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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).
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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
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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
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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
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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
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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.
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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,
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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
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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 SeniorSkills Pre algebra Algebra S Geometry SRegular Algebra S or
Algebra IGeometry 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
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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.
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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
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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-
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task behavior. In contrast, classrooms of perceiving type teachers were more informal,
distinguished by movement, noise and socializing among students.
In addition, some research shows connections between the "SJ" and "NF"
combinations of type and the teaching profession. These results lead to interesting
interpretations. Teachers whose type includes the SJ and NF combinations have been
characterized as "stabilizers" and "catalysts", respectively (Clark & Guest, 1995; Keirsey
& Bates, 1985). Stabilizers (SJ) value tradition and are "conservators and maintainers of
the institutions in which they work" (Clark & Guest, 1995, p.20). They additionally note
that this personality profile does not indicate that a "stabilizer" will particularly welcome
institutional change, or will be likely to adopt radically different instructional methods.
Sparks and Lipka note that "Master Teachers", those that are considered by their peers as
exceptional, are also generally found to have a similar respect for traditional ideas
(Sparks & Lipka, 1992). The implications for curricular and instructional change are
obvious in light of this observation from Keirsey and Bates (1985):
The SJ knows as well as others that change is inevitable, necessary, and even, on occasion, desirable; but it should be resisted when it is at the expense of the tried and true, the accepted and approved. Better that change occurs through slow evolution than by abrupt revolution. As conservator of the heritage, the SJ is an enemy of the revolutionary. (p. 44)
In contrast to the tradition-oriented nature of "SJ" teachers, NF or "catalyst"
personalities are interested in making a difference in the world, and tend to strive for
personal and professional growth. Catalysts may devote significant time and effort to
causes, provided that the cause has deep, lasting significance. However, while open to
innovation, NFs may not be likely to initiate change and need time to process and discuss
potential reforms (Clark & Guest, 1995). It should be noted that these types may tend to
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gravitate toward the humanities and the social sciences, rather than mathematics related
fields (Keirsey & Bates, 1985).
The prevalence of these types among teachers is also interesting. Specifically,
"SJ" and "NF" types are represented in roughly 90% of the teaching force (Hoffman &
Betkoudi, 1981). McCutcheon (1991) reports that the largest subgoup of secondary
teachers in their study were a subset of "SJ" teachers: ISTJ types comprised 14.3%. Their
findings indicate that "SJ" types overall comprised 25% of the secondary teachers
studied, with "NF" at 29% (McCutcheon, 1991). Another study indicates that 42% of
high school teachers are "SJ", and 34% are "NF" (Clark & Guest, 1995). Keirsey and
Bates propose the following: "In any district … over half of the teachers at elementary
and secondary levels pursue an SJ style of life" (1985, p. 46).
Another finding related to the MBTI comes from a study that connects personality
type with teachers' attitudes toward the use of technology (Smith, 1995). This may be
particularly significant, considering that increased technology use has been a feature of
recent mathematics reform recommendations (National Council of Teachers of
Mathematics 1989; National Council of Teachers of Mathematics, 1998). Smith found
that Intuitive/Thinking (NT) types of teachers, those that are creative, analytical, logical,
and imaginative, are more receptive to the use of technology in their classrooms than
"Sensory" types, which are a large proportion of the teaching population. NTs are
characterized as "visionaries", those that are inclined to lead rapid and dynamic change.
Unfortunately, these types make up only 10 to 16 percent of the teaching force, the most
underrepresented of all types (Clark & Guest, 1995).
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So, we should consider that many personality characteristics common among
teachers may potentially conflict with some of the methods of mathematics reform:
cooperative learning, use of manipulatives and technology, investigations, open-ended
problems, whole-class discussions and mathematical discourse. For example, if a greater
degree of noise and disorder may accompany cooperative learning activities and
increased student interaction, one wonders how this might be accepted by many teachers,
who as "judging" types, prefer order and structure (Lawrence, 1982).
Mathematics Teacher Receptivity
Although the research on teacher personalities and tendencies provides useful
background information, it does not speak directly to the beliefs of mathematics teachers
themselves. However, the 1993 Survey of Science and Mathematics Education provides
quantitative data related to the issue of teacher receptivity to reform methods. The study
was based on a national probability sample of science and mathematics teachers and
department heads in the 50 states and the District of Columbia. It provides a wealth of
useful data related to teacher background, textbook usage, and attitudes toward
instructional techniques (Weiss, 1995). Interestingly, the survey was given to teachers
designated as "Presidential Awardees", as well as the national probability sample. These
awardees are teachers who demonstrate the following:
subject matter knowledge and sustained professional growth; an understanding of how students learn mathematics; an ability to generate excitement about mathematics in students, colleagues and
parents; an understanding of the interconnectedness of mathematics and science and the
interconnectedness of all subject matter; an experimental and innovative attitude in their approach to teaching; and professional involvement and leadership (Weiss & Raphael, 1996).
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The data show that Presidential Awardees are much more likely to be in agreement
with the NCTM Standards than their national counterparts (Weiss, 1997). It should be
noted that these awardees were nominated on the basis of these traits, so it should be no
surprise that they strongly agree with the Standards. However, it is interesting to compare
their results with those of the national sample. In examining the two groups' responses, a
number of figures stand out:
Familiarity with the NCTM Standards
Nationally, 56% of 9-12 mathematics teachers considered themselves "well aware
of" the NCTM Curriculum and Evaluation Standards; 40% of 9-12 mathematics
teachers considered themselves "well aware of" the NCTM Professional Standards for
Teaching (Weiss, 1995).
In contrast, 98% of Presidential Awardees teaching grades 7-12 considered
themselves "well aware of" Curriculum and Evaluation Standards; 92% were "well
aware of" the Professional Standards (Weiss & Raphael, 1996).
Nationally, of the teachers who were "well aware of" a particular set of standards,
data were collected regarding their level of familiarity with them:
91% of the national population said they were "well informed about the
Curriculum and Evaluation Standards; 58% considered themselves "prepared to
explain" these standards to colleagues.
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89% said they were "well informed" about the Professional Standards for
Teaching; 55% considered themselves "prepared to explain the Professional
Standards to colleagues (Weiss, 1995)".
From this data, we see that only 33% (or, 58% of the 56% who were "well aware of
the Standards) of national high school mathematics teachers considered themselves
sufficiently knowledgeable about the Curriculum and Evaluation Standards to explain
them to a colleague. Similarly, only 22% could explain the Professional Standards for
Teaching.
However, Stigler and Hiebert (1999) reported a greater percentage of mathematics
teachers being aware of the Standards. Specifically, they found that 95% of teachers
sampled as part of the Third International Mathematics and Science Study (TIMSS) were
"somewhat aware" or "well aware" of the reforms recommended by the NCTM
Standards, and 70% claimed to be implementing these reforms (p. 105). Unfortunately,
after examining videotaped lessons from 81 eighth grade mathematics classrooms, the
authors found that few of these teachers were actually implementing the reforms as
intended. Japanese instruction, in fact, did a better job of emphasizing reform-oriented
ideas such as student thinking and problem solving, multiple solution methods, and
classroom discourse.
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Technology
Nationally, 73% of 9-12 teachers said that "students should be able to use
calculators most of the time" (Weiss, 1995), compared with 93% of the Presidential
Awardees (Weiss & Raphael, 1996).
50% indicated that students working with calculators should "definitely should be
a part of mathematics instruction" (Weiss, 1995), compared with 86% of the
Presidential Awardees (Weiss & Raphael, 1996).
42% of the national population felt well prepared to use computers in instruction;
81% felt well prepared to use calculators (Weiss, 1995).
Views About Mathematics
In contrast to the NCTM Standards, which recommend earlier introduction of
algebraic concepts, Weiss notes that 76% of mathematics teachers indicated that
"students must master arithmetic computation before going on to algebra", whereas only
17% of Presidential Awardees indicated this belief (1996).
Instructional Practices
Nationally, 27% of 9-12 mathematics teachers indicated that students working in
cooperative groups should "definitely should be a part of mathematics instruction"
(Weiss, 1995), compared with 53% of Presidential Awardees (Weiss & Raphael,
1996).
66% of the national population felt well prepared to use cooperative learning
groups.
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62% of 9-12 mathematics teachers said they were well prepared to manage a class
using manipulatives (Weiss, 1995).
Despite these arguably high numbers, Weiss (1997) notes that the survey indicated
that 48% of the time in high school mathematics classes was spent in whole group
lecture/discussion, 14% in small group discussions, and 7% in working with
manipulatives. Furthermore, 94% of high school mathematics classes listened and took
notes during presentation by the teacher at least once a week, and 60% did so on a daily
basis. 98% of high school mathematics classes did mathematics problems from their
textbooks at least once a week, and 86% did so on a daily basis.
In some ways, this research supports the implications of the personality type data.
We have more confirmation that mathematics teachers, as a whole, have not acted on
many of the reform recommendations regarding instructional practices, as was also found
by Stigler & Hiebert (1999). Even when these ideas are accepted, implementation is still
problematic, as case studies discussed below will illustrate. This is not to suggest that
mathematics teachers are incapable of incorporating cooperative learning, whole-class
discussions and mathematical discourse; nor are they universally opposed to these ideas.
However, we see that there is still, at times, a disconnect between these ideas and
teachers' receptivity to using them.
Providing support to the 1993 quantitative data from Weiss, numerous case
studies have been conducted documenting teachers' attempts to implement the NCTM
Standards, as well as their attitudes toward reform-oriented methods (Benbow, 1993;
Chauvot & Turner, 1995; Clarke, 1997; Cooney & Wilson, 1995; Eggleton, 1995;
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Frykholm, 1996). These qualitative studies provides indications of the conditions under
which teachers may be receptive to reform-oriented curricula and instructional practices.
Three themes emerged from an examination of these studies: beliefs about the nature of
mathematics, beliefs about the teacher's role, and reflective practices.
Beliefs About Mathematics
One theme coming from these studies concerns the mathematical beliefs of the
teacher prior to teaching. We see a difference between teachers that consider mathematics
a question of absolute right or wrong, versus those who view mathematics a dynamic
field, influenced by context and personal factors. Specifically, the former group, those
with an “absolutist” view of mathematics, have a more difficult time accepting the
principles of the NCTM Standards than does the latter group, who have a “relativistic”
view (Cooney & Wilson, 1995).
For example, Eggleton reports a pre-service teacher that considered mathematics
a "set of facts", or "number crunching," and himself as the center of mathematical
authority for his students (1995, p. 2). Similarly, Cooney (1995) documents a preservice
teacher that held an "assembly line" view of mathematics, with a rigid filtering system of
what she accepted as evidence of learning and good practice (p. 3). Predictably, these
teachers had difficulties realizing the vision of teaching espoused in the NCTM
Standards, as their initial views about mathematics seem contrary to the spirit of those
documents (National Council of Teachers of Mathematics, 1989; National Council of
Teachers of Mathematics, 1991; National Council of Teachers of Mathematics, 1998).
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Again, this is not to suggest that teachers with an absolutist view of mathematics
are immovable in their beliefs or are incapable of realizing a reform-oriented vision of
teaching. Furthermore, it would be a mistake to assume that one’s beliefs are strictly
absolutist or relativistic. Yet, we see in these studies and others that when teachers hold a
rigid, narrow view of mathematics, that nontraditional teaching develops far more slowly
(Benbow, 1993; Clarke, 1997). Unfortunately, we should also note that this view is not
uncommon in preservice mathematics teachers (Eggleton, 1995).
By contrast, teachers that hold a primarily relativistic view of mathematics have
an easier time implementing the Standards. Another preservice teacher studied by
Cooney had such a view of mathematics, believing in emphasizing the connectedness of
mathematics to the real world for his students. In addition, he saw mathematics from a
broad perspective, as opposed to narrowly defining the field as the sum of his
postsecondary mathematics course topics (1995). Clarke (1997) describes a similar
situation with a veteran teacher. In these cases, the teachers' beliefs proved flexible,
which allowed them to adapt their teaching to learners' needs in the context of
nontraditional methods.
Beliefs About the Teacher's Role
A second notable theme concerned teachers' beliefs about their role in the
classroom, and the ability to critically examine it. For example, Chauvot and Turner
(1995) studied one preservice teacher who saw her role as providing a "non-intimidating,
non frustrating, interesting and motivating" environment for her students. As she
proceeded through her student teaching program, she was encouraged to implement
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reform-oriented methods, which in turn strengthened her beliefs. These beliefs led her to
increasingly use multiple teaching methods and group work and emphasize problem
solving. In addition, this teacher began to negotiate her role in the classroom, specifically
with how much direction she should provide the students.
"Telling" is a theme that appears in other studies as well (Benbow, 1993; Clarke,
1997). Chazan and Ball argue that one aspect of the teacher's role is to use discussion-
intensive teaching to support an atmosphere of "intellectual ferment" (Chazan & Ball,
1995, p. 18). In this environment, "In small and large groups, students are to present their
ideas and solutions, explain their reasoning and question one another" (p. 1). As a result,
teachers must reframe their role as the center of mathematical authority that has
traditionally supported telling the students whether they are correct or incorrect.
Those that were most successful in implementing reform ideas into their
classrooms were those that were able to "step back", in the words of one teacher. Letting
students struggle with problems and resisting the urge to squelch mathematical discussion
by giving the answer became a "liberation" for this teacher. For him, the teacher's role
became a question of "to tell or not to tell" (Clarke, 1997, p. 288).
Of course, Chazan and Ball (1995) argue that mathematical discourse involves
more than "telling or not telling," but it must be noted that the willingness to explore this
aspect of teaching is an important indicator of success with Standards-based materials. In
contrast, we see that many teachers still hold metaphors for teaching that imply an
authority or controlling relationship (Fleener, 1995). This is consistent with personality
type data regarding teachers, but could be problematic for the widespread use of reform-
oriented methods.
22
For example, Gregg documented the difficulties of one mathematics teacher in a
school serving several counties with enrollment of about 1200, located a Midwestern
town of about 5000 residents (Gregg, 1995). The emphasis the teacher placed on
maintaining control in her general mathematics class led to several unfortunate results:
The teacher's expectation that the students would get out of control easily
influenced her behavior, causing her self-described "meanness." The students reacted
to this by being more disruptive. This behavior further influenced her view of the
students, causing her to be more "mean". Thus, a self-perpetuating cycle was
initiated.
When the teacher would have trouble keeping the class under control, she would
stop asking leading questions that she used in other classes to engage students.
Although she was trying to limit the opportunities for breakdowns in control, this
contributed to the students' boredom, which aggravated discipline problems. This
started another contradictory cycle.
Contradictions came into play when the teacher attempted to implement a general
math program written in the spirit of with the NCTM Standards emphasizing problem
solving. The program was discontinued because the "kids couldn't handle the
freedom" (p. 590).
The Ability to Reflect
In virtually every one of the case studies examined, the authors cite the ability to
reflect on teaching practices as a crucial component of success (Benbow, 1993; Chauvot
23
& Turner, 1995; Clarke, 1997; Cooney & Wilson, 1995; Eggleton, 1995). Eggleton
writes, "the difference between instruction and indoctrination was the provision of
opportunities for the individual to radically examine his/her belief systems." (1995, p. 4).
These cases illustrate the truth in these words. Further, Fullan and Stielgelbauer (1991)
write,
Beliefs guide and are informed by teaching strategies and activities; the effective use of materials depends on their articulation with beliefs and teaching approaches and so on. Many innovations entail changes in some aspects of educational beliefs, teaching behavior, use of materials and more. Whether or not people develop meaning in relation to all three aspects is fundamentally the problem. (p. 41)
Case study research suggests that those who were able to reconcile their beliefs
about mathematics and the role of the teacher with reform recommendations were
consistently able to reflect on their practice, and reconstruct it accordingly.
Unfortunately, case studies also suggest that those who had neither the inclination nor the
opportunity to engage in this sort of reflection struggled with their teaching experience.
Departmental Receptivity
Although an examination of the characteristics of individual teachers may be
useful, this knowledge may be of limited importance. Waugh and Punch (1987) suggest
that the change process is additionally dependent on factors such as: organizational
features, the politics surrounding the change, and the type of change being implemented
(p. 241). Furthermore, Baldridge and Deal (1975) found that organizations often have
deep roots in history, and suggest that teachers are not likely to be strongly receptive to
any proposed or attempted implementation of a change that is in direct conflict with the
24
traditional values of a school or school system (p. 15-16). Therefore, when considering
the implementation of a new program at the department level, an examination of the
characteristics of similar departments is appropriate.
Specifically, it may be argued that mathematics reform documents are a response
to a strikingly durable tradition that has evolved in many American mathematics
departments. This is the traditional school approach to mathematics, such as the routine
of checking homework, teacher presentation of material, students working examples, and
assignment of seatwork (Cobb, 1992; Stigler & Hiebert, 1999). One wonders what factors
affect the overall environment of a mathematics department, indicating whether it can
break the stranglehold of the "school mathematics tradition".
Information about mathematics departments and the NCTM Standards is
appearing in the form of both quantitative and qualitative data. Garet and Mills report
results of a 1991 survey given to mathematics department chairs in all public,
comprehensive high schools within 100 miles of Chicago (Garet & Mills, 1995). The
survey asked respondents to provide information about practices in first-year algebra
classes, and several findings provide interesting background for this question.
For example, average consistency with practices recommended in the NCTM
Standards documents among schools is increasing, but the variation between schools is
dramatic. Furthermore, this variation is increasing over time. In part, this is associated
with demographic characteristics in the schools. Generally, suburban schools reported the
level of practice most consistent with the Standards in 1991, but by 1996, urban centers
predicted greater alignment of their instructional methods.
25
In addition, the study documented continued prevalence of ability grouping, a
practice often criticized as inequitable (D'Ambrosio et al, 1992; McKnight et al, 1987;
National Council of Teachers of Mathematics, 1989). These numbers indicate that ability
grouping was associated with school size: 70% of small schools with enrollment under
500 reported just one level of Algebra 1. In contrast, 10% of large schools with
enrollment over 2000 report only one level of algebra. This is probably no surprise, as
smaller schools have fewer resources to expand curricular offerings. Yet, it is interesting
to note that as schools increase in population, the number of course offerings increase.
Curriculum thus becomes less compact, resembling a wider smorgasbord of choices for
students. However, Gutierrez (1996) notes the importance of a compact, less
differentiated mathematics curriculum in encouraging high levels of success in a wider
range of students.
Although these quantitative data are illustrative, specific case studies provide
more detail and allow one to examine the real character of successful mathematics
departments. For example, Meador documents the successes and failures of a
mathematics department that has been engaged in reform efforts since 1977 (Meador,
1995). In addition, Gutierrez studied the beliefs and practices of several mathematics
departments that succeed in encouraging more students to take mathematics at higher
levels by the end of their senior year (Gutierrez, 1996).
Although these studies did not specifically address implementation of reform-
oriented curriculum as defined for this paper, characteristics of these programs are
remarkably consistent with such recommendations. Therefore, it is perhaps no surprise
that these departments have implemented many of the key components of reform
26
documents. In particular, some of these are an innovative curriculum, nontraditional
teaching methods, and the use of appropriate technology (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, 1999). However, the question is, what
aspects of the department allow the implementation and sustained success of these
components?
Professional Activity of Teachers
According to these studies, one common element in successful mathematics
programs is the teachers' involvement in professional activities. These departments allow
and encourage teachers to interact with each other, and with teachers in other schools.
Through workshops, involvement in professional organizations, or informal extraschool
communities, teachers are exposed to new ideas and are able to share information.
Meador writes, "Teachers enjoyed collaboration and interaction with other teachers in
and out of their department, and the network of support they developed outside of the
school was crucial to their continued emphasis on trying new ideas in the classroom"
(1995, p. 13).
Leadership
Related to this, Garet and Mills (1995) note the crucial role played by the
department chair in guiding and supporting change. Specifically, the Department Chair's
involvement in professional organizations is significant. 75% of chairs in the 1991 survey
27
sample belong to the NCTM, and 25% attended that organization’s annual meeting in
1991. However, professional organizational involvement was differentiated by
geography, similar to the variation in practices associated with the NCTM Standards.
Roughly one-half of the chairs in Chicago, Milwaukee, and their suburbs attended the
1991 meeting, but only 14 percent of chairs in smaller cites and 3 percent of the chairs in
rural communities attended (Garet & Mills, 1995).
Similarly, a majority of the chairs in the sample report being quite familiar with
the Standards and strongly agree with the reforms proposed. Yet, while 75% of the chairs
in the suburbs reported being quite familiar with this document, only one third of the
chairs in rural districts reported this. This research suggests that department chairs that
are active in the professional community are more aware of curriculum and evaluation
reforms, and are more likely to report departmental teaching practices espoused by the
Standards documents (Garet & Mills, 1995).
Mathematics for All
Although not easily quantifiable, successful mathematics departments also share
another characteristic supported by reform documents: active commitment to all students
(Mathematical Sciences Education Board, 1989). In these departments, an atmosphere of
"Mathematics for All" is a reality, not an empty slogan. Gutierrez writes, "Despite their
students' weaknesses teachers in (these) departments tended to concentrate on students'
strengths to aid in the teaching and learning process" (1996, p. 511). Furthermore,
teachers held high expectations that all students could handle a rigorous curriculum.
28
In contrast, departments that foster non-constructive, negative conceptions of their
students may be unsuccessful implementing Standards-based curriculum. Examples of
these attitudes may be, “too many students aren’t ready to learn the material we have to
teach them”, or, “students are unwilling to learn” (Gutierrez, 1996, p. 511). This
represents a "clash of cultures" line of reasoning commonly used to explain students' lack
of achievement: the irresistible force of the students' home culture meets the immovable
object of the traditional school culture (Gregg, 1995, p. 598).
Control and Authority
It may also be important to critically examine the school culture and the status
afforded teachers, based on their ability to maintain authority and discipline. In the
successful departments, teaching success is not based on authority relationships, but on
student achievement (Gutierrez, 1996; Meador, 1995). In a study by Gregg, an emphasis
on discipline and control was a main concern with the both teacher and the principal
(1995). Again, the only serious reform effort undertaken was abandoned in less than a
year, as a result of perceived discipline problems.
This is consistent with conclusions drawn from a study that examined the factors
that influenced teacher induction (Schempp, Sparkes, & Templin, 1993). This study
found that teachers were accepted on the basis of their classroom management skills, and
that teachers' control over students is a taken-for-granted assumption in schools. In fact,
one of the beginning teachers studied by Schempp et al. won his mathematics position
because of his classroom management abilities, rather than his meager subject matter
knowledge. He writes, "Success or failure, bluntly put, resided in the teachers' control
29
over students: the greater the level of control, the greater the level of success" (Schempp
et al., 1993, p. 459).
Change Embedded in the Culture
Finally, mathematics departments that are successful at implementing
nontraditional curriculum are described as having change embedded in the culture, as
demonstrated by a department that began implementing reform-oriented curriculum and
instructional practices over ten years prior to the publication of many of the national
mathematics reform documents. Meador writes, "These teachers began their struggle
without the benefit of school reform literature… The motivation to change was theirs and
theirs alone" (1995, p. 1). In this department, change was talked about, looked for and
expected.
Similarly, Gutierrez describes a "commitment to a collective enterprise" common
to departments who encourage high levels of students mathematics success (1996, p.
507). In addition to an emphasis on professional development, teachers in these
departments communicated actively and practiced collective decision-making.
Final Thoughts
After examining mathematics teaching in the United States, Japan and Germany,
Stigler and Hiebert (1999) note a "distinctly American way of teaching" (p. 11), and
suggest that we are often unaware of some of the most widespread attributes of teaching
in our own Western culture. They argue that the cultural nature of teaching may explain
30
why it has so far been so resistant to change, but recognizing this nature may provide
insights as to what must be done to improve education.
Although Stigler and Hiebert (1999) found little variability among American
teaching methods, attempting to change this teaching culture will likely be less
predictable. Conditions under which reform is welcomed, rejected, and when it stalls will
be specific to the local situation shaped by the students, the community, the teachers, and
school administration. Because educational change is such an incredibly dynamic and
difficult process to model, the recurring theme of "culture" could be a crucial organizing
principle. Fullan and Stiegelbauer (1991) write,
Thus the meaning of change for the future does not simply involve implementing single innovations effectively. It means a radical change in the culture of schools and the conception of teaching as a profession…Cultural change requires strong, persistent efforts because much of current practice is embedded in structures and routines and internalized in individuals. Yet cultural change is the agenda. (p. 142-143)
Therefore, if departments do not reflect on and critically consider institutionalized
practices such as tracking, emphasizing grades, and teaching in ways that reinforce the
school mathematics tradition, reform efforts will likely fail. They may never truly be
attempted in more than superficial ways. Since I read of success stories in the research, I
realized there was a possibility that the Adderley mathematics department would add to
these successes.
31
CHAPTER THREE
METHODOLOGY
For this study, my goal has been to collect data related to the culture of the
Adderley High Schools mathematics department, and specifically the receptivity of the
teachers to this curricular and instructional change. To this end, I have noted some of the
departmental history, traditions, and beliefs that have shaped this reform effort. My hope
is that this may help illustrate the conditions under which teachers and departments in
general are ready to accept such a shift in direction.
These data were collected in two stages. The first stage was conducted in
September 1999, and the second in February 2000. A written survey accompanied the
first stage of interviews (see Appendix A). In both stages combined, 15 teachers out of
the 22 in the department were interviewed; six of these teachers were interviewed at both
stages. Because CPMP is not yet fully phased in at all levels, some teachers have not yet
taught these materials, but these interviews were conducted almost solely with Core-Plus
teachers.
The Program
Before discussing the specific nature of these interviews, a description of
Adderley’s program is necessary. Certainly, the foundation is the Core-Plus Mathematics
Program, although the department has modified CPMP considerably by connecting it to
the "Accelerated Mathematics Program". Approximately two days per week, each teacher
suspends CPMP instruction and the class interacts solely with Accelerated Mathematics,
which is a computer-based drill-and-practice instructional program; more detailed
32
descriptions of both of these programs are given below. Another variable to consider
relates to a district-wide technology initiative: the model technology classrooms. This
affects only two mathematics teachers per campus, but the impact could still be
significant, and is further evidence of the many complex changes taking place in and
around the department.
The Core-Plus Mathematics Project
Developed with funding from the National Science Foundation, the CPMP is a
high school mathematics curriculum designed around the principle of "mathematics as
sense making". From the preface to the textbook series, the authors discuss this concept:
"Through investigations of real-life contexts, students develop a rich understanding of
important mathematics that makes sense to them and which, in turn, enables them to
make sense out of new situations and problems" (Coxford et al., 1998).
Guided by this principle, the program has several key features that set it apart
from a "traditional" high school mathematics curriculum. For example, each year does
not focus on one particular mathematical topic, such as algebra or geometry. Instead, four
content strands are woven together each year of the program: Algebra and Functions,
Geometry and Trigonometry, Statistics and Probability, and Discrete Mathematics.
Furthermore, the program emphasizes mathematical modeling through data
collection, interpretation, prediction, and simulation, and central topics are designed to be
accessible to all students. The use of graphing calculator technology is standard at all
levels, and the curriculum developers give a number of reasons for this. First, the
graphing calculator supports emphasis on numeric, graphic, and symbolic
33
representations, a theme repeated throughout the course. In addition, the calculator allows
students to focus on mathematical reasoning rather than computational proficiency in
situations where this is appropriate (Schoen, Hirsch, & Ziebarth, 1998).
Finally, the designers of the curriculum stress that students should be actively
engaged in the material. That is, a high priority is placed on the use of collaborative
activity, as opposed to teacher exposition. The students are to "explore, conjecture,
verify, apply, evaluate and communicate mathematical ideas" (Coxford et al., 1998, p. x).
Therefore, reading, writing, and active mathematical reasoning and modeling on the part
of the students are crucially important to instruction and assessment.
The authors provide a rich description of the intended instructional pattern:
Most classroom activities are designed to be completed by students working together collaboratively in heterogeneous groupings of two to four students…The launch phase promotes class discussion of a situation and of related questions to think about, setting the context for the student work to follow. In the second or explore phase, students investigate more focused problems and questions related to the launch situation. This investigative work is followed by a class discussion in which students summarize the mathematical ideas developed in their groups, providing an opportunity to construct a shared understanding of important concepts, methods and approaches. Finally, students are given a task to complete on their own, assessing their initial understanding of the concepts and methods (Coxford et al., 1998, p. xi-xii).
See Appendix B for specific textbook examples.
Research conducted on the mathematical achievement of Core-Plus students, as
well as on the affective outcomes of these students, indicates that the curriculum can be
successfully implemented in these areas. Specifically, one study found that CPMP
students outperformed traditionally taught students on a number of standardized tests
(Schoen et al., 1998). Other research has suggested that CPMP students are more positive
34
about certain aspects of the curriculum and of their classroom experience than students in
traditional classes in the same schools (Schoen & Pritchett, 1998).
The Accelerated Mathematics Program
In contrast to the CPMP emphasis on exploration and communication, the
Accelerated Mathematics Program is decidedly more skills-based. "Ack Math," as it is
known in the district, is essentially a computer program designed to reinforce basic
mathematical skills. As this is perceived as a weakness of the CPMP curriculum within
the department, the department chair has strongly suggested that teachers split
instructional time between CPMP and Ack Math. As noted above, Mr. Blakey
recommends that students engage with the Accelerated Mathematics Program two full
days per week.
The software has a database of hundreds of skills objectives, most of which are
associated with traditional Pre-Algebra, Algebra I, Geometry and College Algebra course
content. It presents the students with an endless supply of multiple choice problems that
provide practice in each objective. However, the department has made an attempt to
select objectives aligned with the Core-Plus sequence. For example, as students are
engaged in the "Exploring Data" lesson in CPMP, during Ack Math time they might
encounter an objective that reinforces finding the median from a stem and leaf plot. See
Appendix C for specific examples of Accelerated Mathematics objectives and problems.
Unique features of this program are that students work at an individual pace, and
receive immediate feedback on their work. Each classroom is equipped with a laptop
computer that is connected to a scanner that grades the students' worksheets as they
35
complete them. Students are able to practice and receive feedback as often as they like
before they are tested on several objectives at once. Once a student tests at an 80% level,
the computer considers these objectives met, and provides the student practice with a new
set of problems related to new objectives. Since students complete the objectives
individually, a class of 25 students might realistically be engaged in meeting 25 separate
sets of objectives. Therefore, the teacher's role is often to circulate around the class and
provide individual guidance when needed.
Model Technology Classrooms
The Model Classroom Project is a district-wide technology initiative that has
created a number of specialized technology classrooms in both Adderley High School
buildings. The mathematics department currently has one technology classroom per
campus, and two teachers share each classroom. Each room contains seven student
computer workstations, a computer screen projection unit, printer, scanner, and flex-cam
for broadcasting live images on the Internet. In addition, each classroom is wired for
rapid T-1 access to the internet and computers are equipped with standard word
processing and spreadsheet software, presentation tools, web design software, and
internet browsers. The mathematics department has several content-specific applications
such as Geometer's Sketchpad. The 50 teachers in the district who share the 20
classrooms participate in ongoing training, and each is loaned a district-owned laptop
computer for his/her personal and professional use.
The design of the Model Classroom Project was chosen by the district for several
reasons. First, it is viewed as a cost-effective way to fully integrate technology into the
36
learning process. Furthermore, the district believes that the design allows for students to
be actively engaged in their own learning, working in groups or individually to carry out
assignments in research, design, problem solving, and presentation. The mathematics
teachers who share these rooms have periodically enhanced the CPMP/Ack Math
program with the use of spreadsheets and Internet access.
Personal Reflections
The actual introduction of the CPMP materials to the department has a simple
origin. Initially, these materials were discovered and introduced to Adderley by the
current department chair, Mr. Blakey. Before moving into his administrative position,
Blakey was a teacher in the department, and learned about the CPMP curriculum at a
conference. He began to experiment and had some success with elements of the program
in his own Algebra S class. When he became Department Chair, it was a natural step to
begin the pilot program.
However, as encouraged by CPMP as I was, I anticipated problems if the
curriculum were implemented on a department-wide scale. My concerns echoed several
of the criteria teachers use in assessing any given change, epitomized by the question
“Why should I put my efforts into this particular change?” (Fullan and Stiegelbauer,
1991, p. 127). According to Fullan and Stiegelbauer, teachers question how any change
will affect them personally “in terms of time, energy, new skill, sense of excitement and
competence, and interference with existing priorities" (p. 128). For teachers accustomed
to a traditional curriculum, this new curriculum might be extremely demanding in all of
these areas.
37
For example, upon inspection of the course materials, it is immediately apparent
that the CPMP text does not look like a traditional text (see Appendix B). One does not
find a list of exercises and lists of problems to be completed by students during "seat
work". Rather, the textbook is written more in paragraph form, divided into
"Investigations" that emphasize student experimentation and exploration. In fact, the use
of cooperative learning methods is often specified directly in the context of a problem.
While some teachers at Adderley were familiar with these strategies, others were not. In
addition, I knew that the heavy emphasis on the use of the graphing calculator would be
problematic for some teachers, and that extensive training would be necessary.
Furthermore, the program covers different mathematical topics than those covered
in a traditional sequence. For example, the first course begins with a unit in statistics. The
students collect data, construct histograms and box plots, and discuss variation. Later, the
concept of slope is developed in the context of linear regression. Many of the teachers in
the pilot program agreed that they had first encountered these statistical topics in college.
So, this material demands that teachers may initially have to learn along with the
students, which I took as time-consuming, and thus problematic.
In addition, the fact that mathematical topics are organized differently in CPMP
than in a traditional course sequence could have affected teachers' willingness to
implement the curriculum. Again, the text is organized in "strands" that are repeated from
year to year. For example, the "Algebra and Functions" strand extends across several
years of instruction. Therefore, topics traditionally associated with a first year algebra
course may not be developed fully in the first year of CPMP. I predicted that this would
be disconcerting for many teachers.
38
A specific incident illustrates the discomfort some felt with this aspect of the
program. Before implementing the pilot course, the department chair organized a summer
training session with the initial six pilot teachers. It was led by the department chair of a
school in Michigan that had extensive experience with the CPMP materials, having field-
tested the program for the curriculum publisher. During the session, an Adderley teacher
casually noted that he could not find where the "order of operations" was covered in the
first year text. The Michigan teacher thought and replied, "I don't think we taught that last
year. It didn't come up." This is consistent with recommendations that a strict adherence
to teaching a rigid sequence of topics may not be beneficial for all students (National
Council of Teachers of Mathematics, 1998).
Yet, I think that the statement made by the Michigan instructor came as a shock,
and caused quite a lively discussion. The thought that such a common topic wasn't taught
because it "didn't come up" sparked nervous laughter, and was emblematic of the change
that would be demanded of many. It was an uprooting of the familiar sequence of topics
that we had become comfortable with as teachers, and it was intimidating to some.
Another tension perceived at that time by some in the department related to
teachers’ years of experience. Specifically, the teachers were split almost evenly between
those with less than 10 years of professional experience and those with 20 years or more.
Certainly, those that seemed to be open to innovation were not only novice teachers, and
those that tended toward traditional methods were not exclusively the veterans. However,
departmental issues and discussions about instruction and assessment often fell along
these lines. Chapter 4 discusses the implications of this characteristic of the department
more fully.
39
Finally, I was concerned by the assertion that teachers question how changes
affect them in terms of existing priorities (Fullan & Stiegelbauer, 1991). Specifically, I
perceived a philosophy among some teachers in the district that very clearly distinguished
"Honors Students" from "Non-Honors Students". Even if a student displayed a healthy
work ethic and the willingness to be challenged, this philosophy often made moving into
the honors lane more difficult than moving into other lanes. Several of my colleagues and
I joked that a touch of elitism characterized the department at that time; allowing "Non-
Honors" students into "Honors" classes might contaminate the purity of these classes.
Whether this was accurate or not, many influential teachers in the department seemed
satisfied with this tracked, traditional course structure.
This is not to say that the department as a whole was closed to innovation or
experimentation. The Geometry/Chemistry Honors Plus program is an example of this. In
this program, Geometry and Chemistry are team-taught over three consecutive periods,
allowing the teachers to introduce long term projects incorporating Internet website
design. Furthermore, the department had experimented with alternative curricula, such as
the University of Chicago School Mathematics Project (UCSMP), on a larger scale
before 1996-1997. The UCSMP implementation had not survived, although the
department chair during my first year at Adderley strongly emphasized technology, the
use of cooperative learning strategies, and authentic assessment materials. Again, some
teachers in the department agreed that these strategies were beneficial, and employed
them extensively.
Changes are problematic for any mathematics department accustomed to
traditional teaching. My concern primarily stemmed from the perception of mathematical
40
elitism in the department. Again, this curriculum is consistent with a vision of
"mathematics-for-all," rather than "mathematics-for-the-elite” (D’Ambrosio et al, 1992;
Mathematical Sciences Education Board, 1989). I asked myself: Could this really be
accepted as "school mathematics"? Could this program possibly replace a "pure
mathematics program"? Would the teachers in this department be willing to give all
students the opportunity to engage in significant mathematics regardless of their
computational skill? However, despite these questions, I knew that these teachers were
first and foremost professionals, ultimately wanting the best for the students.
Related to the priority placed on doing what's best for the kids, I return to Fullan
and Stiegelbauer (1991). Another criteria teachers use in assessing any change is, “Does
the change potentially address a need? Will students be interested? Will they learn? Is
there evidence that the change works?” (p. 127). I believe that, although these teachers
may have been unsure of how a curriculum change would have affected them personally,
they certainly sensed a need to be addressed. It is possible that concerns about the
traditional curriculum may have been forming within the department for some time. In
the minds of a number of teachers, disturbing trends were beginning to develop, and the
old curriculum was beginning to be cast in doubt. One teacher later noted, "(the
traditional curriculum) just wasn't working for 90% of the students."
Although neither quantitative nor qualitative data were collected at the time, there
was anecdotal evidence that the exisiting curriculum was not leading to the positive
outcomes hoped for by the school. Through conversations with those in the department, I
noted some of these perceptions. According to these teachers, the traditional curriculum
wasn't working because over the last few years:
41
Fewer students were taking more than the required number of mathematics courses.
Fewer students at all levels seemed motivated to complete any class work. An increasing number of students at all levels seemed content to do the minimum
required work to pass. Absences were more common. An increasing number of students failed mathematics classes.
It is important to note that these perceptions may or may not be grounded in fact. It
is not clear, for example, that students were actually less motivated than in years past; no
attitudinal data were collected. Furthermore, if student motivation had actually decreased,
it is still not clear whether was due to the curriculum. Could these same students have
been motivated with the same traditional materials, but with different instructional
strategies? Again, this is not clear, but the fact that these perceptions were common in the
department is significant. It seems that these perceptions did, in fact, influence and
support the decision to move in a different direction.
In particular, one teacher expressed a common perception related to the
ineffectiveness of the traditional curriculum. When this teacher started at East Adderley,
she estimated the campus served some 2000 students. Enrollment later dropped to
roughly 1400, and she estimated it correctly at around 1800 students in 1999. When she
started, the students were primarily of Italian, German, and Polish descent. Now the
student body is 33% Latino, and since the break up of the Soviet Union, the school serves
a larger population of students from Eastern Europe. As a result, she said, "there have
been significant changes in what people bring to us when they come here". That is, some
students have been exposed to little mathematics and little technology. In addition, some
students have a very strong mathematics background but still have had no experience
with technology. Furthermore, far more students speak English as their second language.
42
Many teachers at Adderley see the CPMP/Ack Math program as an attempt to
address needs of this changing student body, an "attempt to level the playing field".
Because the student population at Adderley High Schools is becoming more culturally,
ethnically, and racially heterogeneous, a traditional mathematics curriculum did not
address the needs of these students. The new curriculum is perhaps an attempt to give all
students an opportunity to succeed, in spite of variation among students in English
speaking ability, and knowledge of mathematics and technology. Seen in this light, these
teachers had a very strong impetus to make the new program work. It would "address a
need."
The purpose of the preceding discussion has been to illustrate the background
knowledge I had prior to conducting this research. I had knowledge about the faculty, the
students, the administration, the departmental culture, and about the new program itself.
Equipped with this information, I returned to the department twice to collect data for this
study. The first stage of data collection occurred at the beginning of the 1999-2000
school year, and the second occurred shortly before the end of the first semester.
Data Collection: Stage One
The first stage of data collection was conducted at the beginning of the school
year because I sought to document teachers' perceptions before they had a chance to
become accustomed to their schedule, students, and, most importantly, the new program.
Many of those I interviewed had never taught CPMP, and I wanted to note their
preliminary impressions and expectations. I administered written surveys to 12 teachers
43
of the Core-Plus Curriculum, and I conducted audio taped interviews with 9 of those
teachers.
For this first stage, I focused primarily on teachers' perceptions of one aspect of
the classroom environment. Since a comprehensive discussion and analysis of the many
levels of paradigm shifts that might occur in the classroom would be a monumental task,
I specifically centered the interview questions on their beliefs about authority relations
and control in the classroom.
By “control”, I mean the tendency of many mathematics teachers to micromanage
and act as dictators (however benevolent) of nearly every aspect of the classroom
environment. Traditionally, power relations in the mathematics classroom have been
entirely top-down, where authority is exerted from on high, and all knowledge emanates
from the teacher. All too often, the teachers' desk sits like an altar at the front of the
room, and the students, congregation-like, sit in orderly rows and columns.
Alluding to the control issue, the Principles and Standards for School
Mathematics Discussion Draft states, “Knowing the capacity of the entire class to move
forward from a single student’s idea and juggling the tradeoffs in allowing an
unanticipated digression are part of orchestrating good mathematical discourse” (National
Council of Teachers of Mathematics, 1998, p. 33). Notable here is the suggestion that
students have influence in the direction of the class, that students be empowered in part to
guide their instruction. With this in mind, “classroom management” and “on-task
behavior” take on entirely different meanings. So, it would seem that teaching in the
spirit of the NCTM Standards, for example, implies that the teacher must relinquish some
control and provide the students the opportunity to grasp it.
44
However, the noise and surface disorder of a “student-centered” classroom have
long been the hallmarks of what a good teacher is supposed to avoid. Sizer calls this
unspoken pact between students and teacher to present a facade of sterile orderliness a
“Conspiracy for the Least”, the least hassle for anyone (Sizer, 1992, p. 156). That is, the
teacher demands little effort from the students, and in turn, the students cause the teacher
few discipline problems. Unfortunately, research as shown that these bargains are
common (Cusick, 1983; Powell, Farrar, & Cohen, 1985), and protect the status quo by
"using power to form a negative pact that reduces pressure for reform" (Fullan &
Stiegelbauer, 1991, p. 180).
Furthermore, Romberg notes, “There is an inexorably logical sequence when the
acknowledged work of teachers is to transmit the record of knowledge; the most cost-
effective way to accomplish this is through exposition to a captive audience...And that
exposition cannot happen unless there is control" (Romberg, 1994, p. 7-15).
Because the CPMP is designed to be more student-centered than a traditional
curriculum, I hypothesized that teachers would have to be prepared to relinquish a
measure of authority for a shift toward a reform-oriented vision to be fully realized. In
fact, I hypothesized that the release of a measure of control by the mathematics teacher,
permitting other voices in the classroom to be heard in a meaningful way, is a particularly
crucial element in such a shift.
As a result, I thought it would be fruitful to investigate the relationship between
mathematics teachers’ concepts of mathematics as a static body of knowledge to be
transmitted and classroom power relations. If mathematics itself is considered by a
teacher to be the ultimate non-negotiable authority, a dictatorial relationship between
45
teacher and students may be a natural outward expression of this scheme. Perhaps this
leads to a self-sustaining mathematical culture in which the authoritarian tradition of
mathematics teaching is handed down and recycled generation after generation.
Therefore, these perceptions would directly affect the implementation of the
Core-Plus curriculum. To explore this area of teachers' beliefs, the survey and the
interviews were designed to gather data regarding teachers' intended instructional
methods, beliefs about mathematics teaching and learning, and perceptions of the
teacher's authority in the classroom.
Data Collection: Stage Two
The second stage of data collection was scheduled to occur immediately
following the first semester, and capture whether or not the teachers' attitudes had
changed after actually teaching the materials. Specifically, I intended to interview the
same teachers a second time, documenting their perceptions of the Core Plus classes, and
whether they were able to sustain an environment that facilitated reform-oriented
teaching and learning.
However, the results of the first round of interviews were particularly interesting,
in that I detected attitudes toward authority and control that I never expected. This will be
developed further in Chapter 4, but I suspected that a very intriguing situation was
developing in the Adderley High School mathematics department, and I sensed that the
department's culture was undergoing a palpable change.
Exploring the cultural aspects of the department, Weissglass (1992) defines
culture as follows:
46
Culture is the attitudes, beliefs, values and practices shared by a community of people which they often do not question, are often unstated and which they may not be aware of. (p. 196)
Furthermore, Deal and Peterson (1999) offer this definition of school
culture:
School cultures are complex webs of traditions and rituals that have been built up over time as teachers, students, parents, and administrators work together and deal with crises and accomplishments. (p.4)
Using these definitions, one can see the potential for cultural change in the
department. Weissglass notes the importance of unstated attitudes and beliefs in a
community, and Deal and Peterson suggest that crises and accomplishments can greatly
effect these aspects of culture. Certainly, the Adderley mathematics department faced
obvious crises in the form of a challenging new curriculum, which required new
mathematical knowledge, pedagogical skills, and proficiency in technology. In light of
Fullan and Stiegelbauer’s assertion that “cultural change is the agenda” (1991, p.142-
143), the effect of the new curriculum on the “complex webs of traditions and rituals” in
the department became a more pressing concern than authority and control issues.
Attitudes, overall feelings, and general behavior intentions have been shown to
account for a large percentage of teachers’ receptivity to change, and a causal link has
been established between these three components. That is, overall feelings influence
attitudes, and both feelings and attitudes affect behavior intentions (Waugh & Punch,
1987). Thus, the unstated attitudes, beliefs, values and practices uncovered in the first set
of data led me to focus the second stage on developing a picture of the overall culture of
the mathematics department, as perceived by the teachers.
47
To this end, the work of Gutierrez (1996) had considerable influence on the
second stage of data collection. Her research examined "the beliefs, practices, and general
teaching cultures of mathematics departments that are successful in getting students to
take more mathematics, and higher levels of mathematics, by the end of grade 12" (p.
495). Of course, these were many of the same goals of the Adderley mathematics
department in its reform efforts, and I sought to make connections between this
department and the departments Gutierrez studied.
Specifically, Gutierrez describes a framework she calls "Organized For
Advancement" (OFA). An OFA department is indeed successful in encouraging more
students to take and succeed in mathematics courses, particularly at the higher level. The
key components of an OFA department are: a rigorous and common curriculum; a
commitment to a collective enterprise; a commitment to all students; and innovative
instructional practices (1996). These will be discussed more fully in Chapter 6.
I began to believe that the Adderley mathematics department had begun to change
and reshape itself as an OFA department. Therefore, in the second stage of data
collection, I asked questions of the teachers and observed the departmental culture and
climate, specifically investigating parallels between aspects of the Adderley mathematics
department and the key components of an OFA department. In addition, I explored the
role of leadership, and looked for historical trends that may have facilitated and
foreshadowed the direction of the changes taking place.
48
CHAPTER FOUR
STAGE ONE FINDINGS
Several interesting patterns emerged from the first set of interviews and survey
data. As noted, the first stage of data affected and altered the direction of this study. I
present here the key themes that ran through the survey results and interviews that caused
this. Chapter 5 presents the findings of the second stage of interviews.
Stage One: Survey Data
The surveys I initially administered first asked the teachers for background
information, such as gender, age, and years of teaching experience. Later they were asked
to gauge the extent of their agreement with a number of statements using likert-like
choices (see Appendix A). Results of these data are shown in Table 2.
In response to the statement, "Mathematics is a good field for creative people"
(question b) the average response was 4.33. Similarly, the average response to the
statement "Mathematics education is a good field for creative people" (question c) was
4.42. Every teacher surveyed marked "Agree" or "Strongly Agree", and the mean score
reflected this general agreement.
As far as information related to the presentation of mathematical content and
classroom management strategies, responses to items g and h may also be illustrative.
With respect to these statements, 7 of 12 (58.33%) indicated disagreement or strong
disagreement with item g, and 6 of 12 (50%) indicated disagreement or strong
disagreement with the item h. The inclination in some of these teachers to change their
49
management and presentation strategies is apparent, and the interviews provide further
details regarding these changes.
Table 2
Response Frequencies of Likert Survey Data
Question Response Choices Summary Statistics
1=StronglyDisagree
2=Disagree 3=Undecided 4=Agree 5=StronglyAgree
Mean Response
StandardDeviation
a) Mathematics will change
0 0 4 5 3 3.92 0.79
b) Mathematics is creative
0 0 0 8 4 4.33 0.49
c) Mathematics education is creative
0 0 0 7 5 4.42 0.51
d) Familiar with NCTM Standards
0 0 0 5 7 4.58 0.51
e) Familiar with Illinois State Standards
0 1 2 7 2 3.83 0.83
f) Core-Plus is rigorous
0 0 4 4 4 4.00 0.85
g) Presentation will stay the same
2 5 1 3 1 2.67 1.30
h) Classroom management will stay the same
1 5 1 5 0 2.83 1.11
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Conversely, 4 of 12 teachers (33%) indicated agreement or strong agreement with
item g, and 5 of 12 (41.6%) indicated agreement or strong agreement with item h. It is
not clear whether these teachers believe that their teaching practices are already
consistent with methods recommended by the NCTM and the developers of CPMP, or if
they do not believe such changes are necessary.
Another area of interest is the teachers' familiarity with the NCTM Curriculum
and Evaluation Standards and the Illinois State Learning Standards. In response to the
survey statement, "I am familiar with the National Council of Teachers of Mathematics
Curriculum and Evaluation Standards", the average response was 4.58, where 5
represented “Strongly agree”, suggesting that the teachers felt confident in their
knowledge of the NCTM Standards. Of the 12 teachers surveyed, all indicated that they
"agree" or "strongly agree" with this statement.
This contrasts with Weiss' findings that only 56% of mathematics teachers
nationally considered themselves "well aware of" the NCTM Standards (Weiss &
Raphael, 1996). This level of familiarity with the Standards is more consistent with
Stigler and Hiebert’s findings (1999), and may partly be a result of the periodic Core-Plus
curriculum training provided by the publishers. Most on the staff had participated in this,
as is discussed below. Interestingly, although 100% of the Adderley teachers agreed that
they are familiar with the NCTM Standards, only 75% indicated agreement or strong
agreement with the statement "I am familiar with the Illinois State Learning Standards."
The average response was 3.83.
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Again, it is important to note that these are only perceptions of the staff; actual
familiarity with the Standards or other reform documents is unclear. However, it is
important to note because Weiss' study measured teachers' perceptions as well. The
difference between responses of this department and the national sample of teachers is an
indication of the potential for a reform-oriented direction.
Further in the survey, the teachers were asked to consider several specific
classroom strategies and indicate how frequently they intended to emphasize these
methods (see Appendix A). Table 3 gives the results of these data.
Table 3
Response Frequencies of Intended Instructional Practices
Question Response Choices Summary Statistics1 = Not
emphasized at all
2 = Occasionally,
but not frequently
3 = Emphasized frequently
4 = Almost always
MeanResponse
Standard deviation
a) Passive reasoning
0 9 3 0 2.25 0.45
b) Active reasoning 0 2 7 3 3.08 0.67
c) Mathematical procedures
0 5 6 1 2.67 0.65
d) Mathematical facts
0 5 7 0 2.58 0.51
e) Passive applications
0 6 5 1 2.58 0.67
f) Active applications
1 2 4 5 3.08 1.00
g) Communication 1 0 4 7 3.42 0.90
h) Passive modeling
1 7 4 0 2.25 0.62
i) Active modeling 2 1 5 4 2.92 1.08
j) Multiple representations
0 1 6 5 3.33 0.65
52
Interesting results regarding the control and authority issue in the classroom
related to the teachers' perceptions of passive and active reasoning on the part of the
students, items a and b. Of those surveyed, 9 of 12 (75%) indicated that they would
emphasize passive mathematical reasoning only occasionally. Similarly, 10 of the 12
teachers (83.3%) intended to emphasize active reasoning frequently or almost always.
In addition, intended use of both passive and active mathematical modeling is
indicated in items h and I, respectively. 8 of the 12 teachers (66.67%) indicated that they
would not emphasize passive modeling at all, or infrequently. On the other hand, 9 of the
12 surveyed (75%) intended to emphasize active modeling on the part of the students
frequently, if not always.
The data further show that 11 of the 12 (91.67%) intended to emphasize
mathematical communication, or asking the students to speak and/or write clearly about
mathematical ideas, frequently or almost always. These results suggest that the
department agrees, at least in principle, with one of the central tenets of the Core-Plus
curriculum: active learning (Coxford et al., 1998).
Yet later in the survey, when asked to judge their expertise in using writing in
their classes, none rated themselves as "expert", given the choice between "novice",
"intermediate" or "expert". Furthermore, only three of the 12 (25%) considered
themselves experts in using group work in their classes. This is consistent with the
portrayal of a department that is still learning how to use various instructional methods.
They may not be confident with the means to some of the larger ends of a reform-
oriented curriculum, such as developing the "network of knowledge" held by individual
53
students, as well as nurturing a supportive mathematical community (National Council of
Teachers of Mathematics, 1998, p. 34).
Survey Data: Teacher Interviews
Each teacher was given a list of metaphors for the teaching of mathematics, and
was asked to select the three that best characterized his or her beliefs before each
interview began. Later, they were asked to explain these choices. Figure 2 summarizes
these results, and one can see that the most-selected choice was "coach", followed by
"orchestra conductor" and "entertainer".
Sizer (1992) presents coaching as an appropriate metaphor for the teaching that
should occur, but often does not, in many schools. Echoing recommendations made in
mathematics reform documents, Sizer suggests that the process of teaching and learning
is dependent on active student participation, rather than teacher exposition. He compares
this process to coaching a javelin thrower when he writes, “You throw. I criticize,
suggesting some possible improvements. You throw again. And again I criticize. This is
how skills in a strong athletic program are shaped. The analogies to intellectual training
are powerful and apt” (p. 106).
Similarly, one Adderley teacher noted the following regarding teaching the new
mathematics curriculum:
Well, you're trying to put people in a very rich environment, putting problems and obstacles around them, and trying to get them to deal with those problems. You're trying to coach them in ways that they might approach it, and talk to them about what they're trying to do.
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The choice of "Orchestra Conductor", was interesting, because it reflected a sense
that part of the teacher's job is to keep the class running smoothly and create harmony out
of 30 different voices in the classroom. However, the implications for authority relations
could depend on what one considers the role of conductor. One might see this as a
facilitating role, where the teacher encourages the best performance from each student.
On the other hand, orchestra conductors have been characterized periodically as despotic
and heavy-handed. The following comments support the former interpretation:
Especially in the Core Plus program, where your job as a facilitator is to keep 30 kids working smoothly and cooperatively, it's as difficult or more difficult than an orchestra conductor. In an orchestra, everybody already knows what they want to do, and you have to get everybody going in the same direction and keep them going in that direction.
The teachers were also asked to choose from a similar set of metaphors to
characterize their beliefs about mathematics learning. Overwhelmingly, the most popular
choice was "Working a jigsaw puzzle", followed by "Cooking with a recipe" and
"Building a house," as shown in Figure 3.
In comparing mathematics learning to a jigsaw puzzle, one teacher
remarked,
It's a puzzle because, um, and this is the hardest thing to convey to kids, if you step back and look at what you've learned in all of your math classes, and I didn't get it until I was in college, you can kind of understand what you did after you step back and take a look at everything together. Everything kind of fits together in a whole if you step back.
Although the choices of "building a house" and "cooking from a recipe" might be
interpreted as viewing mathematics learning as rote, according to scripted plans, other
themes emerged from these teachers' explanations. First, these teachers were product-
56
oriented: the end result of mathematics learning should be as useful and real as a house or
a nourishing meal. Second, the teachers noted the artistic and creative aspects of building
and cooking as analogous to mathematics learning. This is reflected in the following
teacher's comments regarding mathematics as a field of study, in which she compared
mathematics to a blueprint and a cookbook:
Mathematics is dynamic...I guess I could say it's a blueprint because you have a structure you're beginning with. But when I think of a blueprint, I think about a house, and the house goes up...but even if it was the same house from person to person, the interiors change and the exteriors change and the design is changed, and there's some flexibility there… And then there's a wall that will be broken down later and the blueprint changes a little bit, so I guess I could say that mathematics is a little like a blueprint to some degree. ...A cookbook too, in that same respect because it does start with so many ingredients. But each cook adds their own flavor, their own personality, their own individualism to whatever they're cooking. And also, cooking is exceptionally creative, and I actually believe that mathematics is very creative, too.
I further posed a number of open-ended questions related to perceptions of
authority. Specifically, I asked, "If someone were to say that you were 'in control' of your
classroom', what would that mean to you?" Most teachers noted that control meant to
them a civil environment, or at least the lack of total chaos. This base definition was often
characterized by statements such as, “students not running around the room”, or “no
objects are being thrown around.”
Yet, deeper interpretations of "control" seemed to have culturally taken-as-shared
qualities that signaled a unique receptivity to this style of teaching within this department.
In nearly every case, these teachers used vocabulary that demonstrated the interpretation
of "authority" and "control" to mean being able to step back and let the students explore.
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Although paradoxical, the teachers make some intriguing statements about this tendency
to "control by letting go". Some illustrative comments were:
Control means that I've set it up well enough for the students to be on their own. That I've designed the lesson well enough that I can let go. I can literally walk out of the room and things are still be going on the way they should be going.
The old teaching style used be that you were in control if no one else was talking except for you. And to this day, if I'm introducing something, I do expect the students, if not listening, at least not talking... But being in control now to me just means that the students are actively involved in doing math, and thy can be talking about the math, they can be, like, measuring each other, or whatever the situation calls for. But if an administrator would walk in and see a class in disarray, and heaven forbid people laughing or enjoying themselves, I think they need to know that actual math is going on, learning is taking place, even if the room is not quiet. Which is the way it used to have to be.
In the olden days, if people were quiet, it didn't matter if you had them (actively engaged) or not. If their eyes were open and they were facing you, you knew that you had control of the class. It's not really that the teacher has control in the class now. Like, it used to be that the teacher had to be totally in charge. I don't feel like I have to be totally in charge anymore. I obviously am the more educated person in the room, but if I can get around to each group and see that they're actively working then that's good. If they're not I need to coerce them somehow to get busy or get on task.
Control...hmm...not a word I use a lot. Control could be that the class is functioning, the students are dealing with some human activity that is an outgrowth of the course we teach, and that they're involved in working on that problem.
Reflections on the Data
These teachers' responses have a number of interesting implications. Most
significantly, it appears that this department does not put a high value on authority in the
traditional sense of the teacher being the source of all knowledge. In particular, a majority
of the teachers expressed a philosophy consistent with the NCTM Standards, and other
59
reform documents. That is, most suggested that they would take a student-centered
approach, emphasizing collaboration and mathematical communication.
In addition, their educational philosophies seemed generally at odds with an
"absolutist" view of mathematics education. As previously noted, I was openly surprised
by these responses in light of my history with the department. In sum, they suggested that
the department may have been particularly receptive to other reform-oriented
recommendations, and perhaps to this curricular change overall. I suspected there were
deeper issues to be uncovered.
Of course, these conclusions are based on the subjective words of the
teachers, and lack the corroborating evidence of classroom observation. Still,
these are significant, as they are indicative of the culture of the department,
revealing beliefs, values, and philosophies. As Fullan and Stiegelbauer (1991)
note, "Ultimately the transformation of subjective realities is the essence of
change" (p. 36).
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CHAPTER FIVE
STAGE TWO FINDINGS
As a result of my interpretations of the first stage of data, I visualized
obtaining a wider picture, a "snapshot" of the department for the second stage of
data. Rather than focusing on specific aspects of classroom dynamics, such as
perceptions of authority, I sought the teachers' perceptions of the overall cultural
context. Weissglass (1992) notes,
One advantage of situating change efforts in a cultural context is that it enables educators to address the complex psychological and political reality of schools and classrooms, rather than focus on the more technical issues (new methods of assessment, curricular or achievement standards, and technology, for example). (p. 196)
During the Stage two interviews, I first asked the teachers their overall
impressions of the department and curricular materials. In general, I asked their
perceptions regarding the new curriculum, the reasons it was implemented and its
appropriateness, and about collegiality and professional development in the department.
Furthermore, I asked them to describe their perceptions in instructional practices and
whether the new curriculum facilitated a commitment to all students at Adderley High
Schools. Again, all of these subjects relate to elements of the OFA framework, and I
asked the teachers to describe any changes in these they may have noted. In addition, I
asked them about the role of leadership in the department, perhaps a key variable in
implementation of the NCTM Standards (Garet & Mills, 1995).
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Significant Cultural Traits
Guided by the definition of culture offered by Weissglass (1992), I interviewed
the teachers, seeking information related to their overall attitudes, beliefs, values and
practices. The teachers were asked first about their attitudes toward the new curriculum.
More specifically, I asked if they believed that the CPMP/Accelerated Mathematics
hybrid was working. Overall, there was general agreement that this program seemed to be
the right direction for the department. Every single Core-Plus teacher interviewed had
positive things to say and most seemed optimistic about the program's future success.
Some examples are:
I’m really high on the Core-Plus, and my reasons are many. I've been around since the time when teachers just lectured to the students for 50 minutes …Core-Plus is excellent in getting them to not only participate in their own education, which I think is the biggest factor, but get to do some critical thinking, and work with another person…The one negative thing is that, I think we're presupposing that kids come into high school with a lot of algebra talent.
…(The Core-Plus students') understanding of slope, their understanding of graphing of a table, their understanding of how the numbers in those tables relate to a story problem is phenomenally better than what is see the (non Core-Plus) juniors doing. But if I ask them to do polynomial work, or exponent work, it's like 'Ack!' They don’t know what to do.
It’s not perfect yet, but it’s better (than the traditional curriculum). Because it’s our first year, we’ll need to sit down at the end of the year and decide what worked and what didn’t.
Core plus, is not only exciting, it works if we take out the piddly stuff and try to figure out what were trying to develop over a 3-4 year period of time.…read, interpret, make sense out of things and have kids really engaged in what they're doing.
The kids enjoyed the class because it dealt with real life things. They learned more in this year than in any other years because there was less emphasis on taking tests, and more emphasis on thinking communicating
62
with another person and not being afraid to answer the question because it's what you think (that's important), not what you do.
The staff is buying into the program for a lot of reasons. Deep down, most of us feel that it’s probably the way to go, but we’re struggling with some of our own insecurities because even some of our young teachers were not trained to teach this way.
Despite this general positive feeling, it is important to note that each teacher
above cited specific concerns, including the department chair who commented,
I think Core Plus, what that did, is put us in the right century. The idea that math being only 'equations', that's gone…. We're at a place now where I think we should be. We're done selling it- it's there. It's my job now to critique it. I don't want people to start thinking that because it's there that it's golden.
This cautious optimism was echoed in every teacher’s comments, although several of
the staff expressed their anxieties more bluntly:
Right now, it's pretty scary. Because we're out alone, and our hearts are not totally there. Because even if what you had wasn't working, it's comfortable. This is not comfortable. This is different.
The program is very time consuming for some teachers who don’t know how to handle it. I've seen three teachers who have been overwhelmed by it. And these are veteran teachers. I don't know what to tell them. You know, 'Just relax, don't get so uptight', but in the same sense, how? Everyone's on different paths, trying different things, and we don’t know what works.
I’ve been teaching 29 years and this is the worst year I’ve ever taught, because the stress level is unbelievable. I have huge classes….It's a major transition, it's a brand new course, with a lot of new technology, and there’s no time to get good with anything…There were just so many things all at once. It has not been a pleasant year. If I don’t teach this again next year, someone's going to die, because I've put in a whole lot of work.
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These comments are indicative of the numerous anxieties being negotiated within the
department as they learn about this curriculum. It is important to note that the Core-Plus
program is designed as a three-year course for all students (Schoen et al., 1998). The
Adderley mathematics department has not yet implemented the third year of the program,
and only began widespread implementation of the second year in 1999. Many teachers
have only seen Year 1 or Year 2, and lack the benefit of perspective over the whole
course sequence. So, some of these uncertainties are understandable and probably natural.
As Lortie (1975) writes, "education is a tenuous, uncertain affair. It is necessary to keep
such uncertainty in mind if we are to understand the psychic world of classroom teachers,
for uncertainty is the lot of those who teach" (p. 133).
So it would seem that the department is at a critical juncture. While many of the
teachers are optimistic, they are mostly unsure where the program will lead. As Fullan
and Stiegelbauer suggest, "Change is full of paradoxes" (1991, p.102). Certainly, this
department might empathize with this sentiment, as the second stage interviews revealed
paradoxes, tensions, and contradictions. Some of the most common themes expressed by
these teachers are discussed below.
Skills and Core-Plus
The debate over the emphasis on mathematical skills in the Core-Plus sequence,
or lack thereof, has had particularly far-reaching results in the implementation of the
program. Selected comments illustrate this:
64
I think there needs to be more of some old traditional-type stuff, as long as colleges are still requiring that they have the basic principles of algebra and geometry.
I am truly concerned that (the students) are getting enough math. There are so many topics that I feel that we don't get to.
The more I teach it, the more I feel that there’s stuff that’s missing, but I feel that they're giving the kids challenges with different kinds of material. I guess I'm torn between this and a more traditional “skill building” course.
My key word this year has been 'balance'….so I'm trying to build some organized chaos as far as getting them to have some of the basic skills in algebra, and yet apply that, and that's the hard thing.
As noted, the Adderley mathematics department has not yet fully implemented
every course in the Core-Plus sequence. Therefore, the debate over the rigor of the
curriculum, and whether CPMP adequately stresses algebraic skills may stem from this
uncertainty. While algebraic skills are a central part of most first year algebra courses,
they are not particularly emphasized in the Core-Plus Year 1 or Year 2 curricula. For
example, students solve equations using the graphing calculator, but there is relatively
little algebraic manipulation. Because CPMP is designed around "strands", algebraic
skills are revisited, increasing in depth and rigor each year. Therefore, many of the
elements that would be crucial to a high school freshman in a traditional course are
indeed present in the third and fourth years of the Core-Plus sequence.
In fact, some research suggests that students having passed through the full CPMP
program score as well or higher than traditionally taught students on nationally known
standardized tests. (Schoen et al, 1998). In particular, this research studied student
performance on the Ability to Do Quantitative Thinking (ATDQT), which is the
mathematics subtest of the Iowa Tests of Educational Development (ITED), and a test
65
styled after the National Assessment of Educational Progress (NAEP) (Schoen et al.,
1998). Because algebraic skills are assessed as part of these tests, the concerns expressed
by some the Adderley teachers may be premature.
Yet, without having examined Core-Plus Years 3 and 4, some Adderley teachers
still perceived a lack of emphasis on mathematical skills. For example, nearly every
teacher interviewed cited “solving for x” as symbolic of this debate. Referring to students'
ability to use an algebraic algorithm to isolate a variable in an equation, the mantra
throughout the department was, "When will they learn to solve an equation?" Coming
from some teachers, this question reflected sincere concern about the rigor of the
program. From others, this phrase was inflected with exasperation and sarcasm. The real
question for the latter group was, "When can we get past worrying about 'solving for x'
and move on to more important issues?" One teacher noted,
There’s still too large of a percentage of people (in the department) who are locked into 'we have to teach (students) how to solve an equation', and order of operations and things like that. Look, if we can replace skills with manipulating (graphing calculator) screens, and not be afraid to make a few mistakes, and teach them to solve in a multiple of ways, we’re doing a better job at educating them.
The Honors Level
The debate over "skills" affects other aspects of the departmental culture as well.
Specifically, the teachers have begun a dialogue centered on the appropriateness of these
materials for the entire student body. The different viewpoints are exemplified by these
comments:
I also don't know if Core-Plus is the type of class...I mean, we're making everyone take it. There's some stuff in there that's deep. Maybe it's just for the honors-type kid. I don't know.
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At this point, it’s going fine. I'm just concerned. And because I'm not familiar with all 4 years, and we haven't taken a group (of students) through all four years, I don't know if they're getting enough math. I'm just concerned that at the honors level…that maybe they're not getting exposed to enough topics.
We do have a (college-intending) population, and those are the people I'm concerned about.... I'm not comfortable doing Core-Plus to the extent that were doing it with the honors-type kids…. I'm feeling that the Core-Plus isn't tough enough in the abstract area, and they're not going to be able to compete….I've always been leery about this top group.
As a result of my personal experience with this department, and the historical
emphasis on classifying students as "honors" or "non-honors", I was not surprised that
this is an unresolved issue. Of course, serious concern about the honors level student is
appropriate. As a result, the curriculum designers have suggested that the fourth year
course under development is intended for preparation of students for college mathematics
(Schoen et al., 1998).
Perhaps the tension is rooted in disagreement between what "honors" students
should be able to do, versus what all students should be able to do. In other words, is
Core Plus ideally suited for honors students because of the deep conceptual content? Or,
is it inappropriate for these college-intending students because it does not stress skills in
the first two years? One wonders whether the definition of "rigorous" mathematics will
similarly need to be redefined. Is "rigor" synonymous only with advanced symbolic
manipulation, or in a reform-oriented environment, does it mean more than mastering
these skills? The department will ultimately have to resolve these issues.
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The Accelerated Mathematics Program
The perceived weakness of the Core-Plus in emphasizing and reinforcing
mathematical skills has had another result discussed at length in this paper: the
implementation of the Accelerated Mathematics Program. The debate in the department
centers on whether Ack Math fills holes in the Core-Plus program, or actually
undermines it. A cognitive psychologist might argue the latter.
The Accelerated Mathematics Program is essentially based on behaviorist
psychology. It is a high-tech version of Skinnerian programmed instruction, under which
knowledge is decomposed into small bits to be mastered individually in a “drill and
practice” format. Much to the chagrin of reform-minded mathematics educators, drill and
practice has had significant impact on American mathematics education since E.L.
Thorndike published The Psychology of Arithmetic in 1922 (Schoenfeld, 1987).
In Ack Math, the small bits of knowledge are presented as multiple choice
worksheets printed by the computer, which are to be completed by the students (see
Appendix C). These problems are similar to those commonly used in standardized tests,
which emphasize predominantly low-level skills (Lacampagne, 1993). Students receive
the positive reinforcement of meeting an objective and having the objective "checked off"
next to the student's name on the computer. One might argue that instructional methods
based on this theory conflict with the emphasis on active learning and constructivism, a
theory that permeates the NCTM Standards, and the Core-Plus curriculum. The NCTM
(1998) alludes to constructivism in the following passage:
Some lines of research suggest that people do not generally learn concepts by building up pieces of knowledge…Instead, they are more likely to plunge headlong into a problem situation, bringing whatever facts, procedures, and understandings they have at hand. Thus, expecting
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students to master as set of so-called prerequisites in a prescribed order before they attempt to tackle challenging problems in a prescribed order is not necessarily productive. Although students taught in this manner may learn facts or procedures well, they often are not sure when or how to apply them, and they may also forget them quickly. More importantly, these students have little chance to experience satisfaction that comes from grappling with complex and interesting tasks. (p. 34)
Certainly, the NCTM is quick to note that the learning of routine skills and facts
has an important place in mathematics education. Yet, they also advocate a coherent
curriculum, which will be visible in the "consistency of style and approach" (p.29). The
issue of consistency seems to be the most significant for the Adderley teachers. These
comments illustrate both sides of the issue:
Basically, what I'm hearing, is that we're not really doing Core-Plus. What we're doing is tinkering with a little bit of this and a little bit of that, meaning the Accelerated Math Component.
To me the essence of whether (the students) understood anything is, are they able to apply their knowledge in a situation they’ve never seen? The more we use the Ack Math, the less that happens...Core math is a great model. It can be supplemented with Ack Math. The biggest interference with the program is the ACT-type question. When we continue to say that ACT scores are important politically, then you have to have those (Ack math) questions in there.
Ack math makes the job of recording what (the students) do very easy. It is much easier for the instructor, but we miss the essential element in education, in my opinion, and that is to read, interpret, make sense out of things, and have kids really engaged in what they're doing. For me to engage them (in an Accelerated Mathematics format), I have to come in with things like, 'Well you have to learn to read a diagram in market, because that's what happens when you look at a blueprint'. But I never bring in a blueprint in, I just bring in these asinine worksheets, and they do their little manipulations, and it just seems to me to be to be a waste of time.
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Other criticisms of the program have been that the objectives are often not aligned
well with the Core -Plus topics. In particular, because the CPMP Geometry strand
stresses different topics than a traditional Geometry course, the department is having
trouble aligning the Ack Math objectives with the Core-Plus topics. The department plans
to correct this problem during the summer of 2000, but this has caused some problems for
the teachers trying to present a coherent sequence of lessons. Furthermore, the technology
used to connect a printer, scanner, and laptop computer has been problematic at times; a
few teachers have experienced problems printing and accessing computer files easily.
On the other side of the issue, some teachers have found that students seem to be
fond of Ack Math. Despite the problems with the Geometry strand, they suggest that it
supplements the Core-Plus materials well, and that students have enjoyed the change of
pace it represents from the Core-Plus instruction.
I like the Ack Math, because it nicely complements the Core math. My students love it. On the couple of times that we haven’t been able to do it, my students have been disappointed. They like the flexibility of it, they like the individualness of it, they like the immediate feedback of it. I like the immediate feedback of it. It gives me time to work with individual students. I feel like I’ve been working with individual students more
I would say that the Ack math at least made classes not teacher centered. It made implementing Core-Plus easier. Just having kids walking around is mind-blowing for some teachers. Now you'll see lectures that are smaller and shorter. People are trying different things, and have a comfort level with the technology. You'll see more in groups and you'll see more conversations between students. It's a much more humanized classroom.
It is important to note that the last comment was made by the department chair,
who has evidently taken the position that Ack Math was significant in "softening the
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blow" to some teachers. He contends that the program is certainly important in
reinforcing basic skills the students may need on standardized tests, or in college-level
mathematics. Yet, the reshaping of classroom interactions to include group work and
student discourse has also been necessary. In his opinion, Ack Math has been a means to
that end.
The Bipolar Department
Earlier, I alluded to a tension I perceived during my teaching tenure at Adderley.
That is, I suggested that the department was split almost evenly between younger teachers
with 10 years or less of experience and veteran teachers who have been teaching 20 years
or more. In fact, Figure 4 illustrates that, in 1999-2000, there were 11 of these "younger"
teachers, and 10 "veteran" teachers. I also suggested previously that significant
differences existed between the philosophies of the older teachers and younger teachers
in the department.
This issue is still prevalent at Adderley, but attitudes toward reform are not so
easily categorized as to neatly fall along the lines of “young” versus “old.”. For example,
when I asked the department chair why he thought the department had been mostly
receptive to the new curriculum, he noted, "A couple things were going on. You had
retirement, and a lot of new people coming in at the same time, and enough senior faculty
who knew what they were doing wasn't working". In other words, it took a mixture of
youth and experience to set the stage for the new program, and no one faction specifically
championed it or took arms against it.
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However, the teachers themselves have conflicting theories regarding the role of
experience in the department and how this variable is affecting the departmental culture.
To some, the veterans are standing in the way of reform, while others believe that the
veterans are actually at the heart of reform. Furthermore, others believe that the younger
teachers are struggling as much as the veterans, rather than driving the changes. These
contradictions are evident in the following comments:
The Core takes us closer in terms of where we want to go in terms of application, analysis and reasoning. We're looking for things that work, that get kids excited about mathematics, and we're dragging along some of the ones that…er... there's been enough changeover in staff that that's become more of what's expected. But there are always the ones that say they’re doing stuff, but go about their same old ways, and go about their 'dittoes', so to speak.
I think that a lot of our younger teachers are more into the working around with the groups rather than standing up there spewing out information, and I think that's very good. They're very computer-literate. They didn't even have computer science majors when I went to college...I think the younger teachers, that's where it's at. Some of us are going to be leaving in the next five years, and we're going to be left with all young teachers.
The teachers that love it have been around a lot longer, and they've seen that the (traditional) stuff just isn’t working. A change needs to be made. (Traditional mathematics) is maybe too dry and boring, and the Core-Plus has a little more spice and real world applications. The ones that hate it, they're also the ones that have been teaching a long time, and love the old stuff and hate to see the change.
The old fogies are set in their ways. They have a to radically change. They’re going along with it, but they're still stuck in their ruts…and I don’t know if they’re willing to give it a chance. I'm not sure if younger teachers are buying into it, either. They're a little 'iffy'.
From these comments, it seems that the department perceives that both younger
and older teachers are actively supporting the reforms, and taking leadership positions in
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the implementation process. In addition, the department has subsets of both groups who
are particularly uncomfortable with the CPMP/Ack Math program, and are moving more
slowly toward compliance with the new departmental standards.
With respect to attitudes toward the new program, perhaps a more appropriate
classification system would not divide the department between "younger" and "older"
teachers. Instead, one might argue that the department consists of some "stabilizers" and
some "visionaries", categories from some personality type research. Again, stabilizers
conserve the traditional values of the institutions in which they work, and visionaries are
inclined to lead rapid and dynamic change (Clark & Guest, 1995).
However, I do not wish to imply that the stabilizers in the department are
uniformly opposed to reforms, and are somehow sabotaging the change effort. As one
teacher suggested, "We certainly have some teachers that aren’t as open to change.
However, I don’t think that we have any that are completely closed to change. They’re all
willing to at least give something a try." Of course, teachers, like anyone else, do not
learn or accept all new ideas all at once or at the same rate. Because these teachers have
moved much more cautiously toward new methods, they may act as a check against
change in the department proceeding too rapidly.
Leadership
Despite these tensions and contradictions, I argue that the department has
generally accepted the new program, and is in the process of implementing it
successfully. Certainly, the faculty deserves much of the credit for this. As suggested by
the department chair, an influx of younger teachers and a core group of veterans open to
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change set the stage for the process to begin. Since then, these teachers make it very clear
that the entire department has worked diligently to make these reforms successful.
However, I argue that leadership, particularly from within the department, has been the
key element to this receptivity.
Upper Level Leadership
As background information, it may be important to note that Adderley High
Schools saw significant changes in leadership following the 1998-1999 school year, the
result of the respective retirement and departure of the previous Superintendent and
Director of Curriculum. The new Superintendent moved into her position after serving as
Assistant Superintendent for several years, and the new Director of Curriculum was hired
from outside the district. The Principal at West Adderley High School then became the
Assistant Superintendent, and the new Principal was also hired from outside the district.
Despite so many changes, this upper level administration has been crucial to the
implementation of the new mathematics program. From all accounts, the mathematics
department has been given a measure of autonomy and license to take risks. In addition,
the reform efforts have been supported financially. For example, the technology
necessitated by the Accelerated Mathematics Program requires a strong financial
commitment, as does the support of continued teacher training. Furthermore, the
department chair has noted a willingness to support his efforts to use standardized test
data and student enrollment data to assess the new program's effectiveness. This
sentiment was not as prevalent under the previous administration.
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As one teacher suggested, "If it wasn’t for administration support and willingness
to take some risks, this never would have happened". Furthermore, the department chair
said, "The new administration has been super. I mean, it’s our ballgame. We don't have
people looking over our shoulder. They'll call us in and ask: what are you doing, how are
you going to do it, and CAN you do it? That's the big question."
Departmental Leadership
As clearly crucial as the upper level administration has been, leadership from
within the department has arguably been an invaluable part of the new mathematics
program. As proposed by Fullan and Stiegelbauer, six interconnected themes related to
successful reform efforts may illustrate the educational leader's role in this process. These
are: vision building, restructuring, staff development and resource assistance, monitoring
and problem coping, evolutionary planning, and initiative taking and empowerment
(1991). Although Fullan and Stiegelbauer discuss these in the context of systemic school-
wide change, these themes still have relevance at the department level. The remainder of
this chapter discusses the department chair's role in the implementation of the Core-
Plus/Ack Math hybrid in relation to these themes.
It must be noted, however, that my data do not support a comprehensive and
detailed analysis of these themes. The examples and illustrations offered here are
presented only as indicators of the significant effect the department chair has had on the
implementation process.
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Vision Building
According to Fullan and Stiegelbauer, vision building "permeates the organization
with values, purpose, and integrity for both the what and the how of improvement"
(p.81). This involves providing direction and driving power for change, as well as
facilitating a shared vision of the strategy for implementation. The following comments
are indicative of the teachers' attitudes toward Mr. Blakey and his efforts to build vision
within the department and among other administrators:
This guy is working. He keeps himself informed, he talks to other people, and he’s got administration that is supporting him... And he is so sure this is the way we need to go. And he's willing to be the forefront leader, he's taken all the classes, he's a year ahead of us, so he can help us. I think that's the big thing: we've had a little more support. (Blakey) is trying to get us to see the long picture of things: to see if we're building a way of thinking (in students), rather than just doing.
The department chair leadership is huge! This program is working because the leader believes in it enough, and the leader has convinced administration that this is what should be done.
Restructuring
This theme refers to the leader's role in shaping the school as a workplace,
shaping the policies that affect time for team planning, staff development, and joint
teaching arrangements. The department chair may have the most impact on these issues
through scheduling teaching assignments. To illustrate this, the proposed course offerings
of three consecutive school years are shown in Appendix D. The differences from year to
year result from the Core-Plus/Ack Math program being phased in, and the traditional
courses being phased out. By the 2001-2002 school year, every course other then
Advanced Placement classes will be "Integrated", or a Core-Plus/Ack Math offering.
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In addition, as the years progress, there are fewer courses offered, and the overall
structure becomes less stratified. Significantly, the number of “tracks” is reduced by one
by 2001-2002. At this time, Adderley High School students will be divided into “College
Prep” and “Accelerated College Prep” groups, rather than “Skills”, “Regular”, and
“Honors.”
As a result of this new curriculum structure, Mr. Blakey was able to design the
schedule so that "two preps is the norm". That is, teachers are in class five periods during
the day, but now only prepare for two separate courses. Because the department offered
more courses to students under the old curriculum structure, three preps was the norm at
that time. This curriculum streamlining affords teachers more time to plan and reflect on
their practice. Furthermore, more teachers have common preps, which has increased
opportunities for collaboration and shared planning. This will be discussed further below.
Staff Development/Resource Assistance
Certainly, collaboration relates to the staff development aspect of the
implementation process as well. Because the curriculum is now less differentiated
between student tracks, the teachers have greater opportunity to collaborate. For example,
all freshman classes are using the same text; the only differences between classes are the
pace and the depth of coverage. So, rather than being divided among three separate texts
and three separate sets of goals, all teachers of freshman students are on the same “sheet
of music". One teacher noted,
The biggest difference that I see is, several years ago, everybody was an island, where everybody did their things their own way. Certain people talked because they chose to. Now there’s a standard for everybody. Everybody has a standard. We only have 2 classes for freshmen, instead of
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5 or 6. When we talk at inservice, it applies to everybody. Consequently, there’s a lot more sharing. I see standards for every teacher, every class. I feel like it’s a real department now.
Another significant example of the support for staff development is the
department's ongoing Core-Plus curriculum training. Every summer, the department
encourages teachers to attend weeklong training sessions conducted by the CPMP
designers; each session focuses on a specific year of the program. Some of these sessions
are held in Michigan, so teachers stay in hotels and attend training for several days.
During the summer of 1998, the summer after the pilot program, the department
sent 14 teachers to Kalamazoo for this type of extended training. For many, it was their
first extended interaction with the course materials, and this period may have been a
significant factor in the department’s receptivity to this program. One teacher observed,
The one thing that changed everybody’s mind was the trip to Kalamazoo. We had 14 teachers, more than half completely jazzed (after the training). There were those that were more reluctant to change, but they walked out saying, ‘Wow! This might work, but I need more training’. (Mr. Blakey) scheduled it so that, for those people, their prep period was when they could go in and sit and visit a class. And some people did that.
The ongoing training, and the ability to observe each others’ classes are
indications that staff development is taken seriously at Adderley, and is important to the
overall implementation process. While this is a staff development issue, it also seems to
cross over into vision building. Fullan and Stiegelbauer write,
Implementation, whether it is voluntary or imposed, is nothing other than the process of learning something new. One foundation of learning something new is interaction (emphases in original). Learning by doing, concrete role models, meetings with resource consultants and fellow implementers, practice of the behavior, and the fits and starts of
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cumulative, ambivalent, gradual self-confidence all constitute a process of coming to see the meaning of the change more clearly. (1991, p. 85)
Also significant has been the resource assistance, particularly in light of the
technological requirements of the Accelerated Mathematics Program. As noted,
supplying teachers with laptop computers, printers, and scanners requires a great
financial commitment of the part of the district. Furthermore, it necessitates technical
support. Although there have been some minor difficulties with this technology in
isolated cases, it seems to be working well for the department. One teacher expressed his
approval with this support by saying,
Blakey is fantastic. He’s got so many great ideas, and he’s getting them done. For example, he wants to do Accelerated Math. Well, that means that all the teachers who are doing Ack math need their own laptops. Done. Got it. “Oh wait! Now we need special printers”. Got it.
Monitoring/Problem Coping
Monitoring and problem coping are particularly complex issues as defined by
Fullan and Stiegelbauer (1991). Monitoring provides access to good ideas and innovative
instructional practices, and it subjects ideas to scrutiny, which helps to weed out mistakes
and develop more promising practices. Certainly, these teachers have no qualms about
subjecting Mr. Blakey's ideas to scrutiny. They are consistently asking questions, and
presenting alternative ideas to him, as I was told on several occasions.
Encouraging staff to visit each others’ classrooms is evidence of healthy
monitoring, as is the overall collegiality in the department, although there is some
disagreement as to how the new curriculum has impacted this.
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Some teachers characterized the East Adderley teachers as historically more
willing to collaborate, and the West Adderley staff as more insular. However, this may be
changing.
Some comments noted from East Adderley teachers were:
I haven’t gotten to meet with other teachers lately. I kind of feel like I’m out of the loop.
I don’t collaborate with other teachers. There are lots of little cliques.” There's tremendous willingness to work together here… I see a lot
more independence now, particularly with the younger teachers. I don't know if they sit down with the same amount of collegiality that we used to.
Some comments from West Adderley teachers were:
As faculty, we are starting to talk a little more. We’re starting to help each other a little more.
(Now) there’s a lot more sharing going on here. There is some collaboration here, although not as much as I'd like to
see. There’s still camaraderie, but not as much as at East.
Reasons for this are not clear. Again, the district has seen a number of changes in
personnel, and the chemistry of the staff at each campus may be a significant contributor
to this.
Another important aspect of monitoring is collecting data in order to assess the
program. As noted, the new upper level administration, and the Director of Curriculum in
particular, is working with the department chair to gather data to assess the new
program’s effectiveness. Mr. Blakey is preparing a “Math Report Card” to measure the
strengths and weakness of the new course offerings. This report will compare statistics
for five consecutive school years in each of the following areas:
Number of students who have failed mathematics courses.
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Percent of student population enrolled in mathematics course.
Advanced Placement (AP) test results.
10th grade students’ Illinois Standards Achievement Test (ISAT) scores.
It should be noted that the ISAT is a statewide standardized test of reading,
writing and mathematics administered to students in 3rd, 5th, 8th and 10th grades.
Results are reported according to four performance levels: Academic Warning, Below
Standards, Meets Standards, Exceeds Standards (Illinois State Board of Education, 2000).
Furthermore, the district plans to compare performance of 8th grade students
according to middle/junior high school attended. Because students from five different
feeder schools attend the East campus, and students from three feeder schools attend the
West campus, there has long been concern about the variability in students’ mathematical
backgrounds. That is, students from particular feeder schools seem to have more
difficulty in high school mathematics classes than others, and the district will attempt to
gather data to support this hypothesis.
Evolutionary Planning
Evolutionary planning relates to adapting the plans to improve the fit between
school conditions and the reform initiatives. The department has demonstrated an ability
to adapt to challenges, as evidenced by the reaction to problems with the second year
curriculum. Specifically, the faculty perceived "major problems" with the Geometry
strand of Core-Plus. A number of teachers of CPMP Year 2 held a planning meeting with
Mr. Blakey to discuss these issues, and he said that the direction of the second year
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curriculum "turned on a dime" as a result. At this meeting, the decision was made to
divert from the Core-Plus sequence in favor of teaching more traditional Geometry topics
for one quarter, and these changes were implemented the day after the meeting. Although
these changes are considered temporary, the resolution of this situation, as well as the
adoption of Ack Math, shows a willingness to adapt the program to the specific needs of
the district.
One issue to consider related to evolutionary planning will be discussed further in
the next section, which discusses Initiative Taking and Empowerment. Fullan and
Stiegelbauer suggest that a blend of top-down and bottom-up participation is common in
successful reforms, and this seems to be one source of tension related to the leadership of
the department.
Initiative Taking and Empowerment
Initiative taking and empowerment involves "power sharing”, delegation of
authority, and developing collaborative work cultures " (Fullan & Stiegelbauer, 1991, p.
83). Although most of the implementation process seems to be in alignment with Fullan’s
suggestions, some tensions should be noted here. Specifically, some in the department
feel that Mr. Blakey's leadership style is primarily top-down, and that there is a need to
encourage more ownership by the teachers in the program, as illustrated by the following
comments:
I know I was not part of the decision making process, so…I don't know where (the curriculum) came from. I can guess, but I’m not sure….
I am seeing a lot of top-down decisions. I understand where (Mr. Blakey) is coming from, trying to find a solution to the detracking
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situation, and you know…I feel like these are all (Blakey’s) decisions. I see him as the “benevolent dictator”. I don't think he's trying to make our lives miserable…. I think his decisions, though, are creating chaos. And I don’t think there’s an ownership going on, like we've said 'We want this'.
However, another teacher flatly challenged this attitude, saying, "(Blakey) is not
jamming instructional styles down people’s throats, although there are a number of
faculty that feel that way." One specific issue seemed to exemplify this tension, and was
raised by a number of teachers. Mr. Blakey, in collaboration with several other teachers,
writes a weekly syllabus for each Core-Plus class, and distributes it to the staff (see
Appendix E). According to Mr. Blakey, the syllabus is meant less as a mandate than as a
guideline, providing the teachers with some sense of the pace of the course and giving
them ideas about particular problems to try with students. "I keep telling (the teachers)
that the syllabus isn’t a mandate", he said, "They should follow the sequence, if not the
timeline. I think we've got reasonable compliance, except being on time, which isn’t the
important thing."
However, there were contradictory perceptions of this issue:
Hmm...the syllabus: is it a general guideline, or is it a mandate? I wish I had more input as to what I do. I mean, I feel what I’m doing is working. If there were more weight placed on what I say, I’d say more.
The syllabus is meant as a guideline. For a lot of the second quarter, I was behind, and (Blakey) never said anything. He also suggests homework problems, but I never look at those. Other teachers and I are doing things differently in our classes, and he doesn't mind. So, I'm surprised that people would think that this is being jammed down people’s throats.
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Final Thoughts
Despite these issues related to management style, the department is dealing with
the challenges positively. They are working constructively with the course materials, and
they are collaborating to a degree with the leadership and each other. I argue that other
departments under similar pressures might react in a far more negative manner, putting
such reform efforts in jeopardy. As Fullan and Stiegelbauer (1991) write,
Innovation can be a two-edged sword. It can either aggravate teachers' problems or provide a glimmer of hope. It can worsen the conditions of teaching however unintentionally, or it can provide the support, stimulation and pressure to improve. (p. 126)
Perhaps at this relatively early stage of the implementation process, one
might argue that the department has felt both edges of this “two-edged” sword. At
times, the teachers have certainly been aggravated with the magnitude and the
pace of the changes, and this uncertainty has caused stressful conditions. Yet, the
department generally seems to see progress, and believes in the direction they are
moving. This is consistent with observations from Waugh and Punch (1987), who
note that meaningful change is a long process influenced by many factors, not an
event.
Evidence of this process is seen in the reflections of one teacher who
noted, "Many people at the very beginning thought that this was just (the
department chair's) ideas, and they weren’t necessarily the best ideas. But as
people are jumping on board, and really getting into this stuff, they’re finding out
that he was right. He was right about a lot of this."
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CHAPTER SIX
SUMMARY AND CONCLUSIONS
An incredibly complex process has begun in the Adderley High Schools'
mathematics departments. The new materials introduced in 1997 are slowly being phased
in at East and West Adderley, and traditions, values, practices and beliefs are
consequently being rethought and reshaped. The culture of the collective department is
changing as the staff, students and administration all adapt to a new paradigm. For
teachers in particular, with new paradigms come new occupational (and perhaps
personal) identities, and educators need to be given opportunities to construct those
identities (Norum & Lowry, 1995, p. 5). With this in mind, I examined some cultural
characteristics of the Adderley mathematics department that may have supported these
changes, which are perhaps the beginning of the construction of these new identities.
Summary
In this paper, I initially discussed some aspects of paradigm shifts among
mathematics teachers and why these changes are important if reform is to occur.
Specifically, I suggested that teachers and departments need to reexamine traditional
practices if mathematics is truly to become "for all". Such a process has begun at
Adderley High School, and I began to describe the situation. Following a brief
explanation of reform mathematics versus traditional mathematics, I discussed details of
the school environment at Adderley, including student characteristics, school climate, and
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the surrounding community. In addition, I began to describe my own history with the
department and how it influenced this study.
The second chapter examined some of the literature relevant to the characteristics
of mathematics teachers, both in general and as these characteristics relate to
mathematics reform. Specifically, this discussion noted mathematics teachers' receptivity
toward reform-oriented curricula and instructional methods, such as new definitions of
the role of the teacher and instructional activities. In addition, I noted research related to
the characteristics, culture and receptivity to change of mathematics departments as a
whole.
This research indicated that there is a wide variation in the psychological states of
teachers, and some of these characteristics may lead some teachers to be uniquely
resistant to change. Consistent with this observation, research shows that mathematics
teachers are not universally receptive or open to the types of reforms recommended by
the NCTM Standards. Three themes were identified as important to a teacher's receptivity
to reform-oriented practices and philosophies: beliefs about mathematics as a field of
study, beliefs about the teacher's role, and the ability to reflect.
For example, if teachers hold a largely absolutist view of mathematics, they may
have a more difficult time implementing or making these recommendations personally
meaningful. Furthermore, teachers that were inclined to promote an open atmosphere of
discussion and exploration in the classroom indicated greater receptivity toward
mathematics reform, and had an easier time implementing reform-oriented methods. In
contrast, holding a control-oriented concept of the teacher's role proved more problematic
in implementing these methods. In addition, the ability and opportunity to reflect on
87
teaching practices was consistently noted as an indicator of receptivity to mathematics
reform.
Departmental receptivity to reform was associated with, and affected by, several
elements. Leadership within the department that promoted practices consistent with the
NCTM Standards was significant, as was a department's overall involvement in
professional activities. This related to outside activities such as conference attendance, as
well as day-to-day interactions with colleagues. Furthermore, departments that held
constructively high expectations for all students were more successful in encouraging the
achievement of a greater number of students in higher levels of mathematics classes.
Finally, mathematics departments that were collectively committed to change were
predictably more successful in implementing change.
The third chapter describes the rationale behind the methodology of this study. It
elaborated on the unique aspects of the Adderley High Schools' mathematics curriculum,
and specific attributes of the Core-Plus and the Accelerated Mathematics programs. I
further elaborated on my history with the department, and my personal reflections on the
departmental culture. The development of this study arose from my interest in these
issues and how they would affect the implementation process at Adderley.
I explored reasons that this department may or may not have been receptive to this
curriculum change, in relation to criteria teachers use in assessing changes as discussed
by Fullan and Stiegelbauer (1991). Despite the fact that this new curriculum would
necessarily affect these teachers in terms of time and energy, and would challenge
traditionally held beliefs, there was a perception in the department that a change was
needed. This chapter then outlines the details of the two stages of data collection.
88
Chapter Four describes the findings of the first stage of data collection, which
involved a survey completed by all Core-Plus teachers at Adderley, and interviews with
nine of them. The surveys asked the teachers to gauge the extent of their agreement with
several statements related to reform-oriented mathematics teaching, and the Core-Plus
curriculum in particular. Furthermore, the teachers were asked to estimate the extent to
which they intended to use certain instructional strategies. The interviews then explored
teachers' metaphors for mathematics teaching, learning, and mathematics as a field of
study. In addition, teachers elaborated on their conceptions of authority and control in the
classroom.
The results of these data changed the direction of the overall study. Initially, the
focus of this work was to be an examination of teachers' conceptions of control and
authority, and the extent to which these affected the implementation of a student-centered
curriculum. However, I was surprised at the level of consistency between the teachers'
responses and recommendations made by national mathematics reform documents. That
is, the department seemed to be quite receptive to recommendations related to classroom
management; most of those interviewed suggested that they control classes by "letting
go". This suggested that a broader examination of the situation at Adderley was
warranted in order to better gauge the level of this receptivity and the elements that have
affected it.
The second stage of data collection consisted only of interviews which focused on
teachers' general perceptions of the new curriculum and their impressions of overall
departmental changes over time. Specifically, they were asked to reflect on leadership,
89
instructional practices, and professional development. As a result of these data, a fuller
picture of the department emerged.
In general, the department is encouraged by the results of the new curriculum,
although some tensions must still be addressed. These are: the perception that the Core-
Plus curriculum does not teach basic mathematical skills; the question of whether the
reform-oriented curriculum is appropriate for honors level students; disagreement over
the compatibility of Core-Plus and the Accelerated Mathematics Program; and tensions
between younger and veteran staff.
However, aspects of the process ultimately point toward successful
implementation. In particular, Fullan and Stiegelbauer (1991) discuss elements crucial to
successful reform efforts, such as vision building, restructuring, staff development and
resource assistance, monitoring and problem coping, evolutionary planning, and initiative
taking and empowerment. Most of these have been dealt with in ways that Fullan and
Stiegelbauer recommend, although the department chair's leadership style has been
challenged as authoritarian. This top-down style is not a universal concern among the
staff, but it should be noted as an area of concern, and may conflict with encouraging an
empowered faculty.
Conclusions
Despite these disputes, I argue that leadership has been the key element to the
implementation of the new curriculum. Not only has the upper level and departmental
leadership been crucial, but leadership by individual teachers has also taken a strong role.
There are a number of passionate advocates of the new program who are committed and
90
excited about this new direction. As one teacher noted, "We won't go back…There are
enough teachers here that nurture, that don't want (the program) to fail." Looking to the
future, these teachers see the importance of encouraging leadership roles among their
peers. One teacher suggested,
Ultimately, individual teachers will become leaders, and they'll move it forward. I don't think we'll go backwards in it (to a traditional curriculum); we'll keep modifying it, and try to bring it closer to…making everybody happy. Hopefully, eventually, the goal of the math dept will be the same as the Prairie State Test, which will be the same as the goal of the ACT test, and we'll all be on the same page and everything will be in tune. Then, we won't have to worry about those leadership issues.
Although the empowerment issue is a concern, this comment indicates that there
are those who are looking toward taking initiative and see it not only possible, but
perhaps necessary for the continued success of the program.
Looking at the entire process, it is not surprising to see that there has been pain
and frustration involved. In fact, conflict and disagreement may be inevitable and
fundamental to change processes (Fullan & Stiegelbauer, 1991). A more colorful
metaphor for this comes from Norum and Lowry (1995), who compare change to a death,
the death of traditions and familiar assumptions. Continuing this metaphor, they suggest
that educators experiencing change go through Kubler-Ross's (1969) stages of grief:
denial, anger, bargaining, depression, and finally, acceptance. Therefore, as with any
family coming to terms with such a loss, one can expect difficulties and painful issues. At
the time of this study, teachers that could be described as experiencing grief for the old
curriculum (many were not) were at different stages of this process. However, the
department as a whole appears to be moving toward acceptance.
91
Furthermore, looking past the uncertainty and tensions that are common so early
in such an implementation process, it is natural to try to forecast the larger implications of
this effort. As I have noted, I see this department on the path toward successful
implementation of the Core-Plus curriculum. Whether or not it remains fused together
with the Accelerated Mathematics Program will be seen in time. However, I believe that
this department has "passed the point of no return", and as the teacher above noted, it
would be difficult for them to return to a traditional curriculum anytime soon.
Beyond the details of implementing Core-Plus or Ack Math, the Adderley
mathematics department may be involved in a more significant process. Figure 5 is a
model adapted from Fullan and Stigelbauer (1991) and Gutierrez (1996) which illustrates
the potential effects of educational leadership on reform issues, departmental culture, and
student outcomes. In Chapter Five, I argued that the departmental leadership at Adderley
High Schools has mostly addressed the six significant aspects of educational reform
presented by Fullan and Stiegelbauer (1991), and this has impacted the departmental
culture as a whole. Furthermore, in Chapter Three I indicated that I see this department as
potentially reshaping its culture as Organized For Advancement (OFA), as outlined by
Gutierrez (1996). Thus, leadership affects culture, which may ultimately lead to student
outcomes similar to those advocated by mathematics reform literature, such as the NCTM
Standards documents.
The OFA Framework
Again, OFA is a framework developed to describe the practices, beliefs and general
culture of mathematics departments effective in encouraging their students to continue
onto high levels of mathematics (Gutierrez, 1996).
92
OFA departments specifically show a high level of overall student participation in
mathematics and a high level of student participation in upper level mathematics courses.
In contrast, non-OFA departments maintained placement policies that ensured only a
select number of students were encouraged to take additional and upper level
mathematics courses. Gutierrez's study has particular relevance here, because the schools
included in her study were very similar to Adderley. That is, each school was mainly
urban, enrolled approximately 1000-2000 total students, served a significant proportion
of low income and minority students, and were identified as schools serving students that
traditionally underperform in mathematics (Gutierrez, 1996).
The framework outlines four components common to OFA departments: a
rigorous and common curriculum, innovative instructional practices, a commitment to a
collective enterprise, and a commitment to all students. These components are explained
more fully below as they relate to Adderley High Schools.
A Rigorous and Common Curriculum
The Adderley program seems particularly consistent with this aspect of the
Organized For Advancement framework. OFA departments generally offer fewer low-
level mathematics courses, and offer a limited choice in courses overall. This
"compressed curricula" (p. 508) leads to students that take more mathematics, and also
higher levels of mathematics. Similarly, the Adderley mathematics department has
reduced the number of courses it offers, and has compressed its "tracking" lanes from
three to two. As noted, all freshman-level courses use the same text. A low-level "Pre-
Algebra" course was eliminated in 1998-1999.
94
Unfortunately, one significant area of difference between Adderley High Schools
and schools that are Organized For Advancement relates to high school graduation
requirements. All of the schools identified as OFA by Gutierrez had a three-year
mathematics requirement; Adderley currently has only a two-year requirement. Gutierrez
writes, "Porter (1994) found that when schools serving traditionally low-achieving
students increased their standards for graduation, their students took more mathematics
and at varying levels during high school" (1996, p.509). This may not be a particularly
surprising finding, but it is important to note. In addition, the fact that Core-Plus has been
designed as a three-year curriculum for all students weighs heavily on this graduation
requirement.
However, this has been discussed within the department and with upper level
administration. Because the State of Illinois plans to implement the Prairie State Test, a
standardized assessment that tests objectives through the junior year of high school, the
department plans to improve the program so that students will voluntarily take an
additional year of mathematics. This issue has not been resolved, and a three-year
mathematics requirement may be adopted at a later date.
Innovative Instructional Practices
In addition, the new mathematics curriculum discussed in this paper seems to lend
itself to this type of innovation. Although the OFA departments studied by Gutierrez still
used lecture format as the primary mode of instruction, new instructional approaches
tended to be used to supplement instruction more often than in non-OFA sites.
Specifically, these approaches are the use of cooperative learning, the use of technology,
95
the use of a variety materials, and content that related to the students' lives. These seem to
be remarkably consistent with the type of instruction encouraged by the Core-Plus/Ack
Math hybrid, as has been discussed at some length in this paper.
A Commitment to a Collective Enterprise
Here, the connection between OFA departments and Adderley is less clear.
Teachers in OFA departments tended to teach courses featuring a wide range of student
abilities and ages, rather than teaching exclusively one segment of the population (such as
the "honors" level). This had the effect of exposing teachers to the entire curriculum and
student body, and rotating course assignments in this way "helps keep you on your toes",
in the words of one teacher (Gutierrez, 1996, p. 514).
Although seniority in the department does not seem to dictate course assignments
at Adderley, there are several teachers in the department that teach only honors courses,
and have done so for years. It is unclear whether this trend will change with the full
implementation of the new curriculum. However, this practice is limited to a select few
teachers, and most others seem to teach a wide variety of students. In addition, a number
of retirements in the next several years may shake loose some of these "honors-only"
positions.
In addition, OFA departments were characterized by "cooperative teacher
autonomy" (p. 515). In other words, the teachers in these departments did not generally
interact with each other on a daily basis about instructional practices, but instead relied
more heavily on a shared conception of the department's goals to guide their work.
Because the curriculum at Adderley is so new, the teachers are still developing their
96
perceptions of the department's goals and mission. As a result, they seem to be relying on
each other for daily support much more than Gutierrez found in OFA departments.
Of course, such reliance is a natural and healthy outcome for a department
implementing a new curriculum. In fact, collegiality, communication and mutual support
among teachers have been identified as strong indicators of implementation success
(Fullan & Stiegelbauer, 1991). In time, the Adderley mathematics department may indeed
develop this level of cooperative teacher autonomy, but for now, they need each other for
support.
Furthermore, OFA departments were noted to be active in professional
development. The extended curriculum training provided by the CPMP publishers
continues to be supported and encouraged by the Adderley administration, as has been
noted. More frequent formalized professional development sessions take place at bi-
weekly "inservice" meetings, which are held in all departments. One teacher had a unique
perspective on how these meetings have changed since the implementation of the new
program, particularly as a result of the existence of a common curriculum for all teachers.
He notes, "Now, everybody has a standard….when we talk at inservice, it applies to
everybody. Consequently, there’s a lot more sharing. Now, at department meetings, I see
growth. It’s constant growth."
Finally, OFA departments practiced collective decision-making. This issue has
already been described as controversial within the department, and several conflicting
reports of the level of collective decision-making allowed by the departmental leadership
have surfaced. This is an issue to be explored further.
97
A Commitment to all Students
This component is in need of the most further study as it relates to the Adderley
mathematics department. Gutierrez suggests several dimensions of a department's
commitment to all students (p. 510):
1. Teachers holding constructive conceptions of students2. Teachers holding flexible conceptions of the learning process3. Teachers sharing the responsibility for learning with students4. Teachers holding high expectations for students5. Teachers actively reaching out to parents6. Teachers being accessible to students (p. 510)
In light of my earlier perception of elitism, this seems the most problematic
component of the OFA framework for this department. However, the first stage of data
collection indicated that these teachers may hold flexible conceptions of the learning
process; a control-orientation seems to have given way to student-centered teaching. In
addition, all Adderley teachers are required to be available to their students every day in
"period 10", a 35-minute period after school in which students can approach teachers for
help in their classrooms on a walk-in basis. Also encouraging is that as of the 2001-2002
school year, the two "tracking lanes" at Adderley are labeled "College Prep" and
"Accelerated College Prep." This indicates that the leadership has made a commitment to
encouraging the attitude that all students are potential college students. Still, the findings
of this study did not adequately assess the level of commitment this staff has to all
students, as defined by Gutierrez. Again, this is an area in need of further study.
Despite these uncertainties, I still contend that the Adderley mathematics
department is a potential OFA department. Because OFA sites were identified as such
based on certain student outcomes, and the Adderley has not had time to measure these
98
outcomes, time will tell whether my hypothesis is correct. However, taken as a whole,
this department is showing signs of reshaping itself as I have argued. Again, it is early in
the implementation process, but key elements of a department that is Organized For
Advancement are in place.
Future Areas of Study
Certainly, the logical next step for future study would be to conduct classroom
observations, as all of the conclusions here are based on the highly subjective nature of
teacher interviews. Specifically, we must see if the department truly features "more
humanized classrooms", as suggested by Mr. Blakey. The perception in the department
seems to be that instructional practices are changing as teachers become more
comfortable with the curriculum. However, this cannot be taken for granted. As Fullan
and Stiegelbauer suggest, understanding reform policies and actually implementing them
are entirely different matters (1991).
In addition, it would be extremely useful to conduct interviews with students.
Because one of the claims made by the CPMP developers is that an outcome of the
curriculum is an improvement in student attitudes and perceptions of mathematics, this
should be explored at Adderley. Although the developers' claim has been substantiated by
some research (Schoen & Pritchett, 1998), it is not yet clear whether this hypothesis holds
true for this situation.
Additionally noted is the necessity of collecting data related to student outcomes.
That is, even if the teachers are implementing the curriculum as designed, and have truly
accepted a reform-oriented paradigm for teaching and learning, do the students actually
99
benefit? Standardized achievement test results, overall student enrollment in
mathematics, student enrollment in upper level mathematics courses, and student
attitudinal data should all be considered as a part of this process. The department has
indicated that this is an important aspect of the overall implementation plan, but these
data must be collected and analyzed thoroughly and effectively.
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APPENDIX A
TEACHER SURVEY
Date _________
Part I: Background information
a. Name ____________________________________________
b. Sex F M
c. Age
d. Total years of teaching experience, not including student teaching:
e. Years of experience teaching Core-Plus materials:
f. Approximately how many hours have you spent in professional development activities geared toward the Core-Plus curriculum?
g. What year(s) of Core –Plus will you be teaching this year?
h. What is your total number of teaching periods?
i. What is your total number of periods teaching Core-Plus classes?
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j. From which institutions do you do you hold post-secondary degrees?
Degree held Institution
Part II.
Express the extent of agreement between the feeling expressed in each statement and your personal feelings. Circle the choice that best reflects your feelings.
a. "Mathematics will change rapidly in the near future."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
b. "Mathematics is a good field for creative people."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
c. "Mathematics education is a good field for creative people."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
d. "I am familiar with the National Council of Teachers of Mathematics
Curriculum and Evaluation Standards."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
e. "I am familiar with the Illinois State Learning Standards."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
f. "The mathematics content of the Core-Plus curriculum is rigorous and challenging."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
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g. " As compared to other classes, my methods of presenting mathematical content will stay the same in implementing the Core-Plus Curriculum."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
h. " As compared to other classes, my classroom management strategies will stay the same in implementing the Core-Plus curriculum."
Strongly Disagree
Disagree Undecided Agree Strongly Agree
Part III. Over the duration of the Core-Plus course(s) that you teach, please indicate how often you intend to employ the following classroom teaching strategies. Circle the choice that best reflects your feelings: a) Lecture: Rarely, if ever Monthly Weekly Several times
per weekEvery day
b) Guided discovery activities:
Rarely, if ever Monthly Weekly Several times per week
Every day
c) Group work:Rarely, if ever Monthly Weekly Several times
per weekEvery day
d) Computer work:Rarely, if ever Monthly Weekly Several times
per weekEvery day
e) Individual work (seat work):
Rarely, if ever Monthly Weekly Several times per week
Every day
h) Graphing calculator work:
Rarely, if ever Monthly Weekly Several times per week
Every day
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i) Class discussion:Rarely, if ever Monthly Weekly Several times
per weekEvery day
Part IV. Over the duration of the Core-Plus course(s) that you teach, please indicate how frequently you intend to emphasize each of the following in your instruction. Circle the choice that best reflects your feelings: a) Mathematical reasoning --passive (YOU show your students justification for mathematical statements; YOU demonstrate deductive proofs of theorems)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
b) Mathematical reasoning --active (your STUDENTS construct justifications for mathematical statements; STUDENTS construct demonstrative proofs of theorems)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
c) Mathematical procedures (your students develop skill in performing routine mathematical procedures, algorithms)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
d) Mathematical facts (your students learn basic and statements of basic theorems)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
e) Applications of mathematics -- passive (YOU show your students how mathematics can be used to solve a variety of "real world" problems, e.g., in science)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
f) Applications of mathematics -- active (your STUDENTS determine how mathematics can be used to solve a variety of "real world" problems, e.g., in science)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
g) Mathematical communication (you ask your students to speak and/or write clearly about mathematical ideas)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
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h) Modeling --passive (YOU show your students how to create and use appropriate mathematical representations of "real world" situations, e.g., in science)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
i) Modeling --active (your STUDENTS create and use their own appropriate mathematical representations of "real world" situations, e.g., in science)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
j) Multiple representations (your students see and work with the same mathematical concept in a variety forms; graphical, numerical [tabular], symbolic, and verbal)Not emphasized at all
Occasionally, but not frequently
Emphasized frequently
Almost always
Part V. Please indicate with a check your level of expertise in using the following resources/strategies in your teaching. Circle the choice that best reflects your feelings:
a) Classroom Lectures/Presentations:
Novice Intermediate Expert
b) Group Work:Novice Intermediate Expert
c) Extended Projects:Novice Intermediate Expert
d) Writing Activities:Novice Intermediate Expert
e) Computers:Novice Intermediate Expert
f) Graphing Calculators:Novice Intermediate Expert
g) Portfolios and other types of non-routine assessments:
Novice Intermediate Expert
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Interview #1
Name____________________________
Date __________________________
1. Select the three of the choices below that best characterize your beliefs about
mathematics teaching. Please explain to me why you made these choices.
“Teaching mathematics is most like being a…”
News broadcaster Entertainer Doctor Orchestra conductor Gardener Coach Missionary Social worker Political leader Other: ____________________________
2. Select the three of the choices below that best characterize your beliefs about mathematics learning. Please explain to me why you made these choices.
“Learning mathematics is most like…”
Working on an assembly line Watching a movie Cooking with a recipe Picking fruit from a tree Working a jigsaw puzzle Conducting an experiment Building a house Creating a clay sculpture A negotiation Other:_______________________________
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3. Select the three of the choices below that best characterize your beliefs about mathematics as a field of study. Please explain to me why you made these choices.
“Mathematics is most like….”
A set of laws A blueprint A piece of art A puzzle A cookbook A language A philosophy A frontier Other: ____________________________
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APPENDIX D
COURSE SELECTIONS AND CURRICULUM STRUCTURE
Course selections, 1999-2000
Grouping Freshman Sophomore Junior Senior
S-Group Integrated Algebra I
Integrated Geometry
Integrated Geometry
Integrated Algebra II
Regular Geometry Algebra II College AlgebraHonors Integrated
Geometry Accelerated
Geom/Chem Honors Plus or Geom. Honors
Trig/PreCalcor Algebra II Honors
AP Calculus or Trig/Precalc
Course selections, 2000-2001
Grouping Freshman Sophomore Junior Senior
S-Group Integrated Algebra I
Integrated Geometry Integrated Algebra II
Integrated Algebra II
Regular Algebra II Integrated College Algebra
Honors Integrated Geometry Accelerated
Integrated Algebra II Honors Plus or Integrated Algebra II Accelerated
Trig/PreCalcor Algebra II Honors
AP Calculus or Trig/Precalc
Course selections, 2001-2002
Grouping Freshman Sophomore Junior Senior
College Prep Integrated Algebra I
Integrated Geometry Integrated Algebra II
Integrated College with an option to take AP Statistics
Accelerated College Prep
Integrated Geometry Accelerated
Integrated Algebra II Honors Plus or Integrated Algebra II Accelerated
Integrated Trig/PreCalc Acceleratedor Integrated College Algebra Accelerated
AP Calculus and/or AP Statistics
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