KNOWLEDGE GROWTH IN MATHEMATICS TEACHING:



TITLE

Knowledge development in mathematics teaching: learning from experienced elementary teachers

PROPOSER

Paola Sztajn

INTRODUCTION

The improvement of mathematics teaching in schools continues to be at the heart of America’s concerns with education. The future well-being of the nation, according to a recent report by the National Commission on Mathematics and Science Teaching for the 21st Century, will increasingly depend on how well we educate the people in mathematics and science. After an extensive, in-depth review of what is happening in classrooms, this commission concluded that, to improve mathematical knowledge, “the most powerful instrument for change, and therefore the place to begin, lies at the very core of education— with teaching itself” (U S Department of Education, 2000). Thus, to better mathematics education in schools, we need to focus our attention on those who teach mathematics to our youth.

The state of Georgia needs direct and immediate action towards better preparation of mathematics teachers. Currently, the state faces a crisis in mathematics education, with students performing bellow national average in standardized tests. Just recently, the state ranked 50 out of 51 in SAT mathematics scores. The University of Georgia, as a leading institution in teacher preparation, has the responsibility of implementing research projects which deal with teachers’ mathematical knowledge and education.

Goals

Those who deal with teacher education ultimately aim to transform the organization and enactment of teachers’ work. To reconstruct it, however, it is necessary to understand it. Therefore, the overarching goal of my research agenda is to better understand elementary mathematics teaching. Towards such goal, it is necessary to identify all types of knowledge elementary teachers construct and use in order to teach mathematics to young children. The underlying assumption is that to understand elementary teachers we need to look beyond content knowledge and beliefs; we need a new way to conceptualize the knowledge teachers need when carrying out their task.

The current study being proposed is guided by a few specific objectives:

• Identify elements through which elementary teachers see the development and growth of their own knowledge for teaching mathematics during their careers;

• Identify factors that, from the teachers’ point of view, were important in the establishment of their ways of teaching mathematics for young children;

• Identify factors (if any) that served as catalysts for change in the teachers’ knowledge for teaching mathematics; relate these factors to perceived attempts of practice modification, revealing the teachers’ perspectives on issues such as continuity and change in the teaching of elementary school mathematics.

Within a broader scope, this project shall also allow the gathering of useful information for conceptualizing and evaluating in-service professional development activities for teachers.

In order to hear elementary teachers’ perspective on issues related to mathematics teaching, I will interview elementary teachers, reconstructing their life and professional stories. Through these “cases”, I will initiate the process of identifying important elements and factors in the development of teachers’ knowledge throughout their experiences in teaching mathematics to children.

Brief Theoretical Considerations on Teachers’ Knowledge

In previous works on mathematics teachers’ beliefs (Sztajn, 1995); Sztajn, 1997a; Sztajn, 1997b; Sztajn, 1998); Sztajn, 2000)I have considered that to study teachers’ beliefs one needs to go beyond what they say—one needs to look at their practices. It is necessary to search for those sets of beliefs that help us understand the way teachers act. However, the study of beliefs is not enough for those who search for an understanding of teachers, and I have recently begun to address the issue of teachers’ knowledge (Sztajn, in press). The study of teachers and their practices requires a much broader approach to the way one looks at them and tries to qualify what serves as justification for teachers’ actions in the classroom. Actually, the blurred separation between knowledge and beliefs already indicates the need for a more comprehensive framework to study teachers.

The theoretical framework that guides the present study is mostly based on the ideas of the Canadian educator Maurice Tardif and his research group (Centre de Recherche Interuniversitaire sur la Formation et la Profession Enseignante—CRIFPE). Tardif proposes an epistemology of professional practice which he defines as the “study of the set of knowledges really utilized by professionals in their everyday working space to carry out all their tasks” (Tardif, 1999, p.15). He highlights the words set, really and all to further qualify the fundamental issues of this epistemology, and explains that his far-reaching idea of knowledge includes content knowledge, aptitude, attitude, and know-how, for example. For Tardif, the ultimate goal of this epistemology is to reveal all such knowledges and understand the ways in which they are integrated in the achievement of professional tasks.

Summarizing studies on teachers from the past 25 years, Tardif (1999) points to three main features that characterize teachers’ knowledge. First, it is time-dependent: it depends on teachers’ life histories and develops throughout their careers. Second, teachers’ knowledge is personal and situated, depending on who is in which context. Finally, it is heterogeneous and has different sources. In his classification of knowledge, Tardif relates different types of knowledges to their social sources—making a contrast with other typologies based, for example, on epistemological principles (e.g. Shulman, 1986). Going beyond his earlier consideration on teachers’ knowledge as a plural concept which includes discipline-based, curriculum-based, professional, and experience-based knowledge (Tardif, Lessard, & Lahaye, 1991), Tardif has come to consider the issue of time. Knowledge coming from the teachers’ personal life, previous education, pre-service professional preparation, contact with curriculum materials and from practical professional experience in schools and classrooms, is not contemporaneous or equally available for the teachers (Tardif & Lessard, 1999).

For Tardif, the concept of knowledge is based on a requirement for rationality. He calls knowledge every idea, thought, judgement, and so forth that can be rationally justified. His rationality criterion is centered on discourse and argumentation. Tardif considers that the study of justifications for action leads to knowledge and that the best way to access the rationality present in any social actor is by asking him or her (Tardif & Gauthier, 2000).

RESEARCH PLAN

Methodology

To understand the knowledge that guides teaching, one needs to realize that teachers know in action (Schön, 1983). Therefore, studying teachers’ knowledge requires a methodological approach that allows representations of action; an approach that can cope with the many ambiguities and dilemmas that emerge from action. Narrative inquiry is such approach. A narrative is a “symbolic presentation of a sequence of events connected by subject matter and related by time” (Scholes, 1981, p. 205). According to Bruner (cited in Carter, 1993,p. 6), narratives are “concerned with the explication of human intentions in the context of action”.

Interest in narrative methodology has grown within the field of education in the past decade, mostly because it “represents a way of knowing and thinking that is particularly suited to explicating the issues with which we deal” (Carter, 1993, p. 6). In mathematics education, however, we have not fully explored teachers’ life narratives and little do we know about elementary teachers’ knowledge development through their careers.

In a recent study, Connelly and Clandinin (2000) have asked what is the link between narrative and teacher knowledge? They explain that teachers’ knowledge can only be thought of in narrative terms. For these authors, teachers’ knowledge is not something “fixed and static to be replaced by something else” (p.316). Rather, teachers knowledge is “something lifelike, something storied, something that flows forward in ever changing shapes” (p. 316).

Discussing the craft of constructing biographies, Smith (1994) lists a few tasks (easily extended to the construction of professional stories). Initially, it is necessary to decide whom to write about—one must select heroes and heroines. The author explains that finding different pieces of information on a person is a jigsaw puzzle, to which he refers as “ethnographic biography” (p. 290). “Life writing as an empirical exercise feeds on data,” Smith (1994, p. 290) claims. By constructing a data file on the person, the researcher should gain “an overview of the life” (p. 291). The following step in writing a life is finding a perspective. Chronology, Smith explains, “is always important, but a simple chronology of birth, education, marriage, career and death won’t do” (p. 291). Finally, the writer must find “the figure under the carpet” (p. 291), capturing the essence of the person and constructing patterns that fit well the data one has about the life of the person studied.

Participants

To talk about teachers’ knowledge that advances from practice, this study will focus on elementary teachers with many years of teaching experience. Dealing with 5 teachers with over 20 years of teaching experience in elementary schools, this project will work on bringing forth the stories of these teachers’ mathematics teaching. Although attention will be given to the teachers’ pre-professional trajectories (family background, schooling, and teacher preparation, for example), the focus of the discussions will be the teachers’ professional lives and development through their careers in schools.

The teachers involved in this study have mastered a vast amount of knowledge during the span of their careers. Nonetheless, research has not looked at their wisdom. Therefore, we are not able to use it to improve the knowledge-base of others who lack the same experience. As we know, two thirds of the current teaching force will retire within the next decade (US Department of Education, 2000, p. 7). It is time we learn from experienced teachers, and research their knowledge.

Procedures and Data Collection

The major volume of data will be gathered from the teachers, especially through individual and group interviews during Spring 2001 semester. Each teacher will meet individually with me at least 3 times for 45-minute, semi-structured, audiotaped interviews. Although participating teachers will agree to meet three times, the exact number of interviews with each teacher will depend on their willingness to continue the dialogue and, hopefully, once teachers get involved in the project they will become interested in meeting more than three times. A couple of instruments will be used in these meetings. One is a life-line (Connely & Clandinin, 1994). Teachers will be invited to construct a chronicle of professional events, recording dates and trying to remember special feelings that were part of those moments. The second one will be a logic-spider. Having their professional growth in teaching mathematics as a central theme, teachers will be presented with different topics such as first years of teaching, important colleagues in schools, participation in teacher development program (which form the different legs of the spider). The purpose of such approach is to give teachers general themes to address without extensively guiding their narratives.

All teachers will also meet as a group at least twice—again, more meetings will be scheduled if teachers want to continue to share their experiences. These group meetings should allow teachers to exchange ideas and possibly raise issues that did not seem important at the individual level. During these meetings, teachers will also address issues repetitively mentioned in many of the individual interviews. Finally, and maybe most important, these group meeting should bring forth episodes and stories that teachers tell each other—which might differ from the ones they chose to tell me during individual interviews.

The tapes from all individual and group meetings will be transcribed. Each teacher will get a copy of the transcription of his or her interviews and will have a chance to further elaborate his or her stories and explain his or her ideas.

Data files on the teachers should not be built only from the interviews. Therefore, teachers will be asked to share objects, letters, and other memorabilia that are part of their professional lives in teaching mathematics. Also, school archives will be searched—when available and accessible—for information on the teacher. The school might have, for example, the teacher’s mathematics lesson plans from different years.

To better understand teachers’ stories, it is also necessary to situate them in time and space.

Stories exist in history—they are in fact deeply located in time and space. Stories work differently in different social contexts and historical times—they can be put to work in different ways. Stories then should not only be narrate but located. This argues that we should move beyond the self-referential individual narration to a wider contextualized collaborative mode. (Goodson, 1996, p. 214)

Therefore, another set of data will be gathered from information on the social, historical and political contexts in which teachers’ experiences took place. As Goodson (1996) once again reminds us,

Lives and stories link with broader social scripts—they are not just individual productions they are also social constructions. We must make sure that individual and practical stories do not reduce, seduce and reproduce particular teacher mentalities and lead us away from broader patterns of understanding. (p. 219)

Data Analysis

Mathematically speaking, the best image for a qualitative study is that of a “progressive problem solving” (Erickson, 1986, p. 136). Although the researcher has a goal in mind, there is no readily available algorithm to answer the research questions. Therefore, one has to be constantly checking on the work done, gathering more information, looking at special cases, planning, trying out tentative solutions (interpretations or explanations), checking on the work, and so forth. In analyzing the data the researcher ought to search for the meaning of each little piece of information and constantly compare every new piece of information with the ones already in hand (Glaser & Strauss, 1967, Strauss & Corbin, 1990). Currently, many software programs are available to help the researcher in such overwhelming endeavor. In this study, I will use Nud-ist.

Although in qualitative studies one should allow for themes to emerge from the field, it is naïve to assume that researchers look at the data with a clean slate. Therefore, in analyzing the data and attempting to reach the goals proposed for the research, I will be attentive to issues related to career phases and life cycles (Huberman, 1993), influences from social contexts (Goodson, 1996), as well as time and knowledge sources (Tardif & Lessard, 1999). Such issues should facilitate my search for patterns of understanding and offer a connection between the data I will have in hand and the current literature on teachers as professionals and teachers’ knowledge.

Generating Stories

Many researchers have reduced the role of generating teachers’ stories to that of giving teachers voice. It is important to emphasize that the purpose of this study is to understand teachers and tell their stories in order to advance knowledge in the field, helping us better educate other teachers. Therefore, the stories generated though this project will be not only theory-guided, but also committed to theory building. The research report will ultimately aim at informing and developing theory on teachers’ knowledge.

Although narratives and stories are very individual accounts, the research literature warns us that one should not reduce the use of stories to the individual level—the researcher ought to look for broader frameworks that take us from the stories toward more in depth understanding. Carter (1993) explains that the issue of generalization is problematic in narrative inquiries because stories resist singular interpretations and their relationship with objective reality is troublesome. Therefore, when utilizing narrative methodology one should not expect to generate social laws to which people must conform. It is, nonetheless, the researcher’s role to frame patterns concerning certain themes. Moreover, the researcher is in charge of searching for “explanatory propositions with which we can make sense of the dilemmas and problematics of teaching” (p. 10).

Concluding Remarks

The development o this project will follow the succeeding timetable:

| Year 2001 |Jan/Feb |Mar/Apr |May/Jun |Jul/Aug |Sep/Oct |Nov/Dec |

|Literature Review |x |x |x |x | | |

|Interviews and Transcripts |x |x | | | | |

|Other data collection | |x |x | | | |

|Data analysis | | |x |x | | |

|Generating stories | | | |x |x | |

|Final account | | | | | |x |

By identifying elements through which elementary teachers see the growth of their knowledge for teaching mathematics, important factors in the establishment of the teachers’ ways of teaching mathematics for young children, as well as possible catalysts for change, the proposed research will allow us to advance our understanding of teachers’ knowledge and its development. The proposed study will also better our current conceptual framework to analyze teachers and teaching. Furthermore, the results of this study will offer a more stable frame of reference to think about teacher education and the development of in-service professional development opportunities in mathematics education—a much needed instrument in the state of Georgia.

BUDGET

A. Faculty Summer salary

B. Other salaries and wages

C. Graduate Research assistant

D. General expenses

E. Equipment

F. Travel

BUDGET JUSTIFICATION

Although most of the data will be gathered during Spring 2001 semester, the heaviest part of the work (data analysis, constructing the stories, and developing patterns of understanding) will begin in Summer 2001. Most important, the data analysis—for which I am applying for this Faculty Research Grant— has to be completed in the first part of the summer for the project to continue into its next phases. Only full-time summer work on the project will allow me to produce the final report on the teachers by the end of 2001.

Data analysis in any inquiry is a very intensive task. In qualitative studies this task is specially demanding due to the wide amount of data one collects in the field. Thus, data analysis requires the researchers’ uninterrupted attention, and cannot be performed with the necessary depth while one is teaching. In order to complete the data analysis part of this project, faculty summer salary is the largest request of this proposal. Without teaching, the researcher will have the opportunity to become fully immersed in the project.

The other items requested are payment for transcription of tapes and miles to go interview the teachers. The former is fundamental given that transcribing tapes is a time-consuming task and part of the success of the project depends on teachers receiving copies of what they said in a timely framework. The latter is due to the fact that, at the moment, the teachers who have agreed to be involved in the project teach in an elementary school in Rockdale County. Most probably, the meetings will happen in the school (after school time), which is 48 miles away from UGA. Both these requests are considering only the minimum of 3 interviews per teacher (5 teachers) and 2 group interviews, adding up to 17 meetings total.

REFERENCES

Carter, K. (1993). The place of story in the study of teaching and teacher education. Educational Researcher, 22(1), 5-12.

Connelly, F. M., & Clandinin, D. J. (1994). Telling teaching stories. Teacher education quarterly (Winter), 145-158.

Goodson, I. (1996). Representing teachers: bringing teachers back in. In M. Kompf, W. R. Bond, D. Dworet, & R. T. Boak (Eds.), Changing research and practice: teachers' professionalism, identities and knowledge (pp. 211-222). Washington DC: falmer Press.

Huberman, M. (1993). The lives of teachers. New York: Teachers College Press.

Scholes, R. (1981). Language, narrative and anti-narrative. In W. J. T. Mitchell (Ed.), On narrative (pp. 200-208). Chicago: Chicago UNiversity PRess.

Schon, D. A. (1983). The reflective practitioner: how professionals think in action. New Yourk: Basic Books, Inc.

Schulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Smith, L. M. (1994). Biographical method. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 286-305). Thousand Oaks, CA: Sage Publications.

Sztajn, P. (1995). Elementary teachers' ideologies in an era of reform: case studies in mathematics education. Unpublished Doctoral Dissertation, Indiana University, Bloomington.

Sztajn, P. (1997). Olhando Teresa e pensando parametros (Looking at Teresa and thinking about curriculum guides). Zetetiké, 5(7), 51-67.

Sztajn, P. (1998). Buscando um Perfil da População: Quais as Crenças dos Professores de Matemática? (Searching for a profile of the population what are mathematics teachers’ beliefs?). Zetetiké, 6(10), 87-103.

Sztajn, P. (2000). Sem óculos e sem mau humor: somos professores de matemática (No glasses or bad humor: we are math teachers). In V. M. F. Candau (Ed.), Reinventar a escola (Reinventing School) (pp. 221-237). Petrópolis: Vozes.

Sztajn, P. (in press). O que precisa saber um professor de matemática? Uma revisão da literatura americana dos anos 90 (What does a mathematics teacher need to know? A revision of the American literature from the 90s). Em Aberto.

Tardif, M. (1999). Saberes profissionais dos professores e conhecimentos universitarios (Teachers' professional knowledge and university-based knowledge) . Paper presented at Rio de Janeiro: Pontificia Universidade Catolica.

Tardif, M. & Gauthier, C. (2000). O professor enquanto ator racional: que racionalidade, que saber, que juizo (Teachers as rational actors: what rationality, what knowledge and what judgement) . Paper presented at Rio de Janeiro: Pontificia Universidade Catolica.

Tardif, M. & Lessard, C. (1999). Le travail enseignant ao quotidien. Quebec: Presses de l'Univesite Laval.

US Department of Education (2000). Before it's too late. National Commission on Mathematics and Science Teaching for the 21st Century. Available: [October 10, 2000].

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