People use the notion of proof incorrectly as synonymous with geometry. Geometry to me is not just about proof. But proving things is something that is a thread through all mathematics, which really boils down to justifying or explaining your reasoning or understanding why you're saying it.
--Josh Berberian
Teachers are the chief delivery system for curriculum in the 21 st century —this despite the prevalence of standards documents and extensive processes to provide for their alignment with resources, practices, and standardized testing, as well as tremendous growth in terms of instructional technology. Remillard notes in her analysis of teachers' enactment of curriculum that, "the curriculum development process does not stop when the textbooks are printed, but continues in the classroom." Indeed, she posits two distinct levels of curriculum development:The first level is what curriculum writers do when they conceptualize plans and write them in resources for teachers. The second level is what teachers do as they alter, adapt, or translate textbook offerings to make them appropriate for their students.
(Remillard 1999, p.318)
This second level of development is the subject of this inquiry. Using Remillard's framework of curriculum development (design or task selection, construction or task enactment, and curriculum mapping), I want to explore how a geometry teacher at the Shipley School develops his curriculum, how that curriculum may or may not align with the NCTM standards, and how, based on observations, his curriculum is experienced by the students in his class. I will ultimately be suggesting that there is a third layer of curriculum development at independent schools, and that it is in part this dimension of the experience that gives independents schools their distinctive character and standing within the larger educational community.
Josh Berberian perceives that he "got into math at an exciting time." He was taught in what Skemp (1978) would call an "instrumental" mode, involving, as Berberian describes it, "more algorithms and manipulations of numbers." This was a curriculum at which he was able to excel and which he enjoyed, but he feels in retrospect that he "didn't really understand what [he] was doing much." Berberian marks the change in the teaching of mathematics as having come when the NCTM released its new standards and the Harvard Consortium published textbooks "that had a new slant on things":
The notion was—and I think it's philosophically where I align myself—that although there is important content, math is more, for me, about understanding mathematical concepts, and to be able to verbalize and communicate those concepts. Mathematical concepts can be just about anything, from practical applications to mathematical theory.
In part because of this philosophical emphasis on concepts and communication over content, Berberian's classroom itself is organized around human interactions. Students enter the classroom talking—and not exclusively about the previous night's homework. Still, they have a clear sense of purpose. Beneath the numbers he has inscribed across the top of the white board, individual students begin writing out their solutions to the homework problems. The rest of the class arranges itself into clustered desks in groups of four and five. They face each other and can easily look across the desks in their group to view the work of their classmates. There are approximately seventeen students in the class.
At the same time that the formal preliminaries are taking place, there is also some discussion of an extra credit assignment that the instructor has given the class the day before. The assignment involves getting pizza for dinner and doing a calculation or visual experiment related to "Areas of Circles and Regions of Circles," the title of the textbook chapter that is currently under consideration by the class. There is some banter about why different students did not wind up doing the extra credit assignment ("My mom said, 'No. We're having pasta.'"), but one student, a fairly quiet young man, says, "I did it." Mr. Berberian's reply is enthusiastic: "You rock . You always do the extra credit!"
As the students at the board complete their solutions, they take their seats, and Mr. Berberian occupies center stage at the front of the room. In this phase of the class, the instructor goes through each problem from left to right across the board. Often, he makes evaluative comments about the problems either at the beginning or end of the solution ("Number 12 starts off hard..."), but for the most part, he has the students describe from their seats how they arrived at their solutions. Occasionally, when an explanation gets off-track or is more detail-oriented than he might like, Mr. Berberian will say something like, "Back up and tell us your big game plan—what's the strategy?" At one point, a student makes a meta-strategic observation of her own: "You have to figure out what you want." Mr. Berberian repeats this insight to be sure that the students pick up on it.
This manner of curriculum construction (or task enactment) points up an issue of which Berberian is mindful in his growth as a teacher:
This is actually something that I've struggled with in ten years of teaching: the reality that there are some students that just want the teacher-centered learning of mathematics. And they'd find it much easier if I gave it to them. That's definitely how I was taught, but I think mathematics—what happens is that mathematics is boiled down to the way it used to be taught in terms of algorithms and manipulations that are pretty much memorized, with less understanding. So it's that notion of being able to communicate what you're doing and, not necessarily being given a formula or given a strategy. But maybe it's slightly constructivist: to work backwards and hear some problems. Develop the strategy, develop the algorithm. It's hard. It's definitely a struggle.
Observing Mr. Berberian in action, one can see the tension between the traditional role of teacher and the constructivist model of facilitator. For example, at one point during the discussion of the problems mentioned above, a student asks a question about one of the problem diagrams in the book (Figure 1). How, this student wants to know, can we be sure that the four circles inscribed in the square are congruent? "I don't know," says Mr. Berberian. He solicits ideas from the class as to how one can tell that the circles are in fact equal. Students chime in with suggestions, which Mr. Berberian draws on the board. "Sometimes the book assumes things," he concludes after some discussion, "that aren't necessarily clear." At that point, having spent no more than a few minutes on the subject the class moves on.
In a later class Mr. Berberian subsequently comes back to the problem with some follow-up information. In explaining his decision to move on in that moment of construction, but to return to the student's question in the more deliberative task selection mode, and finally to complete the process by reporting his results to the class, Berberian has this to say:
There lies an example—and the examples did come up before in the book. Some of them, students had pointed out. Some of them, I had noticed, and I'm unable to sort of talk them through it. But this one, I was convinced that there wasn't enough information, but I didn't have a real proof of it. I really couldn't show it to them, but I appreciated the fact that Danny had picked up on something we had seen before. He was demonstrating a healthy skepticism, which I think is a really good attribute—particularly in math.
Here Berberian frames his decision to return to Danny's question in a subsequent class not solely in terms of desirable mathematics or geometry outcomes, but in terms of the more global goal of cultivating "a healthy skepticism." This approach is consistent with his emphasis during class discussions on larger strategies and approaches. In this respect, the classroom experience in Mr. Berberian's Geometry A section partakes of the same spirit of inquiry explored by Lampert (1990), who quotes a geometry teacher in Lakatos's Proof and Refutations (1976): "I respect conscious guessing, because it comes from the best human qualities: courage and modesty" (Lampert 1990, p.30). Indeed, this passage from Lakatos is strongly reminiscent of the Shipley School's Latin motto: Fortiter in Re, Leniter in Modo. Courage for the deed; grace for the doing.
The motto bears mentioning here because if we are going to understand the process of curriculum mapping and the degree to which it is aligned with standards at an independent school, we must also take into account the dimension of values. At independent schools, values are often communicated via mottoes, missions, vision statements, covenants and so on. At the best independent schools, one can observe in the classrooms both the specific values of the school and also more generic values implicit in academic good practices.
The Shipley School has an extensive formal curriculum mapping process that is undertaken every five years or so as part of the accreditation process for PAPAS (Pennsylvania Association of Private Academies and Schools). According to Berberian, there is no explicit effort to create and maintain alignment with the NCTM Standards, but "if there were some way that we were not aligned with the standards, that is something we would definitely ask questions about." So in the course of the two classes that I observe for this essay, Mr. Berberian enacts the following specific standards for geometry in grades 9-12:
- Analyze characteristics and properties of two- and three-dimensional shapes and develop mathematical arguments about geometric relationships.
- Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
- Apply transformations and use symmetry to analyze mathematical situations.
- Use visualization, spatial reasoning, and geometric modeling to solve problems.
These standards are observable even in the single homework discussion described above. By choosing to frame the tasks as he does—having the students write their results on the board and then verbally walk the class through them while he guides the discussion—Mr. Berberian is able to keep the group thinking on several different levels. For example, in the course of one problem that has two possible modes of solution, Mr. Berberian asks the student at the outset, "Did you need to draw a picture?" This gesture tips to the question that Mr. Berberian later asks when the student has successfully presented a solution following one of the two possible modes: he takes a poll to see whether people did the problem "the proportional way or the algebraic way."
In explaining his process of task selection, Mr. Berberian indicates that ten years as a math teacher have provided him with a significant repertoire of activities and materials. He acknowledges that to a large degree, the syllabus and schedule for the year are governed by the organization of the textbook, but he also says that he sometimes chooses to rearrange the order of presentation according to what he sees as the needs of the class. In providing a short history of textbook selection within the department, however, Berberian describes a process that is decisive in this context.
As noted above, Josh Berberian's mathematical education unfolded along pretty traditional lines. He came into the teaching profession at a time when a more constructivist approach to mathematics education was gaining currency. In our independent school context, where a formal curriculum mapping process includes textbook selection, Berberian was part of a process in which various options were discussed and implemented. The text in use when he came to Shipley six years ago was, according to his description, a very traditional one that emphasized proofs and "favored linear thinking."?Ultimately, although he had originally advocated for an option that was more in the constructivist vein, Josh came to see the wisdom in adopting more of a compromise text.?He explains:
At that time I put together a proposal for Malcolm who was the department head, and I wanted to go back to this old book that I used [in my previous post] which was this incredibly constructivist book where the students would for the most part come up with their own definitions, come up with their own theorems, and I would facilitate... [Malcolm] helped me to see that that's a great way, maybe, to teach, but as a textbook, you need something that students can fall back on. So we really looked at it more as a compromise—that a textbook needed to be not just philosophically in accord with how you teach, but it also had to be a resource for students.
So through this process of curriculum mapping—which included some fairly strenuous philosophical disagreements over textbook selection—a compromise was reached that may have shaped the way that Josh Berberian currently envisions his role as a teacher. This discourse among teachers—the professionals who are also charged with the task selection and task enactment aspects of curriculum—enlarges and deepens the enveloping curriculum mapping context of Remillard's model.?In his talk about teaching and in the context of his planning and enactment choices, Josh Berberian is a mathematics educator who embraces the tension between being a facilitator and being a content resource. His enactment of curriculum values balances these two ideas of what a math teacher should do, and in this he also seeks to reach in a very personal way the many styles of learning and thinking—from the most linear on the one hand to the most discursive on the other—that come into his classroom each day.
In looking at the relationship between reform curricula and equity, Boaler (2002) concludes that "an understanding of the ways in which open-ended approaches promote equity will involve a consideration of the detailed practices of teaching and learning that occur in classrooms" (Boaler 2002, p.255). At the Shipley School, Josh Berberian and the members of his department have in effect been able to eliminate the mediating role of the researcher in this process. Their "consideration of detailed practices" happens among those who understand the details best and who must be most committed to the choices they will ultimately make. At its best, this process is also infused with the values of the institution. When all of these factors are in place, when our organizations are functioning in their most effective, institutionally healthy and coherent mode, independent schools can provide this significant added value, not only in mathematics, but across the curriculum.
We look at NCTM as a standard measure, and we measure where we are with respect to it. But we don't necessarily feel we have to align with it.
Josh Berberian
Understanding how the policy dimension of math curricula influences the students' experience of the curricula at an independent school can be a challenge. Like most secondary school mathematics professionals, the members of the department at the Shipley School are knowledgeable about the NCTM standards, along with other trends in the profession. But the site-specific policy issues that determine their approaches are encompassing of their mathematics pedagogy in ways that both help and at times hinder the mathematical enterprise.
To perceive whether policy has its intended effect one must understand what that policy is and where it is situated. In describing some of the premises of "systemic reform," Knapp (1997) observes that "[t]hough driven by high-level policy action, systemic reform strategies are not incompatible with efforts to enhance local discretion and professionalism; in fact, some formulations of systemic strategies treat governance reforms aimed at maximizing school discretion as indispensable aspects of a fully developed systemic reform strategy" (p.231)." [L]ocal discretion and professionalism" are the policy-making sites for the Shipley math curriculum. However, my conversations with two students in Josh Berberian's Geometry A section (the middle of three ability tracks) suggest that there are other policy elements that scaffold content and method—and help to create what is distinctive about a Shipley education—in math and other subjects, as well. I want to suggest that curriculum is enacted by Shipley math teachers in a way that is decisively shaped by what I will call this "independence scaffolding."
A lot of it's strategy... You have to figure out what you're looking for first... A lot of times you have to start out with common sense and then use your knowledge of math to find out the answer.
Ali Barnes , Shipley '05
Ali Barnes and Edward Donaldson are sophomores at the Shipley School, members of the class of 2005.? I begin my conversations with each of them by asking them to choose a definition of mathematics from Remillard " Geist (2002): " Which description would you say more accurately describes how you view mathematics: 'a set of rules to follow' or a group of 'ideas that [make] sense' (p.19)?" Both students quickly answer that the idea of math as "a set of rules" conforms more closely to their own beliefs. Edward observes that "Math is more systematic... I think of other things as being more of a group of ideas; whereas, math, you have to do this, and then that, and then this—and you get the answer. It's more black and white than, say, an idea."
Neither student embraces math as a favorite or especially successful subject. Edward says bluntly, "I don't like math that much, period. It's just not my strong suit." Ali's response is similar, though not so categorical. Yet in describing her work in geometry and her approach to solving problems, Ali shows a strong awareness of the need for comprehensive strategies, use of prior knowledge (she mentions using information learned in middle school), and deployment of what she refers to simply as "common sense." When I point out to her that her description of her approach to math is at odds with her original observation that math is a "set of rules," she explains, "I usually don't look at math as an idea. I look at it as a problem or an equation."
Both students seem to gravitate toward what Skemp (1978) would call "instrumental understanding."?They prefer this construction of mathematical awareness because it makes math more manageable in important ways. This tendency on the part of the students has interesting resonances with the reflections of their teacher, Mr. Berberian, on his evolving pedagogical orientation. "I feel as though the learning of math happens through discussion, talking about it, and student interaction with the material—although there are definitely points where it's a teacher-centered class. This is actually something that I've struggled with in ten years of teaching: the reality that there are some students that just want the teacher-centered learning of mathematics." Indeed, my interview with Mr. Berberian makes it possible for me to "discern the central intellectual ideas of the lesson and to pay attention to how they are being developed within the classroom's structures and practices" (Nelson & Sassi 2000, p. 574) in the context of this "struggle" or tension that he describes between instrumental and relational approaches.
The related tension between "central intellectual ideas" on the one hand, and "structures and practices" on the other leads to a fuller understanding of what the students experience in Geometry A, as well as the policy implications of that experience. When asked to describe the year in the course (when interviewed during the last week of classes in the spring), Ali and Edward ascribe their successes largely to the high quality instruction they have received from Mr. Berberian. Edward: "I think Mr. Berberian's a really, really good teacher because even though I might not like math, at least it makes sense to me to learn." Edward goes on to describe Mr. Berberian's approaches to the teaching of content as addressing a number of different learning styles (visual, kinesthetic, etc.). This sophistication about learning styles and the pedagogies necessary to address them is shared by Ali, who also notes this resourcefulness as being among the successful strategies of her geometry teacher. At least some of the students' awareness in this regard comes from Shipley's health curriculum (according to Edward), and this circumstance suggests a way in which at another level of policy within the institution, the structures and practices enacted by Mr. Berberian in the service of teaching the ideas are supported by the site-specific independence scaffolding.
My observations of the class also suggest, however, a way in which school structures and practices create challenges for the teaching of math—and may ultimately push students in the direction of more instrumental than relational understandings of mathematics. One class that I observed, the next to last Friday of the school year, was also one of the last days for new material. Mr. Berberian introduced a project that required students to find the volume and surface area of a castle consisting of a variety of three dimensional shapes. On this day, Edward was absent for a crew regatta—a circumstance that occurs regularly at Shipley for all sorts of reasons, most of them legitimate and worthy. But the degree to which students are absent or unfocused tends to undermine more constructivist approaches designed to build relational understanding, because it is impossible to replicate precisely an experience a student misses. This is an issue Berberian notes in reflecting on the department's choice of textbooks: he acknowledges that he came to Shipley with a more constructivist orientation and initially preferred a textbook that reflected that pedagogy. But he came to realize that in an independent school environment, with all of the demands on our students' time and energy, a more traditional text that can serve as a resource is the most useful choice.
Both Ali and Edward dismiss their current geometry book as not much of a factor in communicating the material. Ali says, "The examples go in one ear and out the other."? Edward asserts with characteristic sang-froid, "I'm not a big fan of textbooks." For both students, the teacher is the central resource of the experience in geometry. They cite his efforts to teach to different learning styles, his constant engagement with them personally, and simultaneously, his resistance to simply providing students with answers or solutions to problems that they might come up with on their own. "He really urges us to be independent," is how Ali describes it. This observation is descriptive of intellectual and interpersonal behaviors that the instructor seeks to encourage. But it might also be understood as an articulation of a larger policy principle.
Students who are fortunate enough to attend an independent school that is committed to offering the widest array of resources and opportunities possible, must learn qualities of independence that are not as essential to students in a more regimented, less individualized environment. Because most of the content and pedagogy issues that are usually decided at the policy level—and then not revisited without considerable deliberative process—are decided by teachers within departments, the relationship between policy and practice is a more responsive, immediate and organic one. So, for example, in Lehrer and Shumow's study (1997) the focus is on "aligning construction zones," between parents and teachers within the context of a math reform known as CGI (Cognitively Guided Instruction). For my purposes, what is of interest about CGI is that it attempts to codify and articulate, with a strong emphasis on the work of Vygotsky, ways for teachers "to create a zone of proximal development" (Lehrer & Shumow 1997, p.45). In my independent school environment, these approaches are ones that would seem familiar to faculty in all disciplines—at least from a pedagogical standpoint. The difference is that teachers in our environment are freer to construct these approaches for themselves, and it is not necessary to mandate the approaches and dictate their terms.
The irony, then, is that because of the very extent of resources and opportunities available to our students, their school lives, as well as their lives outside of school, are exceedingly fragmented. And because all of these discrete elements must be funneled through a college process that is relatively inflexible, students are pushed toward understandings that are more instrumental than they are relational.
Neither Ali Barnes nor Edward Donaldson has a particularly robust sense of being a mathematician. When asked if he even likes math, Edward will politely say, "No." Ali says of math class, "Every year, I say that I don't like math as a subject, but the past few years, I haven't minded it at all. So I've had a good experience in high school with math, so far."?Even Edward, when asked to describe his year in math observes, "I would say my experience has actually been pretty good"—a circumstance which he ascribes largely to Mr. Berberian's approachability and skills as a teacher. He continues: "Something has enabled me to do better by myself in math than, say, with a tutor" (emphasis added).
These qualities of independence can be seen as the result of an exceedingly talented teacher's implementation of pedagogy within the independent school policy framework—the independence scaffolding. The trade-off of the policy orientation is that students are less likely to enjoy the kind of sustained, reflective experience of mathematics that might engender relational understanding—or perhaps just an awareness of that understanding. Their conversations with me suggest that Ali Barnes and Edward Donaldson understand more than they realize. Indeed, they seem to be both more invested and more capable than they know.
References
Boaler, J. (2002). Learning from teaching: Exploring the relationship between reform curriculum and equity. Journal for Research in Mathematics Education 33 (4), 239-258.
Knapp, M.S. (1997). Between systemic reforms and the mathematics and science classroom: The dynamics of innovation, implementation, and professional learning. Review of Educational Research, 67 (2), 227-266.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27 (1), 29-63.
Lehrer, R. & Shumow, L. (1997). Aligning the construction zones of parents and teachers for mathematics reform. Cognition and Instruction, 15 (1), 41-83.
NCTM principles and standards for school mathematics. Geometry standards for grades 9-12. Retrieved on May 1, 2003 from http://standards.nctm.org/document/chapter7/geom.htm.
Nelson, B. S. & Sassi, A. (2000).? Shifting approaches to supervision: The case of mathematics supervision. Educational Administration Quarterly, 36 (4), 553-584.
Remillard, J.T. (1999). Curriculum materials in mathematics education reform: A framework for developing teachers' curriculum development. Curriculum Inquiry 29 (3), 315-342.
Remillard, J. & Geist, S. (2002). Supporting teachers' professional development: Navigating openings in the curriculum. Journal for Mathematical Behavior, 5 (1), 7-34.
Skemp, R. R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 67 (2), 9-15.