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

MATHEMATICS TEACHING AS A DELIBERATE PRACTICE: ANINVESTIGATION OF ELEMENTARY PRE-SERVICE TEACHERS’REFLECTIVE THINKING DURING STUDENT TEACHINGABSTRACT.. In this case study I examine the r

MATHEMATICS TEACHING AS A DELIBERATE PRACTICE: ANINVESTIGATION OF ELEMENTARY PRE-SERVICE TEACHERS’REFLECTIVE THINKING DURING STUDENT TEACHING

ABSTRACT In this case study I examine the reflective practices of two elementary service 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 inves- tigate 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 pedago- gical 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

pre-of the act pre-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 movingaway from a technical model of teaching by prescribed methods to onethat regards it as a, complex, demanding practice Two separate butcompatible perspectives have made substantial contributions as to how weview teaching, and correspondingly, to how we approach teacher educa-tion First, viewing teachers as reflective practitioners has underscoredthe 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 thedevelopment of reflective practitioners (Christensen, 1996) This focus

is consistent with a constructivist perspective for teaching and learningthat is the basis of many teacher education programs (e.g., McIntyre,Byrd & Foxx, 1996) Second, the conceptualizing of pedagogical contentknowledge (Grossman, 1990; Shulman, 1986, 1987) as a unique type

of knowledge for teaching has helped researchers, teachers, and teachereducators gain an understanding of the knowledge base that teachers needfor successful practice

Journal of Mathematics Teacher Education 7: 33–61, 2004.

© 2004 Kluwer Academic Publishers Printed in the Netherlands.

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 investi-gate how teachers think about their practice The purpose of my study was

to intersect the constructs of reflective practice and pedagogical contentknowledge (PCK) in order to examine how pre-service teachers use theirmathematical PCK in thinking about their practice, both in planningand classroom teaching I investigated the reflective practices in mathe-matics of two pre-service elementary teachers during their student teachinginternship

THEORETICAL FRAMEWORK

I shall provide a summary of reflective thinking and pedagogical contentknowledge as I applied and merged these two concepts in my work withthe pre-service elementary teachers

Reflective Thinking

While a range of interpretations exists for what is considered to bereflective practice, I referred primarily to the ideas of Dewey (1910) andSchön (1983, 1987) Reflection is a practice that has gained considerableattention in the past two decades, yet Dewey began this discussion early

in the last century Dewey argued that reflective thinking begins whenteachers 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 ofteaching

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 problematicevents and actions, analyzing them, and making decisions Russell andMunby (1991) explained that reflection-on-action “refers to the ordered,

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 wereconsidered in this study

Mewborn (1999) studied the reflective practices of pre-service teachersduring field experience as part of their mathematics methods course.Mewborn examined the elements of mathematics teaching and learningthat 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 evidentuntil they internalized their own authority to generate, reason about, andtest hypotheses in order to examine children’s mathematical thinking.Mewborn argued that to facilitate reflection pre-service teachers need anon-evaluative atmosphere and relationship with cooperating teachers anduniversity 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 puzzlingevent triggering reflection and, thus, I sought to investigate reflection thatarose from problematic situations While it could be argued that reflec-tion might occur during or after a successful lesson (a non-problematicsituation), I chose to adopt the constructs for reflective thinking ofDewey (1910) and Schön (1987) by focusing on pre-service teachers’thinking that was inspired by identified problems I synthesized the variousviews and definitions of reflection considered for this study as a cycle

in teaching practice In this cycle, the problematic event initiates theprocess of reflection Following this problematizing, either during instruc-tion (for reflection-in-action) or after instruction (for reflection-on-action),the teacher analyzes the problem and the options and/or approaches forresolving the problem Next the teacher decides on a resolution or planfor 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 reflectivecycle continues As Dewey (1910) contended, the cycle does not neces-sarily 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

knowledge base to guide their thinking about children’s ways of standing mathematical concepts and process (e.g., Ball, Lubienski &Mewborn, 2001; Borko & Putnam, 1996) Indeed, pedagogical contentknowledge is an important resource for teachers to use as they reflect inpractice Shulman (1986, 1987) first defined pedagogical content knowl-edge as,

under-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 centralcomponents 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 edge of instructional strategies and representations for teaching particulartopics Grossman acknowledges that “these components are less distinct

knowl-in practice than knowl-in theory” (p 9), but this general framework was useful

in thinking about the ways in which the pre-service teachers in this studyused their pedagogical content knowledge

Researchers have found that novice teachers tend to have inadequate orunderdeveloped mathematical pedagogical content knowledge for use inpractice (e.g., Borko & Putnam, 1996; Borko, Eisenhart, Brown, Under-hill, Jones & Agard, 1992) In elementary mathematics education, severalprojects have as a goal the development of teachers’ understanding aboutchildren’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 haveproduced materials (e.g., case studies for exploring teaching and learn-ing) that have the potential to build pedagogical content knowledge forpre-service teachers before entering the classroom However, research

is needed to investigate how these teachers use this knowledge in theirpractice

My study investigated the role of reflection in practice by examiningthe reflections and experiences of pre-service teachers during studentteaching Although Mewborn (1999) focused on pre-service teachers, herresearch was conducted earlier in the teachers’ educational preparation

In my study, I focused on the form of reflective thinking pre-serviceteachers’ exhibited and how they employed pedagogical content knowl-edge in mathematics in their thinking The specific research questions

for service teachers’ mathematics instruction were: (a) When the service teachers demonstrate reflective thinking, what forms does it take(i.e., reflection-in- or reflection-on-action)? (b) During reflective thinking,how do these teachers use their pedagogical content knowledge?

pre-METHODOLOGY

I studied two pre-service elementary teachers, pseudonymous Gerri andDenise, during their semester-long student teaching internship I conductedthis study from a perspective that combined ideas of interactionism andconstructivism in viewing the process of becoming a teacher According tothe perspective of interactionism, people construct and sustain meaningsthrough interactions and patterns of conduct (Alasuutari, 1995; Blumer,1969) This position is in accordance with the constructivist perspective oflearning in that individuals develop understandings based on their exper-iences and knowledge as it is socially constructed (Cobb & Bauersfeld,1995) This framework supported this study in that the pre-service teachersreflected and constructed meanings based on their participation in, andobservation of interactions and patterns of conduct with their students andcolleagues Because I was interested in describing and interpreting thethinking and experiences of pre-service teachers during student teaching, Iselected a qualitative case study as the most promising mode of inquiry(LeCompte, Millroy & Preissle, 1992; Stake, 1995) The cases werebounded by the semester-long student teaching experience and focused

on reflection in regard to their mathematics instruction using incidents ofreflecting-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 programserved pre-service teachers who already held a baccalaureate degree in

a field other than education but desired to become teachers Two primaryobjectives of this program were:

1 To educate teachers to become effective practitioners who areinformed scholars with the leadership and problem solving skills tohelp schools and communities meet the needs of the 21st century and

to enlighten thought and practice by bringing the inquiry method of aresearch university to bear on the entire educational process

2 To empower teachers as reflective practitioners by helping themdevelop the multiple and critical decision making skills essential fortoday’s classrooms (University program description document)This research-based approach to developing reflective practitioners wasevident in the design of the student teaching internship Requirements

of the internship included: 12 weeks in a K–8 school placement; writingweekly in reflective journals; setting goals and reflecting on meeting thesegoals weekly; writing lesson and unit plans; observing and reporting

on other teachers’ instruction; and completing a classroom-based actionresearch project on their own teaching Gerri and Denise completed studentteaching during the spring semester

For the action research project, the pre-service teachers designed theirstudies during the previous semester as part of a course titled “ClassroomFocused Research” Using two texts (Hubbard & Power, 1993; McNiff,Lomax & Whitehead, 1996) as a framework for study, the pre-serviceteachers studied methods of designing and conducting action research, andplanned original classroom-based research projects The action researchproject focused on a specific teaching strategy or approach Each teacherworked with a faculty committee (a supervisor, with expertise in theselected area for research, and two additional faculty members) The pre-service teachers wrote literature reviews in their areas of study as part of

a full study proposal These proposals were submitted to their supervisorsfor feedback and review before submitting a final version at the end of thesemester Then they implemented their studies during student teaching Inthe month following their student teaching internship, Gerri and Deniseanalyzed their data, wrote, and presented reports of their studies to afaculty committee

Given that pedagogical content knowledge is a focus of the study, I shalldescribe the mathematics methods class that served as Gerri’s and Denise’sprimary source of this knowledge prior to student teaching In the firstsemester of the program, Gerri and Denise completed Elementary Mathe-matics Methods for which I was the instructor The primary goals of thiscourse aligned with Grossman’s (1990) four components of pedagogicalcontent knowledge This course emphasized developing understandingsfor: reform-based visions of teaching and learning mathematics (goals

in teaching and learning); how children think and learn about matics, including common misconceptions in elementary mathematics(knowledge of students); the range of resources and curriculum materialsavailable for mathematics instruction (curricular knowledge); and devel-opmentally appropriate strategies for teaching and learning (knowledge ofinstructional strategies)

mathe-Van de Walle’s (1998) methods text was used along with mental research-based readings Instructional approaches included in-classexplorations with textbook activities and extensive use of theoretical

supple-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) Courseassignments included: interviewing children and analyzing their thinking,understandings, and dispositions for mathematics; writing lesson plans thatincluded an analysis of possible children’s approaches and misconcep-tions; and collecting and critiquing problem-centered tasks from reform-

based 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 themaster’s degree program to begin a career in teaching after staying homewith her children for several years While Gerri had volunteered exten-sively in her children’s schools, she did not have any formal teachingexperience prior to entering the program Gerri was regarded by Univer-sity 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 Sheentered the master’s degree program two years after completing her under-graduate program During those two years, she had worked as an educa-tional assistant at an elementary school During this time, she was exposed

to hands-on, student-centered approaches to teaching Denise explainedthat, as an educational assistant, she had recognized the value and bene-fits to student learning in using these approaches However, Denise feltthat she lacked the theoretical foundations and framework needed to planeffectively and to manage student-centered instruction because she did nothave 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 thestudy as their mathematics methods course instructor Additionally, as theirMaster’s degree Committee Chair, I advised Gerri and Denise on theiraction research projects during all phases of their research

Prior to the study and heeding the advice of Mewborn (1999) regardingthe creation of a non-evaluative environment to promote reflection, I

discussed with Gerri and Denise my role as a researcher and as theirsupervisor I explained that the purpose of my study was to understandhow they reflected on their practice I also explained that, as their super-visor, I perceived my primary role to be a resource to them and to providesupport during their internship During my observations and conferenceswith Gerri and Denise, I followed constructivist frameworks for studentteaching supervision that emphasized the developing of self-directed,reflective practitioners (cf., Sullivan & Glanz, 2000) Correspondingly, myfocus was on facilitating the pre-service teachers’ skills in analyzing theirpractice For example, every conference began with my asking the pre-service teachers for their perceptions of: “What went well?” and “Whatwould you like to change?” I encouraged them to identify their strengthsand problem areas; helped them to clarify their thoughts and plans; andasked about their progress with their goals in follow-up meetings.

As is the case with most pre-service teachers in this program, Gerriand Denise were confident they would pass student teaching (only pass

or fail grades were assigned) Therefore, they were more concerned abouttheir professional growth than official evaluations, and Gerri and Denisestated that they also viewed me primarily as a resource Indeed, neitherGerri nor Denise was ever at risk of failing student teaching Throughoutthe semester, both Gerri and Denise commented that they viewed me as

a support and resource rather than an evaluator However, ultimately, asthe university supervisor, I was responsible for evaluating their internship,and thus it would be nạve to think that we had an entirely non-evaluativerelationship

Data Sources

The primary data sources were: audio-taped interviews and conferenceswith the participants; my observations of classroom teaching; reflectivejournal entries and weekly goal statements; lesson and unit plans; andparticipants’ data collected as part of their action research projects andtheir final research reports Nine observations and semi-structured inter-views and/or conferences, occurring approximately every week to twoweeks, were conducted with each participant throughout the semester Irecorded field notes for the observations, and each observation lasted aboutone hour Each interview lasted about 30 minutes and was transcribed Theparticipants wrote at least one journal entry weekly, and wrote lesson plansdaily

Data Analysis

To address the first question, I analyzed the data by analytic induction: Isearched for patterns of similarities and differences for when reflection did

and did not occur and also for the forms of reflection (Bogdan & Biklen,1992; LeCompte, Millroy & Preissle, 1992) I classified events as reflectivethinking if evidence existed that the pre-service teachers completed allcomponents of the reflective cycle To find evidence, I first observed forproblematic events that initiated reflective thinking, and/or I read the pre-service teachers’ journals for their reporting of surprising or puzzlingevents During observations, the pre-service teachers usually demonstratedthat something puzzling occurred through facial gestures, pauses in theirspeech, talking aloud, and/or a lesson that departed from the previouslywritten lesson plan As part of journal writing, the pre-service teachersfocused on problematic events and wrote about them directly.

Next, I investigated whether the pre-service teachers analyzed anddeveloped a possible resolution to the problem I obtained this evidencefrom questioning and from their journal writing about considerations

in making an instructional decision In analyzing the journals, to helpsort out whether their reflections were in-action or on-action, the pre-service teachers placed an asterisk next to any thoughts or ideas thatoccurred to them after teaching rather than just writing about their thoughtsduring the lesson Finally, I examined the data for a resulting action orimplementation of a plan in instruction

It should be noted that the findings were often supported, at least inpart, from the participants’ self-reporting of reflective experiences While Iendeavored 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 operation-alized reflective experiences in a manner consistent with my understand-ings of reflection-in-action and reflection-on-action as presented in thismanuscript We discussed these notions thoroughly in the initial interview(with examples and non-examples of reflection), and then summarizedthese ideas again throughout the semester

For the initial coding of data, I used codes of reflection-in-actionand reflection-on-action to identify and track the forms of reflection.Through the data analysis process, I found a need to distinguish furtherforms 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 1 and describedbriefly below as they were used in coding and analyzing data They areexemplified 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 diate decisions while completely in the act of teaching This form of

imme-Figure 1. Processes of deliberate practice.

thinking was difficult to identify, and thus I relied heavily on focusedobservations and follow-up interview questions to determine if thepre-service teachers identified a problem during instruction, and thencompleted the reflective cycle Moreover, I asked the participants directlyabout occurrences of and their ability to analyze and make decisionabout 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 theact of teaching (e.g., students completing individual work or recess).Similar to immediate reflection, delayed reflection resulted in analysis, adecision, and an action for the lesson in progress or for the plans for theday However, a break in activity distinguished delayed reflection fromimmediate reflection Immediate reflection and delayed reflection bothcorrespond to Schön’s (1987) description of reflection-in-action, but theydiffer in the level of instructional activity and demands occurring duringreflection In Figure 1, the double arrow between Reflection-In-Action andthe Teaching and Learning Episode represents how instruction triggersreflection and consequently, reflection-in-action influence teaching whileinstruction is taking place

With regard to forms of On-Action, Short-Term

Reflection-On-Action (referred to as short-term reflection hereafter) was exhibited

when the pre-service teachers thought back over a short period of time after

a lesson or day was over (e.g., reflecting on a lesson as they drive home

or on the week’s instruction over the weekend) Short-term reflection isdifferent from delayed reflection in that the pre-service teachers were notunder pressure to reflect, resolve, and implement the action to address aproblem immediately during the problematic lesson or day This reflec-tion was often about the success of a lesson in contributing to learninggoals in order to guide planning for the next lesson or unit Long-Term

Reflection-on-Action (referred to as long-term reflection hereafter) was

exhibited when the teachers systematically analyzed and examined theirpractice over an extended period of time for the purposes of understandingand improving practice more globally They looked for emerging patternsand developed personal theories about teaching and learning Most often,the reflective cycle for long-term reflection took place over several months.Deliberate Planning was exhibited when pre-service teachers purpose-fully used existing knowledge, theories, and reasoning about teaching andlearning to design plans for particular students’ learning While reflections

on past experiences may be part of an existing knowledge or theory base,this form is different from reflection Reflective thinking was initiated by

a problematic event Conversely, deliberate planning involved analysis of

instructional options prior to teaching, often preparing to avoid anticipated

problematic events All five forms of thinking compose a framework forthe deliberate practitioner

After establishing this framework for deliberate practice, I used codes

of immediate reflection, delayed reflection, short-term reflection, term reflection, and deliberate planning to classify the forms of thinking,and correspondingly, revised my research questions to consider thinking inthese five forms (versus Schön’s [1987] two forms) Additionally, for moreprecise coding, I used qualitative data analysis software with a combina-tion of open coding (Strauss & Corbin, 1990) and indexing of text (Miles

long-& Huberman, 1994) Once I identified and classified reflective and/ordeliberate events (the first research question), I analyzed these events forhow pedagogical content knowledge was used (the second research ques-tion) That is, I examined how these teachers applied pedagogical contentknowledge (as described in Grossman’s [1990] four component model) inplanning for or reflective thinking about a problematic event Throughoutthis process, the multiple data sources were compared to generate acomplete picture of each event and to confirm or refute emerging patterns

Gerri’s Philosophy and Field Placement

Gerri’s philosophy of teaching seemed to be focused on ensuring thateach student was learning and was forming a conceptual understanding

of the subject areas In an interview prior to beginning student teaching,Gerri made it clear that her foremost interest was in student learning Shewas distressed that current educational environments present barriers tofocusing on students’ understanding of concepts In referring to a fieldexperience prior to the student teaching semester, Gerri stated,

I have seen that there is a real expectation to get a lot of information to the kids

In the practicum that I just completed, I saw the kids pushing forward in the curriculum without really having a good basis of what we were trying to teach Kids don’t really have the concepts (Interview, December 30).

Gerri’s concern for students gaining strong conceptual understandingwas evident in her choice for her classroom-based action research project.She investigated the efficacy of her teaching practices in developing herstudents’ conceptual understanding of multiplication and division

Gerri completed her internship in a third-grade classroom Whileher field specialist, Mrs Baker, was well respected in the school andthe community, she was not perceived as a reform-based teacher Mrs.Baker had a predominately teacher-centered style and focused primarily

on skill mastery Both Gerri and I observed this more traditional style.Gerri described Mrs Baker’s teaching as, “Based on my observations,[Mrs Baker has] a pretty traditional classroom They [the students] areencouraged not to visit with their neighbors, to stay on task” (Interview,December 30) With regard to teaching mathematics, and just before takingover the class, Gerri said, “The students are learning their multiplicationfacts [Mrs Baker] is a traditional teacher, relying on the text1to teachmath Her students have not really worked in small groups or with manipu-latives, so I can expect some problems when I first start teaching math”(Journal, January 3)

Consequently, Gerri understood that Mrs Baker was not likely to

be able to support her in her efforts to implement more reform-basedapproaches to teaching mathematics Additionally, Gerri’s journals and myobservations indicated that Mrs Baker tended to be a reactive mentor,offering suggestions to solve existing problems or dilemmas, but notconsulting with Gerri prior to teaching a lesson

Making Connections: Skip Counting Patterns and Multiplication

The instructional episode Early in Gerri’s student teaching experience,

she taught a lesson on multiples While the students had not necessarilyacquired all of their multiplication facts, Mrs Baker had provided instruc-tion on all of the basic facts for multiplication Gerri’s goals for this lessonwere: “Students will find the multiples of one digit numbers using theirknowledge of 0–9 multiplication facts Students will understand what ismeant by the term multiple in mathematics” (Lesson plan, January 21).The lesson began with Gerri writing, “0 2 4 6 8 10 12” on the board,and asked, “What is the pattern for these numbers?” A student responded,

“Multiples of 2” After another example with fives, Gerri wrote, “0 3

6 9 12 15” on the board, and again asked, “What’s the pattern here?”Another student responded, “Goes up by three” After showing a fewmore examples with similar questioning, Gerri handed out a worksheetwith similar exercises on it (e.g., “Find the first 5 multiples of 6”) Up

to this point, I observed that Gerri had followed her lesson plan exactly.Gerri then went back to her desk, where four students promptly lined upfor individual help (a practice established by Mrs Baker during individualseatwork time) The students did not know where to start on the questionswhen a pattern had not been presented (as the example listed above) As

Gerri worked with individual students, she asked the students how they

found the next number in a pattern When the responses relied on adding

to the last number, she realized that while the students could complete thepattern, they did not relate this pattern to multiplication

Indeed, from my observation many of the students seemed to bring only

a recursive interpretation to the pattern, observing that the next numberincreased by a fixed amount from the last number When one studentsaid, “Multiples of 2”, it was not at all clear that the student linked theword “multiples” to multiplication Perhaps she just used “multiples” as

a word that preceded the quantity of increase in the pattern Gerri did notemphasize this connection, and accepted responses such as “goes up bythree”, without asking for further explanation or exploration Later duringseatwork, Gerri recognized this lack of connection in the concepts, andidentified a problem in meeting her stated goal of developing an under-

standing for the term multiple She explained in an interview immediatelyfollowing the lesson,

We started with the patterns, and they caught right on They were skip counting But then,

I didn’t make the connection very strongly that, besides skip counting, you could also do the multiplication facts I felt like I kind of missed that And when they came back [to my desk] and I was working one-on-one [with students], I realized that I felt like I could have made that [connection] stronger (Interview, January 21)

After realizing the problem, Gerri stopped the students during theirindividual seatwork and began a whole class discussion at the board usingproblems on the worksheet to build the connection she recognized wasmissing

Forms of thinking This episode illustrates aspects of deliberate

plan-ning, delayed reflection during the lesson, and long-term reflection Whilepresenting the patterns, Gerri deliberately planned for the students to

understand the term multiple through patterns However, she did notconsider that simply being able to provide a correct response to the ques-tion as posed (i.e., “What’s the pattern?”) was not sufficient to determine

if her goal of understanding the term multiple had been met Once sheworked with students individually, she began to probe their thinking further

by asking, “How did you find the pattern?” and through this deeperquestioning, she was able to assess that they, at best, had a limited under-standing of the term multiple In not planning to question the students onhow they arrived at their solutions (showing limited deliberate planning),

an approach emphasized in reform-based approaches to teaching (e.g.,NCTM, 2000), she missed an opportunity of on-going assessment and achance to encourage the students to build their understandings However,once a break in the lesson occurred, and Gerri worked with students indi-vidually (without the demands of facilitating a whole class discussion),she had time to consider and assess individual students’ approaches andidentify a problem in instruction: the students were not understanding

“multiple” as referring to multiplication of numbers She analyzed thesituation and implemented a different approach in instruction, emphasizingthe connection between the pattern of products and each product’s corre-sponding multiplication facts Given that this all transpired during a lesson,

it represented delayed reflection Because the problem identification andreflective thinking did not occur until Gerri had a break in action, I did notclassify it as immediate reflection

Long-term reflection also was evidenced in Gerri’s conclusions aboutquestioning and seeking explanations from students This lesson repre-sented one of a series of efforts for Gerri to examine the efficacy of

teaching multiplication and division for understanding as part of heraction research project Based on earlier field experience prior to studentteaching, Gerri identified a problem of students having only procedural orfact-based knowledge of these operations Reflective thinking was exhib-ited as Gerri planned her study on a global level, and then planned eachlesson as part of implementing her study on a daily level As Gerri imple-mented her plans to facilitate her students’ conceptual development ofmultiplication and division, she completed the action of the reflectivethinking cycle.

Use of pedagogical content knowledge In planning the lesson Gerri knew that it is important for students to understand the meaning of multiple

and to look at patterns in different ways (multiples as a pattern and asproducts), consistent with reform goals (e.g., NCTM, 2000) and repre-senting Grossman’s (1990) notion of PCK for goals in teaching andlearning Indeed, once Gerri realized that her goal of understanding had notbeen met, she deemed that this idea was important enough that she shouldadjust the lesson, and return to a whole class discussion to emphasize theconnections Although use of PCK was evident in planning the lesson,Gerri had not yet developed the capacity to apply this knowledge whenprobing the students’ thinking during a whole class discussion Yet, duringindividual instruction (while other students were working at their seats),Gerri demonstrated that she did have PCK for questioning strategies andfor understanding students’ thinking (Grossman, 1990) by recognizing theproblem (students’ limited understanding of the word multiple) Thesefinding are consistent with Corwin (1996) that emphasized the complex-ities involved in listening and questioning students to support mathematicslearning in whole class instruction In sum, Gerri more effectively appliedher PCK when she was not orchestrating the lesson: during planning,during a break in teaching, and after the lesson

Denise

Denise’s Philosophy and Field Placement

Like Gerri, Denise’s fundamental philosophy for teaching and learningwas to focus on students’ understanding in learning This focus was evident

in that her planning and self-analysis consistently relied on students’conceptual learning as a referent over concerns for covering textbook orexternal curricular guidelines, as will be illustrated in the episode below.Moreover, Denise’s action research project was developed around herconcern for students’ attitudes and anxiety as factors in learning math.Denise theorized that by using approaches from multiple intelligences in