graphs" to look for patterns and generate hypotheses and questions. They display and redisplay data, trying different methods and distilling information from these displays.

During the revision phase, they share their conclusions, theories, and questions with small groups or with the whole class. Having an audience makes the generation of further questions particularly exciting. Other students become interested in nuances of analysis, and theories are discussed and debated. Students often go back to collect more data.

Finally, publication of results may involve making recommendations for action. One class in New York City studied the incidence and causes of injuries on their school playground. Their report to the principal included graphs and charts which showed their data clearly and communicated their ideas graphically.

Recommendations

Teachers of elementary school mathematics can adopt more of the process approach to their classroom work, particularly in the area of problem-solving and applications. Statistics and data analysis provide a perfect vehicle for such work. The new Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) recommend much more work in statistics, data analysis, and real-world problem-posing and problem-solving.

Consider mathematics as malleable! Writers shape words to communicate; so do mathematicians. We can and should be using a process approach to mathematics instruction so that students can explore the world of mathematical communication on their own terms.

Just as books and stories exist to be read, and poetry to be spoken or sung, data displays are made to be read, to be understood, to be investigated and explored. There is limited use in being able merely to decode graphs and charts -- what is exciting is being able to understand them. As we learn more about a whole language approach to literacy, we must begin to understand that a whole language approach to mathematics is essential to our students' understanding of number.

I hope for a day when my adult students enter my course with a sense that they can shape mathematics, that they can ask questions, that they have theories and ideas and can invent their own expressive routes through the domain of numbers. Perhaps, with enough attention to helping our younger students make sense of the numerical world, this day will come.

References

Corwin, R and Reinhardt, M. C. (1989) Mathematics education: Learning from the process approach to writing. In Ryan, W. (ed.) Proceedings. Boston, MA: National Educational Computing Conference, June 20-22, 1989.

Ginsburg, H. (1986) Children's arithmetic: How they learn it and how you teach it. Austin, Texas: Pro-Ed.

Graves, D. (1983) Writing: Teachers and children at work. Exeter, NH: Heinemann.

Murray, D. (1984) Write to learn. NY: Holt, Rinehart and Winston.

The National Council of Teachers of Mathematics (1989) Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

Russell, S. J. and Corwin, R. B. (1989) Statistics: The shape of the data. Palo Alto, CA: Dale Seymour Publications.

Acknowledgements:

The work reported in this article was developed jointly by the Used Numbers Project staff. In particular, William Barclay III, Susan N. Friel, Janice R. Mokros, Susan Jo Russell, and Antonia Stone contributed substantially to the development and training on which this work is based .

This article appeared in slightly different form in The Elementary Mathematician, Volume 3, Number 3. Published by COMAP, the Consortium for Mathematics and Its Applications, in Arlington, Massachusetts.

Author Info:

Dr. Rebecca B. Corwin is a Senior Associate at TERC, and an Associate Professor of Education at Lesley College in Cambridge, Massachusetts.