Enhancing mathematics teaching for at-risk students: influences of a teaching experience in alternative high school.

This paper examines a preservice teacher initiative whose overarching goal is to begin to reverse the cycle of educational failure for students labeled "at-risk." Although research in teacher preparation has explored the ways in which preservice teachers learn to teach, few studies have focused on how teaching in an alternative high school interacts with and complicates the process of learning to teach mathematics. The study discussed in this paper investigates the influences of a mathematics teaching experience in an alternative high school on five preservice teachers who planned, designed, and taught an integrated mathematics class for at-risk students who were enrolled in the school. The findings reveal that the teaching experience provided the preservice teachers with a vehicle for reflecting on and reconsidering their conceptions of mathematics teaching and perceptions of at-risk students. The analysis suggests that the experience enhanced the preservice teachers" perceptions of at-risk students, prompted them to set higher expectations for the students, and encouraged them to engage in conceptually-focused mathematics teaching. In addition, teaching in the alternative high school provided the preservice teachers with experiences that fostered an understanding of how at-risk students think mathematically and how to cultivate mathematical thinking in at-risk students.

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This study evolved out of a commitment to improve mathematics teaching and learning in an alternative high school and to begin to reverse the cycle of educational failure for students labeled "at-risk." Although research in teacher preparation has explored the ways in which preservice teachers learn to teach mathematics, few studies have focused on how teaching in an alternative high school interacts with and complicates this process. Paralleling the need for more effective mathematics education programs for at-risk students is the need to improve the preparation of preservice teachers to work with at-risk students and other marginalized students. A goal of this study, to produce instructional change and to improve the conditions for learning, developed out of a recognition that existing instructional practices underestimate and constrain educational achievement of at-risk students (Moll & Diaz, 1987). The teaching experience examined in this study attempted to provide teacher preparation geared for at-risk students; emphasize teaching mathematics through context; encourage the demonstration of the relationships between mathematics and the lives of the at-risk students; and, the use of hands-on materials, small group, one-on-one instruction, or out-of-class experiences (Tobias, 1992). In order to examine what preservice mathematics teachers believe and do in response to a student population composed of at-risk mathematics learners in an alternative high school, this study investigated two research questions:

1. What challenges are faced by preservice teachers as they attempt to teach mathematics for understanding in an alternative high school?

2. In what ways does participating in a mathematics teaching experience in an alternative high school influence how preservice teachers learn to teach mathematics?

Perspectives

At-risk Students

When students' home resources and experiences differ from the expectations on which school experiences are built (McCarthy & Levin, 1992), they are often at risk of not realizing their personal and academic promise. Because the term "at-risk" focuses on individual characteristics, it labels and stigmatizes the learner. However, although objectionable, the term "at-risk" persists because of its wide acceptance and research base (Garcia & Walker de Felix, 1992). In her discussion of the at-risk secondary mathematics student, Vatter (1992) summarizes the characteristics of at-risk learners as: (1) poor self-concept; (2) poor academic performance, high absenteeism, and discipline problems; (3) low aspirations and parents or guardians with low expectations; (4) low family socioeconomic level; (5) nontraditional family life, often with a single or foster parent, or with a stepparent; and (6) inadequate goals or lack of future orientation.

While the literature suggests that learners are at risk due to factors related to their socioeconomic status, family background, or community, it is more likely that learners are at risk because schools are not meeting their specific educational needs (Baptiste, 1992). For example, mathematics for at-risk learners is typically perceived as a hierarchy of skills that are learned in a particular sequence (Carey, Fennema, Carpenter, & Franke, 1995). Learners labeled as less capable than their peers are taught less mathematics and are presented with skill-oriented, direct instruction and practice rather than conceptually focused instruction promoting problem solving and understanding (Campbell & Langrall, 1993). Such instruction often fails because it reinforces learners' negative self-perceptions and deprives them of cognitive stimulation (Silver, Smith, & Nelson, 1995).

Preservice Teachers

Teachers' conceptions of mathematics teaching influence what they do in the classroom and play a significant role in the teachers' characteristic patterns of instructional behavior (Thompson, 1984). In this study, conceptions refer to both knowledge and beliefs (Thompson, 1992; Cooney & Wilson, 1993). A classroom teacher's conception of mathematics has a powerful impact on the way in which mathematics is approached in the classroom (Cooney, 1985). Teaching practices reflect teachers' conceptions which resonate with their own experiences and background (Cabello & Burstein, 1995). Secondary mathematics teachers' knowledge, conceptions, and attitudes about students and student learning impact the way in which these teachers interact with students in their mathematics classrooms (Calderhead, 1984). Because teachers act upon their expectations of students, negative teacher conceptions or low expectations for their students influence classroom practices and may adversely affect student performance (Brophy, 1985).

Teachers' conceptions of mathematics, as well as the social context, have an impact upon teachers' instructional decisions (Ernest, 1991). Studies support the view that teachers' behavior in the classroom is shaped by internal principles based upon conceptions of mathematics teaching. However, it appears that the origins of teachers' conceptions are linked to experiences prior to their participation in mathematics education courses. Prospective teachers enter often mathematics courses with strongly held attitudes and conceptions about how mathematics is learned and the role of the teacher that are derived from their own educational experiences (Crawford, 1992).

Methods

Participants and Context

The participants for this study were five secondary preservice mathematics teachers who were enrolled in a semester-long secondary mathematics methods course at a public university. As part of the field experience component of the methods course, the preservice teachers had volunteered to teach a sixteen-week integrated mathematics class in an alternative high school. The school endeavors to provide a positive forum for students who have experienced difficulties in more traditional academic settings. In addition to having a history of poor academic performance, drugs and alcohol have often played a role in the students' lives. The school offers an individualized curriculum that stresses academic achievement, the development of character and positive societal attitudes, and encourages a thirst for knowledge.

The primary goals of the teaching experience were to enhance mathematics teaching for at-risk students and to improve the preservice teachers' preparation in mathematics and at-risk education. The preservice teachers, who were free to develop their own curriculum, instructional vehicles, and assessments, focused on numbers and operations, geometry, measurement, and algebra. Throughout the semester, the methods instructor and researcher, a former secondary mathematics teacher, provided a mentoring component by offering instructional ideas, activity samples, and critiquing the instructional ideas developed and implemented by the preservice teachers. The preservice teachers also established weekly planning sessions in order to schedule their teaching, refine the curriculum, plan lessons, explore teaching strategies, and engage in a general discussion of the teaching experience.

Data Collection

The researcher spent a total of 190 hours engaged in data collection during the sixteen-week semester. Seeking to understand what participants do and what they know that makes their actions understandable (Spindler, 1982), the researcher played the role of participant-observer, making long-term observations of the mathematics class at the alternative high school and the preservice teachers' planning sessions. The collection and analysis of data took place simultaneously and were guided by the constant comparative method of data analysis (Glaser & Strauss, 1967) and allowed for the testing of themes throughout the study. Findings were shared, discussed, and compared by the researcher and the preservice teachers. For example, in order to provide a measure of external validity (Goetz & LeCompte, 1984), the researcher reviewed transcripts and analyses with the preservice teachers and allowed them to react to analyses and clarify and elaborate on their responses. Consistent with the qualitative research methodology of the study, multiple data sources enhanced the validity of the findings. These included: (1) observations, videotaping, and field notes of the mathematics classes and the methods class; (2) observations, audio taping, and field notes of the preservice teachers' planning sessions; (3) structured and semi-structured, open-ended interviews (Novak & Gowin, 1984; Spradley, 1979) with each preservice teacher focused on perceptions of mathematics teaching and learning, the teaching experience at the alternative high school, and at-risk students; and, (4) collection of classroom artifacts, including the preservice teachers' lesson plans and journal of observations of classroom activities and copies of student assignments.

Data Analysis

The researcher utilized qualitative analytical processes to interpret and find inferences in the data in order to establish emergent patterns and address the research questions (Strauss & Corbin, 1990). As the analysis proceeded, the researcher was able to identify themes embedded within the data. Transcripts of the initial interviews were analyzed in order to identify emerging, related themes and guided both the observations and the formulation of questions for subsequent interviews. Procedures for the coding of the transcripts followed Strauss's (1987) basic work processes in qualitative analysis. Participants' responses were grouped into categories designed to bring together similar ideas, concepts, and themes. Once the data were coded, the researcher wrote interpretive memos in which the meanings of emergent themes in the data were analyzed (Miles & Huberman, 1994). Related ideas and concepts from the data were grouped together to form clusters of related terms and processes. These coding categories based upon the research questions included: (a) conceptions of teaching mathematics; (b) perceptions of at-risk students; and, (c) instructional practices.

Results and Discussion

Conceptions of Mathematics Teaching

The teaching experience provided the preservice teachers with a vehicle for reflecting on and reconsidering their conceptions of mathematics teaching. The descriptions in this section provide examples of how the teaching experience in the alternative high school prompted the preservice teachers to rethink their conceptions of mathematics teaching.

Brad, a preservice teacher who attributed his success in mathematics to following a series of procedures, declared in his initial interview, "Teaching mathematics is giving students a set of skills that they could apply to a variety of situations." Participating in the mathematics teaching experience in the alternative high school initiated several changes in Brad's conception of mathematics teaching. For example, during his final interview, Brad reflected:

   Watching these students work on problem-solving
   activities made me realize
   the fact that math isn't always one right
   answer, there's a lot of, exploration that
   can go on instead of just hammer out
   problems with one right answer. Just as
   more of the ways to teach math, there's
   so many different ways you know there's
   not just one, write it down on the board
   and they learn it, there's so many different
   ways that are much more successful
   as far as teaching students.

At the onset of the study, Mark, a preservice teacher who described himself as a "strong math student", asserted, "From my own experiences, teaching math is making sure that students leave your class with the skills they need to succeed in life." Engaging in the mathematics teaching experience at the alternative high school prompted Mark to reconceptualize his ideas about mathematics teaching as he recognized that teaching mathematics required not only a knowledge of the procedures, but also an understanding of the meaning underlying those procedures and an understanding of the conceptual underpinning of the mathematics being taught. At the conclusion of the semester, he revealed that the teaching experience challenged him to examine his own understandings:

   It's interesting teaching math because,
   it's more than just giving the students
   formulas, I have to go back and re-learn
   it, I'll know the stuff and I can do the
   stuff but to teach math, you need to
   understand all the concepts, it takes up,
   it takes a lot to rethink, ok, "why is
   this?", and, "what is this?", so I learned
   a lot of, and I know I will re-learn a lot
   of stuff in math when I'm teaching it,
   just, stuff that I know, the concept of,
   but then I have to, to really explain, you
   have to know all the concepts so, you
   almost have to re-learn math that way so
   you can you explain it, effectively.

Perceptions of At-risk Students

The preservice teachers embarked upon the teaching experience with feelings of trepidation about the at-risk students' motivation as well as their mathematical abilities. With the exception of two teachers, one of whom identified herself as having been "at-risk" and a second who had previously worked with at-risk students, the preservice teachers initially held negative perceptions of at-risk students. They described at-risk students as, "rebellious", "difficult to work with", and likely to "be a real problem in the classroom." These perceptions found an echo in the preservice teachers' early lessons which were procedurally-focused and teacher-centered. As the semester progressed and the preservice teachers forged close personal relationships with the students, they expressed surprise at the students' mathematical abilities and positive attitudes toward learning. During one of the teachers' planning sessions, Mark commented, "We could be doing a lot more math with these students."

Engaging in the teaching experience in the alternative high school appears to have enhanced the preservice teachers' perceptions of the attitudes and mathematical abilities of at-risk students. At the conclusion of the study, all of the preservice teachers rethought and redefined the label "at-risk." For example, Melissa, one of the preservice teachers, noted, "They can do the work, it's just that they haven't been able to do it in a regular school." Similarly, at the conclusion of the study Kevin, another preservice teacher, stated, "Even though they're at-risk doesn't mean they're not good students and they don't want to learn, it's because they are, and before I didn't really believe that." Brad expressed a similar sentiment:

   The fact that they have a lot of home
   problems, and sometimes school's not
   the number one priority at the time, and
   so it's tough to reach them, you know,
   they come there, they' re there, but, their
   mind's elsewhere, you know, depending
   on whatever happened, and I could
   see that being a problem at times--they
   were, a lot better behaved in the class
   than I thought you know, going in, I'm
   thinking alternative high school and this
   is going to be terrible and they aren't
   going to listen to you one bit, they won't
   do a thing, and then, the reality is the
   actually did, once you gained trust with
   them and they gained trust with you, it
   was, they trusted you to teach them, and
   they paid attention, and that could be a
   good lesson, that just any normal teacher
   going into class, even if you've heard
   stuff about them, have an open mind,
   when you first start out to teach them,
   don't go in with a bad attitude about the
   student.

Instructional Practices

During the early weeks of the study, observations revealed that the mathematics class mirrored the typical pedagogy for at-risk students. For example, the preservice teachers walked the students through the tasks in incremental steps, frequently utilized expressions such as, "just set it up", "plug it in", "what's the next step?", and "what would do you first?", and provided the students with explanations rather than allowing them to articulate their own explanations and strategies. It appears that these directive, teacher-centered lessons were connected to a desire to maintain classroom order. As Brad stated during an early planning session, "We'll want to make sure they're focusing on the math and not messing around. I can imagine this activity having the potential to get out of hand and the next thing you know is we'll lose control of the class."

As the semester progressed, and the preservice teachers revised their perceptions of at-risk students and their conceptions about teaching mathematics, they began to engage in student-centered, conceptually-focused teaching, including the use of collaborative learning groups and mathematical discussions. The preservice teacher had some measure of success in eliciting student explanations and this encouraged them to focus on mathematical concepts, structure, and connections rather than algorithms and procedures. Melissa, noted for her utilitarian view of mathematics and traditional teaching methods, concluded at the end of the semester:

   I think that a good math teacher is also
   going to let students realize that there's
   more than one way to solve a problem,
   they're going to be many ways, they're
   going to use many explanations, I think
   they would present math in ways that
   are more interesting to the students than
   traditional methods.

Although many of the early lessons represented traditional dialogue patterns, with the preservice teachers posing closed-ended questions, later lessons incorporated open-ended questions and student-centered discussion of mathematical concepts, including students responding to one another. The preservice teachers indicated that they made these changes to their instructional strategies because of the at-risk students' positive responses to their attempts to encourage mathematical discussions. Reflecting on an inquiry-oriented lesson she taught on the concepts of area, measurement, and geometric properties, Toni, a preservice teacher, explained:

   I definitely wanted to encourage student
   explanations so I started the lesson
   by asking the students to share their
   strategies for finding the areas of polygons
   when they didn't know the specific
   formulas. Two students saw right
   away that they could break a polygon
   down into shapes whose areas they knew
   the formulas for. I tried to ask prompting
   questions to get the students involved
   and interested and I think I made
   the students feel comfortable with the
   material, rephrasing it for, for students
   who weren't comfortable with the material,
   and then I tried to have the students
   that were uncomfortable, unsure
   of the material, try to explain it.

Conclusion

The present study provides some insight into whether and how a mathematics teaching experience in an alternative high school influences preservice teachers' perceptions of at-risk students and promotes changes in their instructional practices and conceptions of mathematics teaching. An examination of what preservice mathematics teachers believe and do in response to a student population composed of at-risk students revealed that the alternative high school provided a unique environment for the preservice teachers. Within this environment, the preservice teachers engaged in a shared, collaborative process of reexamining and challenging conceptions of mathematics teaching and perceptions of at-risk students. An important implication of this study is that a preservice teaching experience in an alternative high school can foster preservice teachers' understandings of how at-risk students think mathematically and how to cultivate mathematical thinking in these students. The positive effect of the teaching experience on the preservice teachers and the at-risk students demonstrates that educators can begin to reverse the cycle of educational failure for at-risk students.

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Thea K. Dunn, Ph.D., Assistant Professor, Mathematics Education, College of Education, University of Wisconsin-River Falls.

Correspondence concerning this article should be addressed to Dr. Thea K. Dunn, Assistant Professor, Mathematics Education, College of Education, University of Wisconsin-River Falls, 410 South Third Street, River Falls, WI 54022; Email: [e-mail]