Mathematics teaching as a deliberate practice an investigation of elementary pre service teachers reflective thinking during student teaching

AMY ROTH MCDUFFIE MATHEMATICS TEACHING AS A DELIBERATE PRACTICE: AN INVESTIGATION OF ELEMENTARY PRE-SERVICE TEACHERS’ REFLECTIVE THINKING DURING STUDENT TEACHING ABSTRACT In this case study I examine the reflective practices of two elementary preservice teachers during their student teaching internship I extend current views of reflective practice to create a framework for a ‘deliberate practitioner’ With this framework, I investigate the pre-service teachers’ thinking with regard to reflective processes and how they use their pedagogical content knowledge in their practices My findings indicate that the pre-service teachers use their pedagogical content knowledge in anticipating problematic events, and in reflecting on problematic events in instruction However, limits in pedagogical content knowledge and lack of confidence impede the pre-service teachers’ reflection while in the act of teaching They were more likely to reflect on their practices outside of the act of teaching Implications for teacher educators and pre-service teachers are discussed KEY WORDS: mathematics education, pedagogical content knowledge, reflective practice, student teaching, teacher education With the emergence of recent reforms in education in the United States (e.g., National Council of Teachers of Mathematics [NCTM], 1989, 1991, 2000), researchers and educators have re-examined teaching by moving away from a technical model of teaching by prescribed methods to one that regards it as a, complex, demanding practice Two separate but compatible perspectives have made substantial contributions as to how we view teaching, and correspondingly, to how we approach teacher education First, viewing teachers as reflective practitioners has underscored the problem solving nature of teaching (McIntyre, Byrd & Foxx, 1996; Russell & Munby, 1991; Schön, 1983, 1987; Valli, 1992; Zeichner, 1993) Consequently, the focus of many teacher education programs is on the development of reflective practitioners (Christensen, 1996) This focus is consistent with a constructivist perspective for teaching and learning that is the basis of many teacher education programs (e.g., McIntyre, Byrd & Foxx, 1996) Second, the conceptualizing of pedagogical content knowledge (Grossman, 1990; Shulman, 1986, 1987) as a unique type of knowledge for teaching has helped researchers, teachers, and teacher educators gain an understanding of the knowledge base that teachers need for successful practice Journal of Mathematics Teacher Education 7: 33–61, 2004 © 2004 Kluwer Academic Publishers Printed in the Netherlands 34 AMY ROTH MCDUFFIE In studying the knowledge of mathematics teachers, Ball, Lubienski and Mewborn (2001) call for research on “how teachers are able to use mathematical knowledge in the course of their work” (p 450) and “what [teachers] are able to mobilize mathematically in the course of teaching” (p 451) One approach to understanding the use of knowledge is to investigate how teachers think about their practice The purpose of my study was to intersect the constructs of reflective practice and pedagogical content knowledge (PCK) in order to examine how pre-service teachers use their mathematical PCK in thinking about their practice, both in planning and classroom teaching I investigated the reflective practices in mathematics of two pre-service elementary teachers during their student teaching internship THEORETICAL FRAMEWORK I shall provide a summary of reflective thinking and pedagogical content knowledge as I applied and merged these two concepts in my work with the pre-service elementary teachers Reflective Thinking While a range of interpretations exists for what is considered to be reflective practice, I referred primarily to the ideas of Dewey (1910) and Schön (1983, 1987) Reflection is a practice that has gained considerable attention in the past two decades, yet Dewey began this discussion early in the last century Dewey argued that reflective thinking begins when teachers experience a difficulty or troubling event (i.e., a problem) A key aspect of the reflective process is that teachers act on their reflections: Reflection involves not simply a sequence of ideas, but a consequence – a consecutive ordering in such a way that each determines the next as its proper outcome, while each outcome in turn leans back on, or refers to, its predecessors Each phase is a step from something to something There are in any reflective thought definite units that are linked together so that there is a sustained movement to a common end (pp 2–3) Thus, acting on reflections distinguishes reflective practice from just thinking back and may be an important aspect in the development of teaching Schön (1983, 1987) developed these ideas further and separated reflection into two forms, reflection-in-action and reflection-on-action Reflection-on-action is a deliberate process of looking back at problematic events and actions, analyzing them, and making decisions Russell and Munby (1991) explained that reflection-on-action “refers to the ordered, MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 35 deliberate, and systematic application of logic to a problem in order to resolve it; the process is very much within our control” (p 165) Reflection-in-action is a more immediate consideration and resolution of an identified problem in the act of teaching and learning (Schön, 1987) Both types of reflection and the triggering (problematic) events were considered in this study Mewborn (1999) studied the reflective practices of pre-service teachers during field experience as part of their mathematics methods course Mewborn examined the elements of mathematics teaching and learning that were problematic for the teachers and the thinking they engaged in to resolve those problems She found that pre-service teachers did engage in reflective thinking However, this reflective thinking was not evident until they internalized their own authority to generate, reason about, and test hypotheses in order to examine children’s mathematical thinking Mewborn argued that to facilitate reflection pre-service teachers need a non-evaluative atmosphere and relationship with cooperating teachers and university faculty so that they are encouraged to generate hypotheses and to arrive at resolutions to problematic events without fear of judgment Inherent to all discussions of reflection is a problematic or puzzling event triggering reflection and, thus, I sought to investigate reflection that arose from problematic situations While it could be argued that reflection might occur during or after a successful lesson (a non-problematic situation), I chose to adopt the constructs for reflective thinking of Dewey (1910) and Schön (1987) by focusing on pre-service teachers’ thinking that was inspired by identified problems I synthesized the various views and definitions of reflection considered for this study as a cycle in teaching practice In this cycle, the problematic event initiates the process of reflection Following this problematizing, either during instruction (for reflection-in-action) or after instruction (for reflection-on-action), the teacher analyzes the problem and the options and/or approaches for resolving the problem Next the teacher decides on a resolution or plan for action Finally, the plan is implemented in practice and the resolution is tested At this point the process either ends for this event or results in a subsequent problematic event (an unresolved issue), and the reflective cycle continues As Dewey (1910) contended, the cycle does not necessarily move directly from one phase to the next One could, for example, re-define the problem while in the process of planning an action Using Pedagogical Content Knowledge to Guide Reflective Thinking Along with a focus on teachers developing reflective thinking practices, researchers have been concerned about teachers developing a sufficient 36 AMY ROTH MCDUFFIE knowledge base to guide their thinking about children’s ways of understanding mathematical concepts and process (e.g., Ball, Lubienski & Mewborn, 2001; Borko & Putnam, 1996) Indeed, pedagogical content knowledge is an important resource for teachers to use as they reflect in practice Shulman (1986, 1987) first defined pedagogical content knowledge as, The blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction Pedagogical content knowledge is the category most likely to distinguish the understanding of the content specialist from that of the pedagogue (1987, p 8) Building on Shulman’s work, Grossman (1990) delineated four central components of pedagogical content knowledge These components are: conceptions of purposes for teaching subject matter (i.e., forming goals); knowledge of students’ understanding, conceptions, and misconceptions of particular topics in a subject matter; curricular knowledge; and knowledge of instructional strategies and representations for teaching particular topics Grossman acknowledges that “these components are less distinct in practice than in theory” (p 9), but this general framework was useful in thinking about the ways in which the pre-service teachers in this study used their pedagogical content knowledge Researchers have found that novice teachers tend to have inadequate or underdeveloped mathematical pedagogical content knowledge for use in practice (e.g., Borko & Putnam, 1996; Borko, Eisenhart, Brown, Underhill, Jones & Agard, 1992) In elementary mathematics education, several projects have as a goal the development of teachers’ understanding about children’s learning such as Cognitively Guided Instruction (e.g., Carpenter, Fennema, Franke, Levi & Empson, 1999) and SummerMath (e.g., Schifter & Fosnot, 1993; Schifter, Bastable & Russell, 1999) These projects have produced materials (e.g., case studies for exploring teaching and learning) that have the potential to build pedagogical content knowledge for pre-service teachers before entering the classroom However, research is needed to investigate how these teachers use this knowledge in their practice My study investigated the role of reflection in practice by examining the reflections and experiences of pre-service teachers during student teaching Although Mewborn (1999) focused on pre-service teachers, her research was conducted earlier in the teachers’ educational preparation In my study, I focused on the form of reflective thinking pre-service teachers’ exhibited and how they employed pedagogical content knowledge in mathematics in their thinking The specific research questions MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 37 for pre-service teachers’ mathematics instruction were: (a) When the preservice teachers demonstrate reflective thinking, what forms does it take (i.e., reflection-in- or reflection-on-action)? (b) During reflective thinking, how these teachers use their pedagogical content knowledge? METHODOLOGY I studied two pre-service elementary teachers, pseudonymous Gerri and Denise, during their semester-long student teaching internship I conducted this study from a perspective that combined ideas of interactionism and constructivism in viewing the process of becoming a teacher According to the perspective of interactionism, people construct and sustain meanings through interactions and patterns of conduct (Alasuutari, 1995; Blumer, 1969) This position is in accordance with the constructivist perspective of learning in that individuals develop understandings based on their experiences and knowledge as it is socially constructed (Cobb & Bauersfeld, 1995) This framework supported this study in that the pre-service teachers reflected and constructed meanings based on their participation in, and observation of interactions and patterns of conduct with their students and colleagues Because I was interested in describing and interpreting the thinking and experiences of pre-service teachers during student teaching, I selected a qualitative case study as the most promising mode of inquiry (LeCompte, Millroy & Preissle, 1992; Stake, 1995) The cases were bounded by the semester-long student teaching experience and focused on reflection in regard to their mathematics instruction using incidents of reflecting-in and reflecting-on practice as units of analyses Data Collection Context and Participants Teaching program Gerri and Denise were enrolled in a Master in Teaching program at a state university This two-year master degree program served pre-service teachers who already held a baccalaureate degree in a field other than education but desired to become teachers Two primary objectives of this program were: To educate teachers to become effective practitioners who are informed scholars with the leadership and problem solving skills to help schools and communities meet the needs of the 21st century and to enlighten thought and practice by bringing the inquiry method of a research university to bear on the entire educational process 38 AMY ROTH MCDUFFIE To empower teachers as reflective practitioners by helping them develop the multiple and critical decision making skills essential for today’s classrooms (University program description document) This research-based approach to developing reflective practitioners was evident in the design of the student teaching internship Requirements of the internship included: 12 weeks in a K–8 school placement; writing weekly in reflective journals; setting goals and reflecting on meeting these goals weekly; writing lesson and unit plans; observing and reporting on other teachers’ instruction; and completing a classroom-based action research project on their own teaching Gerri and Denise completed student teaching during the spring semester For the action research project, the pre-service teachers designed their studies during the previous semester as part of a course titled “Classroom Focused Research” Using two texts (Hubbard & Power, 1993; McNiff, Lomax & Whitehead, 1996) as a framework for study, the pre-service teachers studied methods of designing and conducting action research, and planned original classroom-based research projects The action research project focused on a specific teaching strategy or approach Each teacher worked with a faculty committee (a supervisor, with expertise in the selected area for research, and two additional faculty members) The preservice teachers wrote literature reviews in their areas of study as part of a full study proposal These proposals were submitted to their supervisors for feedback and review before submitting a final version at the end of the semester Then they implemented their studies during student teaching In the month following their student teaching internship, Gerri and Denise analyzed their data, wrote, and presented reports of their studies to a faculty committee Given that pedagogical content knowledge is a focus of the study, I shall describe the mathematics methods class that served as Gerri’s and Denise’s primary source of this knowledge prior to student teaching In the first semester of the program, Gerri and Denise completed Elementary Mathematics Methods for which I was the instructor The primary goals of this course aligned with Grossman’s (1990) four components of pedagogical content knowledge This course emphasized developing understandings for: reform-based visions of teaching and learning mathematics (goals in teaching and learning); how children think and learn about mathematics, including common misconceptions in elementary mathematics (knowledge of students); the range of resources and curriculum materials available for mathematics instruction (curricular knowledge); and developmentally appropriate strategies for teaching and learning (knowledge of instructional strategies) MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 39 Van de Walle’s (1998) methods text was used along with supplemental research-based readings Instructional approaches included in-class explorations with textbook activities and extensive use of theoretical frameworks and case studies from Cognitively Guided Instruction (e.g., Carpenter, Fennema, Franke, Levi & Empson, 1999) and Developing Mathematical Ideas (e.g., Schifter, Bastable & Russell, 1999) Course assignments included: interviewing children and analyzing their thinking, understandings, and dispositions for mathematics; writing lesson plans that included an analysis of possible children’s approaches and misconceptions; and collecting and critiquing problem-centered tasks from reformbased publications (e.g., Teaching Children Mathematics and Mathematics in the Teaching in the Middle School) Gerri Gerri held a Bachelor’s degree in engineering and consequently had substantial college-level coursework in mathematics She entered the master’s degree program to begin a career in teaching after staying home with her children for several years While Gerri had volunteered extensively in her children’s schools, she did not have any formal teaching experience prior to entering the program Gerri was regarded by University faculty and her field specialist as having strong content background, especially in mathematics and science Denise Denise was a recent graduate and held a Bachelor’s degree in French with mathematics coursework through first-semester calculus She entered the master’s degree program two years after completing her undergraduate program During those two years, she had worked as an educational assistant at an elementary school During this time, she was exposed to hands-on, student-centered approaches to teaching Denise explained that, as an educational assistant, she had recognized the value and benefits to student learning in using these approaches However, Denise felt that she lacked the theoretical foundations and framework needed to plan effectively and to manage student-centered instruction because she did not have an academic background in education prior to entering the program Researcher I served as a participant observer in that I researched the pre-service teachers’ practice while acting as their university supervisor I also had an established relationship with Gerri and Denise prior to the study as their mathematics methods course instructor Additionally, as their Master’s degree Committee Chair, I advised Gerri and Denise on their action research projects during all phases of their research Prior to the study and heeding the advice of Mewborn (1999) regarding the creation of a non-evaluative environment to promote reflection, I 40 AMY ROTH MCDUFFIE discussed with Gerri and Denise my role as a researcher and as their supervisor I explained that the purpose of my study was to understand how they reflected on their practice I also explained that, as their supervisor, I perceived my primary role to be a resource to them and to provide support during their internship During my observations and conferences with Gerri and Denise, I followed constructivist frameworks for student teaching supervision that emphasized the developing of self-directed, reflective practitioners (cf., Sullivan & Glanz, 2000) Correspondingly, my focus was on facilitating the pre-service teachers’ skills in analyzing their practice For example, every conference began with my asking the preservice teachers for their perceptions of: “What went well?” and “What would you like to change?” I encouraged them to identify their strengths and problem areas; helped them to clarify their thoughts and plans; and asked about their progress with their goals in follow-up meetings As is the case with most pre-service teachers in this program, Gerri and Denise were confident they would pass student teaching (only pass or fail grades were assigned) Therefore, they were more concerned about their professional growth than official evaluations, and Gerri and Denise stated that they also viewed me primarily as a resource Indeed, neither Gerri nor Denise was ever at risk of failing student teaching Throughout the semester, both Gerri and Denise commented that they viewed me as a support and resource rather than an evaluator However, ultimately, as the university supervisor, I was responsible for evaluating their internship, and thus it would be naïve to think that we had an entirely non-evaluative relationship Data Sources The primary data sources were: audio-taped interviews and conferences with the participants; my observations of classroom teaching; reflective journal entries and weekly goal statements; lesson and unit plans; and participants’ data collected as part of their action research projects and their final research reports Nine observations and semi-structured interviews and/or conferences, occurring approximately every week to two weeks, were conducted with each participant throughout the semester I recorded field notes for the observations, and each observation lasted about one hour Each interview lasted about 30 minutes and was transcribed The participants wrote at least one journal entry weekly, and wrote lesson plans daily Data Analysis To address the first question, I analyzed the data by analytic induction: I searched for patterns of similarities and differences for when reflection did MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 41 and did not occur and also for the forms of reflection (Bogdan & Biklen, 1992; LeCompte, Millroy & Preissle, 1992) I classified events as reflective thinking if evidence existed that the pre-service teachers completed all components of the reflective cycle To find evidence, I first observed for problematic events that initiated reflective thinking, and/or I read the preservice teachers’ journals for their reporting of surprising or puzzling events During observations, the pre-service teachers usually demonstrated that something puzzling occurred through facial gestures, pauses in their speech, talking aloud, and/or a lesson that departed from the previously written lesson plan As part of journal writing, the pre-service teachers focused on problematic events and wrote about them directly Next, I investigated whether the pre-service teachers analyzed and developed a possible resolution to the problem I obtained this evidence from questioning and from their journal writing about considerations in making an instructional decision In analyzing the journals, to help sort out whether their reflections were in-action or on-action, the preservice teachers placed an asterisk next to any thoughts or ideas that occurred to them after teaching rather than just writing about their thoughts during the lesson Finally, I examined the data for a resulting action or implementation of a plan in instruction It should be noted that the findings were often supported, at least in part, from the participants’ self-reporting of reflective experiences While I endeavored to validate these reports through triangulation of data sources, the nature of this research did rely on participants’ reports of reflection Consequently, I made efforts to ensure that the participants had operationalized reflective experiences in a manner consistent with my understandings of reflection-in-action and reflection-on-action as presented in this manuscript We discussed these notions thoroughly in the initial interview (with examples and non-examples of reflection), and then summarized these ideas again throughout the semester For the initial coding of data, I used codes of reflection-in-action and reflection-on-action to identify and track the forms of reflection Through the data analysis process, I found a need to distinguish further forms of reflection into subcategories (immediate reflection-in-action, delayed reflection-in-action, short-term reflection-on-action and long-term reflection-on-action), and a new category emerged: deliberate practice Each of these forms of reflection is represented in Figure and described briefly below as they were used in coding and analyzing data They are exemplified further in the presentations of the case findings Immediate Reflection-In-Action (referred to as immediate reflection hereafter) represents the thinking when pre-service teachers made immediate decisions while completely in the act of teaching This form of 42 AMY ROTH MCDUFFIE Figure Processes of deliberate practice thinking was difficult to identify, and thus I relied heavily on focused observations and follow-up interview questions to determine if the pre-service teachers identified a problem during instruction, and then completed the reflective cycle Moreover, I asked the participants directly about occurrences of and their ability to analyze and make decision about problems identified during teaching Delayed Reflection-in-Action (referred to as delayed reflection hereafter) represents the thinking the pre-service teachers exhibited when a pause or break occurred in the act of teaching (e.g., students completing individual work or recess) Similar to immediate reflection, delayed reflection resulted in analysis, a decision, and an action for the lesson in progress or for the plans for the day However, a break in activity distinguished delayed reflection from immediate reflection Immediate reflection and delayed reflection both correspond to Schön’s (1987) description of reflection-in-action, but they differ in the level of instructional activity and demands occurring during reflection In Figure 1, the double arrow between Reflection-In-Action and the Teaching and Learning Episode represents how instruction triggers reflection and consequently, reflection-in-action influence teaching while instruction is taking place With regard to forms of Reflection-On-Action, Short-Term ReflectionOn-Action (referred to as short-term reflection hereafter) was exhibited 48 AMY ROTH MCDUFFIE teaching math, her students would learn to utilize their individual strengths and feel more successful in learning math Her plan was to implement a multiple intelligences approach (e.g., Gardner, 1993) in mathematics instruction in order to facilitate her students’ learning Denise was aware that Gardner’s work has been questioned for its scientific merit in the research community In choosing her approach to action research, she acknowledged limitations of the theoretical basis, but still wanted to see the effects of these approaches in practice Denise was placed in a fourth- and fifth-grade combination class that was team-taught Both team members, Mrs Knight and Mrs Earl, served as mentors to Denise, although Mrs Knight was her official field specialist Mrs Knight and Mrs Earl used predominately student-centered, reform-based approaches and materials They focused on concept development and active, hands-on learning, and they frequently used cooperative learning strategies Mrs Knight was a teacher leader in the region Mrs Knight tended to be both a proactive and reactive mentor Mrs Knight discussed, on a daily basis, possible options and considerations with Denise prior to Denise’s teaching lessons and reviewed events and decisions after lessons Developing Meanings for Decimals: Representing Decimals on Grids The instructional episode The lesson was part of a unit that served as an introduction to decimals Prior to this lesson, the students had discussed everyday uses of decimals, decimal equivalents for common fractions (e.g., 1/2 and 1/4), approximate values of decimals (e.g., 0.49 is almost equal to 0.5 which is the same as 1/2), and why we need both decimal and fractional forms of numbers (Lesson plans for March 10 and March 13) Denise developed these lessons in collaboration with Mrs Knight Denise relied primarily on Mrs Knight’s advice and outside resources (e.g., materials from Investigations [TERC, 1998]) and, indeed, I did not ever see Denise using or referring to the district textbook, Addison-Wesley Mathematics (Eicholz, 1991) For this lesson, Denise’s goal was for students to gain an understanding of “representing decimals on grids; reading, writing, and ordering decimals [using grids]; and adding decimals on a grid” (Lesson plan, March 14) Immediately prior to teaching the lesson, Denise and I had a pre-lesson conference Denise explained her goals to me and said that she thought the decimal grids (see Figure 2) were an effective representation for students to gain an understanding of the value of various decimal numbers Denise had been exposed to decimal grids both in her mathematics methods course and as an educational assistant MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 49 Figure Decimal grids Denise used in March 14th lesson Denise discussed her lesson plan with me and said that she planned to have a whole-group discussion on representing various decimals on the grids She planned to ask them to color one tenth (written as “0.1” on the board) on each of the four grids and then to choose the best grid to show one tenth Next, she wanted students to express the decimal of one tenth in fractional forms that correspond to each grid (e.g., 10/100 for the hundredths grid, 100/1000 for the thousandths grid, etc.) Denise then planned to “Write on the board 0.1 = 0.1000 Ask the students to figure out how they know these decimals are equal Have the students share their answers with their partners” (Lesson plan, March 14) Denise planned to 50 AMY ROTH MCDUFFIE end the lesson with students showing various other numbers on the grids (e.g., 0.75, 0.125) and discuss fractional equivalencies for each After Denise shared her plans, I asked what challenges students might have using the grids She replied: “They have to show one tenth, so it’s easy on this one [showing one column on the tenths grid] But I’m afraid they’re not going to know it’s the same shape on here [the hundredths grid] I’m wondering if they’ll color in one little box [representing one hundredth]” (Pre-lesson conference, March 14) At this point Denise revealed that this lesson was the first time her students had seen or used decimal grids Upon hearing this information, I suggested that she may be incorporating too many ideas in a first experience with grids, and that perhaps she should have the students just work with the tenths grid and become comfortable representing various numbers on this grid before moving on to comparing the grids and the multiple representations for decimals As we ended the conversation, Denise decided that she was not comfortable changing the lesson at this point, but she would consider slowing down the discussion if the students seemed to be having difficulties Denise began the lesson as she had planned it and asked a student to come to the overhead and color in one tenth on the tenths grid The student correctly colored one column Next, another student (Maria) colored in one tenth on the hundredths grid by coloring a row of ten of the one hundredths squares, and then labeled it as “0.01” Denise then explained that this second representation was “still one tenth because it was ten hundredths”, but she did not address Maria’s incorrect labeling of one tenth as “0.01” Pointing at the thousandths grid, Denise said, “100 times 100 is a thousand okay?” Indeed, Denise was describing the ten-thousandths grid while pointing to the thousandths grid She paused, looked back at her notes, and said, “Well this is still one tenth”, as she pointed to a column representing one tenth on the thousandths grid She paused again, and corrected herself, “Wait, 100 times 100 is 10,000, right? But this is still one tenth”, again pointing to one column on the thousandths grid Somewhat rushed in her speech, Denise said, “So all of these are one tenth right?” as she pointed to each of the tenths, hundredths, and thousandths grids Uncharacteristic of her typical approach to teaching, Denise did not call on students and the students were silent throughout this part of the lesson Denise, seeming flustered, omitted what she had planned for developing ideas of which grid is most appropriate for which numbers and fractional equivalents of the decimal numbers (as described in her plans above) and quickly began the last part of her lesson plan Denise asked a student to represent 0.75 on a grid The student colored seven columns on the tenths grid Denise questioned further and asked MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 51 the student, “Now what are you going to with the rest?” The student looked again and shaded half of another column, and he explained, “five [pointing to the five in the hundredths place] is half of ten” Next, Denise had another student show 0.75 on the hundredths grid This student colored the same shape as the previous student but on the hundredths grid, coloring in 75 squares Denise asked the class, “Do you agree?” Without much response, Denise asked the student to explain her work Without shading anything on the thousandths grid, Denise asked, “What you think it will look like on the thousandths grid?” Another student responded, “The same thing” Denise asked this student to show this, and asked again if everyone agreed (again little response from the class) Next, she turned to the tenthousandths grid, and asked a student (Carl) to show 0.75 on this grid Carl shaded seven and a half hundredths squares across the top row (showing 0.075) Again, Denise asked who agreed With no response, Denise asked another student (Mark) to show how he thought this should be represented Mark correctly colored the same pattern that had been shaded on the other grids Denise stated that they were both correct (but did not explain how these two representations could be considered correct), and began a final discussion reminding students how to pronounce the numbers correctly (e.g., seventy-five hundredths) From my observations, it was not at all clear that most students understood how to use the various grids While a few students were able to provide correct responses at the overhead, based on facial expressions and lack of response from other students (not typical for this class), the students working at the overhead seemed to be the only students who were confident in their understandings Moreover, all of the students who contributed volunteered to so Maria’s and Carl’s work indicated that not all students understood these representations and the values of the decimal numbers For this second part of the lesson (starting with representing 0.75), Denise’s teaching style seemed to be more representative of what I had observed of her teaching in that she asked students to show and explain their work, rather than Denise doing most of the talking (as in the first part of the lesson) Yet, unlike her usual practice, Denise did not delve into deeper explanations for their thinking and solutions, and skimmed over the idea that both Carl’s and Mark’s representations “were correct” Immediately following the lesson, Denise and I discussed the lesson As soon as we sat down Denise said, “I wasn’t sure how to use that ten thousandths grid! You can tell! I still don’t know if I was completely correct” I asked her to explain what she was unsure about, and she said, “The representation of the examples” She went on to say that she became 52 AMY ROTH MCDUFFIE confused when the two boys represented 0.75 on the ten-thousandths grid I reminded her, “You said that they were both correct” She replied, “They weren’t!” I went on to explain how they could both be considered correct if the first boy defined the unit one to be a row (what had been one tenth in other representations), but that if we were to assume that the unit consisted of the large square (as was the case, but was never directly stated in the lesson), he was not correct We continued to discuss this idea of deciding and declaring the unit one as a first step in using grids, and Denise indicated that she had remembered this kind of discussion in the methods class Indeed, it seemed as if Denise’s vague recollection of the methods class discussion led her to the idea that they both could be correct, but her memory and understanding was not sufficient to recall how both students could be correct Next we discussed her confusion with her statement that 100 times 100 is 1,000 She immediately said, “It’s not one thousand; it’s ten thousand! It gets confusing!” After more discussion Denise said, “This makes sense to me now But you think this is [only] decimals, why should it be so hard! There’s a lot of stuff here, and I didn’t know that until I got up there I guess” Denise continued to think about these issues, and in a subsequent lesson on March 15 and 16, revisited the topics in this lesson, incorporating the ideas from our discussion in facilitating her students’ understanding of and representations of decimals Forms of thinking This episode illustrated deliberate planning, the difficulty of immediate reflection, and short-term reflection In planning the lesson, Denise had clear goals for students to understand the value of decimals (number sense with decimals) and for using grids as a representation to facilitate this learning She was aware of the possible difficulties that students might encounter in using the grids (e.g., seeing that a number would cover the same area on any of the grids), and had specific questions planned to draw out connections among the decimal number, the grid representation, and the fractional equivalent These plans and ideas were consistent with recommendations from NCTM (2000) and from her methods course and text (Van de Walle, 1998) While teaching the lesson, Denise encountered problems in her own understanding of what the grids represented (i.e., her confusion with the thousandths grid representing 100 times 100) and how to use the grids to represent decimal values (i.e., her confusion with Carl’s and Mark’s different representations for 0.75) While she identified a problem, she did not resolve her confusion until after the lesson was completed She admitted in the post-lesson conference that indeed she did not realize the MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 53 limits of her understanding until she was teaching in front of the class In her confusion, she began a short teacher-led explanation of how to represent one-tenth on each grid, without following her plans for developing these ideas with the class In doing so, Denise essentially moved away from her goals and reform-based approaches of facilitating understanding, towards more traditional approaches of presenting material Later in the lesson, Denise’s inadequate understanding of the representation again appeared in trying to sort through Carl’s and Mark’s two ways to represent 0.75 Because Denise had not developed the idea of identifying the unit (either in her mind or in her students’ minds), while she had some sense that both representations could be considered valid (from a vague recollection from the methods course), she was unable to make sense of this in the act of teaching So, she avoided a conceptual discussion and moved on to other parts of the lesson These moves were consistent with the findings of Borko et al.’s (1992) case of Ms Daniels When Ms Daniels was confronted with her own confusion over understanding division with fractions while teaching, she attempted to use a representation she had learned in her math methods course Misapplying this representation, she realized that she had made an error, but while teaching, Ms Daniels did not fully understand the problem With no resolution to the problem, Ms Daniels resorted to a procedural, rule-based explanation Laying my framework for deliberate practice over Borko et al.’s findings, I would interpret this episode as that, when Ms Daniels was faced with a gap or weakness in PCK, her immediate reflection on the problem led her to follow the pull of tradition Similarly, Denise, when challenged to analyze and make sense of her confusions regarding the grid while teaching, evinced inexperienced teachers’ difficulties in accessing instructional options to resolve a dilemma Without a repertoire of options to consider during immediate reflection, Denise opted for more familiar traditional methods Denise resorted to telling her students, “that they all show one tenth”, and indeed omitted a significant portion of her plans to develop and unpack the ideas behind how they each show one tenth That is, when confronting an instructional dilemma, Denise demonstrated immediate reflection and resolved the problem by moving toward the tradition of telling and abandoning plans to implement more innovative approaches To resolve further this dilemma, Denise later exhibited short-term reflection Starting in our post-lesson conference and continuing in her planning for the two lessons, Denise made sense of the representations in her own mind, planned ways to facilitate her students’ understanding, and implemented these plans Thus, given time to analyze the problem, Denise 54 AMY ROTH MCDUFFIE reflected further and returned to her goals for developing the students’ understandings Use of pedagogical content knowledge Denise demonstrated PCK in each of Grossman’s (1990) four areas; however, her underdeveloped PCK caused the problems identified above and her difficulties with immediate reflection in resolving those problems while teaching First, in establishing her goals for the lesson, Denise demonstrated that she understood purposes and strategies for learning decimals, consistent with reformbased goals (e.g., NCTM, 2000) In planning for the lesson, Denise was aware of and chose materials effectively (i.e., drawing from Investigations and using the decimal grids) for facilitating understanding of decimal values, thus demonstrating curricular knowledge and knowledge of instructional strategies Moreover, in our pre-lesson conference, Denise indicated her understanding of students when she explained her concern for how they might misuse the grids and the importance of connecting decimals, fractions, and the grids in learning Denise’s PCK needed further development in understanding fully what each grid represented (e.g., her error in confusing the thousandths and the ten-thousandths grids), and in understanding the role of identifying the unit as part of understanding the multiple representations Her limited PCK was not apparent to Denise until she began to teach, despite her careful planning Moreover, these limitations impeded Denise in her understanding of the complexity of teaching and learning decimals (c.f., Ball et al., 2001), and correspondingly resulted in a lesson plan that had too many concepts packed into an introductory lesson, without sufficient time to explore and develop each of these issues In sum, these limitations in Denise’s PCK led to some confusion and a dilemma during instruction, and upon immediate reflection, she decided to depart from her plans for developing students’ understandings of how to use the grid representation in favor of more traditional approaches of telling what the grid represented DISCUSSION AND IMPLICATIONS The cases of Gerri and Denise illustrated forms of thinking in deliberate practice and how they used each of Grossman’s (1990) four components of PCK during deliberate practice Below, I discuss and suggest implications for the various forms of reflective thinking as they were exhibited, and how the pre-service teachers’ use of their PCK played a role in this thinking MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 55 Supporting Pre-service Teachers through Collaborative Deliberate Planning Using their PCK Gerri and Denise deliberately planned for many of the possible problems of teaching and learning It is important to note that the examples presented here in which Gerri and Denise demonstrated thoughtful, research-based practice were just a sample of the many instances I found It seemed that the use of research-based resources (e.g., Carpenter et al., 1999; Van de Walle, 1998) in their teacher education program prepared Gerri and Denise to approach instruction as a problem solving endeavor, focusing on facilitating students’ understanding and anticipating problems in teaching and learning However, they still encountered problems in teaching that most probably would have been avoided by more experienced teachers For example, in Denise’s case, while her plans indicated strong PCK for decimals, it was not until she actually taught the lesson that gaps were revealed in her PCK Despite the PCK Gerri and Denise had gained through their professional coursework (as evidenced in their pre- and post-conferencing and planning), their understandings were limited without having used it in practice This is consistent with Borko and Putnam’s (1996) argument for inadequate PCK in pre-service teachers In considering implications from this study, the question then became, “Do we just accept these gaps as part of novice teaching? Or, can teacher educators and field specialists better utilize and develop pre-service teachers’ existing PCK and, correspondingly, enhance their skills in deliberate planning?” As I considered Denise’s episode in particular, I realized that if I had had the pre-lesson conference on decimals with Denise prior to the day of the lesson, I could have better supported her in understanding the complexities and issues in teaching this lesson As it was, while I suggested some ideas for her to consider, she was not in the position to act on them just 15 minutes ahead of teaching Mrs Knight, as an expert mathematics teacher, did serve as this kind of resource at times Indeed, Denise’s decision to use the decimal grids was motivated by a planning discussion with Mrs Knight and access to Mrs Knight’s extensive reform-based resources However, even this discussion did not include in-depth, lesson-specific PCK that Denise might need to bring to this lesson Moreover, it is not clear that it is reasonable for mentor teachers to serve in this role Mentor teachers’ primary focus needs to be on supporting pre-service teachers’ in their learning to the daily work of a teacher, and this may mean that developing pedagogical content knowledge is left in the background Conversely, university supervisors could, more aptly, support preservice teachers’ development of pedagogical content knowledge by using the pre-service teachers’ experiences as the context for learning To 56 AMY ROTH MCDUFFIE support effectively our pre-service teachers’ knowledge development and correspondingly their skills in deliberately planning for instruction, university supervisors need opportunities to discuss in depth the teaching and learning issues for specific lessons to pose various problematic scenarios, both after teaching episodes and well in advance of lessons Through these discussions we might be able to provide opportunities for pre-service teachers to consider possible mathematical teaching and learning problems that might not occur to pre-service teachers due to their lack of experience The model used for supervision in this study that emphasized reflective practice in general (Sullivan & Glanz, 2000), using a pre- and post- conference on the day of the lesson, did not allow sufficiently for subject-area specific support prior to a lesson to build on the pre-service teachers’ deliberate planning The pre-service teachers’ experiences did serve as rich contexts for reflection and making sense of mathematics for teaching and learning after the lessons were over, and these discussions were an important part of the pre-service teachers’ development However, it seems that opportunities for teachers’ learning could be expanded by conducting in-depth conferences to plan lessons well in advance of a lesson The purpose of these conferences could be to examine and explore the mathematics and appropriate pedagogical approaches related to a lesson Correspondingly, in order to enact this recommendation, pre-service teachers need subject-area specific supervisors and/or field specialists that could structure and prompt these conversations Using Pedagogical Content Knowledge during Immediate Reflection-in-Action is Difficult In the instructional episodes, I found substantial evidence for the difficulty of immediate reflection Both Gerri and Denise stated that they wanted to teach lessons as they had planned them, and did not have the confidence or flexibility to change these plans in the middle of the lesson Correspondingly, they found immediate reflection to be difficult and it rarely occurred Indeed, I only found one instance of immediate reflection that resulted in a continued effort to develop students understanding (i.e., continue with plans to implement reform-based goals) Instead, for all other observations, when immediate reflection occurred this reflection resulted in a decision to abandon innovative approaches in favor of more familiar traditional approaches (as illustrated in Denise’s case) Delayed reflection as a specific form of reflection-in-action recognized the pre-service teachers’ need for a brief break in activity to provide the opportunity for reflection during instruction, and short-term reflection allowed the pre-service teachers to examine their practice outside of the act of teaching Gerri’s perspective on her reflective practice illustrated how it was easier for her to find opportuni- MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 57 ties for delayed reflection or short-term reflection than it was for immediate reflection: You almost need a quiet time [to reflect] A big time for me was a half an hour before lunch, we’d have prep time That was a big time to think about, “How did the morning go, and what am I going to in the afternoon? And is there something in the morning that’s going to affect what I was planning to in the afternoon? Then, maybe I had to make a quick shift there It’s hard to on the fly because that crunch of time You don’t get much of a chance to say, “Did that work okay?” You have to have that quiet time (Interview, April 26) Gerri’s and Denise’s reflective practices were consistent with Borko et al.’s (1992) report of the experience of Ms Daniels and with Ball et al.’s (2001) notions regarding the complexities of analyzing a problem and its representations during instruction and the need for confidence in one’s own mathematical understanding enough to adjust instruction “on the fly” (p 438) It is important for teacher educators, pre-service teachers, and their field specialists to consider how pre-service teachers find it difficult to reflect in the form of immediate reflection but have the capacity to reflect through delayed reflection and short-term reflection While this study examined only two cases, it seems that reflecting while in the act of teaching might be difficult for many novice teachers, especially given the limits of PCK for novice teachers (c.f Borko & Putnam, 1996) Teacher educators, preservice teachers, and field specialists might benefit from recognizing that this form of thinking could be difficult for novice teachers by: developing strategies for fostering this form of thinking through pre-teaching conferences (as described above with regard to deliberate planning); and/or recognizing that immediate reflection might be beyond the skills of some novice teachers who are wrestling with heavy cognitive demands during initial teaching experiences In those cases where we recognize that immediate reflection might not be possible, we might instead focus on delayed reflection, and help pre-service teachers to structure their instructional plans so that slight breaks in activity are included to allow time for reflection and adjustments to teaching and learning Additionally, we can facilitate short-term reflection through post-lesson conferences and journal writing, as illustrated in Gerri’s and Denise’s cases Encouraging Long-term Reflection for Sustaining Growth The long-term reflection exhibited by the pre-service teachers seemed to be an important part of their reflective practice for future teaching Through long-term reflection, the pre-service teachers put together the pieces of individual reflective episodes to realize a pattern that informed their practices These patterns over time may be even more important 58 AMY ROTH MCDUFFIE in sustaining professional growth than individual incidents of reflection Certainly, while Gerri learned from the multiples lesson that she should not assume that students’ correct use of terms indicates that they understand underlying concepts, it was more important that she learned that deeper questioning and students’ explanations are needed for learning Gerri’s action research project helped her to make these connections in that her project compelled her to look at her practice and her students’ learning (for teaching multiplication and division) over time, and also, to consider approaches from her research-based reading and her university courses in planning Thus, action research should be considered as a means of encouraging pre-service teachers to: experiment with innovative teaching and learning strategies, consistent with those explored in their university courses; focus on students’ understandings and learning process; go beyond teaching approaches modeled by their field specialist; and thereby, stimulate using PCK for long-term reflection on action In summary, by extending Schön’s (1983, 1987) ideas about reflective practice, I presented a framework for deliberate practice that encourages teachers and teacher educators to focus on preparing for instructional dilemmas as well as reacting to them through use of PCK Through awareness of this framework and the complexity of thinking involved in teaching as a problem-solving and a problem-anticipating endeavor, pre-service teachers can be made aware of their thinking and strive to move away from the technician model of teaching that they might have experienced as students More specifically, by thinking about teaching as a deliberate practice, they might focus better on using PCK to anticipate problems in lesson planning, and not simply deliver a curriculum as it is presented from an external source Consequently, pre-service teachers may be more predisposed to enter the field feeling empowered as decision makers, rather than feeling obliged to follow external guidelines without critical analysis Additionally, teacher educators can help pre-service teachers become aware of the complexities of thinking and problem solving in the practice of teaching Teacher educators can support pre-service teachers by helping them to recognize patterns in their forms of thinking, when and how they draw upon their pedagogical content knowledge, and the consequences for teaching and learning from their thinking and decisions Indeed, if Denise were not being observed on the day she used the grid representation, the consequence of reverting to direct teaching my have become a pattern in her practice Instead, through support and discussion with a teacher educator, she became aware of the need for more careful planning when using representations with students and reform-based pedagogical approaches Teacher educators should focus on more than developing reflective practitioners who react to problematic situations, but on devel- MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 59 oping deliberate practitioners who use theory, research, and experience to anticipate problematic situations Further research is needed to examine the applicability and usefulness of this emerging framework for teachers and pre-service teachers In particular, investigations are needed with more experienced teachers that potentially have a stronger knowledge base in place Additionally, research is needed to determine whether experienced teachers exhibit immediate reflection more as a form of reflection and the corresponding implications for fostering immediate reflection in novice teachers NOTE Eicholz, 1991, Addison Wesley Mathematics REFERENCES Alasuutari, P (1995) Researching culture: Qualitative method and cultural studies Thousand Oaks, CA: Sage Ball, D & Bass, H (2000) Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics In J Boaler (Ed.), Multiple perspectives on mathematics teaching and learning: International perspectives on mathematics education (pp 83–104) London: Ablex Ball, D., Lubienski, S & Mewborn, D (2001) Mathematics In V Richardson (Ed.), Handbook of research on teaching (4th edition) (pp 433–456) Washington, DC: American Educational Research Association Blumer, H (1969) Symbolic interactionism Berkeley, CA: University of California Press Bogdan, R & Biklen, S (1992) Qualitative research for education: An introduction to theory and methods Boston, MA: Allyn and Bacon Borko, H., Eisenhart, M., Brown, C., Underhill, R., Jones, D & Agard, P (1992) Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? 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CA: Sage National Council of Teachers of Mathematics (1989) Curriculum and evaluation standards for school mathematics Reston, VA: Author National Council of Teachers of Mathematics (1991) Professional... of a series of efforts for Gerri to examine the efficacy of MATHEMATICS TEACHING AS A DELIBERATE PRACTICE 47 teaching multiplication and division for understanding as part of her action research... complexity of thinking involved in teaching as a problem-solving and a problem-anticipating endeavor, pre- service teachers can be made aware of their thinking and strive to move away from the technician