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2008 Faculty Perceptions of A Calculus Reform Experiment at A Research University: A Historical Qualitative Analysis Douglas M. (Douglas Macarthur) Windham
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FLORIDA STATE UNIVERSITY
COLLEGE OF EDUCATION
FACULTY PERCEPTIONS OF A CALCULUS REFORM EXPERIMENT AT A
RESEARCH UNIVERSITY:
A HISTORICAL QUALITATIVE ANALYSIS
By
DOUGLAS MACARTHUR WINDHAM JR.
A Dissertation submitted to the Department of Middle and Secondary Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Degree Awarded: Summer Term 2008
The members of the Committee approve the dissertation of Douglas Windham defended on June 27th, 2008.
______Leslie Aspinwall Professor Directing Dissertation
______Marcy Driscoll Outside Committee Member
______Elizabeth Jakubowski Committee Member
______Ken Shaw Committee Member Approved:
______Pamela Carroll, Chair, Department of Middle and Secondary Education
The Office of Graduate Studies has verified and approved the above named committee members.
ii
To my long-suffering but supportive wife, parents, friends, and committee members; thank you forever for your undying patience and vigilance.
iii TABLE OF CONTENTS
List of Tables v List of Figures vi Abstract vii
1. INTRODUCTION 1
2. LITERATURE REVIEW 5
3. METHOD 25
4. CASE STUDY A: PROFESSOR VIOLET’S INTERVIEW 42
5. CASE STUDY B: PROFESSOR GREEN’S INTERVIEW 59
6. CASE STUDY C: PROFESSOR RED’S INTERVIEW 70
CONCLUSION 84
APPENDIX A : SURVEY QUESTIONS 114
APPENDIX B: HUMAN SUBJECTS IRB APPROVAL FORM 115
APPENDIX C: HUMAN SUBJECTS IRB INFORMED CONSENT FORM 116
REFERENCES 117
BIOGRAPHICAL SKETCH 125
iv
LIST OF TABLES
Table 1. Conceptually Clustered Matrix—Interview With Professor Violet 58
Table 2. Conceptually Clustered Matrix—Interview With Professor Green 69
Table 3. Conceptually Clustered Matrix—Interview With Professor Red 83
Table 4. Participant’s Acceptance of the Rule of Four—A Rubric 98
Table 5. Correlation of Data Across Case Studies 99
v
LIST OF FIGURES
Figure 1. Professor Red’s Pacing Schedule During the Reform Experiment 81
Figure 2. A Contextual Visual Metaphor for the Calculus Reform Experiment 108
vi
ABSTRACT
From 1999 through 2004, the mathematics department at Research University experimented with using a reform text, Hughes-Hallett et al.’s Calculus, to teach the undergraduate calculus sequence. A historical qualitative analysis was undertaken involving three linked case studies to determine, from the perspective of the professors in the classroom, the success of the experiment in reform. Three professors, one a self-identified reform advocate, one an arch-traditionalist who vehemently opposed reform, and one who professed himself to be in between, gave insight into the results of the switch and the departmental atmosphere that led to the return in 2004 to a more traditional calculus instruction. The results of these case studies include a picture of a department in transition, trying to better serve its students but having difficulty adjusting to the changes in instruction coincident with reform. Each of the participants admitted using the textbook as little more than a delivery vehicle for homework assignments; none of the three participants changed their lecture style or teaching methods to respond to the demands of the reform movement. Calculus reform’s founders and those who have inherited the movement and brought it into the 21st century advocate technological exploration, real world applications, group projects, and conceptual understanding. Each one of the participants admitted to applying some of these in their teaching style, but each in turn rejected other tenets of the reform movement as unusable, or unwieldy. As the department did not change any other aspect of calculus instruction at the university other than the text used, this experiment could have been dismissed as naïve, insincere, or half-hearted. But in fact, the department may have benefited indirectly from the move by even the more traditional text they embraced post-reform, as all participants acknowledged that even traditional texts now contain elements of reform themselves. However, the case studies analyzed in this research would indicate that any reform effort conducted in a research university should expect to meet some resistance of the type exposed at this university. Anyone attempting to reform the teaching of calculus at their college can benefit from reading the perceptions of these professors and addressing them, either with seminars and research that can convince faculty that a change is needed, or at the very least by adjusting curricular structure and pacing so the reforms have a chance to succeed. Also, educational
vii researchers could benefit greatly from a nationwide qualitative/quantitative research focus on the acceptance of calculus reform at mainstream colleges and universities that do not have a vested interest in proving the reforms a success to maintain funding levels. Finally, those educational researchers interested in the perceptions of college math professors at research institutions could further analyze how those professors’ perceptions could impede or enhance efforts at reform, and how those perceptions differ from those predominant at teaching-focused institutions.
viii CHAPTER 1
INTRODUCTION
Calculus has long been the cornerstone of mathematics teaching and learning at the undergraduate college level (Douglas, 1986). For thousands of mathematics, science, business, and economics majors every year, calculus is the gateway to higher mathematics, and a prerequisite for many higher-level courses (Goodfriend, 2006). In the mid 1980s, various groups (including the National Science Foundation (NSF), the Mathematical Association of America (MAA), the American Mathematical Society (AMS), and the National Council on Teachers of Mathematics (NCTM)) independently addressed the issue of calculus teaching and learning (Gantner, 2001; MAA, 1981; Steen, 1987). Although the findings were varied, a common theme of these observations was that calculus instruction in a traditional manner was not seen as effective; more than more than 50% of students who enrolled in calculus courses nationwide did not complete those courses (Douglas, 1986). This period also saw a decline in mathematics majors from the 1970s to the 1980s by 60% and a call from industry to change the way that students are prepared for the post-graduate workplace (Leitzel & Tucker, 1995). At a conference at Tulane University in 1986, efforts began to coalesce into a movement to reform the way that calculus (and concurrently, other math courses) was taught (Douglas, 1986). Whether as a direct result of the Tulane conference or because of simultaneous improvements in classroom technology such as graphing calculators, many different groups soon started research into what would come to be known as calculus reform (Heid, 1997). From the first conferences and research identifying changing views of how students learn (and with the benefit of grants from the National Science Foundation and the Sloan Foundation) sprang such projects as the Calculus & Mathematica Project at the University of Illinois-Urbana/Champaign and the Harvard Calculus Consortium. As a result of this push, by 1994, 68% of mathematics departments responding to a survey indicated some level of movement towards calculus reform (Judson, 1997). Simultaneously, 150,000 students (roughly one third of all calculus students nationwide) by 1994 were enrolled in reform calculus courses (Wilson, 1997). The major focus of the Harvard Consortium group over the course of their $35 Million NSF grants (1987-1995) was to develop a new calculus text that would help instructors and
students to approach calculus in a completely unique way (Gleason & Hughes-Hallett, 1992). Traditional calculus instruction (and textbooks) was seen most commonly in the reform movement as drill- and proof-intensive, forcing students who are unprepared for the level of rigor encountered to simply memorize and regurgitate (Hurley, Koehn, & Gantner, 1999; Tucker, 1999). The reform calculus text was intended to shift the focus to student discovery, a more permanent understanding of the theory behind the calculus, and more dependence on the solution of “real-world” problems (Hughes-Hallett, et al., 1995). The text created by the Harvard Calculus group, Calculus: Single and Multivariable (Hughes-Hallett, et al.,1995), worked on the premise of the rule of three—the concept that equal time should be given to discussing each topic in turn not just symbolically, but also numerically (often with data tables, sometimes with emphasis on calculator and computer applications) and graphically. In the second and later editions (Hughes-Hallett, et al., 1998, 2001), this was expanded to the rule of four, with the inclusion of an emphasis on words, meaning at times both the verbal description by student and teacher of mathematical problems and what are traditionally known as “word problems”—those problems sometimes known as application problems. The resulting text is generally considered the best practical example (and certainly the most widely used) of the theories behind the reform movement (Wilson, 1997). No major discussion of calculus reform would be complete without acknowledgement of the existence of 'misleadingly restrictive' or contradictory definitions of calculus reform itself (Ross, 1995, pg 3). For the purposes of this paper, calculus reform will be defined as any attempt to move away from traditional modes of calculus instruction, including but not limited to the use of a ‘reform’ text (one whose purpose is to pursue the doctrines mentioned above), integration of technology into the calculus curriculum, and innovations beyond a lecture monologue in the delivery of instruction Since the onset of the reform movement, a number of critics of calculus reform, both in theory and in practice, have emerged (Andrews, 1996; Klein & Rosen, 1996, 1997). The resulting backlash against reform has shown itself often in opinion papers by members of mathematics departments, many of who see the reform movement as untested theory-of-the- moment pabulum, while others see it as downright dangerous: The calculus reform movement was a facile response to a real problem: the declining performance of American students in college
2 calculus courses. Without any credible scientific study, many reformers put forth “solutions” such as eliminating theory, making more conceptual problems, decreasing the reliance on algebra, and increasing the use of computers and calculators. But what if the reformers were wrong...? (Klein & Rosen, 1997, p. 1324) The focus of this research is the historical reaction of a predominantly traditionalist mathematics department to the introduction of a calculus reform experiment at the university level. While much of the early research in reform focused on quantitative analyses of student performance in traditional versus reform courses (mostly quasi-experimental pre-test/post-test comparisons), little or no research has been done on the acceptance or lack of acceptance of reform ideology by mathematics instructors at the college level. The university at which the interviews in this research were undertaken will be known for the purposes of anonymity as Research University. The mathematics department at Research University underwent a book change in 1999- 2000 towards a calculus text intended to promote reform in the teaching of calculus— Calculus: Single and Multivariable, the text that resulted from the Harvard Calculus reform research project and the most popular reform text in use (Hughes-Hallett et al., 1998, Wilson, 1997). For the sake of referential simplicity in this research, the text will be referred to as being authored primarily by Deborah Hughes-Hallett, with the understanding that while she is listed as the primary author, the work is actually that of a number of like-minded members of the Harvard NSF calculus reform consortium. In fact, 15 different authors are named, but Dr. Hughes-Hallett is always listed first, even though she is not alphabetically first. The mathematics department at Research University eventually changed course and made a switch back to a traditionalist text (Stewart’s Calculus) by the 2004 school year. Since the majority of college calculus instructors at traditional research universities does not have a background in education (since with very few exceptions, Mathematics and Education departments are separate, most calculus courses are taught by math faculty or math graduate students), and might therefore be less likely to be steeped in the tenets of this mathematics education-sponsored paradigm shift, I became curious as to the departmental environment that would lead first in a direction towards reform and then (in less than five years) return to what to most would be considered a far more traditional calculus focus. Was this switch, and switch
3 back, as innocuous as some observers might insist, or was there something more meaningful in this short-lived experiment with the reform movement? This research would hope to answer the three-part research question: • What were the historical conditions under which Research University underwent a calculus reform experiment? • What was the nature of the reform experiment attempted at Research University? • What were the historical conditions under which the reform experiment was discontinued, and the department returned to a more traditional manner of calculus instruction? Instructors from different sides of the calculus reform issue were interviewed, with the ultimate goal of creating a window for the observer into these faculty members' awareness of reform, their belief in the quality or value of reform, and their self-professed commitment to the reform ideology. This research attempts to give the reader insight into the opinions and beliefs of college faculty as regards to this volatile subject. It is my belief that the use of a reform calculus text by traditional lecturers actually may have created a disconnect, and the combination of reform text and traditional instructional style (or, for that matter, reform-minded instructor and traditional text) may have resulted in a less effective learning environment than the utopian environment anticipated by reform movement proponents (who, in their defense, must have envisioned these reforms being instituted by the reform-minded using reform texts). Interviewing college professors about their perceptions of calculus reform (and specifically their experiences during a reform experiment) gave valuable insight into the effectiveness of that reform as implemented. Finally, it is felt that until such perceptions are exposed, any attempts at future improvements in reform methodology will be met increasingly with hostility and disdain (Wu 1997, Tucker, 1999) from those who teach the course.
4
CHAPTER 2
LITERATURE REVIEW
Research Spawns Calls for Reform The calculus reform movement grew out of the recognition of the changing focus of college math departments from being predominantly the educators of future research mathematicians to being the educators of the many and varied disciplines that use mathematics, including engineering, the natural sciences, and many other fields in which mathematics is considered a necessary prerequisite (Douglas, 1986). In the 1980s, mathematics education researchers were focusing on the different ways in which students learned concepts of high school calculus and algebra; early research focusing on the value of visualization (Presmeg; 1985, 1986) as a means of deeper critical understanding of mathematics was seen in direct contrast to parallel research showing that most students have little in the manner of visual understanding compared to their analytical or algebraic understanding (Vinner, 1989, Tall, 1991). This difference between how students could learn at a deeper level and the type of learning that was actually going on was seen as a direct result of the traditional manner of calculus instruction, which was characterized as focusing almost exclusively on analytical symbol manipulation to the exclusion of graphical and other visual modes of comprehension (Aspinwall and Shaw, 2002). Finally, near the end of the decade, these earlier research efforts led to an analysis of what calculus students were actually learning, including a widely cited article by Selden, Mason, and Selden (1989) titled Can Average Calculus Students Solve Nonroutine Problems? In this article, the researchers discovered that a vast majority of undergraduate calculus students deemed to possess average skill levels were at a loss to solve problems that were not formula-driven. Instead, the research indicated that the manner in which these students had learned calculus had not prepared them for application of those skills outside the narrow constructs of the course. The one major issue with the Seldens’ research was that, as the authors themselves indicated, 40% of C students in Calculus I went on to fail Calculus II. This would imply at least an
5 acknowledgement that a C in Calculus I was not positively correlated with average working knowledge of calculus. Where the course statistics showed that 60% of the students at the school researched in Selden passed Calculus I, one might classify this population as non-representative, based on national statistics. Then, a regrettably small number of those C students were willing to take a test on their calculus knowledge after the fact. Nevertheless, the inability of those students clinically interviewed to show an understanding of how to apply calculus skills to problems unlike those they had seen in the course was seen as an indication that the status quo of calculus learning and teaching was unacceptable. In parallel to (and as an informed reaction to) the research during the 1980s in the mathematics education community that was uncovering differences between teaching, learning, expectations, and outcomes in calculus and other courses, a number of national groups began to come forward with calls for reform and revitalization of the way in which both high school and college calculus (and eventually, math courses at all levels) were taught. The groundbreaking conference where this was first addressed was the Mathematical Association of America’s (MAA) 1981 “Committee on the Undergraduate Program in Mathematics”. The result was a curriculum report that first suggested some change in the design and implementation of calculus curricula nationwide (MAA, 1981). On the heels of this report and the provocatively titled A Nation at Risk, in which the National Council on Excellence in Education (NCEE) exposed publicly the high rate of failure in Calculus courses (more than 50% of those students enrolled in Calculus at that time received a D, F, or W), first secondary schools and later colleges were asked to take a long hard look at the way calculus instruction was being carried out (NCEE, 1983). In 1983, the National Science Board Commission on Pre-College Mathematics Education speculated that technological improvements in calculator technology demanded a change in the manner of secondary school preparation—while this did not directly address college level math, the result was a focus on how these new technologies would effect teaching math at all levels (Schoen and Hirsch, 2003). In 1986, after the need for calculus reform was spotlighted at a Tulane University conference, literature began to emerge that first outlined the proposed nature of that reform. Both Douglas (1986) and Steen (1987) discussed in detail the awareness within the Mathematics Education community of a need for change in addressing this new age of math students. The National Council of Teachers of Mathematics (NCTM) created multiple documents in the late
6 1980s and early 1990s addressing this burgeoning movement—including but not limited to 1989’s Curriculum and Evaluation Standards for School Mathematics, 1991’s Professional Standards for Teaching Mathematics, and 1995’s Assessment Standards for School Mathematics (NCTM, 1989, 1991, 1995; Ferrini-Mundy, 1996). In these conference documents, NCTM called for change in the way that calculus is taught at all levels, and for the first time, a national organization recommended use of calculator technology in the calculus classroom at both the secondary and college level (NCTM, 1989). At the same time as the NCTM was focusing on reform of math at all levels, the National Research Council [NRC] focused specifically on the impact that reform in college calculus instruction would have on an audience that logically included many future educators, since all teachers of math above a certain level would be expected to take college calculus as a prerequisite for their chosen profession (1989). The NRC paper, Everybody Counts: a Report to the Nation on the Future of Mathematics Education, challenged college math departments to look seriously at the way in which calculus was being taught, and justified this attention by pointing out the power of the course in many subject areas: Not only do all the sciences depend on strong undergraduate mathematics, but also all students who prepare to teach mathematics acquire attitudes about mathematics, styles of teaching, and knowledge of content from their undergraduate experience. No reform of mathematics education is possible unless it begins with the revitalization of undergraduate mathematics in both curriculum and teaching style. (p. 39) It may be an issue of semantics, but this NRC paper as well as a later incarnation titled Moving Beyond Myths: Revitalizing Undergraduate Mathematics (1991) used the term revitalization far more than the term reform. It has since been argued that as reform is a word that comes with its own baggage, whether in mathematics education or religion, and that perhaps the movement would have encountered far less hostility from traditionalists had the change been first described as a ‘renewal’ or revitalization from the outset (Smith, 1998). New Math—A Parallel Calculus reform is not the first major revision of mathematics instruction in the last 40 years. Reform is synonymous with education in general, and mathematics education has experienced the reform phenomenon before, sometimes without achieving the intended results.
7 In the 1970s, 'New Math' was a mathematics education-sponsored paradigm shift that attempted to change the way basic math skills were acquired by students in secondary school (e.g., Hill, Rouse, and Wesson, 1979; Hartocollis, 2000). Based in part on attempts to follow the revolutionary work of Jerome Bruner’s The Process of Education (1960), the New Math initiative promoted the awareness of the student to the history and structure of mathematical subjects rather than a modular gifting of skills the usefulness of which was obscured. However, somewhere between Bruner’s profound observations and actual implementation, something went wrong (Hill, Rouse, & Wesson, 1979). Thirty five years later, New Math as a reform can generally be considered a ‘colossal’ failure (Kilpatrick, 1997). Attempting to teach complex axioms to subvert memorization of everything from the times table to the principle of zero products was found to demand more, rather than less, from math students and instructors. ‘New Math’ as a reform was also undermined by a lack of understanding from those who had the largest stake in the reform—the teachers who were tasked with implementing it (Hartocollis, 2000). Many authors have discussed the nature of reform from the perspective of teachers—most commonly, teachers tend to reject the glacial pace of reform, as they tend to be threatened by any change in their instruction, and researchers focused on instructors’ perceptions of change have found that successful change generally demands a pro-reform atmosphere (Tobin, Davis, Shaw, & Jakubowski, 1991). While it is well established that the New Math reform of the 20th century was not successful because of resistance to and impatience with the pace of reform, it remains to be seen whether calculus reform will be given enough time, and enough support outside the mathematics education community, to succeed (Hartocollis, 2000). Reform opponents are not the only ones to compare calculus reform with the New Math reform that predated it; Kilpatrick (1997) suggested that it was unfair to condemn the earlier reform as unsuccessful since “in most classrooms, the reforms were never really tried (p. 957).” The author also tackles the labeling by traditionalists of both reform efforts as misapplications of constructivism, pointing out that the term constructivism “has almost lost its meaning in American mathematics education (p. 959).” Kilpatrick goes on to point out that the use of the term constructivist with negative connotations to imply the misapplication of a theory of learning acquisition to instruction misses the point that many of our best educators, from Plato to Dewey, could similarly be accused of being constructivists, even if they predate Piaget. Kilpatrick goes
8 on to observe that even these labels are shifting, where what was called progressivism thirty years ago is dismissively labeled constructivism now. This article highlights the difficulty of dissecting the sheer semantics of the divide between reform and traditional partisanship. Not all of those opposed to the new reform treat New Math as a cautionary tale: Kleinfeld (1996) criticizes calculus reform for not going far enough towards the original reform’s goals of making math more conceptual and preparing students for abstract thinking, what she calls “the real mathematics (p. 231).” However, the author is opposed to one of the main focuses of the reform movement—the movement away from lecture-based pedagogy towards a project- or problem-oriented approach. Her argument stems from a rejection of the transformation of lecturing as she understands it from the reform community: The calculus reform movement has said we want to replace the “Sage on the Stage” by the “Guide on the Side”. Most people interpret this to mean less lecturing by the professor and more problem solving and projects by the students. Here again, it’s the wrong direction (p. 231). Kleinfeld believes that only by lecturing (albeit in a much different way than that characterized by her opponents) can calculus professors efficiently instill in their students the skills necessary to successfully (and, in a temporal sense, ultimately) master concepts. It is not my desire (or personal belief) to imply that, in general, calculus reform is as poorly implemented as New Math turned out to be. However, distinct parallels exist, and should be subject to comparison. One paper that tackles these parallels head-on is The “New New Math”?; Two Reform Movements in Mathematics Education (Hererra and Owens, 2001). The article draws parallels between the resistance to each reform, the uneven implementation and introduction of curricular change in both reforms, and the ways in which both reforms were seen by many teachers in the classroom as having been introduced from outside. Workable models do exist, however, that would allow teachers to modify their instruction to promote a reformed pedagogy (e.g. Shaw, K. & Jakubowski, E., 1991). However, since all realistic change models represent a phased multi-step process, a teacher would have to want to change before applying such long-term self-improvement techniques—but the vast majority of people in an educational environment tend to believe that change is something their colleagues, and not they, need to engage in (Ornstein and Hunkins, 2004). Reform Movement Picks up Momentum
9 Where the Mathematical Association of America (MAA) led the way in the 1980s with the first calls for reform, and the NRC and NCTM responded to the challenge and carried it into the 1990s, the reform movement really began to accelerate with the influx of grant money from the National Science Foundation in the mid 1990s (MAA; 1981; Steen, 1987; NCTM, 1989; NRC, 1989, 1991; Heid, 1997). Along with the Sloan Foundation, NSF sponsored such groundbreaking projects as the Calculus & Mathematica Project at the University of Illinois- Urbana/Champaign and the Harvard Calculus Consortium (Culotta, 1992). These and other early nineties program-specific reform efforts took as their mandate discussions from a Washington colloquium sponsored by MAA metaphorically titled Calculus For a New Century: A Pump, not a Filter (Steen, 1987). The title for this set of individual opinion papers edited by Lynn Steen was taken from two presentations at the conference. As the opening speaker at the colloquium, Press (1987), the then-head of the National Academy of Sciences, described as he saw it the landscape of college calculus instruction. Press described standard (traditional) calculus teaching as being governed by the philosophy of the calculus course sequence as a filter necessary to weed out those unable to succeed in the rigors of the majors for which calculus is a prerequisite. In fact, this philosophy is not unique to calculus courses—for the initiated, organic chemistry for chemistry and biology majors and thermodynamics for engineering majors have also often been seen as weed-out courses (Reingold, 2005). However, the implication that the philosophy of using calculus to weed out the unworthy was universal to all college math teachers is a generalization by Press that was not backed up with any quantitative evidence. Whether he took an informal poll or was working from anecdotal sources or personal experience is unknown. But the fact remains that the generalization was tied to the aforementioned statistics on failure rates, but without any reference to qualitative or quantitative research to prove a positive correlation between teaching philosophy (what percentage of college faculty really do feel this way?) and success-rate outcomes (would not there be a myriad of reasons why students fail in calculus, just as within any one class students do not all fail for the same reason?). In retrospect, Press’s speculation as to the cause of the struggles within college calculus teaching and learning was little more than anecdotal. The main focus of his speech is the acknowledgement of and dissatisfaction with the state of calculus instruction: the failure rates of more than 50% of students in college calculus nationwide, and the inability of even those
10 students capable of passing to solve nonroutine problems (Hurley, Koehn, & Gantner, 1999; Selden, Mason, & Selden, 1989). Nevertheless, the speech is normally cited later in the reform literature not as an early recognition that something needed to change, but rather as an indictment of traditional instruction as the primary cause of the problem. This interpretation dismisses other possible reasons for low success rates in calculus, including the difficulty of the material compared to the content of other undergraduate courses, or the constantly evolving preparation levels and work ethic of undergraduate college students. White (1989), the then-President of the National Academy of Engineering, tied his presentation at the same colloquium to the implication by Press of culpability of mathematics teachers for the failure rates in the course. White suggested that modern calculus should be reformed in such a way that it would represent a pump, not a filter; a conduit of qualified and conceptually gifted students into all of the majors for which calculus is a prerequisite. As Hurley, Koehn, and Gantner (1999) stated twelve years later, these two opinion papers at the 1987 Washington colloquium became a ‘rallying cry’ for the reform movement (p. 804). Unfortunately, while their statements were the opinions of leaders in their respective areas, they were not based on a preponderance of research showing that these teaching philosophies were pervasive, or even that teaching issues were the direct or sole cause of the negative outcomes associated with calculus instruction (Andrews, 1996). Reform—Overviews One of the more comprehensive overviews of the reform movement from the nascent period is attributable to Ferrini-Mundy and Graham (1991), in which the authors outline both the justifications of reform and also the structure of a reform calculus course curriculum. This reform curriculum involved a lesser dependence on lecture, and also a balancing of analytical manipulation (what some call mechanics) with the newly championed areas of visual (graphical) discovery and group project work. The inclusion of these facets to a curricular redesign followed the aforementioned work of researchers on mathematics student learning like Krutetskii (1976)—specifically, his work on the psychology of math students and analytic versus geometric reasoning—as well as the groundbreaking work on multiple representations by Presmeg, Aspinwall, Shaw, and others. Only later in the 1990s did a further research-driven dimension, sometimes described as applications or word problems, elsewhere designated as reading and
11 writing problems, become a fundamental part of the reform curriculum (Aspinwall & Miller, 1997, 2001; Hughes-Hallett et al., 1998). Schoenfeld (1995) contributed a 'biography' of calculus reform summarizing the movement's first 10 years, with a shot-by-shot discussion of many of the historic conferences mentioned earlier in this review. Schoenfeld also discussed a working definition of calculus reform (contrasted with the generalized view of traditional instruction) repeated often in the literature: "Calculus reform is more hands-on [than traditional instruction], students are expected to understand complex mathematical concepts in more connected ways, and new approaches to content and understanding must be supported by a reformed pedagogy (p. 4)." The connected nature of students’ conceptions mentioned by Schoenfeld calls to mind Bruner’s work on the symbolic mode of cognitive representation, and the insistence on change in the pedagogy parallel to change in the calculus itself indicates that the author realized the futility of change in the teaching materials without changing the manner of instruction (Driscoll, 2000). The central focus of this research is a reform experiment built around the introduction of Hughes-Hallett et al.’s Calculus: Single and Multivariable (1995, 1998, 2001), a best-selling collaboration of 15 authors from the Harvard NSF-sponsored Calculus Reform Consortium. This book addressed all of the major outcomes of the Tulane conference, the Washington colloquium, and the work of the many national organizations that had sponsored the move towards reform. In the body of the text itself, the authors describe how they feel their text can improve the teaching and learning of calculus: One of the guiding principles is the ‘Rule of Three’, which says that whenever possible, topics should be taught graphically and numerically, as well as analytically. The aim is to produce a course where the three points of view are balanced and students see each major idea from several angles (1995, p. 121) The three facets described in the quote reflect prevalent research on multiple representations (e.g., Janvier, 1987, Vinner, 1989, Tall, 1991) and the Seldens’ research showing that predominately analytical skills-based calculus instruction was not generating average (C) students with a clear conceptual understanding (Selden, Mason, & Selden, 1989). Graphical understanding involved visual and geometric presentation of the topics, whereas numerical models allowed the introduction of technological improvements (calculator and computer usage), which statistics showed had been slower to make their way into collegiate math instruction. Only
12 7% of college calculus courses in 1991 used calculator or computer technology, in spite of the efforts of NCTM and NRC to call for their inclusion (Ferrini-Mundy & Graham, 1991) The Harvard calculus reform was not the only project to result in a reform text; in another key example at Purdue University, Dubinsky and Schwingendorf created Calculus, Concepts, and Computers with Cooperative Learning (C4L), a reform curriculum with not only a new text but also homework assignments and laboratory activities. As the title of the effort implies, this effort involved some of the same focuses as the Harvard reform with a built-in laboratory group- work element (Dubinsky, 1992). Schwingendorf, McCabe, and Kuhn (2000) did quantitative analyses of the success of this NSF-sponsored reform initiative showing that students enrolled in their C4L program received better grades in Calculus and performed equally well in follow-on coursework, when compared with students in a traditional course. Nevertheless, the program met with such hostility from within the mathematics department at Purdue that Dubinsky left the university in 1997 (Wilson, 1997). It would be beneficial to the research community to analyze why a program whose creators had deemed it such a success would not be accepted by the math department in which the reforms were piloted—to date, no independent quantitative or qualitative analyses of the C4L experiment have been attempted. Reactions to the Growth of Reform Soon after the creation of the first Calculus reform texts, journal articles in Mathematics journals like American Mathematical Monthly and Primus, and in the Chronicle of Higher Education began to focus on the resistance to these changes from within the college teaching community. In this section, a few of the earliest and most influential of these reactions to reform are outlined, with others discussed more comprehensively under a later sub-heading entitled Debate Literature. The first indication that all was not perfect in the move towards reform came from England. Calculus reforms had by the mid-1990s been successfully introduced in the United States at (among others) the University of Connecticut, Dartmouth, the Naval Academy, Baylor, and Purdue with results that, when quantitatively analyzed, tended to show student acceptance of and benefits from reform (Hurley, Koehn, & Gantner, 1999). England had rapidly introduced the calculus reforms in the 1980s at the secondary school level involving use of technology, building conceptual understanding through geometric processes, and a de-emphasis on proof and analytical computation. However, in 1995, a report before the London Mathematical Society
13 titled Tackling the Mathematics Problem, compiled by mathematicians, scientists and engineers in higher education, cited what the report deemed “serious problems” with the entry-level preparation of undergraduates in math, science, and engineering (p. 3). The authors of this report summarized the problems: (i) a serious lack of essential technical facility—the ability to undertake numerical and algebraic calculation with fluency and accuracy; (ii) a marked decline in analytical powers when faced with simple problems requiring more than one step; (iii) a changed perception of what mathematics is—in particular of the essential place within it of precision and proof. (p. 3) In the same way that some of the founding fathers of reform had earlier made a causal connection between generalized calculus teaching philosophy and the success rates of college calculus without citing research to support this, the anonymous authors of this report were quick to acknowledge that they offered little data to back their claims: “Various universities have offered us data supporting the case we have put forward. We have chosen not to reproduce, or even to try to summarise, all these (p. 11).” It is important to note that the report to the London Mathematical Society was a criticism of calculus reform at the secondary level, not at the college level. However, within a year or two, a growing number of American articles citing this report began to appear, calling to arms the mathematics community in response to the perceived encroachments or inadequacies of calculus reform. If the earlier presentations of White and Press (1987) had been a rallying cry for reform, this paper and its offspring can be seen as the earliest evidence of an anti-reform (later to be termed traditionalist) backlash. In The Mathematics Education Reform: Why You Should be Concerned and What You Can Do (1997), Wu accused the reform movement of “advocating pedagogical practices based on opinions rather than research data (p. 946).” He admitted both merits in the reform movement and flaws in the traditional manner of calculus instruction where some depend too heavily on proof and ignore the fact that 80% of their student audience will not go on to graduate-level mathematics. However, Wu stated that Hughes-Hallett’s (1995) reform calculus text (then in the first edition) passed college students through calculus without requiring any deep understanding of the analytical or algebraic structure of the subject. He further objected to the mutation of constructivist learning theory into a theory of instruction.
14 Wu cites not only the report to the London Mathematical Society, but a then-online article, Applications and Misapplications of Cognitive Psychology to Mathematics Education (Anderson, Reder, & Simon, 2000) by three psychology researchers who summarized their analysis of the application of constructivism to mathematics education reform thusly:
The evidence for such information-processing approaches to education, however incomplete, is enormously stronger than the evidence for the opposite approaches, supposedly based in cognitive psychology, that are currently dominating discussions of mathematics education. And this is our main message: A program of educational reform is being adopted with weak empirical and theoretical bases while a better, and better validated, program stands ready for further development and application, and that is a situation that should be and can be altered (p. 1)
As Driscoll points out, “there is no single constructivist theory of instruction” (2000, p. 375). In fact, what is misidentified by Wu as constructivism in general is more probably associated with radical constructivism, whereas the more universally accepted constructivist view is a conglomeration of aspects of theoretical work from Piaget, Bruner, and Vygotsky, among others (Driscoll, 2000). The objection of the cognitive researchers quoted above isn’t that the reform movement is based on constructivist theory, as Wu asserts, but rather that the connection between the empirical research and the particular brand of constructivism espoused is tenuous. With calls from within the mathematics community and from even researchers in cognitive psychology casting doubt on the structure of calculus reform and the science behind it, more math journals began publishing articles from mathematicians/college math teachers dismayed at what they perceived as the interference of reform with their autonomy as college math professors. Andrews (1996) cited the failure of new math and posited that the new calculus reform, like its predecessor, was less a reform than a revolution—the outright “rejection” of all traditional manners of teaching, to be replaced by facilitation of discovery at a level where this may not be feasible from a curricular standpoint. Andrews further identified what he saw as zealotry within the reform
15 community and a desire to classify “all who oppose them as extremists of the traditional approach—“Back to Basics” troglodytes (p.343).” Finally, Andrews uses the distinction, derived from the work of environmentalist E.F. Schumacher, that the educational reform problem is a divergent one. Schumacher (1978) had noted that many problems do not have a single perfect solution, but rather multiple solutions that might not be analytically disprovable. Andrews felt that the reform advocates viewed their solutions to the identified concerns with collegiate math instruction as the only valid way to improve calculus, and that this view of an inherently divergent problem had led to a polarization of those for and against reform. At the same time as Andrews and Wu were criticizing reform in general, studies first appeared that showed that individual calculus reform experiments were not universally successful: Alarcon and Stoudt (1997) detailed a Mathematica-based reform experiment at the University of Pennsylvania that was abandoned within five years as being untenable. Mathematica is a computer algebra software (CAS), one of the first and still most commonly used of a series of such software programs designed to advance both the computational and symbolic manipulation of mathematics. The software has facilities for graphing in multiple dimensions, numerical integration and solving subroutines, and even does symbolic derivatives, albeit with approximately the same success or accuracy as first-generation grammar checkers in word processor programs. The authors concluded that the failure of the reform experiment at the University of Pennsylvania was the result of a number of factors, including an inability to fit all elements of a reform calculus pedagogy into the time constraints of a normal calculus sequence, and a perceived lack of preparation for post-requisite coursework. Also, the early versions of Mathematica used in the course demanded a level of coding skill not normally expected from freshman Calculus students. It should be noted that in the reform experiment at the University of Pennsylvania, curriculum transformation and technological inclusion were major factors in both the construction and the demise of the reform initiative. This has some bearing on the reform experiment analyzed in the present study. In 1990, Turner had predicted in The Chronicle of Higher Education that before the end of the century, sweeping reform in the manner of Calculus instruction at the college level would be the norm, not the exception. By 1997, Wilson in the same forum
16 used terms like “pessimism”, “haggling”, “divisive”, and most strongly, “disaster” when describing the state of those reforms. While this is not a research journal article, it does give an indication of how far things had gone in less than ten years. Wilson’s article described anecdotal instances of college professors angry with reforms poorly conceived, executed, and forced on them. But the author also described an atmosphere of hostility towards the minority contingent of reformers in some departments, many of whom felt abandoned. Included among these reformers were Osgood of Stanford, who planned to leave the mathematics department for engineering because of perceived acrimony from traditionalists, and Dubinsky, whose C4L reform program at Purdue was still being cited as a success by Schwingendorf, McCabe, and Kuhn (2000) and Hurley, Koehn, & Gantner (1999) two years after Dubinsky had left for Georgia State University because of the negative reaction to his reform curricular restructurings at Purdue. Dubinsky is quoted in Wilson’s article as predicting that calculus reform would be short-lived, as the changes involved in a successful curricular and mindset change would be far more difficult than the status-quo of traditional lecture-based instruction. Kaput suggested that part of the difficulty with the reform movement is that “the renovation is taking place while the owners and stakeholders continue to inhabit the institution” (1997, p.1). Kaput describes the resistance to change in the community as the result of a lack of distinction between the knowledge of calculus and the institution of calculus. In his mind, reformers running roughshod over the institution, and traditionalists worried about its protection, are the cause of the divisiveness by then prevalent between reformers and traditionalists. Kleinfeld (1996) struck out at one of the fundamental tenets of calculus reform: the use of examples from other disciplines: "Don’t let them browbeat us into teaching every subject but our own! We have calculus for engineers, calculus for biological sciences, calculus for business— why can’t we have calculus for mathematicians (p. 231)?" This paper is really half of a debate- paper coupled pair (see the later sub-heading below), a type of article that occurs frequently in mathematics journals and attempts to give equal time to both sides of the reform issue. Kleinfeld does not mention in her paper that historically, at least, it was math departments (in an attempt to segregate majors) that came up with specialized calculus courses as a direct answer to pressures from other departments less enamored of having 60% of their students fail a math prerequisite (Ralston, 1981). A calculus-for-mathematics-majors course would be a small class in most
17 universities; most mathematics departments spend a majority of their time teaching students from other majors (Seymour, 2002). Many from within the reform movement arose at this time to combat the traditionalists: Kilpatrick (1997), in the oddly-titled Confronting Reform, is actually an article that critiques Wu’s article, before in summary asking for more support for one another from between the two camps. Kilpatrick summarizes the reform effort succinctly: ”Change in education is notoriously complex, difficult, and unpredictable. Reform movements in mathematics education turn out neither as advocates hope, nor as detractors fear (P. 961).” It would appear from the literature that most efforts at analyzing the success of reform efforts can be categorized as attempts to quantitatively measure these hopes or fears. It is interesting, as a semantics exercise, to analyze the usage of the unequal terms “advocates” (to denote those who are pro-reform) and “detractors” (to denote those who are anti-reform) in Kilpatrick’s article. One could undoubtedly do a fascinating meta-analysis on the usage of partisan language in reform literature. Hirsch (1996) theorized that perhaps the reform movement in math education and elsewhere was plagued by a misapplication or misreading of constructivist learning theory into what he called romanticism, a championing of student-constructed critical thinking skills that actually denigrates efforts by teachers to impart knowledge. The author argued that without a background in mathematical knowledge (literacy), students would stand to gain very little from conceptual discoveries. This is not entirely at odds with constructivism, as the theory of learning does not imply that the learner’s constructions be externally valid (Driscoll, 2000). The term romanticism actually predates Hirsch’s work, originating in the writings of Italian educational theorist Gramsci (1932), who at the time of this quote had been jailed by Mussolini for being a communist: The new concept of schooling is in its romantic phase, in which the replacement of “mechanical” by “natural” methods has become unhealthily exaggerated. Previously pupils at least acquired a certain baggage of concrete facts. Now there will no longer be any baggage to put in order. (p. 42) Hirsch, who has been labeled as a neo-conservative by some within the reform community for his embrace of drill and mechanics, openly rejected the concept that analytical knowledge is conservative: “to denominate such methods as…memorization of the multiplication table as “conservative,” while associating them with the political right, amounted
18 to serious intellectual error (1996, p. 7).” In essence, however, Hirsch would certainly be aligned with the traditionalists, and he succinctly breaks the argument between reformers and traditionalists down to a question of direction: reformers, or as he denotes them, progressives, feel that understanding, through a naturalistic discovery in line with constructivist theory about learning, leads to knowledge, whereas traditionalists feel that knowledge of facts, ‘rote’ consolidation of mechanical skills, is the foundation of understanding. These polar opposite (what Schumacher called divergent) views of the correct approach to mathematics education lead the purveyors of reform and its opponents to view themselves as aligned with partisan camps, with labels as volatile as the political or ideological labels of liberal and conservative. The recognition of this at-times acrimonious relationship between two groups led to a cottage industry of journal-based debate literature starting in the 1990s.
Debate Literature
Debate literature, often of the “he-said—she-said” variety seemed in the 1990s to be a very common method of presenting the issue of calculus reform. There is something of a cottage industry of journal articles and papers from the mid-1990s that acted as a forum for the debate between mathematics education faculty and mathematics faculty analyzing (at times heatedly) the pros and cons of the reform movement. The earliest example found of this article style appeared in The College Mathematics Journal, in which Mathews (1996), a proponent of the reform movement, parried with Andrews (1996), a traditionalist. In a pattern repeated throughout the literature, the reform proponent [Mathews] cites pro-reform literature such as the National Research Council’s (NRC) reports Everybody Counts: A Report to the Nation on the Future of Mathematics Education (1989) and Moving Beyond Myths: Revitalizing Undergraduate Mathematics (1991), while the anti-reform Andrews (1996) cites what he deems the weakness of the pro-reform research and the vagueness of citation from that research: “These large-scale [NRC] studies” are filled with numerous pronouncements that begin “Evidence from many sources shows that...” and “Research shows clearly that...” There are no footnotes and no citations to speak of, so we must take all this on faith.... In other words this is a document for the already-convinced. (p. 351)
19 It is a commonly recurring argument for the anti- reform traditionalist: the belief that none of this reform has been properly thought out, that “evidence” that lecture is “bad” and any deviation from it is “better” is not backed up by sufficient verifiable research. Other examples of this literary archetype include Cargal (1997), in which the author acts as mediator between reform proponent Tucker and traditionalist Swann, and Mumford (1997) and the inevitable rebuttal from Klein and Rosen (1997) in Notices of the AMS. In each case the calculus reform advocates portray traditional teaching methods as “drill and kill” and traditional instructors seeing as their primary function the weeding out of the unworthy, and the traditionalists counter that reformers give their movement a benefit of the doubt when reform fails that is not forthcoming towards traditional program failures (Steen, 1987). Any interested observer of the reform movement will have a vested interest in perusing this and subsequent debate literature on the reform movement; however, the vary nature of that literature (it is not research) gives it limited application to the research at hand other than to catalog the prevalent opinions on the reform debate. Most of the actual research on calculus reform involves ground-level analyses of reform experiments through qualitative (student and TA surveys and clinical interviews) and qualitative pre-test post-test analyses of assessments in reform versus traditional course. Quantitative and Qualitative Analyses Primary research in the field of calculus reform is more readily found in dissertation databases: relatively few articles in research journals have reported on calculus reform research experiments. Most of the research measuring the success of reform experiments is quantitative in nature, consisting of pre-test and or post-test comparisons between traditional calculus sections and reformed sections (e.g., Alexander, 1997; Bready, 2000). One of the inherent difficulties in making such comparisons is that the very nature of calculus reform demands different methods of assessment than are done in traditional courses. This leads to a fundamental flaw in quantitative analyses of this type: does one give a reform-taught group of students a traditional exam, or alternately, a traditional class a reform exam? Even proponents of reform note that this less controlled experimental environment leads to less purity in the results of quantitative studies than most scientists would be comfortable with, as Tucker (a self-identified member of ‘the reform camp’ (1999, p. 910), pointed out:
20 Most of the comparisons I know between traditional and reform courses at the same institution are not controlled experiments…When reform courses come out looking better on common exams, perhaps it is because the instructors who choose to teach the reform versions are not typical instructors (p. 913). In fact, many of these quantitative studies give results that favored whichever group’s pedagogy was emphasized. It would be an interesting (if entirely different) research project to analyze whether a correlation exists between who is funding the research and what the results of that research are. Some qualitative analyses exist, predominately focused on students or teaching assistants (TAs) as opposed to tenured faculty and survey- or observation-driven as opposed to case study or interview-based. Speer (2001) looked qualitatively at the beliefs of teaching assistants (TAs) in reform- oriented calculus classes and how those TAs' beliefs affected their willingness to buy into reform. She found that factors such as TAs’ beliefs about what learning is and how it happens were influential in the manner of the support that they gave to their students. Bready (2000) did an exhaustive quantitative analysis of students who took advance placement calculus in high school and then took reform or traditional calculus II in college. She found that college calculus students who had had reform methodology in high school were more likely to be successful in reformed college calculus, but there was not a statistically significant difference in reform versus traditional overall. Similarly, Alexander (1997) had earlier found no statistically significant difference in retention or grades in calculus post-requisite coursework between a reformed calculus track and a traditional one at the University of Arizona. After interviewing volunteers from both tracks, Alexander states: "A pattern of comparisons emerged which showed that consortium students somewhat outperformed traditional students. The patterns were indicative of better teaching and cannot be attributed to [reform] materials (p. 347)." He goes on to state that the “outperformance” was not statistically significant. Armstrong (1997) focused her doctoral research on the effects of introducing the curriculum espoused by the Harvard Calculus consortium on the achievement of community- college calculus students. She found that the students’ algebra knowledge and engagement in the course were the greatest indicators of success in both reform and traditional calculus sections,
21 and further that non-Asian minority students were more likely to be hindered by enrollment in a reform curriculum. Miller (2000) looked at reform methodology and attempted to find a correlation between reform teaching and conceptual understanding. However, her research indicated that there was no significant relationship between reform or traditional methodology and conceptual performance and/or van Heile level (used to measure the subjects’ geometric reasoning). In a summary of mathematics education research through 1998, Lubienski and Bowen pointed out that roughly 2% of the 3000 articles they analyzed even had calculus as a descriptor (2000). When adding to this the focus on K-12 levels within mathematics education research, one can readily surmise that the area of analysis of reform efforts at the college undergraduate level is a fertile but under-tapped area of research. Further, since the small number of existing studies focuses primarily on students’ perceptions to the exclusion of the perceptions and views of tenured research faculty on reform in their teaching, it was felt strongly that research on this important group of informants was under-represented in the literature. One might speculate that one reason for the dearth of such analyses is logistics—it is hard for mathematics education researchers to conduct research within mathematics department (much harder than it is to conduct such research in hybrid or mixed departments) (Selden, 2002). Also, since much of the research that spawned the reform movement was research on learning rather than teaching, it is inevitable that much of the follow-on research of the effectiveness of reform experiments would focus on measuring improvements in student learning resulting from reform-based curricular or teaching changes. Finally, it has been speculated from both within and outside the reform movement that it may not be effective to rely exclusively on quantitative analyses to measure the success of reform efforts, as there are many variables involved in the process that cannot be controlled, including instructor quality (Tucker, 1999). Summary There is only one study that truly works in the same arena as this proposed research: Matney, Hurtado, and Ziskin (1999) presented a paper at the annual Association for the Study of Higher Education (ASHE) conference in 1999 outlining a case study they accomplished with three faculty implementing calculus reform at their various universities. The language of the document points to a conference atmosphere in which reform is seen as an extremely promising
22 concept, and the choice of three teachers at teaching-focused universities, all of whom are positive about reform, adds to the potent optimism of the paper. The authors state: In our study, we found faculty who were excited about the growth in student learning, especially in those students who ordinarily would not be expected to succeed in calculus. In selecting institutions, we sought to ensure both a focus on teaching and a rich comparison between institutions... All three campuses reported high levels of innovative activity in teaching across disciplines (p. 2). The selection process described above results in a study that, while a fascinating insight into the minds of faculty who embrace change, might bear little resemblance to the reaction one would expect from a less teaching-and-learning-centered institution. For example, how would faculty at a research university less focused on teaching, or less universally “excited about the growth in student learning”, differ in their perceptions of such a reform experiment? The university where this experiment was conducted was not the first or last where calculus reform was attempted—in fact, in 1995, when the Harvard consortium text was just in its first edition, UCLA experimented with reform by using the Hughes-Hallett text (Cipra, 1995). Within a year, due in large part to student and faculty protests, the text was abandoned. No known analysis of that calculus reform experiment was ever conducted, but while the text used is related, the second and third edition used at Research University (where this study was conducted) are considered more traditional than the first edition, wherein much of the algebraic language of calculus was removed. The transformation of the various editions of the Hughes- Hallett text is a research topic of a profound nature on its own, as is the more recent transformation of traditional texts to integrate the more openly embraced facets of the reform movement. To summarize, there has been a relatively small number of journal-focused studies in the field of calculus reform since its introduction 30 years ago. What research has been done fits into two main classes: quantitative quasi-experimental pretest/posttest analyses, and, to a lesser extent, qualitative analyses of students and teaching assistants. Opinion papers dominate the literature; these are often frustrating in their relative lack of data-related citations. This research is an attempt to document for the first time the perceptions of mainstream university professors as they implemented calculus reform--not supported by grants or external funding, which would demand reports of success, but simply from a desire to better meet the needs of their students. It
23 is hoped that their responses are recognized as more open, candid, and credible than if they were molded by external factors like grant renewal. The analysis of the literature focused on calculus reform lead to the following historical- qualitative research questions: • What were the historical conditions under which Research University underwent a calculus reform experiment? • What was the nature of the reform experiment attempted at Research University? • What were the historical conditions under which the reform experiment was discontinued, and the department returned to a more traditional manner of calculus instruction? During the research, an emergent research question sprang from the data: • Is it possible that in failing to align the pacing, course curriculum, and instructional style of faculty with the intended calculus reform ideology, the department created an unintentional dissonance between text and delivery? In the research questions above, the term experiment is used repeatedly. This is not to imply that any experimental method was used (quantitative or qualitative) by the mathematics department to measure the success or failure of the calculus-reform-based pedagogical change at Research University. This is not an empirically measured experiment, but rather a casual experiment derived from the parallel goals of having a single departmental text for all faculty, and the desire (by some, at least) to attempt to better serve the student population. The majority of the faculty on the book committee at Research University in 1999 felt that if student failure rates were unacceptable in Calculus locally, as they were nationally, that a modicum of reform introduced into the course materials might, in time, lead to better retention and completion rates. As this research data suggests, the experiment was abandoned without any measurements to determine its success or failure beyond the perceptions of the faculty themselves. These research questions are specific to the calculus reform experiment at Research University, but they also represent an opportunity to enrich the existing body of literature about reform, college mathematics instruction, and perceptions of reform from outside the mathematics education community. The research accomplished herein not only seeks to answer these questions and paint a picture of a department in transition, but also should set the table for other future efforts in an area pregnant with research opportunities.
24 CHAPTER 3
METHOD
Literature On Research Methodology—Qualitative Analysis The qualitative analysis techniques undertaken in this research are derived from work done in this area over the past thirty five years by such noted authors as Lincoln and Guba, Corbin and Strauss, Miles and Huberman, and Bogdan and Biklen, among others. Lincoln and Guba (1985) first outlined the manner in which qualitative research (or as they termed it, naturalistic inquiry) could be undertaken in such a way that it would be equal to, and in some cases superior to, the more prevalent, and more universally accepted (at least at the time of their original text) quantitative research techniques. These authors describe the inherent danger of the establishment view that there is only validity to research if human judgment is totally removed from said research. The authors go on to outline ways in which qualitative analysis can be vetted (although they caution that any attempt to guarantee validity are as futile as they are misdirected), including debriefing by peers. Greene, Caracelli, and Graham (1989) describe a conceptual framework for mixed- method analysis that differs from what they themselves term the purism of Lincoln and Guba’s qualitative-only approach. Among the various justifications they offer for a mixed method (i.e. quantitative + qualitative) combination, they include not only triangulation but also complementarity and initiation. They define these terms as follows:
o Triangulation seeks convergence, corroboration, and correspondence of the results from the different methods or sources.
o Complementarity seeks elaboration, enhancement, illustration, and clarification of from one method with results from the other.
o Initiation seeks the discovery of paradox and contradiction and the recasting of questions from one method with questions or results from the other method. Greene and her co-authors categorize using mixed-method research for cross-checking purposes as triangulation, but while their framework treats these justifications as discrete for organizational purposes, their own meta-analysis of research projects revealed that many such
25 projects use more of a hybrid, blend or superimposition of these foci. In the language of Greene, this research’s mixed-method historical case study analysis would seek to combine elements of triangulation and complementarity, but with an emphasis on initiation. Gretchen Rossman, in her paper Building Explanations Across Case Studies: A Framework for Synthesis (1993), provided what Greene described in her work as a middle ground between her own mixed-method approach and what she termed Lincoln and Guba’s purist reliance on solely qualitative analysis. Rossman’s framework for synthesizing across multiple case studies involved 8 steps, each of which was useful in creating a structure for this research:
1. Beginning
2. Bounding the Scope of the Synthesis
3. Inventorying the Cases
4. Reading [each] Case Study
5. Developing an Interpretation of Each Case
6. Juxtaposing the Cases
7. Synthesizing the Cases, and
8. Writing the Synthesis
Finally, an analysis of the literature tapped in the development of the coding of each case study in the research to follow should be identified. Miles and Huberman (1994) first introduced me to different styles of coding, as well as the benefits and detriments of each. The first level of coding accomplished in the analysis of each case study follows the accounting method of Bogdan and Biklen (1992). They outline 10 main coding categories, of which 7 were useful to this research (reprinted, from Miles and Huberman (1994), in no particular order)
o Perspectives—Ways of thinking about setting shared by informants
o Context—General information about surroundings
26 o Definition of the Situation—How informants understand or perceive the research topic
o Ways of Thinking About People and Objects—Informants’ understanding of each other, outsiders, and their world
o Process—sequence of events, flow, transitions and turning points, changes over time
o Strategies—ways of accomplishing things—tactics, methods, and techniques
and
o Relationships—unofficially defined patterns such as cliques, coalitions, friendships and enemies.
All of these general categories for coding spoke to inherent organizations of the data within and between the case studies for this research. A secondary level of coding was more related to the inductive approach of Strauss and Corbin (1990), which agrees with Lincoln and Guba in that both insist that really meaningful codes can not be developed from someone else’s existing prior codes, but rather, that they derive directly from the data once collected. Miles and Huberman (1994) summarize this secondary level of coding: “Data get well molded to the codes that represent them, and we get more of a code-in-use flavor than the generic-code-for-many-uses generated by a prefabricated start list.(p. 58).” To summarize, the analyses of the case studies in this research is grounded in case study frameworks created by Greene and Guba, Rossman and Wilson, and coding analysis based on prior frameworks of Glaser and Strauss, and Bogdan and Biklen. The Research Problem Since calculus reform was first implemented in the classroom in the late 1980s/early 1990s, little or no major research has been undertaken to qualitatively determine the reaction of tenured college faculty teaching calculus (but not necessarily motivated to reform) to this innovation. This research focused on college faculty at a Research One university where such a reform was implemented, and attempted to determine the perceptions of those faculty as regards the reform movement, their beliefs as to the effects of the reform on teaching calculus at their
27 own university, and how much (if at all) their teaching style reflects the reform ideology. The university I chose for this research from 1999 to 2004 attempted to reform the manner of calculus instruction. During this reform ‘experiment’ the mathematics department chose as the departmental text the aforementioned Hughes-Hallett text Calculus: Single and Multivariable (1994) created by the Harvard calculus reform consortium. In 2004, the department abandoned this calculus reform experiment and returned to a more traditional calculus instruction, replacing in the process the Hughes-Hallett text with Stewart’s Calculus: Early Transcendentals (2002). The main focus of this research was the following three-part question spawned by this departmental switch: 1. What were the historical conditions under which Research university made a transition from traditional to reform calculus? 2. What was the nature of the reform experiment attempted at this university? 3. What were the conditions under which the reform experiment was discontinued, and the department returned to a more traditional manner of calculus instruction? Instructors from different sides of the calculus reform issue were interviewed, with the ultimate goal of creating a window for the observer into these faculty members' awareness of reform, their belief in the quality or value of reform, and ultimately, their self-professed perceptions of reform and reform textbooks. Before conducting the interviews, I developed a list of foreshadowed or anticipated problems that might result from the interview-based historical research being conducted: 1. Some of the faculty interviewed might have had little or no concept of the reform methodology and little or no training in how to implement it. 2. Further, whether the faculty identify themselves as proponents of reform or not, most likely teach in a manner more related to traditional teaching style--in other words, the number of faculty who are teaching calculus in the manner outlined in reform literature (less lecture, less rote or drill, more student exploration and group project work) will be considerably less than the number holding on to the status quo.
28 3. Finally, it was my expectation that faculty at the research institution (where research and publication dominate any lecturer’s agenda, and most faculty identify themselves as mathematicians first and math educators second) would be more hostile to reform than the education researchers who first called for this paradigm shift, or those mentioned in the literature review invested in reform, and thus more motivated to show success in reform efforts.
An issue anticipated at the outset of this research as the result of using interviews to assess these joint research questions was the knowledge that it is human nature to say what you think the interviewer wants to hear: if the interviewer is a mathematics education graduate student, perhaps some interviewees would portray themselves as more reform-minded, while in classroom practice they may behave more traditionally than they are willing to let on (Wiersma, 2000). There is research that supports this phenomenon: in the area of teacher training, for example, researchers have shown that future math instructors may be passionate about new ideas or teaching philosophies in education classes, but most will revert back to previously held beliefs about teaching once they are in the classroom (Book et. al, 1983). Therefore, questions had to be chosen that would not give participants the feeling that they were being put on the spot or judged in such a way that they would feel required to respond in a politically correct manner. Further, it would not have been informative, or truly qualitative, for example, to simply ask the faculty members “Do you like this text?”, as the question presupposes that mathematics instructors would ever admit liking a text, even if they might prefer it to others. It seemed more promising and insightful to simply ask the instructor if he would prefer one text to the other, allowing the instructor's own description to give insight into his preferences and perceptions, and how they align (or do not) with his self-identification. One reason for the length of the quotes in these case studies was a desire to allow the reader of this research to judge for herself the candor of the responses—undiluted and uncut, they are potent data. The participants/interviewees were faculty at Research University. The department was first informally surveyed through a pilot study to determine the lay of the land. The pilot study survey questions were intended to determine how the department (or at least those that taught calculus regularly) would self-identify as either reform-minded or traditional in their teaching approach. Some excerpts from the survey respondents appear at the end of this chapter. The results of the
29 survey were not statistically analyzed (nor could the results have been successfully quantitatively analyzed, as only 70% of the calculus-teaching faculty responded). It was accomplished mainly as a way of providing background, as complete as possible, for the environment in which the interviews were to take place. The entire research project was qualitative-descriptive and historical, with an emphasis on giving insight (a snapshot) into the status of calculus reform in a typical institution via in-depth interview-based case studies. After the pilot study survey, three participants were chosen, representative of the main archetypes of the indicated departmental views on calculus reform. Each participant agreed to be interviewed; the interviews were transcribed, and thick descriptions of the context of the interview were undertaken—this is the key to the transferability of this research. The results of the pilot study gave me the detailed image of the mathematics department to be studied, that which Lincoln and Guba called the mélange of descriptors (1985, p. 125). The survey showed that of the 14 faculty members who responded, only one described himself as using solely a reform model for teaching, while 4 self-identified as traditionalist in their teaching model (the majority of respondents to the survey professed a personal model of calculus instruction that was a combination of both reform and traditional elements). The major issue of difficulty with the pilot study was one of access and participation—only 14 of the 20 faculty who taught calculus at Research University were willing to complete the anonymous survey, even when assistance was solicited on multiple occasions. However, the pilot study was nevertheless useful in developing the questions that would be asked to each participant in the qualitative case-study analysis that represented the main focus of this research, and also gave the interviewer insight into how volatile some of the faculty opinions would be. As with most qualitative analysis, no attempt was made to generalize from three faculty members what the national trend is in undergraduate mathematics. The outcomes of this research should be transferable; but it would be dangerous to imply or infer that all discoverable calculus- teaching faculty perceptions and reactions to reform could be addressed or included in any single qualitative historical analysis of a single university, and such a claim would distract from the most valuable contribution of interview-based research—insight into the perceptions of professors of mathematics actively engaged in teaching calculus as they implemented calculus reform in this heretofore untapped area of a research university.
30 The method of inquiry undertaken in the research conducted in this dissertation involved multi-case studies of a qualitative-historical nature, and an emergent design. Lincoln and Guba, who wrote an influential and oft-cited book on qualitative analysis, discuss in great detail the nature of interview-based qualitative research, and the ways in which this research differs from the fundamental truths of classic quantitative research (1985). It is important when doing such research to keep in mind their warnings about just how different naturalistic research is from traditional quantitative positivist models: [The Naturalist researcher] elects to allow the research design to emerge (flow, cascade, unfold) rather than to construct it preordinately (a priori) because it is inconceivable that enough could be known ahead of time about the many multiple realities to devise the design adequately; because what emerges as a function of the interaction between inquirer and phenomenon is largely unpredictable in advance; because the inquirer cannot know sufficiently well the patterns of mutual shaping that are likely to exist; and because the various value systems involved (including the inquirer's own) interact in unpredictable ways to influence the outcome…(p. 41) This is in line with the inductive researcher’s qualitative coding outline forwarded by Glaser and Strauss (1970) and later, Strauss and Corbin (1990)—the style is inductive in that the researcher allows the data to generate meaning rather than attempting to force preconceived assertions onto the data. When Lincoln and Guba talk about the inquirer's own value systems, they make an extremely cogent point. As an interviewer of mathematics instructors immersed to varying degrees in the reform of calculus instruction, I must not cover up my own status as a mathematics instructor both at a four- year college and community college who has taught in the last eight years from both traditional and reform-based calculus texts. However, it is not my views that are the focus of this research; rather, it is the perceptions of tenured faculty at a research university, as the authors themselves point out: [the inquirer] prefers to negotiate meanings and interpretations from the human sources from which the data have chiefly been drawn because it is their constructions of reality that the inquirer seeks to reconstruct; because inquiry
31 outcomes depend on the nature and quality of the interaction between the knower and the known, epitomized in negotiations about the meaning of data…(p. 41) Thus, if this research is to be credible, my interpretations of the data must come from source material in the data itself, and the negotiations about the meaning of the data may be made more transparent if the data is exposed to the reader as well as the interpretations and negotiations thereof. Bias is one of the major limitations of any research project. All information is funneled to some extent through the eyes of the researcher. Thus it was fundamentally important to make sure that the interviews were conducted without leading questions, or any attempt to massage the data to fit the researchers' expectations. One way to create transparency is to let the reader know who the researcher is, and what baggage that researcher brings to the research question. I am a college mathematics professor, and I have spent the last 14 years teaching mathematics at the high school, community college and university levels. I have taught calculus many times, using both reform and traditional texts, and I believe wholeheartedly that the effective teaching and learning of calculus is fundamentally important to a quality undergraduate education in a number of majors. While an undergraduate engineering student, I struggled with some aspects of traditionalist lecture-based calculus instruction, most notably the abhorrence of technology (calculators were often forbidden, and otherwise ignored) and the use of pure mathematical proofs to define such calculus concepts as the limit. The epsilon-delta definition of the limit was a complete mystery to me until I had to teach it myself, so it seems logical that I would embrace the reform movement, wherein for the first time the efficacy of teaching limits in this way was called into question for students ill-prepared to grasp the refinements of this method. Nevertheless, I do not feel passionately aligned with either the reform movement or the traditional camp, preferring to straddle the fence and cull from both philosophies those tenets that can best serve my students (an attitude that was prevalent among the departmental faculty at Research University who responded to the pilot study survey). While I feel that undergraduates would benefit from a greater awareness of reform among faculty (particularly those who still look askance at calculator usage), I believe the single most dominant factor in student preparedness and success in this course sequence is the understanding that calculus demands a different work ethic than that required to succeed in prior levels of math. Calculus, like organic chemistry, represents a huge increase in what is expected of the average
32 freshman or sophomore. In the 14 years that I have taught, I have many times seen the demands of a calculus course overwhelm the academic preparation and maturity of many enrollees. In this respect, at least, I see calculus reform as an attempt to solve a problem that may have no solution—calculus by its very nature may always be destined to have lower retention and completion rates than other, less challenging courses, regardless of how innovative or student- centered the reforms we introduce may be. Still, I have introduced many of the main thrusts of the reform movement into my teaching (including visual and verbal representations, an emphasis on understanding the underlying concepts and not just rote or drill, and student exploration of those concepts as opposed to didactic monologue) with some successes in both student engagement and retention/success rates. Finally, personally teaching from the Hughes-Hallett text Calculus: Single and Multivariable Calculus (1998, 2001), I felt it necessary to supplement student assignments from the text with additional algebra-based mechanics exercises, feeling the text went too far in eliminating practice of integration and differentiation skills so necessary to students continuing on in applied mathematics, engineering, or the natural sciences. When developing this research, an anticipated cause for concern was the tension acknowledged to exist between mathematics educators and mathematicians (Selden, 2002). If participating mathematicians, as the primary deliverers of instruction, see an interviewer as a mathematics education researcher bent on 'exposing' the differences between these two areas, the participants may be inclined to be more careful in their choice of language, and that care may affect the honesty of responses. The participants therefore needed to understand the nature of the research, and they had to be made aware that their responses would not be "judged" or manipulated to pursue an agenda; further, they could be put at ease by knowing that their identities would be kept private through the use of pseudonyms. Initially, a 2003 pilot study in the form of an anonymous paper survey with 9 questions (multiple choice and Likert-scale only) on calculus reform was distributed to faculty at Research University. The survey was informational for the researcher and served as a snapshot of the departmental makeup. The questions from the survey appear at the end of this chapter. A majority of the faculty teaching calculus (70%) responded to the survey, and those that responded were evenly split between self-identifying as reform advocates, traditionalists, and a third group that saw themselves as somewhere in between. The answers to the free response
33 question “Which model for teaching do you use, reform or traditional? Which model do you favor?” were as varied as they were informative. A sampling appears below: 3 2 • I use and favor a mix: reform + traditional. 5 5 • I currently [favor] traditional—at least compared to Hughes-Hallett. • I tried whole-hearted reform first in 1999, have trended away from it. • I’ve done both, but I’m more traditional at the moment—still do group projects. • Mixture of the two—but mainly traditional, and • I have been reforming my calculus courses every year since I began…yet I am probably considered a “traditionalist” rather than a “reformer”. Stupid, isn’t it? (In my view, the calculus reform movement has done nothing but waste a lot of people’s time and effort while making some people rich). An unexpected result of the pilot study was the discovery that all but two of the respondents found the Hughes-Hallett text at least somewhat useful in their approach to calculus instruction. Only one faculty member found the text not very useful and one other (the respondent whose quote is listed last above) found the text to be useless. This was unexpected because the department was already anticipating moving away from using Hughes-Hallett as the departmental text, yet most survey respondents admitted being influenced to at least some extent by the soon-to-be abandoned text. A main result of the pilot study survey was to reformulate the interview questions based on the responses to the survey. This resulted in questions that were more provocative and to the point, as will be seen in the quality of the responses to those questions in Chapters 4-6. After the pilot study surveys were collected and analyzed, attempts were made to contact faculty who represented all three of the groups represented in the survey. Faculty members were chosen to participate based on a number of criteria, including willingness to participate, availability, and whether they had taught from both reform and traditional texts. Purposive sampling is far better than random or representative sampling in this type of interview process because it "increases the scope and range of data exposed…as well as the likelihood that the full array of multiple realities will be uncovered (Lincoln & Guba, 1985, p. 40)." A faculty member who self-identified on his web page as a traditionalist agreed to be interviewed first. Then, a
34 reform advocate (one of the faculty who had first suggested the use of the reform text) agreed, and finally, a faculty member self-identified as non-committal (and identified by one of the other participants as “sitting on the fence”) agreed to be interviewed. The participants were then interviewed in their offices (with interviews audio-taped). The faculty members were asked to talk at length about their perceptions of the calculus reform experiment at Research University, as well as their opinions of the reform movement and its future in higher education. More importantly, they were asked to give insight into the atmosphere that led the department to first undertake a calculus reform experiment, and later to switch back. The interview style was semi-structured: a set of interview questions was prearranged, but interviewees were allowed to pursue each question in whatever direction it took them. Using the terminology of Lincoln and Guba (1985), the interviews were classifiable as depth interviews, in which the interviewer and participant have a peer relationship, or the interviewer is seen as petitioner and participant as sage (asymmetrical trust). Below is a compilation of the interview questions developed to allow the interviewees to give insight into their perceptions of reform: Interview Questions 1. What is Calculus Reform? What are the pros and cons of the movement?
2. To what extent do your teaching practices reflect the calls from the mathematics education community in the past two decades for reform in the teaching of calculus?
3. Describe the major distinctions in a reformed-calculus textbook? Describe your specific impressions of the Hughes Hallett reform calculus textbook used by your department. What are the benefits of this text? What are its drawbacks?
4. Describe the major distinctions in a traditional textbook? Describe your specific impressions of the Stewart traditional textbook used by your department. What are the benefits of this text? What are its drawbacks?
5. Describe your pedagogy when you teach calculus. Would you consider yourself a reformer or a traditionalist? Explain what these terms mean to you.
35
5. How important is the text in your teaching style? How important is lecture? How important are student participation, group work, tools like graphing calculators, and dialogue? Do you use multiple representations of concepts, and if so, for how long have you been using this approach?
7. What does it mean: • to teach calculus? • to learn calculus? • to (truly) know calculus?
8. What changes (if any) do you believe are necessary in calculus instruction?
9. Do you believe that with the advent of technology in the classroom there is still a value to the mechanics of drill, rote, and by-hand calculation?
10. Having returned to a more traditional calculus textbook department-wide, do you feel the department is teaching calculus better than it was 5 years ago? Would you ever go back to a reform text? 11. Which of the following is most important to your teaching calculus? • The right text • The right lecture approach • The right students
The data for the purposes of this research refers to the transcripts of the clinical interviews. The introduction of the pilot study survey data in the analysis of the interviews was useful as a continuing effort to give "thick description", the richness of depth and background that can allow the reader to determine whether these results are transferable to a different environment (Lincoln and Guba, 1985, 2007). Analysis of the data consisted of transcription, coding, and thick description related to the interviews as well as a reporting of the survey results to show the extent to which similar opinions exist within the department. An exhaustive transcription of each of the
36 interviews was completed. Preliminary coding was conducted using an a-priori first-order coding scheme along the outline of Bogdan and Biklen (1992) with codes in 4 major categories defined below:
1. Context/Events (C/E): information that gives the reader insight into the departmental atmosphere in which the interview was taking place, or providing historical insight into the chronology of the change in text.
2. Personal Philosophy (PP): the interviewee sharing information about himself and his teaching philosophy.
3. Awareness (AW): statements showing the interviewee’s conception of calculus reform, and his familiarity with the research that led to the movement.
4. Partisanship (PA): the relationship of the interviewer with either the reform or traditional camp. After organization of the data according to codes like the ones above, a second and more in- depth level of codes emerged from the data, representing subcategories and hybrids of the above, including codes like • PP-Text: The participant discussing his perceptions or opinions of the calculus reform or traditional text • PA-Language: The participant uses language that implies his alignment with one of the camps (calling the other group “they”, for example) • C/E-Department History: The participant describes an event central to the experiment or its termination, and • AW-Reform Tenets: The participant is aware of the fundamental tenets of the reform movement. At times, many of the participant statements were loaded with multiple codes, as in the following excerpt from Professor Violet, the case analyzed in Chapter 4: “…I don’t have any real interaction with the math education people, and I don’t even know their stance on this.” This statement (a response to a question of the participant’s familiarity with the research that led to
37 the movement) was simultaneously coded as PA-language, AW-Lack of interaction, and PA-Us vs. Them. Overlap between the codes was understandable, but an attempt to organize the data in a different and more compelling way led to the construction of a conceptually clustered matrix for each case, comparing key statements from each subject to the research question. A tertiary level of codes was temporal, focusing on the three time periods in the life of the reform experiment: moving towards reform, during the experiment, and the shift back. This structure paralleled that of the three-part research question. Finally, a cross-case analysis of the three informants and their superimposed reactions to the calculus reform experiment was undertaken; this is the focus of the Synthesis section in the Conclusions chapter of this study. Triangulation is a major issue in qualitative research--it is imperative that the researcher be able to show his data as credible and interpretable. This research is triangulated through interviewing multiple sources all of whom personally lived through the experience of calculus reform at Research University. Also, two different collection styles (interview and survey) provide another degree of triangulation (Denzin, 1978, in Lincoln and Guba). Finally, it is the responsibility of the researcher to show, as only richly detailed interview data grounded in the context of this particular instant in time can, that the results of this research will be applicable to an understanding of teacher reactions to reform elsewhere, and that the information analyzed within this research will be consistent with results that will hopefully be gathered in other locations with other respondents in the future. A learning community involving mathematics educators, faculty, and peers (the doctoral student research group) has read this research in an ongoing manner and provided insight, suggestions, and ultimately consensus on the data and conclusions. This form of vetting provides another facet of the triangulation process. Classroom observations were deemed impossible, as it was felt that college math professors would not be enamored of having education researchers observe their teaching from on high (Selden, 2002). One point should be addressed before discussing the interview data in depth. While this historical case-oriented qualitative approach depends on first-person observers (the participants) for insights into all aspects of the calculus reform experiment, past research has shown that self-reported data (for example, that on teaching methods used in the classroom) can be suspect without independent confirmation by observation. Observation
38 of mathematics faculty by a mathematics education graduate student was rendered infeasible, as the focus of the interviews was not the instructors’ teaching methods themselves, but rather, their reaction to the reform experiment undertaken by the department. The instructors were open and willing to participate when the focus was on the reform experience or their post-experiment opinions of reform; it was made clear that a focus on what the faculty were or were not doing in the classroom would not be as willingly tolerated. However, contrary to the findings of researchers whose interviewees “tell them what they want to hear”, the openness of the responses (and the rather shocking nature of some of them) should reassure the reader that the participants were uninterested in impressing the interviewer with their knowledge of reform, or application thereof in the classroom. Instead, the petitioner-sage relationship of asymmetrical trust allowed the interviews to expose actual teaching methods employed—even when they were unapologetically anti- reform. Since the interviewer was actively pursuing a doctoral degree in education, a sure sign that the participants’ responses were not to be trusted would be if they tried to portray themselves as more attuned to this educational reform than they actually were. As can be seen in the analysis of all three case studies, this was not the case. In retrospect, even the most ardently self-portrayed reform advocate interviewed did little to show an embracement of reform instructional practices. Since even the most ardent reform advocate admitted to very little in the way of reform-based methodology or curricular adjustment, and in fact all admitted to using the reform text at the center of this experiment for little more than assigning homework, abandoning or never attempting to do group work, and even expressing disdain for technology in the hands of students, the triangulation and confirmation of the validity of their self-identification comes from one another, and from their own words. It is hoped that the results of this research can be seen as an insight into the minds of faculty teaching calculus in this era of calculus reform, and that this insight will be edifying to the mathematics education community and to the mathematics departments invested in calculus reform. Reliability and Validity It is important to address a few key points about the conclusions that are drawn from this research. The reliability of the research depends on the synchronous nature of
39 the interviews, all of which were conducted within the same time period, using the same questions and interviewer, and which were the results of purposive sampling (Kirk and Miller, 1986). Miles and Huberman (1994) caution that a danger to reliability could exist if all of the informants gave “monolithic party-line” answers—this was definitely not the case in each of the three cases of this research. Informants regularly varied from the stereo-typed scripts of reform advocacy or traditionalism, to the extent that at times even their own self-identifications became curiously suspect. Internal validity, also described as credibility, depends on the ability of the data to “ring true” to the reader, enabling the reader to feel the “vicarious presence” of the informants by reading their interview data (Miles & Huberman, 1994, p. 279). Also, triangulation between the three distinct sources led to convergent conclusions which represent another measure of validity, as will be seen in Chapter 7,. Finally, the data descriptions and excerpts are context-rich and thick in nature (Denzin, 1989). The result of these elements is a research project that is internally reliable, or credible. Finally, the external validity of this research depends on its congruence with the available literature. All of the opinions and perceptions of reform expressed in the interviews echoed opinions expressed on one side or the other of the reform debate (sometimes, both sides at the same time) The findings were further consistent with the researcher’s own experience, and most importantly, the outcomes and conclusions are general enough to be applicable anywhere that mathematics departments institute reform without a vested interest in proving that reform successful. Finally, the narrative elements of the data are exposed and a cross-case theory emerged from the data (Miles & Huberman, 1994). Because this research focused not on the advocacy of any ideological position but rather on a historical event and its consequences, it is more valid than any paper trying to either refute or support a position by creating complementary research. This research is not intended to answer the unanswerable (divergent) problem of reform or traditional calculus instruction’s superiority, but rather to expose the ground-level reactions of college professors to a reform they themselves undertook and later abandoned. The best proof of the transferability of this research would be the similar but remote analysis of some other department’s reform experiment—a quantitative analysis of members of a
40 department that had more or even less success with reform. How would the attitudes or perceptions of those faculty members differ from those analyzed herein; how would the curriculum of a successful reform experiment differ from the curricular changes accomplished at Research University? Only with complimentary qualitative analyses undertaken elsewhere can the prevalence of these perceptions nationwide be judged—it is clear from the data discussed in chapters 4 through 6 how prevalent such perceptions are at Research University.
41 CHAPTER 4
CASE STUDY A: PROFESSOR VIOLET’S INTERVIEW
During the 2003-2004 school year, the Mathematics department at Research University (all university and participant names have been changed to comply with Human Subjects interview requirements) re-evaluated the departmental text used for all calculus courses. After having used a reform calculus text, Hughes-Hallett’s Single and Multivariable Calculus (Hughes-Hallett et. al., 1998, 2001), through the second and third editions, the department chairman and book committee, with input from the faculty who taught the course, returned exclusively in the 2004-2005 school year to a more traditional Calculus text, Stewart’s Calculus (2002). While math departments are constantly going through such book adoption processes, Professor Les Aspinwall and I felt that there were a number of cogent research questions that this particular calculus reform experiment raised. We believed that a further investigation of the conditions under which the change occurred would provide insight into what reform elements calculus teachers at a research university find important. During the prospectus defense, my committee agreed that a qualitative case study involving interviews of members of the department might shed light on the historical event of a department in transition from a reform approach to calculus instruction towards a more traditional focus, and the causes and effects of this calculus reform experiment at this university. A strong majority (between 80 and 95%) of the calculus lecture courses taught at the research university level are taught by “professorial faculty” (Maxwell & Loftsgaarden, 1997, p. 216) with degrees in mathematics, whereas TAs are normally the purveyors of recitation sections. While no statistical analysis exists to support the following speculation, it is unlikely based on the requirements of a mathematics graduate degree that a majority of that faculty teaching calculus have backgrounds in such mathematics education topics as curriculum or learning theory. When I arrived at Research University as a graduate student in Applied Mathematics in Fall term, 2000, I was actually surprised to find that the mathematics department (that, as an undergraduate ten years ago, I had thought archetypical of
42 traditional math instruction) was using as its standard calculus text a reform text. During my first year teaching in the department, I heard many faculty members voice complaints about the text itself, and the need for supplements that these faculty members felt the text’s omissions demanded. I taught Calculus I and Calculus II out of this text during my time in the mathematics department at Research University, but as I was graduating in 2003, the department was reassessing the value of the text in the preparation of advanced pure and applied mathematics students, as well as engineering and physics students. I was aware that some of the teachers I knew who were teaching out of this reform text were, by their own admission, not reform-minded, and I wondered whether the clash between teaching a “traditional” calculus class with a reform textbook was as difficult as it seemed. I was also interested in what the instructors who had used that text had taken away from its usage—were the teachers who used the text changed by the experience? I decided, and my committee agreed, that I should attempt to collect some rich qualitative interview data with willing subjects in the department. The interview coded in this chapter was done with a tenured faculty member who felt passionately about the teaching of mathematics and also about the change of texts and the reform movement. I also interviewed an advocate of the reform text, Professor Red (Chapter 6) and another faculty member who professed himself to be in the middle, neither a committed reform advocate nor a traditionalist, Professor Green (Chapter 5). The self-identification of each informant led to the naming of each as a color represented in relative position on the color spectrum. Calculus reform has created less of a continuum of opinions and advocacies than it has a modular set of camps, metaphorically assigned spectral colors, where the red and violet are extreme or polar and green is somewhere in the middle. From the analysis of this interview data, I have attempted to answer the following related research questions (restated from Chapter 3): • What was the effect of introducing a reform experiment into the undergraduate calculus course at a research university? • What was the nature of the reform experiment?
43 • What were the circumstances under which the faculty chose to abandon the reform experiment and return to a more traditional approach to teaching calculus? The data collected in the interviews gave insight into the reaction of a mathematics department to the use of a reform-focused text, as well as the reasons for their subsequent return to a more traditional text. Biographical Sketch—Professor Violet Professor Violet is a tenured professor of mathematics at Research University. He was also at one time an administrator within the department. He is a passionate lecturer and an oft-published researcher. He was the most adamantly opposed of the three participants to the reform movement—I actually became aware of his feelings about the reform movement from reading an intensely negative manifesto he had written about the reform movement and had posted on his personal webpage while the department was using a reform text. All of the major statements first espoused in this online manifesto were reflected and expanded in the interview process. Professor Violet appeared at ease throughout the interview, but at times both his passion for teaching and his disdain for the reform text and the reform movement that spawned it were evident in his voice and responses. He was not averse to ardent and adamant criticism of both the motives behind the reform movement and the implementation of it in the Hughes-Hallett text in use by the math department. I approached him about an interview only after a number of faculty suggested that he would give great insight into the book change and the motivations behind it, since he was involved in the decision-making process, and he immediately agreed to be interviewed on these topics. With the assistance of my major professor, I created a list of questions that I thought would get to the heart of the matter and speak to each of the research questions in turn. As it turned out, Professor Violet was extremely willing to open up at length about teaching calculus and the calculus reform movement. The questions asked appear in Chapter 3. The interview took place in the instructor’s office. The interview was audio- taped and transcribed at a later date by the interviewer. In the next section, excerpts from Professor Violet’s interview are discussed and interpreted in detail, giving thick
44 description (Lincoln & Guba, 1985) of (and insight into) this moment in a department’s history. Assertions and Excerpts—Professor Violet An analysis of the interview begins by focusing on assertions about Professor Violet and his beliefs about the reform movement and the reform text, followed by confirmations of these assertions reflected in excerpts from the data—the subject’s own words. While normally, even in qualitative analyses, interview excerpts are truncated greatly so that emphasis can be placed on the researcher’s interpretations, many of the excerpts contained in Chapters 4-6 are what some might consider long. The main reasoning behind this inclusion of longer quotes is that this is historical research, and as such, part of the concept of thick description and addressing validity is to allow those who witnessed this departmental transformation and return to explain in their own words what the experience meant to them. Movement Towards Reform As an openly hostile reform opponent, Professor Violet might give insight into why some within the mathematics community are so resistant to reform. Within the debate literature discussed in the literature review, this opposition was characterized by a kind of partisan disregard for the research that led to the reform movement and a suspicion of the efficacy of this reform in its translation from ideological conception to curriculum reality. Advocates of reform discussed in the literature review also implied that traditionalists (as Professor Violet and reformers would both agree he represents and archetypical example) would be ignorant of the reform movement and its theory. My primary assertions about Professor Violet based on his interview data were that he represented an example of the passionate traditionalist, even someone who was openly opposed to the reform movement, but that he was more knowledgeable of the movement, and more analytical of his own teaching, than the stereotype of the traditionalist from the literature would appear. Also, that as a self-identified opponent of reform, he objected to this movement because of cynicism about its foundations. A tertiary assertion was that he would be unlikely to extensively use any text in his teaching, preferring to use his lectures as the primary source and seeing the text as a repository for homework problems.
45 Professor Violet was asked during his interview to give insight into the decision- making process that led to the department-wide usage of a single reform-focused text for the teaching of calculus. He complied by giving a step-by step history of the formation of a committee that was intended to represent the different groups perceived to exist within the department. In the following excerpt, he described in his own view the makeup of the committee: My impression from talking to [Professor Green]—I was not on the committee—but my impression is that there were three people on the committee that were diehard reformers, in fact I know them to be that because I know these people: Professor Red, Professor D, and E, and there were…2 people who were open to pretty much anything—Professor K and G, and then there was one person, Professor M, who was a diehard traditionalist, and then Professor Green felt like he had to be sort of the mediator. Note that at this relatively early stage of the interview, Professor Violet had no problem using labels to describe the partisan nature of alignment within the department: diehard reformer and diehard traditionalist are terms used to describe calculus teachers within the department who are on the book committee that ultimately made the decision to switch to a reform text. It is interesting to note that the reform text had already been used in the department by roughly a third of those teaching the course, and that the main focus of the meeting was actually just to find one text everyone in the department could agree on. Prior to this meeting, an individual instructor of calculus could use whichever text he chose, which made it difficult for students taking the sequence of Calculus I through Calculus III, as they could conceivably be forced to use a different text each term. Professor Violet described in the next passage why the calculus reform experiment almost never happened: … and from day one the three reformers said that Hughes-Hallett was the only book and they stood by that and the final vote was 4 to 3 to keep Stewart and not use Hughes-Hallett . That was to be turned in to the chair, and the day they were going to turn it in was the day before 2nd edition of
46 Hughes-Hallett came out and was sent to some people in the department, including some on the committee, and so the reform people asked that the committee hold off on its report and consider the 2nd edition, because it had addressed some of the concerns that had been raised against Hughes- Hallett by others on the committee. And so they brought in the book and said “look, we now have integration by parts, there’s more given to this, this is better written, blah, blah, blah. And the result of that was that 2 of the people who had voted against using Hughes-Hallett and for using Stewart changed their votes and so in 1999… we adopted Hughes-Hallett. This excerpt not only shows Professor Violet’s disdain for the arguments of the reform advocates in his use of the dismissive “blah, blah, blah”, but it also show objections reflected in the literature about the degree to which the first edition of Hughes-Hallett’s Calculus (1994) was rejected by mathematics professors teaching calculus. The first edition of the reform text had eliminated much of the analytic and mechanics rigor of traditional calculus texts (including the integral concept of integration by parts), and the book committee’s refusal to adopt that edition may reflect an attitude so prevalent in the mathematics community that the Harvard Consortium’s second edition was forced to reintroduce many of these analytical techniques (Klein & Rosen, 1996). Further, it is interesting to observe from Professor Violet’s recollection that even at the time that the committee ultimately mandated a change to Hughes-Hallett et al., there was already a well-defined group who resisted such changes: [Former administrative official Professor W], who was in my position when all this took place, had pushed the issue of choosing one calculus book (because we had been using two). We had allowed the reformers to use Hughes-Hallett, but it created real problems, though, when students went from one course to the other, form Calc I to Calc II or II to III (particularly from Calc I to II) and he was, I think he was very tired of dealing with those problems and just wanted to get this one book… From this candid observation from a (former) administrator in the mathematics department at Research University, it is clear that the department, even while making a change intended to be for the greater good of their students and faculty, did not feel
47 overwhelmingly supportive of the text chosen—that in fact, that text was a compromise that only a minority contingent of the faculty teaching calculus strongly felt was a superior text. This is important to note—unlike the research from the literature review, in which departments showed a readiness to implement reform (c.f.a. Matney, Hurtado, & Ziskin, 1999), the faculty at Research University perceived such an experiment with much trepidation and more than a little protestation. This is not just Professor Violet’s personal recollection of what happened—while involved in the interview, he got onto his office computer and pulled up the voting record for the book committee at the time, and gave statistics to support his observations about the departmental makeup by philosophy. He eventually found the voting tally not only for the first book committee, but for the later return to a more traditional text as well, wherein he found overwhelming evidence of the voters’ rejection of the reform calculus text. Personal Teaching Philosophy and View of Reform In just the excerpts above, Professor Violet gave great insight into his personal opinions of calculus reform as a movement, as well as his frustration with the members of his department he called “reformers”. At more than one time during the interview, he challenged the assertions that led to the movement as well as the philosophy underpinning the reforms themselves: Well, I think this is such a silly…I’m very cynical on this, I believe the NSF came out in the mid 90s with a ton of money for education reform, and I believe some of these smart people saw that that was a good way to get funding and so that’s why we have...that’s the real reason we have calculus reform. That’s a cynical statement but…and that’s a statement I have no evidence for [Laughs]. At a later point in the interview he tackled a topic central to the criticisms in the literature of the reform ethos (see chapter 2 for a summary of articles related to this topic); the idea that drill and rote are less valuable than a conceptual understanding of the fundamental tenets of calculus: I don’t know any good calculus teacher—someone who is traditionally considered good—who first of all did not stress geometric reasoning, did not stress concepts, but at the same time, did not stress that you learn
48 through calculations, and this is the thing that these—that the calculus reformers sold without any real basis for selling it. They said that (you know) a mindless monkey can do calculations now because you’ve got computers so there’s no need to do them; let’s stress concepts. Professor Violet’s usage of descriptors like “they” bolsters the belief that the reform movement and traditionalists are truly partisan ideological camps. However, despite the fact that he was at the time vehemently disagreeing with one facet of the reform movement, Professor Violet was simultaneously admitting that other foundational elements of reform (stressing geometric reasoning and conceptual understanding) were important to quality instruction. The implication that the reform text ignored mechanics and calculations was probably more a reaction to the first edition of the Hughes-Hallett text—never used by the department as a group, this text edition was so different from the traditional calculus text that it completely abandoned such fundamental topics as L’Hospital’s rule, integration by parts, and partial fraction decomposition (Klein & Rosen, 1996). Many of the passionately held opinions critical of this reform text and reform in general are possibly a reaction to this first edition’s omissions. Each of the second and third editions brought back more of the traditional calculation elements so prevalent in calculus texts before reform (Hughes-Hallett, 1998). Professor Violet continued to decry what he saw as the abandonment of the way he learned calculus: doing calculations and then as a result of the repeated mechanical processes, coming to some kind of conceptual understanding about why the processes worked they way they did—and what the big picture means. My claim is that’s not how people learn mathematics. And see, this is probably a testable hypothesis, but I believe my claims until someone disproves it, but they just decided that my claim has no merit. My claim is that 99% of the people in the world through all levels of math, most of the levels of undergraduate math in particular, and certainly all the levels of high school math, algebra and trigonometry and so on, learn mathematics by learning how to do calculations and becoming comfortable with them. This concept, that mechanics are far more fundamentally important than the “drill and kill” stereotype deemphasized by the reformers, is not exclusive to traditionalists like
49 Professor Violet. Even Jerome Bruner, the godfather of many of the constructivist ideas championed by the reform movement, noted that “adults typically require a certain amount of motoric skill and practice before they are able to develop an image representing their actions” (Driscoll, 2000, p. 226) This is an issue fundamental to the differences between those in the reform camp and those advocating traditionalist instruction: surely there are many faculty teaching calculus who advocate drill too much, as the reformers point out, but for some learners at least, some mechanical skill might be a necessary precursor to a deeper understanding, as Bruner noted. From the passages excerpted above, one could glean that Professor Violet was suspicious of the motivations for calculus reform, questioned the experimental justification of much of the reforms as enacted, and doubted the benefits this could have for his students. What was also clear, however, is that Professor Violet. was constantly striving to be a better calculus instructor himself, even at points agreeing with elements of calculus reform where they mesh with his own personal beliefs. The following excerpt is fundamental to an understanding the seemingly contradictory relationship between Professor Violet’s opinion of reform and the manner of his own instruction, which appears to contain elements of reform ideology: I introduce the topic, I give a big geometric picture, almost always, I talk about what it means…I try to make the connection to the real world, and then I do a lot of simple calculations, I go to the homework, and work a lot of homework problems, and I tell them, you know, “You’ve got to master these calculations....” And I’ve always done that, for 20 years I’ve done that, I wouldn’t change the way I’ve taught calculus based on all the stuff I’ve heard because I’ve always incorporated all the stuff I’ve heard that I thought made sense. Professor Violet’s acknowledgement within this quotation of two of the elements in the aforementioned reform text doctrine—what Hughes-Hallett et al. termed the rule of four (geometric understanding and real world applications) is perhaps a response to the forward of the Hughes-Hallett text, where these strategies are deemed part of the new method of teaching calculus (2001).
50 It is important to point out that part of the frustration with reform Professor Violet makes no attempt to hide is the feeling that he had been reforming and evolving his curriculum his entire career, including incorporating what he saw as universal truths co- opted by the reformers as part of their packaging of the movement. Still, the fact that even as he was openly critical about the movement’s motives, he was confirming the value of some of their “innovations” points out just how difficult it is to restrictively define someone (or even to allow them to so define themselves) as either traditionalist or reformer. Professor Violet sees his text as a way to supplement his lectures with concept- reinforcing homework problems, rather than as a practical working aid to help teach the material. It is quite possible that if this dismissive attitude towards any text was found to be a common theme for faculty at research universities, this attitude could shed light on how ineffective a change of text can be without the fundamental shift in philosophy that would seemingly have to accompany it. Professor Violet essentially admitted on multiple occasions within the interview that he tried to do things differently from the text, as he felt he had arrived at a better understanding of what his students needed: My claim is that students get a lot out of lecture because I don’t do the stuff in the book. I don’t do the examples in the book, I don’t necessarily follow what the book’s done in terms of the way they give a geometric picture of things, because I have ways that I think are good. If, as all three of the interviewed participants claimed, the book has little effect on the style of their instruction, and if this trend could be seen to exist for research universities elsewhere, it would demand from those intent on reform at the college level a different tack than just creating a reform text—skeptical college professors would seem to require more convincing that the changes the reformers desire are worthwhile and well researched. The move back Finally, Professor Violet addresses the remaining research question concerned with the return to a traditional (Stewart) text and departmental focus after five years. He again cites faculty concern over the success of students taught with Hughes-Hallett et al. in more rigorous follow-on coursework (although it appears this was only anecdotal):
51 I believe it was in the fall of 2001, [Professor] X sent a letter to the faculty saying we had used that for two years and he felt like it was just a disaster- he was now getting the people who had gone through Hughes-Hallett and he was making these claims...that these people did not know what they needed to know. They didn’t know how to do simple derivatives, they did not know how to do simple anti-derivatives and he said I would like to raise this issue with the faculty, and let’s talk about this. And so what I did (we didn’t have a committee) I just sent out a note to the faculty—to everyone who had taught calculus, to please give input on what they felt we should do. Earlier, when describing how the 1999 committee had approved use of the reform text, Professor Violet. had expressed that some faculty were making similar claims or voicing some concern about the preparation students received from those using the reform text in calculus. But by late 2001, these voices became loud enough that the department felt the shift worth revisiting. However, department-wide change is generally a slow process, and it was 2003 before a straw poll confirmed that a majority of those teaching calculus in the department were dissatisfied with the reform text: Look, the reason this is happening is because I’ve had—at that time I had had about 8 or 9 people write me saying get rid of this book, and so I said the question before us is do we keep Hughes-Hallett or not, and if we do we do. That’s all we have to do, and if we don’t, then we will choose another book and most people accepted that; a few didn’t. 20 out of the 30 faculty members polled wanted to get rid of the reform text. The result was a lengthy book search that resulted in the usage of Stewart’s Calculus: Early Transcendentals (2002) by 2005. This text is considered a much more traditional text in contrast to Hughes-Hallett’s text (Professor Violet addresses this below) (Wilson, 1997). But while Professor Violet never shied away from portraying this department-wide calculus reform experiment as completely over, he does acknowledge that as a result of the movement even the Stewart text has changed, moving more towards NCTM standards of inclusion of calculator technology:
52 I would say [Stewart and other traditional texts] were traditional texts that have moved toward the reform...they had incorporated a lot of stuff in them that reformers like. For instance, Stewart (the current Stewart) has a lot of graphing calculator problems, and he has a little calculator or something beside them so you can tell those apart. I don’t use any of those, but… It is fascinating that even in acknowledging the blurring of the lines between reform and traditional texts, Professor Violet still admits that he will not be taking advantage of these incorporations: I don’t care about the graphing calculator, period. I don’t care about it. I don’t use it; I don’t use it for anything I do. If they want one and they know how to use it, that’s fine. I don’t care; if they don’t know and if they don’t want to get one, that’s fine. They won’t need it; for anything I do in my class, they won’t need the graphing calculator. I’m teaching calculus, I’m teaching concepts—I’m the one teaching concepts, okay [laughs]? Professor Violet was wonderfully frank and unapologetic about his disdain for calculus reform and his dislike of the Hughes-Hallett et al. text. While one might gather from his disdain for the text and his preference of his own lectures to any present text, he did give insight into his interview about the type of text he would be excited about using: I like to use surface theory as an example. I mean, Felix Klein and a guy named Fricke wrote a 2000 page book in the 1890s on surfaces, 2000 pages, no index, in German [laughs] and people have been writing papers on...surface theory is a big thing today, and people have been writing papers on things, and sometimes they’ll write it, and someone else will look at it and say “Yeah, you know, Fricke and Klein did some of that, and you should go and look at what they did.” And you go and look at Fricke and Klein, and Fricke and Klein were...they were walking around on these surfaces. Professor Violet shows in this excerpt that the things that impress him about a text are not the pretty pictures and annotations of the modern high school or college math text, but rather a book that shows a concrete rather than abstract understanding of the subject
53 matter—even if the text is 2000 pages long. Truly, this is a traditionalist view of the quality text. However, Professor Violet did appear to have invested quite a lot of thought and energy into his own well-respected and award-winning teaching philosophy, and he implied that should quantitative experimentation prove the reformers right, he would be willing to change. Critics of reform are often seen as close-minded and unwilling to accept even warranted change (Kaput, 1997), but Professor Violet appeared to have expended much time and energy to arrive at his personal teaching philosophy. It is interesting to note that his reluctance to use any textbook as other than a collection of homework assignments posits a fascinating topic for further research: how prevalent is the view among college faculty that the text is simply a homework bank? In other words, if some qualitative analyses of instructor dependence on text echoes Professor Violet’s opinions, and shows them to be less than unique, perhaps reform must be approached from some other perspective than primarily a textual revision. Analysis of Coding of Interview—Professor Violet Professor Violet spoke to all the major research questions of this dissertation at length; the summary of his positions on those questions is given below, with titles taken from emerging preliminary coding organizations that, while self-explanatory, are discussed in detail: • Moving Towards Reform (What was the nature of the reform experiment?) Even when the department was moving towards reform, many (Professor Violet included) were opposed to some extent, feeling the championed reform text ignored important fundamental teaching practices (rote and drill, as well as concepts the reform text’s authors had downplayed as less important to modern students with the advent of calculator technology) in favor of a less easily quantifiable conceptual understanding (what he called “putting the cart before the horse”). • During the Experiment (What were the historical conditions of the reform experiment?) Professor Violet admits that within two years of the implementation of the new text, efforts were under way to abandon the usage of a reform text for (anecdotal)
54 concerns over student preparation and success in higher mathematics post-requisites of calculus. • The Shift Back (What were the conditions under which the department discontinued the reform experiment?) Professor Violet took the lead in the surveying of the department, the analysis of the results (more than 66% of the faculty voting to reject the reform text and return to something more traditional) while acknowledging that calculus reform has influenced even those traditional texts, although he did not embrace those moves toward a common ground wholeheartedly. While generalizability is not necessarily a focus of qualitative research, continuity is. During the interview, Professor Violet himself named 2 persons directly involved in the same decision-making process described in his interview: the head of the initial book committee, Professor Green, and a “true believer” (Professor Violet’s words) in calculus reform, Professor Red. It was clear to me that for a more nuanced picture of this historical event, I would have to seek out these other central figures and interview them, with the hopes that they could be as insightful and candid as Professor Violet himself had been. The general codes used above to categorize responses in a historical timeline are worthy of contextual explanation. Moving towards reform categorizes events leading up to the department-wide switch to a reform text (1994-1999), while During the Experiment describes events related to the department wide usage of the Hughes-Hallett calculus text (1999-2003), and The Shift Back covers the period of return to a traditional text department-wide (2004-Present). These general codes also parallel the original research questions. Within each of these temporal coding categories, secondary codes included those related to self-identification as traditionalist or reformer, which were labeled partisan codes, and codes related to teaching style. Perhaps most interesting were the secondary codes that emerged from Professor Violet’s interview—what Lincoln and Guba called bridging codes (1985). At heart they are partisanship codes related to Professor Violet’s suspicion of the amount of research that went into the reform movement (Partisan-Lack of Research) and his rejection of what he saw as the reformers’ de-emphasis of drill (Partisan-Mechanics).
55 Overall, the most frequently recurrent code for Professor Violet’s interview was Partisan-Language. 19 times during his interview, Professor Violet used language that cast the relationship between traditional and reform elements of calculus instruction as an “Us versus Them” battle, while self-identifying as a firm traditionalist with no small amount of disdain for the reform movement. The second-most frequent code (with 8 mentions) was Partisan-Mechanics. Professor Violet continually returned to the claim that there is not enough respect in the reform movement given to the mechanics of drill as a method of strengthening student skills and conceptual understanding. Finally, I return to the primary assertions of the interview with Professor Violet— that he represented a partisan example of a traditionalist, and that he was opposed strongly to reform The assertion is based on Professor Violet’s language, his open and repeated hostility towards what he viewed as the reform movement’s attempts to tell him how to teach, his suspicion of the grant-based research that supports the movement, and most convincingly, his distaste for the text that he sees as representative of reform. Professor Violet consistently equated in his interview the calculus reform movement with the Hughes-Hallett calculus reform text. Both, he stated, show little respect for the beauty of calculus as he feels the traditional approach captures it. His hostility in part appears to have derived from the reformers’ assumption (or his interpretation thereof) that traditionalists do not constantly attempt to improve their teaching. Professor Violet’s interview showed that he felt the reformers had pigeon-holed traditionalists as all alike; however, he did not feel that these reformers deserved the same acknowledgement of diversity from him. The second assertion, that Professor Violet showed a familiarity with the reform movement, was immediately evident from the data. Professor Violet showed in his interview an awareness of the reform movement uncommon in the stereotyping of traditionalists normally associated with the reform debate literature mentioned in the literature review. He knew of the rule of four, and at some time during his interview referred to each of those four foci. He also knew about the NSF grants that had sponsored the Harvard Calculus reform. However, he admitted that he had not read a large amount of the research that spawned the reform movement. Most surprisingly, the data revealed that while Professor Violet was opposed strongly to the reform of calculus as undertaken
56 by the Harvard consortium, he admitted to thinking constantly about the way he taught and revising his lecture style as he felt this was needed. In other words, it was surprising to find that Professor Violet was not opposed to reform in general—but rather, this one specific reform movement that he felt ignored much of his own experience as a college math professor. All of these assertions will be discussed further and their relation to the data amplified in the Conclusions section of this research. I have attempted to distill the highlights of the interview into a contextual matrix. I feel that part of what Lincoln and Guba describe as thick description is letting the participants’ words speak for themselves, un-distilled and full force. I will include my further perceptions and interpretations of the data (the big picture, taking in all three cases) in chapter 6—for now, it was enough to let Professor Violet speak for himself.
57 Table 1. Contextually Clustered Matrix—Interview with Professor Violet SELF-REPORTED AWARENESS REACTION TO RETURN TO PARTISANSHIP TEACHING OF REFORM HUGHES- STEWART PHILOSOPHY MOVEMENT HALLETT (TRADITIONAL) TEXT TEXT
Moving I don’t know a good You won’t find They said that a Dr, X sent a letter to We have some true Towards teacher who doesn’t anyone out there mindless monkey the faculty saying he believers here… Reform stress geometric who will embrace can do had used this book reasoning or stress the whole gamut of calculations—let’s for two years…and concepts. calculus reform. stress concepts he felt like it was just a disaster.
During the I don’t do the Stress the ideas I give a big I said the decision is I claim that calculus Experiment examples in the behind calculus, geometric picture, I between do we keep reform gets it book, I don’t stress geometric talk about what it teaching reform or backwards—they necessarily follow reasoning, stress means, I try to make do we all find a book say let’s not worry what the book’s numerical a connection to the that allows people to about done. approximation, real world, and then do what they want. calculations…but stress what they call I do lots of simple they just decided my real word problems. calculations claim has no merit. The Shift Back I think it’s right to Stewart has a lot of I cannot teach the I had 8 or 9 people I wouldn’t change criticize the way a graphing calculator kind of course I writing to me saying the way I’ve taught lot of people have problems, and he want out of this get rid of this calculus based on taught calculus. has a little book. book…My claim is all the stuff I’ve calculator or that that’s not how heard because I’ve something besides people learn always incorporated them…I don’t use mathematics. all the stuff that any of those. made sense.
58
CHAPTER 5
CASE STUDY B: PROFESSOR GREEN’S INTERVIEW
Biographical Sketch—Professor Green Professor Green is a tenured professor of mathematics at Research University, where he was also a college administrator. He has received 5 different teaching awards during his 9-year tenure at the university, and was one of the first faculty members to approach me (in 2001) about the Hughes-Hallett et al. reform text we were both using to teach Calculus at the time. Professor Green was in my experience a gentle, thoughtful, soft-spoken man, and would represent a different point of view from that of Professor Violet, if not an entirely different self-alignment with regards to the reform movement. I had planned on interviewing him as a part of this research even before Professor Violet named him as a principle figure in the move towards a reformed text. However, Professor Violet’s words led me to view my interview with great anticipation, as this excerpt makes clear: Well, that committee was rather an interesting committee [chuckles]. [Professor] Green chaired it, you might want to talk to him if he’s willing—I don’t know if he would be willing to… My anticipation was that Professor Green., based on Professor Violet’s description, would represent someone on the fence about Calculus reform—maybe not a “diehard traditionalist” as Professor Violet labeled himself, but not a “true believer” in reform either. I assumed, based on his reputation as an award-winning instructor, that he would be open to reforms intended to benefit his students and more willing to embrace reform in general than his traditionalist colleague. Assertions/Excerpts From Interview—Professor Green My primary assertions from the interview with Professor Green were that Professor Green represented a less-hostile, or at the very least less-partisan reflector on the reform experiment at Research University. Also, unlike Professor Violet, I noted from his interview that as someone generally identified by his colleagues as on-the-fence about
59 reform, he was more inclined to see both the benefits and drawbacks of the curriculum change represented by the use of a reform text. I had forecasted about Professor Green was that he would more willingly have integrated the text philosophy of combining group work, projects, and other curricular innovations into his teaching. But the data from his interview led to the assertion that while Professor Green experimented with more facets of the reform movement than just the text change, he did not feel that such reform elements as group projects and discovery exercises were feasible in the course as presently constructed. Finally, the data from Professor Green’s interview revealed him as more willing than his earlier colleague to acknowledge a positive outcome from the reform experiment at Research University—that is, the residual influence of the reform movement on the department post-experiment. Personal View of Reform Movement Professor Green was incredibly at ease throughout the interview, never displaying the obvious frustration Professor Violet had shown through his tone and volume about the reform text or elements of the reform ideology. However, Professor Green was just as unsure as his colleague about the motivations for the reform movement, and he did not avoid expressing that suspicion, as the following excerpt bears out: I believe that there are some legitimate issues, or there were some legitimate issues about calculus and the way it was being taught that were raised, and maybe had been raised over the years many times. I think that some of those folks made suggestions on the way to change the delivery of the material, but I also think that there was a movement that took over. The movement was driven largely by funding, to try and search for something larger than what it had initially started as, what the calculus reform movement started as, and that it perhaps grew beyond where the founders, if there were founders, from where they began. The feeling that calculus reform was driven by infusions of grant monies from the NSF is inseparable in both case studies of Professor Violet and Green from an implicit subtext that this somehow corrupted the research that resulted. What was different about Professor Green’s approach was that he actually acknowledged that the issues behind the movement were legitimate. However, while Professor Green tackled the corruption of
60 funding issue more gently than Professor Violet, he was no less adamant about its importance: That calculus reform project…became a bit tainted, as all research does in our system, by funding trends. Funding of a certain research area or a certain scholarly activity tends to draw people flowing in who want some of that funding. Personally, I think that the reform calculus movement was tainted a bit by that. The people who started it all, I think as I said at the very beginning, there were people who genuinely felt that there were some danger spots in the way that calculus appeared to be taught by most people teaching it. I think they may have been wrong on that assumption. This statement appeared to contradict the early assertion of legitimacy about the beginning of the movement. However, it is possible that Professor Green could have simultaneously believed that concerns about calculus failure rates could be legitimate while still feeling that the assumption of teacher responsibility could be spurious. Professor Green admitted in the interview that he had not studied the “goals of the leading reformers carefully, but [he had] heard things. There is much to be gleaned from his choice of words, as in quote below, when he was asked how his teaching might reflect the calls for reform: Well, I embrace some of what they preach. They talked about this rule of three. I mean, I think--in a way--that was kind of a fad that they put together, because that was a convenient way to try and convey in a package what they were trying to do. The use of the word “preach” parallels the religiously-themed comment from Professor Violet that the calculus reformers are “true believers”. Both of these participants appear to be consciously or unconsciously echoing the criticisms of reform from the Chronicle of Higher Education and other journals labeling the reform movement a disaster in the debate literature outlined in the literature review (Wilson, 1997). Calling the rule of three a “fad” is a less-subtle jab at the authors of the reform text used throughout the experiment at Research University, as is this comment about the structure of the text itself:
61 I did not like the text. It was not a book that I felt was particularly effective for what I wanted to do in the classroom. I thought the problem sets were for the most part unsatisfactory. There were good problems here and there, but there were not enough problems that I would consider training problems. There were not enough problems that showed off the power of calculus, which I think is important. The book seemed to me to suffer from being written by committee. There were lots of authors on it, and I felt that there were inconsistencies within the text because of that, from one section to the next. So I struggled with that. There were chapters, sections, where I ended up rewriting, giving out all of my own exercises, rewriting completely the presentation of the material because I just wasn’t pleased with what I saw. I didn’t want to follow the book. The last statement in this quote is extremely telling; unlike Professor Violet, who disregards the text altogether, in favor of his own presentations, Professor Green expressed disappointment that the text was not one that he could use. In a way, this is more revealing than Professor Violet’s comment, since it appears from his words that Professor Green would have felt comfortable lecturing in parallel to the text if he had felt better about the contents. His criticism was one echoed by many faculty members about the first edition of Hughes-Hallett et al. (and to a lesser extent, the second and third edition)—the assertion that the homework sets were not comprehensive enough, and demanded supplements (Klein & Rosen, 1997). This represents a common traditionalist criticism of a seeming disconnect between reform theory and practice, one that cuts to the heart of curriculum structuring. An Emergent Question The founders of reform envisioned a reform text functioning within a reformed curricular structure of calculus—what Douglas labeled a lean and lively calculus (Douglas, 1986, Steen, 1987) What this reform founder implied by the use of the descriptor lean was a calculus course wherein whole areas of traditional focus were cut away so that there would be more time for open-ended exams and projects, along with opportunities for student growth and discovery aligned with a more constructivist approach to calculus learning. But what happened along the road between the ideological
62 structuring of the reform movement and the actual implementation of calculus reform in experiments like the one conducted at Research University is significant, and wide- reaching beyond this specific research. Instead of changing the curriculum of calculus courses to conform with the changes in a reform text, and the mandated NCTM suggestions for reform in the manner of teaching calculus, this reform experiment at Research University attempted to retrofit a reform text to a traditional calculus syllabus, pacing, and teaching model. In effect, this disconnect is the most profound discovery of this research, and explains and informs the departmental reaction to (and abandonment of) the Hughes-Hallett text. The return to a traditional text at the end of this experiment may have less to do with an outright condemnation of the reform movement (although it parallels in both Professor Violet’s and Professor Green’s interviews a suspicion of the movement), than it does a dislike of a specific text edition and its deviation from the traditional calculus text model. An emergent theme can be summarized by the following question: Is it possible that in failing to align the pacing, course curriculum, and instructional style of faculty with the intended calculus reform ideology, the department created an unintentional dissonance between text and delivery? I followed up on the question about text usage, as it sounded as though Professor Green had a completely different view of its usage than Professor Violet (who had only used it for homework assignments). The text is important to me because I know that it’s important to the students. They need to have some place, some guideline they can follow, and some other source to read about the material. The topics and the way they’re laid out are important. At that level, the calculus level, I think the problem sets are important. If the problem sets are inadequate in the text that I happen to be using, I supplement. The book is important in that it makes more work for me if I’m not happy with those exercise sets. This excerpt shows a major difference between the two participants. Professor Violet did not feel that his students can or will use the text to supplement his lectures, while Professor Green believed his students would use the text, and so it was important for it to be comprehensive. His main objection with the reform text then, unlike his colleague,
63 was not that he felt the text was misinformed or wrong in the ideology that spawned its structuring. Rather, it was that he felt the inadequacies of the text made his teaching harder. Effects of the Reform Experiment Because of concerns about airing departmental administrative laundry, Professor Violet preferred not to discuss the actual book search committee he chaired that first championed the switch to a reform text. However, it did seem odd to me that although Professor Violet had earlier considered him to be a neutral figure, his views at this later time were far from neutral. Take, for example, his comments about the text and the claims that it had not prepared the students well for follow-on coursework: “It’s hard to find the perfect book, but I thought that this one fell short in more areas than our previous books, and one we’re using now.” While his rhetoric in regards to these criticisms of the reform text was less fiery than Professor Violet’s, it was equally damning. I probed further, in an attempt to draw out of Professor Green a reaction to the assertions of Professor X, who had criticized the reform text’s ability to prepare engineering and math majors for the rigor of advanced math courses like Partial Differential Equations. The excerpt below is Professor Green’s response to a question about whether there was merit to the idea that students who had taken the calculus sequence during the reform experiment were less prepared for their subsequent math courses: Yes…at least for the students who came through a course where it was taught strictly using this text and the ideas or the philosophy surrounding the text. Now I think an instructor, if he or she decided to supplement, those could be overcome. I think they were in many cases. But I think if you just follow that text and what it’s trying to do, that you miss out. So I think there is merit; I think it’s a problem area for it. Like Professor Violet, Professor Green felt that the computational facet (use of numerical approximation techniques instead of algebraic manipulation) of the rule of four devalued the importance of by-hand calculations: I think there’s value in doing some computation. I do believe that calculators have changed it, there’s no question about it. And then…I think certain things, integration techniques--that’s one area that the
64 reformers loved to attack—I think integration techniques have value, or something like integration techniques. It sharpens those algebraic skills is really why. He also reflected on another element of the rule of four—applications or word problems, which he felt were too far removed from the math itself: I think one of the problems with the reform calculus movement was that it tried to make calculus more than it was. It tried to apply it in what it thought or what it advertised as “real problems” in all of these different areas. Well, as soon as you do that, you’re trying to do too much. You’re teaching the calculus--that’s what you should stick to. It’s OK to give applications, but to drive it only with these applications from the other areas, that means you’re teaching biology, you’re teaching physics, you’re teaching chemistry, you’re teaching statistics, economics, whatever your application happens to be. I think the reformers tried to make calculus bigger than it actually should be at that level. The language of this response echoes very closely the language of an article about calculus reform by Kleinfeld (1996) entitled Calculus: Reformed or Deformed? One might imply that although Professor Green had not read much in the way of reform- sponsored research, he had almost certainly read some of the reactions to it. By this point in the interview, I was beginning to suspect that while Professor Violet might consider Professor Green to be “sitting on the fence” on calculus reform, Professor Green was in fact just as adamantly opposed to the reform movement ideology as his more vitriolic colleague. His opinions and counterarguments against the reform movement and the reform text employed at the university paralleled Professor Violet’s in many cases, which is understandable for professors who work together and talk regularly. He simply expressed his disappointment with the reform text more gently. Also unexpected was the extent to which both participants had analyzed reform. Looking only at the excerpts presented here, one can see that both faculty members had invested considerable time and energy in learning much more about reform than might be expected. In fact, in the literature mentioned in Chapter 2, reformers often painted
65 traditionalists as ignorant of the philosophy of reform. In fact, both men were well aware of the key components of the reform movement—they simply failed to be convinced. Finally, when asked about the change back to a Stewart text and whether this was a positive thing, Professor Green’s reply was cast in terms of the text in the hands of younger or novice teachers—he admitted that more experienced teachers would have no trouble (as he did) supplementing the text with whatever they needed: So, I think that’s where the biggest impact is going to be, on people who are new faculty or people who haven’t taught calculus for years, or graduate students who are teaching. They’re going to at least have perhaps a little more of a middle of the road approach. It’s not an extreme traditionalist approach at all; it’s more middle of the road. I think the Hughes-Hallett text was extreme the other way, and it forced that on the instructors. Thus, like Professor Violet before him, Professor Green felt that those texts considered by the reformers and supporters of Hughes-Hallett et al. to be traditional are actually more mainstream and more valuable than the “extreme” reform text itself. The use of the term “extreme” to describe a reform textbook underscores the passion inherent in this ideological divide. In retrospect, Professor Green’s interview, while more polished and dispassionate than that of his colleague, was loaded with the kind of language that casts the debate between reform and tradition in calculus instruction in partisan terms. Professor Violet had expressed outrage at what he felt was the reformers’ dismissal of rote or drill in favor of conceptual understandings he felt the students were unprepared to understand, while Professor Green disliked the text more for the difficulty he had in applying it to his teaching. While Professor Violet refused to even consider the value of calculators, Professor Green admitted their power for a new generation of students. And yet, in spite of his adherence to many of the best practices sponsored by the ideology of calculus reform, when asked how he would label himself, Professor Green said: I would consider myself more of a traditionalist than a reformer. I don’t know the exact definition of these two. They’re individual. I’m more
66 traditional than I am reform…[and] I do believe that some of the traditional components are certainly part of the way I teach. In the next section, a contextual matrix is presented in the same manner as at the end of chapter 4. When the two are overlaid, one can see that while their individual objections may differ, there is much more agreement between the participants than was anticipated. Professor Violet and Professor Green are colleagues of long standing—it is quite possible that neither of them developed their opinions about reform in a vacuum, but rather through years of dialogue with one another and fellow faculty on both sides of the issue. It remains to be seen if any common ground can be found between the perceptions of these participants and those of a self-proclaimed advocate of reform—Professor Red. Analysis of Coding of Interview—Professor Green The most remarkable aspect of coding Professor Green’s interview was the discovery of the sheer volume of statements that could still be coded as partisan. While he self-identified as middle-of-the-road and accepted the value and legitimacy of some elements of reform, Professor Green still managed to isolate the reform movement with his language. Professor Violet called advocates of reform “they”, as in “they just dismissed my claim”. Professor Green called the same nebulous group “the reformers”. Essentially, by identifying this group as some outside entity, both participants distanced themselves from both the reform movement and its advocates. This is remarkable only because, based on Professor Green’s reputation among students and faculty as a top-rung educator, one might expect that he would be patently accepting of any right-minded reforms intended to improve student success. That he did not feel that the reform movement as it had evolved had merit, even after 3 or 4 years using a reform text, could be seen by many as implication that the reform movement may need to do more to educate those who implement reform as to its value, as it appears that the target audience of mathematics instructors is not universally convinced. But based on the obvious amount of thinking Professor Green had done about his teaching, this is more likely a case of an established and experienced educator taking from the movement those elements which years of teaching have taught him might help his students—and dismissing those parts of the calculus reform ideology he felt are unworkable.
67 Like Professor Violet, Professor Green expressed suspicion that money drove calculus reform in a different direction than originally intended, and implied that some of the conclusions drawn by the creators of the reform movement were faulty. Like many of the traditionalists from the paired opinion papers mentioned in Chapter 2 from the early days of the reform movement, these men believed that the reason calculus retention rates are perpetually low has less to do with flaws in the delivery of this subject than the rigor and difficulty of the subject itself, or the lack of preparation of the modern student. Both men expressed that the early calculus reform movement assumed that if retention/completion rates were low in this course sequence, there must be a causal relationship with quality of instruction. Like Professor Violet, Professor Green saw calculus reform and the reform text used at Research University as an overly simplistic and not wholly thought out response to a real, but nuanced problem. That these two participants, seemingly so different, shared a common distrust for the reforms implemented in the Hughes-Hallett Calculus text, was quite surprising.
68 Table 2. Conceptually Clustered Matrix—Professor Green
SELF- AWARENESS REACTION TO RETURN TO PARTISANSHIP REPORTED OF REFORM HUGHES- STEWART TEACHING MOVEMENT HALLETT (TRADITIONAL) PHILOSOPHY TEXT TEXT Moving Certainly, the I think there was a I did not like the text. Proving the I believe that there Towards conceptual, movement that took It was not a book that fundamental theorem are some legitimate Reform numerical and over—the I felt was particularly of calculus, even. I’ve issues, or there were geometric… I movement was effective for what I always felt that that some legitimate embrace those largely driven by wanted to do in the is all lost on the issues about concepts. funding classroom. audience if that’s calculus and the overdone at that way it was being stage, so it’s a real taught that were balancing act raised During the Now, it isn’t clear I’ve never felt that There were In my view, there is a Over time, I think it Experiment to me that the the intensive chapters…where I danger in a calculus kind of blew up…as discovery calculator ended up rewriting, course for the simple “OK this is related component has-- problems that are giving out all of my ideas to be presented, to math education. should have, really, common in the own exercises, and then for there to {it} is ailing, you a major role in a reformed method-- rewriting be a movement down know, at all levels. beginning calculus I’ve never thought completely…because the path of rigorous We should move in class. that they were that I just wasn’t pleased presentation of those and try to contribute useful. with what I saw main ideas to this The Shift Back I don’t really think What students need It’s hard to find the It’s not an extreme Personally, I think it’s a good idea to to take out of the perfect book, but I traditionalist that the reform withhold great class are the main thought that this one approach at all; it’s calculus movement amounts of ideas… and then a fell short in more more middle of the was tainted a bit by technical certain amount of areas than [the] one road [funding]. machinery from the technical mastery we’re using now. students. of the material.
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CHAPTER 6
CASE STUDY C: PROFESSOR RED’S INTERVIEW
Biographical Sketch—Professor Red The third and final interview covering the historical arc of the text change from traditional (Stewart) to reform (Hughes-Hallett) and back again was conducted with a full time faculty member at Research University who described himself, and was described by the others interviewed, as well as by colleagues in the department, as a reform proponent. Professor Red, as he will be described, was mentioned in the interview with Professor Violet as someone who had used a reform text (Hughes-Hallett, et al., 1995) before the department made the move to universal usage, and as a member of the book committee who voted in favor of that move. Professor Red is an Ivy League alumnus, a tenured professor of mathematics, a respected pure mathematics researcher (as was Professor Violet—Professor Green being the only participant interviewed whose background was Applied Mathematics), and the only one of the three participants whom I had actually had as a professor, while I was completing a Master’s Degree in Applied Mathematics. He was willing to meet with me during his office hours, assuring me that we would not be disturbed, and that only rarely do students show up. Professor Red was a thoughtful and quiet man, who at times spoke so softly that I had to move the tape recorder closer to him. During the interview, he seemed the most guarded and cautious of the three participants, perhaps because he knew me in some other capacity than interviewer/fellow instructor. Nevertheless, he did admit to fatigue and disillusionment with his colleagues for their failure to embrace reform. Professor Red was the faculty member who set up the course pacing schedules when I started teaching Calculus at Research University—I learned from Professor Violet that he had been chosen to do this in part because of his prior familiarity with the text. Assertions and Excerpts—Professor Red The questions asked were the same as those in the first two interviews, although Professor Red appeared less comfortable both with the questions themselves and with answering most questions at length. He had to be re-prompted on a number of occasions,
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and asked for the questions to be broken into smaller parts. He appeared to be more careful in his responses, and at times trailed off. My primary assertion from that interview data was that Professor Red, a self-professed reform advocate, still taught in a more traditional lecture style, although he alone of the three cases felt uncomfortably restricted by the curriculum as designed. While I had initially forecast that Professor Red was the most likely participant to still be integrating the reform structure into his teaching—group projects, technology used for exploration and discovery, open-ended assessments, and less dependence on mechanics and drill than his colleagues, this did not turn out to be the case. While Professor Red did feel that technology made some of the analytical rigor of calculus unnecessary and that group projects would help his students, he questioned the feasibility of such reform strategies from within a recognizable calculus curriculum. Finally, I felt that Professor Red gave tremendous insight into what it was like to be pro-reform in a department that had overwhelmingly rejected the continued use of a text he fought for. Positive Views of Reform Professor Red appeared to be well aware for one of the fundamental motivations for the reform movement, which he describes as a contrast between manipulations (or what the other participants called rote or drill) and content knowledge: So, I think calculus reform was some movement that just came up because people were reacting to the way calculus classes were being taught for a long time, and I think that a lot of what they were reacting to was that people were learning to do, certain manipulations, but without having very much understanding about the significance of the manipulations that they were going through, so, you know, the common anecdote was, you’d go up to a student who had been through a Calculus I class and say, “what’s a derivative?” and they’d say “n times x to the n-1 power” or something like that, and the implication was that they had learned some formulas for taking a derivative, but had forgotten the definition of a derivative and its interpretation as a rate of change, which definition is really the link between Calculus and its applications.
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However, unlike the other participants, Professor Red felt that these concerns were still valid. When asked if he felt the reform movement had merit, he answered as adamantly as one might expect a self-identified reform advocate to: Yes, Yes, I do; I think those problems were real, and still are real to a certain extent, and I think that a lot of the things that were incorporated into some of the calculus reform textbooks really did address those questions in significant, effective ways. This question may be considered by some to be anticipatory or leading; however, my intent was for this to be an opportunity to discern whether the participant was pro-reform by self-alignment. While both of the other professors interviewed felt to varying degrees that calculus reform was responding to a real concern, neither stated that they felt, as Professor Red did, that the reform movement made headway in tackling the problem. Neither did the others express that the problem still existed, even after the integration of a reform text and possibly, to a lesser extent, reform philosophy into the department. This excerpt (less than a minute into the interview) exposed Professor Red’s self-alignment as pro-reform. A Self-professed Reform Advocate The interview questions had been developed so as to allow for confirmations or contradictions of such statements, as when I asked point-blank which camp Professor Red was aligned with: Well, I think I’ve certainly taken to heart a lot of the things that calculus reform movement had to say. I do really believe that it’s important for students to understand the definitions and not burn their bridges to applications. I think I’ve adopted a lot of the ideas that the reform movement advocated. Another thing they advocated was the approach emphasizing the graphical approach to a certain extent, using graphing calculators, and that’s been an influence. While he answered to the point questions about facets of the reform movement, he had not answered the question as asked, i.e. “Would you consider yourself a reformer or a traditionalist?”, so I repeated the question, to which he replied: “I think I’m more of a reformer. I’m more in the reform camp.”
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Another instance of this self-alignment appears when Professor Red addresses technology. A founding tenet of NCTM’s calls for reform, technology in general and graphing calculators in particular were addressed with hostile indifference by Professor Violet and cautious acceptance by Professor Green. Professor Red saw them instead as a way of changing the focus of what is taught in the Calculus sequence: I definitely don’t believe there is much of a value to arithmetic calculations done by hand… I don’t think there is a lot of value to be gained by them. I guess I believe less that there’s a whole bunch of value to doing a whole bunch of complicated integrations by parts or rational function decompositions. I think I’d like the students to be familiar with that sort of thing, but I’m less in favor of a lot of energy expended on a lot of the red exercises in that area than some people are. Because he tended to trail off, and at times seemed reticent to go into detail, I attempted to clarify this by tackling a hypothesis from the reform movement refuted by Professor Violet earlier—that the graphing calculator made some rote or drill (what Professor Red had earlier called manipulations) obsolete. In reply to the question “Do you think that the use of a graphical calculator has made any particular types of problems obsolete?” he replied: I don’t know—I may waver on that. I certainly think that students should know how to do the derivative on a piece of paper, to make a rough sketch of the way a graph looks, but on the other hand, a graphing calculator can certainly enhance that. It would be nice if a student understood how to do some easy integrals by substitution, but I’d—I don’t see anything wrong with using a TI-89 to whack out complicated integrals once you are in Calculus III, for example. Using what fits Professor Red had earlier expressed that some elements of calculus reform were less effective in the classroom because of time constraints and curriculum concerns…it appeared that he used those facets of reform that were easily integrated into a traditional lecture style, but had a harder time integrating those parts that are more time-intensive, like group-work and class participation:
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I think theoretically they are important. I don’t use those things as much as I did a couple of years ago; probably, in full point of reason that’s laziness and also some lack of time. Group discussions--that was never something I was into very much in a calculus class, although I am a real advocate of having them work together. I used to assign group projects, tried group quizzes…although it’s a little bit difficult to dictate that… Professor Red’s honesty here was telling: he would have liked to try some of these innovative elements of a reformed calculus curriculum, but felt that this was impossible, or at the very least extremely difficult, while still keeping to the department-mandated coverage of topics. This response gave insight not only into the extent of his personal experimentation with calculus reform (he is close to Professor Green here, in that he used what he could and ignored the rest) but also into a fundamental cause for the slowness of change in the reform of collegiate calculus instruction. Calculus reform experiments have only truly been successful, as the research in the literature review revealed, when they are instituted small-scale, in individual courses or departments where the change was considered more important than the adherence to the traditional curriculum. When, at Research University, the changes that were made were forced to align with what most would consider a traditional calculus curriculum, the results were less demonstrable: a hybrid of the components of calculus reform that are easily digestible within a framework more recognizably traditional, with lecture still far more the norm than group work or projects, and with homework still the primary tool of student exploration and discovery. A Different Us; A Different Them An immediately recognizable difference between Professor Red and the others was his non-alignment with criticisms of the text. Note in the following excerpt that he does not personally address concerns with content, but rather attributes these concerns to “other people” and “those people” [Note: this is a response to the question “Do you see any downside to using a reform textbook?”]: A lot of the criticisms that have been offered have been that there is only so much time in a calculus class, and if you are going to spend more time on interpretations and things then you are going to have to spend less time on certain other things, and that’s sorta true. So a lot of people have
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bridled when they have seen their favorite topics omitted or de- emphasized in a calculus book. And then, even more so than the usual calculus textbooks, calculus reform books tend to soft-pedal the theoretical aspects, the epsilons and the deltas. And for those people who are in favor of those in a calculus course of course there is going to be opposition to them… I think there’s some merit on both sides of those arguments. While his last statement would tend to put him on the fence, he never actually says “I felt” or “I did not like”, implying that he had not a single objection he could lay personal claim to—anyone who has been to college would have a hard time recalling any professors who expressed universal agreement with their text (unless they themselves wrote it). Professor Red, however, at no time criticizes any aspect of the reform text. Instead, he refers to all criticisms as anecdotal references to third parties. This was a recurring theme throughout the interview—he even uses his wife as such a proxy, when asked about the relationship between the Mathematics Education community and mathematicians: Well, my wife’s a mathematics educator. I think they [math education people] have some good things to say, to people like me. My wife does tend to think that a lot of the people in the math education department and in the school of education could profit by a lot more time spent in the classroom. Since I felt that he was vaguely implying by proxy that mathematics education researchers were out of touch, in spite of the fact that he himself interacted with mathematics educators on a daily basis, I followed up: “Do you think mathematics education researchers are out of touch with actual mathematics instruction?” I don’t have a lot of personal experience in that realm, but my wife does, and she does definitely believe that, and that’s something we talk about a lot, and I’m willing to take her word for it. This care by Professor Red to distance himself from personal opinions must be seen as an individual trait—it is not implied nor should it be inferred that all who are pro-reform would choose their words so obliquely, any more than it should be assumed that all traditionalists are as bluntly passionate as Professor Violet. However, Professor Red does
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show what he personally feels was a Mathematics Department presence behind what is generally acknowledged to be a Mathematics Education movement towards reform: I might argue just a little bit that the call for the reform of calculus came so much from the math education community…you know, a lot of the authors of those Calculus…the reformed texts were actually in mathematics departments and not associated with the math ed. community. It is notable in this excerpt that Professor Red equates the reform textbook he championed with the reform movement that led to the formation of that textbook. In fact, as was noted in the literature review, the textbooks related to reform were a second generation phenomenon; the research leading to their formation was done almost exclusively within the mathematics education community—mathematicians do not tend to do research in the area of mathematics instruction. The Disillusioned Reformer Even with the care he took in his responses, Professor Red still gave honest and insightful data—even to the point of agreeing with more traditional colleagues that the text is less effectual a component of instruction than other facets: The text is not very important. I use it to assign problems. I will tell them to read it, hoping that they will read it, but knowing that not very many of them will. …just to get perhaps a different point of view. An interesting revelation in this quote repeated throughout the interview with Professor Red is his belief that his students are not as a whole well motivated, well- prepared or very studious. This was contrary to my personal expectations that a professor who was pro-reform and therefore more sensitive to student needs would have a better opinion of his students (in contrast to the stereotype of traditionalists not caring about their students—an explanation for wash-out rates that is little more than generalization). However, while all three participants felt their students were less well-prepared than they would have preferred, Professor Red returns to this multiple times: I think we do a pretty good job of teaching here, but I am not so sure our students are doing a real good job of learning, and I think that part of the responsibility for that has to fall on the students. I think it’s not
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unreasonable to put a significant part of the blame, I don’t know how it is at other universities, but on the higher administration of this university, who gets upset when they see a large percentage of people failing a particular course, but it’s really true that a lot of the students we get in Calculus I are unprepared for Calculus I, and to a lesser extent, up the line. This belief that it is students and administrators that are most responsible for a lack of success in higher level math courses is one that, in the debate literature mentioned in chapter 2, was commonly associated with traditionalist instructors. Herein lies a major problem with compartmentalizing people as traditionalists or reformers: while it is innately human to categorize (this is something the brain does innately and involuntarily) (Hawkins & Blakeslee, 2004) , real attitudes—especially those of people whose whole life has been spent in the sciences, in research and analysis of research—are much more subtle and nuanced than the one-dimensional partisan molds created in the early 1990s to pigeon-hole reformers and traditionalists alike. So while Professor Red might self-report as a reform-advocate, he does not fit into either camp’s stereotype of one. In two excerpts from the Professor Red interview in particular, he perhaps sheds light on why he is not as passionate an advocate of reform as he might once have been. In the first, he answers a question about the significance of the department’s move back to a more traditional text: You know, Professor Violet was the architect of that change, and he’s always been a real advocate of the traditional text and has not liked the Hughes-Hallett text very much. He’s in a position as [administrator] where he has a lot of power to do that sort of thing. And I think that through a combination of the fact that those of us who fought the battle to fend that off had just gotten tired, and a combination of that and the fact that the traditional text, the Stewart text is a lot more like a reform text--it has moved significantly in that direction, but you know, what the heck, there was more fighting this time. I went back and listened to this again recently, and it really does sound like the admission of someone who had decided to quit fighting the powers that be, but still believed he was right. The word “tired”, as in “tired of fighting”, could give a greater understanding of
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what it was like to be pro-reform in a department that, by its own tally, was dominated by traditionalists. Professor Red did not acknowledge that a majority of the department wanted to switch back, but rather, felt that one person in a position of authority was responsible. The second excerpt that showed Professor Red as still pursuant of reform, albeit on an individual level, was when he was asked if he wished he was still using a reform text: I don’t think I wish now that I was using it—you know, I took a lot away from it, a lot of what was in it, but…a lot of that stuff is in Stewart now, which may be explained as just them trying to get more market share, but more idealistically maybe, which shows that maybe those people are open- minded too. It should comfort those who continue to work for reform in college level math instruction that even when those reform efforts are deemed less than completely successful, as the book-change-only experiment at Research University ultimately was, that the effects of that experiment may continue to be felt in the long term. The changes in the Stewart text towards reform noted in all three interviews may be seen as a recognition that change is not always seen as a negative, even by those who opposed the change, and that the technology and critical thinking elements championed by those in the mathematics education community since before the Tulane conference as being lacking in traditional texts are slowly (some might argue glacially) making their way into the Calculus course of the twenty-first century. However, one of the issues that the innovators who created reform-minded Calculus texts must be made aware of is the manner in which those texts are used, and misused. Echoing the statements of both self-identified traditionalists, Professor Red exposed a primary reason why introducing a new text alone cannot have a more dramatic impact on a department: “It’s not unusual for me not to read the section…to look and see what it covers, to pick out the problems, and present it in a way that I want to present it.” If even a self-professed reform advocate is unwilling to familiarize himself with the method of instruction presented in a text he himself fought to use, one must wonder about the efficacy and effectiveness of reform based entirely on textbook change. All three participants interviewed made statements to this effect—one could see this as an
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under-anticipated stumbling block for acceptance of any reform that is not undertaken more wholeheartedly. For college mathematics professors to embrace calculus reform experiments centered around a reform text, they must first know what is in the text, and how the text intends the material to be taught. If instead, as the interviews showed, they simply use the same instruction methods but with a revolutionary text, a dissonance is created that can result in abandonment of the half-hearted reform experiment, as occurred at Research University. In order for reform to be embraced in areas where traditional teaching styles predominate, the professors themselves must be educated as to the benefits of the reform movement, and the philosophy and research behind it, as well as the less popular notion that their present teaching style might be part of the problem. This education could include seminars, discussion groups, and the like—and might prepare them to be better teachers, and prepare them to create better students—but it must have a grounding in research that can show faculty convincingly that the methods espoused have been proven more effective than the status quo of lecture-centered delivery. While instructions existed for how to use such a text, in the form of teacher aids and support literature, Research University provided only the text and homework solutions manual, and not those support materials for faculty. Furthermore, the pacing of the calculus course was left in the hands of Professor Red because of his prior knowledge of the text; but as figure 1 below shows, he did provide days or even note on his departmental pacing schedule that time could or should be spent engaged in group work or discussions, classroom projects or lab exercises. The only reform-oriented activity mentioned on the pacing schedule was an optional project, that Professor Red himself no longer used by the time of the interview. While an individual instructor with knowledge of reform pedagogy might have the foresight to try to juggle these activities with the daily material assigned for each lecture, it would be very difficult for a novice teacher or one unfamiliar with reform teaching activities to gain any insight into how the book was to be used from the pacing schedule. Professor Green himself had indicated this in his data. Analysis of Coding of Interview —Professor Red
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Professor Red was the most challenging of all the interviews to transcribe (he tended to mumble, trailed off, constantly asked for questions to be repeated or broken down) and to code. In spite of the fact that he had agreed to be interviewed, he seemed unable or unwilling to speak for himself, preferring to begin his replies with “I have heard” or “Some people say”. Nevertheless, it was possible to infer from the expressions he used that he was not happy with the department’s move away from a reform text. He consistently referred to any faculty member who criticized reform as “those people”, while the only criticism he could offer for the text itself was that it was too hard to fit all facets of the rule of four into his lectures while maintaining the departmental pacing, which he himself, as a long-time user of the text, had created. Moving towards reform Professor Red had searched for some reason why his students were less than universally successful in Calculus. The Calculus reform movement appeared to provide the perfect explanation. While some in the department saw the reform movement and the representative text as a danger to traditional teaching methods (Professor Violet) and others reserved judgment (Professor Green), Professor Red led the move to change the standard department text. When criticisms of the first edition of Hughes-Hallett et al.’s text made the chances for such a paradigm shift unlikely, he was instrumental in delaying the departmental book committee vote until the second edition was released and available for study. The second edition changed from the rule of three to the rule of four, but more important for the math department book committee, some of the more intense criticisms of the text were addressed—more of what Professor Red called “manipulations”, or drill exercises and a return of integration by parts, which had been touched on only lightly in the first edition (Hughes-Hallet et al., 1998). In the most candid of his responses, Professor Red admitted “I am in the reform camp.” During the Experiment Professor Red does maintain his partisanship throughout the time that the textbook is used department-wide. He speaks with regret about not being able to use more of the philosophy behind the text, citing time constraints and laziness as reasons for not using group work and exploration, cornerstones of the reform movement and the text. He admitted, like the more traditional participants before him, that his use of the text was at
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best superficial—he assigned homework from it, and hoped that some student might read it.
Figure 1. Professor Red’s Pacing Schedule During the Reform Experiment
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He felt that students and administrators were more responsible for the quality of instruction than faculty. Finally, he felt that the reasons for a move to reform still exist within the department and felt that the reform movement “got it right”. The Shift Back As an advocate of reform, Professor Red did not try to hide his disappointment or bitterness about the department’s overwhelming support of the move back. Even as he admitted that he sees no reason to return to usage of a reform text (all three participants acknowledged that traditional texts look more and more like the reform text used in the department), he was extremely upset at what he portrayed as one administrator’s decision to reverse the reform. He admitted to being tired of fighting for reform, but in his view the administration was more responsible for students being unsuccessful than any flaws in delivery of instruction. This was the strangest point that he made—the reform movement’s original catalyst was low success in calculus at all levels, and yet Professor Red felt betrayed by administrators who criticize him for failing a certain percentage of his students. His blaming of social promotion, and implicit acceptance of failing a significant percentage of his students, was almost completely antithetical to the catalysts of the reform movement whose values he so espoused. For this reason alone, Professor Red’s interview, while the most frustrating to conduct and transcribe, evolved into the most fascinating and surprising. His self-perception of his beliefs and his avowed partisanship with the cause of calculus reform (and his championing of the reform text in the department) appear in direct contradiction with the way he actually acknowledged teaching and his opinions about where the blame lies in student success. On the next page, in the manner of those undertaken in Chapters 4 and 5, is a conceptually clustered matrix of the key excerpts from the interview with Professor Red categorized by their relationship with the broad temporal and subject codes developed while transcribing the interviews. In the next chapter, I will undertake a summary of the results of the three interviews, analyze them in parallel, present conclusions and suggestions for the direction of future research in this fascinating and underexposed area of qualitative analysis
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Table 3. Conceptually Clustered Matrix---Interview with Professor Red
SELF- AWARENESS REACTION TO RETURN TO PARTISANSHIP REPORTED OF REFORM HUGHES- STEWART TEACHING MOVEMENT HALLETT (TRADITIONAL) PHILOSOPHY TEXT TEXT
Moving Well, I think I’ve [Professor Red was Calculus reform If you are going to I think I’m more of Towards certainly taken to principle in getting textbooks really did spend more time on a reformer…I’m Reform heart a lot of the the department to address those interpretations and more in the reform things that calculus move to universal questions in things then you are camp. reform movement usage of a reform significant effective going to have to had to say. text, having used it ways. spend less time on since 1998] certain other things During the The text is not very I think I’ve adopted I think that a lot of I definitely don’t but you know, what Experiment important. I use it a lot of the ideas the things that were believe there is the heck, there was to assign problems. that the reform incorporated into much of a value to more fighting this movement some of the calculus arithmetic time (against the advocated. reform textbooks calculations done reform) really did address by hand… I don’t those questions in think there is a lot significant, effective of value to be ways gained by them. The Shift Back I don’t use group I don’t think I wish I think those a lot of that stuff is those of us who work or exploration now that I was problems were real, in Stewart now, fought the battle to as much as I did a using it—you know, and still are real to which shows that fend that off had couple of years I took a lot away a certain extent maybe those people just gotten tired, ago; from it, a lot of are open-minded and the traditional probably…that’s what was in it, too. text, the Stewart text laziness and also but… is a lot more like a some lack of time. reform text
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CONCLUSION
The three interviews analyzed in Chapters 4 through 6 might seem on the surface to have been describing different historical events at times. Professor Violet described calculus reform, in the form of the Hughes-Hallett text piloted by Research University, as a mathematics education-sponsored movement taking a real concern (the traditional teaching of calculus and how best to do it) and making the cure worse than the disease. Professor Green suspected calculus reform as insincere and funding-driven, the issues on which the reform movement was based being warped into something far different by grants and the perpetual need for their continuance. Professor Red felt that mathematicians were the true creators of calculus reform, and felt abandoned by his department for having to switch back to a traditional text. And yet there are remarkable parallels between the data given by these three participants. In this chapter, the assertions first addressed in each of the interview chapters will be re-addressed and expanded, with data from each interview. Then, literature first mentioned in the literature review is discussed that parallels the beliefs and perceptions exposed in the interviews. Part of the focus of this chapter will be the similarities between the three interviews, as they relate to the original research questions, and one other that emerged during the analysis of the interviews. Finally, possible directions of future research triggered by the analysis of these most robust interviews are covered. The original three-part research question focused on the historical event of the department’s decision to switch texts, which spanned a 5-year period from 1999 to 2004. The original research questions were: o What were the historical conditions under which Research University underwent a calculus reform experiment? o What was the nature of the reform experiment attempted at Research University?
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o What were the conditions under which the reform experiment was discontinued, and the department returned to a more traditional manner of calculus instruction? During the research, an additional research question emerged from the data: o Is it possible that in failing to align the pacing, course curriculum, and instructional style of faculty with the intended calculus reform ideology, the department created an unintentional dissonance between text and delivery? Each of these questions will be addressed in turn. But first, assertions are connected from Chapters 4-6, and then supported by interview data. Assertions And Supporting Data—Professor Violet My primary assertions about Professor Violet were that he represented something of a polar extreme example of the passionate traditionalist, but that he was more knowledgeable of the movement than some reformers would give a traditionalist credit for. The interview data also exposed that he was passionately anti-reform: I believe the NSF came out in the mid 90s with a ton of money for education reform, and I believe some of these smart people saw that that was a good way to get funding and so that’s why we have...that’s the real reason we have calculus reform.[Interview, Professor Violet] Combine this excerpt, in which Professor Violet casts doubt on the motivations behind the reform movement, with this second excerpt, where he describes his own teaching style: I wouldn’t change the way I’ve taught calculus based on all the stuff I’ve heard because I’ve always incorporated all the stuff I’ve heard that I thought made sense. [Interview, Professor Violet] From these two excerpts and in many others from Chapter 4, a picture emerges of Professor Violet as a math professor resisting a change that he sees little value in, while simultaneously admitting that if he is aware of something that could better help him teach, he would use it. At least in his mind, the Hughes-Hallett reform- based text (1995) is not such an improvement. A reform proponent might counter
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that as a traditionalist, Professor Violet was criticizing something he had never bothered to understand, but in fact, he shows an awareness of the tenets of reform (and even mentions the rule of four) in the excerpt below: …stress the ideas behind calculus, stress geometric reasoning, stress numerical approximation, stress what they call real world problems, data, real world data, where you don’t have a function, you just have some data points, and how do you come up with a function and approximate the derivative, stress technology, stress the fact that we have machines to do a lot of the calculations…let’s stress concepts [Interview, Professor Violet] However, knowledge of the movement’s aims and goals did not translate into acceptance for Professor Violet. He did not object to change, per se, but rather to a change he did not consider educationally valid: I claim that calculus reform gets it backwards; they say “Let’s think critically about what it means first and really explore that in detail, and let’s not worry about calculations”, and I claim that it’s very difficult (feeling) comfortable with concepts in mathematics that way, that it’s the cart before the horse. Now the things that don’t make sense to me is “let’s not stress calculations, let’s not do calculations, because machines can do it”—I think that’s stupid. [Interview, Professor Violet] Compare this criticism to one authored by reform critic George Andrews: “The reformers believe that they will get around the roadblocks of basic arithmetic so students can get to higher-order skills. But to learn piano, you must learn scales and chords before you move to the ‘Moonlight Sonata’” (Andrews, in Wilson, 1997). It comes as no surprise after reading the prevalent literature about the reform movement that many traditionalist instructors feel that the reform movement has not done enough to prove it own assertions about the validity of largely eliminating what the reformers call drill and kill, and traditionalists like Professor Violet call calculations (Andrews, 1996; Klein & Rosen, 1997, Cargal, 1997, Wilson, 1997). In fact, in a scathing critique of the reform movement
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provocatively titled Calculus Reform—For the $Millions, Klein and Rosen in a mathematics journal cited many of Professor Violet’s suspicions: The calculus reform movement was a facile response to a real problem: the declining performance of American students in college calculus courses. Without any credible scientific study, many reformers put forth “solutions” such as eliminating theory, making more conceptual problems, decreasing the reliance on algebra, and increasing the use of computers and calculators. But what if the reformers were wrong...? (Klein & Rosen, 1997 p. 1324) There is certainly evidence in the data excerpted herein that these attitudes have not disappeared within mathematics departments, although they are less evident in the more recent literature. Perhaps the great debate between traditionalists and reformers is no longer being fought on the pages of the Chronicle of Higher Education, but the evidence exists in these interviews that it is not yet over (Wilson, 1997). What makes these interviews so striking is that they represent professors openly discussing their unvarnished reaction to calculus reform after actually having attempted to teach for almost 5 years using a reform text, if not reform methodology. It is also worth noting that Professor Violet’s criticism of reform as a movement is often directed at the Hughes-Hallett text in particular. The equating of the movement as a whole with the lone reform instrument introduced into the curriculum at Research University calls into question whether what is actually being measured is these professors’ reaction to reform as an ideology or the specific representative reform text. However, the participants at the very least make little distinction between the movement and the text. Professor Violet’s interview data led to the assertion that was common to all three case studies—namely, that text usage is restricted to assigning homework: The only thing I use the book for—there are only two reasons to have a calculus book—one is to give me a pacing (two reasons as a
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professor, I mean)…to give me a pacing...so I said “oh, chain rule today, I’ll do the chain rule today”—and the other reason is to have a good set of problems for the students to work on. And a good reason for the students to have a calculus book is number one to have a good set of problems to work on, and number 2 to have a good set of examples worked out in the text, and number 3 to have something they can read that explains the theory, but that’s the least important, because no one ever reads it anyway. [Interview, Professor Violet] From this quote, one can infer that there can be little expectation that any text would have a considerable influence on Professor Violet’s teaching, and that the honesty of his responses opens up a powerful question that could be answered by future quantitative research: what percentage of college math professors view the text as nothing more than a problem bank? If this percentage is large, the reform community would have to approach reform from a more broad curricular focus. No reform that involves student exploration and discovery through innovations in text will be successful if professors do not assign readings, or do not encourage their students to read the text. Further, what percentage of traditional college math professors assign readings to their students? If this number is low, how can the message of the reformers (or any reformed text author) be delivered to their ultimate audience, the college students taking the courses? Assertions And Supporting Data —Professor Green My primary assertion about Professor Green was that he was not at all middle-of- the-road (as Professor Violet had identified him) in his language, using words like ‘they’ to describe the calculus reform movement, ‘preach’ to describe their representation of reform ideals, and ‘fad’ to describe the way the focus of the reformers was presented to students: Well, I embrace some of what they preach. They talked about this rule of three. I mean, I think--in a way--that was kind of a fad that they put together, because that was a convenient way to try and
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convey in a package what they were trying to do. [Interview, Professor Green] That Professor Green calls the rule of four the rule of three is significant, as this statement implies reference to an earlier edition of the Hughes-Hallett (1995) text, the one which Professor Violet described as “terrible” for its omission of various traditional calculus topics the authors deemed obsolete. The earlier edition had not included a focus on integration by parts or partial fraction expansion, two topics traditionally taught in calculus. Although Professor Green mentions all four of the main facets from the expanded later edition’s rule of four, it is possible that in both cases the participant had a defining impression of the Hughes-Hallett text already cemented before the book’s authors could come out with the compromise of the later edition. Just as Professor Violet had, Professor Green saved some of his sharpest criticisms of reform for the Hughes-Hallett reform text itself: I did not like the text. It was not a book that I felt was particularly effective for what I wanted to do in the classroom. I thought the problem sets were for the most part unsatisfactory…The book seemed to me to suffer from being written by committee. There were lots of authors on it, and I felt that there were inconsistencies within the text because of that, from one section to the next. So I struggled with that. There were chapters, sections, where I ended up rewriting, giving out all of my own exercises, rewriting completely the presentation of the material because I just wasn’t pleased with what I saw. I didn’t want to follow the book. [Interview, Professor Green] The closest Professor Green came to embracing anything related to reform was when he admitted to incorporating some elements of reform methodology into his teaching (although, like Professor Violet, he explained that he used those long before the reform movement championed them): On the other hand, I think that from the start I’ve taught calculus in a way that embraces certain parts of the reform movement. I think I was always one of those people who thought you have to know
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your audience when you teach this course. You cannot teach as if you’re teaching to people who all want to be mathematicians. [Interview, Professor Green] Still, this excerpt seems in direct contrast to his criticism of at least one of the facets of Hughes-Hallett’s rule of four: I think one of the problems with the reform calculus movement was that it tried to make calculus more than it was. It tried to apply it in what it thought or what it advertised as “real problems” in all of these different areas. Well, as soon as you do that, you’re trying to do too much. [Interview, Professor Green] This rejection of the ‘applied‘ nature of the reform approach is surprising coming as it does from a professor whose background is applied mathematics. It echoes very strongly an article titled Calculus: Reformed or Deformed?, mentioned in the literature review, in which the author objects to evolution of her department’s mission from the teaching of mathematics students to the teaching of Physics majors, engineering majors, and others in math-based fields (Kleinfeld, 1996). In Professor Green’s interview, as in the Kleinfeld article, which appeared in American Mathematical Monthly, a mathematics professor laments calculus reform as a blurring of the purity of his/her department’s main mission. This reaction to reform was actually anticipated by Kaput (1997), who predicted that it would be difficult for the then-nascent reform movement to take a foothold in college mathematics departments because of an innate resistance to change among traditional mathematics faculty. Kaput used an analogy of trying to renovate a house while the owners are still inside. To follow this further, Professor Violet (as are those faculty members who share his views) was the homeowner who never agreed to the renovations in the first place, while Professor Green realizes that renovations may be necessary, but does not agree with all of them. Perhaps this duality, the simultaneous attempt to see the value of some elements of reform while being strongly opposed to certain aspects of it, is what separates Professor Green most from his friend, colleague, and fellow self-identified traditionalist, Professor Violet. While Professor Green still sees calculus reform as a faddish
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movement driven by funding, he can see elements of the reform ideology that parallel his own self-reported teaching methodology: The conceptual and the geometric really go hand in hand, because when you try to present the basic ideas, say of Calculus I, the way to do it is geometrically if you’re trying to get at the concept. The numerical, there’s a question in my mind what that means, exactly. I think computational skills are important, but those are oftentimes algebraic skills that the students-- those aren’t really numerical. I’ve never felt that the intensive calculator problems that are common in the reformed method-- I’ve never thought that they were that useful. [Interview, Professor Green] So in effect, the last two excerpts show that Professor Green accepts two of the four main facets of the reform rule of four ideology forwarded by the Hughes- Hallett text (conceptual understanding and geometric presentation) while rejecting the two others (real world applications and calculator-based numerical approximation problems). Granted, this represents only a half-hearted embrace of reform embrace, but two out of four is still two more facets than Professor Violet had accepted before him. Finally, Professor Green, although able to see some benefits of the reform ideology, still preferred those reform elements to be delivered to his students through a more traditional presentation. He stressed the importance of lecture as the primary means of delivery of instruction, and he far preferred the traditional text that the department returned to post-experiment, as when he was asked to contrast the contents of the two texts: [Stewart] forces the people who teach the course to at least spend some time on some of these areas that I think are important for a solid calculus course…. It’s not an extreme traditionalist approach at all; it’s more middle-of-the-road. I think the Hughes-Hallett text was extreme the other way, and it forced that on the instructors. [Interview, Professor Green]
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After these first two interviews were analyzed, it was clear from the data that the opinions held by Professors Violet and Green were in keeping with both the results of the survey and the departmental vote that returned the department to a traditional text by a more-than-2-to-1 margin. But more fascinating was the similarity between these participants. Both had a greater understanding of the reform movement than I had anticipated; both had a working knowledge of the structure and presentation of the Hughes-Hallett text—and yet both felt that the text did not benefit, but actually could harm the successful education of their student audience. Compare these excerpts: I thought the problem sets were for the most part unsatisfactory. There were good problems here and there, but there were not enough problems that I would consider training problems. There were not enough problems that showed off the power of calculus, which I think is important. [Interview, Professor Green] and The original [Hughes-Hallett text—2nd edition] I thought was (personally) I thought it was terrible, it didn’t have integration by parts, for example, I mean this is a thing that our applied mathematicians were appalled at. [Interview, Professor Violet] Both participants object to the text contents in the reform text; the only real difference is in the vehemence of the language—the adamant anti-reformer calls the omissions “terrible”, the more tact-conscious ‘fence-sitter’ simply calls them unsatisfactory. Assertions And Supporting Data —Professor Red Professor Red both self-identified, and was identified by the other participants, as a reform advocate—in his own words, he was “more in the reform camp”—while in the words of Professor Violet he was a “true believer”. A primary assertion derived from his interview data was that while he was as knowledgeable as his fellow participants about the aims and foundation of the reform movement, he was not teaching with any more emphasis on reform innovations than his more traditional colleagues. The following excerpt is in
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response to a question about the importance of group work, calculator technology, dialogue, and class participation in his teaching: “I think theoretically they are important. I don’t use those things as much as I did a couple of years ago; probably, in full point of reason that’s laziness and also some lack of time.” [Interview, Professor Red] The criticism of the lack of time to try these reform innovations is a common one to anyone familiar with curriculum—state and federal guidelines determine the breadth of subjects to be taught, the number of days in a semester are calculated, and pacing schedules are created. Professor Red had actually been tasked with this responsibility at the outset of the reform experiment at Research University, as he was the most familiar with the Hughes-Hallett text. But while the pacing schedule that he created for use with the reform text was almost identical to that used before and after with traditional texts, one might expect that as a reform proponent he would find time for some group work, exploration, and discovery. However, by his own admission, he did not do this. As shown in Figure 1 on page 81, a pacing schedule with annotations hand-written by Professor Red shows that the only reform initiative related to the course pacing (an individual student project) is left up to the instructor, so as to allow for autonomy in the degree to which each instructor embraced even a key tenet of reform. Also, the pacing schedule gives a section number out of Hughes- Hallett’s 3rd edition (this representing the first semester that the 3rd edition was used) for each class meeting, with no single section ever receiving two full lectures (usually about an hour and a half long) to cover. No attempt is made on the pacing schedule to give over whole lecture periods to discussion, exploration, or discovery exercises, or even to work in concert or in parallel on group work or projects. It is apparent to anyone who has ever taught from a pacing schedule that few faculty members teaching from such an outline would feel that they had time to simultaneously get through all the expected sections and still accomplish the seemingly extra-curricular activities that would represent a synthesis of lecture with the other components of a reformed teaching/learning ideology.
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Some might counter that a motivated proponent of reform could have found ways to integrate reform teaching and learning strategies into the pacing shown in Image 1. But it is abundantly clear from his responses that even a self- identified reform advocate like Professor Red did not feel that he could juggle these innovations while still teaching in the manner to which he was accustomed, and so over time his teaching during the reform experiment tended by his own admission towards traditional lecture-based instruction with a reform text. I next asked Professor Red about his personal use of the text, and was surprised to find that he did not actually emphasize the text or its usage by his students any more than his admittedly traditionalist colleagues: The text is not very important. I use it to assign problems. I will tell them to read it, hoping that they will read it, but knowing that not very many of them will. …just to get perhaps a different point of view. It’s not unusual for me not to read the section…to look and see what it covers, to pick out the problems, and present it in a way that I want to present it. [Interview, Professor Red] This is practically a word for word agreement with the teaching philosophy of Professor Violet as regards text usage. That a self-identified reform advocate would forego group work, projects, and calculator-based exploration is odd; that he would not rely on the text or expect his students to read it calls into question why he considered the reform text so fundamental to reformed teaching in the first place. But when probed about his understanding of the philosophy behind the reform movement, Professor Red does show (as the other participants had) a comprehensive knowledge of the reform movement’s philosophy and innovations. What is most unique about this participant is that he really saw no downside to reform—in fact, he did not utter a single criticism of the movement, and responded in an equal but opposite partisan fashion when talking about the move away from reform: [Professor] Violet was the architect of that change, and he’s always been a real advocate of the traditional text and has not liked the Hughes-Hallett text very much. He’s in a position as associate
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chair where he has a lot of power to do that sort of thing. And I think that … those of us who fought the battle to fend that off had just gotten tired. [Interview, Professor Red] Professor Red appears from this excerpt to have laid the responsibility for the return to traditionalist text application in the department on Professor Violet, whereas Professor Violet had tallied votes against/for the Hughes-Hallet text at 20-3. There is documentation in the literature of reform proponents who were hassled and intimidated by traditionalist colleagues; a mathematics professor at Stanford who left the mathematics department for engineering after reforms he sponsored where quashed calls the strife between traditionalists and reformers a “battle for the soul of the profession” (Osgood, in Wilson, 1997). But for Professor Red, the battle is apparently over. He appeared in the interview at times resigned, at other times bitter, and most often just disappointed that the experiment had failed to take. At times, he appeared to take umbrage to questions about the movement itself, as in the following exchange: Interviewer: To what extent do your personal teaching practices reflect the calls for reform from the math education community in the last few years…? Professor Red: I might argue just a little bit that the call for the reform of calculus came so much from the math education community. Interviewer: Where would you say that it came from? Professor Red: But, I think that the—you know, a lot of the authors of those Calculus…the reformed texts were actually in mathematics departments and not associated with the math education community, but that’s not the point of the question, so… I need the question again. Like his fellow participants, Professor Red equates the reform textbook and its authors to the reform movement and its founders, whereas the question referred to the contributions from the NCTM and other such education-related research groups. I tried to further explore his feelings about the relationship between
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mathematicians and mathematics education, and while he answered the question by proxy, the answer was still striking: Well, my wife is a math educator…My wife does tend to think that a lot of the people in the math education department and in the school of education could profit by a lot more time spent in the classroom. [Interview, Professor Red] Intrigued, and a little frustrated that he was not willing to proffer his own opinion, I asked a follow up: “Do you think that mathematics educators are out of touch with mathematics instruction?” I don’t have a lot of personal experience in that realm, but my wife does, and she does definitely believe that, and that’s something we talk about a lot, and I’m willing to take her word for it. [Interview, Professor Red] While these excerpts would be more significant if Professor Red actually discerned a math education presence behind calculus reform, they nevertheless bring up a fascinating scenario. While it is normally considered taboo to even discuss interdepartmental rivalries or other anxieties, literature does exist about the rift between mathematicians and math educators (Selden, 2002). Perhaps the only way that Professor Red could rationalize his advocacy of reform was by associating it with his department, and not his wife’s. Professor Red’s knowledge of and devotion to the ideology of calculus reform did not translate into a markedly different teaching style, which he chalked up to pacing constraints. From the interview data excerpted here and in Chapter 6, Professor Red could be seen as a reluctant traditionalist, or a traditional instructor in reform clothing. Synthesis There were fascinating connections between the data when analyzed— among the most startling was the degree of alignment between the participants. All three professors admitted to using the text in a very limited scope, as essentially a homework bank. All three instructors felt that many of the strategies associated with reform were not effective within a traditional calculus course curriculum. None of the three felt that curricular change was necessary to go
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along with the usage of a reform text. All three felt that mainstream traditional texts had moved towards reform. Finally, all three participants equated the reform text used at Research University with the reform movement. Thus, while the self- identification of each as reform advocate or traditionalist was fascinating for the partisan undertones involved, it is in fact the shared perceptions of these faculty members that flavored the reform experiment at Research University. Since the department these men represented overwhelmingly voted to return to a traditional text at the end of the experiment, it can be inferred that these perceptions were by no means exclusive. The most exciting thing about this research is the direction it represents for future analyses—how prevalent are these views on reform nation-wide, or even globally? What percentage of college math professors (calculus or otherwise) use the textbook in this manner? What percentage of traditionalists views reform textbooks as representative of the face of reform? This research represents the tip of the iceberg, not just for a new generation of qualitative analyses but for quantitative and hybrid projects as well. Below, in Table 4, the individual participants’ alignment with the main facet of Hughes-Hallett’s “Rule of Four” is analyzed in a rubric. One can see that all participants acknowledged the application of traditional analytical and even conceptual or geometric emphases, but none applied the verbal component. On the following pages, a matrix of excerpts from each of the three participants is created to show this synthesis and correlation on a number of topics related to calculus reform and the specific research questions of this study. In the next section, each of the original research questions, as well as the emergent one, will be addressed by interview data and the literature.
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Table 4 – Participants’ Acceptance of the ‘Rule Of Four’-A Rubric
Professor Professor Professor Violet Green Red
Analytical X X X (Mechanics, Drill)
Geometric X X X (Conceptual, Visual)
Numerical Ø X X (Technology, Data)
Verbal Ø Ø Ø (Discussions, Dialogue)
Legend: X – Participant applied this facet Ø – Participant did not apply this facet
98 Table 5a. Correlation of Data Across Case Studies—Part I
Text Usage By Students Text Usage by faculty Primacy of Lecture
Professor Violet No one ever reads I don’t use it at all. The only reasons to My claim is that students get a lot them anyway. have a text are pacing and a good set out of lecture because I don’t do the of problems. stuff in the book.
Professor Green The text is important to me The book is important in that it To me, the lecture is very important because it is important to the makes more work for me if I’m to my teaching style. students. not happy with the exercise sets.
Professor Red The text is not very important It’s not unusual for me not to read there’s hardly time to give a good I will tell them to read it…knowing the section—to look and see what explanation and have time to go over that not very many of them will. it covers, pick out the problems, and homework problems in the traditional sense then teach it the way I want to. in the time that’s allotted for a calculus class.
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Table 5b. Correlation of Data Across Case Studies—Part II
Flaws in Reform Flaws in Hughes-Hallett Text as Homework Bank
Professor Violet “let’s not stress calculations Personally, I thought it was terrible— A good reason for the students to have because a machine can do it” it didn’t have integration by parts. a calculus book is to have a good set of –I think that’s stupid problems to work on.
Professor Green The group projects that appeared It was not a book I felt was particularly At the calculus level, I think the problem in the reform calculus—I never effective—I didn’t want to follow the sets are what’s important. convinced myself those were book. I thought the problem sets were important to the way I was for the most part unsatisfactory. trying to teach the course
Professor Red There is only so much time in a Calculus reform books tend to soft- The text is not very important—I use Calculus class, and if you spend pedal the theoretical aspects, the it to assign problems. more time on interpretations, then epsilons and the deltas. you are going to spend less time on other things
100 Research Questions What were the historical conditions under which Research University underwent a calculus reform experiment? The department chose a reform text (over some objection) through the workings of reform advocates on the book committee. The department did not begin the reform experiment as part of a concerted effort to change the way calculus was a taught but rather as a way to make sure everyone in the department taught from the same text. No seminars or teaching workshops accompanied the change in text. In fact, the curriculum and pacing of the course were the same as before the experiment. The experiment at Research University did not heed the warnings of Schoenfeld (1995), derived from Bruner’s instructional model; Schoenfeld had cautioned that pedagogy, not just materials, must be changed if calculus reforms were to be successful. But the mathematics department at Research University did not institute any long-term changes in the manner of instruction—in fact, within a few years of the introduction of the reform text, none of the participants was using group work, discussion, discovery, or projects to supplement a lecture-based pedagogy. To summarize, the mathematics department at Research University did not move into the direction of reform because of a department-wide mandate for change, or even because the department was concerned about the failure rate of the course, but rather because a small but vocal group of reform advocates on the book committee felt that the reform text was “the only choice” (Interview, Professor Violet). The majority of the department had previously used a traditional text, and there were already (before the experiment) department members vocally opposed to and distrustful of the Hughes-Hallett text (in large part through experience with the first edition, that which even some of the book’s authors later admitted was overzealous in its omissions (Gantner, Lewis, & Hughes-Hallett, 1998). The atmosphere described by all three participants was almost the antithesis of that prevalent in the pro-reform qualitative experiments described in the literature. What was the nature of the reform experiment?
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The reform experiment was 1-dimensional. Traditionalists and reform advocates alike were expected to use the reform text Calculus (Hughes-Hallett et al., 1998), but received no training and had no oversight on how they applied the text to their traditional lecture-dominated instruction. The faculty who considered themselves traditionalists (Professors Violet and Green) felt that the reform text chosen by the book committee was inferior to traditional texts used before and since, that it did not prepare their students properly for post-requisite math coursework, and most importantly, with suspicions about the merit of the reform movement itself, they felt the text confirmed all of their doubts and suspicions about reform. Of course, they did not feel it necessary to change their lecture styles to conform to the new text, and in all three participant cases, admitted to using the text as little more than a repository of homework problems. Even the self-identified reform advocate Professor Red admitted that he did not expect his students to read the book, nor did he feel obligated to do so, preferring after years of teaching the subject to just familiarize himself with which topic related to which section before beginning his lecture. If the faculty had read the text, they would not have had to read very far to find a wealth of supplemental materials for students and faculty designed to ease the transition into a reformed pedagogy. The third edition of the Hughes-Hallett Calculus text, for example, offers in its preface suggestions for how to use the text effectively, including an introduction to the rule of four, discussions of the development of mathematical thinking, the development of mathematical skills, and an introduction of the supplemental materials available to faculty, including an instructor’s manual with sample tests, a resource CD-ROM, and access (through the publisher) to a Faculty Resource Network with peer suggestions for integrating technology (Hughes-Hallett, 2001, p. viii). When the department switched back to a traditional text in 2004, there was no public outcry, although a few faculty members aligned with reform (Professor Red, for example, and according to the vote tally, at least 2 others) felt that the reform was not given a chance to succeed. Whereas other reform experiments that failed (like the ones at the University of Pennsylvania (Alarcon & Stoudt, 1997)
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and Purdue University (Dubinsky, 1992) mentioned in the literature review) could be judged on how faculty and students reacted to the changes in instruction, it is clear from the data in this study that no long-term changes in instruction actually occurred among the participants at Research University. When the faculty in the department (30 in all, 20 of whom actually taught the course) returned to the traditional Stewart (2002) text for the 2004 school year, it was by a straw poll vote of 20 to 3 with 6 abstaining. This shows something heretofore unexplored in the research of calculus reform: most, if not all, of the available qualitative and quantitative research of calculus reform has been accomplished within departments or at colleges where the reform was accepted by a welcoming group of motivated faculty (Matney, Hurtado, and Ziskin, 1999, Bready, 2000, Speer, 2001). But if the movement has gained less of a foothold in research universities, as these cases suggest, it may be as a result of perception and implementation rather than feasibility. Faculty must be educated as to why the change is necessary, and educated about its effectiveness, and the research behind it, in order to buy into the process. One could not expect researchers who teach to embrace the trappings of reform in the form of a textbook, if a majority was suspicious of the reform in the first place, and devalued the usefulness of any text in the first place. This is one of the overarching assertions that can be drawn from the data in these cases. The main assertion to derive from the data in all three interviews is that the faculty interviewed did not really embrace the text, and as a result, no paradigm shift ever really occurred at the local level: • The original [Hughes-Hallett text—2nd edition], personally, I thought it was terrible .[Professor Violet] • I did not like the text. It was not a book that I felt was particularly effective for what I wanted to do in the classroom. [Professor Green] • The text is not very important. I use it to assign problems. [Professor Red] To put it another way, the text change brought on no paradigm shift in teaching, because full-time tenured research faculty like the three participants do not appear
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to be externally directed to teach according to the given text. The instructors introduced the new text, but in all three cases proceeded by their own admission to teach in the same manner they had before. Even those who implemented changes in the lecture style, like Professor Red, found those changes in contrast or direct contradiction to the departmental curricular restraints. In short, they did not feel they had time to do group work, projects, exploration and discovery while still trying to teach the skills they knew their students would need up the line. Under what conditions did the department complete the reform experiment and return to a more traditional manner of calculus instruction? The reform experiment went on from 1999 to 2004. Eventually, faculty teaching higher level courses complained that the students arriving in their classes after the introduction of the reform text were less prepared and performed worse than those before: [Professor X] sent a letter to the faculty saying we had used that for two years and he felt like it was just a disaster- he was now getting the people who had gone through Hughes-Hallett and he was making these claims...that these people did not know what they needed to know. They didn’t know how to do simple derivatives, they did not know how to do simple anti-derivatives [Interview, Professor Violet] While such complaints were never followed up with research and were thus nothing more than anecdotal, they represent a window into the departmental perception of the reform as something from the outside that was introduced and having a negative effect. How else can one explain the fact that the reaction to this was not a discussion of how the book was being used, an attempt to determine how to teach best with the text in hand, or even an analysis of a quantitative nature, of which there are examples in the literature, but rather an immediate push to drop the book altogether (c.f.a Bready, 2000). It is obvious from the lack of attention to this facet that the participants and the department did not feel a change in teaching style was necessary to improve the teaching of calculus at Research University, but rather that the usage of the book was seen as a casual experiment, and that the minute that the experiment failed to reduce a positive
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goal (whether that goal was improved retention and completion without any loss of comprehension at a pre-reform level was never publicly addressed, or analyzed) it was abandoned. The fact that only 3 of the 30 faculty who teach math at Research University voted to retain the reform text after 5 years says as much about how this casual experiment was undertaken as it does about the qualities of the text itself [Interview, Professor Violet]. As much as anything, it is a mirror into group- think and resistance to change, and the uphill battle reforms must endure (Kaput, 1997). In a Curriculum textbook in use in the Education department at Research University, I found the following quote, one that resonates for this research: “Most people, when they talk about change, and say they welcome it, often would prefer that other people within the organization change” (Ornstein and Hunkins, 2004, p. 301). The fact that all three participants acknowledged the movement of traditional texts towards implementation of facets of reform would be more comforting if they universally saw this as a positive phenomenon. Professor Violet, for example, admitted that traditional texts like Calculus (Stewart, 2002) have integrated elements of reform—but did not feel it necessary to take advantage of them: …They (traditional texts) had incorporated a lot of stuff in them that reformers like. For instance, Stewart (the current Stewart) has a lot of graphing calculator problems, and he has a little calculator or something beside them so you can tell those apart. I don’t use any of those. [Interview, Professor Violet] Professor Red also acknowledges the movement toward reform of traditional texts: “The traditional text, the Stewart text is a lot more like a reform text--it has moved significantly in that direction” [Interview, Professor Red] Even the literature has acknowledged this shift, as in this quote: At least one publisher, Brooks/Cole ITP, is capitalizing on the discontent [between reformers and traditionalists]. James Stewart, who wrote the most popular traditional textbook, is now working on a new one that will use some principles of reform. But it also
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will include many of the basics that critics say other reform books have left out. (Wilson, 1997) Although none of the three participants considered the reform experiment a success, all three admitted to using one or more of the “rule of four” facets in their teaching, and to the logic of some of the foundational ideas that began the reform movement. Professor Violet, while adamantly opposed to reform to such an extent that he wrote an on-line manifesto opposing it, nevertheless felt that teaching epsilon-delta definition of the limit and proofs is lost on an undergraduate calculus student, as did the reform movement’s founders. Professor Green expressed that in spite of the taint of funding pressures on the growth of the reform movement, that the influences reform has had on the traditional text now used are beneficial (specifically, visual representations), and that the result of the experiment could be positive. Professor Red, who invested the most energy in convincing his colleagues to try reform, stated that technology usage in the classroom department-wide, which he attributes directly to the reform experiment, is better preparing his students for a 21st century world. Is it possible that in failing to align the pacing, course curriculum, and instructional style of faculty with the intended calculus reform ideology, the department created an unintentional dissonance between text and delivery? This question emerged from the analysis of the interviews—it is undeniable that the implementation of the reformed text at Research University was not done in the manner envisioned by the founders of the reform movement and the reform text’s authors. It will undoubtedly be quite shocking to most mathematics educators to learn that some mathematics instructors (all three participants) do not read the text from which they are teaching, preferring to use their own tried and true methods. If this is widespread, it begs the question of how to implement reform in a research university, or any college, for that matter. One must expect based on the data from these participants that this is not just true of traditionalists; Professor Red echoed his more traditionally aligned colleagues when he stated “. It’s not unusual for me not to read the section…to look and see what it covers, to pick out the problems, and present it in a way that I want to present it. ”
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[Interview, Professor Red] If, as the founders of the reform movement speculated, teaching methodology is in large part responsible for the woes of calculus, then perhaps curricular redesign and teacher training should predate or replace reform textbooks as the primary delivery of this innovation in teaching (Press, 1987). The interviews with the participants in this experiment revealed that mathematics faculty at a research university can be suspicious of results derived from a teaching college or education department, and would quickly point out that their faculty and students are different from those other environments. Care must be taken in preparing a department for such a profound change of mindset. The most dramatic realization related to this emergent research question occurred when the participants admitted that they did not use the textbook in the manner outlined by the book’s authors, i.e. that they did not do group work, projects, dialogue, and other innovative tenets of reform methodology. • Sometimes it seems like there’s hardly time to give a good explanation and have time to go over the homework problems in the traditional sense in the time that’s allotted for a calculus class. Group discussions--that was never something I was into very much in a calculus class [Interview, Professor Red] • Group projects, I’ve never been a fan of that. The group projects that appear in the reformed calculus, I never convinced myself that those were important to the way I was trying to teach the course. [Interview, Professor Green] It gives the casual experiment undertaken at Research University a totally different contextual background to consider that this was, in fact, a predominantly traditionalist department, teaching calculus in a traditional lecture and homework style, trying to accomplish this with a reform-oriented and –structured text. The fact that the department attempted to supplement the text with more drill exercises, and moreover that they made no attempt to adjust the curriculum pacing to allow for the reform-based innovations like group work and discussion, cuts to the heart of the disconnect and dissonance between these two competing philosophies—it recalls nothing so much as an attempt to hammer a square peg
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into a round hole. The square peg is representative of the ill-fitting reform attempt, the round hole standing in for the department’s lack of flexibility in curriculum. Figure 1 gives a visual impression of this interpretation of the data— the ill-fitted calculus reform experiment at Research University, which attempted to measure improvement of calculus instruction anecdotally by forcing a reform text on a predominantly traditional pedagogy, was as futile and ultimately unworkable as hammering a square peg into a round hole.
Figure 2. A Contextual Visual Metaphor for the Calculus Reform Experiment at Research University: A Square Peg in a Round Hole
Many times during the course of the data collection, the participants equated the reform movement with the reform text in use in the department. In discussing the merits of the reform movement, for example, Professor Violet focuses his entire response on the perceived flaws of the ‘radical’ early edition of Hughes- Hallett These are the things, by the way, that some reform calculus threw out—most of them put the chain rule back in, and most of them finally said “okay, well, we do need at least integration by parts,” and then say “well, we do need to do a little bit more of trig substitution/rationalizing substitution,” and now some of them are (now) saying that “well, we do need to teach more than just power series,” there are reasons for that and convergence of series and in
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particular tests for convergence. They seem to be coming back, but originally they just threw all of that away—and without having any evidence that just because machines can do a lot of this stuff that we shouldn’t teach it.. [Interview, Professor Violet] The use of them and they are partisan terms for the other, but most important in this excerpt is that Professor Violet never actually talks about anything but what the specific authors of Calculus: Single and Multivariable threw out of their first two editions (the latter of which he later described as terrible). For the majority of the mathematics instructors at Research University, it is quite possible that the Hughes-Hallett text itself was the first representation of the reform movement they experienced outside of journal articles. Unfortunately, they may have been queered to the reform movement from the very start by the incomplete nature of the text’s earliest edition, which completely removed such traditional topics as integration by parts and had even less of the drill and rote problems (less volume of homework altogether) than the 2nd edition (Wu, 1997). The 2nd edition also changed from the rule of three to the rule of four, but the damage may already have been done. It is consistent from interview to interview that rightly or wrongly, the participants equated the Hughes-Hallett text with the reform movement. Even the textbook’s primary author has admitted since that perhaps too much was removed in the early editions (Ganter, Lewis, and Hughes-Hallett, 1998) The inability to distinguish between the movement and the movement’s iconic text is completely logical, as it is the only evidence of the reform movement the participants had admitted encountering. This leads one to suspect that if a later edition is more in line with the demands, or rather addresses the concerns of the traditionalist base, the experiment could be salvageable at some later date. This will be addressed in the next section when directions for future research are discussed—and is dependent on the survival of the reform movement as a conscious entity. If reform is to forever be seen as other by mathematics faculty, however, then the best one can hope to achieve is the continual realignment of traditional texts towards a consensus or hybrid text.
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Limitations This research would not be complete without an acknowledgement of certain limitations that resulted from the sensitive relationship between interviewer and participants. While the interviewer was seen as a petitioner and the participants as sages, attempts to observe classroom teaching methods were not allowed by the administration of Research University. The department official from which access to the classroom was solicited felt that since the nature of the research was a historical analysis of faculty perceptions, that this was unnecessary; while there is some logic to this, it is also true that almost no department would feel comfortable about outside entities ‘judging from on high’ the teaching going on in the classroom, even if only to confirm the results of qualitative data. When my status changed from mathematics master’s student to mathematics education doctoral student, there was definitely a change in the access that I was granted. One anticipated hurdle that all similar future research projects must encounter is this: access to any university math department that is not “excited about reform” will be a challenge for any would-be-petitioner (Matney, Hurtado, & Ziskin, 1999). A second limitation was that member checks were deemed unwise. Due to the brutal honesty of the interviews, there was the fear that should the faculty see their own responses, no matter how exactingly they were transcribed, they might impede the completion of the research. It was therefore left to the individual interviews to check each other, rather than to depend on the participants after the fact to vet their own statements. Future researchers will have to individually determine the wisdom of confronting their participants in such an analysis with such volatile data—in this case, a pleasant working relationship between departments was deemed more important than such an (even academic) confrontation. Directions for future research After years of rigorous inquiry, a rather intimidating realization dawned on this researcher—this is just the tip of the iceberg. Among the most promising follow-ups to this research is the obvious but necessary task of doing similar
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qualitative analyses of other research universities and their faculty. While one of the things that made this research so powerful was the unique availability of its faculty to participation (this was possible because I was an active participant in both the mathematics and education departments at the same time, at ground-zero, so to speak), other universities must assuredly have had similar experiments, and undoubtedly with varying degrees of success. Among others, UCLA, USC, and the University of Iowa have all at one time or another scrapped efforts at reform—surely, there are some fascinating historical data sets at these and other institutions. It would be extremely beneficial to see research done on a department that was more successful in the implementation of such reform—what was done differently? But a number of such interview-based projects must be undertaken, so as to paint a broad picture nation-wide of the state of calculus reform in research university mathematics departments. Studies have been accomplished at the state and national level, as well as research focusing on entire countries’ acceptance or rejection of reform, but that attention needs to be focused inward as well (Hurley et al., 1999). A meta-analysis or overview, and the big picture that will result, can only be accomplished with the co-operation of many researchers bent on the same goal: an insight into the delivery of instruction in the average college mathematics department. Even the founders of the reform movement have long called for empirical qualitative and quantitative analyses of college mathematics programs (Gantner, 1994) Another area rife with possibilities, but difficult to broach, is the relationship (or lack thereof) between mathematicians, as the primary deliverers of college math instruction, and mathematics education researchers, the primary researchers on instruction. The participants in this study did not open up at length on this subject, which has been addressed by such prominent mathematics educators as Annie Selden (2002) and Willi Dörfler (2003). In fact, only Professor Red actually responded to a question about the relationship between mathematics educators and mathematicians, and he refused to share his own opinions, preferring to cast them as his wife’s opinions that he agreed with. This is problematic, as the openness of the willing participants in this study might not be
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repeated for this somewhat taboo topic. However, it should be obvious to any involved person in either field that the absence of, or ban on such research has not helped to bridge the distance between two such fundamentally related departments. There is an inherent dissonance, similar to that between traditionalists using a reform text in a traditional manner, if mathematics researchers research the methods and manners of instruction but mathematicians who are the primary deliverers nationwide of collegiate math instruction are not involved in that research. The only way to attack this topic, and the taboo of discussing it, is head on. Brave researchers, and brave participants, need to be willing to go on record stating that they do not communicate with one another (as all three participants in this research admitted) before bridges can be successfully built between these two important and related departments. The significance of this study to the scientific community goes far beyond the gap that it fills in assessing the success of a calculus reform experiment in a research university. For students of methodology and curriculum, this research represents a cautionary tale of a resistance to change and a departmental experiment that was not constructed so as to succeed. For the proponents of reform, this research can show how not to institute reform in a predominately traditional department (and might provide insight into what would be more successful; namely, a research-rich introduction of faculty to the benefits with what is called in industry the ability to ‘buy into’ the reform. For myself, this research helped me to think about my teaching and the effects (good, bad, and indifferent) that my years using the Hughes-Hallett text had on my pedagogy. And finally, for mathematics faculty, this research represents a snapshot of a research university math department in transition, and might provide math faculty with a confirmation or contrast of their own departmental relationship with reform. Perhaps further qualitative analysis of active or historical reform experiments can expand the cross section of perceptions that this research has added to the voices of reform in the literature. Sadly, and this may be indicative of the state of reform, there are less articles written now about reform than ten years
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ago—it may be that this research represents a microcosm of a similar disconnect occurring in math departments around the world. In conclusion, the most profound result of the three independent case studies accomplished within the math department at Research University is the snapshot they provide of a department in transition. At first, the transition was between business as usual and the status quo of traditional teaching of calculus. But as the reform experiment ended (with overwhelming support for the return to a text viewed as more traditional), the case studies provide a view of a department post-reform, as well as providing important insights into the limitations of change without planning, and curricular reform and implementation on a surface level. It is not a revelation to anyone who reads this research that college professors, like most people, resist change. However, any faculty preparing to make a text change for philosophical reasons could benefit from analyzing these case studies, as could the publishers of future reform, traditional, or hybrid texts trying to learn more about their audience. This research has forced me to analyze my own teaching, and the extent to which I actually integrate reform elements into my teaching philosophy. It will be deemed successful if it has a similar effect on any other reader.
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APPENDIX A SURVEY QUESTIONS
1. What calculus courses do you teach? 2. How many times have you taught using a traditional text? How many times using a reform text? 3. How useful was the Hughes -Hallett textbook in your approach to calculus instruction? 4. For how many years have you been an educator? 5. In what capacities have you taught? 6. Which model for teaching calculus do you use, reform or traditional? Which method do you favor? 7. If you teach, or have taught, reform calculus, did you first receive training in how to teach in a reform manner? 8. How would you rate the quality of calculus instruction at this institution? How responsible (from 1 to 5; 1 is least responsible, 5 is most responsible) are students, faculty, and the departmental text for the quality of calculus instruction at this institution?
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APPENDIX B
HUMAN SUBJECTS IRB APPROVAL LETTER
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APPENDIX C
HUMAN SUBJECTS IRB SAMPLE INFORMED CONSENT FORM
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REFERENCES
Alarcon, F. E., & Stoudt, R.E. (1997). The rise and fall of a Mathematica-based calculus curriculum reform movement. Primus, 7(1), 73-88.
Alexander, E. H. (1997). An Investigation of the results of a change in calculus instruction at the University of Arizona. Unpublished doctoral dissertation, University of Arizona.
Anderson, J.R., Reder, L.M., & Simon, H.A. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review.
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BIOGRAPHICAL SKETCH
Douglas MacArthur Windham, Jr. is the offspring of generations of educators. His father was a distinguished-service university professor of Economics of Education, and his mother was a high school English and Art History instructor. His great grandfather was the sole teacher in a one-room schoolhouse on an island in Nova Scotia, where his grandparents met. After completing 5 years of active duty service in the United States Air Force as a Russian linguist, Douglas matriculated at Florida State University, the alma mater of his parents, with a B.S. in Electrical Engineering and Russian. He then completed a M.S. in Electrical Engineering at the University of Illinois at Urbana- Champaign. His thesis research was entitled “Optical Spectroscopic Electron Temperature and Density Diagnostics Side-on on an Electromagnetic Railgun Plasma with Pellet in-situ”. After receiving his first Master’s Degree, Douglas returned to Florida to teach Physics, Mathematics, and Film at an alternative Adult High School at Seminole Community College. In 2000, Douglas returned to graduate school, this time at Florida State University, where he received a second Master’s Degree in Applied Mathematics with the ultimate goal of teaching at the college level. His thesis was entitled “A Geometric Modeling of 4-image Gravitationally-Lensed Quasars”. Since 2003, he has been a full-time Professor of Mathematics at Tallahassee Community College. In his spare time, Douglas frolics with his lovely wife Kimberly and their cats. He enjoys biking, hiking, traveling, and tennis, and is a DJ at a college radio station. He hopes that on completion of this degree, he will never have to do anything this difficult ever again.
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