Academic Exchange Quarterly Summer 2007 ISSN 1096-1453 Volume 11, Issue 2
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Process pedagogy, engaging mathematics
Irene M. Duranczyk,
professor, and Lee, associate professor and chair, Department of Postsecondary
Teaching and Learning,
A process-orientated pedagogy was used in a learning community to engage first-year college students as writers and mathematicians. Students were involved in exploring and experiencing how writers and mathematicians work. Emphasis was on the processes used by professionals to achieve their end products—a final manuscript or solved problem. This article presents examples of mathematics activities modeling a process-orientated pedagogy.
Mathematics and writing are unlikely bedfellows in most minds, yet these subjects in first-year college curricula have more in common than one might assume. Post-secondary education research asserts that these subjects are key portals to success in college (National Research Council, 1989; Venezia, Kirst, & Antonio, 2003). First-year students enter college with below college-level competencies in writing (32%) and mathematics (59%) (ACT Newsroom, 2005). Lack of confidence in these competencies often creates anxiety that informs first-year students’ overall academic identity (Astin, 1993; Pascarella & Terenzini, 2005). Thus, first-year math and writing courses are an ideal setting for enhancing students’ sense of confidence and competence.
The process approach, developed in composition over thirty years ago, demystifies writing by focusing on the composing process and by positioning the finished product as one stage in that process. Whether professionals or students in a composition course, writers do not magically create publishable texts. Rather, they engage in a sequence of activities (brainstorming, collaborating, seeking feedback, re-reading, stepping away from the text, reading other texts, creating outlines and multiple drafts) to generate, shape and improve their final product. Process pedagogy invites students to experience what writers do: drafting ideas and responses, soliciting feedback, reflecting on rhetorical choices, and revising (Britton, Burgess, Martin, McLeod, & Rosen, 1975; Emig, 1971; Faigley, 1986; Tobin, 2001).
Process pedagogy can empower students in mathematics too. Mathematicians go through a process of thought and activities to approach mathematical problems: framing a problem, generating solution paths, re-conceptualizing, and revising. They do not simply “arrive” at an answer. Mathematicians seek feedback, conferring with colleagues for viable alternative approaches and explanations. In the same way that process pedagogy in a composition class encourages students to identify as writers, a process approach in a mathematics class can shift the focus from “doing problems” to being mathematicians—doing the work mathematicians do (Blair, 2006; National Council of Teachers of Mathematics, 2000). The work of ethnomathematicians (Powell & Frankenstein, 1997), social justice mathematicians (Bishop, 1985; Gutstein & Peterson, 2005) and critical mathematics education researcher (Skovsmose & Valero, 2002), emphasize the need for connections between mathematical ideas and processes, culture, real world activities, empowerment of the individual and the academic mathematics classroom. Process pedagogy, applied within a first-year mathematics course, offers a practical set of tools for enacting this sense of engagement and empowerment.
Setting the stage
Our learning-community students were primarily first-generation college students in under-represented racial and ethic groups (Native American, Hmong, East African, or Chicano). Many students were also from cities, recent immigrants, and economically disadvantaged (50% TRIO participants). All had completed high school with four years of English and at least three years of mathematics but our institution’s placement mechanism, based on class rank, grade point average and ACT scores, identified them as students with writing and mathematic deficiencies. Our courses were designed to help students demonstrate mathematical and rhetorical knowledge and find their voices in the respective disciplines and the academy. Our recruitment materials and syllabi explicitly communicated a voice approach to mathematics and writing as a means of empowerment and agency. The use of the word “scholar” instead of “learner” was deliberate to inform students that we would acknowledge, legitimize, and incorporate students’ lived experiences within the academy. The explicit expression of shared power and for engaging in the production of knowledge was grounded by our understanding that participants would need to negotiate relationships between teaching and learning so as to build a community of scholars committed to reflection and purposeful social action.
The process approach in action
The sociocultural agency framework of Freire (1973, 1974), Kolb (1984), Bishop (2002), Frankenstein (1989) and Moses & Cobb (2001) informed our pedagogical practices. Constructing our curricula and pedagogical approach around process; concrete, real-world activities; and social issues enabled our students to tap into mathematical knowledge in a deeper and more substantive way (Boaler, 1998; Gutstein, 2003; Skovsmose & Valero, 2002). No longer “playing school,” mathematical skill and thinking became a way to understand and act in and on the world.
We began with mathematics-based group activities exploring how Hurricane Katrina’s
devastation exposed the extent of housing, poverty, and race issues in
We examined the
individual neighborhoods in
After reviewing the
news stories and demographic data, students prepared draft reports for peer
review. Their reports drew on personal context to make sense of and connect to
numbers and come to a more intimate understanding of the situation in
Student reports varied in mathematical content depth but created a starting point grounded in unfolding news stories. With peer review and feedback, students demonstrated that (a) in mathematics there is not one right answer nor one correct way to arrive at a conclusion, (b) mathematics is a not solitary activity, and (c) mathematics can add to our information and understanding of real world situations. Bishop (1985) theorized that development of mathematical skills would only occur when students are involved in the communication, negotiation, and development of shared meanings. We were witnessing this transformation.
Students began looking at
Our focus on
Have you ever lived in a community
with 93.5% African American, and . . . the population being 16,955 and 38% are
living in poverty? Seventh Ward in
Students used their identity and experiential knowledge to gain and apply mathematical skill, and engage in a deeper look at the socioeconomic issues of neighborhoods. This project relied on a process approach, to read, reflect, write, peer review, and revise in an attempt to evaluate assumptions about community based on numbers, percents, and attitudes. This exercise assisted students in taking a more critical view on how numbers frame our world and world views, and in thinking through how multiple experiences inform our understanding of the world, thus yielding different answers or interpretations, even when viewing the same data.
The Building Experience
We extended this connection between mathematical skill acquisition, housing and neighborhoods through a hands-on, community service learning project. Together we participated in a Habitat for Humanity building project. The experience provided another concrete, opportunity to apply some basic algebraic knowledge, while shifting perceived roles in the class. Some students had expansive experience with construction. We had no experience and were thus positioned as learners alongside the students. Shifting our context from the classroom to the building site enabled us to become a genuine community of learners, each member having skills to contribute and that needed development.
Work such as framing and placing walls provided a need for real-world use of algebraic formulas: measuring distance, determining corner angles and parallel walls, plumbing walls, and comparing/proofing the work with detailed floor plans. In some cases, initial work became merely a first draft, and the importance of paying attention to even seemingly insignificant numbers was made clear. An entire internal wall had to be deconstructed, and repositioned when the site supervisor discovered a discrepancy.
Our building experience followed a unit on relative error and rounding principles. Only someone being picky, perhaps, would care about rounding 1/16 of an inch in a 1,300 square foot house. But at our Habitat build, this error impeded further construction. Students were faced with a context wherein “errors” and attention to exact numbers were crucial. Some students complained that the building site supervisor was merely “too picky” (and what math educator hasn’t heard similar complaints about homework or exams?). The supervisor explained that this rounding error in the wall placement would have subsequent effects–it was not an isolated, self-contained error. This was a kitchen wall, and it had to be perfect for the cupboards and countertops to fit. Our error in erecting this wall did have significant consequences and a second, corrected “draft” was necessary.
The location of the
project site became a point of discussion in class, allowing us to revisit the
intersection of personal experience and individual vantage points in shaping
one’s truths. The city offered land
that was once public housing in
Two students offered a critical perspective of the situation by sharing their personal experience with this land re-allocation. Their families had been dislocated to clear the site for this development—which was to favor families with higher incomes than past occupants, and for single family housing instead of multiple dwellings. A secondary effect of the housing development was to increase city revenue by generating higher property taxes and improving the neighborhood. A multi-billion dollar baseball stadium was planned for a site nearby. City officials believed that the new stadium would lose support and funding if it was in an impoverished area. For our students, their personal relationship to this site led to a different interpretation. As they saw it, this was not simply a “win-win” improvement of their neighborhood, it was a gentrification – the direct result was forced relocation. It was no longer their neighborhood. Once again, students had to contend with more than one right answer – propelling them to sort through conflicting and legitimate takes on a problem. In this instance, an algebraic formulation didn’t motivate our discussion of competing approaches and solutions, but similar processes of interpretation, formulating, peer dialogue, and ultimately revision were in play. The mathematics we envisioned for the class became secondary to the mathematics of city planning and the social and political issues experienced by our students. Thus, the project illustrates the possibility for fostering students’ development as mathematicians in conjunction with developing “students’ social and political consciousness, their sense of agency, and their social and cultural identities (Gutstein, 2003, p. 42)”.
Mathematics Test as a first draft
Because the course was informed by a process-oriented pedagogy, tests and quizzes took on a revised purpose. These traditional tools served not only to test students’ knowledge of algebraic content, but also became a vehicle for students to reflect on their learning, and to apply algebraic knowledge to real and abstract data. Using tests as a gatekeeping device would limit their value within our course, and simultaneously conflict with the terms and values at the center of our pedagogical approach. A process approach, wherein the tests are one stage of a developmental process, allowed us to revise our design and deployment of exams. In test design, we prioritized not only the application and demonstration of mathematical knowledge, but the demonstration and application of mathematical process. Tests required students to represent their approach to problems, their conceptions about how to solve them, and their reflections on the how and why of their ultimate choices about solutions.
In addition to their design, the mid-term and final were administered differently, positioned as first drafts much in the way that a paper project in a writing class would have a first draft. The test results counted toward the final grade, and provided us with a snapshot of a student’s progress and performance. As a test and a product of a testing moment, they were final drafts. Following the testing moment, however, they became first drafts that went on to receive comments from peers and faculty, and were then revised into second drafts.
The first two tests were followed by an individual conference with each student. Students discussed their answers, and related conflicts or questions that occurred in the process. These sessions allowed students to further explore and critique their thinking. We gained insight into the obstacles, sticking points, and misperceptions that impacted students’ understanding and performance. We used this insight to design subsequent class sessions to clarify concepts, and extend knowledge. If a student was still struggling with resolution, they could go on to produce a third or fourth draft. In this way, power shifted from positioning the test as a sole arbiter of student knowledge, to positioning the student as able to work to develop and demonstrate knowledge.
“To have more than a surface understanding of important social and political issues, mathematics is essential” (Gutstein & Peterson, 2005, p.2). A process-orientated pedagogy within the mathematics classroom expanded students’ ability to link their lived experiences with academic mathematics. Our process approach and incorporation of projects increased attendance rates, retention rates, and student engagement. Students saw the work of mathematicians—solving problems that do not have neat answers or only one right answer—solving problems that matter and inform our thinking and vision of the world.
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