Running head: Teaching Elementary Mathematics



Running head: Learning to Teach as Assisted Performance

Learning to Teach as Assisted Performance

Denise S. Mewborn

David W. Stinson

University of Georgia

Please address all correspondence to the first author at

Department of Mathematics Education

105 Aderhold Hall

Athens, GA 30602-7124

Phone: 706-542-4548

Fax: 706-542-4551

e-mail: dmewborn@coe.uga.edu

Paper presented at the Research Presession of the Naitonal Council of Teachers of Matheamtics Annual Meeting, April 21, 2004, Philadelphia.

Submitted to Teachers College Record.

The research reported in this manuscript was funded by the Spencer Foundation. The views expressed in this paper are those of the authors and do not necessarily reflect official positions of the Spencer Foundation.

Abstract

Using ideas that can be traced to Dewey, Feiman-Nemser (2001) questioned the deeply-rooted assumption that teacher education, in general, and field experiences, in particular, are supposed to provide future teachers with an opportunity to practice the things that they will be expected to do as teachers. In contrast, she proposed that teacher education programs should provide an opportunity for future teachers to engage in “assisted performance” (p. 1016) that enables them to “learn with help what they are not ready to do on their own” (p. 1016). In this manuscript we provide evidence of how preservice elementary school teachers learned to teach mathematics via assisted performance.

Learning to Teach as Assisted Performance

Feiman-Nemser (2001) proposed that teaching should be viewed as a continuum with specific tasks to be accomplished in the preservice, induction, and continuing professional phases of a teacher’s career. She identified five tasks that are central to the preservice years:

1. Examine beliefs critically in relation to vision of good teaching

2. Develop subject matter knowledge for teaching

3. Develop an understanding of learners, learning, and issues of diversity

4. Develop a beginning repertoire

5. Develop the tools and dispositions to study teaching. (p. 1050)

In discussing the first of these three tasks, Feiman-Nemser (2001) suggested that the images and beliefs about teaching and learning that preservice teachers bring to their teacher preparation programs act as filters for new learning. Therefore, they must critically analyze “their taken-for-granted, often deeply entrenched beliefs so that these beliefs can be developed and amended” (p. 1017). Feiman-Nemser also noted that this examination of beliefs should be coupled with the formation of new “visions of what is possible and desirable in teaching to inspire and guide their professional learning and practice” (p. 1017).

In order to develop an understanding of learners and learning, Feiman-Nemser recommended that preservice teachers need to learn what children are like at different ages, how they make sense of their worlds, how their thinking is shaped by language and culture. These types of understandings enable teachers to design age-appropriate, meaningful instructional activities, make and justify pedagogical decisions, and communicate with others.

Feiman-Nemser said that preservice teachers need to develop the tools to study teaching because “learning is an integral part of teaching and …serious conversations about teaching are a valuable resource in developing and improving their practice” (p. 1019). She proposed that teachers could develop the tools to study teaching by analyzing student work, comparing different curricular materials, interviewing students, studying other teachers, and observing their own instruction. Feiman-Nemser argued that when these activities are carried out in the company of other teachers they advance norms for professional discourse as teachers gain confidence in critically examining their own and their colleagues’ teaching. Feiman-Nemser also questioned the deeply-rooted assumption that teacher education, in general, and field experiences, in particular, are supposed to provide future teachers with an opportunity to practice the things that they will be expected to do as teachers. For example, is the purpose of a field experience to immerse future teachers in the daily life and routines of classrooms so that they become comfortable with such tasks as taking attendance, collecting morning seatwork, preparing bulletin boards, and conducting calendar time? In contrast, Feiman-Nemser proposed that teacher education programs should provide an opportunity for future teachers to engage in “assisted performance” (p. 1016) that enables them to “learn with help what they are not ready to do on their own” (p. 1016).

The prevailing apprenticeship model of teacher education often leaves preservice teachers with a feeling that the only way to learn to teach is to wait until they have their own classrooms and are able to devise their own methods of teaching. Lanier and Little (1986) caution that this “wait and see” approach makes it difficult for preservice teachers to see the range of possible decisions and actions available in teaching, resulting in a continuation of the teaching practices by which they were taught and a tendency to see these patterns of teaching as the only options. This restricted view of teaching makes it “less likely that they will escape from intellectual dependency and begin to take responsibility for decisions about curriculum and students” (p. 552).

Feiman-Nemser identified three characteristics of high quality teacher preparation programs that provide opportunities for future teachers to engage in the five tasks identified above via assisted performance. First, the program must have conceptual coherence by being built upon a “guiding vision of the kind of teacher the program is trying to prepare” (p. 1023). This vision includes a particular view of learning, the role of the teacher, and the goal of schooling. Second, the program includes purposeful, integrated field experiences that provide preservice teachers with opportunities to “test the theories, use the knowledge, see and try out the practices” (p. 1024) they have learned in their courses. Third, the program treats preservice teachers as learners, recognizing that they bring experiences and knowledge with them and that their learning is continuous and dynamic.

In this manuscript we provide a description of a mathematics methods course that was coherent, included purposeful, integrated field experiences, and was based on a view of teachers as learners. We substantiate this claim by showing how the preservice teachers addressed three of the central tasks identified by Feiman-Nemser: examine their beliefs, develop an understanding of learners, and develop the tools and dispositions to study teaching via assisted performance.

Methodology

Setting

The study took place in the context of the professional education program for elementary education majors (certification PreK-5) at the University of Georgia. Prior to entering the program, students completed a 60-hour liberal arts curriculum, including a mathematics content course designed specifically for elementary education majors. The professional education program was a 60-hour, four-semester program that included two mathematics methods courses (3 semester hours each). These courses were taken during the first two semesters of the program, along with methods courses in other content areas. The final two semesters consisted of additional methods and content courses and student teaching.

The first author was the mathematics methods instructor for all of the students in the study. The mathematics methods courses focused on eliciting, understanding, and crafting teaching practices based on children’s mathematical thinking. The goal of the instructor’s teaching was to help novice teachers craft practices that were consistent with current reform efforts but that were also respectful of their personal theories and the institutional constraints of their teaching situations. The instructor’s goals were to help teachers see unexpected student responses as teachable moments rather than as management problems (Floden & Buchmann, 1990), learn to encourage students to continue to seek understanding rather than concede defeat (Floden & Buchmann, 1990), challenge students’ mathematical misconceptions (Ball & McDiarmid, 1990), and focus on pursuit of meaning rather than pursuit of truth or fact (Grimmett, 1989). Experiences with children were coupled with an examination of appropriate research and theoretical literature (i.e., constructivism, children’s solutions to addition and subtraction word problems, children’s fraction knowledge). Novices had structured opportunities to use their experiences with children to make sense of the theory and vice versa. The intertwining of theory and practice also provided the novices with opportunities to examine their personal theories in light of actual teaching practice and research-based theories.

Participants

The data reported in this paper came from a larger five-year research project entitled Learning to Teach Elementary Mathematics. The goal of the project is to develop conceptual frameworks for understanding teaching and learning in elementary mathematics teacher education by studying how novice teachers craft their teaching practices across time as a result of their personal theories, teaching experience, and teacher education programs. Specifically, the goal of the larger study resonates with Feiman-Nemser’s (2001) call for teacher educators to view preservice teachers as learners because we seek to build a theory of how preservice teachers learn to teach mathematics.

The participants for the project were selected from two cohort groups who began the four-semester teacher education program in the fall of 2000 (Group A) and 2001 (Group B) and remained together as a cohort for the four semesters. Some data were collected on all students from each cohort, but the majority of data collection focused on two target subsets—six students from Group A and nine students from Group B. The target students were selected by purposeful sampling (Bogdan & Biklen, 1992) to represent a range of personal theories about mathematics. And to the extent possible, the target students were reflective of the diversity of students enrolled in each cohort (i.e., gender, race, age, etc.). In this manuscript we use data from a further subset of these participants—two participants from Group A and five participants from Group B–to illustrate the ways in which our program implemented an assisted performance model of teacher education.

Data Collection and Analysis

Data were collected and analyzed using the interpretive paradigm for teacher socialization (Zeichner & Gore, 1990). This interpretive approach involves an attempt to understand the nature of a social setting at the level of subjective experience. The purpose of this approach is to gain an understanding of the situation from the perspective of the participants and within their levels of consciousness and subjectivity. The goal is to “capture and share the understanding that participants in an educational encounter have of what they are teaching and learning” (Kilpatrick, 1988, p. 98). Eisenhart (1988) noted that the purpose of the research questions posed by researchers using the interpretive paradigm is to first describe what is “going on” and second to uncover the “intersubjective meanings” (p. 103) that undergird what is going on in order to make them reasonable.

To conduct interpretivist research, it is essential for the researchers to “be involved in the activity as an insider and able to reflect upon it as an outsider” (Eisenhart, 1988, p. 103). It is important for the researcher to be able to “take on the views of those being studied” (Eisenhart, 1988, p. 105) and also to be able to “step back from the immediate scenes of activity and to reflect on what is occurring from the perspective of someone who is aware of other systems and of theoretical perspectives”(Eisenhart, 1988, p. 105). Our research team attempted to accomplish these objectives in several ways. As mentioned previously the first author was the instructor for all mathematics methods courses taken by the participants, thus making her an insider. A graduate student who contributed to the data collection and analysis for this manuscript was a participant-observer in every class period. The second author joined the research team after the two mathematics methods courses were completed. He assisted with data collection during the remainder of the study and participated in data analysis. Thus, the researchers had varying degrees of involvement with the participants.

For research studies in the interpretivist tradition, Eisenhart (1988) advocated the use of ethnographic methods of data collection. Ethnographic methods have traditionally been used to study populations that are radically culturally different from the researcher, but these methods can be adapted for use in educational research. Ethnographic studies have often involved researchers literally living with a group of people over time to come to understand their lives. In the educational setting, the researcher “lives” with the participants within the community of the teacher education program. This study made use of all four methods of data collection attributed to ethnographic study: collection of artifacts, participant observation, ethnographic interviewing, and researcher introspection. Each of these methods provided a different perspective and allowed for triangulation of data.

Data collected for the larger project includes a mathematics beliefs survey (Integrating Mathematics and Pedagogy, 2004) and all written work produced by the students during both mathematics methods courses. In addition, the six students from Group A and the nine students from Group B were observed four times teaching a mathematics lesson and individually interviewed on four occasions. The observations occurred during field experiences in the second and third semesters of the program (one observation each) and during their student teaching experience (two observations). The four interviews were semi-structured and took place at the end of every semester of the teacher education program. Currently, the students from Group A and B are being observed monthly and interview bi-annually as they progress through their first 2 years of teaching. The data analysis for this manuscript was chiefly focused on the participants’ written work produced during their first elementary mathematics methods course and transcriptions of their first four interviews.

All data were transcribed and organized for coding purposes, and the research team defined an initial set of 25 codes. Data coding and sorting was done using a computerized qualitative data coding tool. Line-by-line coding of data took place chronologically for all 15 participants with different researchers coding data for different participants and then sharing summaries, called “data stories.” Finally, the data stories and original data for the subset of seven participants were coded across participants with respect to the research questions.

Findings

We identified three aspects of the mathematics course that provided preservice teachers with an opportunity to engage in assisted performance. We describe each one in turn and provide data from the participants to indicate the ways in which these aspects of the course helped them develop their thinking about mathematics teaching and learning. We being this section with a brief description of the participants’ initial views of mathematics teaching and learning.

Participants’ Initial Views about Teaching Elementary Mathematics

The seven participants whose data we use in this manuscript include Adam and Sara from Group A; and Emily, Erica, Erin, Katie, and Stacie from Group B. All were Caucasian with Adam being the only male in the group. (Adam was one of two males in the combined group of 60 students.) We deliberately showcase both their more traditional views and their more constructivist-leaning views so as to capture the complexity of their beliefs upon entering the program. In general, their views, whether traditional or constructivist, were not well developed as evidenced by statements not supported by evidence, examples, or elaboration.

The participants’ reflections on their own mathematical learning generally contained references to liking or disliking the subject, their motivation to learn mathematics, and their perceptions of the value of mathematics. Adam said what he most remembered from elementary school mathematics classes was just paper-and-pencil drills. From this experience he learned that math was simply rote memorization: “There was no purpose to the math, except to get a certain grade.” Sara, on the other hand, experienced a mathematics teacher who made mathematics “fun and entertaining, as well as educational. It was a class where we would learn in so many different ways.” Stacie remarked that she liked learning mathematics to be like a cookbook where she knew exactly what to do and how to do it in order to be successful. Similarly, Emily reported that “One quality I loved about some of my teachers was that they would not only tell me what I was doing wrong, but they would also tell me how to fix it. It is much easier to correct yourself when you are given advice or possible solutions.” Katie noted, “I don’t like math! It isn’t that I don’t like math in general—in fact, I enjoy certain ‘practical’ math applications. Frankly, I believe the reason that I don’t enjoy math is because I don’t see the point to a lot of it.” The composite portrait that these participants painted of their mathematics learning experience was one that seemed to be devoid of an emphasis on conceptual understanding or the application of mathematics to real-world problems. However, the participants did report several examples of caring teachers who made mathematics learning fun.

The importance of caring teachers who make learning fun was echoed throughout their descriptions of ideal teachers and their reasons for wanting to teach. Emily remarked, “In my opinion, the most important qualities [of an ideal teacher] are being kind and creative.” Erica described a former teacher who she considered the ideal teacher because she made “math fun, interesting, and beneficial for everyone in the class” and was “compassionate, intelligent, honest, fair, fun loving, energetic, and caring.” Katie said the ideal teacher should “focus on creating an environment where children look forward to learning and being challenged.” Sara described a teacher who turned her on to mathematics in middle school as “clear, concise, and to the point. If there ever came a time that we didn’t understand a problem, she would go over it again and again until we did. She never talked down to us or made us feel stupid.”

Many of these preservice teachers entered the program with some sense of the importance of listening to children and allowing children to think for themselves. For example, Erica noted that the teacher should provide the students with the basics, “the things to get going and let them work with it and figure it out.” Sara said that the ideal teacher “is a listener who cares about what their children think and feel. She must take the time to listen, not just hear.” Stacie said that in her experience teaching in a preschool she “tried to turn a child’s idea into a learning experience,” which helped her realize how challenging it is to “process in your mind how to respond to what children are saying and to react quickly to it before the moment is lost.” Although most of these comments are not specific, Erin did provide a more detailed description of why it is important to listen to children. She wrote, “Teachers need to listen and find out what students do not understand, why they don’t understand, and how the explanations affect the student. Every student is different; therefore, it is very important for teachers to listen to each one.” The participants’ initial comments suggest that at the beginning of their mathematics methods courses they were aware of the role that children’s thinking and actions could play in shaping instruction. However, their views lacked depth or specificity of what it meant to apply this notion to the teaching and learning of mathematics.

Participants’ Learning to Teach Elementary Mathematics

We now turn our attention to how components of these preservice teachers’ initial mathematics methods course that engaged the preservice teachers in assisted performance and helped them add more depth to their ideas about using children’s mathematical thinking and actions in instructional decision-making This discussion is organized (roughly chronologically) around three specific components of that course: critiquing an essay written by a teacher as she reflected on her teaching practice, working one-on-one with a child for an extended period of time, and observing an experienced teacher teach an elementary mathematics lesson. Throughout the discussion we draw parallels to three of the five tasks that Feiman-Nemser (2001) argued are components that “form a coherent and dynamic agenda for initial preparation” (p. 1016): (a) analyzing beliefs and forming new visions, (b) developing understandings of learners and learning, and (c) developing the tools to study teaching.

Critiquing an essay by Vivian Paley. Early in the semester of the first mathematics methods course, preservice teachers were asked to read and respond in writing to a book chapter titled “On Listening to What the Children Say” by Vivian Paley (Paley, 1987). Paley’s chapter is her first-person account of how she became aware of the importance of listening to the children in her kindergarten classroom in order to uncover the meaning behind their words. The chapter does not address mathematics teaching; rather it describes classroom activities such as show-and-tell and the housekeeping center in the classroom. Throughout the chapter, Paley provides numerous examples of things she heard children say, how she initially interpreted their words, and what she learned about the children’s meaning when she continued to listen to and probe them for further explanation.

One particularly vivid example of listening and probing that Paley recounted involved a discussion of birthdays in which a child asserted that his mother no longer had birthdays. This would be an easy comment to dismiss or to respond by saying, “Of course your mother has a birthday. Everyone has a birthday.” However, Paley asked the child with genuine sincerity why his mother did not have a birthday. The child replied that when it is your birthday, your mother bakes you a cake. Since his mother’s mother was no longer alive, there was no one to bake her a cake, so she could not have a birthday. By probing the child’s statement and by listening carefully and sincerely to the child’s response, Paley uncovered a very logical, but incorrect, conception the child had about birthdays.

In the methods course, we had a lively discussion about the chapter. Some students thought that Paley had gone to extremes to make her point, and they noted that it is simply not possible or desirable to probe every single thing that each child says. Many students, however, found the chapter very enlightening because it provided specific examples of situations in which an adult had the potential to underestimate, misinterpret, or dismiss a child’s reasoning, but because she took the time to listen to the child, she came away with a much better understanding of and appreciation for the child’s thinking. Following the discussion in class, students wrote reflective commentaries on the chapter.[1] These commentaries provide the first evidence of the preservice teachers’ deepening understanding of what it might mean to listen to children in an instructional setting. It is especially notable that many of the teachers related the Paley chapter to their prior or current teaching experiences.

Erin’s initial reaction to the birthday story was that Paley was wasting valuable class time, and she should just tell the child that everyone has a birthday. After the class discussion, Erin reflected on the birthday story differently and noted that if Paley had tried to explain to the child why his mother does have a birthday without understanding the child’s reasoning, her explanation likely would not have made sense to the child or changed his views because it did not connect to what the child was thinking. Although Erin wrote in her autobiography at the beginning of the semester that it was important for teachers to listen to students, her reflections on the Paley chapter suggest that she did not have a very robust idea of what it meant to listen to children. Erin described the influence the chapter had on her this way:

When we discussed this topic in class, it helped open my eyes even further. I realized that students have their own point of view. I know this sounds ridiculous that this realization had never occurred to me. I always thought of children as empty sponges ready to be filled with new knowledge. I never thought about them having already formed their own definitions and opinions. I just thought students would take the information I gave them and soak every bit of it up. I did understand not every student would grasp this information, but now I see that in some cases, it is because they already have a different understanding about the information I am presenting to them.

Erin went on to say that it is important that teachers not get “stuck” in their lesson plans because it is the teacher’s responsibility to “see the world through [the child’s] eyes and attack problems and questions as if you were in the mind of that child.”

Erica initially dismissed Paley’s ideas, thinking they were extreme and that the message was that children should discover everything on their own. However, after a second reading of the chapter and the class discussion, Erica said she realized that Paley was simply reminding teachers of the importance of allowing children to learn from themselves. She said that the impact of Paley’s chapter became clearer to her when she was observing a first-grade classroom as part of an assignment for another course. As she reflected on that experience, Erica noted:

As I began watching the teacher, I immediately found myself feeling critical of the way that she was simply telling her students how to solve the math problem. I just wanted to jump up and say, “Please, just let the children experiment first and see if they can solve the problem on their own! Let them see how far they can get without being forced to do it your way!”

Erica then turned the mirror back on herself and wondered how many times someone had observed her working with children and thought the very same thing about her interactions. Erica concluded her essay by stating new goals for her teaching:

I want my students to know that I am listening to them, and I also want them to listen to themselves and each other. I hope to be able to have class discussions and lessons filled with open-ended questions and multiple right answers.

Katie connected the Paley chapter to both her previous experience working in a pre-kindergarten classroom and her experience in the methods class working one-on-one with a child (discussed later in the paper). Regarding her experience with PreK children, Katie said that she often walked away from interactions with children wondering if she had caused more confusion. She said the children often did not seem to understand her answers to their questions, which puzzled her because she knew her answers were “correct.” She said the Paley chapter helped her realize that her answers did not fit with what the children’s existing ideas, so the children could not “work what I said into the thought processes they already had going in their minds.”

Katie related the Paley chapter to her first-hand experience with teaching mathematics to a child, noting that one of her goals for the teaching experience was to “keep my mouth shut” and “stay out of the problem solving process.” Katie quickly discovered that this was easier said than done for her, both because she wanted to jump in and “give” the child the solution and because she had a hard time adjusting her thinking to the child’s thinking. In her final portfolio recounting her field experience Katie wrote:

When we were discussing teachers who just jumped right in and gave a child the answer, or one who only responded positively to correct answers, I knew I had that problem. I have the one right answer in my mind, and that’s the only answer for which I’m looking. I also have a bad habit of leading children to the answer instead of letting them discover their own way. I also want to jump right in with my own suggestions during the problem solving process.

Stacie said the Paley chapter challenged her to put more effort into trying to understand children’s thinking by interacting with them and asking open-ended questions. Stacie said that in the past she had often dismissed children’s ideas as cute or funny, and now realized that these ideas generally spring from a well-formed web of logical ideas. In her commentary about the chapter, Stacie admitted to having beliefs about the value of directive teaching at the beginning of her mathematics methods course, and the Paley chapter challenged these beliefs. In response to the chapter she wrote:

Sometimes when I interact with students I find myself teaching as I have been taught. It is a hard habit to break because for some reason it feels natural. Paley talks about, as a teacher, she wanted to supply students with knowledge and correct answers because it gave her the surest feeling that she was teaching. …So many times we are so wrapped up in successfully teaching a lesson that we are not truly observing the thoughts our students are concluding on their own.

As is clear from the excerpts above, the preservice teachers explicitly analyzed their beliefs in light of Paley’s descriptions of children’s thinking and its impact on her instruction. In some cases, these analyses were more thorough and well-connected to the participants’ teaching experiences. In other cases, these analyses were more of a surface-level nature, making it difficult to determine whether the participants had actually developed “visions of what is possible” (Feiman-Nemser, 2001, p. 1017). The specific examples provided in Paley’s chapter seemed to present the preservice teachers with “compelling alternatives” and “powerful images of good teaching” (p. 1017).

The ideas contained in the chapter laid the groundwork for the preservice teachers to better understand learners and learning in their subsequent field experiences, particularly their work with their Barrow Buddies (discussed later in this paper). Although reading the chapter, per se, did not help them develop a better understanding of children’s learning in mathematics, it did seem to provide a concrete example of what the instructor meant by her continuous references to listening to students’ thinking and building instruction on that basis.

The experience of reading the chapter and discussing it with peers also provided the preservice teachers with an opportunity to develop the tools to study teaching. The chapter gave the preservice teachers an example of how one studies one’s own teaching and that of others. Paley’s open and explicit reflection on her own teaching exemplified the manner in which one can deliberately observe one’s own teaching and analyze it later. Although some preservice teachers initially dismissed what they had read, discussing the chapter with peers with support from the instructor helped them take a more open-minded approach to the text of the essay, to find something of value in the chapter, and to relate it to their own experiences. It is this aspect of the experience that can be viewed as assisted performance. Left on their own to read the chapter and write a reflection, most preservice teachers were not yet ready to take Paley’s ideas seriously. However, assisted by their peers and the instructor, the preservice teachers were able to engage in a meaningful discussion of what Paley meant, how it worked in her classroom, and the implications for mathematics teaching and learning. The fact that the preservice teachers continued to refer to the Paley chapter in their assignments for class and their interviews suggests that this was a powerful learning experience for them.

Barrow Buddy Field Experience. Approximately four weeks into the mathematics methods course, each preservice teacher began to work one-on-one with an elementary school-aged child in mathematics once a week for eight weeks. The field experience took place at Barrow Elementary School and is hereafter referred to as the BB experience.[2] The goal of the field experience was for the preservice teachers to learn to listen to and assess children’s mathematical thinking and to plan subsequent instruction based on this assessment. The preservice teachers were encouraged to engage the children in problem solving rather than in computational work. Accordingly, the children who were selected for the experience were working at or above grade level so that the preservice teachers would not feel pressure to “cover” certain mathematical content.

Throughout the BB experience, as the preservice teachers were working with the children, the instructor/researcher and two teaching/research assistants moved from pair to pair using a coaching model to provide input. For example, if a child solved a problem correctly and the preservice teacher was ready to move on to the next problem, the instructor interjected and asked the child to explain her/his solution method. In this manner, the instructor was able to model for the preservice teachers what she was teaching in the university classroom: the importance of uncovering children’s mathematical thinking in order to inform subsequent instruction. Moreover, because the instructor and her assistants observed each preservice teacher in an instructional setting each week, they were able to tailor classroom instruction on campus to reflect and enhance what happened at Barrow Elementary School. The Barrow Buddy experience was deliberately set up to be an opportunity for assisted performance.

For Sara, the BB field experience motivated a critical examination of her beliefs about children and their capabilities. In an interview shortly after the BB field experience ended, Sara made the following statement regarding her newfound understanding of children:

I've never really understood how much kids can understand at such a young age. I was just amazed to watch Sion [her Barrow Buddy]. He was actually thinking in his little head, and I just somehow looked at him in awe. He was just so bright, and if something didn't work, he'd try another way, and he'd try another way. They're just so persistent, and I just never realized kids could grasp that much. I mean, they were at such an advanced level for how old they were. They're just like miniature adults, I guess. It was really, that just shocked me.

Sara remarked that she learned so much about herself and her style of teaching from the BB field experience. She believed that the experience taught her the importance of being “ready to adjust what [she] had planned at anytime during [her] lesson” in order to capitalize on the understanding, thinking, and persistence of children.

The BB field experience also helped Sara develop an appreciation for the value of questioning students. In her final portfolio for the course she noted,

I learned how to ask better questions. I learned that sometimes the way a problem is phrased is the only thing keeping a child from solving it. I learned that if you just rearrange the wording of some questions, a child would understand it more clearly. …We need to build off of what they already know and understand. So often, all a child needs, is a little hint. My Barrow Buddy helped me learn how to give better hints. He helped me to see that sometimes all it takes is a little help (from the teacher) to make the problem “click.”

Coming to see the value of effective questioning and how it can help the teacher build instruction from the child’s existing knowledge base was a reoccurring result of the BB field experience for Sara and most of the project participants. Throughout her written assignments from the first methods course and during her initial interviews, Sara made several references to the BB field experience and how it changed her attitudes about children and children’s learning and her conceptions of teaching and planning.

The BB field experience affected Adam’s conceptions of mathematics teaching and learning—particularly the value of multiple solution strategies. On several occasions when Adam presented his Barrow Buddy with a mathematics problem, Adam was amazed by the solution strategies used by the student because they were ones that had not occurred to Adam in his planning. In an interview, Adam remarked that when he posed a mathematics problem for his Barrow Buddy he had in mind “pretty much a set pattern” of how to do the problem; however, his Barrow Buddy would “do it a completely different way.”

Adam articulated how he developed a better understanding of learners from working with his Barrow Buddy. He noted that the experience allowed him “to visualize first-hand how children solve problems and where difficulty may lie.” He also came to recognize that children could work independently to successfully solve problems using their own strategies and not those imposed by the teacher. Adam became more aware of the limiting effects of making assumptions about what students could or could not do mathematically, recognizing that students could teach teachers as much as teachers teach the students.

The BB experience prompted Stacie to shift her focus as a teacher from what she was teaching to what the student was learning. Stacie initially began planning for her BB experience sessions by seeking “first grade activities” from books and internet sites because she held some stereotypes about what children of a particular age could do mathematically. However, her first-grade Barrow Buddy surprised her with what she was able to do:

Maria used her basic knowledge of counting, adding, and subtracting to do the higher math that she had not learned yet such as place value, multiplying, and dividing. The thing that was amazing was that I did not even instruct her on how to use her basic skills to do the higher math. She figured it out on her own.

After Stacie realized the thinking of which her BB was capable, she shifted her planning from searching for first-grade activities to building her next lesson based on the child’s thinking in the previous lesson. In addition, Stacie noted that she tried to ask more questions sought to phrase explanations “without giving [the answer] away.”

The BB experience caused Katie to reexamine her focus as a teacher from a focus on her thinking to her Barrow Buddy’s thinking. After reading the Paley chapter, Katie said that she wanted to base her teaching on children’s ways of thinking, but she found it very difficult to listen to her Barrow Buddy and respond accordingly. Katie said that figuring out students’ thinking was difficult “because I have a hard time seeing things except in the way that I would have done it.” She acknowledged that it was important to be able to approach a problem “from another angle” if children did not understand the first approach, but this was a significant concern for Katie: “I could only think of my way, one way. I don’t know how to do math from any other angle.”

Katie’s initial focus on her way of thinking resulted in problematic interactions with her Barrow Buddy. She became fixated on “getting through” her plans and on helping her Barrow Buddy get the correct answer to the problems she posed. She was not very patient and tended to interject suggestions at the first sign that her Barrow Buddy did not know exactly how to proceed with a problem. The instructor and assistants gave Katie a great deal of feedback, both in the form of written feedback on her plans and in the form of coaching during her lessons, to alert her to these tendencies. She was very receptive to this feedback and was already somewhat aware of these tendencies. She made a deliberate effort to “back off a little bit and let [the child] figure out problems on her own.” Katie found this extremely difficult, however, and her Barrow Buddy resisted Katie’s attempts to “back off.” Because of her earlier habit of providing directive feedback to her Barrow Buddy every time she was stuck, Katie created the expectation that she would “give her the answer,” and the child became upset, frustrated, and even withdrawn when Katie tried to change tactics and withhold her input in favor of allowing the child to explore solutions on her own. In some sense, Katie went from one extreme—telling the child what to do at every turn—to another—telling the child nothing. This second extreme was reflected in Katie’s comment: “It was a lot easier to say that I was going to stay out of the problem solving process than it was to actually do it!”

Through instructor coaching and peer feedback, Katie came to understand that her goal was to find a middle ground in which she could provide encouragement, hints, and suggestions without railroading the child into her way of thinking. Katie’s final reflection on her BB experience showed that she began to mediate the extremes of “backing off” and directing the child’s thinking. She wrote, “I learned from working with my Buddy to keep my mouth shut! It took a while, and it was difficult, but I think I’m better about not giving so many suggestions. I think I am realizing when my Buddy needs help and when she really just needs me to let her think.”

Erica developed a deeper understanding of student motivation and the pedagogy of questioning by working with a Barrow Buddy who was not particularly motivated. Erica initially struggled with finding mathematical problems that were both accessible and challenging for her Barrow Buddy. In order to motivate her Barrow Buddy, Erica began by playing mathematical games. Erica, however, quickly saw that her Barrow Buddy was fixated on wining and lost interest in the game if he did not win; consequently, she began letting him win in order to keep him engaged. She summarized this experience by writing, “This was probably not the best approach to take, but I am struggling to find a way to make my meeting with Terrence a fun, positive, learning experience.” The instructor suggested that Erica should make the mathematical game playing less competitive by explaining the strategy that she was using while playing the game. Erica began to develop an appreciation for the complexity of student motivation as evidenced by her reflections upon implementing this approach:

I tried to be more interactive with him. I asked lots of questions about why he did things the way he did, and I tried to get him to see the endless extensions and connections of everything we did together. I learned to allow Terrence to discover the answers on his own, but I also learned how to effectively ask him questions that would make him think and go that next step. …This “new” way of solving problems made both Terrence and me feel much more successful. Terrence felt more confident because he knew that he could apply strategies that were familiar to him from previous problems, and he became more comfortable experimenting with manipulatives and different ideas.

Through this approach Erica also began to see the link between effective questioning and student engagement. One year later, Erica was still reflecting on the effect of the BB field experience.

Towards the end of last year (her second semester in the program), it was “Wow, questioning is really important.” And I was really seeing the benefits of it. Whereas with my Barrow Buddy I’m not quite sure I knew then what I was looking for when I was questioning. I was like “Okay I’ll ask lots of questions and listen to his answers” But I didn’t really realize as much why or what to do with it I guess. I think personally I grew, like as a teacher in that aspect, if that makes sense.

As the data reported above suggests, the Barrow Buddy field experience provided preservice teachers with on-going and sustained opportunities to analyze their beliefs about teaching and learning and to develop understandings of learning and learners (Feiman-Nemser, 2001). Because the preservice teachers worked weekly with the same child for a period of two months with on-campus class sessions interspersed with the fieldwork, they had an opportunity to engage in a protracted and deliberate study of the mathematical thinking and learning of a single child with support from their instructor. By writing weekly plans, enacting them, and writing a reflection on the session (that was shared with the classroom mentor-teacher and university instructor), the future teachers gained experience with designing appropriate instructional tasks, justifying pedagogical actions, assessing student learning, and communicating with other educators. The teaching sessions themselves, coupled with the support and challenge from the instructor and teaching assistants, enabled the preservice teachers to experiment with various methods of teaching, questioning, and assessing student learning in order to develop a preliminary understanding of what works best under which conditions.

A component of the BB experience that assisted the preservice teachers with developing the tools to study teaching (Feiman-Nemser, 2001) was the writing of and reacting to short cases about their Barrow Buddies. After the fourth week of the BB experience, each preservice teacher wrote a pedagogical case in which she described a challenge she was facing with her Barrow Buddy. They exchanged their cases with two peers (selected by the instructor) and gave a copy to the instructor. Each person was then responsible for reading and responding to their two peers’ cases. The peers also received a copy of the instructor’s feedback to the writer of the case. Through this activity, the preservice teachers were able to experience professional discourse that called for evidence, openness to questions, and valuing alternative perspectives.

The BB experience provided a deliberate forum for assisted performance in the areas of linking theory and practice, for developing teaching skills and pedagogical strategies, and for learning to analyze and reflect upon student learning and teaching. The preservice teachers were assisted in their performance by their Barrow Buddies, their peers, and their instructor.

Observing an Experienced Teacher. Toward the end of the first mathematics methods course, preservice teachers were given an opportunity to observe a mathematics lesson taught by a mathematics specialist in a local school. In some cases, the preservice teachers were able to discuss the lesson with the teacher afterward, but in other cases this was not possible. Observing the teacher was one of several options for a particular assignment, and all but two of the participants (Adam and Sara) chose to observe the teacher, Patti Huberty. Ms. Huberty had been identified as an exemplary mathematics teacher by university faculty and by fellow teachers in the school district. (See Mewborn, 1998, Mewborn & Huberty, 1999 a description of this teacher’s practice.) The purpose of having the preservice teachers observe Ms. Huberty was to offer them an example of how the things they were learning in their methods class and with their Barrow Buddies might be implemented with a full classroom of diverse learners. The specific assignment that was given to the preservice teachers was to observe a lesson and write a 500-word paper about the teacher’s actions, the students’ learning, and the connections between the two.

While reflecting on Patti’s lesson, Erica wrote,“ the entire lesson was filled with questions for the students, [Patti] was constantly asking about [the students’] thought processes, their ideas, and even their behavior.” Erica made a connection to student behavior by noting, “The classroom environment seemed so student-centered and productive that there was very little need for discipline.”

Observing Patti teach demonstrated to Katie how the teacher can use the thinking and actions of the children to develop the mathematics lesson.

I was shocked by how little [Patti] said while the children were trying to decide what to do. I’m sure I would have jumped in there and tried to give them a hint and ruined the whole process!

Because Katie was struggling with her role as a teacher in her BB experience, the observation of Ms. Huberty seemed to reinforce her self-awareness. Further, Katie had an opportunity to see the rich thinking of which children are capable—something that she did not experience much with her Barrow Buddy because she fluctuated between telling her what to do and telling her nothing. Katie noted, “The depth and clarity of the children’s explanations [in Ms. Huberty’s class] were amazing! …Without her explaining much, they picked up the concepts and basically taught one another.” Katie explicitly contrasted Ms. Huberty’s teaching with her own: “I benefited from watching [Ms. Huberty teach] because I have a bad habit of leading students …to the answer I’m thinking of, and I may be stifling their own ideas.”

Erin remarked she was impressed by how “Patti took the time to call on every student, even if the first person gave the correct answer.” Erin believed the communication between Patti and the students “allowed Patti to understand where the students were coming from, and it encouraged the students to participate.” Similarly, Stacie noted the value of listening to children’s ideas. She said that during Patti lesson children “were able to learn about measurement from each other’s responses and mistakes.” She further explained that the questions Patti asked encouraged the children to “use their common sense and reasoning skills” to arrive at answers.

Feiman-Nemser (2001) suggested that studying the ways in which different teachers work toward the same goal could help preservice teachers develop the tools to study teaching. In this case, the preservice teachers were able to compare and contrast their own teaching of one child with Ms. Huberty’s teaching in a classroom setting. In fact, several of the preservice teachers drew explicit connections between their teaching or beliefs and what they observed in Ms. Huberty’s classroom. Also interesting to note is the fact that approximately half of the preservice teachers’ comments in their write-ups about their observation of Ms. Huberty pertained to the teacher’s actions while the other half pertained to student learning. This balance of attention suggests that the preservice teachers were beginning to appreciate the synergistic relationship between student learning and teaching. We view this experience as an example of assisted performance because the preservice teachers were assisted in developing and articulating their ideas about the role of the teacher. They were not yet capable of doing this on their own based on their limited teaching experiences and observations. The assistance in this case came from the experienced teacher, who was carefully selected, and from the students in the classes that were observed. The observation and the assignment that went with it were carefully structured by the instructor to facilitate this kind of assisted performance. This is certainly a less robust type of assisted performance than the two examples noted earlier, but it is, nonetheless, suggestive of the ways in which preservice teachers can stretch their thinking with assistance.

Discussion

The three components of the mathematics methods course illustrated above, reading the Paley chapter, the BB experience, and observing an experienced teacher, supported the preservice teachers’ learning about children’s mathematical thinking, which was the central focus of the course. The Paley chapter gave the future teachers an opportunity to read about children’s thinking, while the Barrow Buddy field experience gave them a chance to experiment with eliciting children’s mathematical thinking. Observing Ms. Davis enabled the preservice teachers to extend these experiences by seeing what it might look like for a teacher to orchestrate a classroom so as to build instruction on children’s mathematical thinking. These three components dovetail with three of the central tasks of teaching advocated by Feiman-Nemser (2001)–analyzing beliefs and forming new visions; developing understandings of learners and learning; and developing the tools to study teaching.

We believe that the three course components detailed above–reading the Paley chapter, working with Barrow Buddies, and observing an experienced teacher–provide images of what assisted performance might look like in a preservice methods course. Although Feiman-Nemser used “performance” to mean tasks of teaching such as planning and lesson implementation, we have expanded the notion of performance to include such things as analyzing one’s own teaching and that of others. In different ways, each of the course components helped preservice teachers do something that they were not ready to do on their own when they started the program. The assistance part of assisted performance came from a variety of sources. In reading and reflecting on the Paley chapter, the assistance came mostly from peers and somewhat from the instructor. In the Barrow Buddy experience, the assistance came from the children and the instructor. In the observation, the assistance came from the experienced teacher and the children.

Feiman-Nemser reviewed case studies of teacher education programs from the report of the National Commission on Teaching and America’s Future (1996) to identify promising practices in teacher education that may lead to preparation of teacher candidates who are able to meet the challenges of the reform agenda. The three promising practices identified by Feiman-Nemser are the conceptual coherence of the program; the use of purposeful, integrated field experiences; and attention to teachers as learners. We believe that the mathematics methods course we described above provides an illustration of these ideas.

Although Feiman-Nemser’s quest is for entire teacher education programs that have a salient conceptual frame, it is logical to start at the level of individual courses and ask what it would mean for a single course to be conceptually coherent. We have described aspects of the course that we believe show evidence of a conceptually coherent approach to elementary mathematics teacher education. The course had a clear focus on children’s mathematical thinking, and course activities were designed with this goal in mind. Other aspects of the course not described in this paper that supported this goal included watching video tapes of individual children solving problems and of whole class lessons and reading research about children’s ways of solving particular types of problems or using tools such as calculators or manipulatives. Furthermore, the goal of the course was explicitly stated in the syllabus and by the instructor on numerous occasions. Preservice teachers also had repeated opportunities to articulate their emerging understanding of the course goal throughout the semester.

The BB experience is an example of a purposeful, integrated field experience. The BB field experience had a clear purpose that was linked to the on-campus course: to provide preservice teachers with an opportunity to learn to elicit and respond to children’s mathematical thinking. The BB field experience was also integrated with the coursework in that the instructor and assistants were actively present in the field, using it as an extended instructional site, and the subsequent class period on campus reflected events that happened in the schools.

The instructor of the course had an explicit commitment to viewing preservice teachers as learners. She assumed that preservice teachers, like children, had well-reasoned explanations behind the ideas they brought with them to their mathematics methods courses. Thus, class sessions often included questions posed by the instructor, multiple answers from students, followed by in-depth discussions of those answers. Rather than dismissing preservice teachers’ prior experiences as simply leading them toward more traditional views and practices of teaching, she used the aforementioned components of the course to help preservice teachers examine the knowledge and beliefs that they brought to the teacher education setting. Further, class activities, such as the BB experience, provided preservice teachers with opportunities to engage in assisted performance–to do with help what they were not yet ready to do on their own.

Implications and Conclusions

In this section we consider further what it might mean to pay attention to teachers as learners and to view teacher education as an opportunity for assisted performance. To do so, we challenge some of the prevailing practices in teacher education that are linked to an apprenticeship model of learning to teach.

We echo Feiman-Nemser’s (2001) concern that instructors of both content and methods courses often feel pressured to “cover” the curriculum so that future teachers will be prepared to teach any topic they may encounter. Recognizing that it is not possible to provide preservice teachers with content knowledge, activities, and connections between every important topic at every grade level, we generally confine ourselves to the “big ideas,” which in elementary mathematics education typically include such things as prenumber work, place value, the four basic operations with whole and rational numbers, geometry, measurement, and data analysis. As with K-12 teachers, there is an ever-increasing pressure to put more and more content into this curriculum; the latest push is for algebraic thinking at the elementary level. What would it mean to free ourselves from any sense of having to cover the curriculum? What might a course look like if it was not organized by content topics but instead by aspects of teaching (such as planning and assessment) or processes in which we want students to engage (such as conjecturing and generalizing) or by some other categorization and used examples from content to illustrate these organizing ideas? Again, a look at what we want preservice teachers to do while they are with us is in order. Do we want them to practice the things they will be doing when they leave us, or do we want them to take advantage of the support they have from us in order to engage in challenging tasks for which they may not have the resources to engage after they leave us? How do we assist them in performing these tasks?

A second area to question is the tasks in which we typically engage preservice teachers. This logically connects to the prior notion of the goals and topics around which our courses revolve. For example, if a central course goal is for future teachers to understand the link between instruction and assessment, what kinds of tasks would we provide for them? What kinds of tasks would qualify as assisted performance in this arena? A first step would be to consider the purpose of the myriad typical tasks in which we engage preservice teachers, such as finding or writing activities, lesson plans, and unit plans; writing reflections; critiquing textbooks; and peer teaching or microteaching.

Field experiences provide a rich ground for questioning why we do the things we do and how we might do them differently if we are serving the goal of creating opportunities for preservice teachers t engage in assisted performances. For example, we might investigate such practices as only assigning one preservice teacher to one mentor teacher, using only experienced teachers as mentor teachers, and staying with the same mentor teacher for an extended period of time. We might investigate the affordances and limitations of a field experience that is conducted for full days during a one-month block in comparison to weekly field experience on Tuesday and Thursday mornings. Beyond the configuration of field experiences, we could also question the tasks in which preservice teacher engage in the field, such as observation, teaching small groups, and teaching the whole class. We might look to co-teaching (with peers or with mentor teachers or with university faculty) as a form of assisted performance. We might consider having preservice teachers co-teach different subjects with different teachers because not all teachers are equally strong in all content areas. Or we might consider having a preservice teacher observe and co-teach with several different teachers–all in one subject area. Co-planning is another avenue for assisted performance. A mentor teacher and preservice teacher might co-plan a lesson that is then implemented by the mentor teacher. Or a university supervisor and a preservice teacher might co-plan a lesson that is implemented by the preservice teacher with support from the university supervisor.

We might also question the role of the university supervisor in field experiences. The role is typically one of evaluation and supervision, but if field experiences are an opportunity for assisted performance, we could consider the possibility that an observation is not a chance to sink or swim but a chance to be coached. Similar to our description of the Barrow Buddy experience, the mentor teacher and/or university supervisor might intervene in a lesson to ask a question that better reveals student thinking or to offer a management suggestion or to make a connection to the previous day’s lesson. Rather than teaching entire lessons, preservice teachers might teach portions of lessons, such as the introduction and connection to yesterday’s lesson or the main body of the lesson, or the summary and wrap-up of the topic. On a grander scale, we might ask why we have a semester devoted to student teaching. Is the typical two weeks of “solo teaching” a sacred cow whose time has come for reconsideration?

Freeing ourselves from the idea that teacher preparation is just that–preparation for the future in the form of practice–and opening ourselves to the idea that teacher education is about structuring learning opportunities for future teachers using the same principles we use to design educational opportunities for children provides a wide variety of interesting avenues for future scholarly work. However, the unarticulated and untested assumption within the idea of assisted performance is that somehow the teachers will be able to “pull it all together” when they leave us in order to do the things that teachers do. Feiman-Nemser (2001) takes the next step in her article by suggesting what the central tasks of the induction years and professional development are and identifying key features of successful programs. Similarly, we plan to follow the participants in this study in order to develop robust descriptions of their learning trajectories and to ascertain the staying power of the teaching practices they began to develop in their preservice program.

References

Ball, D. L., & McDiarmid, G. W. (1990). The subject-matter preparation of teachers. In W. R. Houston (Ed.), Handbook of research on teacher education (pp. 437-449). New York: Macmillan.

Bogdan, R. C., & Biklen, S. K., (1992). Qualitative research for education: An introduction to theory and methods. Boston: Allyn and Bacon.

Eisenhart, M. A. (1988). The ethnographic research tradition and mathematics education research. Journal for Research in Mathematics Education, 19(2), 99-114.

Feiman-Nemser, S. (2001). From preparation to practice: Designing a continuum to strengthen and sustain teaching. Teachers College Record 103, 1113-1055.

Floden, R. E., & Buchmann, M. (1990). Philosophical inquiry in teacher education. In W. R. Houston (Ed.), Handbook of research on teacher education (pp. 42-58). New York: Macmillan.

Grimmett, P. P. (1989). A commentary on Schön's view of reflection. Journal of Curriculum and Supervision, 5, 19-28.

Integrating Mathematics and Pedagogy. (2004). IMAP web-based beliefs survey. Available: .

Kilpatrick, J. (1988). Editorial. Journal for Research in Mathematics Education, 19(2), 98.

Lanier, J. E., & Little, J. W. (1986). Research on teacher education. In M. C. Wittrock (Ed.), Handbook of research on teaching (pp. 527-569). New York: Macmillan.

Mewborn, D. S. (2000). Meaningful integration of mathematics methods instruction and field experience. Action in Teacher Education 21(4), 50-59.

Mewborn, D. S., & Huberty, P. D. (1999). Questioning your way to the Standards. Teaching Children Mathematics 6(4), 226-227. 243-246. (Contribution percentages: Mewborn 50%, Huberty 50%)

Mewborn, D. S. (1998). The quarter quandary: An illustration of NCTM’s Professional Teaching Standards. Teaching Children Mathematics 5, 160-163.

National Commission on Teaching and America’s Future. (1996). What matters most: Teaching for America’s future. New York: Author.

Paley, V. G. (1987). On listening to what the children say. In M. Okazawa-Rey, J. Anderson, & R. Traver (Eds.) Teachers, teaching, and teacher education (pp. 77-86). Cambridge, MA: Harvard Education Review.

Zeichner, K. M., & Gore, J. M. (1990). Teacher socialization. In W. R. Houston (Ed.), Handbook of research on teacher education (pp. 329-348). New York: Macmillan.

-----------------------

[1] I made a deliberate pedagogical decision to have the students write their commentaries after the class discussion rather than before it. I wanted the students to hear their peers’ reactions to the chapter and to have an opportunity to articulate their own views before writing the commentary. I believe that the commentary had a more reflective nature to it because of this sequence.

[2] See Mewborn, 2000 for more details about this field experience.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download