Boaler (2002) FLM

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The Development of Disciplinary Relationships: Knowledge, Practice and Identity in Mathematics Classrooms

Abstract

The article does not have an abstract.

Outline of Headings

  • The relationships between knowledge and practice
  • Relationships between knowledge, practice and identity
  • Developing relationships with the discipline of mathematics
  • Discussion and conclusion

Summary

Boaler wrote this article as a reflection of how she'd come to see the learning experience and "relationships between teaching and learning, between knowledge and practice, and between learning and believing" (p. 42). Boaler's experience in classrooms coincides with the "cognitive revolution" (Schoenfeld, 1999; Resnick, 1991) that superseded behaviorist views of learning that claimed knowledge was stable, individual, and transferrable. Mathematical learning was seen to rely on opportunities to acquire knowledge that was taught and practiced to reinforce behaviors. Constructivists believed that knowledge was not acquired by students, but that students actively organized and reorganized ideas as part of their own constructions (Lerman, 1996). While different, both of these perspectives see knowledge as characteristic of the individual and something that can be taken from place to place.

Learning can also be seen from a situated perspective where knowledge is distributed among people, activities, and their environment (Lave, 1988; Greeno and MMAP, 1998; Boaler, 2000; Cobb, 2000). Situated learning reflects how people use and co-construct knowledge in ways that depend on their environment. Therefore, while cognitive processes may still be important (Greeno, 1997), they are seen as occurring as part of a broader context in which they occur (Greeno and MMAP, 1998) and knowledge is no longer viewed as exclusive to the individual.

In her research of two contrasting high schools, "Amber Hill" and "Phoenix Park," Boaler (1997) investigated the relationship between teaching, student beliefs, and student understanding. She showed that in Amber Hill, where students spent much of their time working on textbook exercises, students came away with an ability to do textbook-style problems but struggled in more open or applied problem-solving situations. In contrast, students at Phoenix Hill spent most of their time in project-based approaches to mathematics. Those students outperformed the Amber Hill students on the national examination, despite similar scores prior to the curriculum intervention, and despite the exam looking mostly like textbook exercises. A cognitive interpretation of these results would be that the Phoenix Park students simply learned more. Boaler, however, acknowledged that the Amber Hill students learned a great deal but struggled to apply that knowledge beyond the practice of completing prescriptive textbook exercises. Therefore, the practices students engage in as they do mathematics are important — not because it necessarily affects the amount learned, but because the practices shape the students' knowledge itself (Dowling, 1996). Boaler concluded that for students to use and apply their mathematics knowledge flexibly, they would have to engage in mathematical practices that engage them in similarly flexible learning experiences.

Along with this more situated view of mathematical learning, Boaler included student identity as a dimension along with knowledge and practice. In a study of AP Calculus students (Boaler & Greeno, 2000), students in four of the six high schools in the study engaged in a more traditional, procedure-focused mathematics similar to students at Amber Hill. In the other two high schools, there was a greater focus on student discussion and collaboration around problem solving. Boaler related the traditional approach to the idea of "received knowing" (Belencky, Clinchy, Goldberger, & Tarule, 1986) where knowledge is presented to students in a way that gives mathematical authority to the teacher and the textbook (Ball, 1993), making the students "received knowers." Students in these classes, while generally successful, reported not liking mathematics and feeling too constrained in practices that "left no room for their own interpretation or agency" (p. 44). Many of these students wished to not take mathematics in the future, even though the pedagogical practices of their mathematics experience were unrelated to the nature of mathematics itself (Burton, 1999a, 1999b). Some students in these classes reported being satisfied and motivated, apparently pleased to be a "received knower":

Jerry, Lemon School: I always like subjects where there is a definite right or wrong answer. That's why I'm not a very inclined or good English student. Because I don't really think about how or why something is the way it is. I just like math because it is or isn't. (p. 44)

In the discussion-oriented classes, students formed a relationship with mathematics that did not seem to conflict with their identities as creative problem-solvers:

Veena, Orange school: Sometimes you sit there and go 'It's fun!' I'm a very verbal person and I'll just ask a question and even if I sound like a total idiot and it's a stupid question I'm just not seeing it, but usually it clicks for me pretty easily and then I can go on and work on it. But at first sometimes you just sit there and ask — 'What is she teaching us?' 'What am I learning?' But then it clicks, there's this certain point when it just connects and you see the connection and you get it. (p. 44)

Student identity, which includes the relationship students had with mathematics, was an important factor in Boaler's study even though student achievement in the two types of schools was similar. In the discussion-oriented classes students described their participation in mathematics in ways consistent with how Wenger (1998) described learning as a process of "becoming":

Because learning transforms who we are and what we can do, it is an experience of identity. It is not just an accumulation of skills and information, but a process of becoming — to become a certain person or, conversely, to avoid becoming a certain person. Even the learning that we do entirely by ourselves eventually contributes to making us into a specific kind of person. We accumulate skills and information, not in the abstract as ends in themselves, but in the service of an identity. (p. 215)

With this Boaler had reflexively related knowledge and practice, and seen how practice influenced identity, but had yet to connect knowledge to identity. For this she engaged in new research involving about 600 students in three high schools. (See note.) As students engaged in either traditional or reform-based courses, student identity and agency played an important role. In reform classrooms, students developed agency as they proposed "theories" and critiqued each other's ideas in ways that did not center mathematical authority with the teacher or the textbook. Boaler acknowledged that such agency raised concerns (Rosen, 2000) that students will spend much of their time and effort unproductively and are left with a "fuzzy" mathematics (Becker & Jacob, 2000). But in her study, Boaler found that the nature of the agency bore similarities to what Pickering (1995) observed in the work of professional mathematicians. Pickering saw two main types of agency at work when professional mathematicians made mathematical advances: a human agency to propose new approaches and ideas, as well as a disciplinary agency that seemed to prescribe certain standard methods or approaches. In Boaler's reform-oriented classrooms, students could more regularly engage in both types of agency, whereas traditional classrooms were dominated by disciplinary agency. This "dance of agency" in the reform classrooms was evident as students described proposing new approaches, testing those approaches, and evaluating their success as they found ways forward.

While other researchers have described the importance of student beliefs and dispositions (Schoenfeld, 1992; McLeod, 1992), Boaler saw a connection between knowledge and identity she described as a "disciplinary relationship" where knowledge is not so much mastered as it is appropriated. Herrenkohl & Wertsch (1999) described the appropriation of knowledge as a student's need to not just develop skills, but to develop a disposition to use them. Boaler believes the students at Phoenix Park and in other reform classrooms had developed a "positive, active relationship with mathematics" (p. 46) that made them able to apply more knowledge in different contexts. In the case of professional mathematicians, Boaler believes that a mathematician without some particular knowledge will make progress on a problem because she would know a set of practices for problem-solving and have a productive disposition towards doing so. This combination of practices, identity, and agency will put her in a position to find the resources needed to fill the gaps in knowledge.

Boaler admits that the relationships between knowledge, practice, and identity need to be more well-defined, and more needs to be known about if or how learners appropriate particular knowledge. Also, more needs to be known about how advantages for learners vary in the "dance of agency" and how factors like gender, race, and class intersect with students' relationships with mathematics.

Note: This is almost certainly what became Boaler's well-known Railside study, described in Boaler & Staples (2008). --Raymond Johnson (talk) 03:52, 21 November 2013 (UTC)

Also

APA
Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms. For the Learning of Mathematics, 22(1), 42–47.
BibTeX
@article{Boaler2002,
author = {Boaler, Jo},
journal = {For the Learning of Mathematics},
number = {1},
pages = {42--47},
title = {{The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms}},
url = {http://www.jstor.org/stable/40248383},
volume = {22},
year = {2002}
}