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CHAPTER ONE

INTRODUCTION

At the national level, the grassroots level, and all levels in between, a dialogue is underway regarding directions for mathematics education. The nature of curriculum, the roles of teacher and learner, and appropriate uses of technology are but a few subjects of lively debate. A great many sacred cows are in danger of being slaughtered; one wonders which shall be spared. Yet the more things change, as they say, the more they stay the same. When discussing educational matters, this has particular weight, considering that the core operations of schooling have changed little since the nineteenth century (Tyack & Cuban, 1995).

Still, this could be a watershed moment in mathematics education. Perhaps this point in history is a nexus of critical events including, but not limited to: advances in technology, developments in cognitive and social psychology, the emergence of highly-publicized international studies of school achievement, the acknowledgement of educational inequities, and the purported intentions to address these inequities, however sporadically. In whatever light history casts on these events, the door to a different mathematics education is open; who passes through it remains to be seen.

Of course, one might argue that any real change must happen from the ground up. That is, the determination whether the status quo is rejected or reaffirmed will be made in the not-so-routine daily decisions of the classroom teacher in the context of a particular mathematics department (D'Ambrosio et al, 1992). As Fullan and Stiegelbauer (1991) suggest, "Educational change depends on what teachers do and think-it's as simple and as complex as that" (p. 117).

Although issuing more government reports or state requirements could lead to some surface change, to a teacher these acts may only represent hollow political posturing, lacking real transformative power in the classroom. Such outside mandates may be treated as simply another educational fad which will pass in time, as have so many others.

For example, the Mathematical Sciences Education Board has suggested that school mathematics must be equitable, and should not filter students out of scientific or professional careers (1989). Yet, while Volmink (1994) agrees that such filtering is undesirable, he argues that this has in fact been the historical function of mathematics education. He writes, "Mathematics is not only an impenetrable mystery to many, but has also, more than any other subject, been cast in the role as an 'objective' judge, in order to decide who in society 'can' and who 'cannot'" (p. 51).

So, the rejection of this gatekeeping function of school mathematics, as well as the acceptance of other "reform-oriented" recommendations, would indicate a significant shift in philosophy by the mathematics education community. Certainly, this sort of shift could not be mandated for the profession by the state, for the principles underlying different models of school mathematics are deeply rooted, and beliefs are often passionately held by educators. Instead, such a philosophical transformation would require an intense period of debate and soul-searching by the profession as a whole Again, the present time may be such a period.

When such a transformation occurs for an individual teacher, it might be likened to a professional “Copernican Revolution”, a complete paradigm shift regarding the fundamental assumptions of mathematics education. Whether this shift is a painstaking evolution or a sudden awakening, the discovery that one’s profession has been spinning on a different axis may be simultaneously exciting, intimidating, and perhaps frightening. As Norum and Lowry (1995) suggest, "When a change occurs, for some, it will be uncomfortable but manageable. For others, it may be downright terrifying" (p. 4).

Statement of the Problem

Although the process of change is uncomfortable, mathematics departments may be forced to reexamine the traditional habits, methods, and attitudes that have guided practice for many years, and which may no longer be appropriate for a changing American populace. Yet, as Gutierrez (1996) suggests, if we are to improve opportunities for all students in mathematics education, we need to examine the learning environments of students who traditionally underperform in mathematics. This paper is a snapshot of a mathematics department in a large, comprehensive, suburban public high school that has begun such an effort, and has actively undertaken steps to move toward a reformed vision of mathematics education.

Specifically, the mathematics department at Adderley High School (a pseudonym) has begun to phase out a "traditional" mathematics sequence in favor of the Core-Plus Mathematics Project (CPMP), a curriculum designed to encourage teaching consistent with such documents as the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards (1989), and Principles and Standards for School Mathematics (2000). As a result of the curriculum change, and turnover in teaching staff and district leadership, the department has been in a period of substantial restructuring since1998. That is, this mathematics department has been forced to critically analyze its instructional practices, methods of student placement, use of technology, and professional relationships. In short, they are operating in a new paradigm.

In this paper, I discuss the successes and challenges experienced by this mathematics department in transition, and I analyze some cultural traits that have facilitated the department's receptivity to this educational paradigm shift.

Explanation of Terminology

It is essential to clarify what is meant by "reform-oriented" versus "traditional" mathematics instruction in this paper. Many reforms of mathematics education have been proposed, but the most appropriate to this discussion are those described in the NCTM Curriculum and Evaluation Standards (1989), Professional Standards for Teaching (1991), and Principles and Standards for School Mathematics (2000). However, I do not wish to suggest that "traditional" instruction and "reform-oriented" teaching are mutually exclusive. Yet, compared to "traditional" teaching, a "reform-oriented" vision of mathematics education is supported by very different principles and generally employs very different methods. These underlying differences are fundamental and should be made explicit.

Explanation of "Reform-Oriented Mathematics"

Some crucial differences between reform-oriented mathematics and traditional mathematics are found in the roles envisioned for both the teacher and the students. In a reform-oriented mathematics paradigm, students are both actively and passively involved in mathematical problem-solving and application of mathematical ideas (National Council of Teachers of Mathematics, 1989). The teacher's role, therefore, is more of a guide or facilitator of student activity rather than a lecturer. Of course, there will necessarily be some presentation of material, but the use of lecture as an instructional strategy is certainly de-emphasized (D'Ambrosio et al, 1992; National Council of Teachers of Mathematics, 1989). In addition to these roles, other hallmarks of reform-oriented mathematics include:

  • Increased student interaction and classroom discourse
  • The use of a variety of instructional techniques, such as cooperative learning and projects, as well as individual work
  • Activities arising out of problem situations
  • The position that all students should have access to learning significant mathematics, as opposed to a select few
  • Heterogeneous student grouping
  • Increased use of appropriate technology, such as calculators or computers
  • Covering fewer mathematical topics at greater depth
  • Employing multiple methods of assessment

( Mathematical Sciences Education Board, 1989; Mathematical Sciences Education Board, 1990; National Council of Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 1991; National Council of Teachers of Mathematics, 1998). These suggestions stem in part from important research in cognitive development, the need to meet the demands of an information society, and the duty to provide an adequate, equitable education for all students.

Explanation of "Traditional Mathematics"

In contrast, "traditional" American mathematics instruction has been characterized by a familiar daily routine: the teacher checks the answers to the previous day's homework, works problems on the board, introduces a new concept, works some examples, and assigns seatwork (Cobb, 1992; Stigler & Hiebert, 1999). In contrast to reform recommendations, teacher exposition is the primary method of instruction, and the student has the passive role of listening and taking notes. Furthermore, rote memorization of facts and procedures dominates instruction, characterized by paper-and pencil skill work (National Council of Teachers of Mathematics, 1989).

In addition, traditional mathematics has been characterized as mathematics-for-the-elite, and significant mathematics is taught only to the top-performing students (D'Ambrosio et al, 1992; Volmink, 1994). Therefore, sorting students into ability groups, or "tracking", is a common practice.

Volmink (1994) presents this rather pessimistic view of traditional mathematics:

There are strong hegemonic forces in our society, that impose a certain view of mathematics on us all. Our schooling in many ways has encouraged us to accept as unproblematic, that the traditional mathematics curriculum somehow embodies uniquely powerful knowledge and eternal truths which should be taught and learned in a catechistic fashion. Furthermore, this draconian body of knowledge is not only infallible but also universal. (p. 52)

Stigler and Hiebert (1999) found that mathematics instruction is far from "universal"; Japanese, German, and American teachers differ drastically in their assumptions, methods, and goals. Certainly, these strong words from Volmink reflect the passion that mathematics education can inspire. I have deliberately drawn stark distinctions between reform and traditional mathematics, but this is not to inflame emotions or to oversimplify the choices teachers make. Classroom teaching is rarely black-and-white; most teachers probably operate in shades of gray. Yet, the distinctions help explicate the current situation unfolding at Adderley High Schools.

School Context

Adderley Township Community High Schools consist of two large public institutions: East Adderley and West Adderley. These buildings serve approximately 1,700 and 1,500 students, respectively. The pupil-teacher ratio in the district is 18.8 to 1, and there are 22 total mathematics teachers employed. Although several miles apart, both schools are considered one unit. That is, students at East and West Adderley compete on the same athletic teams, and the respective academic departments have one department chair travelling between campuses. Still, each school has an administrative team consisting of a principal, an assistant principal, and two deans of students.

Despite the fact that the district is in suburban Chicago, both schools are located in urban environments. Industrial and retail areas have built up around the schools, providing the district with a significant tax base. Despite being well-funded, the district confronts many of the same problems facing urban schools. For example, a zero-tolerance policy toward gang activity on school grounds is strictly enforced. To this end, students may not wear clothing that bears the insignia or colors of certain popular sports teams, as these are associated with gangs in the area. This is not to suggest that the schools are dangerous or in some way chaotic. Student discipline is handled efficiently, and the school grounds are well-kept and orderly.

Student Information

To provide further background, some student demographic information may be illustrative. According to the Illinois School Report Card, the district has a sizable Limited-English-Proficient (LEP) population. In fact, 12.8% of the student body is LEP, which is exactly double the Illinois state average. The student population is also racially and ethnically diverse, although this is not uniformly distributed throughout the district [See Table 1]. The district-wide average class size is between 21 and 22 students, slightly above the Illinois State average of 18.3.

Table 1

Student Racial/Ethnic Background

School / White / Hispanic / Black / Asian/Pacific Islander / Native American
East Adderley H.S. / 68.3% / 26.7% / 0.8% / 3.6% / 0.5%
West Adderley H.S. / 52.4% / 40.7% / 2.5% / 4.3% / 0.1%

Personal History with the District

I began my teaching career at West Adderley High School in 1996, after graduating from the University of Illinois at Urbana-Champaign with a Bachelor's Degree in the Teaching of Mathematics. At the time, the department had a traditionally constructed curriculum that sorted students into "skills", "regular" and "honors" tracks. Typically, a student would take a sequence of algebra as a freshman and geometry as a sophomore. If this student decided to take mathematics beyond the district's two-year mathematics requirement, some form of advanced algebra was generally next in the sequence. Figure 1 illustrates this course sequence and structure.

While Algebra I and Geometry were offered at the "regular" level, Pre Algebra, Algebra S, and Geometry S were remedial-level courses taken by the generally lower-achieving "skills" students. Honors students started their high school mathematics education with Advanced Algebra Honors, and some proceeded as sophomores to Geometry Honors Plus, a course team-taught with Honors Chemistry. From the figure below, one notices a fairly stratified system. Yet it must be noted that this was not an entirely rigid structure; students often moved between tracks with a teacher's recommendation.

Grouping / Freshman / Sophomore / Junior / Senior
Skills / Pre algebra / Algebra S / Geometry S
Regular / Algebra S or Algebra I / Geometry S or Geometry I / Algebra II / College Algebra
Honors / Advanced Algebra Honors / Geometry/Chemistry Honors Plus or Geometry Honors / Trig/PreCalc or Algebra II Honors / AP Calculus

Figure 1. Course selections, 1996-1997.

The seeds of this study were planted when a new department chair, Mr. Blakey (a pseudonym), took over at the start of the 1997-1998 school year. He immediately replaced the remedial Algebra courses (Algebra S) with the Core-Plus Mathematics Program (CPMP) as a pilot program. I was one of six teachers in the department who taught CPMP at this time. Of course, this was only the start of the changes that form the basis of this paper.

I was personally intrigued by the reform-oriented perspective of mathematics teaching and learning encouraged by the CPMP materials. Students who might ordinarily have been disconnected from a traditional algebra course seemed interested in the Core-Plus content because of the emphasis on presenting mathematics in a real-life context. Many were able to transcend a lack of computational skill through the use of graphing calculator technology, a key feature of the curriculum. In addition, perhaps related to the issue of computational skill, many told me that they were able to overcome a measure of "mathematics anxiety", the fear of failure they had felt taking past courses. As a result, I saw a great deal of potential in this program for all students. I believed that it would offer opportunities for not only skills students, but also regular and honors students. That is, they could all engage in real, meaningful mathematics while developing critical thinking and problem solving skills. Details of the Core-Plus curriculum are discussed further in Chapter 3.

After the 1997-1998 school year, the first year of the pilot program, I took a leave of absence from Adderley. However, I was still interested in the program, which had been successful enough to merit wider implementation. This research arose from keeping contact with the department during my leave and graduate studies, and observing the extent to which the Core-Plus curriculum was being phased into all classes and at all levels. In my opinion, such a rapid and sweeping change in direction warranted further examination.

CHAPTER TWO

REVIEW OF LITERATURE

For this study, my focus has been to identify characteristics of a mathematics department as a reform-oriented curriculum in the beginning stages of implementation. As a result, I have sought documentation of the receptivity of mathematics teachers, individually and collectively, to curricular and instructional change. That is, under what conditions are teachers and departments ready for such a shift in direction?

Much has been documented on traits of teachers as a whole, and this is discussed below. From this general investigation of teacher characteristics, the discussion moves to research on indicators of mathematics teachers' receptivity toward reform-oriented curricula and instructional methods. Specifically, this is research that notes teachers' attitudes regarding the role of the teacher and instructional activities. Finally, I investigate the characteristics, culture, and receptivity to change of groups of teachers: the mathematics department as a whole.

Teacher Traits

It has been found that personality types are not distributed evenly among professions (Lawrence, 1982). Certainly teaching is no exception. Research has shown wide variation among teachers’ conceptions of control, motivation, self-actualization, and desire for change (Ashton & Webb, 1986; Hopkins, 1990; Huberman, 1992; McKibbin & Joyce, 1980; Rosenholtz, 1989). One scale often used to measure personality traits of teachers is the Myers-Briggs Type Indicator (MBTI). Based on the work of C.G. Jung, this inventory defines personality based on 16 types. Four bipolar scales are used in this measure: Extraversion-Introversion (EI), Sensation-Intuition (SN), Thinking-Feeling (TF), and Judging-Perception (JP).

The EI scale measures one's tendency to obtain information through the external world of people and things, or through the inner world of ideas. That is, extroverts are more outgoing, and introverts tend to be more reflective. The SN scale reflects ways of perceiving the world, either through the senses or by intuitive judgements. Sensation-oriented individuals base their perceptions on real, concrete, objective data, as opposed to the hunches and unconscious information that tend to guide Intuitive perceivers (Kent & Fisher, 1997). The TF scale is a measure of the preferred method of judging experiences. Experience may be judged mainly by logic (T), or by subjective, personal assessments (F). Finally, JP refers to one's attitude toward the outside world. Judging types prefer to have things decided according to an organized, rational plan. Perceiving types are naturally prone to flexibility and spontaneity (Lawrence, 1982). Within each individual, the four preferences interact to determine one's personality type.

For example, as measured on this scale, a person might be identified as more predisposed to introversion over extroversion, intuition over sensation, feeling over thinking, and perceiving over judging. Therefore, this person would be classified as "INFP". However, some research has explored trends among subsets of these 16 personality types, specifically as they relate to teachers

For example, Lawrence notes that 67% of high school teachers are "judging" types, whose classrooms are more likely to be orderly and governed according to structure and schedules. This characterization is supported by Kent and Fisher (1997), who suggest that judging type teachers see themselves as encouraging high levels of on-task behavior. In contrast, classrooms of perceiving type teachers were more informal, distinguished by movement, noise and socializing among students.