Highlighting the Humanistic Dimensions of Mathematics Activity through ClassroomDiscourseAuthor(s): Beatriz S. D'AmbrosioSource: The Mathematics Teacher, Vol. 88, No. 9 (DECEMBER 1995), pp. 770-772Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27969588 .

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Highlighting the Humanistic Dimensions of Mathematics

Activity through Classroom Discourse

c

Teachers

engage students in

defining what the

curriculum will be

lassroom discourse is often understood as the

process of engaging the members of the classroom

community?students and teachers?in talking with one another. In this discussion I use the term classroom discourse to mean the process of engag ing the classroom community in real dialogue, wherein meaning is negotiated and assumptions are questioned. An underlying assumption through out the discussion is that classroom discourse can

help shape the views of the nature of mathematics that are held by students. The goal of this article is to raise questions and invite the readers to reflect on the issues raised rather than to answer any spe cific questions.

In this article I discuss three ways in which class room discourse can promote and nurture students'

understanding of mathematics as a humanistic

activity, that is, an activity engaged in by people in a community. First, classroom discourse can serve to involve students in defining the curriculum. Sec

ond, classroom discourse can help students build a

personal relationship with mathematics while

engaging in authentic mathematical inquiry. Third, a look at the history of mathematics can serve as a

capstone experience as classroom communities strive to understand how mathematical thought develops in society.

STUDENTS DEFINE THE CURRICULUM Most teachers know that as students grow older, more and more of them find mathematics uninter

esting and unpleasant. They become increasingly less motivated to participate in mathematical activities in school and more and more alienated from what is considered mathematics in schools. This estrangement happens in large part because an environment has not been created in which stu dents build a "personal relationship" with mathe matics. Although I could cite many causes for such

alienation, I do not take the time to analyze this situation here but instead work with the basic

assumption that for too many students, school mathematics is an uninteresting, irrelevant

subject.

As mathematics teachers, most of us have estab lished a love of mathematics and find it difficult to understand why so many of our students are strug gling and resisting engagement with it. We make the following comments about our students: "If they would only put more time into it! If only they would do the homework! If only they would not miss class

_* Such comments reflect our beliefs that what we are doing in class is appropriate and that stu dents' struggles are due to their own disinterest and lack of motivation. But let us imagine taking the perspective of the students who have led us to form these beliefs. We can come to understand their perspective only if we create a classroom envi ronment in which the students' voices are heard, in which the students' interests are explored, and in which the students are called on to give direction to the classroom activities.

Noddings (1993), who has written eloquently about this subject, proposes that mathematics teachers engage students in defining what the cur

riculum will be. She contends that the curriculum can be negotiated between teachers and students to involve students in decisions about the direction

Prepared by Beatriz S. D'Ambrosio Indiana University-Purdue University at Indianapolis Indianapolis, IN 46202 bdambro@indyvax. iupui.edu

Edited by Fernand J. Pr?vost 13 Guay Street Concord, NH 03301 fjp@christa. unh.edu

Beatriz D Ambrosio of Indiana University-Purdue Uni versity at Indianapolis, IN 46202, is an associate profes sor of mathematics education. Her interests lie in engag ing teachers and future teachers in inquiry-based learning.

The Editorial Panel welcomes readers' responses to this article or to any aspect of the Professional Standards for Teaching Mathematics for consideration for publication as an article or as a letter in "Reader Reflections."

770 THE MATHEMATICS TEACHER

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their studies will take. To do so, she insists that we reconsider our narrow focus on subject matter and instead direct attention to issues that are of con cern to the students.

Recently I worked with a talented high school teacher. Upon opening a discussion with her general

mathematics students, she was surprised to find that they were very interested in learning algebra. She proceeded to design an algebra course for them and found their motivation higher than that of her traditional algebra class. A key component in this situation was the initial dialogue that this teacher established with her students. A true dialogue, that

is, a two-way communication, occurred between her and her students as she put her agenda aside and

negotiated with the students the object of study during their year together.

STUDENTS ENGAGE IN AUTHENTIC MATHEMATICAL INQUIRY A view of mathematics as a discipline full of engag ing and intriguing questions worthy of exploration does not characterize the view of mathematics held

by many students. Understanding the evolution of mathematical ideas, raising questions, and chal

lenging what is accepted as standard mathematical

knowledge are issues deserving attention in the school curriculum. Our students need to under stand mathematics as an ever changing and grow

ing field. They need to understand the changes that occur within the field as new questions are asked, as assumptions are challenged, and as new conven

tions are accepted. Borasi (1992) suggests that approaching mathe

matics through inquiry is crucial in helping stu dents understand the nature of mathematics and mathematical activity. In her book Learning Math ematics through Inquiry, she presents several

examples of activities in which her students were

acting like "real mathematicians," engaging in authentic mathematical inquiry. In one of her

examples, Borasi and her students engaged in

defining the familiar notion of a circle. Definitions were proposed and submitted to the scrutiny of their "mathematical community." In this setting Borasi was able to replicate the activities of research mathematicians as they participate in

generating knowledge. Her examples reflect many

aspects of her view of teaching, which she expresses in the following way (1992, 3):

[Learning mathematics through inquiry involves] pro viding necessary support to students' own search for

understanding by creating a rich learning environ ment that can stimulate student inquiry and by orga nizing the mathematics classroom as a community of learners engaged in the creation of mathematical

knowledge.

In Borasi's example, discourse occurred as learn ers negotiated meanings and understandings. As a student proposed a definition of a circle to be consid

ered, the group searched for examples and counter

examples that would support or deny the definition. The dialogue was established as the students sought to align the proposed definition with their personal understandings and interpretations of a circle.

Although Borasi's work is a recent example of a teacher's engaging her students in a learning envi ronment wherein student inquiry drives the cur

riculum, the idea is not new. In the 1930s, Harold Fawcett used the same approach with high school

geometry students in a course called "The Nature of Proof." In this course, Fawcett's students behaved like "real mathematicians," determining basic assumptions, defining concepts, and exploring their own theorems. The experience is described in the Thirteenth Yearbook of the NCTM, titled The

Nature of Proof (Fawcett 1938). Fawcett's experi ence is a clear example of the use of discourse to build a community that engages in authentic mathematical inquiry, wherein students negotiate meanings and understandings and construct mathematics from their personal explorations and

investigations. Thus far I have described experiences in which

discourse is used to help students build meanings and personal understandings of mathematical ideas. In the next section I suggest using the histo

ry of mathematics to illustrate how mathemati cians have struggled with mathematical ideas and how discourse has played a role in their personal development of meanings and understandings.

THE HISTORY OF MATHEMATICS: A CAPSTONE EXPERIENCE Another dimension of understanding mathematics as a human endeavor lies in pursuing a greater understanding of the history of mathematical ideas. It is important for students to view mathematical

knowledge as culturally based and to understand the complexity of the process through which the mathematics community determines what becomes

accepted as "legitimate" knowledge. Studying the

history of mathematics can reveal parallels between the evolution of mathematical ideas and the experiences of students who have sought to

engage in authentic mathematical inquiry, as

described in the foregoing section.

Through a critical look at the history of human

thought, our students can better understand the role of human beings in constructing mathematical

knowledge and accept that they, too, can engage in the inquiry process with ideas that reflect original thought. Unfortunately, the history of mathemati cal thought, as it has been traditionally portrayed, tends to present the finished form of mathematical

The group searched for

examples and counter

examples that would

support or

deny the

definition

Vol. 88, No. 9 ? December 1995 771

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exploration, giving the learner a false concept of

mathematical activity. We can raise many ques tions about pieces of the history of mathematics

that are missing from typical history textbooks

through reflecting on P?lya's (1954, vi) description of "mathematics in the making":

Mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and fol

Studeilts ^ow ana^?^es' y?u ̂ave t? fry and try again. The result of a mathematician's creative work is demon

ana teachers strative reasoning, a proof; but the proof is discovered

are partners Plausible reasoning, by guessing.

in the This exploratory phase of mathematics in the

inauirv making is typically not a part of our students' mathematical experiences or of the study of the

history of mathematics. Instead, students are pre sented with, and expected to learn the finished form of, mathematics.

I am not advocating that students reconstruct mathematical knowledge as it was constructed over

the course of many centuries. Nonetheless, I con

tend that it is important for the learner of mathe matics to understand the development of the field over time through a greater focus on a history of mathematics that explores the cultural roots of ideas (Joseph 1991; D'Ambrosio 1985) and the cre

ative process through which ideas are explored and advanced (Lakatos 1976).

IMPLICATIONS FOR INSTRUCTION Classroom discourse is at the heart of supporting students as they build a relationship with mathe

matics and construct an understanding of mathe matics as a humanistic activity. The nature of the discourse to promote these two levels of under

standing of mathematics needs to be inquiry based. In an inquiry-based approach, the curriculum is

promoted by student inquiry. The direction of the students' inquiry is enhanced by the teacher's con tributions to the inquiry process. Thus, students and teachers are partners in the inquiry and both contribute to the process by posing problems, by proposing solutions, and by extending explorations and investigations beyond the accumulated knowl

edge of either students or teachers. Some of the most exciting and memorable teaching moments in

my career occurred when my students' inquiry went beyond my own accumulated knowledge and we became true partners in the investigation. I remember a recent teaching episode in which my students were exploring ideas in non-Euclidean

geometry and I did not recognize one student's

conjecture as a common theorem. Our subsequent explorations and attempts to prove or deny the con

jecture was an exciting and memorable experience. Often in the context of exploring students' conjec tures, mathematics comes alive in my classes and

my students and I become true partners in learn

ing?engaging in authentic mathematical inquiry. The difficulties inherent in an inquiry-based

environment are numerous. The demands on the teacher to be a lifelong learner, to serve as a

resource, to share authority for knowledge, to set

the curriculum agenda aside when necessary, and to question and learn with the students necessitate a major shift in focus on what constitutes the teacher's role. The new role suggests that the rela

tionship between the teacher and students be one

of collaboration and dialogue, where both teachers and students work toward their own growth in

understanding. Finally, this classroom environment can promote

success for all students. All students explore the mathematical ideas to the degree that reflects their interests and excitement and to the degree that

explorations become relevant and important. All contributions should be valued and respected. Stu dents' personal and collective histories shape the curriculum as their interests are reflected in their investments and engagement in learning.

This article has sought to raise questions rather than answer them. Within these perspectives, I invite readers to consider their use of discourse and

verify whether it is aiding them in portraying a

view of mathematics as a discipline that is deserv

ing of students' emotional engagement and person al commitment.

REFERENCES Borasi, Raffaella. Learning Mathematics through

Inquiry. Porstmouth, N.H.: Heinemann Educational

Books, 1992.

D'Ambrosio, Ubiratan. Socio-Cultural Bases of Mathe matical Education. Campinas, Brazil: UNICAMP, 1985.

Fawcett, Harold P. The Nature of Proof. Thirteenth Yearbook of the National Council of Teachers of Mathematics. New York: Teachers' College, Colum bia University, 1938. Reprint, Reston, Va.: The

Council, 1995.

Joseph, George G. The Crest of the Peacock: Non

European Roots of Mathematics. London: Penguin Books, 1991.

Lakatos, Imre. Proofs and Refutations. Cambridge: Cambridge University Press, 1976.

Noddings, Nel. "Politicizing the Mathematics Class room." In Math Worlds: Philosophical and Social Studies of Mathema tics and Mathematics Education, edited by Sal Restivo, Jean Paul Van Bendegem, and Roland Fischer, 150-61. Albany: State University of New York Press, 1993.

P?lya, George. Induction and Analogy in Mathemat ics. Princeton, N.J.: Princeton University Press, 1954. @

772 THE MATHEMATICS TEACHER

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