Classroom Culture, Mathematics Culture, and the Failures of Reform: The Need for a Collective View of Culture
by Michele Gregoire Gill & David Boote — 2012
Background/Context: Despite the tremendous amount of effort devoted by many mathematics educators to promote, defend, and implement reform-based mathematics education, procedural mathematics, which locates mathematical correctness in the procedures learned from textbooks and teachers, persists. Many researchers have identified school and classroom culture as the source of the problem; however, the exact meaning of school culture and its influence on teachers’ practices remains unclear. What is needed is a clearer understanding of classroom culture and how it influences practice.
Purpose: The purpose of our study was to examine how the aspects of a culture reinforce each other (and how they resist aspects alien to the cultural system) to understand the sui generis nature of culture. We use five aspects or indicators of culture—language usage, standard practices, tools and equipment usage, ongoing concerns and values, and recurring problems—to describe how they work together to create a culture.
Population/Participants/Subjects: The primary participant in this study was an eighth-grade mathematics teacher renowned for being a good teacher whose teaching conformed to the intentions of the reform-oriented National Council of Teachers of Mathematics (NCTM) standards, with a particular emphasis on problem solving.
Research Design: An ethnographic case study was conducted.
Conclusions/Recommendations: Although Ms. Bryans appropriated some of the rhetoric and practices of reform mathematics, her goals and assessment methods and most of her instructional methods were inconsistent. The analysis of the case shows that three conceptions of culture—individual, interactive, and collective—lead to quite different understandings of the problem. This case suggests the importance of differentiating each of these three conceptions of culture and discusses their implications for educational reform policies and professional development efforts.
Although the community of mathematics education researchers has reached broad consensus about the goals of mathematics education and at least some agreement about the kinds of curricula, instructional methods, and assessment methods that do and do not support those goals, mathematics education in U.S. K12 schools remains largely unchanged (Ross, McDougall, & Hogaboam-Gray, 2002; Stigler & Hiebert, 1999). A number of studies have investigated why teachers are either unwilling or unable to implement these reform practices (Ambrose, 2004; Ma, 1999; Orrill & Anthony, 2003), and among such studies, school or classroom culture is often identified as the cause of the problem (Cohen, 1990; Hiebert & Wearne, 1993; Turner & Meyer, 2004).1 Yet what these authors mean by culture or the role that culture plays in mathematics education reform is at best vague.
We first review the literature on the challenges of implementing problem-solving-based practices in mathematics classrooms. We then identify three quite different notions of cultural representations: individual, interactional, and collective (Nicolopoulou & Weintraub, 1998). This framework of three conceptions of cultural representations is then used to present and reanalyze an ethnographic case study of one experienced middle school mathematics teacher who is recognized in the local mathematics education community as an exemplary problem-solving mathematics teacher. By reanalyzing this previously unpublished case study, our purpose is to investigate the nature of classroom practice and how it is supported by the culture of a classroom.
CULTURE AND CURRICULUM CHANGE
Reform mathematics education had been characterized many different ways, but leading mathematics educators and researchers agree that its curricula and methods of instruction should promote deep conceptual understanding via problem-solving (Hiebert et al., 1996; Mayer & Wittrock, 2006; National Council of Teachers of Mathematics, 2000).2 Problem-solving may or may not involve authentic real world problems, but it must be taught, according to a consensus document of the leading experts in the field of mathematics education, so that the subject matter is problematized (Hiebert et al.). In other words, rather than have children work on arithmetic exercises, teachers should engage students in trying to view the world mathematically, subsuming computation and memorization to deep conceptual understanding of mathematics. For instance, if subtraction is being taught, a problem might be presented asking the class how much taller Keisha is compared with Taylor. Instead of giving students the subtraction problem or the algorithm for solving it, the class engages in trying to figure out the problem on their own, using a ruler and multiple methods for determining the difference in height. Conceptually focused mathematics is the foundation of so-called constructivist mathematics reforms, particularly the influential National Council of Teachers of Mathematics (1989, 2000) standards. In contrast to earlier academic mathematics curricula, the idea that mathematical understanding is best achieved through problem-solving is supported by philosophical, sociological, and anthropological perspectives that view mathematics as fallible and quasi-empirical (Tymoczko, 1986). These social constructivist ideas hold that justification for knowing is centered in the community of learners, not in the authority of the text or teacher (Palincsar, 1998). As a result, those reformers associated with problem-solving mathematics share an ontological presupposition that mathematics is primarily a problem-solving activity and should be viewed as a dynamic, continually expanding field of human creation and invention, a cultural product. Mathematics is a process of inquiry and coming to know, not a finished product, for its results remain open to revision (Ernest, 1989, p. 250). This ontological belief, in turn, leads to the epistemological belief that mathematical knowledge is constructed within communities of learners (A. L. Brown, 1997) and what Sfard (1998) called the participation metaphor of learning. In contrast to purely academic traditions of mathematics education, one goal of problem-solving mathematics is the Deweyian ideal of democratic participation, presupposing that mathematical understanding is essential for everyone in contemporary society (e.g., Hersh, 1986; Kitcher, 1986; Lakatos, 1986; Lerman, 1983).
Despite the tremendous amount of effort devoted by many mathematics educators to promote, defend, and implement reform-based mathematics education, procedural mathematics persists. It locates mathematical correctness in the procedures learned from textbooks and teachers (see also Battista, 2001; National Council of Teachers of Mathematics, 1989, for a comparison of traditional and reform perspectives). The persistence of procedural mathematics in U.S. schools vexes many reformers. Although the NCTM standards (National Council of Teachers of Mathematics, 1989, 2000) have been in existence for over two decades and have received widespread support from mathematics educators and researchers, there is little evidence that practice has significantly changed (TIMSS Video Mathematics Research Group, 2003). Meanwhile, several studies have documented that procedural mathematics teaching is associated with numerous negative outcomes for students, from a lack of conceptual understanding, to the inability to solve real-world problems, to compliance and overreliance on the teacher as authority figure (Battista, 2001; Ross et al., 2002). Considering the negative outcomes associated with the procedural mathematics curriculum, many want to know how teachers can implement more effective practices.
Surprisingly, many teachers already believe they are implementing reform-based mathematics. In a key cross-cultural study of mathematics education, although 70% of U.S. teachers said that their videotaped lessons aligned with the NCTM standards to at least a fair degree, most of the observed lessons were inconsistent with the intent of the standards (Hiebert & Stigler, 2000). For example, 96% of U.S. students time during seatwork was spent practicing procedures, compared with Japanese students, who evenly divided their time between creative mathematics (inventing procedures and original problem-solving) and procedural practice. Further, U.S. teachers (78%) were about as likely to simply state concepts as develop them, whereas teachers in Germany (77%) or Japan (83%) were more likely to develop concepts. Finally, whereas 53% of Japanese lessons contained instances of deductive reasoning or working through proofs, no lessons did in the U.S. sample. After extensive analysis of the cross-cultural videotape data, Hiebert and Stigler concluded that teaching is a system embedded in a cultural context bound by an invisible script that determines the course of daily lessons and thus is difficult to change.
A number of researchers in mathematics education have sought to understand the persistence of the procedural mathematics curriculum in the face of widespread support of problem-solving-based mathematics and nearly two decades of curriculum changes in in-service and preservice education (see Schoenfeld, 2006, for a review). Many of these studies have identified school and classroom culture as the source of the problem. For example, Stigler and Hiebert (1999) asserted, Teaching is a cultural activity, not learned deliberately, but through scripts based on experience and is consistent with the beliefs and assumptions of a culture (p. 87). Most studies of the persistence of the procedural mathematics tradition have centered on novice teachers. Cooney and Shealy (1997), for instance, found that the teacher they observed did not teach in a way that reflected the reform-minded beliefs he expressed during his preservice program (see also Ball, 1990; Gregg, 1995). More generally, Ernest (1989) suggested there are two causes for mismatch between teachers beliefs and practices: (1) social context, including the expectations of students, parents, colleagues, and administrators, and the institutionalized curriculum, including textbooks and standardized assessments (see also Borko & Putnam, 1996; Joram & Gabriele, 1998), and (2) teachers level of consciousness of their own beliefs and extent of self-reflection (Ernest, 1989; see also Philipp, 2007; Raymond, 1997; Stipek, Givvin, Salmon, & MacGyvers, 2001; Thompson, 1992).
Notice that in all these invocations of the importance of culture, the exact meaning of school culture and influence on teachers practices remains unclear. Culture, it seems, is magicalbeyond clear description but very powerful. This problem is not limited to mathematics education reforms, but pervades the sociological and anthropological literatures in education and beyond. Latour (2005) sees this problem every time people invoke culture as an explanation rather than what needs to be explained. Using school or classroom culture as an explanation for the persistence of teachers mathematics instruction is no more helpful than saying that magical spells or aliens determine teachers practice. What we need, by contrast, is a clearer understanding of classroom culture and how it influences practice.
THREE CONCEPTIONS OF CULTURAL REPRESENTATIONS
To understand the limitations of the existing literature on the effect of school and classroom culture on attempts at mathematics reform, we can use Nicolopoulou and Weintraubs (1998) typology of individual, interactional, and collective representations. The individual perspective on cultural representations is central to the psychological analysis of mathematics education reform and is concerned with understanding how individuals understand important ideas and practices so as to better influence how teachers think about mathematics and mathematics education. We see this in Cooney and Shealys (1997) research on preservice teachers, in which school culture is portrayed as overpowering individuals beliefs about effective instruction. This analysis seems plausible, but how exactly school culture is learned or manifested in the individuals beliefs remains unclear. Such studies tend to use interviews and surveys, collecting data about teachers beliefs and attitudes. The current zeitgeist of research on teachers resistance to reform is situated in this perspective, most notably research on mathematics teachers beliefs and affect (see Phillip, 2007, for an exhaustive review of this literature), and research on teachers professional identity (e.g., Lasky, 2005). This analysis of individual representations reinforces the traditional notion that changing mathematics instruction should focus on changing teachers and students thinking. Moreover, this analysis of individual representations reinforces the belief that learning to teach mathematics correctly is primarily a matter of somehow overcoming school culture to enable teachers to think correctly about mathematics and mathematics education so they can teach correctly.
In contrast to the individual perspective, a number of sociocultural researchers have argued for an interactional perspective on mathematics education and educational research (e.g., Greeno, 1998). Simply, mathematics and mathematics education are not purely psychological or individual phenomena; teachers and learners must interact with each other, their materials, and so forth. These interactions are situated socially and physically, and over time clear patterns of behavior emerge among these interactions. To understand mathematics education, it is crucial to understand the patterns that emerge from these communicative and physical mediated interactions. Communicative and physical mediation will be particularly evident and amenable to examination during prolonged engagement and observation in mathematics classrooms, such as Greenos analysis of how a new mathematics game can change the communicative dynamic of a classroom. In Nicolopoulou and Weintraubs (1998) taxonomy of representations, these are interactional representations. They are representations that arise from the specific, situated interactions among people and objects, within a specific context. Unfortunately, these representations cannot explain, for example, why most mathematical reforms regress to procedural mathematics over time or what is different in the few cases in which the changes persist (Engström, 1998).
The work of Cobb and colleagues also relies on an interactional perspective on cultural representations (Cobb, McClain, de Silva Lamberg, & Dean, 2003; Cobb & Yackel, 1996, 1998). This program of work explores the relationships among sociomathematics norms, classroom norms, mathematical practices, mathematical and curricular objects, communities of practice, and so on. Their analysis emphasizes the problems with reducing the problems of mathematics education reform to individual cognition or motivation, and instead emphasizes the importance of the ways that people interact with each other and with material objects within their communities of practice. Unlike researchers who emphasize individual cultural representations, especially language and values, Cobb and colleagues examined the interrelationships among these aspects of culture.
Our purpose is not, of course, to suggest that either individual or interactional representations are unimportant. Indeed, these kinds of representations deserve close empirical examination and have received it. However, focusing exclusively on individual and interactional representations, as does the existing literature on understanding teachers reluctance to adopt problem-solving-based mathematics, misses the irreducible sociality of mathematics education.
In his foundational work in sociology, Durkheim (1897/1997) argued that the study of culture is the study of social facts and that social facts are sui generis, irreducible to another level of analysis (see also Garfinkel, 2002; Latour, 2005). For example, by arguing that suicide is a cultural phenomenon, Durkheim tried to help us understand that suicide could not be reduced to solely psychological explanations (i.e., individual thoughts or situated interactions). By looking at the broad sociological patterns of suicides, Durkheim saw that people in certain religions, occupations, and economic standings were more likely to kill themselves because their cultures did not stress healthy social norms, relationships, integration, or support. Although Durkheim did not deny that psychological reasons played a role, he argued that psychological reasons alone were insufficient to understand individual behavior. He proposed the following law: The determining case of a social fact should be sought among the social facts preceding it and not among the states of the individual consciousness, followed by the codicil: The function of a social fact ought always to be sought in its relations to some social end (Durkheim, 1895/1950, p. 110; see also Star, 1989). The individual and interactional views of cultural representation ignore that schooling in general and traditions of mathematics instruction are also cultural, emerging from social activity and directed toward social ends.
Procedural mathematics has a long history as a social institution; procedural mathematics predates the recent problem-based conceptions of mathematics by several millennia. The procedural mathematics education tradition is a social institution, thousands of years old and spanning the globe; it cannot just be reduced to the thoughts, beliefs, and identity of individuals. It survives as a social institution because of the instrumental value of learning mathematics, perpetuated by the common belief that mathematics is an accumulation of facts, rules, and skills to be used in the pursuance of some external end (Ernest, 1989, p. 250; see also Gregg, 1995). Babylonian students learned procedures for accounting, ancient Egyptian students learned procedures for surveying, and early Chinese scribes learned methods of calculating dates (Katz, 1993). There is no suggestion in the scant documentation available that these accountants, surveyors, and scribes were taught to understand why these procedures worked or to inquire into the mathematical ideas involved. So it was until very recently, until the U.S. curriculum reforms of the 1960s and similar reforms around the world: For most of history, students were taught mathematical procedures in school because those procedures were presumed to be useful in adult life. Only a small percentage of students were even thought capable of understanding or inquiring about mathematical ideas, and even they did not learn anything resembling an academic curriculum until late adolescence. To believe that only problem-based mathematics is cultural is either naïve or disingenuous. Following Durkheims (1895/1950) injunction and methodological program encouraged us to investigate the social ends served by the culture of procedural mathematics education.
Innumerable specific claims about the social ends of schooling have been forwarded, but the most general goals of schooling are to (1) prepare children for what we imagine their adult lives to require, including the skills and knowledge to be useful to (or at least not a burden on) society; (2) ensure that they learn the norms and values expected of them as adults; and (3) sort them into social categories (Apple, 1979). In turn, the purposes of procedural mathematics have mirrored many of the purposes of schooling, ensuring that children learn the skills that society values and learn their place in society. Several authors have argued that the resilience of the procedural view of mathematics is tied to its ability to sort students and inculcate hegemonic culture, perpetuating existing social stratifications (Apple, 1992; Nasir et al., 2008; Noddings, 1994; Secada, 1989). Although empirical evidence clearly supports that these are the outcomes of mathematics education (summarized in Schoenfeld, 2002), no one has yet provided an entirely satisfactory explanation of how these values are perpetuated through established classrooms norms and practices. Nonetheless, the procedural mathematics tradition has served the traditional goals of schooling.
Although understanding the goals of schools and procedural mathematics is crucial, it is inadequate. The failure to see the sui generis nature of cultural representations is the great weakness of much of the current culture talk in educational research. By focusing on, say, teacher beliefs, attitudes, identity, or the formal structures of a school, as much of the literature does, we miss the interconnections among these aspects of culture. This collective concept of cultural representations leads to our methodological strategy, adapted from Latours sixth rule of method (1987). He suggested that when we are presented with interpretations of behaviors as naïve or even irrational, researchers should instead study the peculiarities of the setting and their practices. Only when the researcher cannot explain behaviors from the social contexts should they rely on individual or psychological explanations.
For our purposes, we use five aspects or indicators of culture: language usage, standard practices, tools and equipment usage, ongoing concerns and values, and recurring problems (adapted from Denning & Dargan, 1996; see Figure 1). The language used by individuals helps us to understand their thinking, as the individualist perspective on culture does, but patterns of language use in a culture also helps us to see what they value and (by omission) do not value. How people use tools and equipment indicates a cultures values and its perceived problems, and shapes what people spend their time doing and not doing. The recurring problems within a culture give us insight into their values, standard behavior, and language usage. Thus, by examining how the aspects of a culture reinforce each other (and how they resist aspects alien to the cultural system) we can see the sui generis nature of culture. Following Durkheim (1895/1950), the point is not just to enumerate the factors, but instead to describe how they work together to create a culture. This ethnographic case study tries to do that.
Figure 1. Relationships among the five aspects of culture
To find a counterexample to the many published studies of unsuccessful implementation of problem-solving-based mathematics, the goal of the original study3 was to document how an experienced problem-solving-based teacher helped low-level students with cognitively challenging material. Two university-based mathematics educators independently identified Sylvia Bryans4 as a good teacher whose teaching conformed to the intentions of the NCTM standards; her principal corroborated this assessment. Ms. Bryans, as she preferred to be called, is certified to teach mathematics in Grades 612 and has been teaching mathematics for 20 years. By choosing an experienced teacher, the first author also hoped to see how her teaching had consolidated into coherent practice.
Ms. Bryans sponsored Mu Alpha Theta, a competitive math club, and designed the eighth-grade pre-algebra curriculum, the class observed for this study. She also taught algebra 1 honors and algebra 1 to eighth graders. In her initial interview, Ms. Bryans explained that her teaching changed dramatically 15 years earlier, when she attended a weeklong problem-solving workshop taught by a professor from the local university. She described her classroom as pretty traditional before attending the workshop. After the workshop, she began to use these problem-solving activities . . . for mental math and front-end activities. . . Because of him [the university problem-solving expert], I changed my style of teaching.
During the study, Ms. Bryans was in her first year at Latham Middle School (LMS), having transferred from another middle school because the latter wanted her to teach a prescribed curriculum that she believed was opposed to her more activity-oriented curriculum. LMS is located in a suburban area of a midsized city in the southern United States and has a teacher-student ratio of about 1:19. It serves sixth through eighth graders, and 28% of its students are non-White. Many of the policies and practices implemented by LMS are those that characterize exemplary middle schools (George & Alexander, 1993). For instance, they have block scheduling of four periods per day that last for 80 minutes each, with the addition of a half hour for lunch. Each grade is housed in its own building, creating a school-within-a-school atmosphere. There are no bells to signal the end or beginning of classes; students are trusted to get to class on time (although tardy policies exist for students who are consistently late to class). Parent volunteers are very active in the school, and the administrators keep in close contact with parents via the Parent Teacher Association, informal principal coffees on several Saturdays throughout the year, conferences, and an annual school climate survey that parents are asked to fill out. Other LMS practices that are representative of good middle schools include a teacher advisor program, exploratory programs for students, and teacher teams.
The first author formally observed Ms. Bryanss classroom 16 times over a 4-month period, usually visiting twice a week for approximately 2 hours each time. Total observation time equaled 27 hours and 21 minutes. In addition to the informal discussions during these visits, the first author conducted four semistructured interviews with Ms. Bryans, two with the principal, Dr. Cannell, and one with the assistant principal, Mr. Previce. The purpose of these interviews was to confirm or explore issues observed during classroom visits. Total interview time equaled 3 hours and 35 minutes. These observations and interviews generated 103 pages of expanded field notes, 28 pages of transcribed interviews, and 172 analyzed domains, according to the methodology established by Spradley (1980). In addition, artifacts such as teacher tests, school newsletters, test scores, and homework grades were also collected.
Themes emerging from the observations and interviews were discussed weekly with an external consultant (a university expert in ethnographic observations) and became the focus of subsequent observations with Ms. Bryans. Whenever possible, other forms of triangulating data were used to confirm emerging themes. During the course of the study, it emerged that Ms. Bryanss teaching was not what the first author expected. Although she identified herself as a problem-solving instructor, her practice was more consistent with a procedural understanding of mathematics. A preliminary analysis of the study was presented to Ms. Bryans at the last interview; she agreed with the description of her practice but strongly disagreed with being interpreted as an essentially procedural mathematics teacher. To explain these discrepancies between our expectancies, Ms. Bryanss beliefs, and her observed practice, we will analyze the data using a model that we believe accurately depicts her classroom culture.
We describe her teaching by presenting a typical classroom observation and note the few exceptions observed in her routine. We then use the typical class and other data to show the relationships among the aspects of her classroom mathematics culture, showing how her practices, materials, values, communication, and problems all support each other. It is these mutually reinforcing aspects that we call her classroom culture. We then briefly situate Ms. Bryanss classroom culture within the broader school culture within which she works, showing how its aspects, in turn, support the maintenance of her classroom culture.
In 88% of the classes observed, Ms. Bryanss teaching followed the same pattern during the 75-minute class: a 15- to 20-minute warm-up activity (usually comprising problem-solving or mental math) while Ms. Bryans recorded students homework in her grade book, 20 minutes reviewing the previous days homework, 20 minutes of direct instruction, and about 1520 minutes for students to begin their homework (a break for lunch came either after the homework or after the introduction of new content). The only exceptions to this pattern occurred when there was no homework to review (on Mondays, for example), when she spent more time teaching new information, or on days when she gave an exam.
Ms. Bryans claimed that she was committed to teaching problem-solving. Yet the only evidence of anything resembling problem-solving occurred in the first 1520 minutes of class time and does not resemble what most researchers mean by problem-based mathematics teaching. Every class observed began with an introductory activity, usually a mathematical problem displayed with an overhead projector. These problems were presented according to the heuristic strategy used to solve them: make a table, logical reasoning, estimation, guess and check, or draw a picture. On five occasions, she also started the class with a mental math exercise, a short check of previously learned skills, or a review for a standardized test.
On this day in Ms. Bryanss pre-algebra class, the problem of the day was displayed on the screen, handwritten in bright green marker. Most of the 31 students were in their seats arranged in neat rows facing the front of the room, but some were still settling. Walking into the room from her duty as hall monitor, Ms. Bryans headed directly to the overhead projector and started talking over the students,
All right, here we go .Shhh .One more time. This is problem-solving. The name of the problem is The Gambler.
Some students were still socializing, talking, and getting their books out. The electric sharpener loudly grinded Toms pencil to a suitable point. In a loud voice Ms. Bryans said,
Thats ONE! When I am talking that means youre not supposed to be talking. One of the problems is youre trying to pass your homework forward, so go ahead and pass your homework forward quietly, then Ill go over this.
Students settled down at the warning, and there was now only limited subdued talking. After waiting about 30 seconds, Ms. Bryans told the class that she would read the overhead problem to them as they passed their papers forward.
A gambler has two kinds of chips. The red ones are worth $5 and the blue ones are worth $8. What is the largest bet he CANNOT make?
As soon as she finished reading the problem, Ms. Bryans began to break it down into more manageable steps.
Lets see. Lets make a chart of the ones he cannot make and the ones he can make and maybe if we do this in some organized fashion, we can determine how many coins he has. . . . How many total coins do we have?
A few students said they dont know. She echoed this:
T: We dont know. Im going to tell you he has an infinite number of coins.
S: What does that mean?
Ms. Bryans explained what infinite means. She then drew two columns on the overhead with the headings cannot and can. She placed an asterisk next to cannot to highlight that this was the column of interest. She then began calling out the counting numbers, starting with one. After each number, she asked the class, Can I make a bet of ____? The class answered yes or no to each. She repeated each answer and wrote the number in the appropriate column. The overhead looked like this:
Ms. Bryans then told the class:
I want you to continue in the same fashion until you find one he cannot make. . . . It is a number, and its a lot less than 100. So continue on with this chart and find the one he cannot make. Spend a couple of minutes and see if you can find the one. . .
She then returned to her desk to grade their homework assignments while students worked on completing the chart.
After about 5 minutes of individual problem-solving, Ms. Bryans walked around the room, handing back the homework. Classroom noise had been steadily increasing as students either solved the problem or gave up on trying. She told the class:
All right, when I come around, show me your work.
Students chatted socially among themselves while Ms. Bryans walked up and down the orderly rows, checking their answers. After checking each students paper, Ms. Bryans walked to the overhead projector.
All eyes should be up here; all mouths closed. (Turning to the overhead chart, marker in hand) Can we make the number 14?
Some students called out No. She continued like this, with students (about five or six voices at a timemostly the same voices) chorally calling out yes/no as appropriate to fill in the chart.
T: Yes, cause I can make 18 and 10.
T: All right, 13 and 16 good . . . 30, can I make 30?
T: Yes. 32?
Class: Yes. No.
T: Yes or no? Someone tell me why 32 will work?
S: Four eights.
T: Four eights, thank you. (Ms. Bryans stops writing numbers on the chart and asks the class, pointing to the list under can:) Would you think if I could make 3040, I could make all the rest of the numbers? . . . because all I would have to do is add a 10 to them?
T: So, what is the largest number I cannot make?
T: Mr. Johnson is the only student who got this right so be proud of it.
Sam Johnson blushed as a few of his classmates acknowledge this accomplishment by calling out whoa. Ms. Bryans told students how to record this in their notebooks. She asked them what strategy this problem used, then repeated their answer. Make a table is correct.
This is typical of how Ms. Bryans taught problem-solving.
REVIEW OF HOMEWORK
After having students record their strategies in their notebooks, Ms. Bryans began to read out the answers to the homework problems as students checked their papers. She had already recorded whether they completed their homework and given those students full marks, regardless of correctness.
T: All right. Questions on ones you do not understand?
She offered students who did not do their homework a way to stay engaged in this review:
T: Once again, if you didnt do your homework, Im writing these down; you can at least write these down.
Students who turn in their homework late get half marks, again, regardless of correctness. She continued to write each homework problem on the overhead, showing the class how to solve the problem, while asking questions. Students raised their hands if they knew the answer, though she occasionally called on students who did not raise their hands.
A moment later, another student muttered to herself, Im all confused, in response to Ms. Bryanss explanation. Ms. Bryans did not respond to her; instead, she drew a circle on the overhead, cutting it into fourths.
T: How many 3/4ths?. . . . Lets try the next one. 3/4ths divided by 1/8th. This is a hard one to draw. . . As you divide by smaller and smaller numbers. . . Tommy turn around . . . what happens to the quotient as the divisor?. . . So what would happen?
After Ms. Bryans completed that question, she declared,
All right, that was good. This is going to help you in todays game. If you didnt pay any attention, then it will hurt you in the game.
Ms. Bryans continued with the next problem, and the class continued this pattern of question and response until, because of block scheduling, the class broke for lunch.
As the same students filtered back from lunch, Ms. Bryans spoke to individual students, telling them to have a seat. She then completed the review of homework problems.
Throughout this review, though she occasionally paused for students to answer; Ms. Bryans mostly stated the question and then showed the class how to answer it. The focus was always on the correct procedure. Even though the homework review consumed a very large portion of class time, Ms. Bryans rarely entertained or demonstrated alternative procedures, stopped the flow of questions and answers to identify and remediate students conceptual problems, used materials other than the overhead projector and marker, or engaged students in collaborative inquiry.
When the review was over, without pausing, she told the class to get a slip of paper.
T: You are going to do the following. You are going to roll the 2 dice. For instance, Im going to roll the 2 dice.
She rolled the dice.
T: It says 1/2 and 1/3. What you are to do is, you are going to pick the operation that will give you the largest answer. If I add, subtract, multiply, or divide, which one do you think will give you the largest answer?
She called on a few students in turn, stopping when she did not get the correct answer.
T: First off. . . I have to find a common denominator.
Ms. Bryans stood at the overhead projector, going through the various algorithms while students watched and called out answers. She notified the class that they were going to work in cooperative groups, which received a collective groan from the class. As she moved toward the cupboard with the supplies, she further explained,
T: Sorry, but that is all the dice I have. You will have to get in your groups
Ms. Bryans often used cooperative groups as a means for sharing limited resources. When she continued to give directions, several students complained that they had not yet received their supplies. Ms. Bryans responded, I know . . . Please listen! All right, ONE! She started counting. A few students said, Shhh! As she finished giving directions, students pushed their desks together in haphazard groups of three and four, and then they started the game. Seven or eight students were not engaged in the activity, and two groups did it incorrectly; none was corrected. After about 10 minutes, Ms. Bryans said, All right, there are enough people who have finished that were going to . . . bring the dice to me. . . . Put the room back together.
She concluded the activity by telling the class the answers to the dice game they just played. She mentioned in passing that students might think adding or multiplying will give them the largest number, but added, With fractions that is not necessarily true. Division will make things actually larger. Ms. Bryans provided no further explanation or elaboration of this potentially difficult concept.
As the class neared completion, she gave the homework assignment. By custom, students understand that they can use the remaining class time to start their homework. Ms. Bryans walked around the room offering help, sharing some of her responses on the chalkboard for the benefit of all students.
T: Yeah, you divide the denominator into the numerator. . . Dont leave your answer like that, it either has to repeat or keep on going.
Many students began their homework, but many others chatted quietly about other things. When it was time to leave, students stood up, packing their stuff.
Of the 16 observations of Ms. Bryanss class, all but one followed this routine. The exception to this pattern occurred on the day after students took the statewide high-stakes standardized achievement test, when Ms. Bryans allowed students to work on coloring and cutting tessellations for several days. In her words, This week were going to do stuff thats a little unusual because your brains are tapped. On the day of the test, students worked on tessellations the entire day. At the end of the class, Ms. Bryans was concerned that students would tell parents about the class:
T: The only problem is youre going to go home and tell your parents, Hey we did this really neat art thing in math. Who can tell me how does this relate to math?
A student calls out that it is geometry.
T: There you go, theyre geometric figures.
Two days later, Ms. Bryans slightly changed the pattern by starting with a review of the homework from the previous day and then allowing students to work on their tessellations. She followed this typical pattern, even when reviewing for the two main standardized tests of the year, by giving them sample problems to work on to start the class.
Ms. Bryans used this same instructional pattern in both her pre-algebra to algebra I classes, according to the first authors observations. She also used the same classroom organization, the methods of assessment, classroom communication, management, and discipline, suggesting that her pattern of instruction was largely invariant.
CULTURAL ANALYSIS OF MS. BRYANSS MATHEMATICS CLASSROOM CULTURE
Ms. Bryanss typical day suggests vaguely that she is engaged in some sort of cultural practice, but by focusing on the connections among her practices, her professed values, her language, her use of materials, and her perception of her problems, we begin to see the social nature of that practice. These facets of culture support and reinforce each other in a system. Moreover, these aspects of culture are situated in broader cultural contexts that support and reinforce each other. It does not matter where we begin our analysis of culture so long as we trace how each part of the culture is related to the others (Latour, 2005). By doing so, we see that her instructional practices emerge from the collective cultural representations of procedural mathematics, and we can better understand why her practice is as it is.
Because problem-solving5 is central to Ms. Bryanss description of her own mathematics teaching and it is a central practice in problem-solving-based mathematics education, our analysis focuses on Ms. Bryanss practice of teaching problem-solving. Through this analysis, we show that her teaching of problem-solving both reinforces and is reinforced by other aspects of culture in her classroom and, in turn, broader school culture. Through this analysis, we show that Ms. Bryans did not merely misunderstand problem-solving or have misplaced values, as the individual perspective on cultural representations would have us believe, nor is it merely the result of communicative and physical interactions, as interactionist perspectives would assert. Only a collective conception of cultural representations will help us to understand her practice.
We adapted Denning and Dargans (1996) five aspects of culture to provide a more robust lens to examine the culture of Ms. Bryanss classroom: language usage (L), standard practices (SP), materials (M), ongoing concerns and values (V), and recurring problems (RP). The dynamic relationship among each of these components is depicted in Figure 2. In the following discussion, we use Figure 2 as a heuristic to frame these different aspects of culture and self that influenced Ms. Bryanss teaching and to highlight different examples of each that emerged from the observation of Ms. Bryanss teaching. Specifically, we use parenthetic codes to identify the connections between the different aspects of culture (e.g., Ms. Bryanss understanding of problem-solving is represented by the code L1).
Figure 2. Model of the relationships between the five aspects of culture with examples of each from our analysis of Ms. Bryanss classroom
Problem-solving, is like, there are strategies to solve it, like drawing a picture. (Ms. Bryans, in her interview with the first author)
Echoing the title and perspective of Polyas (1957) classic book, How to solve it, Ms. Bryanss teaching of problem-solving (L1) consisted of practicing and demonstrating specific heuristic strategies using the overhead projector (M2), with the occasional mental math or logic puzzle while students sit at their desks (M1), during the first 1015 minutes of class while she checks attendance and homework (SP6) (a practice that other people call bell work SP1). Selected excerpts from interviews with the first author (MG) are presented next to give insight into Ms. Bryanss understanding of problem-solving.
MG: Could you give some examples of how you teach differently, like how you used to teach and how you teach now?
T: In the past my teaching (L3) was, umm, pretty traditional. You go over the homework (SP6), then you have the lesson (SP2), then you give them work to do. Umm, then I changed it to doing something different at the beginning of class, like problem-solving (L1). And eventually that led to doing mental math and logic puzzles. . . and with block scheduling (SP7) that makes it really easy to do everything (V2) where in regular schedule I was really pushing <pause> to get it all done.
For Ms. Bryans, her teaching was no longer traditional once she added problem-solving (L1) to her curriculum. And Ms. Bryans is considered by many, including herself, to be very good at teaching problem-solving. Problem-solving is central to many conceptions of reform mathematics, but her meaning of problem-solving (L1) only partially reflects the meaning of most reformers. This is seen mostly clearly in her narrow understanding of problem-solving, which excludes both word problems and open-ended problems:
MG: . . . (discussing her own teaching) Word problems for example, were the hardest thing to teach.
T: But see, but word problems and problem-solving arent the same thing.
MG: OK, whats the difference?
T: Theyre close but theyre not the same. Problem-solving (L1), is like, there are strategies (L6) to solve it, like drawing a picture. Word problems can be that way, but theyre questions like, a frog goes up the well, seven feet, and drops down four feet. How many days? If the well is 10 ft high, how many days will it take him to get out of the well? As opposed to, theres three apples over here and four apples there, how many apples are there all together? . . . . Its much more involved.
Problem-solving for Ms. Bryans was distinct from solving word problems in that the problems were more complex, but both are based on specific strategies, (e.g. drawing a picture) and could be taught and learned, with practice. This was a different view of problem-solving than one advocated by most problem-solving-based math educators who argue against explicitly conveying specific heuristic strategies to students (for a review, see Schoenfeld, 2006). Ms. Bryanss understanding is also distinct from problems that might require students to work in groups to solve a complex problem and might take more than a day to complete (Schoenfeld, 1992).
T: . . . theres only so much undirected activity you can do until its time to get down to business and solve equations. So its a combination of hands-on (L4) and didactic instruction (SP2).
G: What if someone said that its never time to get down to business? You should never ever teach them those equations?
T (interrupting): This person is obviously not a teacher. Because you wouldnt get anywhere (V2). You would never get anywhere, and so youd never get past that.
Ms. Bryanss reaction suggests that she is frustrated at the suggestion that students could learn though extended cooperative problem-solving; such a practice would contradict her beliefs and values. Although cooperative problem-solving might provide a nice introduction to a topic, the business of mathematics teaching should be done through direct instruction. And business is procedural (V2):
M: How do you think kids become good problem solvers? I mean, they dont start out being good problem solvers, so how do they become good problem solvers?
T: They have to practice it (V1).
Her understanding of problem-solving is framed by what she values about mathematics. When asked what were the minimal skills she expected her students to master, she did not list problem-solving. Instead, the list included adding, subtracting, multiplying, and dividing the following:
T: Whole numbers, fractions, and decimals. . . (and) integers. And I would like them to be able to solve one-step equations, and two-step equations, but it has become apparent that they cannot solve equations with variables on two sides. They cannot do that. We did that, and it did not work. So, we decided that this group couldnt do that. That maybe they hadnt reached Piagets level of <pause> thinking yet, that they couldnt do that. Um, and we would like them to be able to graph lines. (L6)
Not only was problem-solving not something that Ms. Bryans expected all of her students to achieve at the end of class, but deeper understanding was not emphasized either. Instead, the focus of her pre-algebra curriculum was to get students to be able to perform those math procedures that were presented in the pre-algebra textbook (M4) as appropriate for eighth grade. Moreover, problem-solving was still not something that she assessed in class even though it was assessed on the states standardized achievement test (SP3). Ms. Bryanss views of mathematics generally, and problem-solving specifically, are procedural.
It is clear from students comments and interactions that the primary focus is on following established procedures (V2). Neither Ms. Bryans nor the students valued understanding mathematical concepts. This is further seen in an analysis of her quizzes (SP3), which focused entirely on students using the procedures presented and practiced in class. For instance, in the pre-algebra course, a midchapter quiz on decimals and fractions was typical. Using a black-line master from the teachers edition of the textbook, Ms. Bryanss quiz includes multiplication and division of fractions and decimals that required no more than rote application of previously taught procedures. Mathematical concepts were not assessed directly, and none of the questions required (or seemingly allowed) problem-solving. Thus, Ms. Bryans can feel that she is an effective math teacher because the mean grade in her classroom is fairly high, and most of her students seem to be learning exactly what she has taught.
But Ms. Bryanss values as a math teacher also depend on her understanding of students who struggle in her class. We see this in an incident in which Ms. Bryans had to change her practice of reporting student class rankings with their grade reports. She justified the practice because competition (V4) is a part of the real world. Students compared their class rankings, a practice that Ms. Bryans did not discourage, but this bothered one of her honors students. His parent complained about the practice, so she eliminated the rankings (RP2) (it remains unclear if her reaction would have been the same if one of her less able students had made the same complaint). Now, if students wanted a ranking, she gave it to them individually or to their parents. Thus, she tacitly endorsed the view that competition is not only acceptable but even natural; Kids have a right to fail, (V4) she told the first author in the hall after class one day.
By naturalizing the idea of competition (V4) as representing the real world and by attributing students poor performance to a lack of individual effort (V5), Ms. Bryans justifies to herself that she is not responsible for the achievement of lower performing students. Her assessment (SP3) and grading practices (SP4), in turn, support these beliefs. By having relatively low student performance expectations, most of students can perform fairly well on classroom assessment, thus not causing parents or administrators to question her practices. However, despite her notoriety as a math teacher, the relatively high grades her students earn, and her predominantly White and affluent student population, her standardized test scores are only average for the state. In addition, she is able to marginalize low-performing students through her use of class rankings and her emphasis that poor performance is due to lack of effort (V5).
Yet even her emphasis on procedural mathematics needs to be seen in a larger perspective. By focusing on instructional strategies that emphasize procedures and downplaying mathematical understanding, she can reduce the chance of students blaming their poor performance on her teaching. For example, if she really valued procedural understanding, she might require students to share their work with their peers on the board (a practice that can be used to assess and provide feedback for procedural competence or to identify and remediate students conceptual misunderstanding):
T: Yeah, see I have real problems with kids coming up to the board because I dont want them, I dont, if a kid doesnt feel comfortable coming up to the board, I dont want a kid to do that cause I dont wanna lower their self-esteem in their own eyes and their classmates eyes (V3).
Ms. Bryanss concern with the development of students self-esteem might be seen as consistent with problem-solving mathematics teaching, but it is incongruent with the rest of her classroom practice and her explanations. In all the observations and interviews, this is the only time she expressed any concern for students self-esteem.
By contrast, her instructional practice seems to be primarily organized to ensure that students were:
T: Sitting in their seats (M1), and Im in control, theyre usually quiet and doing what theyre supposed to be doing, but . . . if they have an activity to do, theyre gonna get a little louder because theyre communicating to each other which is more than just one person communicating to me, so the class will be louder. And then there are kids who cant handle that, you know, group behavior (SP5), and there are kids that can handle it, so I would expect it to get louder. And some would be off-task (RP1), but how many kids are on task in the classroom when youre lecturing to them to begin with anyway?
This quote reflected Ms. Bryanss belief that group tasks (SP5) werent good for all students, but then again, neither was lecturing to them. This belief may at first seem in favor of reform-oriented practices, but for the classes observed, cooperative groups comprised only about 3 lessons out of the 16, and those that were labeled cooperative activities (SP5) were not true collaborative groups, but primarily independent hands-on activities (L4) in which students sat next to their peers and shared supplies (M3). Furthermore, the products of the group work were assessed on the same basis as homework grades (SP3): 100 points for completion, 50 points for late work.
So even Ms. Bryanss emphasis on procedural mathematics taught primarily using direct instruction is, in turn, consistent with her desire to minimize classroom disruptions (V2) that can occur when she uses other instructional methods. To teach without giving a formula is just really really slow (V2). The routines of her typical class seems to be another quite intentional strategy for avoiding disruptions: Everything that youve seen they already have a pattern going (V2) .
What emerges from this analysis, then, is that Ms. Bryanss understanding of problem-solving is not an isolated psychological phenomenon, nor is it simply an emergent product of the culture of her school. Instead, her understanding of problem-solving is a coherent part of her mathematics classroom culture that cannot be meaningfully isolated from the other aspects of her classroom culture.
At this point in the analysis, we might choose among several ways of discussing the ways that culture influences how Ms. Bryans teaches problem solving. Taking an individual perspective on culture, we could emphasize how Ms. Bryans seems to be playing out an invisible script. This analysis would show the ways that her beliefs and values about mathematics education were acquired during her enculturation as a student of mathematics and in her early experiences as a teacher, and how those beliefs and values continue to be reinforced in her current school. Taking an interactionist perspective on culture, we would emphasize how her mathematics education practices are afforded and constrained by the materials in her classroom and in the habituated interactions between herself and her students. These affordances, constraints, and habituated interactions are, in turn, clearly biased toward traditional approaches to teaching problem-solving and would make it more difficult for her to use those materials or interact with students in different ways. Taking a collective perspective on culture, finally, we would emphasize the sui generis nature of classroom culture that is always being recreated. Each of these analyses would highlight and obfuscate different aspects of Ms. Bryanss classroom practice. We return to this comparison later in the article.
However, Latour provided (2005) an additional caution about the ways that these theoretical perspectives are deployed by most researchers:
This unproblematic use of the word [social] is fine so long as we dont confuse the sentence Is social what goes together? with the one that says social designates a particular kind of stuff. With the former we simply mean that we are dealing with a routine state of affairs whose binding together is the crucial aspect, while the second designates a sort of substance whose main feature lies in its differences with other types of materials. We imply that some assemblages are built out of social stuff instead of physical, biological, or economic blocks, much like the houses of the Three Little Pigs were made of straw, wood, and stone. (p. 43)
This quote highlights the unfortunate tendencies of researchers, in the field of mathematics education and beyond, to use culture and cultural for the aspects of the situation that they intend to villainize or valorize. Researchers working in the individual tradition of cultural analysis tend to study the practice of teachers who have failed to adopt reform-based practices. In doing so, they emphasize the cultural origins of teachers values and beliefs that they view as negative, because they discourage teachers from adopting more enlightened educational practices. Doing so ignores that all values and all beliefs have their origins in a persons culture; designating only those values and beliefs that we disagree with as cultural is disingenuous. By contrast, researchers who use an interactionist perspective on culture tend to study the practices of teachers who have successfully adopted reform-based practices. In doing so, they emphasize how the interactions in the classroom have been reshaped though efforts to change the available materials or socio-mathematical norms. Again, however, they tend to ignore that all interactions are shaped by culture, including those with which we disagree.
Finally, Latours quote clarifies the fallacy of restricting analysis of culture to certain kinds of data or certain aspects of those dataincorrectly labeling as social only certain kinds of things. Instead, Latour told us, the social analysis of culture is an attempt to describe how things are bound together into routine states of affairs. This approach seeks to describe as fully as possible the full state of affairs and the processes by which its elements are bound, and it avoids the unfortunate tendency to assume that we know a priori which aspects of the culture will be important to include in the description or how they bind each other.
BROADER SCHOOL CULTURE AND THE INADEQUACY OF A PURELY SOCIAL EXPLANATION OF CULTURE
Just as problem-solving is a coherent aspect of Ms. Bryanss classroom culture, so too is her classroom culture a coherent aspect of the broader school culture. However, although we were able to provide a purely social explanation of problem-solving in her classroom culture, we were unable to explain the relationship between the culture of her classroom and the broader school culture without invoking individual representations of culture. One of the organizational characteristics of schools is that they are loosely coupled systems (Weick, 1976), meaning that the activities and actors in the school are responsive to each other, but that their attachments may be circumscribed, infrequent, weak in [their] mutual affect, unimportant and/or slow to respond (p. 3). This loose coupling means that an analysis of the relationships among language usage, recurring problems, standard practices, values, and materials by themselves provided a very thin description of the relationship between Ms. Bryanss classroom culture and the broader school culture.
From a collective perspective, the parents, the administration, and the other mathematics teachers of the school valued procedural mathematics, albeit for different reasons. Parents of students at the school valued success in school generally and mathematics in particular as important preparation for college, and they seemed to value procedural competence in mathematics because it was important for their own preparation for college (RP2). In addition, many valued Ms. Bryanss emphasis on competition and her use of class ranking for the same reason (RP2). However, direct communication between the parents and Ms. Bryans was infrequent. The only direct communication noted in the data that resulted in a change in practice was when parents complained to the principal about Ms. Bryans posting class rankings, a practice that also violated the U.S. Federal Family Educational Right and Protection Act. She also seemed anxious to avoid parent complaints when she reminded her students that her tessellation activity was geometry and not art. Students average grades were fairly high, and, although the standardized tests she was required to give to her students did require problem-solving, Ms. Bryanss students did well enough to not draw attention to her instructional methods.
The administrators valued their time and very much appreciated that Ms. Bryans almost never had to refer a student to the principals office to deal with behavioral problems (RP1, V2, EB3). Further, her principal stated that Ms. Bryans was a good teacher because We know what good teaching practices are. These are the same types of behaviors that are measured by the FPMS [Florida Performance and Measurement System]. He highlighted Ms. Bryanss ability to orchestrate student movement. She didnt let students pull her off task. She builds success for kids by offering a variety of learning activities. The principal, like most principals across Florida, used a variant of the FPMS as a standard material for assessing successful teaching during his contractually required annual observation of all teachers who have been granted continuing contracts (a.k.a. tenure). We note, however, that the FPMS is based on the process/product research of effective teaching that was popular in the 1970s and 1980s (MacMillan & Pendlebury, 1985), especially Hunters model of direct instruction (Hazi & Garman, 1988; Hunter, 1982). Although the FPMS does not preclude problem-based mathematics teaching, the principals evaluation materials and practices and the values they embody do not direct his attention to aspects of Ms. Bryanss practice that are congruent with more recent research on effective models of mathematics instruction (discussed earlier).
Indeed, the heart of a reform-oriented math curriculum is student problem-solving, but the principal valued Ms. Bryanss approach to teaching problem-solving in a procedural way by presenting the material clearly, logically and sequentially. He did not seem to notice that she did not allow students time to actually solve problems, discuss alternative solutions, or develop deeper conceptual understanding and reveals that the principal, too, values straightforward explanation of how to solve problems rather than promoting a more inquiry-oriented learning environment. Consistent with the research by Spillane, Diamond, and Jita (2003), the administrators in this school did not use different observational practices or materials when assessing the teaching of different subjects and tended to defer to subject matter leaders in the school to provide subject-specific instructional leadershipfor mathematics, that was Ms. Bryans. The result of this evaluation process and distributed leadership practice was that Ms. Bryans was praised as an effective teacher, assigned to write major curriculum frameworks for the school, and assigned as a mentor for newly hired teachers.
Although there were few substantive conversations between the administrators and Ms. Bryans, the mathematics teachers met in weekly planning sessions to review progress against the schools curriculum framework and discuss instructional methods (also described in Gill & Hoffman, 2009). Although the more experienced teachers in the group sometimes disagreed about the best ways to accomplish their goals, the primary goal of helping students gain procedural competence was never questioned, and methods of helping students to gain deeper conceptual understanding were never discussed. Instead, the textbook was the primary material used for organizing the content to be covered in the curriculum, although a frequent recurring problem was that the recommended textbook activities were too difficult or boring, resulting in student frustration and misbehavior. To avoid classroom disruptions, the teachers often discussed alternative activities that might be easier or more enjoyable. These planning meetings supported traditional teaching practices. However, even after these issues were discussed in the planning meetings and the teachers agreed to use particular instructional methods, there was no further direct interaction or observation to ensure that everyone was teaching the same way. In short, although the broader school culture supported traditional mathematics instruction, the attachments [were] circumscribed, infrequent, weak in [their] mutual affect, unimportant and/or slow to respond (Weick, 1976, p. 3).
Notwithstanding Latours (1987) methodological injunction to limit the use of psychological explanations, we were not able to completely explain the loose coupling (Weick, 1976) between the broader school culture and Ms. Bryanss instructional methods without describing the important role of her beliefs, especially her very strong, explicit self-efficacy beliefs.6 Although it may seem odd to combine individual and social perspectives on culture in the same analysis, we were unable to explain the relationship between the broader school culture and Ms. Bryanss classroom culture without acknowledging the role of her self-efficacy beliefs as the mediator. Although we do not believe that teacher self-efficacy beliefs can be reduced to a collective explanation of culture, we also do not believe that teacher self-efficacy beliefs are sufficient to explain the persistence of procedural mathematics. Instead, we suggest that self-efficacy beliefs are additional factors that influence Ms. Bryanss practice, and the persistence of traditional methods of mathematics instruction, along with language usage, standard practices, tools and equipment, ongoing concerns and values, and recurring problems.
Both teachers self-efficacy beliefs and their understanding of mathematics have often been used to explain the persistence of procedural mathematics. The relationship between self-efficacy beliefs and conceptual beliefs has only recently become an area of academic study (Gregoire, 2002; Pintrich, 1999), but it seems likely that they are related. When a persons beliefs are supported by a sense of positive self-efficacy but his or her beliefs are challenged, it is easier to ignore or explain away the challenge rather than directly confront the challenge (Chinn & Brewer, 1993; Gregoire, 2003). Both teachers beliefs about mathematics and mathematics education and their various efficacy beliefs seem to play some important role in maintaining procedural mathematics. Pace Durkheim and Latour, these psychological phenomena are also irreducible to cultural explanations.
It is important to notice that her beliefs about mathematics and her self-efficacy beliefs emerge from and are reinforced by her experiences as a student of mathematics, her 20 years as a mathematics teacher, and the specific culture of LMS; that is, her beliefs are cultural. Judging from the interviews and observations of Ms. Bryans, she had a strong sense of efficacy as a mathematics teacher. She told the first author that she liked students, she liked teaching math, and she has never experienced being burned out. She said she thought it was because of her attitudeYou cant really tick me off. In addition, her expressed efficacy and achievement could also be due to the fact that what she teaches conforms to her view of how math should be learned, and she received praise from other teachers and her principal for her teaching.
Her sense that she is being successful as a math teacher in turn depends on her selective attention to student achievement and her attributions of students who struggle. For example, her assessment of students is based primarily on chapter tests that focus on students procedural ability, in addition to notebook tests (SP3), portfolio grades (SP4), and completion of homework (SP6). By the end of the semester of the study, students grades on four assessments (SP3) (two chapter tests, a notebook test, and their portfolio) averaged 79%. About 71% of students turned in their homework (SP6) on average (they were scored on turning in homework on time, not for correct answers). The following scene shows students reactions to receiving the grade (SP4) for their notebook test:
Ms. Bryans handed back their quizzes. She said she would go over the answers with them after lunch. A student questioned Ms. Bryans on points he lost, asking if she took off points for a certain error.
T: Yes, and I specifically mentioned that before you took the quiz.
Some of the student conversation about their quizzes included the following dialogue:
Tommy asked a neighboring student: What did you get Sheree?
Allen said, Tonya, I must have cheated.
Daniel to Alex: What did you get?
Another girl in back called out, in a happy voice, Oh my God. 101!
Tonya got her paper back. She looked at it sadly and folded it in half so no one else can see it. It was a 64%. Once Ms. Bryans finished distributing the quizzes, she told the class, If you didnt put your measurements, its one point off. . . I have written all the right answers so you can check your work. Theres just some minor things which well check after lunch. Ok. See ya.
This interaction was typical of students receiving grades on any assignment. Students received a wide distribution of grades, yet errors were not discussed, let alone used as an opportunity to explore why students made those errors. Ms. Bryans also made no effort to comfort or encourage students who received low grades, consistent with her attitude that students have a right to fail. It was simply assumed that students who received low grades were either less capable or did not try hard enough. Either way, she was not at fault, so her self-efficacy beliefs were not threatened.
The beliefs and values of procedural mathematics teaching and learning dominate the beliefs and values of teachers, students, and the public in the United States (Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999). These ideas represent a complex of related beliefs that result in teachers, administrators, and students preferring procedural instruction practices such as direct instruction, rote drill and practice, and objective assessment. One central outcome of maintaining a belief in the procedural mathematics tradition is that it maintains teachers, students, and administrators sense of self-efficacyit is easier to believe and value those things at which we think we are good. The procedural mathematics tradition allows teachers to feel that they are accomplishing something positive in their classroom (Smith, 1996). In particular, the belief that mathematics is a fixed set of facts and procedures contributes to teachers sense of self-efficacy in five ways:
restricts the content that teachers must master to a manageable range;
provides a relatively detailed model of what teachers should do in their teaching;
accentuates the sense of knowledge transfer in teaching;
defines what students should do to learn: listen, watch, and practice; and
provides a rough outline of a typical days instruction and simplifies issues of planning and classroom management. (Smith, 1996, pp. 392-393)
In other words, the shared beliefs that support procedural mathematics create clear expectations among teachers, students, and others associated with mathematics education. These clear expectations in turn reduce disagreement and interpersonal friction and allow classrooms and schools to function more smoothly.
The trouble with problem-based mathematics reforms, according to Smith (1996), is that they undermine the sense of efficacy teachers feel with procedural mathematics in two important ways. First, repositioning teachers as facilitators of problem-solving experiences, rather than as knowledge providers moving neatly from topic to topic in the textbook, makes it much more difficult for teachers to see their progress. Second, designing problem-based lessons is much more difficult and may be beyond the immediate capabilities of most traditionally trained mathematics teachers. Additionally, Sfard (1998) noted that in problem-based mathematics, it is difficult to find the appropriate real-life situations on which to base curriculum, leading to a gradual disappearance of a well-defined subject matter (p. 10). Without a strong sense of efficacy tied to promoting students deep conceptual understanding, teachers are likely to experience negative emotional consequences (Ashton & Webb, 1986; Cherniss, 1993).
According to this individual perspective on school culture, then, the primary benefit of perpetuating procedural mathematics is that it contributes to teachers sense of efficacy and the seemingly smooth, orderly operation of classrooms and schools. Procedures are important because they allow mathematical tasks to be completed efficiently. Procedures can be executed quickly and with relatively little mental effort (Hiebert & Carpenter, 1992, p. 78). Yet this purely individual perspective on culture is inadequate because it fails to notice the problematic nature of the broader school culture. A good procedural mathematics teachers classroom seems to operate like a well-oiled wheel, rolling toward expected outcomes. Teachers, students, and schools may resist many educational reforms, including problem-based mathematics, because they seem unnatural and inherently more difficult to implement. We cannot but agree with the sentiment when Nasir and colleagues (2008) wrote that classrooms should be organized to circumvent or disrupt the societal power structures that leak into classrooms (pp. 200201). But these practices only seem more difficult to implement because of the values of teachers, parents, and administrators who prefer fewer conflicts over deeper conceptual understanding.
Our analysis leads us to the tentative interpretation that Ms. Bryanss self-efficacy beliefs can be understood as one factor in the sui generis complex of interrelating factors that make up culture of her classroom. Decades of research on social cognitive theory (e.g., Bandura, 1989a; Bandura, 1989b, 1997; Christou, Phillipou, & Menon, 2001; F. Pajares, 1996; M. F. Pajares, 1995; Pintrich & Schrauben, 1992; Zimmerman, 1989) has shown that people learn their self-efficacy beliefs from observation of others behavior, their own experiences, and vicariously through various media. Of course, these behaviors, experiences, and media are cultural. In turn, a plethora of research (Friedman & Kass, 2002; F. Pajares, 1996; Soodak & Podell, 1996; Tschannen-Moran, Woolfolk Hoy, & Hoy, 1998; Usher & Pajares, 2008; Wheatley, 2002) has shown the various ways that self-efficacy beliefs influence peoples behavior, including the research discussed earlier that suggests how self-efficacy beliefs may influence the practice of mathematics teachers.
Consistent with prior studies (e.g., Spillane & Zeuli, 1999), we found that although Ms. Bryans appropriated some of the rhetoric and practices of problem-solving-based practice, her goals and assessment methods and most of her instructional methods were not consistent with common ideas of problem-solving mathematics. The case of Ms. Bryans can be described and interpreted within each of the three conceptions of culture, depending on the purposes of the researcher.
From an individual perspective on social representations, this case study provides a detailed analysis of her beliefs and values about mathematics education. Like other researchers who have used this perspective, we might note the virulence of these beliefs and values, persisting through dozens of hours of professional development. We might speculate that she learned these beliefs and values, and the practices that accompany them, during years as a student of mathematics and during her formative years as a preservice and early in-service teacher. During this apprenticeship of observation (Lortie, 1975), she learned the cultural beliefs and values that define her profession. After acknowledging that she must have learned these beliefs and values from her culture, we might then use other ideas from the psychological literature to try to understand the persistence of her beliefs and values or to speculate about methods of professional development, supervision, or other policy instruments that might help to change her thinking, such as those recommended by Gill, Ashton, and Algina (2004).
From an interactional perspective on social representations, this case study provides a preliminary analysis of the social interactions and discourse within Ms. Bryanss classroom. Like other researchers who have used this perspective (such as Bartolini Bussi, 1998; J. S. Brown, Collins, & Duguid, 1989; Gauvain, 1998; and Turner et al., 1998), we might focus on the discourse in the classroom and whose voice receives legitimacy in particular instructional events. We might also analyze the norms established by the teacher, both explicitly and implicitly (along the lines of Yackel and Cobb, 1996), and how these norms influence students and teachers learning opportunities in the classroom.
Finally, from a collective perspective on social representations, this case study provides a detailed analysis of the sui generis nature of a mathematics classroom culture. This perspective helps us to understand her misappropriation of the language and practices of problem-solving mathematics education. The complex, interdependent relationships among language, values, materials, practices, and problems support Ms. Bryanss belief that she is engaging in problem-solving mathematics practices and do not present her with any evidence that her practice is contrary to the intentions of the NCTM standards. As a result, it is very unlikely that she will change her practice without significant intervention.
Each of these perspectives has advantages and disadvantages. Both the individual and interactional perspectives suggest ways that Ms. Bryanss practice may be changed. In the individual perspective, change occurs through teacher professional development. In the interactional perspective, change occurs through changing communities of practice (e.g., see Stein, Silver, & Smith, 1998). By contrast, the collective perspective is more interested in describing and explaining the persistent characteristics of a culture. This perspective reminds mathematics education reformers that classroom mathematics culture is neither arbitrary nor capricious. Classroom culture emerges from mutually supporting interactions of the history of mathematics education and the contemporary conditions of schooling. Unfortunately, mathematics reformers have casually dismissed cultures of schooling, leading to vague descriptions of culture and its effects on mathematics curriculum and instruction. Mathematics educators specifically, and all educators generally, must understand the complex, dynamic, interdependent nature of school culture and resist the temptation to reduce or ignore the challenges of school culture if we are to help educators adopt and adapt practices that are more effective.
Ms. Bryanss classroom culture, situated in the broader culture of her school, supported a set of pedagogical practices that have particular consequences for her students and herself. Although the literature shows that procedural mathematics leads to a number of negative consequences for students (Schoenfeld, 1988), our analysis of Ms. Bryanss class shows that it also provided a stable and coherent environment for Ms. Bryans, her students, their parents, and school administrators. Her stable classroom routine made lesson planning much easier than planning and adapting to the challenges of problem-based mathematics (see Nuthall, 2005, for further discussion of how routines perpetuate the culture of a classroom). Her instructional and classroom management practices supported a quieter and calmer learning environment, which was appreciated by administrators and students. Her assessment practices reduced the amount of time that she had to devote to grading student work and appeared to be more objective, decreasing the potential for time-consuming and emotionally fraught disagreements. Parents (at least a vocal minority) valued her willingness to rank students performance in this school subject that is widely seen as a predictor of academic prowess and lifetime earning potential. We could go on tracing these connections among aspects of her classroom culture and its connections to broader school and societal culture, but our point is clear: Ms. Bryanss persistent use of procedural mathematics is neither aberrant, nor naïve, nor irrational; it is an entirely sensible way of behaving when understood within the culture of schooling.
In addition, this collective analysis of classroom culture suggests two crucial differences in the way we theorize the widespread resistance to changing mathematics education. First, we are situating k12 teachers as participants who are acting intentionally and thoughtfully, looking at the various educational practices that are being recommended to see what serves their intentions. We see this in the ways that Ms. Bryans appropriates the practices of heuristic problem-solving from the problem-solving tradition and the way she appropriates the social prestige of academic mathematics. Second, in contrast to the perceptions of mathematics reformers who view themselves, tacitly or explicitly, as policing the boundaries of what constitutes good mathematics education, we see that it is Ms. Bryans who really has the power to police the boundaries of practice in her classroom. Problem-based mathematics, with its norms and practices that support the social construction of mathematical meaning and meritocratic worldview, serve the purposes of the applied mathematics communityengineers, technologists, computer scientists, applied mathematicians, and so onbut these norms and practices are extremely disruptive to public school teachers, students, and schools. What we are seeing is the clash of two cultures, each coherent its own terms, and until those interested in seeing mathematics reformed accept that procedural mathematics runs according to its own logic-in-use and represents several thousand years of development, we predict that reforms continue to be frustrated.
Beyond these conclusions, this case study of Ms. Bryanss classroom practice provides others interested in reforming mathematics education specifically, and education more generally, with a more coherent and rigorous conception of culture and methodology for studying it. Consistent with Latour (2005), we approached this case by viewing culture as the thing that needed to be explained, not as an explanation. We have argued that only a collective understanding of Ms. Bryanss classroom culture provides an adequate explanation of her misappropriation of the notion of problem-solving from the problem-solving tradition of mathematics education and her ability to attend ohours of professional development7 yet remain unaware of that misappropriation. Although both individual and interactional perspectives provide useful insights, neither is adequate to understand the resilience of her practice and her classroom culture.
Our interest in analyzing Ms. Bryanss case focused on how Ms. Bryanss classroom culture was organized around problem-solving. However, other researchers with different agendas might wish to ask: Why doesnt Ms. Bryans make more effective use of educational technologies? Why doesnt she use more effective instructional methods? Why does she seem so uninterested in addressing the needs of students who belong to cultural minority groups? There are multiple research questions that could be analyzed with this data. Although each such analysis of the case would start with a different focus and perhaps draw our attention to different aspects of her classroom culture, the analysis of her classroom culture would yield much the same results.
We are not simply reiterating that school culture influences practice. To do so, as Latour argued (2005), is to mistake culture as an explanation rather than explaining the culture. Ms. Bryanss mathematics classroom culture may be slightly or significantly different from the classroom culture of Ms. Jones, another mathematics teacher across the hall (see Gill & Hoffman, 2009). Both Ms. Bryanss and Ms. Joness practice have settled into coherent, mutually reinforcing patterns of language, practices, materials, values, problems, and efficacy, but students and adults who enter those classrooms for any period of time can see and feel the differences. A student behavior that gets a cold, hard look from Ms. Bryans gets a warm, affectionate smile from Ms. Jones. Yet both Ms. Bryans and Ms. Jones are viewed by the school administration, other teachers, parents, and students as good math teachers. Both classroom cultures are successful within the broader school and parent cultures, so it does not seem plausible that their classroom cultures could be completely different. These are complexities that must be left for future research.
Specifically, we suggest two ways of extending this research. First, comparative case studies will enable future researchers to retain the conceptual and methodological advantages of this approach to studying classroom culture while enabling researchers to see different configurations of classroom culture in the same department, same school, and so on. Second, longitudinal studies of teachers trying to implement new mathematics education practices or teachers who had successfully made changes to their practices would allow for a more robust understanding of how classroom culture and school culture interact when there is a potential incongruence between them. The data for Ms. Bryanss case were collected in the early years of the statewide high-stakes standardized tests to assess student problem-solving abilities and before the U.S. federal legislation called No Child Left Behind demanded annual learning gains for all subgroups. It would be interesting, to say the least, to see how Ms. Bryanss classroom culture has changed (if it has) and how she has integrated these changes into her classroom culture.
What seems clear from this analysis of Ms. Bryanss mathematics classroom culture is that many of the suggestions to improve the teaching of mathematics seem at best naïve. The individual focus of most research leads us to believe that reforming mathematics education is simply a matter of encouraging teachers to change their beliefs and values, leading to frustration when we find that those beliefs and values are only likely to change in the direction of procedural mathematics. Young (2008) perhaps captured this most clearly when he argued that curriculum is not just a product of the practices of teachers and pupils or even government policies but a social institution that needs to be understood independently of the individual actions of teachers and policy makers (p. 11). By contrast, when researchers have pointed vaguely at cultural issues in the past, their underspecification of culture led to a belief that culture is immutable and resilient to all change. In turn, frustrations with this sense of determinism has led to support for high-stakes standardized testing as a means of breaking the cognitive and cultural blockages, which has led to many negative consequences for students.
A more sophisticated, collective view of the culture of schools and classroom life suggests alternative or at least additional ways of influencing educational practice (Cobb et al., 2003; Engström, 1998). Recognizing the complex interplay among aspects of culture suggests more subtle ways of recognizing teachers professional discretion over curricular and instructional decisions and the individual and organizational factors that influence those decisions (Boote, 2006). Curricular and instructional policies and materials, local and high-stakes standardized assessment systems, professional development and collaboration, to name but a few, all can and should play some role. But coordinating these efforts can only happen once we develop a corpus of work that recognizes the complex interrelationships influencing school and classroom culture.
The first author would like to extend her thanks and appreciation to Ms. Bryans for graciously allowing the first author to spend a semester in her classroom. Ms. Bryans remained open to the study and continued to participate in a further study by the first author the following year, despite our differences of interpretation regarding what constitutes problem-solving based instruction.
1. Our concern with culture in mathematics education is distinct from the research and scholarship that seeks to understand the effect of students ethnic culture on their experiences during mathematics instruction (e.g., Diversity in Mathematics Education Center for Learning and Teaching, 2007; Nasir, Hand, & Taylor, 2008; Presmeg, 2007)
2. A similar focus on problem-solving and conceptual understanding has emerged in the other subject area domains (for science, see Brown, 1997; for reading, see Campione, Shapiro, & Brown, 1995; for medical education, see Camp, 1996), and they form the basis of the new curriculum standards in science and language arts as well (e.g., see Holt-Reynolds, 2000; Winitzky & Kauchak, 1997). Taken together, these curricula represent a shift in the traditional notion of academic curricula because they see a more active role in students learning but nevertheless stress the importance of student understanding over instrumental value. Thus, although the instructional and assessment methods are different, the goals and values remain unchanged.
3. The current article reanalyzes data from an unpublished study conducted by the first author (Gregoire, 1999). Although the original study focused solely on the role of Ms. Bryanss self-efficacy beliefs in the persistence of her traditional teaching methods, the second author suggested that the persistence of these practices would be more fully explained by reexamining the original data through a more robust cultural lens.
4. All names and some nonrelevant details have been changed to protect the confidentiality of the teacher, students, and the school in this study. In addition, all direct quotes are symbolized by either double quotation marks () or presented on a separate line, and indirect quotes by a single quotation mark ().
5. In this section, we distinguish Ms. Bryanss understanding of concepts such as problem-solving from the more commonly accepted understanding of that concept seen in the literature by using quotation marks to distinguish her use of the term.
6. Throughout the review process, our efforts to introduce self-efficacy beliefs into our social explanation of Ms. Bryanss behavior has received significant attention from the reviewers and the Editor. We thank them for encouraging us to clarify our analysis. Theoretically, our use of an idea that was developed by psychologists and became a mainstay among psychologists seems to be taken either as an affront to the intellectual autonomy of sociology as a discipline (Durkheim, 1897/1997) or as a failure to recognize that social and psychological explanations cannot be mixed. However, we knew that social-behavioral and, later, social-cognitive theory, from whence the idea of self-efficacy beliefs emerged, were efforts to provide social explanations for behavior and learning. Moreover, methodologically, the data suggest that Ms. Bryanss beliefs play a critical role in the culture of her classroom, including her beliefs about her ability to organize and execute actions to accomplish her intended goals. Unfortunately, the data only allow tentative claims, though we intend to continue examining this issue.
7. One plausible explanation of Ms. Bryanss poor understanding of the meaning of problem-solving in mathematics education would be that her professional development was only cursory or superficial. However, she participated in an intensive summer weeklong workshop on problem solving with a leading research and nationally recognized authority in mathematical problem-solving. In addition, this researcher was hired as a consultant by her school.
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