Shattering Stereotypes: Changing to Qualitative Research

 Sheryl A. Maxwell
 
 

Abstract

This presentation traces the journey of a mathematics educator who has become a qualitative researcher. Affected by studying and participating in the mathematics reform efforts, she subtly starts the research transformation. Remaining in tension with others by reason of stereotypical perceptions, she chooses to change paradigms becoming more congruent to her developed educational philosophy in the process.

Background

Research is inaugurated when there exists a tension between what we believe we know and our awareness of how little we know. Related to this tension, the question/problem of significance is formulated, background information is sought, the solution of either theoretical or practical importance is discovered. Within the field of mathematics education, reform efforts in teaching and learning mathematics differently have been underway during the past decade. Initially, most research pertinent to the salient issues, tended to be quantitative. However, the conceptual framework of much recent research is changing from quantitative to the use of qualitative and interpretive research paradigms. Mathematics educators are learning that qualitative research traditions "appear to be more amenable to providing answers to contemporary questions [problems] about teaching and learning than the more traditional quantitative approaches" (Lederman & Niess, 1996, p. 393).

"Shattering Stereotypes" is the narrative autobiography of the on-going personal, yet potent paradigm shift involving me, a woman who is mathematics educator in the postmodern era. My roles as a mathematician, a student, a teacher, a colleague and a researcher are entwined creating the tension in my real working world. Although aspects of my research problem/questions change according to my respective roles, the one consistent research question is "How can stereotypes best be shattered to facilitate change?" Viewers or readers will be invited to interact with me in scenes prompting the narrative to evolve through these brief encounters. The purpose of each story is to invite you into my experiential world and draw you into exploring its problems and its possibilities. This endeavor will link us to where we are, where we have been, and where we are headed professionally. It is my hope that by being a reflective practitioner, you will connect these narratives to your past while you continue your professional journey.

SETTING:

The principal actor: Entering stage right is a mathematics educator, who will share her autobiographical research, disclosing her on-going problem of growth relative to her chosen field of interest. This paradigm shift in becoming a qualitative researcher, thirty years after her initial indoctrination into teaching, creates tension.

Other actors. Sprinkled throughout the stage will be participants who will be invited to contribute qualitative data, to share their tensions, and to assist in the mathematics educatorís/researcherís on-going narrative journey. Perhaps this paper/presentation will help you to explore your own journey of how you became and are developing as a qualitative researcher.

ACT I

Prologue

Stereotypes are powerful images resulting from encounters with past individuals or situations that leave indelible perceptions. To determine the typical perception of an educator, a person should write three words that characterize the noticeable, general, and distinguishing aspects. A request of this type regarding most past mathematics teachers yield such words as regimented, fear, dread, or ridicule--all distinctly negative connotations. Consequently, the stereotypical mathematics educator is viewed as a traditional, behaviorist, and regimented individual who teaches a dreaded, fear-producing, non-understandable subject. This perception continually bleeds over into my mathematics methods classroom. Iíve had students state with disdain "I really hate math! Iíve dreaded taking this class from the moment I decided to become a teacher. I hope I never have to teach math!" In puzzlement, I question why someone would pronounce this to a teacher educator who clearly enjoys teaching mathematics.

Scene I

The Mathematician: Constructing understanding.

To me, the stereotypical mathematician is a male individual, generally introverted, wearing thick glasses, who can spout off, at an instant, some equation or algorithm, when the need arises. He works independently to solve the problem at hand, often using unique methods which he seemingly pulls out of some archive of the past. Working rapidly with his mechanical pencil, erasing an error from time to time, he prods along. Eventually, he yields his chosen proof of the solution. With a smile on his face, he waves the volumes of papers indicating his distinctive proof. Although a woman, I have fit the role of a proficient mathematician comfortably in the past.

Although I became competent as a mathematics specialist by rotely learning mathematical procedures, I intuitively desired to understand WHY specific math properties worked. I began constructing my own understanding long before "constructivism" was in vogue. The mathematics reform efforts occurred within me as I sought to determine such aspects as why canít you divide by zero, what would a geometry be like where no parallel lines exist, or what math systems are not commutative. Although on the outside, I continued to procedurally perform numerous routine mathematics skills, internally, I explored the hidden world of understanding mathematics from within by examining the "what if" scenarios. Procedural knowledge of mathematics plays an important role in both the learning and in doing mathematics. However, by itself, these step-by-step routines are little more than rules without reasons. Mathematics can no longer be equated with doing mundane computational skills.

According to mathematics educators, learning mathematics conceptually is "intrinsically rewarding, enhances memory, requires that less be remembered, helps with learning new concepts and procedures, improves problem-solving abilities, can be self-generative, and has a positive effect on attitudes and beliefs" (Van de Walle, 1998, p. 29). Did these seeds of questioning, reflecting and constructing knowledge assist me in becoming aware of and experiencing initial aspects of constructivism and action research?

Scene II

The Student: Questioning

To me, the stereotypical student in education is a bubbling, young woman, eager to please and conform to the educational ways within the classroom. I envision the student dutifully respecting the expert professor, or experienced cooperating teacher, engaged in a pattern of teacher-pleasing , absorbing the culture of the classroom and learning from the mentoring process.

The tumult of early 1960s was occurring when I learned how to teach mathematics using the "New Math" format. This "modern math", known by such acronyms as UICSM or SMSG, stressed that students understand WHY behind mathematical operations through a procedural approach. During the fall of 1964, I student taught in a large, suburban high school. My experiences in teaching ALGEBRA showed me that modern math teaching tactics WERE NOT applicable for ALL students. Consequently, I wrote a position paper, that was later published, revealing how various math properties could easily be misunderstood by the average student. I questioned if ALL students really needed to understand mathematics to the extent stressed in the Modern Math reform efforts. Was this Modern Math really the BEST approach to teaching mathematics to ALL students? In a time in which many mathematics educators were jumping on the band wagon to promote understanding the WHY behind mathís regimented and procedural methods, I was striking out against the norm. This incident represented one encounter where my actions showed a visible rupture from the stereotypical student teacher. I questioned or challenged ideas and teaching strategies. I risked trying different teaching tactics: yielded control to students; moved carefully positioned rows of studentsí desks into clusters; allowed students to present unique solutions; and gave partial credit for conceptual understanding even when the answer was incorrect. Why should I continue to promote the boring methods I had encountered if something else seemed more plausible, beneficial and interesting?

Scene III

The Teacher: ConstructivistCapacity

To some, the stereotypical image of a teacher is a caring but control-oriented individual who plans, organizes, and presents content material to others in an intriguing manner.

When I first began teaching, I taught in a very procedural style. Knowing intuitively that mathematics was only a set of patterns, I sought to discover the pattern, write the correct recipe and share my discovery with my students in a prescriptive fashion. I sought to rescue my charges from the struggle of creating their own understanding of mathematics. However, as the blend of the mathematics reform facets occurred, I began to undergo a paradigm shift from the behavorist teacher to the constructivist teacher. Although being an excellent mathematician, I recognized the value of the investigating struggle, for I always learned much during the effort expelled often desiring to share my findings with others. The hierarchical nature of mathematics makes this content material a ripe medium for a behaviorist approach. As such, the traditional approach to teaching mathematics is viewed as the teacher precisely states the objectives, plans instruction by identifying the prerequisites needed, uses these building blocks to accomplish the goal, then helps students practice the desired learning outcome. Students are "shown" algorithms and mathematical relationships are "illustrated." Teachers hope students have mastered specific, measurable skills through much practice.

In response to reports in the early 1980s about studentsí lack of mathematics understanding, mathematics educators began reform efforts. Learning theories suggest that learning mathematics is active, internally monitored, and a process of understanding through discovering and constructing meaning from previously learned knowledge, thought, perceptions and even feelings. This constructivist framework shifts away from a behaviorist paradigm associated with traditional direct instruction toward an interactive process. Both skills (procedural) and concept (relationship) knowledge can and should be developed by students for meaningful learning of mathematics to occur. In my elementary mathematics methods classroom, I desire the preservice teachers to confront what it means to do mathematics, to grapple with problem solving and to investigate various mathematical concepts using manipulatives and technology. In short, I invite them to participate on an exciting adventure of learning mathematics through a refreshing approach.

ACT 2

Scene I

The Colleague: A Quandary

Realistically, an educational colleague is the individual housed in an adjacent classroom or office, that one cautiously interacts with during the academic year. Although much of our professional teaching lives occur within the isolated walls of our classroom, recent change efforts have begun to alleviate this stance. Consequently, some day a colleague will be an associate in a personís professional arena who performs several entwined roles: A supporter, a confident, a collaborator, and a friend.

As a colleague on a college/university faculty, I believe I am viewed cautiously by differing groups. By mathematicians, I am admired for my zeal in helping students make sense of mathematics. Yet, I have been criticized for helping my students discover, for themselves, basic conceptual knowledge of elementary topics. These mathematicians believe students should be thinking abstractly during college years. By the generalist teacher educators, I am viewed as a stifling specialist, interested in the prescriptive approach to teaching mathematics. They question with puzzlement why a mathematics specialist embraces constructivism, hands-on manipulatives, and qualitative research rather than numerical interpretations of statistical studies. I believe that through positive interactions with others, the stereotypical views of colleagues can change. By becoming aware of educational trends, available resources, and by experiencing professional development, a person grows professionally. Can we risk to listen to others, read about innovations in the other classrooms, and view professionals who are trying to align their behaviors with current reform changes?

Scene II

The Researcher: Qualitative Inquiry

Of the two differing frameworks of research, quantitative and qualitative, one assumes that a mathematics educator will obviously choose quantitative research since he/she is already comfortable with numbers. Clearly, mathematicians who think about or experience collective terms provided through quantitative methods, will excel in the interpretation of quantities of data, and enable the research community to learn from their calculated analyses.

When deciding to obtain my Ph.D., I wrestled with the fact that I would be doing research. Since I had taught statistics to undergraduates, most people determined that educational statistics used in quantitative research would be right up my alley. However, I disliked categorizing students in norm-related fields. Not only could statistical parameters be easily manipulated, I questioned the "ethics of caring" associated in withholding effective methods just to have a control group. Of more interest to me was qualitative research, particularly case studies using narrative format. I believe that qualitative research invites educators to understand themselves instinctively. I discover that by writing narratives, I nurture my own voice, release my consciousness, and in the process learn and grow. The narrative format has become a mirror. As I learn about my transforming life, a window opens to examine and help others explore their beliefs and behaviors. Unlike quantitative research that ends with the analysis of the specific data collected, I believe that qualitative research continues to grow and develop, sometimes exponentially. To me, qualitative research provides the opportunity to study people in their natural settings, uses more flexible techniques for collecting, analyzing, and interpreting data and allows me to creatively report findings that bring individual voices into examination.

ACT 3

Continuing Tension:Crystallization Element

As an individual I experience tension due not only to stereotypical perceptions, but also to stresses of change and growth within me relative to the five aforementioned elements. For example, as I learn more about mathematical connections, this automatically crystallizes into wondering how I can best translate these findings into a mathematical task, where in a constructivist classroom, students can grapple with their own understandings. In turn, this may promote action research that can encompass a case study of preservice teacher. As I design and share findings with a colleague, she can support my efforts and I can encourage her to investigate similar aspects within her classroom. The seemingly two-dimensional, rigid pentagon encompassing the five elements of mathematician, student, teacher, researcher and colleague, becomes a three-dimensional crystal when the mathematics educator driven by a qualitative stance becomes one of the focus elements.

The central image becomes the CRYSTAL, which combines the six interconnecting focus points into an ever changing three-dimensional object. The crystal can be a prism that reflects and refracts light within itself, creating different colors and patterns. Additionally, the crystal can be studied through examining the planes created by any three elements. For example, examining the elements "mathematics educator" in combination with "colleague" and "student" may provide insight into how I can become a more effective classroom preservice teacher.

Finale

As a reflective practitioner, I recognize that tensions exist among the various roles I undertake. Stereotypes can continue to plague me, or they can be modified and changed to reveal to others a surprising, and dynamic creation. Through this endeavor, I recognize the following:

Although change likely entails discord, disequilibrium, dissension, discomfort and even disgust, it can also be enlightening, energetic and exciting. The choice to shatter stereotypes and accomplish change remains mine.

References

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Jalongo, M. R. (1995). Teachersí stories: From personal narrative to professional insight. San Francisco, CA: Jossey-Bass Publishers.

Lederman, N. G. & Niess, M. L. (1996). Problem"less" research: "Less" is not more!  School Science and Mathematics, 96(8), 393-4.

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

Reys, R.E, Suydam, M.N., Lindquist, M.M., Smith, N.L. Helping children learn mathematics. (5th ed.). Boston, MA: Allyn and Bacon.

Richardson, L. (1997). Writing: A method of inquiry. In N. K. Denzin & Y. S. Lincoln (Eds.), The handbook of qualitative research (pp. 516-29). Thousand Oaks, CA: Sage.

Van de Walle, J. A. (1998). Elementary and middle school mathematics: Teaching developmentally . (3rd ed.). New York: Longman.