Teachers' Reflections on Their Reform-Based Teaching in Mathematics: Implications for the Development of Teacher Self-Efficacy

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Teachers' Reflections on Their Reform-BasedTeaching in Mathematics: Implications for theDevelopment of Teacher Self-EfficacyAnthony J. Gabriele a & Elana Joram aa University of Northern Iowa , USAPublished online: 03 Jan 2012.

To cite this article: Anthony J. Gabriele & Elana Joram (2007) Teachers' Reflections on Their Reform-Based Teachingin Mathematics: Implications for the Development of Teacher Self-Efficacy, Action in Teacher Education, 29:3, 60-74,DOI: 10.1080/01626620.2007.10463461

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Teachers’ Reflections on Their ReformMBased Teaching in Mathematics : Implications for the Development of Teacher Self-Efficacy Anthony J. Gabriele University of Northern Iowa

Elana Joram University of Northern Iowa

ABSTRACT The purpose of this study was to examine the sources of efficacy of experienced teachers who are making the transition to reform-based mathematics teaching, by examining the criteria that they use to evaluate their teaching effectiveness. The quantity and quality of 10 elementary teachers’ verbal reflections on the success of their lessons were compared as a function of the length of time that they had participated in a professional development project. Results suggest that the sources of information upon which teachers base evaluations of their teaching change in important ways as teachers shift from traditional to reform-based mathematics teaching. Implications are discussed for understanding how teacher self-eff icacy for reform- based teaching can be enhanced through professional development.

Supporting teacher learning, either by helping teachers implement new pedagogical techniques in the classroom or by creating long- term collaborative networks focused on inves- tigating one’s own classroom practices, is recognized as being critical to achieving meaningful educational reform (Cohen & Hill, 2000). Researchers have learned that ef- fecting substantive transformations in teaching practices is a slow process, often character- ized by much resistance at both the individual and the systems level. There is interest, therefore, in understanding the teacher learning process and those factors that support or im- pede its progress (Fennema & Nelson, 1997).

A teacher’s self-efficacy is one factor that may play a key role in the process of initiating and sustaining changes in teaching practices.

Teacher self-efficacy, which has been defined and measured in a variety of ways, is primarily thought of as a future-oriented construct that relates to an individual’s belief in his or her ca- pability to perform instructionally related actions that are likely to bring about student learning (Ross, 1998; Tschannen-Moran, Woolfolk-Hoy, & Hoy, 1998). Although detailed process models have not been fully developed to explain the links between teacher self- efficacy beliefs and desirable teacher behaviors, extrapolation from work on self-efficacy (Ban- dura, 1997) suggests that it is the effect of efficacy beliefs on the motivation to expend effort, on the willingness to set challenging goals, and on the persistence through difficulty that a teacher’s self-efficacy beliefs exert their power- ful influence on teacher learning. From this

Address correspondence to Anthony J. Gabriele, 651 Schindler Education Center, Department of Educational Psychology and Foundations, University of Northern Iowa, Cedar Falls, IA 50614; e-mail: [email protected].

60 Action in Teacher Education Vol. 29, NO. 3

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Teachers’ Reflections on Their Reform-Based Teaching in Mathematics 61

perspective then, teacher self-efficacy primarily influences a teacher’s adoption of new ways of teaching by increasing her or his willingness to take risks and persist through the difficulties and setbacks that accompany the implementation process.

In general, empirical research on the rela- tionship between teacher self-efficacy and implementation of change in classroom practices indicates that teachers who are high in self- efficacy are more favorably disposed to adopt new pedagogical practices (e.g., Berman, McLaughlin, Bass, Pauly, & Zellman, 1977; De Mesquita & Drake, 1994; Evers, Brouwers, & Tomic, 2002; Ghaith & Yaghi, 1997; Guskey, 1988; Stein & Wang, 1988). However, in these studies, teacher self-efficacy is treated as a static individual difference variable. As such, the findings are of limited value for those interested in the design of professional development activities, which seek to involve teachers who are not already high in teacher self-efficacy (Fritz, Miller-Heyl, Kreutzer, & MacPhee, 1995; Labone, 2004). Although several researchers have speculated on the importance of understanding the sources of teacher self-efficacy and its role in teacher change (e.g., Labone, 2004; Smith, 1996), little empirical research has examined the sources of teacher self-efficacy. Furthermore, enhancement of teacher self-efficacy has rarely been a target of professional development (Ross & Bruce, 2005).

In the study reported here, we empirically examined one source of teacher self-efficacy that the literature suggests is likely to play a key role in facilitating the enhancement of teacher self-efficacy beliefs: a teacher’s reflection’ on his or her teaching performance and the criteria used in evaluating success (Labone, 2004). As we describe, studying the criteria the teachers use when reflecting on past performance and examining how this potential source of teacher self-efficacy changes over time should provide insight into the development of teacher self-efficacy and how it can be directly enhanced by professional development strategies. We argue that the focus on sources of teacher self-efficacy may be particularly important for professional development

efforts where new pedagogical methods are being introduced and where their desired goals, outcomes, and time frames are expected to differ significantly from those of conventionally used practices. In these situations, the cues that teachers rely on to evaluate their success in enacting conventional teaching practices are not well aligned with new practices. With- out a corresponding shift in these cues and criteria, when teachers attempt to enact new practices, they will find it difficult to notice and appreciate progress. Cut off from this vital source of teacher self-efficacy enhancement, they may become discouraged and likely to dis- continue implementation efforts. One area of professional development where this appears to be the case involves reform-based mathematics teaching (Smith, 1996). Therefore, in this study, we focused on experienced teachers at different points in their transition from traditional to reform-based mathematics teaching and compared the way that they reflected on their teaching performance.

We begin by briefly reviewing the sources of teacher self-efficacy and the studies that have examined the conditions under which changes in teacher self-efficacy have been observed. We then discuss the challenges faced by teachers who are making the transition to reform-based mathematics teaching. Finally, we discuss the purpose of this study and how the design and methodology used address the study’s questions.

Sources of Teacher Self- Efficacy: The Importance of Reflection on Past Performance

Researchers interested in sources of efficacy have typically drawn on Bandura’s (1997) social cognitive theory as a foundation (e.g., Lent, Lopez, & Bieschke, 1991). According to Bandura, there are four sources of self-efficacy: past success in performing the behavior, physiological and emotional arousal, vicarious experiences, and verbal persuasion. Whereas each of these sources may play important roles in the development of self-efficacy, according

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62 ANTHONY J. GABRIELE AND ELANA JORAM

to Bandura, the formation of self-efficacy beliefs is based primarily on reflection and interpretation of past performance (also referred to as enactive mastery experiences). Previous experiences in which a particular performance was enacted by an individual and was perceived by that individual as successful will tend to raise self-efficacy beliefs related to this performance; those experiences perceived as unsuccessful will tend to lower self-efficacy beliefs.

As it relates to teacher self-efficacy for implementing a new curriculum and pedagogical technique, the research on sources of efficacy suggests that a teacher’s reflection on his or her teaching performance in using the new teaching method should play a crucial role in the development of a teacher’s self-efficacy. In this sense, teacher self-efficacy is related, but not identical, to teacher effectiveness in that perceived evidence of past successes and fail- ures strongly influences a teacher’s expectations about her or his future performance (i.e., teacher self-efficacy). However, it is the interpretation of performance, not the performance itself, that Bandura considers primary in the development of self-efficacy.

Studies of Changes in Teacher Self-Efficacy

Empirical research that examines the conditions under which teacher self-efficacy under- goes change generally supports the importance of teachers’ interpretations of their performances in the classroom as a crucial source of information in the development of teacher self- efficacy. For example, Ross, McKeiver, and Hogaboam-Gray ( 1997) interviewed four high school teachers as they made the transition from teaching streamed mathematics classes (ability grouped) to destreamed mathematics classes (mixed ability). The researchers found that although there was an initial negative im- pact on teachers’ self-efficacy, over the course of 1 year, the confidence that teachers expressed during the interviews about their teaching seemed to return. The authors con- cluded that teacher self-efficacy rebounded

when the teachers believed that use of the new methods resulted in student learning and that their judgment was affected by collabora- tion with others who gave positive feedback about their teaching. Similar results were ob- tained when Mulholland and Wallace (2001) tracked the development of one science teacher as she made the transition from preservice to beginning teaching. Interviews with this teacher (“Katie”) revealed that near the end of her 1st year of teaching, important changes began to occur in the way that she evaluated her success as a teacher. Rather than base her evaluation of success on the degree of control that she held over the class, Katie focused on her students’ learning as the main criterion for judging a successful lesson. Mul- holland and Wallace noted that this shift was accomplished through the encouragement of older teachers who showed her “how to rein- terpret the efficacy information her teaching experiences provided” (p. 257).

Sources of Teacher Self-Efficacy in Traditional Versus Reform- Based Mathematics Teaching

Bandura’s sociocognitive theory and studies of teachers who were making significant transitions in their teaching, such as those described earlier, suggest that when there are important changes in teachers’ ability to see evidence of success, teacher self-efficacy is enhanced. This may be particularly problematic for reform efforts where the desired goals, outcomes, and teaching practices differ from those of conventionally used practices. For example, alternative instructional programs that are aligned with the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (2000), which we refer to as reform-based mathematics teaching, differ from traditional mathematics teaching in several ways (Anderson & Piazza, 1996; Hiebert, 2003). According to Heibert (2003), reform- based mathematics teaching emphasizes instructional goals related to student understanding of mathematical concepts and

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procedures. Teachers who are engaging in reform-based mathematics teaching use instructional practices that build on students’ informal knowledge of mathematics, provide opportunities for students to contribute and invent their own methods during problem solving, emphasize classroom discussion and analysis of multiple methods for solving a given problem, and expect students to provide explanations for their answers. Hiebert notes that in contrast, the typical pattern of traditional classroom practice primarily focuses on developing procedural skill in executing steps in conventional algorithms. Teachers who engage in traditional mathematics teaching rely extensively on ( 1) teacher-directed demon- stration and explanation as well as close- ended questioning as a means of communicat- ing procedural information and (2) follow-up with student practice in the form of applying taught procedures to similar problems.

Given these differences in goals and pedagogical methods, Smith ( 1996) speculates that one reason why the transition from traditional to reform-based mathematics teaching may be particularly difficult for teachers is that typical sources of evidence that teachers use to judge their teaching success, which in turn support their senses of teacher self-efficacy, are no longer operative. As Smith points out, traditional mathematics teaching provides an explicit model of what teachers should do in their teaching and how they should judge the quality of their teaching. As discussed earlier, the content to be taught involves providing the proper sequences of procedures and problems for students to work on. The basic struc- ture for teaching-that is, demonstrating a procedure, guiding practice, correcting student work, and reteaching when necessary- provides teachers with a manageable amount of content to “deliver” and explicit criteria for judging whether it has been covered and whether students are successful.

In contrast, teaching mathematics in reform-oriented ways provides fewer explicit criteria for judging success (Smith, 1996). Be- cause the primary target of such teaching is the development of student thinking and understanding, the criteria for judging success are far

less visible. Issues of coverage, completion, and success require different interpretations and cues when using reform-oriented, rather than traditional, teaching practices. For example, for teachers who are pursuing this type of instruction, evidence of success may come in the form of seeing a few students shift from using a less- sophisticated strategy to a more-sophisticated strategy without any difference in the accuracy of the results. Or it may come from observing a few students’ willingness to embrace the challenge of a problem that they cannot initially solve, by reworking their wrong solutions and persisting through initial failure. The ability to notice these small steps toward deep understanding and engagement are accompanied by a set of developmental expectations and belief that all students-with the proper encouragement, instructional scaffolding, and time-will continue to make progress.

Smith (1996) speculates that a teacher’s ability to interpret her or his performance as being successful and so develop the confidence for teaching reform-based mathematics (i.e., teacher self-efficacy) may be impaired because success is defined differently when teaching reform-based mathematics. This suggests that an important step in understanding the process by which teachers develop teacher self-efficacy in situations where the goals, methods, and outcomes are sub- stantially different from their previous teaching practices-such as when they make the transition from traditional to reform-based mathematics teaching-is to examine the shift in the criteria used by teachers to evaluate their teaching success.

Purpose of the Study

The purpose of this study was to investigate the sources of efficacy among teachers who are making the transition to reform-based mathematics teaching, by examining the criteria that they use to evaluate their teaching effectiveness. Specifically, first- and second-grade teachers involved in a reform-oriented mathematics teacher development project for 1 and 5 years, respectively, were asked to teach a

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reform-based mathematics lesson and then to evaluate the success of that lesson. To study this source of efficacy information, we used a talk-aloud methodology. This type of methodology, in which participants are prompted to talk aloud as they engage in a task (in this case, reflecting on a lesson just taught), is particularly well suited for revealing what information teachers selected and attended to, the meanings associated with this information, and how this information is used to evaluate the relative success of a lesson (Chi, 1997).

Two questions were the focus of this study: First, do teachers with more and less experience with reform-based mathematics teaching differ in their evaluations of the success of a lesson? Second, do teachers with more and less experience with reform-based mathematics teaching differ in what events they attend to and use as evidence to support their evaluations of success? According to Smith (1996), we would expect teachers with more experience with reform-based mathematics teaching to not only perceive their lessons as more successful but also attend to and use as evidence different events in the lesson to support their evaluations of success. This finding would provide support for the idea that a key obstacle in developing teacher self-efficacy for reform- based mathematics teaching involves shifting from criteria used for judging success in a traditional math lesson to criteria appropriate for judging a reform-based lesson.

Alternately, if we find that there are no differences in the types of evidence that teachers use to judge the success of their lesson hut, rather, differences in the amount of success that they report, it would suggest that the difficulty in making the transition to reform- based mathematics teaching may lie elsewhere and have little to do with shifting criteria and noticing success.

Method

Participants

Ten first- and second-grade teachers were selected to take part in this study from a group

of 22 who had volunteered to participate in an innovative teacher development program focused on helping teachers change their mathematics teaching practices. All selected teachers were female and had over 10 years of experience in teaching in elementary schools. Half were (‘veterans” with respect to the teacher development program (over 5 years of involvement), and half were “newcomers” (involved for 1 year). Selection of teachers for inclusion in the study was based on the program developers’ recoinmenda- tions of those who were most successful at implementing changes in their classroom among those classified as veteran and newcomer. The teachers taught in a variety of schools, ranging from urban schools primarily attended by students of low socioeconomic status to rural schools with students of middle-range socioeconomic status.

Teacher Development Program

The participating teachers in this study were involved in a reform-based program led by mathematics educators at a midsize university in the Midwest region of the United States. As described by the mathematics educators who designed and carried out this program, an effort was made to create a professional development program that would encourage teachers to view themselves as teacher-researchers:

From the beginning we sought to establish a collaborative, shared-expertise approach to our work. That is, we believed both teachers and project leaders possessed spe- cial insights and expertise that would contribute to the learning of each other. We felt that through sharing our perspectives and experiences, both groups would grow. (Trafton & Thiessen, 1999, p. 87)

Throughout this program, teachers were encouraged to make sense of, take ownership of, and manage the implementation of new instructional approaches. Although the project leaders accepted that teachers would iinple- ment ideas in their own ways and on their own schedules, a shared core belief about the im-

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portance of children’s thinking, learning from children, and making curricular decisions based on observations in the classroom provided a common framework for all teachers (Trafton & Thiessen, 1999).

Like that of other reform-based mathematics programs (e.g., cognitively guided instruction), the pedagogical philosophy was constructivist, and teachers were encouraged to focus on children’s thinking and strategies. However, a distinctive feature of this teacher development project is the focused attention given to helping teachers not only notice but value and enjoy the idiosyncrasies of student mathematical thinking. This objective was primarily accomplished through modeling and feedback provided during monthly meetings, which were approximately 2 hours and typically began with teachers’ sharing their classroom implementation experiences for 30 to 40 minutes while the project leaders and other teachers gave supportive feedback and sugges- tions. Meetings were followed by the project leaders’ summary insights and by any questions from the teachers, and in the remaining time, the project leaders introduced new material- for example, research articles describing new student thinking strategies, mathematical insights into content, new mathematical tasks for teachers to try out with students, and so forth. Teachers’ experiences in trying out the new materials and ideas then formed the basis for discussion at the next meeting. Between monthly meetings, project leaders facilitated conversation through e-mail discussion about experiences and concerns made throughout the month.

Lesson Reflection Talk-Aloud

To examine the cognitive processes that teachers use when they reflect on their teaching performance, we developed the lesson reflection talk-aloud procedure. The talk-aloud method-also referred to as the think-aloud method-is a member of a class of verbal report methods that is particularly useful in the collection of data on cognitive processes (Ericsson & Simon, 1993). The goal of the talk-aloud method is to collect as accurate

and as complete a record as possible regarding the thinking that is used to accomplish a particular task. Unlike other verbal report methods where individuals are asked to describe their thinking retrospectively, the talk-aloud method seeks to capture thinking processes concurrently, by having the person talk aloud as he or she engages in a task. Interruptions, suggestive prompts, and follow-up questions by the researcher are avoided during the collection of verbal data. Instead, participants are typically given brief instructions to think aloud while engaging in a task; they are encouraged to say anything that comes to mind as they are doing the task; and they are occa- sionally given a warm-up task in which the researcher provides additional training as needed to help the person become comfort- able with the experimental setting and procedure. The verbalizations reported during the talk-aloud procedure are audiotaped and later coded and subjected to verbal protocol analysis. According to Ericsson and Simon (1993), the talk-aloud method does not lead to much distortion of thinking processes, nor does it interfere with task performance except under a limited set of conditions.

For this study, we used the talk-aloud method to examine the contents of teachers’ thought processes as they reflected on a lesson that they just taught. To initiate lesson reflection and provide a task about which teachers could verbalize their thinking, we utilized a talk-aloud procedure that consisted of two prompts: “How do you think the lesson went?” and “How do you think your students responded to the lesson?” These prompts were designed to simulate the kind of questioning that teachers might ask themselves when reflecting on a lesson.

In addition, after a response to each prompt, the researcher provided a follow-up question: “Is there anything more specific you can tell me about . . . ?” (i.e., “how the lesson went,” “how your students responded to the lesson”). As is used with talk-aloud methods, these prompts were intended to encourage complete verbalization of the process under study-in our case, teachers’ reflections on their lesson. Beyond these minimal prompts,

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66 ANTHONY J. GABRIEL€ AND ELANA JORAM

however, the researcher purposely did not ask additional follow-up questions, so that the procedure remained standardized and so that differences in the number of comments made by respondents would primarily reflect differences in their memory of events and their reflections about those events.

Using the talk-aloud methodology has several inherent strengths when studying the criteria that teachers use to evaluate their teaching (e.g., sources of efficacy information). By asking teachers to evaluate a lesson that they just taught-rather than, for example, asking them directly what criteria they use to judge the success of lessons in general (e.g., “How do you know if your lesson has been successful?ll “What do you look for when trying to assess whether your lesson has gone well?”)- we hoped to ground teachers’ beliefs in specific events rather than have them speculate on and reconstruct what these criteria might be. The kind of social desirability bias that plagues semistructured interviews was a particular concern for our study because of our participants’ involvement in a teacher development program, and we did not want the data to consist of their parroting back to us reform- based philosophy. In addition, by having teachers evaluate a lesson that they had just taught without the aid of reviewing it again on videotape-in contrast to, for example, using a stimulated recall procedure-we capiralized on teachers’ selective memory processes as indicators of what was most salient to them about the lesson. Although we could have asked teachers direct questions such as “What did you think of the level of student thinking in the lesson?” we assumed that our technique, in revealing what was most salient to them about the lesson, would indicate what was most valued by the teachers when judging their performance. Perhaps most important, we argue that this methodology closely simu- lates the actual reflective process that a teacher might experience when thinking about his or her own teaching. As suggested, because this is likely a primary site at which efficacy beliefs are constructed, it made sense to try to capture this evaluative process as accu- rately as possible.

Verbal Analysis of Talk-Aloud Data

Transcripts were subjected to verbal analysis, a method of quantifying the qualitative coding of the contents of verbal utterances. Ac- cording to Chi (1997), verbal analysis methods seek to blend the richness of context provided by qualitative methods with the ob- jectivity and replicability of quantitative methods. Chi suggests that verbal analysis methods are particularly valuable for studying mental representations of knowledge used in cognitive performances. Although the me- chanics of the verbal analysis technique de- pend on the research hypothesis in question, at least four functional steps are performed on the verbal data: data reduction, segmentation of the reduced protocol, development or determination of a coding scheme, and application of the coding scheme. We describe each of these steps as it relates to our coding of the interview data.

Data reduction. Because we were interested in teachers’ evaluations of enacted performance as a potential source of efficacy information and not in their general beliefs and philosophy of teaching per se, only comments that referred to events that occurred in the enacted lesson were coded in this analysis. For example, comments such as the following were included in the analysis because they clearly refer to an event that occurred in the lesson that the teacher was evaluating: “When Katie presented her problem, it was like they were warmed up; the other two problems were just warm-up problems. I thought that whole process went very well.” However, general comments related to pedagogical philosophy, anecdotes referring to other unrelated lessons, and so forth, were removed from the transcripts and not coded-for example, “The more work they have, the more sense they tnake out of fractions and the better understanding they’ll have down the road in third, fourth grades.” Although such comments revealed interesting information about this teacher’s pedagogical beliefs and strategies, they were not included in the analysis, because they were not tied to events that occurred during the enacted lesson.

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Teachers’ Reflections on Their Refm-Based Teaching in Mathematics 67

Segmentation of the reduced protocol. Teacher verbalizations were divided into separate lesson-related comments, or idea units. A lesson-related comment in this context was defined as a comment that referred to a single referent, where a referent was either a recalled lesson event or a distinct reflection about that event. Lesson-related comments varied in length (in terms of the number of words); some were only a few words long whereas others were several sentences. To distinguish between teachers’ lesson-related comments, we looked for boundaries between separate comments, defined by distinct shifts in focus and change in topic. Redundant comments, in which the teacher referred to the same event without adding any new information, were coded only once. A total of 197 lesson-related comments across the 10 teachers were identified. Interrater agreement in segmenting transcribed interviews was established by having two coders each segment approximately one-third of the data. Percentage agreement was 89%.

Development or determination of the coding scheme. Two literatures informed the design of the coding scheme used to analyze the verbal data: research on teachers’ criteria for judging ‘‘good” lessons (e.g., Dunkin, Pre- cians, & Nettle, 1996) and research on self- efficacy (e.g., Bandura, 1997). Research on teachers’ evaluation of their own teaching indicates that several criteria are used by teachers to judge teaching effectiveness and that what is considered most salient varies with teaching experience (Dunkin et al., 1996; McLeod, 1988). In these studies, three sets of criteria were reportedly used by teachers: pupil based, having to do with the behavior of students during or after a lesson; teacher based, having to do with teacher behaviors before, during, and after a lesson; and other based, evaluative feedback from students, other teachers, parents, and so on. For our purposes, we adopted criteria having only to do with behaviors occurring during the lesson, which led to the formation of several categories, loosely based on those identified by Dunkin and colleagues (1996): lesson goals and outcomes, pedagogical behaviors, student com-

prehension, student mathematical thinking, student affect, and student conduct.

Bandura’s (1997) work on self-efficacy suggests that emotional and physiological arousal are important internal sources of efficacy information that can influence the interpretation of enacted mastery experiences. We tried to capture this aspect by coding in- stances in the verbal protocol in which teachers verbalized emotional reactions to the events they were describing. In addition, Bandura stressed the importance of noticing progress as an important source of efficacy information. We therefore included a category to code teachers’ comments related to change over time. Table 1 provides definitions of these categories, with examples.

Application of the coding scheme. A list of segmented lesson-related comments for each teacher was generated, and each comment was coded with respect to the following four di- mensions:

1. Content of lesson event. Each comment was coded with respect to the seven mutually exclusive categories listed in the table, to assess potential sources of efficacy information, or evidence, that might be used by the teacher to judge the relative success of the lesson he or she just taught. For example, the comment “The dice part they really enjoy- they’re really enthusiastic about that” would be coded as student affect because the main point of the statement was a reflection on how the students felt about a part of the lesson. The statement “The beginning of it, with the dice . . . I was real pleased with the way the children were thinking” was coded as student thinking because the main point of the statement was a reflection on student thinking.

2. Evaluation of the lesson event. Each of these comments was coded as either a successful or unsuccessful instance of the lesson event described. In most cases, teachers provided explicit positive evaluations (“I thought that went well”) or negative evaluations (“They had

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Table 1. Definitions and Examples of Categories Used to Code Content of Teacher Comments

Definition Example

Lesson Goals and Outcomes

An evaluation of the success or partial success "So. overall, I think the problem went well in terms of the learning they showed me and the kind of learning they were doing."

in achieving the main objectives and intended outcomes of the lesson

Pedagogical Behaviors

An evaluation of the quality of teacher actions (e.g., presenting material, asking questions, providing feedback)

"I thought I answered student questions well."

Student Comprehension

An evaluation of students' relative understanding "I reminded Kayla to have just 12, because she had 14. That in itself, maybe she didn't get the gist of the whole thing."

of a concept, procedure, idea, and so on, emphasized in lesson

Student Thinking

A description of a student's solution strategy, "I'm pleased with how they can add multiple numbers approach to problem solving, idea, or an explicit mention of a mental state term (e.g., thinking, strategy, process)

in their head by looking for tens or groups of something that would make it easier to count."

Student Affect

An evaluation of students' enjoyment, frustration, or other motivational dispositions (e.g., persistence, initiative)

"They find the estimation thing fun. They're really getting into it."

Student Conduct

An evaluation of students' on-task, off-task, or disruptive behavior that does not contain information about a student's mental state (e.g., comprehension, affect).

"They were busy attacking the problem."

Change Over Time

A comparative statement that focuses on how "That maturity level and that opportunity for Patrick to show what he knows and to work through problems wasn't there at the beginning of the year. It's exciting for me to see that he's developed into that kind of student."

something that happened in the enacted lesson (related to the student, curriculum, or teaching practice) has changed over time.

trouble with that") of the events to which their comments referred. In those cases where no explicitly evaluative comments were provided, we inferred whether the event was successful or problematic based on common standards of performance. For example, for the category student lesson comprehension, if the teacher commented on a stu-

dent's having difficulty understanding a part of the lesson, we coded it as an unsuccessful instance of lesson comprehension. Similarly, if a teacher described a student's enjoying a particular aspect of the lesson (student affect), using a successful strategy (student thinking), staying on task (student conduct), and so on, we coded the event as successful.

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3. Specificity of ksson event. A lesson- related comment was coded as specific if the lesson event was described with de- tail and precision with respect to time, place, and subject-for example, “When I had Shawn go over and ex-

ply the code scheme to approximately one- third of the segmented data. Percentage agreement was 89%.

Procedure -

plain it to that far group, Shawn sur- prised me-he was the first one to get it and I didn’t think he would have it the whole time, but he had it right away. I wanted to reinforce that and have him explain what he was doing so in his own mind he could say it out loud.” In contrast, a lesson-related comment was coded as nonspecific if the referent described in the lesson event was vague and described in general terms. For example, the comment “They seemed to stick at their work” was coded as a nonspecific lesson event because it did not communicate specific information about who, what, or where the event took place.

4. Emotion associated with lesson event. To code affective reactions, which may play an important role in efficacy formation, we examined teacher expres- sions of their feelings of satisfaction and frustration that were associated with comments about events in the lesson. We examined each comment about a lesson event (e.g., lesson goal and outcome, student thinking) and coded whether the teacher verbally expressed a positive, negative, or neutral emotional reaction to it. For example, the comment “The beginning of it, with the dice . . . I was real pleased with the way the children were thinking,” would, in addition to being coded as student thinking, be coded as a positive emotional reaction because the teacher explicitly expressed her personal pleasure in seeing the students’ thinking. If no discern- able reaction was verbally expressed, the comment was coded as neutral.

Interrater agreement in applying the codes was established by having two coders each ap-

Participating teachers were contacted and scheduled for observation and interview. Teachers were asked to teach a lesson that they thought exemplified their current reform-oriented style of teaching and that they felt confident teaching. Two researchers who were not involved in the teacher development program each observed lessons and collected lesson reflection talk-alouds, all of which were conducted in the teachers’ classrooms and took place immediately after the teachers enacted their lessons. Lesson reflection talk-alouds were audiotaped and later transcribed for analysis. Each researcher observed and collected lesson reflection talk- alouds on roughly equal numbers of teachers designated as newcomers and veterans. Data was collected in April, near the end of the school year.

Results

Table 2 shows the breakdown of teachers’ comments by content. As expected, veterans in the program more frequently recalled successful events in evaluating their lessons (89%) than did newcomers (68%). Analysis of the data disaggregated by teacher revealed that for 4 of the 5 veteran teachers, more than 85% of the lesson events recalled were successful. In contrast, for only 1 of 5 newcomer teachers, 85% or more of the lesson events were recalled as successful events. Instead, for 3 of 5 newcomer teachers, 40% or more of the lesson events were recalled as unsuccessful events. Thus, with respect to our first research question, the data indicate that our two groups of teachers differ in their perceptions of their success in teaching a reform-based lesson, sug- gesting that they are likely to be at different points in their transitions to reform-based mathematics teaching.

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Table 2. the Lesson Event and Level of Experience in the Professional Development Program

Successful Unsuccessful Content of Lesson Event Lesson Events Lesson Events

Veteran Ne wcomer Veteran Newcomer

Frequency and Percentage of Teacher Comments Coded as Successful or Unsuccessful by Content of

Lesson goal and outcome Pedagogical behavior Student comprehension Student thinking Student affect Student conduct Change over time

Total

n = 7 (6%)

17 (16) 29 (26) 26 (24) 11 (10) 8 (7)

98 (89)

0 (0)

Differences in Veterans’ and Newcomers’ Sources of Efficacy Information

Our second research question asked whether there were differences in the events that teachers with varying degrees of experience with reform-based mathematics teaching attend to and use as evidence to support their evaluations of success. To address this question, we examined differences in the frequency of events that veteran and newcomers reported during the lesson reflection interview. If veterans and newcomers judged their enactments of lessons using similar criteria, then we would expect the type of events reported to be distributed similarly among them, with the main difference being whether they are related to successful or unsuccessful events. As shown in Table 2, differences were observed in the extent to which veterans and newcomers commented about the categories of events within the lesson. Veterans made more statements about student thinking (28%) than newcomers did (10%). In contrast to this pattern, newcomers made more comments about lesson goals and expected outcomes (22%) than veterans did (7%).

No striking differences were found in the extent to which veterans (67%) and newcomers (61%) described events in the lesson with specificity. However, an analysis of these comments with respect to content indicates that differences in the extent to which veterans

and newcomers made detailed references were most pronounced with respect to descriptions of student thinking. Veterans made more specific statements about student thinking (2 1 %) than newcomers did (5%). In addition, 4 of the 5 veteran teachers, when commenting on lesson events involving student thinking, described these events more frequently (i.e., greater than 50% of comments made) in specific, rather than nonspecific, terms. In contrast, only 1 in 5 newcomers made more fre- quent specific, rather than nonspecific, references to student thinking. In no other category of event did we find a consistent pattern of differences in the extent to which veterans and newcomers described events in specific o r nonspecific terms. This finding suggests that for the teachers whom we interviewed, differences in the degree to which lesson events were described specifically versus nonspecifically were not a function of global stylistic differences in the way that they tended to reflect on their lessons but, rather, were the result of differences in the way that they spoke about student thinking.

Differences in Affect Associated With Lesson Events Mentioned by Veterans and Newcomers

Analysis of affective reactions associated with the reporting of lesson events revealed that across both groups of teachers, frustration and

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Teachers’ Reflections on Their Reform-Based Teaching in Mathematics 7 1

disappointment were rarely explicitly expressed when evaluating lessons just taught (only 2 of the 197 coded comments were identified as expressing negative emotions). Roughly three-fourths of the comments ana- lyzed were coded as neutral, indicating that for the most part, when teachers talked about their teaching, they did not overtly express feelings of satisfaction or frustration. However, we found differences in the frequency with which veterans (33 of 110 comments, or 30%) and newcomers (17 of 87 comments, or 19%) expressed positive emotion. Further analysis suggests that veterans and newcomers may differ with respect to the types of lesson events that serve as the basis for feelings of satisfaction and enjoyment. Examination of these comments with respect to content indicates differences in which events appeared to evoke the most spontaneous emotional reactions from teachers. For newcomers, positive feelings were most frequently reported when the teachers described achieving lesson goals and desired outcomes (8 of 17, or 47%, of the newcomers’ comments coded as expressing positive emotions); for veterans, this category was rarely associated with positive feelings (1 of 33, or 3%, of the veterans’ comments coded as expressing positive emotions). In contrast, for veterans, student thinking was most associated with positive feelings expressed during the interview ( 12 of 33, or 36%, of comments coded as expressing positive emotions); for newcomers, only 29% (5 of 17) of their comments coded as expressing positive emotions mentioned student thinking. Thus, with respect to our second research question, the data suggest that different sets of events were salient to veterans (student thinking) and newcomers (lesson goal and objectives) when evaluating a lesson just taught.

Discussion

The purpose of this study was to explore the reflections of teachers who are making the transition to reform-based mathematics teaching on the events in a lesson they just taught. As we argue, studying the criteria that teach-

ers use when reflecting on past performance and examining how these potential sources of teacher self-efficacy change over time should provide insights into the development of teacher self-efficacy. Such insights promise to be particularly valuable in the design of professional development.

As expected, we found that veterans in the teacher development program differed from newcomers, not only in the proportion of successful events they reported, but also in events they chose to emphasize as evidence for success and in the way they chose to describe those events. Specifically, relative to newcomers, veterans focused more on student thinking and described it in more precise and specific terms. In contrast, newcomers tended to talk more about whether they had met their curricular goals for the lesson. Newcomers made fewer comments about student thinking, and when they did, they were just as likely to describe student thinking in less-precise terms. We also found trends in our data to suggest that not only did the veteran teachers more frequently mention student thinking and describe it in specific terms, but they more often expressed some pleasure and satisfaction when making these comments than did newcomers. Newcomers, however, more frequently reported positive affective reactions when describing their progress toward achieving instructional goals and outcomes.

One interpretation of this pattern of results is that different sets of events, some of which are connected to positive feeling states (student thinking for veterans and lesson goals and outcomes for newcomers), serve as power- ful sources of efficacy information for veterans and newcomers. That is, different sets of criteria for judging success are salient to these teachers. To the extent that criteria related to student thinking are aligned with the instructional goals and practices of reform-based mathematics teaching, we expect that over the long term, teachers who rely on these criteria for judging success of their teaching would be better able to notice their progress in teaching reform-based mathematics and therefore even- tually develop high teacher self-efficacy for teaching in this way. In turn, this developing

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teacher self-efficacy should contribute to the ongoing motivation to learn the intricacies of reform-based mathematics teaching practices and, ultimately, lead to the successful implementation of reform-based mathematics teaching practices.

Limitations of the Study

Whereas the approach used in this study to examine sources of teacher self-efficacy pro- duced promising results, several limitations should be noted. First, although the cross- sectional design was a useful way of gaining insight into changes in sources of teacher self-efficacy information as a function of greater experience with professional development, longitudinal studies that examine differences in reflections as teachers progress through professional development would lend themselves well to our research questions. Interviewing the same teachers multiple times, although leaving open additional threats to internal validity, would allow for the documentation of changes within individuals as they progress through professional development programs.

Second, we focused on only first- and second-grade teachers, and the sample size was relatively small. Thus, the results must be interpreted with caution. A larger sample of newcomers and veterans of different kinds of professional development programs in mathematics at different grade levels would be desirable and so enhance the external validity of the study. Notwithstanding these limitations, it is promising that differences were found between the newcomer and veteran teachers, and it suggests that this area will be fruitful for future research.

Third, although we argue that the talk- aloud methodology has certain strengths, other methods-such as in-depth case studies and unstructured interview techniques- might complement data from talk-alouds and thus reveal additional information on sources of teacher self-efficacy not easily captured by the talk-aloud methodology. Such methods can be used to expand on information collected during the talk-aloud and to focus on

other possible factors, such as the role of colleagues, students’ parents, and so on, in the development of teacher self-efficacy.

Finally, the logic of our design and the data reported suggest that teachers who engage in more professional development are likely to be higher in teacher self-efficacy for teaching reform-based mathematics than are teachers who engage in less-professional development on such reform. Replication studies that identify teachers who are higher and lower in teacher self-efficacy, by independently assessing teacher self-efficacy using conventional paper-and-pencil measures (see Tschannen-Moran et al., 1998), will help to further establish the linkages for which we found initial evidence.

Implications for Professional Development in Reform-Based Mathematics Teaching

One implication of the contention that an interest and ability to notice student thinking during classroom instruction may lead to deep and sustained transformations in teachers’ self-efficacy for teaching reform-based mathematics instruction is that teacher development projects might explicitly focus some of their efforts on scaffolding teachers’ fascination with student thinking. For example, the teacher development program in which the participants in this study were involved actively promoted an increasing valu- ing and enjoyment of student thinking. Anecdotes were often shared by teachers in which they gave examples of students’ thinking about mathematics problems that the teachers were all trying out. Reactions by other teachers to these anecdotes were often affectively charged and positive-this cher- ishing of student thinking was frequently modeled by the program leaders. Over time, the comments made by veteran teachers in the group setting would help to enculturate newcomers and promote in them an interest in student thinking.

Other professional development strategies beyond informally discussing student thinking

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may help to develop teacher self-efficacy for teaching reform-based mathematics. In a re- cent study, Ross and Bruce (2005) designed a short-term professional development program that targeted enhancing teacher self-efficacy by redefining teachers’ criteria for determining the success of instruction. To do so, they provided teachers with detailed rubrics and then modeled standards-based mathematics teaching practices, having teachers observe and evaluate the modeled practice using the rubrics. Debriefing sessions followed in which an explicit discussion of the standards for judging success were discussed in relation to the teaching practice. Although both of these professional development strategies were specific to reform-based mathematics teaching, they could easily be adapted to other subject matter teaching reform efforts.

Conclusions

In conclusion, understanding how teacher self-efficacy develops and can be enhanced through professional development is an important yet neglected feature of efforts that seek to support teacher learning through the difficult transition process that reform entails. When these reform efforts involve substantive changes in the goals and criteria for defining success and not just in the teaching methods to be used, it is likely that professional development activities would benefit from focusing particular attention to helping teachers notice and interpret successful experiences during such transit ions.

In the case of reform-based mathematics teaching, the ability to notice and appreciate various examples of student thinking during a lesson provides the teacher with a critical source of efficacy information for continued motivation to teach the kinds of lessons that are characteristic of reform-based mathematics teaching. Helping teachers to value the intricacies of student thinking and having a clear picture of how it develops are likely to play a key role in the successful shift from teaching in traditional ways to teaching in reform- oriented ways. E

Note

1. We use the term reflection to refer to the de- liberate process of analysis that a teacher engages in when trying to understand and improve his or her teaching performance. This more limited use of the term is related to, but distinct from, other con- ceptualizations of reflection discussed in the extant literature (see Davis, 2006, for a brief summary of these differences).

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Anthony J. Gabriele is an associate professor in the Department of Educational Psychology and Foundations at the University of North- ern Iowa. His research interests include teacher learning, motivational influences on cognition, and achievement gaps in tnathe- matics performance.

Elana Joram is an associate professor in the Department of Educational Psychology and Foundations at the University of Northern Iowa. Her research interests include preservice and practicing teachers’ beliefs about learning, instruction, and mathematical cognition.

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Documents

Teachers' Reflections on Their Reform-Based Teaching in Mathematics: Implications for the Development of Teacher Self-Efficacy