a structure to enable preservice teachers of mathematics to reflect on their teaching

ALICE F. ARTZT

A STRUCTURE TO ENABLE PRESERVICE TEACHERS OFMATHEMATICS TO REFLECT ON THEIR TEACHING

ABSTRACT. This article presents a conceptual framework for studying the relation-ship between cognition and instructional practices of preservice secondary mathematicsteachers. It describes how the framework was used as a basis for activities in which pre-service teachers engaged in structured reflection on their teaching as a means towards theirprofessional growth. The approach required student teachers to engage in both prelessonand postlesson reflective activities. These activities are described, and details of two casesare given. This article demonstrates how this approach can facilitate the progression ofpreservice teachers’ pedagogical techniques and conceptions.

In his discussion of the current state and future directions of researchon teacher education, Cooney (1994) acknowledged the challenge ofpreparing future mathematics teachers. He raised the question of whattypes of experiences preservice teachers would need in order to becomeeffective teachers of mathematics. Research suggests that teacher reflectionis central for the improvement of mathematics teaching (Jaworski, 1994;Kemmis, 1985; Schön, 1983). Furthermore, within the last two decades,researchers have emphasized the importance of teacher cognition as theyhave recognized a well-defined link between teachers’ cognition and theirinstructional practices (Artzt & Armour-Thomas, 1998; Brown & Baird,1993; Ernest, 1988; Lappan & Theule-Lubienski, 1994; Shavelson, 1986;Shulman, 1986). In addition, projects that have resulted in change in bothteachers’ cognition and instructional practice have placed emphasis on theteachers’ experiences as a focus for reflection (Cooney & Shealy, 1997).Such research suggests that teacher education programs should includestrategies and activities that engage teachers in reflection on their ownthinking and on their instructional practice.

Researchers who have described changes in mathematics teachers’beliefs and practices (e.g., Cooney, Shealy, & Arvold, 1998; Fennema,Carpenter, Franke, Levi, Jacobs, & Empson, 1996; Schifter & Simon,1992; Schram, Wilcox, Lappan, & Lanier, 1989; Thompson, 1991) agreethat there may be several developmental stages of teaching. For example,the initial stage can be characterized by traditional instruction: The teacheris driven by the belief that students learn best by receiving clear infor-

Journal of Mathematics Teacher Education2: 143–166, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

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mation transmitted by a knowledgeable teacher. Subsequent stages ofinstruction are characterized by instruction that is more focused on helpingstudents build on what they understand and less focused on helping themsolely in the acquisition of facts. The instruction is grounded in theteacher’s belief that students should take greater responsibility for theirown learning. The final stage can be characterized by instruction in whichthe teacher arranges activities that involve both thehows and whys ofmathematical concepts and processes. The teacher is motivated by thebelief that, given appropriate settings, students are capable of constructingdeep and full mathematical understanding.

Goldsmith and Schifter (1997) suggested that motivational and indi-vidual dispositional factors may significantly affect the course of ateacher’s development. It is possible to examine the dispositions andmotivations of preservice teachers through writing assignments. Goldsmithand Schifter supported the use of writing, in that it is one possible socio-cultural transition mechanism that has the potential to enable the growthfrom one stage of teaching to the next. Writing, they suggested, helps one“to hold an idea or experience still for reflection” (p. 43). This form ofwriting played a significant role in the project presented in this article.

This article describes a model of structured reflection intended topromote reflective behavior of preservice secondary school mathematicsteachers. As part of this process the preservice teachers had to write aboutspecific elements of their thoughts and their practices and about the rela-tionship between both. I examine how this process of reflection served as ameans for facilitating their development from one stage of teaching to thenext.

RELATIONSHIP OF TEACHER COGNITION ANDINSTRUCTIONAL PRACTICE

Recent research that uses a conception of teaching as problem solving hasbegun to indicate that teachers’ cognition drives their instructional practice(Artzt & Armour-Thomas, 1993; Carpenter, 1989; Fennema, Carpenter, &Peterson, 1989). The method presented here for studying teacher cognitionis based on a framework that was developed by Artzt and Armour-Thomas(1996). This framework represents one way to view teaching as an inte-grated whole in which cognition plays a well-defined role in instruction(see Figure 1).

Specifically, the framework suggests that teachers’ knowledge, beliefs,and goals directly impact their instructional practice. The different factorsaffect the nature and quality of teachers’ thoughts and actions before,

A STRUCTURE FOR TEACHER REFLECTION 145

Figure 1. Framework for reflection on teacher cognition and instructional practice.

during, and after their lessons. They affect the way teachers design alesson, the way they monitor and regulate their instruction during theteaching process, and the way they analyze the lesson after it has beenconcluded. The framework served as a conceptual basis for the work withthe preservice teachers. A more detailed description of the frameworkfollows.

The framework presented in Figure 1 outlines the relationship betweenteachers’ mental activities and their lessons. Teachers’ knowledge, beliefs,and goals appear to drive their instructional practice.Teacher knowl-edge includes knowledge of subject matter, knowledge of pedagogicalstrategies, and knowledge of the students (Shulman, 1986). These compo-nents of teacher knowledge can affect instructional practice and studentlearning (Ball, 1991; Fennema & Franke, 1992; Peterson, 1988).Beliefsrefers to teachers’ integrated system of personalized assumptions regardingthe nature of mathematics, of students, and of ways of learning andteaching. Syntheses of the existing literature on beliefs by Ernest (1988),Kagan (1992), Pajares (1992) and Thompson (1992) suggest that teachers’

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beliefs influence their instructional practice.Goals refers to teachers’expectations about the intellectual, social, and emotional outcomes forstudents as a result of their classroom experiences. TheStandards(NCTM,1989, 1991) have set forth goals for all students which are expected to bereflected in instructional practice: that students value mathematics, becomeconfident in their ability to do mathematics, become mathematical problemsolvers, learn to communicate mathematically, and learn to reason math-ematically (NCTM, 1989, p. 5). The goal of teaching for conceptual as wellas procedural knowledge has been addressed by Hiebert (1986), and Silver(1986), as well as in recent reform initiatives (MSEB & NRC, 1991).

Jackson’s (1968) distinctions of preactive, interactive, and postactivestages of teaching are useful for examining teacher cognition before,during and after teaching a lesson. Clark and Peterson (1986) andShavelson and Stern (1981) have done comprehensive reviews of researchon teacher thought processes. The major components that appear to impactinstructional practice are (a) planning during the preactive stage (Clark& Elmore, 1981; Clark & Yinger, 1979), (b) monitoring and regulatingduring the interactive stage (Clark & Peterson, 1981; Fogarty, Wang, &Creek, 1983), and (c) evaluating and revising during the postactive stage(Ross, 1989; Simmons, Sparks, Starko, Pasch, Colton, & Grinberg, 1989).As Shavelson (1986) pointed out, however, these aspects of thinking arenot conceptually distinct, but rather interconnected components of theteaching process.

Findings from these studies suggest that the cognitive components ofteaching play a critical role in shaping a teacher’s instructional prac-tice. There are many lenses through which instructional practice can bestudied. The perspective described below was developed by Artzt andArmour-Thomas (1996) and was used to examine the instructional prac-tice of secondary school mathematics teachers. It is grounded in theaforementioned literature.

A STRUCTURE FOR REFLECTION

The framework provided the conceptual basis for the approaches used withpreservice teachers during their entire year-long program of preparatorystudy. That is, based on the fact that preservice teachers have observedand participated in the teaching and learning process as students for mostof their lives, the program built on the teachers’ existing knowledge andbeliefs. The methods course focused largely on these existing systemsof knowledge and attitudes as well as on the current goals of secondarymathematics instruction as outlined in the Standards (NCTM, 1989, 1991).


This focus served as a basis for the teachers’ reflective activities duringthe following semester of student teaching. During the student teaching,students were required to teach at least one class per day in a secondaryschool. In addition to being supervised by cooperating teachers in thefield who worked with them on a daily basis, the student teachers wereobserved by their college supervisors four times during the course of thesemester. The supervisors analyzed and assessed the student teachers usinga structure based on the framework. Prelesson and postlesson reflectiveactivities were used to facilitate preservice teachers’ analysis of theircognition and instructional practice before, during, and after their lessons.Specifically, before teaching their lessons, in addition to writing lessonplans, the student teachers were required to submit a paper in which theydescribed their prelesson thoughts and concerns. After the lesson, thestudent teachers engaged in a conference with the supervisor and cooper-ating teacher in which the student teacher gave his or her impressions andanalysis of the lesson. During the latter part of the conference, the cooper-ating teacher and supervisor shared their impressions of the lesson andgave suggestions for further thought and improvement. After the confer-ence, the student teachers were required to write a paper describing theirpostlesson thoughts. This paper was submitted to the supervisor at the nextclass meeting. In addition, student teachers documented their thoughtsand experiences through weekly entries in their journals. Each of theseactivities was structured in a way that was consistent with the conceptualframework previously described. A detailed description of these reflectiveactivities follows.

Prelesson Reflections

In order to help student teachers formulate their broad goals and activatetheir knowledge they were required to provide a written account of theirprelesson thoughts. This reflective writing assignment was in addition tothe actual lesson plan. The following instructions served as a guide for theprelesson reflection:

As you begin to think about constructing your lesson plan, write down all of your concernsand the steps you are taking to account for these concerns. Use the list below as a guide:

1. Goals for students2. Knowledge of students (e.g., ability levels, interests)3. Knowledge of content (e.g., its place in the curriculum)4. Knowledge of pedagogy (e.g., alternate ways of teaching the lesson)5. The teacher’s role in the lesson6. The students’ role in the lesson7. Anticipated difficulties8. Sources used to get ideas and criteria for selection

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The purpose of this list was to make explicit to the student teacher andthe supervisor the knowledge, beliefs, and goals that drive the design andimplementation of a lesson. The hope was that the suggested points of theprelesson reflection could help the student teachers advance more quicklyfrom the initial stage of teaching with content as the main focus to a morestudent-focused stage. In the first item on the list, the student teachers wereasked to consider their goals in terms of students only. Because my experi-ence suggests that beginning teachers’ overriding concern is often contentcoverage, my intention here was to convey the message that contentcoverage has value only in relation to student understanding. The nextthree items on the list (2–4) were designed to encourage student teachersto activate the knowledge they had about the students, the content, andpedagogical strategies as a means for informing their lesson design. Thenext three items on the list (5–7) were meant to help student teachers envi-sion the engagement and interaction of both the teacher and the studentsduring the lesson and to anticipate and plan for difficulties that might arise.Item 8 was designed to encourage the student teachers to use differentresources as a means for increasing their knowledge and expanding theiralternatives with respect to the design of the lesson.

Although beliefs are one of the key components in the model, they werenot directly addressed in the student teachers’ assignment for prelessonthoughts. There was an important and interesting reason for this. Afterextensive study of the NCTM Standards and other contemporary mate-rials in the methods course that preceded student teaching, the studentshad a clear idea of the latest philosophies that were valued for mathe-matics instruction. By this time in the year, if asked about their beliefsregarding mathematics instruction, the student teachers knew what theirinstructors wanted to hear and were very adept in providing it. Artztand Armour-Thomas (1998) suggested that actions are a more reliablereflection of teachers’ beliefs than are words. Specifically, they found thatseveral teachers gave lip service to their beliefs about student-centeredinstruction, yet contradicted these ideas during their instructional practice.In fact, when interviewed while they viewed the videotape of their lesson,they often contradicted their prelesson espoused beliefs in order to justifytheir action during the lesson. Therefore, the supervisor placed a greatervalue on the beliefs expressed by student teachersafter their lessons asthey reflected on their practice than on beliefs they would describebeforethey taught their lesson. Thus, the discussion about beliefs was postponeduntil the lessons were concluded and the teachers were asked to justifytheir actions.


The descriptions of their prelesson thoughts provided opportunities forthe student teachers to activate and make explicit their knowledge andgoals that drove their lessons. In addition, the supervisor, who observedthe lesson, gained insight into the decision making and reasoning that wasbehind the lesson. The knowledge of the student teachers’ reasons for whatthey did made it easier for the supervisors to understand and assess theinstruction they observed.

Postlesson Reflection

During the postlesson conference, student teachers reflected on theirinstructional practice as well as on their thoughts and decision-makingrelated to that practice. Following the conference, the student teacherswere required to submit both a written analysis of their lesson and adescription of their thinking about the lesson. Details of this method ofpostlesson reflection follow.

Self assessment.After the student teacher had completed his or her lesson,time was set aside for a conference with the supervisor. At that time,the student teacher was called upon to reflect on and assess the lesson.The supervisor encouraged the student teacher to do all of the talking,but helped structure the student teacher’s thoughts by asking questions.Specifically, the student teachers were first asked to examine the relation-ship of their prelesson thoughts, their lesson plans, and their teaching. Forexample, they were asked to recall their original goals for the lesson, howthese goals were addressed in the lesson plan, and the extent to which theybelieved they had accomplished the goals. They were asked to comparewhat they planned to do in the lesson with what they actually did. Second,the supervisor asked the student teachers to explain the decisions that influ-enced them to deviate or not deviate from their original plans. That is,they were asked to explain their monitoring and regulating actions duringthe lesson. For example, they were asked to describe what feedback theyreceived from students and how this feedback informed their practice.

In some cases, student teachers were then asked to further assess theirlesson with a focus on the three elements of tasks, learning environment,and discourse. They were asked to think about, describe, and evaluate thenature of these elements as they unfolded in their classrooms. They wereasked to consider factors that might have influenced their teaching. Finally,they were asked to consider suggestions they might have had for improvingthe lesson if they were to teach it again.

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TABLE I

Sources for data collection

Time Activity

Before the lesson Prelesson activities:

Written prelesson thoughts

Written lesson plan

During the lesson Enactment of the lesson

After the lesson Postlesson activities:

Self assessment at conference

Exchange of Ideas with supervisor at conference

Written postlesson thoughts

Ongoing Journal entries

Supervisor feedback.After the student teachers completed their ownpostlesson reflections, the supervisor shared her thoughts regarding thelesson. It was at this time that the supervisor could request clarificationof issues and help the student teachers think about what new knowledgethey had gained or beliefs that may have changed as a result of teachingthe lesson. The supervisor asked such questions as: As a result of teachingthis lesson what have you learned? What new ideas did you learn aboutthe content, the students, or best ways of teaching the lesson? What beliefsdid you have about the content, the students, or best ways of teaching thelesson that changed in some way?

Postlesson thoughts.After the conference with the supervisor the studentteachers were required to write about their postlesson thoughts regardingthe lesson and the ideas discussed during the postlesson conference. Theywere asked to write about the strong points and the weak points of theirlesson. They were asked to suggest how the lesson might have beenimproved. This written evaluation served at least two purposes. First, itgave the student teachers the opportunity to reflect on the lesson once againin light of all they and the supervisor had said. Second, it gave the super-visor the opportunity to assess what the student teachers had learned fromtheir experience and whether they had internalized any of the ideas thatwere discussed. Table I summarizes the above data collection activities.

Examples will be given of the prelesson reflections, the instructionalpractice, the postlesson reflections, and the journal entries of two preser-vice teachers, Ms. Carol and Mr. Wong. These two students were selected


to exemplify how the structure was used in contrasting situations. Althoughboth were novice teachers, each began student teaching with a differentdisposition which seemed to account for their different progressionsthrough the course of the semester. With each example a discussion willfollow regarding how these reflective activities informed both the teachereducator and the preservice teachers about aspects of professional growth.

THE REFLECTIVE STRUCTURE IN ACTION

Ms. Carol

Ms. Carol had been an accountant who had decided to make a careerchange into teaching. Unfortunately, during the methods course hercooperating teacher would ask her to teach his class five minutes beforethe class started. She felt obligated to say yes, despite the fact that shecertainly was not required to do so. This on-the-spot teaching resultedin several upsetting experiences. She found herself completely “flusteredand unable to teach.” Therefore, at the beginning of the student teachingsemester she revealed that she was insecure about her teaching abilitiesand was almost ready to change her mind about entering the teachingprofession. She questioned her knowledge about mathematics and her ownbeliefs about whether she really wanted to be a teacher. She wrote:

I spent the entire Christmas break stressing about student teaching. All I know is that Ireally don’t want to student teach, and I am having second thoughts about teaching ingeneral. My self-esteem is at an all-time low, and I am more depressed than I can begin toexpress. I dread coming to school every day. I feel I have no strengths as a teacher. I don’tknow the progression of the curriculum. I don’t know the precise definitions of things.(February, Journal Entry)

Through this writing assignment the supervisor was able to see Ms.Carol’s low self-esteem. This low self-esteem may have created the moti-vation that Goldsmith and Schifter (1997) suggest is an important factorfor progressing through the stages of teaching. In fact, as will be shownfrom the excerpts of her work, Ms. Carol did progress rapidly.

During the course of the semester, Ms. Carol learned that by explicitlyformulating her goals and giving written explanations of what she knewabout the content, the students, and methods of pedagogy, she could planand implement better lessons. It appeared that by reflecting on her lessonsin a structured manner she learned how to improve on her own teaching.In the following lesson, which she taught toward the end of the student-teaching semester, we can see how Ms. Carol had progressed well beyondthe initial stage of teaching. Her student-centered focus is evident in her

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written expressions as well as in her instructional practice as evidenced inher planning, the tasks she designed, the learning environment she created,and the discourse she orchestrated.

Prelesson reflection.Ms. Carol’s lesson involved a review of graphingsystems of linear equations and systems of inequalities. In her prelessonthoughts Ms. Carol was asked to state the broad goals she had for herstudents. She wrote that her goals were to get her students to “really thinkabout what they’re doing and understand the differences between graphinga system of equations and [a] system of inequalities.”

The prelesson activity also prompted Ms. Carol to activate and makeexplicit her knowledge regarding the nature of her students’ level ofunderstanding. She wrote:

The students seem to have a lot of trouble with graphing in general. They don’t seemto understand the underlying concepts of what the graph of an inequality, equation, orline really represents mathematically. I had noticed the day before that when the studentswere asked to label the solution set to a system of inequalities, they marked the point ofintersection, as if they were asked to find the solution to a system of equations. They woulddo the shading, but it meant nothing to them. They don’t understand why the solution to asystem of equations is (generally) one point, and the solution to a system of inequalities is(generally) a region consisting of many points. I wanted them to have a little more practicewith both types of graph systems, and then be able to sum up the differences at the end.(May, Prelesson Thoughts)

Unlike an initial stage teacher, Ms. Carol was not satisfied withstudents’ procedural knowledge alone. She was not satisfied that studentscould complete the shading, because she sensed underlying conceptualweaknesses. The lesson she designed was student-centered. She createda learning environment in which students were encouraged to interact withone another as they worked in small groups to complete four problems(see Appendix 1). Two of the problems involved systems of inequal-ities; one problem involved a system of equations; and a fourth problemasked students to highlight the differences between graphing a system ofequations and a system of inequalities.

Because Ms. Carol had recorded her anticipated difficulties, therebyactivating her knowledge about the students, she arrived at a plan for anappropriate instructional strategy. She said:

They [the students] have the tendency not to think, and the tendency to question them-selves to the point where they’re constantly raising their hands to ask, ‘Is this right?’ I’mhoping that if they work together, they’ll askeach otherif they’re ‘right.’ (May, PrelessonThoughts)

Because she used a small-group learning environment it appeared thatMs. Carol believed in the value of students sharing ideas and relying on


each other for feedback. She was also willing to try new techniques, asneither she nor her cooperating teacher had ever used small groups.

Instructional practice.Ms. Carol began her class with an interestingproblem in which students were shown a graph of a system of inequalitiesand were asked to discover the inequalities that comprised the system.This was the reverse of most problems, in which students are given thesystem of inequalities and are asked to draw the graphs. As the studentsworked on the problem, Ms. Carol circulated around the room encouragingsome students to begin working and observing how other students wereapproaching the problem. One student agreed to come to the front of theroom to show his solution and to answer any questions posed by otherstudents. Most students appeared to be on task and involved in the work.When students made incorrect comments, Ms. Carol did not interfere,but rather allowed the students to debate the issue until they resolved themisunderstandings.

Students were then given a worksheet that consisted of the four prob-lems. In her directions to the class, Ms. Carol explained that a member ofeach group would be randomly selected to explain his or her work to therest of the class. As the groups began working, Ms. Carol noticed that therewas not enough time for each group to complete all four problems anddecided to let each group work on one problem only. Some confusion aroseas she changed the directions and assigned a single problem to each group.Because the students had already begun drawing their graphs on graphpaper, further confusion was created when she distributed one transparencyto each group on which they were to draw their graphs for presentation tothe class on the overhead projector. As the students settled into their groupwork, Ms. Carol walked around overseeing what the students were doing.Occasionally the students would ask her a question, but rather than answerit herself, she referred the student to the other members of the group.

With only ten minutes remaining in the class, one student was randomlyselected to describe her group’s work and display the transparency onthe overhead projector. Ms. Carol allowed that student, as well as othermembers of her group, to field questions from the class. With only oneminute remaining, Ms. Carol engaged the students in a brief summarydiscussion of the differences between a system of inequalities and a systemof equations.

Although far from perfect, the nature of this instructional practicewas well beyond that of a teacher in his or her initial stage of teaching.The student focus, evident in the prelesson reflection, was evident in thelesson as well. By allowing students the opportunity to work in groups

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and requiring them to answer each others’ questions, she encouraged themto take responsibility for their own learning.

Postlesson reflection.In her postlesson conference, Ms. Carol was askedto reflect on and assess her lesson. Specifically, she was asked to reviewher prelesson comments to see how they might have accounted for thenature of her lesson enactment. For example, Ms. Carol was asked todetermine whether she had accomplished her goals: that students arriveat an understanding of the concepts and rely on each other for feedback.She noted that she had allowed students to explain their group’s work atthe board and that, when they did or said something incorrect, she hadwaited for other students to make the correction rather than commentingon it herself. She noted how she had walked around the room and listenedto what students were saying to one another and that when they asked hera question, rather than answering it herself, she referred the question backto the group. She also pointed out the conceptual questions she asked thestudents: “Where would you find a point that satisfies the first inequalitybut not the second? How could you change the system of equations so thatthe solution set is empty? How could you change the system of inequalitiesso that the solution set is empty?” However, she acknowledged that shewas unsure as to how much conceptual understanding her students haddeveloped because only a few students had responded. She also ran outof time and never reviewed the last problem that asked for a synthesis offindings.

Ms. Carol was then asked to account for the difference between whatshe planned to accomplish in the lesson with what actually took place. Ms.Carol’s immediate reply was that she ran out of time. When asked why thathappened she explained how both the tasks she designed and the instruc-tional strategy she used prevented the class from getting to a discussion ofthe last, most important problem. When asked what she learned as a resultof teaching this lesson, Ms. Carol replied:

By wandering the classroom I got to see the things that really trouble students; aspectsof mathematics that are completely obvious to teachers are not necessarily comprehendedso easily by the students. I learned that instructions that are clear to me might not beclear to them. For one thing, I said “work together,” but my definition of “work together”isn’t necessarily theirs. I also thought that saying something once was enough, but theysometimes need constant reminders. Also, their tasks not only have to be meaningful, theyhave to be more structured. (May, Postlesson Conference)

In her written postlesson analysis she wrote:

The strong point of the lesson was that most of the students eventually got to workingtogether and arguing their ideas. They got the chance to explain their ideas at the overhead,


using the transparencies enabled the students to get an accurate description of what wasgoing on. I asked questions that really made the students think. But since that was a goal ofmine, I should have put fewer graphing examples on the worksheet so the students coulddo all of the problems and then spend the majority of the time working on the thoughtquestions. I think that with a few minor adjustments, this lesson could have been a powerfulone for the students. I really like the cooperative learning aspect. The period goes by muchfaster, I talk less, and the students get more out of it. (May, Postlesson Thoughts)

The postlesson reflection prompted Ms. Carol to analyze the lessonin light of her original goals for students and her beliefs about the roleher students should play in their own learning. From her statements, itis clear that she was able to make the connection between the elementsof her lesson and their role in helping or impeding her efforts to accom-plish her goals for students. It is also clear that she used the studentsas her barometer for determining her level of goal accomplishment. Herthoughtful, coherent, and logical comments suggest that Ms. Carol hasthe ability to assess her lesson carefully and critically and to generateconstructive ideas for revision. These self-reflective abilities suggest thatMs. Carol has the tools for continuing her professional growth after thesupervisor has left the scene.

More importantly, Ms. Carol was able to give an insightful evaluation ofher own competence as a teacher. This was evident in Ms. Carol’s final self-evaluation at the completion of the student teaching semester. As suggestedbelow, Ms. Carol had learned that the ingredients for success include bothher ability to reflect and her underlying beliefs about students and how theylearn. She wrote:

As far as evaluating my ability to teach at the present time, I can only say that I’m betternow than I was when I started. The only thing that makes me think that I will be a goodteacher is that I can see what it is that I’m doing wrong, and I am really concerned if thestudents are gaining conceptual understanding of the underlying mathematics of a problem.I respect their thoughts, and I wait and expect them to explain themselves fully. I insist thatthey work together and listen to each other. I value the importance of getting them touse reason and speak like mathematicians. I put a lot of work into my lessons so that thestudents will benefit and get the most out of it. Most importantly, I really care about them.I want to see them all do well and appreciate mathematics. (June, Journal Entry)

Mr. Wong

Mr. Wong had been a computer programmer who, like Ms. Carol, hadmade a career change into teaching. His beliefs at the beginning of thesemester were characteristic of a first stage teacher and quite contrary towhat was being promoted in class. In his journal at the end of the studentteaching semester he wrote:

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At the beginning of the student-teaching semester, my perception of teaching mathematicswas rather naïve. I thought that all you needed to do was to think of an interesting problemto get the students motivated in the beginning. Then, just present the lesson pretty much asthe textbook has it written. I was imagining that the students were attentive and ask ques-tions. I love to deal with children and I love to be the one to provide the correct answers.I had the image of me just teaching and talking for the entire period, like a preacher, withthe students absorbing all the knowledge that I impart to them. (June, Journal Entry)

In contrast to Ms. Carol, Mr. Wong’s strong beliefs about the “right”way to teach left him inflexible and unmotivated to learn new instructionalapproaches. According to Goldsmith and Schifter (1997) someone whosedisposition is so contrary to the kind of teaching envisioned by the reformmovement may need extra support in changing teaching practice. Inthis regard, the structure for reflection was particularly useful since itchallenged Mr. Wong to reexamine many of his beliefs about his role as ateacher and his students’ roles as learners. In the following example wewill see how Mr. Wong was encouraged to take on a more student-centeredfocus by gaining more knowledge about his students and the content ofhis lesson.

Prelesson reflection.Mr. Wong’s lesson involved probabilities ofcompound events. Contrary to Ms. Carol’s detailed comments about whatshe wanted her students to understand and how she wanted them to beactively involved in that process, Mr. Wong tended to speak in generalterms, reflecting a lack of knowledge regarding both the content and hisstudents. He wrote:

The goal in this lesson is to help the students to understand the concept of compoundevents, independent events, the counting principle and finally how to use the aforemen-tioned knowledge to compute the probability of compound events. The students hadreviewed some of the basics of probability during the previous lesson. In that lesson, thestudents reviewed the definition of outcome, sample space, and event. They also reviewedthe basic rules of probability.

Based on the experience of the previous class, there is, in general, a fair level of interestamong the students for the subject matter. Maybe because they were exposed to this matterin their last course, the students seem to be able to pick up on the material without toomuch difficulty. (March, Prelesson Thoughts)

Although Mr. Wong made mention of wanting his students to under-stand the material, he gave no indication of what aspects they might finddifficult nor how he would render it understandable to them. He suggestedthat students did not find the material particularly difficult. Given that prob-ability is generally considered a very difficult topic for students, suspicionarises that Mr. Wong has not been monitoring his students’ understandingvery well, if at all. Furthermore, although Mr. Wong mentioned several


topics, he never discussed them in any detail. His lack of student focusand apparent lack of awareness of the trouble spots entailed in teachingsuggested that he was still at the first stage of teaching. The supervisor usedMr. Wong’s prelesson thoughts to point out to him in a tangible way that heneeded to increase his knowledge of the content, of his students’ intuitiveknowledge of the content, and of ways of helping students understand thecontent.

Mr. Wong’s lesson plan consisted of an interesting and potentiallymotivating problem which required students to take a multiple choice quizcontaining the following questions: What is Neil Armstrong’s Birthday?How long is the Nile River?, and What is the diameter of Jupiter? Thequestions had 3, 4, and 2 choices, respectively, and the choices were closein value. In his postlesson thoughts Mr. Wong admitted that he had outlineda tight script for his lesson. He wrote:

I was going to give them the quiz and then prod them to recognize the compound eventsby asking them if they noticed any difference between what they did yesterday (rolling adice and spinning the spinner) and what they are doing now. I was expecting the studentsto come up with the correct answer – compound events. I would then write the “AIM.”Following the AIM, I then planned to ask the class what was needed to find the probabilityof getting all three questions correct in the quiz. I was expecting the class to reply that weneeded to know the sample space. This would have provided the opening to investigate thesample spaces of this activity. The sample space investigation was to be a tree diagram toprovide the necessary clarity. This was to be followed by asking the students for an easierway of getting the sample space – the counting principle. After establishing the countingprinciple, the plan was to then compute the probability of getting all three quiz questionscorrect by knowing the number of elements of the compound event or the number of waysthat the compound event can take place. As a finale to the lesson, I was going to cover theCounting Principle with Probabilities by asking the students if there is an easier way tocompute the probability of the above compound event. (March, Postlesson Thoughts)

Based on what he wrote, it was not surprising that the lesson enactmentwas problematic. That is, Mr. Wong’s focus was mostly on whathe wasgoing to do, with specific expectations for what the students were goingto say. This rigid disposition precluded the flexibility and explorationcharacteristic of later stages of teaching.

Instructional practice.The lesson did not proceed as Mr. Wong had envi-sioned. The first problem he encountered was with the discourse. Afterthe students took the quiz, one boy yelled out the correct answer to theprobability question, “What is the probability that you will get all of themultiple choice questions correct?” which Mr. Wong had not yet asked. Mr.Wong told the student that he was correct, and then he was at a loss for whatto do. For the remainder of the lesson he lectured, giving the definition ofa compound event and writing out a tree diagram of the sample space for

158 ALICE F. ARTZT

the students. The students were passively taking notes and the potentialmotivation of the interesting task was all but lost.

Mr. Wong’s inflexible disposition and limited knowledge impeded hisefforts to salvage the lesson and presented a challenge for the supervisor tohelp him understand the nature of his difficulties. Fortunately, Mr. Wong’spostlesson reflection activities seemed to help him focus in a systematicand objective way on how and whether he was indeed facilitating hisstudents’ understanding of mathematics.

Postlesson reflection.In his postlesson conference, Mr. Wong was askedto review his original goals for the lesson and determine whether they hadbeen realized. When he stated that he covered all of the material he hadplanned, he was asked whether he “helped the students understand” theconcepts as he had stated in his original goals. He said he had tried butwasn’t sure whether the students understood or not. He was asked why hedidn’t know, and he said, “The class was quiet. It was just my voice.” Hewas then asked about his beliefs regarding the best ways to help studentsunderstand. Did he think it was by explaining everything to them as he haddone? Through this questioning Mr. Wong began to recognize that to reallyhelp students understand and be able to know if they understood therewould have to be more informative discourse in the class. He admittedthat he would have to “ask more probing questions,” that he would haveto “hear students explain their answers,” and that students would need to“respond to other students’ questions.” Mr. Wong was then asked to recon-sider his knowledge regarding students. He was asked if he still thoughtthe students did not have much difficulty with the topic, as he had statedin his prelesson thoughts. He stated again that the class was very quiet. Hewas asked to consider the reason for their silence. Mr. Wong was askedto consider the following questions: Did the fact that one student knew theanswer to the unasked question at the beginning of the class mean that eachof the students in the class knew the answer as well? Even if they knew theanswer, did it mean that they understood why that was the answer? Perhapsit was a lucky guess? How would the teacher know if it was a lucky guess?

Mr. Wong was also asked to examine the nature of the task he haddesigned. He thought the quiz was very motivational but realized that themotivation ended when the student gave the probability. He was asked howhe thought the motivation might have been maintained. How could the taskbe sustained in such a way that the other concepts such as sample spacecould be developed? Because the class had been so quiet, he realized thatthe learning environment had been awkward. He was asked how he mighthave actively engaged the students. After a lengthy discussion, Mr. Wong


suggested that he might have recorded each of the students’ answers on thetest. This would have helped in developing the idea of sample space and atthe same time engaged each of the students.

After considerable discussion, Mr. Wong was asked what he hadlearned as a result of teaching this lesson. Had any of his beliefs changed?He said that he realized that “a good problem is not enough to motivatestudents.” He expressed the remainder of his thoughts in his postlessonreflective activity. The following comments suggest that Mr. Wong hadlearned a great deal.

One of the major problems with my lesson was my inability to capitalize on the make-believe quiz to motivate the class to learn the subject matter. My approach was too dry andtoo inflexible. The class was not motivated. Some students were heard asking what wasgoing on. I was too anxious to get the word “compound event” out of the students’ mouthso I could write the AIM. Why not stir up enthusiasm by asking the class what is theprobability that anyone in the class has answered all three questions correctly? Challengethe class to determine how many students answered all three questions correctly. Thisshould lead naturally into the examination of sample space of the activity. When I did askthat question on sample space, one student answered correctly (not understanding why,however), and I applauded him for the “correct” answer and then went on to obtain thesample space by doing the tree diagram. I DID ALL THE EXPLANATION, AND ALLTHE TALKING. Big mistake! I should have used this opportunity to assess the studentsas to their understanding of sample space. Instead of applauding the student’s answer, Ishould have asked why and how he arrived at the answer. (March, Postlesson Thoughts)

This reflective activity appeared to enable him to systematicallyexamine all elements of his lesson: the tasks, the learning environment,and the discourse.

Throughout the semester, Mr. Wang was called on to constantlyexamine his teacher-centered approach but it seemed difficult for him toleave the security of lecture-driven lessons. He spent the entire semestergrappling with his beliefs about how students learn best. He was affectedby his own positive experiences in China with learning through listening.Only after extensive experience with examining the learning environmentand discourse that took place in his class did he begin to get the feelingthat his students were not paying attention, much less learning, while heengaged in “chalk and talk.” Through the reflective activities he beganto realize that he was not monitoring student understanding sufficiently,which was one of the reasons that he had rarely regulated his instructionalpractice to fit the needs of his students. In his final self-evaluation at thecompletion of the student teaching semester he wrote:

I must get rid of my old habit of tightly following the script of my lesson plan. It is a habitthat I still follow in the classroom setting although I am much more aware of it now thanbefore. In a classroom setting, I have got to learn to use the lesson plan only as a guide. I seethat I am not assessing the understanding of the students continuously. Interestingly, being

160 ALICE F. ARTZT

aware of my bad habits, I tried to observe my tutoring technique for the past few days.What I have noticed is that I do not make many of the important mistakes as I have donein the classroom setting. During tutoring, I asked a lot of questions to assess the learninglevel of the students, and I tended to allow the students to learn by prodding them withquestions instead of just “teach” by talking away. (June, Journal Entry)

Judging from Mr. Wong’s comments about his tutoring, the super-visor got an indication that his comments were authentic and that thereflective activities facilitated his efforts to examine and question the valueof his teacher-centered approach to instruction. The structure for reflec-tion helped serve as a mechanism through which he could dislodge someof his inflexible attitudes and approaches and begin to progress in hisdevelopment.

FINAL REMARKS

Working with student teachers is a challenging and often mysterioustask. I have often found myself frustrated by my inability to help preser-vice teachers develop as I wished. I have often sat in the back of theirclassrooms bewildered by the things I saw. I used to ask myself whetherthese students had really attended my classes for the previous six months.When interacting with them I focused on their instructional practice andaddressed the issues that I felt needed attention. I tried to hide my frustra-tions to protect their fragile egos and often left the conference disturbed.Since I used the approach described herein, much has changed. I do notmean to claim that my students are now all final stage teachers. What I doclaim is that, by getting into the minds of my students in a structured way,I am better able to make sense of what they do and am therefore betterable to help them progress. Additionally, I have the impression that bybeing required to probe, express, examine, and question their own thinkingprocesses, my students are better able to understand and improve theirinstructional practices.

Goldsmith and Shifter (1997) suggested that teacher change is charac-terized by (a) qualitative reorganizations of understanding, (b) orderlyprogressions of stages, (c) transition mechanisms, and (d) motivationaland dispositional factors. The approach presented in this article facilitatesteacher change by engaging students in activities that served as transitionmechanisms for professional growth. It encouraged preservice teachersto reveal their motivations and dispositions and to organize their under-standings of the relationship between their thoughts and their instructionalpractice. When students engaged in thinking and writing about their goals,


their knowledge, and their beliefs in relation to their instructional prac-tice, their motivational and dispositional factors became apparent. Ms.Carol initially revealed that she felt insecure about her knowledge, herabilities, her ideas about teaching. Her feelings of discomfort and dissat-isfaction served as motivation for her to be open-minded about learningnew approaches. In contrast, Mr. Wong’s writing revealed that he enteredteaching with fixed beliefs that the best way to teach was through themedium of lecture. He initially resisted learning new approaches. Ashis supervisor, I realized that the only way to help him progress was toencourage him to focus on his students and challenge him to examine theextent of their learning. By knowing a preservice teacher’s motivationsand dispositions the supervisor is in a better position to facilitate change.The reflective activities served as the vehicle through which change waspromoted.

According to Goldsmith and Schifter (1997) a developmental modelfor mathematics teaching needs to account for a qualitative change in thebelief-behavior complex. The framework used in this study served as theconceptual basis for the structure that enabled preservice teachers to relatetheir thoughts and actions. Without the reflective activities, which requiredMr. Wong to confront his beliefs and focus on his students, it is possiblethat Mr. Wong might never have changed his teaching style. In fact, inhis written prelesson thoughts, he did not focus on student understandingand ways of learning; rather, he focused on whathe was going to do inthe lesson. It was only after the conference, in which he was continu-ally asked to examine student learning, that he was able to make someimportant connections. According to his postlesson thoughts, he came tounderstand that his rigid approach to teaching (using a script and transmit-ting knowledge) did not allow him the flexibility to monitor his students’understanding or to adapt his instruction as it became necessary.

Since I used this structured approach I have learned much about howmy students think, and have thereby changed many of my ideas and myapproaches. I am now less rigid in my views and therefore less frustratedabout what I see as I sit in the back of a classroom. I have come to realize,more than ever, that there are reasons for how preservice teachers teach.These reasons are a reflection of the knowledge and beliefs they haveabout the mathematical content, the particular students they teach, and theway students learn. Their reasons also reflect the goals they have for theirstudents, which are often influenced by the cooperating teachers’ goals.Preservice teachers need help and support in order to construct new mean-ings about what it means to be an effective mathematics teacher. Theymust continually be called upon to share, reexamine, and question their

162 ALICE F. ARTZT

knowledge, their beliefs, and their goals for students. They must be madeaware of the need to monitor student understanding during the lesson asa means for reconstructing their own meanings about what is going onin their classrooms. I have found that when preservice teachers are mademore aware of the monitoring they do of their students, they become moreconscious of the need to change their instruction accordingly. Rather thanlooking at these interactive changes as a negative aspect of their lesson(“I couldn’t do what I planned”), they realize that they are indeed actingon behalf of their students and are engaging in more effective teaching.Finally, when they reflect on their lessons in a structured way I have theassurance that they have attended to the most critical facets of classroominstruction: tasks, learning environment, and discourse. Whereas initiallythey tended to focus on more efficient ways to cover the content and main-tain control, they now tend to focus on better ways to involve all of thestudents.

For preservice teachers the structure for reflection appears to be apowerful tool for facilitating their continual professional development.They are encouraged to be more analytical about their teaching. Theyare encouraged to examine and attend to the underlying assumptions andbeliefs that drive their practice. They are encouraged to think about whythey make the decisions they do in light of their goals for students. Theyare encouraged to think about how their knowledge and beliefs regardingthe content, their students, and methods of teaching impact the design oftheir lessons. Hopefully, these experiences will enable them to develop ahabit of reflective thinking about their teaching. Ms. Carol’s comment inher evaluation of the course suggests that this habit of reflection is possible.

Most of the work done as a student teacher was essential for self awareness. I think bybeing focused in on prelesson and postlesson thoughts I became able to focus on themsubconsciously. These thoughts and my awareness of them were brought constantly to myattention, and now they always will be. I wasn’t always thrilled to write them, but I nowrealize that it will be to my benefit always.

Most importantly, I hope that the prospective teachers will view them-selves as authorities who can evaluate their own classroom instruction interms of their own knowledge, beliefs and goals for students and be flex-ible enough to modify their beliefs as the evidence indicates. According toCooney and Shealy (1997) such autonomous behavior is critical for teacherdevelopment. In this article I have attempted to show how an approachfor examining teachers’ thought processes and instructional practice canbe used as a tool for the professional growth of preservice mathematicsteachers. Teacher educators can use this approach as a vehicle throughwhich preservice teachers are made aware of their underlying thinking and


how it impacts their instructional practice. By being provided a compre-hensive structure for self-reflection, preservice teachers can be empoweredto assess and improve their teaching.

ACKNOWLEDGEMENTS

The author is grateful to the many people who have contributed to thedevelopment of this work. Thanks go to Naomi Weinman who assisted mein supervising the student teachers, to Eleanor Armour-Thomas who gaveextensive feedback in the writing of this article, and to my students whoworked so hard.

APPENDIX A

Class Work Assignment to be done in Groups

1. a. On the same set of coordinate axes, graph the following system ofinequalities:

y + x< 5y ≥ 2x + 3

b. Based on the graphs drawn in part a, write the coordinates of:

i. A point in the solution set of the system of inequalities.ii. A point that satisfies the first inequality, but not the second.

c. Explain in a few brief sentences how you would check your answers topart b algebraically.

2. a. On the same set of coordinate axes, graph the following system ofinequalities:

y < 12x + 2

y ≥ 12x − 1

b. Label the solution set of the system of inequalities S.c. Change the inequalitites so that the solution set is empty. Explain your

answer.3. a. On the same set of coordinate axes, graph the following system of

equations:

y − 2x = 7x + y =−2

b. What is the solution of the system of equations? Check your answer.4. In your group discuss the differences between graphing a system of equations

and a system of inequalities.

164 ALICE F. ARTZT

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Queens College of the City University of New YorkDepartment of Secondary Education and Youth ServicesFlushing, NY 11367-1597Email: [email protected]

a structure to enable preservice teachers of mathematics to reflect on their teaching

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