IIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII I; ,,,,, rut-:91! 310532 5272 ! «Li if a,“ .. y F This is to certify that the thesis entitled CRITICAL MOMENTS IN THE TEACHING 0F MATHEMATICS: I'IHAT MAKES TEACHING DIFFICULT? presented by JANET CAROLYN SHROYER has been accepted towards fulfillment of the requirements for Ph.D. degree in Education dig/m Major professor Date August 14, 1981 0-7 639 MSU LIBRARIES ”- RETURNING MATERIALS: P ace in 00 rop to remove this checkout from your record. FINES wiII be charged if book is returned after the date stamped beIow. CRITICAL MOMENTS IN THE TEACHING OF MATHEMATICS: WHAT MAKES TEACHING DIFFICULT? By Janet Caronn Shroyer A DISSERTATION Submitted to Michigan State University in partial fquiIIment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of EIementary and SpeciaI Education 1981 Copyright by JANET CAROLYN SHROYER 1981 ABSTRACT CRITICAL MOMENTS IN THE TEACHING OF MATHEMATICS: WHAT MAKES TEACHING DIFFICULT? By Janet Carolyn Shroyer This study investigates how teachers cape with unpredictable student performance which is inevitable in classrooms, particularly when teaching emphasizes problem solving. Prior research on teacher thought indicated that teachers experience problems, but this is the first use of student difficulties and insights as a strategic research site. The focus is on understanding the relationship between student performance, teacher thought and behavior and the subject -- mathematics -- being taught at critical moments. Using a process-tracing technique and stimulated recall to gather teachers' thoughts and feelings, case studies were conducted of three teachers as they taught a six- or seven-day unit on rational numbers. Teachers from the upper elementary grades were selected because of their experience with the Unified Science and Mathematics Elementary School project and their good reputations. Teachers exhibited variation in the content emphasized,instructional materials, strategies and style which were indicative of their different instructional goals and conceptions of learning mathematics. Teaching Janet Carolyn Shroyer ranged from traditional mathematics instruction to a problem-solving approach recommended by the National Council of Teachers of Mathematics. The impact of the instructional environments was evident in the variations in frequency, proportion, and density of student difficulties and insights. Unsolicited student contributions were more prevalent with the less traditional instructional approach, and student difficul- ties outnumbered student insights. Teachers' elective actions also varied. Those used with student difficulties were similar when classi- fied according to the broad categories of exploiting, alleviating, and avoiding moves, but techniques for alleviating student difficulties varied. Teachers also responded differently to student insight. Elec- tive actions varied in accordance with teachers' understanding of mathe- matics, instructional goals, conceptions of learning and attributions for student difficulties. Likewise, their negative emotional reactions varied in intensity and type. A distinction had to be made between critical moments of a short- term nature and those based on pervasive problems. Critical moments arose over specific incidents of student difficulty or insight, while what was eventually labelled critical discrepanCies reflected a more pervasive pattern of student performance. There were four types of con- ditions that produced cognitive difficulties and emotional discomfort for teachers: student difficulties, student insights, instructional pace, and unanticipated success. Further subdivisions according to the particular difficulties teachers reported suggest a taxonomy of critical moments. Janet Carolyn Shroyer Although teachers processed student occlusions somewhat differently than pervasive patterns of student performance, the same processes were involved: interpreting, goal setting, searching for elective actions and evaluating the effectiveness of the actions. Once the goal was set, teachers attempted to find, execute and test the results. Critical moments typically occurred with more distinctive student occlusions. Teaching became difficult when teachers had trouble achieving their goals which happened by reason of default -- being unable to think of something else to try -- or by interference from antigoals -- situations to be avoided. Some goal had to be satisfied; if not the initial one, then a secondary one. When the threat was to the teacher's activity goal, the switch was more difficult. The primary use of teachers' critical moments,however, was their limited knowledge of mathematics and occasionally their limited repertoire of pedagogical techniques. To help cope more effectively with critical moments, they need task- and topic-specific information about the types of student occlusions to expect and prescriptions for elective actions. Recommendations were made for the use of critical moments in training teachers and for pre- paring curricular materials by activities and units. I dedicate this dissertation to my parents, John and Claire Shroyer. Throughout my educational endeavors, they have been steadfast in their Tove, support, and encouragement. In addition, they have been model educators: effective and dedicated. No one could have asked for more. ii ‘p '5 cu ACKNOWLEDGEMENTS I wish first to express my admiration and appreciation to the three teachers who participated in this study. Without their courage, honesty, and cooperation in divulging their thoughts and feelings and the time they willingly gave, this study could not have been done. Nor could so much have been learned about the mental life of the teacher. The members of my doctoral committee have been supportive through- out the study, even when progress was slow. Special thanks go to Perry E. Lanier for serving as my doctoral advisor, to William M. Fitzgerald for continuing our work on the development of problem- solving units, to Edward L. Smith for giving advice, and to John Wag- ner for helping me to maintain a historical perspective. I am par- ticularly indebted to my dissertation director, Lee S. Shulman, for his repeated efforts to focus my attempts to complete this project. His insistence on quality has been both an inspiration and a challenge. This study was planned and the data collected while serving as a Research Intern at the Institute for Research on Teaching at Michi- gan State University. The Institute was funded primarily by the Teaching Division of the National Institute of Education, United States Department of Health, Education, and Welfare (Contract No. 400- 76-0073). I am grateful for the support, resources, and research en- vironment which the Institute provided. My friend and typist, Barbara Reeves, has helped and encouraged me to complete this paper. To her, no expression of appreciation is adequate. My thanks also to Sandra Gross and Susan Battenfield for their contributions. To my husband, Joe Byers, I offer my appreciation for his en- durance and support; and to my good friends, I thank you for your understanding. Lastly, I thank my special friend as well as my oftentimes distraction, Tasha. iv TABLE OF CONTENTS List of Tables ..................... vii List of Figures .................... ix CHAPTER I: INTRODUCTION ................ 1 Problem ...................... 1 Purpose ...................... 6 Assumptions .................... 6 Study and Questions ................ 13 Overview of Study ................. 15 CHAPTER 11: REVIEW OF THE LITERATURE , ......... 16 Overview ..................... 16 Research Findings ................. 18 Methods of Studying Teacher Thought ........ 30 Conclusion .................... 39 Summary ...................... 41 CHAPTER III: METHOD .................. 43 Teacher Selection ................. 45 Data Collection .................. 47 Data Analysis ................... 52 Summary ...................... 56 CHAPTER IV: INSTRUCTION ENVIRONMENTS ......... 57 Introduction . .................. 57 Nature of Instructional Flow ........... 58 Nature and Distribution of Student Occlusions . . . 87 Nature and Distribution of Elective Actions . . . . 99 Conclusions .................... 109 Summary ...................... 111 CHAPTER V: CRITICAL MOMENTS .............. 113 Introduction ................... 113 Type A: Student Difficulty ............ Type B: Student Insights ............. Type C: Pacing Dilemna .............. Type D: Unexpected Success ............ Summary ...................... CHAPTER VI: TEACHERS' MENTAL PROCESSING ........ Introduction ................... Teachers' Cognitive Processing of Student Occlusions .................. Teachers' Cognitive Processing of Pervasive Patterns of Student Performance ....... Summary of the Factors Which Influenced Teachers' Cognitive Processes ............. Teachers' Emotional Responses to Unexpected Student Performance ............. Summary ...................... CHAPTER VII: DISCUSSION ................ Teacher Thought and Teaching ........... Methodology .................... Implications and Directions for Future Research on Critical Moments ............. Teaching Mathematics ............... Directions for Training Teachers and Development of Curricular Materials ........... Summary ...................... CHAPTER VII: SUMMARY AND CONCLUSIONS ......... Appendix ........................ References ....................... vi 169 175 175 178 208 220 253 258 LIST OF TABLES Instructional Time and Activity Data Per Teacher ..... 70 Minutes of Class and Instructional Time Across Days and Teachers .................. 7l Types of Student Occlusions ............... 88 Distribution of Proportions and Frequencies of Student Occlusions by Types and Teachers ....... 93 Proportions and Frequencies of Student Occlusions by Initiator and Teacher ............... 95 Density of Student Occlusions by Types and Teachers . . . 96 Elective Actions ..................... lOO Proportions and Frequencies of Elective Actions by Type and Teacher ................... lO3 Conditional Probabilities of Alleviating Moves by Types of Student Difficulties and by Teachers . . . . 107 Distribution of Critical Moments by Type and Teacher . . . ll7 Summary of Type A Critical Moments: Student Difficulties ..................... I46 Summary of Type 8 Critical Moments: Student Insights ....................... ISO Summary of Types C and D Critical Moments: Pacing Difficulties and Unexpected Success ......... l70 Factors Influencing Teachers' Cognitive Processing of Student Occlusions ................ I99 Skemp's Model of Emotions as Signals of Change with Regard to Goal States and as a Consequence of Perceived Ability to Effect Change ........ ZOl Martha's Fraction Unit: Time in Minutes and Content by Activity ..................... 253 vii A.2 A.3 A.4 A.5 Zelda's Fraction Unit: Time in Minutes and Content by Activity ..................... Ralph's Fraction Unit: Time in Minutes and Content by Activity ..................... Distribution of Instructional Time in Minutes by Content and Teacher ................. Frequency and Density of Student Occlusions by Teacher and Activity ................. viii 254 255 256 257 1.1 2.1 4.1 4.2 6.1 6.2 LIST OF FIGURES Model for teaching: a Critical Moment .......... Model of a teacher's cognitive processes during teaching (Peterson & Clark, 1978) (labeling of paths added) ................... Cuisenaire rod key .............. Conditional probabilities of elective actions for student difficulties and insights by teacher ..... A model of a teacher's mental processing of student occlusions and critical moments ..... A model of a teacher's cognitive processing of pervasive patterns of student performance ix 12 23 75 106 179 196a CHAPTER I INTRODUCTION Problem The National Council of Teachers of Mathematics' recommendations for school mathematics in the 19805 call for teachers to "create classroom environments in which problem solving can flourish" (NCTM, 1980, p. 4). Students need experience in applying mathematics which includes formulating questions, analyzing and conceptualizing prob- lems, discovering patterns and similarities, suggesting explanations, experimenting and transferring skills and strategies to new situa- tions. These recommendations come at a time when the council has more information than ever before about what students have learned about mathematics and how teachers teach mathematics (Suydam & 05' born, 1977; Easley, 1978; Stake & Easley, 1978; Weiss, 1978; Fey, l980). Despite the curricular projects of the sixties and teacher train- ing efforts over the last twenty years, little change has occurred in the way mathematics is taught. The data indicate that mathematics instruction limits students to "...routine computation at the ex- pense of understanding, applications and problem solving...[leaving] little hope of developing the functionally competent student that all desire"(NCTM, 1980, p. 5). In addition, the use of manipula- tives to model and apply concepts is rare. The Council acknowledges dc " " that "...explanation, practice and directive teaching are important but should not diminish the time necessary to achieve this priority [em- phasis on problem solving]" (NCTM, 1980, p. 11). To assist teachers in making the change, NCTM calls for the de- velopment of appropriate curricular materials to teach problem solv- ing, identification and analysis of effective teaching strategies and techniques, and improvement of teacher training programs. Past fail- ures to produce substantive changes in the teaching of mathematics through similar efforts indicate that something has been omitted. It is the teacher and how s/he actually functions in the classroom. Efforts to improve the teaching of mathematics must be based on a much greater understanding of teachers and the teaching process if they are to have any chance of success. One aspect of teaching which needs to be considered is how teachers process and respond to unexpected student performance. The NCTM recommendations call for learning environments which foster un- predictable student performance. Errors, misconceived conjectures, and insightful ones should be commonplace in activities where ques- tioning,conceptualizing, experimenting, and discovering are the de- sired learner behaviors. Teachers need to be flexible and respon- sive to these student difficulties and insights, but how much is known about how teachers actually cope with unexpected student per- formance? How teachers behave, think, and feel about such unpredict- able events is not really understood. It has only been in recent years that the "mental lives"1 of teachers, not only classifications and tabulations of their behavior, have received attention. Interest in the mental life of the teacher was spurred by the National Conference on Studies of Teaching (l974) and the panel report on Teaching as Clinical Information Processing (Shulman, 1975). The goal of this panel was ...to develop an understanding of the mental life of teachers, a research-based conception of the cognitive processes that characterize that mental life, their an- tecedents and their consequences to teaching and student performance (p. l). The report encouraged the view of ...teacher as agent, rather than as a passive employer of teaching skills or techniques, and a commitment to understanding the ways in which teachers cope with the demands of the classroom (p. 2). It is the relationship between teachers' thoughts and actions which become crucial in the formulation rather than thoughts or actions studied in isolation. The few studies which have been conducted on the mental life of teachers during interactive teaching have focused on identifying, classifying, and quantifying teachers' decisions and on identifying principles or "implicit theories" by which teachers operate. Work by Peterson and Clark (l978) suggests that much of what teachers do is habituated "business as usual." Teachers often have little con- scious thought to report during stimulated recall sessions.2 1It was William James who apparently first referred to psychology as "the science of mental life." 2Stimulated recall is a research method in which audio or video tape replay is employed to assist participants to recall their thoughts and feelings while they were involved in the taped occur- rences. Teachers do report thoughts about unexpected student performance, particularly if their lessons are going badly. Morine-Dershimer (l979) found that teachers were able to handle some unexpected stu- dent performance with "inflight" decisions, but that other situa- tions were far more discomfiting. Teachers rarely made changes even when they felt a situation demanded that something be done. Expectations produced by considering teaching as decision mak- ing (Shavelson, 1973) have not been supported. Not only were teach- ers making decisions infrequently, but they were rarely considering alternatives. This was particularly evident in the quantitative data of Peterson and Clark (1978). Peterson, Marx, and Clark (l978) found that teachers considered alternative strategies only when teaching was going poorly as determined by student participation and involvement. Teachers did not report thinking of alternatives when things were going well which is not surprising unless teachers are expected to continually strive to improve the learning experience. Research which has been conducted on teacher thought provides little understanding of teachers as they deal with unpredictable student performance. Part of the problem is the way in which the research has been conducted. In the current infancy of research, teacher thought has typically been examined independently of both teacher and student behavior. The researchers have paid little at- tention to the context in which these thoughts occurred. Studies have been conducted in both laboratory and natural settings with small and large group instruction. Investigation of teacher deci- sion making was even limited to small segments of lessons, although other studies have traced teacher thought for entire lessons. If F‘ '0 vi - 4.3, V I ,. A H.‘ I.. VFW the antecedents as well as the consequences of teachers' cognitive processing are to be understood and the relationship between teach- ers' thoughts and actions are to be found, greater attention must be paid to the instructional environment and the behaviors of students and teachers. To obtain sufficient numbers of incidents of unex- pected student performance and to be able to isolate antecedents and consequences, instruction also needs to be traced over longer peri— ods of time. One thing research on teacher thought does is to support unpre- dictable student performance as a strategic research site. The con- cept of a strategic research site was first articulated by Merton (1959). In discussing the history of sociology, he indicated that "...problems could be brought to life and developed by investigat- ing them in situations that strategically exhibited the nature of the problem" (p. xxvii). Merton cited multiple discoveries in science as one example. Shulman (1978) elaborated the notion of a strategic research site for understanding the process of teaching. Discontinuity or incongruence is the primary criterion for identity- ing these sites. Striking discontinuities which are perceived by the teacher were recommended as appropriate sites for understanding not only the behaviors of students and teachers, but their judgments and intentions. Incidents which interrupt the teacher's internal stream of consciousness and the action they take to recover it also provide an opportunity to examine what occurred before and what follows. Research on teacher thought does indicate that such dis- continuities occur as teachers face unpredictable student performance. NJ: I u. bob, D “foz‘ Thus the student difficulties and insights which occur during instruc- tion provide a research site both as occasions to investigate teachers' thoughts and feelings and as desirable events in the teaching-learn- ing experience which teachers sometimes find problematic. Purpose The purpose of this study is to investigate incidents of unex- pected student performance during teaching of a mathematics unit and to identify and describe those which prove troublesome for teachers of mathematics which we label "critical moments." From an examination of these critical moments, it is anticipated that much will be learned about how and why they make the teaching of mathematics difficult for teachers. In addition, a more general characterization of teachers' mental processing of critical moments will be sought. Before examin- ing the questions of interest to this study, two assumption on which this study is based need to be examined: one is the assumption of teacher as information processor, and the other has to do with a par- ticular theoretical conception of instruction. Assumptions Teacher as Information Processor The information processing perspective has been adopted for this study of teacher thought and behavior. According to Newell (1973), to predict or even understand the behavior of a teacher coping with unex- pected student difficulties and insights would require knowledge of three things: the first is the gggl§_of the teacher, the second is the struc- ture of the task environment in which the teacher is operating, and the third is the invariant structure of the teacher's processing mechan- isms. There is much that has been learned about the invariant limi- tations of human mental processing capacities. According to Simon (1969), ...evidence is overwhelming that the system is basically serial in its operation: that iS can produce only a few symbols at a time and that the symbols being processed must be held in special, limited memory structures whose content can be changed rapidly. The most striking limits on subject's capacities to employ efficient strategies arise from the very small capacity of the short-term mem- ory structure and from the relatively long time required to transfer a chunk of information from long-term memory to short-term memory (p. 53). The consequence of these processing limitations is that the teacher must function with less than full awareness and knowledge of all that might be known or recognized about the student's performance. The teacher must make choices on the basis of his/her own simplified re- presentation of the situation. Simon (1957) refers to this as "bounded rationality" which he explains in the following manner: The capacity of the human mind for formulating and solv- ing complex problems is very small compared with the size of the problems whose solution is required for objectively rational behavior in the real world--or even a reason- able approximation to such objective rationality--the first consequence of the principle of bounded rational- ity is that the intended rationality of an actor re- quires him to construct a simplified model of the real situation in order to deal with it. He behaves rational- ly with respect to this model, and such behavior is not even approximately Optimal with respect to the real world (Simon, 1957, p. 198). Similarly, to understand and even to model what teachers do when con— fronted with unexpected student difficulties and insights, it will be necessary to try to reconstruct their simplified representations of events, the "problem spaces" in which their choices were made. To do so information will come primarily from the thoughts teachers are able to a", l- flue-r Uri I I: THIN nIV Q l :1... IE: .71 . a .6 u o l t v D- .11 or, 6 h .\ Pd o .6 v T. i .c .Iv C Q , ‘3 k‘ III... .1 as,” l.‘ c All I .u 1 LL r- A.» .~ . .V; . .. a) Aniv I .51.; ' Man 6 a u u a; q a. 1. '- rl .' .. v ‘4‘ $.~ .- Q: I o\.v 4 c r us I \1- 1 report through the process of stimulated recall. Inferences might also be drawn from their behavior patterns and from other informa- tion they supply about their beliefs, goals, and knowledge. The task environment of interactive teaching is the most diffi- cult to characterize. It is substantially more complex than the re- latively simple tasks which have been used in other process tracing studies. The task environments, even when extensive are more de- finable for some tasks such as chess (de Groot, l965) and the Mis- sionary-Cannibals problem (Jeffr Polson, Razran, & Atwood, l977). Teaching also has a more complex task environment than required by the investment broker managing a portfolio (Clarkson, 1962) or a physician diagnosing patient illness (Elstein, Shulman, & Sprafka, 1978) because of the number of people who form part of the task en- vironment in the classroom. The complexity of classroom teaching is largely a consequence of the number of people involved; students and their actions confound the task environment because of their di- versity and because of the dynamic nature of teacher-student inter- actions. Concept of Instruction This study of critical moments is based on a model of instruc- tion stimulated by discussion with Schwab during a staff session at the Institute for Research on Teaching at Michigan State University. In this mode, the teacher is viewed as an information processor. Interactive teaching is assumed to be a purposeful activity, one which a teacher ap- proaches with some plan in mind. According to Schwab, the teacher seeks to generate an instructional flow for an activity, one which provides . f'Nf-p "c...:, .II‘V‘Ir" ". .Ir, m b 4 . ..-.I ”'3‘ “frr bf" ‘5 h 0 9;.“ ‘r: W" ~ 61'» I [~“ I”_1 _ § 5‘ ‘.- ' - h " :}_~ . u h I“c -. . ‘S 1': -_‘:c '.'\ n.- . \ o (I! p '.\ FF u the momentum to both attract and maintain student involvement. Instruc- tional flow denotes more than the pace of an activity; it also refers to the routines for performing the task and for presenting the mathe matics. Instructional flow includes the teacher's style, strategy, ques- tioning pattern and the content. In the interactive phase of teaching, the orchestration of an activity and the dynamic interactions between teacher and students sometimes require a teacher to respond in ways that cannot always be planned or even routinized. The instructional flow, then, is susceptable to occlusions, unpredictable events which can impede or even stop the intended flow. The primary source of occlusions is unexpected student performance, unexpected in the sense that student performance does not fall within the expected limits of the planned instructional flow. Student diffi- culties and insights represent unexpected student performance; they are observable §tudent occlusions. An overt student occlusion is not necessarily viewed by the teacher as interrupting the instructional flow Thus, a distinction is needed between observable student occlusions and the student occlusions which make teaching difficult, the critical mo- meptg. Critical moments are student occlusions for which teachers re- port some cognitive difficulty or emotional discomfort; to classify an event as a critical moment requires both objective and subjective evi- dence that the instructional flow has been interrupted. Student occlu- sions constitute potential critical moments. Occlusions of student in- sight or difficulty are not necessarily signals of undesirable events in the interactive classroom. Occlusions also provide teachers with "teachable moments." The purpose of identifying and describing 3 mm?!“ .uvv v ’6‘ :r‘ .-i and» e\.- P6 (u a» If” .u A v ‘ I u. A~\ F ~ d O I n i- I ‘ICI" _. :5 u \ r i a a Aux .. .u a: t 5 .~ .. .. . . 10 teachers' critical moments is an attempt to understand how coping with student occlusions makes teaching difficult. A similar distinction could be made between the potential oppor- tunities for and actual "sacking" of the quarterback that the game of football affords the defensive linebacker. The opportunity for a "sack" does not ensure its occurring as most fans know. The line- backer can only try to capitalize on those opportunities which he en- counters and recognizes. In the meantime, like the teacher, he will function with the habitual behaviors he has learned for his position. If this analogy has a flaw, it is that teachers might not be looking for unexpected student performance to exploit as much as defensive linebackers are looking for a chance to down the quarterback. The teacher's role is not only to establish the initial instruc- tional flow for an activity, but to maintain it. When confronted with a student occlusion, a teacher elects to respond with some action. The term elective action was chosen as an alternative to decision because decision has traditionally implied the consideration of alternatives, a process for which research on interactive teacher thought has found little support (Peterson and Clark, 1978). For teacher behavior to be labeled as an elective action requires only than an outside observer would judge that different action could have been taken, regardless of whether the teacher considered such alternatives. Elective actions can vary from little or no shift in the prevailing flow to a radical change in activity. However, it is assumed that after the elective action has been completed, the teacher would either return to the same instruction- al flow which preceded the student occlusion or establish a new one. fi u vl‘u Ill 12/? (D If) ‘1 m :9- .‘ . 3". . n \ \ .- ... .op ~'- ‘ ‘14 o 11 'Thus, the critical moment is followed by a subsequent flow, unless the cycle repeats itself with yet another student occlusion. A proposed model of critical moments is shown in Figure 1.1. It begins with the prevailing flow of instruction which is interrupted by a student occlusion. Some mental processing is assumed to take place followed by the teacher's elective action and the consequences of the critical moment. Consequences can be overt or covert. They may be evident in the subsequent flow of instruction when changes are made or they may be evident in the emotional reactions and thoughts of the teachers. With the limitations in the human processing capacity, it would be unrealistic to expect teachers to process each student occlusion with the same degree of awareness. Research on teacher thought sug- gests this to be the case. The construct of a critical moment, however, refers to a psychological state in which the teacher senses a discon- tinuity or occlusion in the instructional flow. For critical moments, teachers should be more likely to have some thoughts and feelings to report. Since cognitive processing takes place within the teacher's mental "problem space"--a simplified version of reality--an investiga- tion of critical moments should provide some evidence of the con- straints and influences under which teachers operate. For example, if teaching is a purposeful act, then evidence of the teacher's goals and plans influencing his/her thoughts and actions should be detected in the examination of critical moments. 12 Prevailing Figure 1.1. Instructional \ / FTOW STUDENT OCCLUSION Recalled Cognitive Processing and Apprehension \/ ELECTIVE ACTION CONSEQUENCES Subsequent Instructional Flow \ y Model for teaching: a Critical Moment. Critical Moment 13 Study and Questions This study of the critical moments associated with unexpected stu- dent performance calls for rich descriptions, interpretation, and theo- retical conjecture about the instrdctional environment, the unpredictable student performance, and the teacher's reactions, both overt and covert if the phenomenon is to be understood. Critical moments will be inves- tigated through observations, videotaping of teachers engaged in in- struction of the total class, the use of stimulated recall to obtain teachers recalled thoughts and feelings, and an analysis of all these data. Case studies of three teachers will be conducted. In order to provide adequate opportunity for natural variations in the amount and types of unpredictable student occlusions and for criti- cal moments to occur, these teachers will be followed as they teach a unit of mathematics. The set of related lessons in a unit provide a cohesiveness not otherwise available in the brief "snapshots" of teach- ing one lesson or segments of a lesson. The sequence of lessons also makes it possible to establish behavioral patterns and to search for antecedents, causes, and consequences of critical moments. The questions to be addressed in this study fall into four cate- gories: instructional environment, critical moments, mental processing, and implications. 1. What is the nature of the instructional environment? A. What is the nature of the instructional flow? 1. The mathematical content of the units and how they are organized 2. The instructional style and strategies of the teachers 3. Teachers' instructional goals and conceptions of the mathematics to be learned 99, Q“. . o'- I ‘ l . 7‘“ v . O‘o . ._ ‘1 ‘- lh '5 ‘I .i r3. 4 O 14 B. What is the nature and distribution of the unpredictable student performance? 1. Student occlusions of difficulty and insight 2. Teacher elective actions 11. What is the nature of the critical moments? A. What are the overt conditions which produced them? 1. Characteristics of the prevailing flow 2. Student occlusions and teacher elective actions 3. Characteristics of subsequent flow B. What are teachers' covert reactions to them? 1. Cognitive thoughts 2. Factors influencing their mental processing 3. Emotional consequences III. What is the nature and function of teachers' mental pro- cessing of unexpected student performance during critical moments? A. What model(s) best describe teachers' cognitive pro- cessing? B. What are the characteristics of each process? C. What factors influence each process? IV. What has been learned about and what are the implications for the methodology of research on critical moments? What are the implications of the findings on critical moments for teachers, teacher training, and pre- paration of curricular materials for mathematics? Understanding critical moments--what precipitates them, how teach- ers process them, and what consequences they have for teachers--will provide evidence about what makes teaching difficult. It is only a first step in trying to find ways to help mathematics teachers be more a”. 1.. I_' d.. 15 effective in coping with unexepcted student performance. Nevertheless, this study of critical moments will have implications for research on the teaching of mathematics and research on teacher thought. Overview of the Study Chapter 11 consists of a review of the pertinent literature on in- teractive teacher thought with attention to both the methodology and the results of the research. In Chapter III, the methods employed in this study will be described, including the selection of the teachers, descriptions of their classrooms and schools, and the manner in which data were collected and analyzed. Chapter IV offers a description of the instructional environment in which the teachers were teaching. The focus is on the nature and characteristics of the prevailing instruc- tional flow and the nature and distribution of student occlusions and teachers' elective actions. In Chapter V, the critical moments which have been identified from teacher reports of cognitive difficulty and emotional discomfort will be described in detail. In Chapter VI, teachers' mental processing of unexpected student performance and models of their processing will be discussed. A discussion of the im- plications for methodology, teachers, teacher training, and the prepara- tion of curricular materials will follow in Chapter VII. A summary and conclusions are presented in Chapter VIII. CHAPTER II REVIEW OF THE LITERATURE Overview In proposing a research paradigm for studying teacher decision making, Clark and Joyce (1975) were able to report that ...there are virtually no reported studies of teachers' in- teractive decision making. That is to say, we know very little in any scientific sense about the kinds of informa- tion or cues that teachers use when making decisions "on the fly." We have not even developed major categories for classifying the acts of decision making that we know must go on regularly in the course of teaching (p. 3). Since that time, there have been few studies of interactive teacher thought processes. By 1977, when he addressed a conference about research on mathematics teaching, Kilpatrick made the following ob- servation: ...research on the teacher's thoughts and behavior in teach- ing mathematics to elementary school pupils is such a wil- derness! Researchers have spent a lot of time during the past two decades sitting in classrooms and watching teachers teach, and yet almost no one has considered how teachers ap- proach elementary school mathematics (p. 1) Interest in teacher as decision maker was spurred by the ad0ption of this view in educational literature (e.g., Shavelson, 1973; Whit- field, 1971; Bishop, 1970). The decision making paradigm is based on a rational model which assumes (a) goals are set, (b) alternatives are formulated, (c) outcomes are predicted for each outcome, and (d) al- ternatives are evaluated in relation to goals. Shavelson (1976) elaborated on this paradigm for teacher decision making by 16 17 incorporating teachers' judgment of the states of nature and the util- ity (value) of an alternative. Research on decision making sought evidence of the cues teachers used to make their judgments, alterna- tives they considered, and reasons for their choices. To their dis- appointment, the researchers found little evidence that teachers per- formed as decision makers--at least not rational decision makers. The contrast between the types of decisions teachers were ima- gined to make and those that are revealed by research exemplifies two different views of humans. Simon (1957) dispenses with Barnard's (1938) conception of "economic man” as a rational decision maker who "...selects the best alternative from among all those available to him." Instead, Simon substitutes "administrative man” as a "satisfi- cer" who "looks for a course of action that is satisfactory or 'good enough'..." (pp. xxv-xxvi). Thus, as the evidence accumulated, Peter- son and Clark (1978) were led to characterize teachers as satisficers rather than optimizers in the classroom. Elstein, Shulman, and Sprafka (1978) came to the same conclusion about the diagnostic process of physicians. In this chapter the field of research on teaching which has fo- cused on interactive teacher thought and decision making will be re- viewed. In the first section, a portrait of the teacher in the inter- active classroom is presented. Findings of these studies provide in- formation about how teachers function; the nature, substance, and types of decisions; their lessons; their mental problem spaces; their princi- P165; and their instructional approaches. Methods for studying teacher thought are also reviewed with details on the manner in which stimu- lated recall sessions were conducted. Mention is also made of some of the (flassification systems used to code teacher thought and behavior. ..u. 41v .\- out. '1 eh h V! '4 u 18 Research Findings Marx and Peterson (1975), Clark and Peterson (1976) and Peterson and Clark (1978) in their Stanford study'examinedteacher decision mak- ing in a laboratory setting using social studies units with junior high students. Morine and Vallance (1975), in their Beginning Teacher Effectiveness Study (BTES), conducted by the Far West Laboratory, in- vestigated teacher and pupil perceptions of classroom interaction us~ ing reading materials. They compared teachers with varying effective- ness determined by their pupil gain scores and their teaching styles. Sutcliffe and Whitfield (1975) used physiological data in England to study the process of teacher decision making. Joyce, Morine, and McNair (1977), Joyce and McNair (1979), and Morine-Dershimer (l979) in their South Bay study examined interactive teaching thought using ele- mentary school teachers and reading lessons. [These reports may be found in McNair and Joyce (l978-79) and Morine-Dershimer (1978-79) in a shortened form.] Marland (1977), conducting research at the University of Albertas Center for Research on Teaching, used elementary school students to investigate the kind of information teachers processed and the ways in which they processed it during language arts and mathematics lessons. A year later, Conners (l978) replicated Marland's study of teacher thought processes, beliefs, and principles in the same academic areas. In 1980 Heymann did research on the modeling and conceptualizing of teaching, learning, and thinking processes related to mathematics. His study from Bielefeld's Institute for Didactics in Mathematics is the only study which has the teaching of mathematics as its prime concern. AU E .rlh l. r.” P. (a p11 .«I c» .Q. A “U an» r ‘IHI ”Orb - n D.- I n 5 , t u :— as .VI I. re 6 v ‘1“ F ' Q.“ ”P. Aux. 0r. AI— ~_a . .fi . A: . c. ‘1 I ‘4.- » v A H ahh .1 u A_. r n h ..Iv nus P o . .Fu 1‘. a v a u a :L U. \ Au» VI A\v 6 v F. o\o - . -I.- .9 v !\.i d auc F S A.. 1.. n\- ‘II p U a) a I n t I“ I. ES '1 ul’ 19 Findings from these research efforts provide the following por- trait of teachers' decisions while engaged in classroom instruction. Researchers found that teachers operate in a relatively'habituated manner for which they are often unable to recall any conscious thoughts or feelings. Teachers often report doing what they planned to do or what they usually did. This led Clark and Peterson (1976) to charac- terize much of what teachers do as business as usual. Nature of Decisions Teachers rarely mention having considered alternative actions or strategies for their tactical moves (Clark & Peterson, 1976). Instead, as Marland (1977) indicated, alternatives seemed to "pop up" spontane- ously as if they were ready-made habits. Whitfield (1971) came to the same conclusion, saying, "Our explorations so far suggest thai:teachers do not generate all the options open to them before making their choice." He speculates that either "they are insensitive or inflexible...or“they have well-developed value systems as a result of experience which en- ables them to filter out options which they might have generated in early career" (p. 164). Teachers reported considering alternatives only when their lessons were going poorly (e.g., Marland, 1977). There was little evidence of Conscious thought when lessons were going smoothly. Marland indicated that little of teachers' interactive thinking conformed to the problem- s01Ving paradigm. Teachers encountered problems when students failed to SJrasp a point, when teachers were unable to find ways to help a Student understand, and when the teacher's expectation for the indivi- dual (did not match his/her performance. Heymann (1980) found that. teachers emphasize behavior when they are confronted with I ,yb .9- 950 ahA..‘ an“. a she I- ‘H‘ ~nd -‘ at my. VII a any a u u... P\ Ow .5 u t. “1‘ I. l‘r“ .. . c v as. S..- \J- v guy '1 n\.~ 20 learning difficulties as well as when introducing mathematical con- cepts. Marland added that teachers did not usually seek explanations for unexpected student performance. Morine-Dershimer (l979) remarked about a related problem. Teachers sometimes postponed taking action, but she was unable to follow through to see if postponement meant a delayed response or the absence of a response. Teachers' judgments about the effectiveness of their lessons were made primarily on student involvement or attention and not the quality of the classroom discussions (Clark and Peterson, 1976). Teachers in the Stanford study did not mention the performance of individual stu- dents since they were in a laboratory setting and they had not met their students prior to the study. Teachers did not appear to be very flexible with regard to their teaching. Few instances were noted when teachers made changes in their plans (Morine and Vallance, 1975; Marland, 1977). Evidently teachers teach the lessons they plan. Marland (1977), however, focused on evi- dence of customized teaching behaviors in the protocols of all six of his teachers. Teachers do alter their behavior for some unpredictable events, whether they do so consciously or not. Substance of Teacher Thought The substance of teacher's interactive decisions and thoughts were r91£3ted to instructional moves. Most had to do with pupils and what they were doing and saying. Morine and Vallance (1975) indicate the bulk: of teachers' decisions had to do with their interchanges with stu- dents and the activities they planned. Joyce, and McNair (1979) and hH1itfield (1971) discuss how teachers constantly make 21 "inflight" or "on-the-spot" decisions about who to call on, how to pro- vide corrective feedback, when to discipline a student, and how fast to proceed through a lesson. More than anything else,teachers focused on what students were learning. Much less concern was shown for students' attitudes or teachers' affective goals. Teachers also mentioned se- lecting, organizing, presenting, and reviewing content (Morine & Val- lance, 1975; Joyce & McNair, 1979; and Conners, 1978). In the South Bay study, Joyce and McNair (1979) determined that teachers of less successful students were more concerned with student behavior, the effectiveness of their own directions, the appropriate- ness of the content for the students, the content being learned, and pupils' attitudes than were their counterparts with more successful students. Marland (1977) reports that teachers' lesson plans indicate little attention was given to teacher-student interactions; teachers do not anticipate unpredictable student performance before teaching a lesson. Teachers did not, as had been anticipated, engage in fre- quent or systematic consideration of ...their own teaching style, its effectiveness and impact on students....Teachers seldom checked the accuracy of their interpretations...[or]...inferences about the covert cogni- tions of students or their affective states. They operated on the basis of hunches and intuitions (Marland, 1977) Ilfiugs of Decisions The fact that teachers did not consider alternatives as often as the «decision making models predicted led to different ways of classify- ing oz mcwoaw>o__< to .mowow_ananoca Facowowucou .m.a opoae NEE. mpmwmmm szuw>wncH mpmwmmm Ppmp cwopaxm\mvwzm\mnoca xmmmg\puwcmvmm mm>os mcwpoe>mpp< 108 sum to 100 in those categories for which teachers used avoiding or exploiting moves. Most important is how the conditional probabili- ties for the same elective actions varied from all student occlu- sions (Table 4.3) to the different types of student difficulties (Figure 4.2). Although Martha used the redirect/re-ask technique for only a third Of the student occlusions, she was more selective when the nature of the student difficulty is taken into account. She used this techni- que for more than half the errors, but not for student confusion. Sim— ilar contrasts can be noted for Zelda and Ralph with the same elective action. The relatively Small frequencies in other categories did not lend themselves to analysis of such internal variation for a given teacher. The predictability of teacher behavior is indicated when these data are examined by type of student occlusions. When confronted with patterned errors, Zelda was most predictable using redirects and re-asks for almost every error. Martha was less predictable, although she used these same moves for approximately one— half of the students' patterned errors. Ralph's behavior was more dif- ficult to predict because he used redirects and re-asks with only one- third of the patterned errors. He used some form of probing, guiding, or explaining almost as Often. For nonpatterned errors, redirects and re-asks were most likely to be used by all three teachers, but with varied predictability. For nonpatterned errors, Ralph chose redirects or re-asks three times as Often as he had for patterned errors. Zelda was less likely to use the same techniques as she had for pat- terned errors while Martha chose these techniques with the same prob- ability as she had patterned errors. Confusion stimulated a tendency 109 to probe, guide, or explain for Martha and Zelda, while Ralph con- tinued to re-ask. The varied distribution of elective actions by types and teachers demonstrates the selectivity Of teachers in choosing their actions. The nature of the student occlusion does matter, although to a greater or lesser extent according to the variability of the actions used. Ralph's behavior was the most predictable because he varied his op- tions so little. For patterned errors, however, he was less consis- tent. The range Of Martha's and Zelda's elective actions was more extensive which meant their choices were not as predictable as Ralph's at least not for all types of student occlusions. Conclusions The key argument for investigating teachers' critical moments was the fact that instruction of the type recommended by NCTM would neces- sitate teachers' having to contend with more unpredictable student oc- clusions. The concern was that typical instruction in mathematics as described by Fey (1980) and Easley (19 78) does not "create classroom environments in which problem solving can flourish" (NCTM, 1980, p. 4). In other studies about interactive teacher thought, the teachers have been more like the "typical" mathematics teacher. For this reason, it is important to make some assessment about the nature of the instruc- tional focus for each of the teachers in the study. A general charac- terization would be inappropriate as teachers in this study did not teach in the same manner nor did they encourage the same type of learn- ing. Instead, they provided examples of different types of teaching 110 which are located at different points on the continuum between the typical mathematics teacher and the one characterized by NCTM. Ralph represents the typical mathematics teacher using typical methods. He focused on skill-oriented and symbolic tasks using ex- plaining and questioning techniques to teach decimals. Students were asked to supply numerical answers for the various subroutines in con- verting decimals to fractions or vice versa. Lessons were planned and taught in accordance with the procedures and problems in the textbook. Ralph's goal was for students to be able to do the problems; it was a goal for instrumental understanding. There was less variation in student occlusions and in the types of critical moments than in the other classes. Student errors and alleviating moves dominated; Ralph's behavior was also the most predictable. Martha's instructional approach would take her in the opposite di- rection from Ralph. Her teaching came closest to examplifying the NCTM recommendations. Martha's focus was on developing concepts and apply- ing them through the use of models (Cuisenaire rods and pictorial re- gions). Posing problems and asking questions, she relied on student answers and their unsolicited contributions to convey the ideas. Her instructional strategy was one of exemplify and try. As the tasks be- came more complex, however, she did more to direct her students through the problems. Her goal was understanding, which was a goal for rela- tional understanding. She developed her own unit and did not follow any textbook. Her instructional approach to teaching fractions was also more in keeping with research findings (Suydam, 1978). Student - 111 occlusions were more plentiful and diverse with students' Offering a number of suggestions. Zelda taught in ways which had some of the characteristics of the traditional and the NCTM-type instruction. Her intent was to help students span the gap between the concrete models and the more skill- oriented tasks of the textbook. She did this by briefly introducing concepts and relationships through concrete tasks and then shifting to more symbolic tasks. First, students developed procedures based on patterns exhibited through example and then more traditional algor- ithms were introduced. In this manner she encouraged a relational understanding while also fostering instrumental understanding. The nature and distribution of student occlusions and teacher elective actions in Zelda's class were similar to but less varied than those in Martha's. Summary In this chapter, the instructional environments in which three teachers experienced critical moments have been described. Descrip- tions were given for the instructional flow that was generated and the student occlusions and elective actions that occurred during the flow. Despite the similarities in the fraction and decimal units planned for and taught by these experienced teachers for a period of six or seven days, there were numerous differences in the instructional flow: stu- dents, nature of the topics and tasks, and teachers' instructional styles. Many of the differences were a reflection of the teachers' goals for how and what mathematics students were to learn. 112 Differences in the instructional flow also produced some differences in how students performed. Distributions based on the frequencies, proportions, and densities of student occlusions indicated differences across the three classes, both in the types of occlusions and the rate at which they occurred. Student difficulties--errors in particular-— were far more prevalent than insights,just as there were more solicited responses than unsolicited contributions or expressions Of confusion. Teacher responses to student occlusions also varied. In keeping with the patterns of student occlusions, teachers more frequently used alle- viating actions to handle the difficulties, most often with redirects or re-asks. Exploiting and avoiding moves were used more sparingly. The following brief characterization of each teacher's instructional environment will illustrate some of these differences. The varied instructional environments which resulted provide the backgrounds in which to search for critical moments for each teacher. To understand what precipitated the cognitive and emotional discomfort that teachers report with their critical moments, it is necessary to be able to view the events in perspective. Knowledge of the patterns in the instructional flow, student occlusions, and teacher elective actions will provide some clues to what caused and what influenced teachers' critical moments. In Chapter V, the critical moments that were identified will be described in detail. CHAPTER V CRITICAL MOMENTS Introduction The three teachers in this study experienced critical moments while teaching mathematics in the instructional environments described in Chapter IV. The model of critical moments (Chapter I) assumed they would arise over unpredictable student difficulties or insights. Identification of these teaching difficulties, therefore, necessitated that teachers report, through the process of stimulated recall, the problems and discomfort they experienced while coping with the unpre- dictable student behavior. The purpose of this chapter is to describe and exemplify the nature of the critical moments revealed by the teachers. Descriptions will include information about the relevant characteristics of the prevailing flow of instruction, the student be- havior, teacher moves, the thoughts and feelings of the teacher, con- sequences of teacher behavior, and possible causes of critical moments. Written transcripts and audiotapes of the stimulated recall ses- sions were examined for cognitive or emotional evidence Of teaching difficulties. When appropriate, these teaching problems were identi- fied with student occlusions. However, not all critical moments arose over specific incidents of student difficulties or insights. Some arose over more general patterns of student performance. These pat- terns created pervasive teaching problems which extended over time and 113 114 and across students; they characterized student performance for acti- vities, not for specific student occlusions. Even when patterns Of student performance were set by student occlusions, it was not any specific occlusion that troubled the teacher; it was the overall pat- tern. Teachers in this study did not report teaching difficulties in the absence of external cues from students. Critical moments described in this chapter arose over two different sources of unpredictable stu- dent performance: isolated and momentary student occlusions and per- vasive patterns of student performance. In all, only twenty critical moments were identified during the six or seven days of instruction by the three teachers in this study. Fourteen occurred over specific student occlusions and six over per- vasive patterns of student performance. Compared with the 421 student occlusions and 41 activities listed in the previous chapter, a rather small proportion resulted in critical moments. Fifteen percent of all the activities became pervasive problems for the teachers, but 85% did not. The number of critical moments associated with specific student occlusions represents three percent of the total number Of student oc- clusions, but this figure is somewhat misleading. Critical moments about specific student difficulties or insights often occurred over episodes involving several related occlusions rather than a single interchange whichsuppresses the actual percentage. Furthermore, for a pervasive teaching problem, teachers reported recurring thoughts about their students' performance during an activity; it was not a single experience. Also, unpredictable student performance for which teachers recalled thoughts and feelings about their successes were not included. 115 To qualify as a critical moment, there had to be some evidence the teacher was experiencing cognitive difficulty or emotional discomfort. Why so few of the student occlusions and activities resulted in critical moments is, at least in part, explained by what has already been established in other studies of teacher thought (Clark and Yinger, 1979). Much of what teachers do is "business as usual." Teachers im- plement their plans and rely on routine behaviors for much of what they do unless they find things are not going as expected. Routine beha- viors include not only the typical responses to students, but routines associated with the task and content of an activity. The advantage of routine behavior is that it enables teachers to reduce the complexity of their tasks and to be more selective over what they do give their attention to. It was the more distinctive student occlusions, perfor- mance patterns, and teacher elective actions that teachers tended to report through stimulated recall. Such things as student errors which signaled clear misconceptions or recurring incidents of the same dif- ficulty were noted. These did not occur often. The same was true for teachers' elective actions. The less frequently used exploiting and avoiding moves were quite prevalent in critical moments involving stu- dent occlusions, as were moves which demanded mathematical knowledge not being used during an activity (see Appendix for time and content of activities). These events were distinctive; they were not common. The scarcity of these twenty critical moments in no way diminishes the need or importance of studying them. If anything, it increases the possibility of learning something of value about teachers and their teaching Of mathematics. Tracing three teachers' units in mathematics provided the background necessary for viewing their critical moments in 116 perspective. Critical moments provided the focal points around which to organize and interpret the otherwise massive amounts of data. Dis- tinctive features of critical moments were more apparent when viewed in relation to the patterns of student and teacher behavior and charac- teristics of the activities. Furthermore, with access to so much data, information is incorporated from all four Of Schwab's common places-- teacher, learner, curricdlum, and milieu (Schwab, l978)--a1though the perspective of the teacher is central. Situations which produced critical moments for the teachers in this study are universal teaching problems and, therefore, Offered no surprises. Critical moments arose over essentially four different situations: student difficulties and insights, pacing dilemmas, and unexpected success. For purposes of simplification, the author labeled these situations Types A, B, C, and D. A brief description of what caused the teacher trouble in each type of situation follows: Type A: Student Difficulty: Teacher was unable to cor- rect misconception, diagnose difficulty, or elicit desired response Type B: Student Insight: Teacher was unable to accept or satisfactorily execute the elective action Type C: Pacing Dilemma: Teacher had problems maintain- ing the instructional pace while trying to meet the needs Of one or more students who had difficulty Type D: Unexpected Success: Teacher was confronted with unexpected student success This is not intended as an exhaustive listing of all possible teaching problems, although it is a beginning of such a taxonomy. The distribution of critical moments across these four situations and three teachers is shown in Table 5.1. Most student difficulties and insights arose over single student occlusions, 117 or clusters of related occlusions while pacing dilemmas and unexpected success were most frequently pervasive teaching problems associated with activities. More critical moments were reported by Martha partly because of her four pervasive teaching problems in contrast to one for each Of the others. Pervasive critical moments accounted for 20% of her activities, eight percent of Zelda's, and about 12% of Ralph's. Martha also gave total class instruction for more time and with more activities. She provided only one separate work period for the stu- dents during her introductory unit on fractions. Although more student occlusions occurred during Martha's instruction, it was Ralph who had the highest proportion of critical moments involving specific student occlusions--12% compared to an average of three percent. Table 5.1. Distribution Of Critical Moments by Type and Teacher. Iype_ hghtha Zelda Ralph Igtgl A: Student Difficulty 4 (1%)3 (lg) 9(2P) B: Student Insight 2 l 3 6 C: Pacing Dilemma (2%) ' 1 3(2p) 0: Unexpected Success (2%) ' ’ 2(2p) -1T;— -——3— -_—7— 20 a (#P)refers to the number of pervasive critical moments Experiences teachers reported and circumstances which led to their critical moments revealed similarities as well as differences within 118 and across types. To highlight these similarities and differences, the critical moments are described by types and, when appropriate, by subcategories within types. Type A: Student Difficulty The student difficulties which resulted in critical moments were correcting misconceptions (A-l), eliciting desired responses (A-2), or diagnosing difficulties (A-3). The somewhat different nature of these tasks define the subcategories into which these critical moments are clustered and described. In the first subcategory (A-l),Martha proved to be the most consistent, reporting four similar experiences. Each was an isolated episode in which her attempts to help a student under- stand a concept or procedure failed. There was only one other criti- cal moment in this cluster; it was reported by Ralph. Related, but quite different with regard to the covert teacher experiences, was the second subcategory Of student difficulty (A-2). Zelda and Ralph found they could not elicit the desired response to a question. The third subcategory (A-3) contained two critical moments; both were pervasive problems caused when teachers could not determine why students were exhibiting difficulties with an activity. Zelda noticed student dif- ficulties with the task, and Ralph had trouble figuring out why the students were not more willing to volunteer. Type A-l: Teacher Unable to Correct Student Misunderstandjhg So similar were the experiences Martha reported about four criti- cal moments in this first subcategory that one general description and one detailed example characterize them quite well. Brief remarks about 119 the other three critical moments are offered only to point out varia- tions in circumstances, their impact on Martha, and causal agents. The four critical moments occurred when Martha found herself un- able to help a student understand a concept or procedure despite her efforts to do so. A chain of interactions began with what Martha be- lieved to be a signal of student difficulty, a patterned error, or a specific question about a procedure. The initial indicators of stu- dent difficulty included student-initiated confusions and responses to teacher questions. One initial speculation was coded as insight in- stead of difficulty because the question raised had mathematical va- lidity. Whether the student had intuitively recognized this or not was never certain, but Martha's interpretation was. She reported thinking the student was confused and had behaved accordingly. These initial signals presented Martha with tangible evidence of a particular mis- conception or concern over the task. The initial student difficulties in these four critical moments were precipitated by some new aspect Of the content or task. Students were being asked to perform a new varia- tion Of a previously introduced task, work with a new fraction concept, or both. - In response to these distinctive signals of misunderstanding, Martha chose elective actions which offered some form of assistance through an exploiting or alleviating move. She illustrated an erro- neous interpretation of an improper fraction in contrast to the cor- rect one already in evidence, she asked the class to build an appro- priate representation of an improper fraction before asking what frac- tion was modeled, she gave the class another problem to work so that a student could try to figure out why it was that white rods were used to 120 identify the sum of two fractions (teacher error as it depends on the choice of unit), and she probed to try to uncover how a student was interpreting the fractional pieces. Both the student signals which precipitated the critical moments and Martha's techniques for dealing with the student difficulties were relatively unusual behaviors. Each of Martha's Type A-l critical moments was preceded with a related incident which may have sensitized her for what was to follow. In these related student difficulties, she may have prepared herself or her prior elective actions may have set a precedent which was easily followed. As with the prevailing flows in which these critical moments took place, Martha never offered these prior incidents as reasons for her actions or covert responses. These critical moments occurred at the end of the lesson during activities that had not been planned. She never mentioned time as having influ- enced her, nor did she seem terribly aware of it at that moment. The lack of preparation, on the other hand, may have contributed to Mar- tha's difficulties. By themselves, the initial student occlusion and elective action did not produce sufficient evidence of a critical moment. It took evidence that the student misunderstandings persisted. Recurring sig- nals were made possible as Martha provided additional Opportunities for the individuals to respond. She did so by continuing to question the student or by checking back with the individual after a demonstra- tion or by picking up on a less direct cue indicating a student was still struggling. In the face Of the second student difficulty in- dicating that her efforts had not been successful, Martha continued to try in three of her four critical moments. She used some of the same 121 techniques already described and the more direct one Of telling, a move she seldom used. She never continued after four inappropriate responses from the same student. Martha broke Off her attempts to help the students with an avoid- ing move, knowing that the students still did not understand. First, she indicated that the idea or task would be dealt with again, a positive gesture. Then, she either suggested that the student con- tinue to "puzzle it out" or indicated her belief that the student would eventually "get it." Her second remark suggests the conse- quences of differential expectations for her students. The challenge to continue "puzzling" was given to the two students for whom she had high expectations, and her supportive comments went to two low expec- tation students. A significant critical moment, both in its duration and how it affected the teacher, occurred during the third day of the unit during the last activity (Activity 14). Students had been so successful with the planned activities that Martha felt another change was indicated. Switching the task from Offering fractions with like denominators, she began giving two fractions with like numerators for students to com- pare. For the second problem Martha inadvertently wrote 6/5 before the class had been introduced to improper fractions. As might have been foreseen, students objected to this number, and Martha exploited the opportunity to demonstrate how the improper fraction could be re- presented. After two "pies" had been drawn and divided into fifths with six of them shaded that. theinitial student occlusion for this critical moment occurred. 122 Max wanted to label the pictorial representation for 6/5 as 6/10. Martha's choice of action was to counter by illustrating 6/10. There was ample precedent for this action during the activity, as she had used the picture presentation to justify the comparison of 3/4 and 3/5, the first problem, and then again to demonstrate 6/5. Cut- ting the circle into tenths and shading six Of them, Martha said, "Tenths are a lot smaller than fifths." Unfortunately, she had drawn the circle for tenths much larger than she had the circles for fifths which made a realistic comparison of the two models impossible. The distortion of the circles did not appear to contribute to Max's argu- ment, however, as he was focused on the number of pieces into which the circles had been divided rather than the relative sizes of the pieces. The distorted unitscficlsuggest Martha may not have thoroughly understood the necessary conditions for representing and comparing fractions. After having illustrated 6710, she asked the class if 6/10 and 6/5 were the same. As "no" came from a number of students, she erased both representations and asked if anyone were confused. Stu- dent response was mixed. Martha responded with, "We'll try a few more and maybe that will help you.” The antecedent to "few more” was not clear, however, as she gave another comparison problem, one with no improper fraction. At this point Martha was not bothered and, had Max not been so persistent, the critical moment might never have occurred. She was not "let off the hook" so easily as Max continued to wonder and, evi- dently, mumble about the picture for 6/5 being 6/10. Noticing this, Martha provided Max with another Opportunity to use an improper 123 fraction with a somewhat easier task. Writing down two fractions with like denominators, including one improper fraction, she asked which was larger, 5/4 or 3/4. At first, Max said he didn't know, although he later gave the correct response. In the meantime and after wait- ing for him to respond, Martha tried once again to demonstrate the concept with another picture, taking suggestions from the class. After several student comments (unfortunately inaudible on the tapes) Martha went on to explain, "I have to have 4/4 and 1/4 to make 5/4." Once again, she turned to Max to probe, "What about that puz- zles you?” Still struggling over the picture for 6/5, he repeated his belief that it represented 6/10. Sounding exasperated she replied, "You can't do it that way!" and proceded to recapitulate what had already been said. Her last remark pointed to future chances to deal with the concept and encouraged him to "keep puzZling over it." Her own recollection indicates why Martha discontinued the dis- cussion: I was feeling inadequate here because I couldn't explain it to him! I wanted to make it clear. I've got to come at it in a new way, and I'm really getting frustrated here! ...All I can think to say at one point is that you just don't do it that way! [Activity 14] Several things became clear here in addition to her feelings of in- adequacy and frustration. It substantiates her conscious desire to help the student, a goal consistent with her learning and instructional goals for the unit. She valued and wanted to be flexible and respon— sive to her students as they developed an understanding of the frac- tion concepts. Martha's inability to think of a different explanation because of her limited knowledge forced her to abandon her goals. 124 Peterson and Clark (1978) found approximately 40% of their teachers' decisions demonstrated an inability to think Of an alternative. In Martha's other three critical moments from this category simi- lar consequences were reported. Frustration was the most commonly re- ported emotion, although the intensity of the emotion varied as did the amount of effort expended. She also reported having difficulty in figuring out what a student was thinking or what could be done to help to explain the ideas. One source of her difficulty was her own knowl- edge Of fractions and the concrete approaches she was using to teach them. Evidence of her limited knowledge of fractions came from Martha's own admission and several mistakes she made during the unit. She was asked if she realized she had drawn unit circles of different sizes for comparing tenths and fifths to see if it were accidental or that she had insufficient understanding. She had not realized she had drawn different sized circles, nor did she immediately grasp the signifi- cance of that fact. Martha appeared confused as she first said it didn't matter and then that it did. It was only after the investiga- tor interfered, posing an analogous question about cutting different sized pizzas, that Martha was certain she needed to use the same sized units for comparing fractions. It was not that she had never known the concept, but that she had not thought about it ahead of time and was not familiar enough with it to have been able to remember itunder pressure. Similar difficulties were mentioned by all the teachers at one time or another as they talked Of needing more time to think some things through or to come up with another idea. They did not, however, talk about the need to think something through ahead of time or to plan ' 125 more meticulously. Research on teacher planning demonstrates that teachers do not typically prepare their lessons with such care (Clark& Yinger, 1979; Smith & Sendlebach, 1979). The reason Martha was so interested in trying to find out what students were thinking in these critical moments can be attributed to several things. It was compatible with her instructional goals. She also expected her students could learn the concepts. This was un- doubtedly enhanced by her perception Of student effort; several times she mentioned they were trying. For these same reasons, Martha also saw the student difficulties as indications they did not understand, and she felt that could be corrected with some effort on her part. In addition, Martha believed the fraction concepts were "hard for students to understand." She was not attributing blame to the students, but Offering a plausible reason for their difficulties. Martha may have persisted in trying to help Max as long as she did because of her expectations or perceptions as much as her desire to help him. "I don't know quite why he is not seeing it. That's why I keep questioning him. I want to see where he was coming from. He iS 0” the verge 0f getting It!" Her last comment may have been due to a perceived change or an anticipated one. Research on the effects of teacher expectations have shown a greater willingness on the part of teachers to pursue an idea with high-expectation than with low-expecta- tion students, and Max was rated in the top group of mathematics stu- dents. A third suggestion of teacher expectation effects in Martha's Type A critical moments was her reported fear of confusing the stu— dents, a fear expressed for the students for whom she had lower 126 expectations. This mention of fear was confounded with Martha's ap- parent understanding Of the mathematics in these critical moments. She seemed to know what the representations and concepts were even though she had difficulty figuring out what the student was thinking orin finding a way to explain it. When dealing with the better math students, Martha appeared to have trouble understanding the concepts, and she was aware of it in one of the two cases. Martha's sensitivity to confusion students might be experiencing was apparent in the following comments from the two critical moments with her less able math students. Boy, this is screwing up her head [imprOper fractions] 1 can just tell that! I can just feel her confusion! [Activity 17] [What] I worry about here is that he has his way Of un- derstanding it [unit fractions] and by forcing him to say other things, I might be confusing him...I just wanted to get off it so it didn't shake him up. [Activity 6] Her fear Of confusing students was based her own experiences in learn- ing mathematics which were not always satisfying. "Math is like that for me [confusing]. And when someone tried to explain it to me, it just got worse because I was so upset I wasn't getting it! My whole mind was turned like an eggbeater!" Martha had good reason for backing off when she sensed or anticipated that students were or might be get- ting confused. Ralph experienced the only other Type A critical moment which was similar in the evidence of student misconception, the teacher's im- pulse to Offer some assistance, recurring evidence of student diffi- culty, and admission of being unable to think Of an alternative ap- proach. Both teachers mentioned not wanting to confuse students as 127 supporting their choice to offer no more help, but Ralph's experience was quite different from Martha's. He never really attempted to ex- amine the student's misconception or to provide additional Opportuni- ties for the student's confusion to surface. Instead, he merely told the student what the correct response was and hoped that it was suf- ficient. A nonverbal cue indicated it was not. The critical moment occurred during a review of the first day's assignment during the first activity of the second day (Activity 3). The content, then, was still relatively new,and the problem which triggered the patterned error was a decimal in which a zero was in the last place and after the decimal. It was the first time such an example had been covered in class, al- though Ralph had selected another problem with a zero between two other non-zero digits. In expressing 5.490 as the sum of whole and fractional numerals, Bill answered incorrectly saying 90/100. Ralph responded by asking what place the nine was in and was given the correct response. At this point, he told Bill how he would have responded (9/100 + 0/1000 or, simply, 9/100) and then moved right on to the next problem. For this critical moment, Ralph detected the student's continuing con- fusion from his behavior. "He just kind Of gave me a blank expres- sion on that. I could see he wasn't clear on that, but I hoped I made it clear when I said don'tforget the zero in the thousandth's place." This was the only occasion for which Ralph revealed having wanted to offer an explanation and found himself unable to do so. Knowing what he had done to assist the student was insufficient and that he was unable to think of anything else to say clearly made an impact on him: 128 I was very conscious of whether I should speak Of [stu- dent's] mistake and try to clear that up in his mind... but then I thought, because I was on very shaky ground there, I am going to confuse others by doing that. I had the tendency to want to [clear it up on the spot] real quick and then I caught myself. I thought, I don't know what I would do on the board or what I could say... [so] I am just going to give the correct answer. Unlike Martha, Ralph did not label his emotions, but his reactions seemed to be a mixture of fear and frustration. Ralph's fear of con- fusing students was directed not at the student, whose confusion he was trying to alleviate, but at the rest of the class, which is similar to his pacing dilemma. Also, there was no preceding incident of a re- lated nature, though something else may have served to heighten his sensitivity to student difficulty with the problem. As it was a re- view of the previous day's assignment, Ralph deliberately selected problems with greater potential for error. Type A-2: Teacher Unable to Elicit Desired Responses A somewhat different situation led to Zelda's only Type A-2 cri- tical moment. She was unable to elicit a desired response to a planned question from anyone in the class, even after repeated attempts to do so. As with the Type A-l critical moments, students were experiencing difficulty with new content. There was recurring evidence that the difficulty had not been resolved, and the final teaching move of this episode was to leave the question using a delay tactic. The episode occurred during the first activity of the unit before the clear pattern of student response or teacher questioning had been established. Zelda was using the numberline to motivate the students to think about numbers between 0 and 1. She first divided the segment 129 (0,1) in half and obtained 1/2 as the name of that point. Then, after marking the other two quarter points and getting 1/4 as the name of the first one, she sought to obtain 2/4 as the name of the next point. It was over this question that the student difficulty and cri- tical moment occurred. Zelda agreed to the first response of 1/2 and asked if it could not be something else. One-fourth was offered next, and Zelda coun- tered by pointing to that position on the numberline. By this time, more students were trying to call out answers, but no one called out 2/4. Again, she acknowledged that 1/2 was correct, reemphasized where 1/4 was located, and asked for the value of the next point. A repeat of the inappropriate callouts was rebuffed as Zelda encouraged them to think about it and not just call out answers. One-fourth was offered again, and this time she agreed that the second segment was another fourth and that two l/4s reached the middle. At this stage, Zelda altered her question somewhat, asking what 1/4 and 1/4 would be. An error of 1/3 was essentially ignored as she asked how the answer should be written. After seven unsuccessful attempts, Zelda finally aborted her quest, saying to the class, "Let's not worry about that right now; I don't want it to get confusing." Differences between this and the previous critical moments are apparent--some due to differences in instructional style. It was not one individual exhibiting this difficulty; it was the class. A num- ber of different students tried unsuccessfully to give Zelda the re- sponse she was seeking. Some she solicited from individual students, and others were callouts. While this was a fairly common situation for Zelda, in all the other cases there was at least one correct response 130 for her to select. In this critical moment, this was not true, and her covert reaction was different. Zelda seemed to be more annoyed than frustrated that the students had failed to come through on this ques- tion. She was not bothered by the recurring evidence Of student dif- ficulty, although the number was large, but that no one was able to give her the response she was seeking. Her discomfort was aggravated by two things: she had planned to pursue this line of questioning throughout the activity, and she had reason to believe thatiflmestudents were capable of giving her the answer. Her choice to abandon seeking the answer Of 2/4 was total. She did this recognizing that she had other things to do and there would be other opportunities to deal with the concept: I really didn't consider anything else because I knew I wanted to do something with Cuisenaire rods today. I thought it really wasn't worth it, and we are going to get there anyway. It wasn't a matter Of I'll do this, this, and this; it was a matter of I want to get to these things; and if I'm not going to, I'll just dump it. Another comment made nearer the end of the recall session, however, suggested that the incident had made a more lasting impression on Zelda and that she may have been more disappointed than she had rea- lized at the time, "That really limited what we accomplished today!" She made it clear why she preferred not to spend too much time on this concept: "There are so many things to learn about fractions that it seems a shame to drag it out forever." Zelda's surprise over the inability of the students to respond was evident when she spoke of her plan. "I was thinking about going 1/4, 2/4, and 3/4; but the next thing that happened is I tould nofl because no one could rename 1/2!" The expectation that students should have 131 been able to answer her question was based on a previous year's learn- ing experience with some of the students in her class: I was thinking I had a hard core [about six or seven stu- dents] that had done fractions before, and we could start building equivalent fractions which would lead us right into that the next day; so it would be very easy then to deal with equivalent fractions. Unlike Martha who attributed her students' difficulties to a lack of understanding and difficult concepts, Zelda suggested an inability to remember-~a mental bhmfle-andenvironmental conditions as the inhi- biting influence Of large group instruction. Neither required a change in her expectations about the capability of her students: ...Obviously they don't remember it. Also, we had already named it 1/2, and they probably thought that you can't name it anything else because a half is a half is a half! ...the ability range is interesting because it stops kids that can handle it in small groups. [Student] could han- dle it last year...but she couldn't today. Both of these conditions, one internal and one external to the students, were unstable. She did not mention the task as a possible source of difficulty as did Martha,even though the numberline task was distinctively different from what the students had experienced before. With Cuisenaire rods, individual rods can be counted as 1/4, 2/4, and so on, while the numberline requires that a value be assigned to a position. Whether or not the task difference contributed to the stu- dents' difficulty cannot be stated, but the distinction existed. Zelda's reports of her cognitive experience during this time were in conflict. She spoke of trying both to think of an alternative way to ask the question and about not thinking of any alternatives. If both reports were valid, it would be reasonable to assume that the search took place during the time she was engaged in her alleviating 132 actions, still seeking the desired response, and that the absence of search was concurrent with her decision to abandon the question. In much the same manner as Martha did, she described her search as "kindm Aw. :ucmmm v, Fmoc pom [pmcacmch compu< m>vpompm m m H m z cowua< L zm KI \ on A 30pm . i i i - i u a : m>wpomFm - i u u u - i . comma—duo 30pm «coacmmpam . - i i i - - m H musomxm . i - u - i i . pcmuapm mcwfiwm>mca / m m z a 180 goals. In this model, critical moments would occur when teachers changed their initial goals for responding to a student occlu- sion. Not all student occlusions were processed in exactly the same man- ner. The small number of critical moments was an indication that teachers cope with numerous incidents of student difficulty and even insights without havingeuw conscious thoughts worth retelling. Conse- quently, in modeling these events, it was necessary to incorporate a mechanism by which teachers could process student occlusions automati- cally as well as with increased awareness. The following questions, one for each of the cognitive processes, provide such a mechansim: 1. Is the student occlusion routine (not novel)? 2. Is a goal available to handle the occlusion? 3 Is an elective action available? 4. Is the goal satisfied? Affirmative responses to all the above questions indicate the teacher was able to process the event with a low level of consciousness using routine behaviors. A negative response to any of these questions, how- ever, indicates the corresponding process would be performed with an increased awareness by the teacher. Nature and Function of Cpgnitive Processing Intenpretation Process. Teachers' interpretation of student oc- clusions were basically interpretations about the content of student occlusions. Of interest to teachers were the validity of student re- sponses or contributions and, in some cases, a diagnosis of their mis- conceptions. For the most part, teachers were correct in their judg- ments about the validity of student difficulties or insights with some 181 notable exceptions. Teachers' judgments, regardless of their correct- ness, served as a filtering system by which they exercised control over the direction their lessons would take. For the first few problems in which fractions with unlike denomi- nators were being added, Zelda used only unit fractions, thus exempli- fying the pattern ;—+ [1' = a a+bb . When one of the students tried to express this rule, Zelda was not receptive. She did not seem to re- alize the validity of the rule or that her choice of fractions had precipitated the offer. Her rejection of the offer was not in keeping with her usual strategy of selecting problems for the purpose of stim- ulating and rewarding pattern recognition, but this was also not a rule she had intended to elicit. In this instance, Zelda's lack of expectation and unfamiliarity with the pattern biased her interpreta- tion of the student's insightful offer. Student occlusions for which teachers were more aware of their in- terpretive thoughts had distinctive characteristics. Patterned errors and unsolicited contributions were perceived as being novel, particularly if they were being offered for the first time. So, too, were student occlusions which ran counter to the overall pattern of student response. Zelda reported her conscious thoughts about the rule being offered in the previous example just as Ralph did about Judy's inability to answeraisimple review question. Less distinctive stu- dent occlusions were processed more routinely. Judging the correct- ness of a student response was typically done with little conscious thought unless the teacher had trouble determining the correctness of the response or the response proved interesting. 182 Repetition Of student occlusions influenced teachers' judgments about their distinctiveness; it either enhanced or suppressed teachers' awareness of some occlusions. Repetition made Offers of equivalent fractions seem more commonplace to Martha until she no longer men- tioned them during the recall process. Skemp explains this process in the following way: ...when these [student occlusions] are encountered for the first time...consciousness is heightened. As a director system gradually becomes more adept in a new situation, its functioning becomes more routine and less conscious...If the initally novel components of a situation are enountered re— peatedly, they gradually cease to be experienced as novel. Less conscious attention is then needed for the director system to function in what has now become a familiar situa- tion (p. 15, 1979, bracketed words added). Repetition could also heighten teacher awareness. During one of Mar- tha's lessons, the same error pattern occurred in responses from several students over several problems. Martha did not recognize the pattern immediately; it was the repetition of the same error pattern which alerted her to the misconception. Similarly, it was not Max's initial error with the improper fraction that produced a critical mo- ment for Martha; it was his persistent misinterpretation of the con- cept which caused her problems. Another factor influencing teachers' perceptions of the novelty Of student occlusions was interactive planning, which refers to thoughts about the teaching of an activity while it is in progress. Concentra- tion on the presentation of content and the formulation of a question- 183 ing pattern for an activity were two forms of interactive planning which either suppressed or enhanced teachers' awareness of student occlusions. How interactive planning could prevent a teacher from con- sciously interpreting a student response is easily imagined, but an example is needed to illustrate how interactive planning could heighten a teacher's awareness of an occlusion. In planning the first activity of her unit, Martha had been un- certain how students would grasp the concept of fractions. She was still not sure when the first question was asked. Students were to hold up a Cuisenaire rod which represented one-third of the rod she displayed. Out of numerous correct displays, Martha noticed an incorrect one. It suddenly occurred to her to exploit this error in order to clarify the concept, and the move proved most effective. Martha's incomplete plan caused her to notice and exploit an incorrect response that could easily have been overlooked. It was an error that definitely would have been ignored if the correct answer had been all that she was seeking. Goal Setting Process. Goals which directed teachers' responses to student occlusions were momentaryggoals. Teachers were conscious of having chosen some goals while others were revealed through their ac- tions. Conscious goal setting occurred with distinctive student oc- clusions: patterned errors, insightful suggestions, and student re- sponses which deviated from the overall class performance or the ex- pected difficulty of a question. 184 Goal setting was an automatic process by which teachers relied on general operating goals such as seeking correct responses to their questions. The process was also simplified by repetition of an occlu— sion, when a teacher continued to operate with a goal set for an earli- er occlusion. This was true for Martha as students continued to volun- teer equivalent responses. Using a goal set for one of the earlier in- stances, she responded to subsequent offers in much the same manner. Some goals deliberately incorporated subsequent occlusions. Martha chose to delay feedback until different answers had been given for the first addition problem involving fractions with unlike denominators. Of the nine answers given, she indicated having only thought about one or two. Teachers reported goals were either impulsive or reasoned choices. Impulsive goals reflected their interpretations of distinctive student occlusions. Students' exhibiting difficulty were to be helped, and in- sightful ideas were to be used. Rppggpg were offered for goals which did not coincide with these interpretations. For example, teachers were reluctant to explore mathematics they considered inappropriate for their classes. When a student tried to justify the representation of a fraction by division, Martha simply acknowledged the suggestion and moved on to Unenext problem. Some students, she explained, had not yet started to work on learning how to divide and were unfamiliar with the operation. Teachers' momentary goals were not complete plans of action. They merely indicated the directions teachers wanted or be— lieved they ought to follow. Similarities in teachers' goal statements did not reflect the full nature of their intent, nor did they indicate 185 the specific moves teachers would use. Simple goal statements,such as wanting to help a student.had different meanings. For Ralph it meant providing hints and clues until the student could come up with the cor- rect response, while for Martha it meant assisting the student to ac- quire a proper understanding of a concept. These substantially dif- ferent goals reflected teachers' more stable instructional goals (being able to work problems in the text versus developing and applying concepts) and their different conceptions of learning (paying attention versus constructing concepts through concrete experiences). Teachers sometimes functioned with more than one goal in mind, al- though one eventually dominated. On several occasions, Martha re— sponded to patterned errors by probing and asking related questions. She seemed to be trying both to find out what caused the students to make their errors and to help them understand. For some episodes which ended without any clear evidence that the students understood, Martha expressed pleasure over having figured out possible causes of their errors. Her desire to know what the students were thinking may have overshadowed her desire to help them and, possibly, her awareness of whether she had. In this manner, goals influenced her judgments. Once teachers were able to satisfy their initial momentary goals, they reestablished the instructional flow. When they were unable to satisfy these goals, however, they abandoned or converted them to se- condary goals. Secondapy goals were reported in all critical moments involving student occlusions. This was a form of retreat as it was typically followed by an avoiding move. Skemp describes the changing or unsetting of a goal as a defense mechanism by which a person is 186 released from the obligation of an existing goal. While this mechanism had immediate benefits for teachers, it also had emotional consequences as will be discussed in a later section. Secondary goals were set by default, interference, or some combi- nation of the two. To default meant that a teacher either had to anti- cipate being unable to effect a desired change or to feel unable to retrieve the necessary knowledge, mathematical or pedagogical The switch in goals could occur before any action had been taken or after numerous attempts had been made to satisfy the initial goal. It was by default that Zelda finally gave up trying to elicit 2/4 as the name of a point. After seven attempts she could no longer think of anything to try. In this case the change was for more than a momentary goal. Zelda abandoned all questions dealing with non-unit, proper fractions for the rest of the lesson, thereby limiting the amount of practice students received on this task. When Martha finally broke Off her attempts to help Max understand 6/5, she was exasperated at not being able to think of yet another way to approach improper fractions. Having tried and failed caused Martha to report feeling inadequate, but not Zelda. She admitted feeling limited, but not inadequate. To describe how secondary goals came to be set by interference, it is necessary to make a distinction between desired and undesired goals. Skemp refers to them as goals and antigoals. The assumption is that teachers would actively seek to effect movement in the direction of de- sired goals and away from undesired or antigoals. Teachers learned to avoid certain situations from unpleasant teaching experiences which they had translated into principles or "rules of thumb." Martha had learned it was best to back off once she sensed students were becoming 187 confused because excessive teacher explanations had only confused her when she was learning mathematics. Similarly, Ralph had found that when attempting to explain something for which he was unprepared, his lesson deteriorated. Fear of these antigoal situations interfered with teachers' willingness to try to satisfy their initial momentary goals. Fear prompted teachers to switch to secondary goals. Teachers did not begin their lessons thinking about these antigoals; their awareness of antigoals was activated by their recognition of the potential for such situations to develop. Search Process. Once a momentary goal for coping with a student occlusion was set, an elective action was chosen for carrying out that goal. This was the function of the search process. Most elective ac- tions were chosen and executed with little conscious thought on the part of the teachers. These were commonly used behaviors which could be handled with content-free moves such as a redirect or with no more knowledge than required by the mathematical task and routine of an activity. At most, teachers had to rely on knowledge needed for recently completed activities. These were relatively simple plans for action, many of which could be attributed to habit. Habits, ac- cording to Skemp, ”are learnt, not innate, but with repetition they be- come so automatic that one might almost think of them as 'wired-in...'” (1979, p. 168). This is not to say that all such teaching behaviors were found without conscious thought, but that most were processed automatically, thereby enabling teachers to press on to the next ques- tion or problematic situation. 188 Just as distinctive student occlusions received more conscious at- tention from the teachers, so did distinctive elective actions. Teach- ers reported thoughts about moves which did not conform to their peda- gogical or mathematical routines. This occurred before the routine of an activity had been established or when exploiting and avoiding moves were being used in a new situation. Unusual elective actions simply received more conscious effort than routine moves. Teachers reported their unexpected successes as well as their problems. They were acutely aware of an unproductive search for an elective action. In some instances they were aware Of this immediate- ly following the student occlusion, while in others several attempts to satisfy a goal were made before they realized their options were being depleted if not exhausted. Ralph, for example, realized immediately that he did not know how to respond to an insightful suggestion or how to explain the decimal concept. On the other hand, Zelda talked of the difficulties she had in trying to think of ways to elicit the name of 2/4 as she continued to ask one question after another. The nature of the search process was relatively unexplained by what teachers described. Their plans seemed to be formulated from scattered bits of information they were able to retrieve. Teachers re— ported sudden recognitions of what they might do with comments such as, "Oh, I know what I can do!" There was little description of how or even why they chose the actions, nor was there much evidence of their having considered alternatives. Whatever plan of action their search produced was executed. In fact, some moves were initiated before the teachers actually knew what they were doing. This was apparent when ~ 189 Martha and Zelda became confused while trying to exploit students' in- sights. The effectiveness of their actions was determined after the fact. Once a momentary goal had been set, teachers sought to find, execute, and Egg: an elective action. This was similar to Miller, Galanter, and Pribram's TOTE (1950). The processes were performed in rapid-fire succession with the sequence repeated until the teachers satisfied their momentary goals or until they switched to a secondary goal by default or interference. The largest number of moves used to pursue a goal was seven, which occurred when Zelda failed to elicit 2/4 as the name of a midpoint. Teachers usually made two to four attempts when students did not understand or respond. The search process just described resembles what Skemp terms in- tuitive path finding.” It is a form of planning in which connections between concepts are established. The process itself might not be con- scious, but the products tend to "erupt into consciousness." A more conscious process is reflective planning which requires a more compre— hensive structure in which the connections have already been estab- lished. If teachers were to engage in reflective planning, they would be able to examine these relationships. Very little of what teachers described, however, suggests reflective planning or even competent problem solving (de Groot, 1965). There was none of the conditional "trying out"of‘pflans prior to taking action that de Groot found with experienced chess players; however, these teachers were not experts in mathematics. All three claimed that social studies was their best subject. 190 Teachers encountered difficulties with the search process, pri- marily because oftheirinsufficient understanding of mathematics and a limited repetoire of pedagogical moves. ...it is hard to have a good idea if we have little knowl- edge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formally acquired knowledge (Polya, 1957, p. 9). Teachers freely admitted when they had difficulty thinking of something to try and when they knew they lacked understanding. Inadequate knowl- edge was also apparent when teachers made mistakes or became confused. Another indication of their limited knowledge was the extent to which they were able to draw upon mathematical knowledge not already being used with the task and routine of the activity. Sometimes teachers were only able to repeat what they had already said; other times they incorporated very different mathematical approaches. The variety of pedagogical moves, establiShed in Chapter IV, showed Martha was the most versatile, and Ralph was the most consistent. When faced with similar situations in which neither knew how to offer a mathematical explanation, they relied on different pedagogical techniques. Ralph told how he would have answered the question, while Martha posed another problem to give the student another opportunity to figure it out. Another example illustrates the differences of teachers' goals and knowledge as well as their pedagogical preferences. In almost identi- cal circumstances, students in Martha and Zelda's classes asked how two fractions with unlike denominators could be added. The questions were asked during activities in which addition of like-denominator fractions had been introduced with Cuisenaire rods, and students were solving 191 problems with great success. Both teachers chose to exploit the oppor- tunities, but in somewhat different ways. Martha immediately switched to a simpler problem than the student had been given and used the rods to find the solution. Zelda had students put away the rods before at- tempting the fractions. It was only after she made several mistakes that she substituted simpler fractions and solved the same problem Martha had posed. Evaluation Process. The function of the evaluation process was to determine the effectiveness of the teacher's elective actions by moni- toring change with respect to the goal in effect. Teachers made judg- ments about their own actions, about students' reactions to their elec- tive actions, and about causes of student difficulties. These judg- ments were used to determine what teachers would do next. When it was determined that a goal had not been reached, teachers' mental process- ing was recycled to search for a different move or to change goals. The evaluation process was relatively automatic for many incidents involving student occlusions which had already been processed with little awareness or which had ended satisfactorily. Teachers comments suggested a more conscious awareness of the evaluation process and indicated their surprise or disappointment. They reported their own errors and confusion when they were aware of it. They also reported their judgments about students' reactions in those incidents when they had already begun a more conscious monitoring of a student's difficulties or when the student continued to have trouble understanding. An example of how a student's reaction to an elective action could capture the teacher's attention, even when all that pre- ceding it was handled routinely, comes from one of Ralph's critical 192 moments. When Judy did not respond after Ralph had repeated a ques- tion, he was stunned because the question had been a simple review task Teacher evaluation of student reactions to their elective actions was not unlike their interpretation of student occlusions. Mathematical knowledge was a major factor influencing the teacher's judgment about the content of the student's response. It was also important in deter- mining how long teachers were able to search for other approaches as was their pedagogical knowledge. Differences were evident hitheinterpreta- tion and evaluation processes. Evaluations were based on more informa- tion than the interpretation of initial student occlusions. Knowledge of the subsequent interchange between students(s) and teacher was also available, although teachers did not always request or receive overt student reactions to their elective actions. In some instances teach- ers told of noticing nonverbal cues or of simply hoping what they had done would be helpful. To avoid this uncertainty about the student's understanding when other student input had intervened, Martha had adopted a habit of checking back with the student whose difficulty she was trying to resolve. Thus, the history of the event was another fac- tor influencing the evaluation process. Teachers' expectations also played a role in determining how long teachers would persist in their efforts to help a student. Past experi- ences and lesson plans contributed to Zelda's tenacity (seven attempts and failures) in trying for the elusive 2/4. She believed some stu- dents were capable of answering the question because of a prior learn- ing experience, and she had planned to ask this and similar questions. Expectations about individual student ability were also found to have some influence. When Martha was trying to help Max understand improper 193 fractions, for example, she kept thinking he was "just about to get itJ' This expectation undoubtedly contributed to her willingness to keep trying just as her fear of confusing the less capable students con— tributed to her reluctance to continue. Research on teacher expecta- tions has also shown that teachers are more likely to persist in their efforts with students they regard as being more capable (Brophy and Good, 1974). Although they were not formally asked to do so, teachers spontane- ously Offered their Opinions of what caused their students' difficul- ties, attributing most student difficulties to unstable causes inter- nal to the students: students did not understand, they were not paying attention, or they were forgetting--experiencing a mental block. Ex- ternal conditions mentioned by teachers were the inhibiting influence of a large group and the difficulty of the content. All these causes except the temporary state of not understanding could be found in at- tributional literature (Cooper and Burger, 1980). Teachers did not cite themselves as causes of student difficulties, at least not with- out prodding. This was consistent with Cooper and Burger's finding that participants report playing a larger role in success than failure. To designate teachers as causes required investigator influence. Attributing cause was not exclusively the function of the evalua- tion process, but of the interpretation process as well. When Martha first diagnosed Max's error, she attributed his difficulty to a lack of understanding. At this stage in the processing of student occlusions, her judgment undoubtedly had some impact on her choice of goal and actions. 194 Teachers in this study attributed student difficulties to causes which conformed with their own conceptions of how students learn and their own instructional goals. Ralph, who believed students would learn as long as they paid attention and followed what he was saying and demonstrating in class, attributed student difficulties to a fail- ure to pay attention. When he did mention student understanding, his concern was that the student could not do the problem. Of the three teachers in this study, Ralph was the most concerned about student at- tention. Zelda attributed student difficulties to "forgetting," "not remembering,“ and "mental blocks." This reflected what she believed about learning--once presented it was retained--and her belief that students had already been given Opportunities to learn. Martha, who wanted students to develop their own understanding through concrete experiences, attributed students' difficulties to a lack of understand- ing. To her this meant that a correct or complete concept had not yet been constructed or means Of its application understood. Martha's assessments were made with the interpretation of the student occlu- sions and did not change when students continued to exhibit difficulty. Attributions cited in this study also differed from those reported by Cooper and Burger (1980). In trying to link high and low expecta- ' tions with attributions, they found that ...unexpected events led to greater use of internal and un- stable causes, whereas expected events led to greater use of internal stable causes. Bright student failure was more often attributed to immediate effort while slow student failure was perceived more often as ability causes (1980, p. 108). Since teachers in this study offered internal and unstable causes for both high and low expectation students, this expectation effect was not 195 supported. As previously mentioned, some relationships were indicated between teachers' expectations and their willingness to persist in helping students. Summary. In summary, then, teachers were able to process student occlusions by interpreting, goal setting, searching for and executing elective actions, and evaluating the effects. When teaching problems were encountered, the processing was recycled for another search or to replace the present goal. The same four processes were used in teach- ers' cognitive processing of critical moments for pervasive student behavior patterns associated with activities. A model and description will indicate how this deviated from teachers' processing of student occlusions. Teachers' Cognitive Processing of Pervasive Patterns of Student Performance Ml Teachers' cognitive processing of critical moments involving per- vasive patterns of student performance was similar in many ways to what has just been described for student occlusions, but there were significant differences. A student occlusion caused momentary inter- ruption in the instructional flow of an activity. Teachers had only to respond to the individual(s) and content of the occlusion by setting a momentary goal and taking action. If they had problems satisfying their goals, they could disengage by switching to secondary goals. The choice was irreversible on only three occasions when the momentary goal was to change activities and explore a new concept or task. 196 Pervasive student behavior patterns associated with critical mo- ments challenged the appropriateness of ongoing activities. They re- presented threats to teachers' goals, plans, and expectations for on- going activities. Even when change was indicated, activities were not easily altered or discarded, particularly without something to suggest direction for change. When teachers became conscious of a pervasive pattern of student performance, they had to evaluate it in relation to the goals Of the activity. If it were determined that change was needed, the means to do so still had to be found. The sequencing of cognitive processes for pervasive patterns Of student performance, therefore, required a somewhat different processing model than the one already proposed for student occlusions. A model of teachers' cogni- tive processing of pervasive patterns of student performance is shown in Figure 6.2. According to this model, no teaching problems were reported when teachers' responses to the first two questions in the model were affir- mative: 1. Is the pattern of student performance routine (not novel)? 2. Does it satisfy the activity goals? 3. Is a change indicated? 4. Is an alternative activity or routine available? Recognition that students were not behaving as expected forced an evaluation of the pattern. Thus, when teachers acknowledged the stu- dents' performance was not acceptable, a search was automatically ini- tiated, with change dependent on finding an alternative. This process- ing sequence was repeated several times during an activity. 196a .mocmEcowcmn acmcsum mo mccmuuma m>wmm>cma we mcwmmmuoaa o>_u_:mou m.cm;ummu a we Pocoz .N.c mc:m_m “monsoon“ I amm room muczpw>m zucmmm oz mpno__c>< umamuwu=_ m>wumccmu~< mmcmzu mcwuzom mm> mm» ,/// ozx/JL oz mm> muemscoucoa acmuzum we ccmuuoa o>_mm>cma 36pm mcw__m>ms¢ mpmoo\mcm~a xum>wuu< mmcmgu 197 The proposed model is actually a modified version of the Peterson and Clark (1978) model shown in Figure 2.1 (p. 23). The major difference is the reversal of two processes: goal setting and searching. Peter- son and Clark's model assumed that teachers would search for an alter- native action before deciding to make a change. Instead, teachers concluded a change was needed before searching for the means tO bring about the change. In all probability, the two processes are not separ— ate and sequential, but interwoven. Nevertheless, the distinction seems to be important. Teachers implemented whatever actions they thought of, be that responding to student occlusions or modifying activities. Nature and Function Of Cognitive Processes When teachers became aware of these unanticipated and undesired but pervasive patterns of student performance, they exhibited one of two coping mechanisms. They suppressed conscious awareness while continu- ing the activity or they took action to alter the condition. They did not institute change immediately as recognition of these behavior pat- terns seemed to fade in and out of their consciousness until they were ready to act. Teachers allowed an activity to come to a natural con- clusion, modified the routine of an activity, or terminated it prema- turely. Each activity in which critical moments arose over pervasive student behavior was followed by another activity. Although these ac- tivities had been planned, at least partially, there were reports of further planning or alterations being made to implement them. Change in the routine of an activity could be obvious or subtle. By specifying which fractions to compare rather than having students come to the board to spin for them, Martha was able to quicken the pace 198 and avoid giving more individual assists. It was a minor but obvious change. During an activity in which his students were not being very responsive, Ralph increased the amount of explanation. It was a subtle change made apparent only by his own report. Teachers found it difficult to make changes in their activities. This can be explained by their planning practices and by the scarcity of available Options. Teachers planned their lessons antiCipating stu- dent success. They did not consider or plan for excessive difficulty or remarkable success. No alternatives were available unless teach— ers planned for other activities. Spontaneous plans were not easily formulated with the teachers' limited knowledge of how to teach the content. Summary of the Factors Which Influenced Teachers' Cognitive Processes The more stable factors found to influence teachers' cognitive processing of critical moments included teachers' mathematical knowl— edge, pedagogical options, instructional goals, conceptions of mathe- matics and learning, beliefs about teaching, antigoals and expectations Somewhat less stable influences were teachers perceived ability, attribution for student performance, and the history of the event itself. The complexity of the interaction between these fac- tors and teachers' mental processing is difficult to untangle. As Clark and Peterson (1976) have indicated, teachers are not able to explain just why they do what they do. Nor is circumstantial evidence enough. to establish the extent of influence by any factor. Never- theless, some relative measure of the importance of the factors can be recognized by considering which processes these factors were noted 199 to impact. Table 6.1 helps to emphasize which factors influence more processes and the complexity of some processes. Table 6.1. Factors Influencing Teachers' Cognitive Processing of Student Occlusions Interpretipg Goal Setting Searching Evaluating Mathematical Mathematical Mathematical Mathematical Knowledge and and Knowledge Pedagogical Pedagogical Knowledge Knowledge Interpreta- tion Repetition Instructional Instructional (history of Goals and Goals and prior Conceptions Conceptions events) of Learning of Learning Perceived Ability Principles Expectations (Antigoals) Attributions History of History of Event Event The importance of teachers' mathematical knowledge is more ap- parent as it has some relationship with each of the four processes. Teachers' repertoire of pedagogical Options and their instructional goals are also important factors. They influence two processes: goal setting and evaluation. Goal setting and evaluating are influenced by more factors than are the interpreting and searching processes. Goal setting is par- ticularly complex during critical moments when teachers' goals are challenged or changed to secondary goals. Similarly, the evaluation 200 process is influenced by more factors when teachers find it necessary to reprocess an event because their goals are not being satisfied. Teachers' Emotional Responses to Unexpected Student Performance Critical moments were characterized by their negative affect as well as the cognitive difficulties teacher were having. When teachers did not verbalize their emotions during recall, their remarks were Often indicative of the discomfort they felt. Skemp regards emotions as signals of the progress an individual is making in relation to goal and antigoal states. In an effort to begin building a theoretical framework in which emotions are incorporated, Skemp designated relationships between emotions and movement with respect to the two types of goals. In Table 6.] emotions of pleasure and relief are linked with movement towards a desired goal state and away from an an- tigoal state, respectively. A general state of displeasure is indi- cated when movement is away from a desired goal and when there is fear of movement towards an antigoal. Teachers in this study reacted as Skemp's model predicts. How- ever, the model in which only change and emotion are related to goal states does not account for differences in the intensity and nature of teachers' emotional reactions. Martha's frustration was qualita- tively different from Zelda's. Zelda was only a little bothered while Martha was very frustrated. Yet, both emotional reactions occurred when they had failed to satisfy a momentary goal after mul- tiple attempts. Skemp anticipated this variation: But even more important to an organism than whether it is, at a given time, moving towards or away from goal or anti- goal states, is whether it is able by its own efforts to bring about these changes (1979, p. 12.) 201 Table 6-2.. Skemp's Model of Emotions as as Signals of Change with Regard to Goal States and as a Consequence of Perceived Ability to Effect Change. . Perceived Goal Emotion Ability States Change Emotion Confidence Capable Towards Pleasure Desired Goal Frustration Incapable Away From Displeasure Anxiety Incapable Towards Fear Anti- goal Security Capable Away From Relief By incorporating the teacher's perceived ability into his model, Skemp was able to describe the emotions more definitively. These relation- ships to goal states and perceived ability are also shown in Table 6.2. Different emotional reactions associated with movement away from a desired goal can be explained by differences in the teachers' per- ceived abilities. Martha reported frustration came with the admis- sion that she felt incapable, while Zelda, who was only slightly bothered, made a point of saying that she did not feel incapable, only limited in her knowledge of mathematics. When antigoals inter- ferred with teachers' willingness to pursue their goals, emotions were more difficult to identify. Perhaps teachers failed to label their emotions because of mixed reactions: relief at having been 202 able to prevent or terminate an antigoal situation and displeasure at not being able to do as they set to do so. When Ralph broke off efforts to help a student because he feared the loss of attention from his class, he appeared to be comfortable with his choice. However, when students were not responding as he felt they should, he appeared anxious. It was more of a helpless reaction, an indication of his lack of confidence or ability. Zelda was the teacher least bothered by her critical moments. Ac- cording tO Skemp's theory, this was because she was able to maintain her perception of being capable. This suggests a further relationship between teachers' attributions and their perceived abilities. When students were having trouble answering a question or performing a task or when Zelda became confused while exploiting a student suggestion, she attributed it to forgetting something which neither she nor the students could control. With forgetting as her causal explanation, Zelda's capability was not threatened, nor was she pressed to seek other explanations for the student difficulties. Attention, the attribution most used by Ralph, is generally viewed as being under the control of the student. Nevertheless, when Ralph attempted to gain student attention and was not successful, he seemed to feel helpless to do otherwise. From his comments it was clear that Ralph accepted responsibility for restoring attention when he could not teach in a manner which maintained it. Thus, failure to control what he believed to be a controllable condition produced a negative reaction as was the case with Martha. She attributed student difficulties to a lack of understanding and accepted the responsibility for correcting their misconceptions and confusions. Like Ralph, this left her little 203 room to escape blame or to deny her incapability. Quite predictably, Martha was frustrated when she was unable to help students and angry when she made the mistakes. Attributions teachers Offered were also indications of their beliefs about the controllability of the causes they cited. Those causes over which teachers had little control re- sulted in less negative emotions, while those for which they judged themselves to be responsible and able to control produced the most negative emotions. The relationships between attributions and emotions for both suc- cess and failure have also been investigated by Weiner (1980). He found a link between the attributions about ability and the emo- tional reactions of different outcomes associated with confidence or incompetence. The same relationship was indicated for the general ascription of internal locus of control and confidence. Martha and Zelda, however, were most explicit about the anger they felt when they made mistakes exploiting student insights; and Weiner linked anger with attributions to others and external conditions. While the relationships among emotions, attributions, and per- ceived ability, efficacy, or control is evident, the direction of these relationships and their effect on teacher behavior is not. What does seem clear is the relationship of emotions to the outcomes with regard to goals and antigoals, as Skemp proposed. It is further evi- dent that some factor(s) influence the degree of comfort or discomfort a teacher feels in relation to any of the four goal states. Each of the factors mentioned can be shown to have some possible influence, but they cannot yet be ordered if, indeed, ordering is even appropriate. 204 Weiner (1980) has been examining the role of affect in attributional approaches to human motivation and suggests that (a) emotions are re- sponses to particular attributions; (b) emotions, rather than causal ascriptions, are motivators of actions; and (c) effects can function as cues guiding self perception (p. 4). The verification of this model, however, has yet to be completed. It should be remembered that attri- butions in this study were also shown to reflect the teacher's instruc- tional goals and conceptions of learning, so the circle of influences may be ever widening. For now, the relationships between emotions and the cognitive aspects of teaching are still being established. For critical moments over pervasive patterns of student perfor- mance, one of the defense mechanisms which enabled teachers to con- tinue the activity was the suppression of their conscious thoughts. Skemp calls this“withdrawal of consciousness"which he offers as a defense mechanism for escaping the negative emotions. While feelings such as frustration and anxiety are still present, the teachers are no longer fully aware of them. Skemp (1979) suggests that "a residual aware- ness may persist and be experienced as 'tension.'" That was evident in Martha's remarks about several of her experiences; she mentioned feeling tense. Skemp admits he can offer no explanation for what he believes to be an involuntary process, but he does describe the con- sequences: ...no further adaptation takes place in a director system from which consciousness is withheld. So, once established, such a situation may be self-perpetuating, since a return Of consciousness to this area is inseparable from a return of the experienced frustration and anxiety (1979, p. 17). It was the teachers' recurring awareness of the undesired performance patterns of their classes that produced the critical moments. 205 Summary Through the investigation of critical moments, much was learned about teachers' mental processing of unexpected student performance. In this chapter the focus has been on the nature and function of the cognitive processes including factors which influenced them, on models of the cognitive processing, and on the emotional component. Because of differences in the ways teachers processed the two types of critical moments, separate models and descriptions were required for their cog- nitive processing of student occlusions and pervasive patterns of stu- dent performance. The corresponding models were based On four cogni- tive processes: (a) interpreting student performance, (b) goal setting (c) searching for and executing elective actions, and (d) evaluating the effectiveness of their actions. Teachers' cognitive processing of student occlusions proceeded in sequence. Teachers set momentary goals to cope with student dif- ficulties or insights and followed with a fairly rapid find-execute- test routine. Elective actions were found by habit or intuitive planning with little evidence of a more rational approach. Teachers were more aware of their thoughts and feelings for distinctive stu— dent occlusions and elective actions. Distinctiveness was determined by deviation from the norm, either for content or student performance, and by repetition of similar occlusions. Teachers experienced critical moments over student occlusions when they were unable to satisfy their momentary goals. Failure to do so was due to default, being unable to come up with an effective elective action, or to interference of an antigoal. Antigoals were associated with perceived or anticipated situations which teachers preferred to avoid. To disengage themselves from an existing goal, teachers changed 206 their initial, momentary goals to secondary goals which could be more easily satisfied by avoiding, telling, or even modifying an action. With pervasive patterns of student performance, the processing order was somewhat different as was the duration of the cognitive ex— perience. Critical moments began with teachers' recognition of an un- desirable pattern of student performance. These patterns were observed over time and across students. An evaluation of the pattern was made in relation to activity goals. Once a need to change was recognized, the teacher's problem was what to do. Overt changes came slowly as teachers' awareness of these undesired situations faded in and out of consciousness. Eventually, some change was made either in the routine of the activity or with the ending of it. Relationships were noted between teachers' cognitive processes and the factors influencing them: mathematical and pedagogical knowledge, conceptions of mathematics learning, expectations, prin- ciples and instructional goals. Relationships were also indicated between teachers' emotional reactions, their progress with respect to their goals, their perceived abilities to produce change, and their attributions about student difficulties. Emotions teachers reported with critical moments reflected the progress that was being made with respect to their goals and antigoals. The variation in emotions further reflected teachers' perceptions Of their ability to bring about change--an example of Skemp's theoretical model. Also indicative of teachers' emotional reactions to these incidents were their attributions for student difficulties. Knowledge about how teachers' cognitive processing of student occlusions and pervasive student performance patterns should suggest 207 ways in which teachers might be helped to improve their teaching of mathematics. Implications from these findings will be discussed in the next chapter. CHAPTER VII DISCUSSION This study of critical moments has investigated teachers' reac- tions to unexpected student performance which occurs during instruction of the total class. The goal has been to understand teachers when they are faced with incidents of unexpected student performance and critical moments. Much has been learned about the three teachers in this study and what made teaching difficult. The purpose of this chapter is to discuss some of the findings in relation to existing research, highlight the significant findings, and to critique the method Of research. As there are some differences in the points of interest, this chapter is divided into three sections: teacher thought and teaching, methodology for studying critical moments, and teaching mathematics. Teacher Thought and Teaching Research Findings Much of what was learned in investigating teachers' critical mo- ments lends support for the portrait of the teacher given in Chapter II which reviewed research on teacher decision making and teacher thought. Rather than repeat the same results, attention will be given to a few of the more significant findings: 1. Different types of critical moments and teachers' cog- nitive processing 2. Distinctive and observable characteristics of student occlusions associated with critical moments interrelatedness goals of teachers' activities, actions, goals, conceptions, and attributions 208 209 3. Variation in teaching and student occlusions Critical moments were initially conceived as arising over inci- dents of unpredictable student performance which were designated as student occlusions. Student occlusions were described as momentary but observable interruptions in the instructional flow of an activity identified either by students' inability to provide correct responses to teachers' questions or by students' unsolicited contributions. While such behavior might observably occlude the instructional flow Of a student, it did not always intrude on the teachers' mental life. For this reason, stimulated recall was used as the means of obtaining evidence of which events teachers consciously processed. Student oc- clusions for which teachers indicated having cognitive difficulties and emotional discomfort were designated as critical moments. Criti- cal moments, therefore, represented momentary crises which were ex- perienced by the teacher as problematic or disruptive. In analyzing the data, another form of unexpected student perfor- mance was found to produce similar reactions among the teachers. In addition to reporting their subjective problems with isolated student occlusions, teachers also reported their difficulties and discomforts over more pervasive patterns of student performance which were ob- served across students and over time. These pervasive patterns of student performance were designated and reported as critical moments. The two experiences, however, were so significantly different that two models of teachers' cognitive processing of unpredictable stu- dent performance were proposed: one for isolated student occlusions and another for pervasive patterns of student performance. Although 210 the cognitive processes were the same, the sequences in which they were arranged differed. Different, too, were the immediacy with which teachers had to respond to the two types of critical moments, their goals which were in jeopardy, and the types of action required to re- solve the teachers' problems. Teachers reported having only fleeting thoughts and discomfort over undesirable patterns of student performance, but the recurrence of their thoughts over time belies the meaning of a momentary experience. In the future, then, it would be helpful to distinguish between the two types of problems teachers experience with different labels. The term critical moment should be reserved to indicate teachers' momen- tary crises over student occlusions. Another term to refer to teach- ers' more enduring problems regarding the pattern of student perfor- mance associated with an activity might be critical discrepancy. Morine-Dershimer (1979) labeled lessons in which more than 50% of teachers' decision points were both unexpected and bothersome with this term. Her criterion does not apply to all the teachers' critical moments associated with the pervasive patterns of student performances as will be discussed later. Two models of teachers' cognitive processing were given. One was for teachers' processing of pervasive patterns of student perfor- mance associated with activities and another for their processing of student occlusions. Neither model had been used before to investigate or classify teacher decisions, although the one for pervasive patterns of student performance was quite similar to that used by Peterson and Clark (1978). The model for student occlusions was sufficiently 211 different from the model for pervasive patterns Of student performances. Similarly, teachers' experiences of the two types of critical moments were different enough to have implications for future research and analyses of teachers' decisions. Student performance, associated with critical moments, was quite distinct from student performance associated with student occlusions. As this point will be repeated in several sections throughout the chapter, these differences will not be reviewed here other than to say that the observable characteristics for student occlusions were much more ap- parent than were those for pervasive patterns of student performance. A good deal of variation was evident in teachers' activities, actions, intentions, and attributions. There was also noticeable variation in student performance across activities and teachers. As much of the variation was already detailed in Chapter IV, only a brief description of the teachers will be given. Ralph represented a more typical mathematics teacher, and Martha one teaching more in the style recommended by NCTM. Ralph's goal was for students to be able to do the problems, and Martha's was for them to be able to experience or understand the concepts. The difference in their goals was essentially the difference between what Skemp (1978) has termed instrumental and relational understanding. There is also a difference in the mode of learning they' encouraged (Skemp, 1981). Differences were apparent in the causes to which teachers at- tributed student difficulties. Ralph cited lack of attention for stu- dents' difficulties, while Martha talked more about their lack of understanding. Ralph was also teaching more in accordance with the direct in- struction model for which teachers are encouraged to seek low error IEIIII 212 rates. The direct instruction model has most frequently been applied to the teaching of facts and skills rather than discovery learning or problem solving. The benefits of direct instruction have been demon- strated by a number of researchers including Stallings and Kaskowitz (1974). .In reViewing research on direct instrdction, Peterson (1979) and Rosenshine (1979) were able to report that a direct or tra- ditional teaching approach was somewhat more effective in increasing student achievement than an open approach. Nevertheless, Peterson does not support direct instruction as the sole model of instruction. To me, the picture of direct instruction seems not only grim, but unidimensional as well. It assumes that the only important educational objective is to increase measurable achievement and that all students learniri the same way and thus should be taught in the same way.... Educators should provide opportunities for students to be exposed to both [direct and open instruction] (1979, pp. 66-67 . Implications and Directions for Research Implications of the findings from this study and directions for future research will be discussed in relation to the (a) role of the activity, (b) role of student performance, and (c) teachers' responses to critical moments. Role of activity. The activity--the task, questioning routine and instructional style of the teacher--plays an important role in determining student performance and teachers' mental problem space. The variance of student occlusions could well be attributed to task differences in the activities. Most incidents of unexpected student occlusions came during activities in which new content or tasks were being presented while review and slightly modified tasks resulted in 213 fewer student occlusions. Thus, the nature of the task and student's familiarity with it were two important characteristics of the activity. A teacher's questioning routine and instructional style for an activity were related to the nature Of the tasks and, therefore, also influential in determining student performance. Ralph's emphasis on skills and questions seeking numerical responses to subprocedures may have had as much to do with the lower frequencies and densities (rates) of student occlusions as the tasks themselves. In contrast, the concrete tasks for which students were asked to model, identify, and apply concepts and the symbolic tasks for which students were to recognize patterns from examples produced more occlusions, particularly more insightful suggestions. Ralph's aVOidance of insightful stu- dent suggestions may have discouraged others from offering their ideas while Martha'ssnpport and exploits of unsolicited student contributions probably encouraged others to try. Similarly, by allowing students to call out their responses to some questions, Zelda increased the number of student guesses. The importance of the task in influencing student behavior is central in the ecological perspective (Doyle, 1979). This view has already begun to influence research on teaching. After extensive re- search on classroom instruction, Soar and Soar (l979) acknowledged the importance of the learning task and what might be termed the in- structional goal of an activity as influencing student behavior and thought. Their acknowledgement of the importance of task parallels Kounin's shift from the more general characteristics such as "withit- ness" (Kounin, 1970) to the more specific aspects of the task signals (Kounin & Doyle, 1975; Kounin & Gump, 1974). Doyle (1979) acknowledges 214 the teachers' role in selecting tasks, saying they "are not simply directors of activities or contingency managers, but rather organi- zers of task systems." So far, the researchers have characterized classroom tasks into fairly general categories. This study suggests a need to classify activities on several dimensions because of the interdependence of the task and the teachers' questioning routine and instructional style. Joyce, Morine, and McNair (1977) characterized the activity as defining the teachers' problem space with which they processed unex- pected student performance. This notion will be discussed further in relation to the findings about teaching mathematics. At this point, it need only be said that teachers were able to process more routinely or more successfully the student occlusions for which the mathematical content was within the task, numerical domain, and questioning routine of the activity. Teachers were more likely to experience critical mo- ments when the mathematics they needed was not within this space. They also had more difficulties when needed pedagogical techniques were not within their repetoire. When teachers did not know mathematically or pedagogically how to respond to students, they often cited their fears as to what might happen if they tried to respond. 215 Role of student performance. The nature of student performance had a definite effect on the manner in which teachers processed student behavior and the actions they took. Different models of teachers' COgnitive processing were required for different types of unpredictable student performances. The type of student performance also had impli- cations for teachers' choice of elective action. When frequencies of teachers' elective actions were examined as conditional probabilities according to the nature of occlusions--difficulties, insights, and their subcategories--patterns of teacher preference were quite apparent. If the data were reanalyzed so that routinely processed student occlusions were separated from the more consciously processed, the patterns might become even clearer. The distinctiveness Of the episodes involving student occlusions for critical moments suggests that a reasonably accurate list of poten- tial critical moments can be identified from a more extensive list of student occlusions. The routinely processed events are identifiably different than the more consciously processed events. The critical moments described in this study offer the beginnings of a taxonomy for student occlusions with the potential for producing critical moments. The completeness of this taxonomy and its ability to be generalized across teachers and situations, however, has yet to be determined. Characteristics of critical moments which arose over more pervasive patterns of student performance associated with activities rather than student occlusions, are not as easily recognized from observable cues. Research is needed both to confirm and refine the criteria for identi- fying potential critical moments and critical discrepancies. Some uses for the taxonomy will be indicated in the sections that follow. O IE ' 216 Teachers' responses to critical moments. Teachers varied in their responses to critical moments. Ralph and Martha's behavior represents qualitatively different responses to their critical moments. Martha found ways to use them to her advantage; they became benchmarks by which to monitor the effectiveness of her activities and to tailor her plans to the needs of the students as she perceived them. Student difficulties were often explored while she attempted to resolve them, and student insights were often exploited as she attempted to capital- ize on appropriate ideas. This is not to suggest that a critical moment was not discomforting for Martha, but that she responded to them dif- ferently than Ralph by using them more constructively. Ralph did not want students to exhibit any difficulties or con- tribute any insightful ideas during his presentations. Instead, he wanted this phase of the lesson to progress according to plan with little or no disruption from student occlusions; he preferred to deal with individual student difficulties during the work periods. Ralph was simply not receptive to and did not value the unpredictable student occlusions which occurred during his lessons. Nor did his teaching benefit from the existence of critical moments. He viewed the critical moments as threats and problems to be overcome or avoided. The difference between these teachers was not so much a matter of one's having a much better understanding of mathematics than the other, as both Marth and Ralph exhibited a limited understanding of the mathematics they were teaching. Differences were also apparent in their instructional goals and beliefs about the teaching of mathema- tics and in the manner in which they taught. 217 The third teacher in this study, Zelda reacted to the critical moments more like Martha than Ralph, but she did not use the critical moments as benchmarks for planning subsequent lessons. She was so tolerant and unperturbed by most unexpected student performance that she reported few critical moments. In two of her three critical moments, Zelda terminated a question or activity, which suggests her acceptance or constructive use of critical moments was less than Martha's. Zelda was also of two minds about her instructional objec- tives focusing on both relational and instrumental learning. She attributed learning difficulties to forgetting or mental blocks, which was similar to but not the same as Ralph. With regard to the differences in their activities, instructional approaches and goals, Ralph and Martha support what Bussis, Chitten- den, and Amarel (1976) reported. They found that as teachers' cur- ricular constructs progressed from a total focus on grade-level facts and skills to an orientation toward more comprehensive priorities, their actions demonstrated an increasing willingness to experiment or change. Ralph and Martha's different responses to critical mo- ments may have been exaggerated by the particular units they were teaching. Martha lamented being unable to teach other mathematics topics in the same manner as fractions. Ralph offered verbal support for problem solving experiences, but admitted he was unable to and did not have the time to plan such activities. Thus, the differences in these two teachers' responses to the critical moments may have been a function of the activities--task and instructional approach. 218 Zahorik (1975) and Peterson and Clark (1978) suggest that teach- ers' planning can be counterproductive to their sensitivity to stu- dent ideas if it is too extensive. During this study, five factors were observed to reduce teacher responsiveness to student ideas and needs within the ongoing lesson: availability Of a detailed plan, provision for time in which students were to work on their assign- ments, difficulty in instantly formulating a new plan, instructional goals which stressed doing and not understanding mathematics, and teachers' limited knowledge and understanding of mathematics they were teaching. Planning, which was one of the factors, would also reflect differences in teachers' goals. A teacher like Ralph becomes more stressed as the frequency or intensity of critical moments increases, whereas a teacher like Martha is able to constructively use a number of critical moments without their becoming too stressful. For Martha, stress became greater when critical moments were too infrequent or when the amount became quite excessive. On the other hand, Ralph was upset by most of his critical moments, particularly the one in which he was displeased with a per- vasive pattern indicating less eagerness on the part of the students to participate. Teachers of the same type as Martha and Ralph might be expected to respond in a similar manner unless differences could be achieved by having them teach activities of a different nature. The question of stability of a teacher's instructional style and goals, conceptions of how students learn and even their attributions have yet to be tested across different types of taSks and content. 219 If teachers are to teach in a problem solving mode which fosters unpredictable student performance, it may make a difference as to how they respond to critical moments. A teacher with the same goals and beliefs that Ralph held during the unit investigated might be expected to respond in a similar manner. An increase in the number of critical moments would expose a teacher to more difficult and stressful situa- tions, making it unlikely that the teacher would willingly continue to function in such an environment. If the teacher could not adjust to the increase in the number or intensity of critical moments, then s/he would have to modify the type of instruction to reduce the stress. It is easier to imagine a teacher like Martha being able to implement the NCTM recommendations, than it is one like Ralph. These remarks are not meant to imply that teachers like Ralph can not learn to use critical moments in a more constructive manner. Be- fore any generalizations about flexibility and constructive use of cri- tical moments can be determined, however, the same and different teach- ers must be examined across situations. One study to show how the same teachers from this study responded while teaching a problem sol- ving unit is already underway. If it should turn out that the teachers do, in fact, respond in accordance with the materials being taught, there can be some optimism about the possibility of implementing the NTCM recommendations with more traditional teachers. If, instead, the teacher who is resistant to unpredictable student performance is unable to adjust for different types of instructional materials, the possibility is slim and the future for improving mathematical teaching is bleak. 220 Methodology For this study, a great deal of time was spent observing teachers both in their classrooms and repeatedly through the use of videotapes. Time was also spent in gathering teachers' thoughts through the process Of stimulated recall and then using these data to identify and describe the behavior of students and teachers and teachers' critical moments. From these experiences, the investigator has formed some impressions and drawn some conclusions about the methodology for studying critical moments and some implications of these conclusions. This section is a critical analysis of the methodology for studying teachers' critical moments. Attention will be given to (a) identifying criteria for critical moments, (b) classification of stu- dent and teacher behavior, (c) method and effect of stimulated recall, and (d) investigator bias. This will be followed by some remarks as to the implications and directions for future research on critical moments. Criteria for Identifying Critical Moments In order for a student occlusion to be designated a critical moment, it had to be associated with teachers' reports of cognitive difficulties and felt discomfort. These criteria made it possible to identify and describe critical moments, although the choice was not always clear. The task was made simpler by the indication of secondary goals being set to replace teachers' initial goals or im- pulses, even when they took no action to implement them. Separate criteria for recognizing critical discrepancies had not been estab- lished prior to the analysis. 5R3, when teachers indicated similar 221 reactions to more pervasive patterns of student performance instead of student occlusions, the occasions were also designated as critical moments. Excluded by these criteria were incidents in which teachers faced momentary crises over what to do, but were pleased when their actions led to satisfactory results. Examples of successfully re- solved student occlusions proved useful in characterizing and modeling teachers' cognitive processes. The question is whether or not the concept of a critical moment should exclude these experiences. TO extend the notion of critical moment to include both satisfactorily and unsatisfactorily resolved student occlusions would require a revised set of identifying criteria. Although the choice should depend on the purpose of an investigation, in the opinion of this investigator, the modification is necessary if teachers' subjective classroom experiences are to be captured adequately. Classification Of Student and Teacher Behavior All student occlusions were identified and coded,as were teach- ers' elective actions from the videotapes and transcripts of each lesson. Stimulated recall data about these occlusions were used to provide evidence of teachers' critical moments. In Chapter IV the data on student occlusions and teacher elective actions were not re- ported with reliability measures; there was no systematic cross vali- dation of the classification of student and teacher behaviors. As some subjective judgment was required in identifying and coding these behaviors, some comments about the process and the appropriateness of the classification scheme is in order. 222 Student occlusions were easily recognized by inaccurate responses to teacher questions and unsolicited student comments. It was more difficult to identify students by name from the videotape and tran- scripts. When available, this information was used in describing critical moments because teachers' Opinions of the individual students were considered. The distinction between student difficulty and insight was not difficult to make since the criterion was mathematical validity which did not require any inference about the student or teachers' interpretations. There was some concern, however, over what should be included as student insight. Unsolicited and valid contributions were included so that alternative names for fractions were coded as insights in Martha's class. There were few instances when teachers asked students to describe patterns. Instances for Which teachers asked for new patterns and students could describe them were designated as student insights. Subcategories, which were determined after the occlusions had been identified, were also reasonably apparent. The most subjective choice was whether student errors were patterned or nonpatterned. Classification of teachers' elective actions was more difficult as a pre-existing scheme was not used, at least not systematically. In retrospect, a modified version of the Brophy-Good Dyadic System would have been satisfactory. Initially, an attempt was made to de- fine a new coding scheme. Eventually, the categories of exploiting, alleviating, and avoiding moves emerged, but classification was highly subjective until specific behaviors were designated and as- signed a category. The coding task was still somewhat difficult, 223 however, when teachers used several behavioral techniques in response to the same student occlusion. Distinguishing between alleviating and exploiting moves was sometimes difficult since it required a judgment about the task and questioning routine of an activity. Some student occlusions occurred early in an activity before routines were well established. This made it impossible to determine what the routines would have been had the occlusions not occurred. If this coding system is to be used in other studies, reliability measures need to be estab- lished. A few remarks are also in order about the teachers' stimulated recall data. Each thought unit and decision point was not coded even though distinctions were made between recalled thoughts and feelings or other information. The emphasis of this study was on the substance of teachers' remarks and not on quantifying or classifying them. The loss of such detail was compensated for by the interpretations of teach- ers' intentions, the relevant antecedents to and causes of their Criti- cal moments, and the consequences. Method and Effect of Stimulated Recall on Teachers The method of conducting stimulated recall proved to be a reason- ably effective means of obtaining teachers' thoughts and feeings. How- ever, asking teachers to view the entire lesson for roughly two hours on a daily basis had both advantages and disadvantages. One advantage was that most incidents of unpredictable student performance which teachers processed at a more conscious level were reported. Had seg- ments of the lessons been preselected for viewing, some might have been missed as so few actually occurred. The sustained use Of 224 stimulated recall also reduced the amount of verbal discourse that teachers sometimes use to explain their actions (e.g., Conners, 1978). Teachers realized what things had been said before and avoided repe— tition. They also became conditioned to the process as they focused on recalling their covert experiences. Nevertheless, teachers found it difficult to restrict themselves to reporting recalled thoughts and feelings, as Marland's (1977) analysis of teachers' com- ments showed. Interfering with teachers' suspended state of recall were interactive diversions, retroactive thoughts, fatigue or boredom, and the probes of the investigator. The process was an interactive experience for teachers. Viewing the lessons again provided them with Opportunities to notice things they had missed and to plan what they would do next or what they would do differently another time. Retroactive interference occurred when knowledge of a future event interfered with a teacher's recollection of the incident being shown. Some events were mentioned earlier than they appeared on the screen so that recalled thoughts did not always occur in juxtaposition with what was being observed on tape. Nor were teachers always certain just when their thoughts did occur. Fatigue or boredom was particularly noticeable during long stretches of a lesson when there was little of interest for teachers to recall or examine. During these periods, teachers either remained quiet or began to talk about other things. Fatigue was evident as a consequence of participating in the daily stimulated recall sessions after about a week. All three teachers reached a point where they were no longer interested in continuing, so they ended the units. The lengthy recall sessions interfered with their planning time. They 225 were tired of preparing lessons for the substitute teachen,who relieved them for one class,and of student complaints about the substitute and changes in their schedules. Investigative probes to elicit teachers' thoughts may have dis- tracted teachers on occasion, particularly when teachers continued to talk as the videotape played and when the investigator sought to learn more about the teachers' understanding of mathematics. On a couple of occasions, the probes bordered on intervention or helping moves, par- ticularly when teachers appeared to be at a loss for what to do or when they had made glaring errors. These probes may have been perceived as a threat. In such instances, the problem for the inves- tigator was in maintaining a detached and nonevaluative or noninstruc- tive role. The investigator tried to remain supportive of the teachers--they seemed to need it--in much the same manner as described by Conners (1978). From the perspective of the investigator, the re- lationship between the investigator and teachers was friendly and non- threatening. Teachers exhibited different styles in response to the stimulated recall process. While all the teachers cooperated, some found it easier to recall thoughts and feelings. The same is apparent in the quantitative measures of teachers' comments in Marland's (1977) study. Zelda reported the fewest thoughts; little seemed to bother her during her lessons. As a result, the investigator's repeated inquiries about her thoughts and feelings may have been irritating. Her primary dis- traction during recall was with what she might have done to improve the activities. Martha was particularly vocal about her thoughts and feel- ings so that probes were made more to clarify than to elicit. She used 226 the opportunity afforded by viewing the lesson again to examine her students' behavior and thinking. Both Martha and Zelda were comfort- able in talking while the videotape continued to play, particularly if their remarks were brief. Ralph willingly reported his thoughts and feelings, but in a manner which indicated resistance to interference by the investigator. He retained more control over stopping and starting the videotape than the other teachers. While the tape con- tinued to run, he concentrated on remembering the experience, but once he began talking, he wanted the tape stOpped. He was more verbal about his concerns and his own teaching than were the other two teachers, which was most apparent in his long passages in the typed transcripts of the recall session. Investigator Bias Observing a teacher in action is quite different from combining the observation with the experience of and data from stimulated recall. At the time the classroom Observations were made, Zelda's classes were the most exciting, Ralph's seemed dull and boring, and Martha's were good. In the process of analyzing and reporting data, however, the investigator's Opinions underwent some change. Martha emerged as the most sensitive, caring, and competent teacher, despite her own reser- vations about her ability and knowledge of mathematics. Zelda came across as a confident teacher, responsive to the students, but always to their mathematical needs. Ralph remained the least interesting with his emphasis on skills for doing problems. If he is a true represen- tative of teachers in typical mathematics classes and research on teacher thought, then it is no wonder that so little variation is 227 found in these classes. One of the distinct advantages in the choice of teachers for this study was the variation in their teaching and goals for instruction which helped to highlight characteristics of teachers' actions and mental processing of unpredictable student per- formance. Implications and Directions for Future Research on Critical Moments The coding system used in this study for describing student occlu- sions and teachers' elective actions needs to be checked for reliability, modified if necessary, and compared with other classification schemes. Separate, but nevertheless similar, coding systems emerge from different studies because of different researchers' goals. Reanalyses of the same data set by various coding and analysis procedures would provide more direction about the differences of each. In future studies of teacher thought where stimulated recall is to be used, some modifications may simplify the procedure without loss of relevant data. First, because some episodes were so salient in teachers' minds, they wanted to begin talking about them immediately following the lesson. Some of their spontaneity may have been lost or recollections distorted by asking them to wait until the episodes ap- peared on the screen. A debriefing interview prior to the showing of the tape might be advisable, although this procedure could alter the recall process once the videotapes are shown. A debriefing interview might even be adequate for gathering teachers' recollections without viewing a videotape, although that remains to be investigated. Second, if an entire lesson is to be shown, fast forward and pause controls on the videotape deck would be helpful. The tape could be speeded up when 228 there was little of interest to report and stops could be made easily when teachers were speaking. A third suggestion is to preselect por- tions of the lesson for viewing based on the criteria for identifying potential critical moments discussed earlier in the chapter. After teachers' critical moments and the models Of cognitive processing had been identified and described, it became clear that the separate coding and counting of each student occlusion did not reflect Unateachers'subjective experiences during critical moments. The behaviors which occur while a teacher is attempting to satisfy a momentary goal should be considered as one event. Thus, the event can consist of only one student occlusion and teacher elective action or it can include a chain of several such interactions. As long as the teacher is attempting to elicit a particular response, help an individual overcome a misunderstanding,or use a student suggestion, the sequence of student and teacher behaviors should be considered as a whole unit. The coding Of these episodes should reflect the number and nature of the student occlusions and the sequence of teaching moves--the teacher's strategy. The goal state of the teacher is a natural umbrella under which to cluster student and teacher behavior. Episodes involving more than one student occlusion were recalled as a single experience, since teachers were not always able to distinguish when particular thoughts and feelings occurred. A benefit of coding the unpredictable student behavior in this manner is that potential sites for critical moments can be more readily identified. Disappointed in the lack of consistency which he found between teachers' differential expectations for student performance and 229 their behavior, Cooper (1979) recommended a similar approach for ana— lyzing teacher behavior: Undeniably, classroom interactions are not independent events; any exchange is highly dependent on what has gone before. Closer concentration on behavioral sequences, as Opposed to ag- gregated behavioral frequencies, would undoubtedly produce a wealth of new information. Models which predict behaVioral se- quences ought to be viewed as more powerful than models dealing with total or averaged behavioral occurrences (p. 405). If prOportions or frequencies are to be used, the number of criti- cal moments should not be compared with the total number of student occlusions; instead, the number of episodes involving student occlusions should be clustered according to the teachers' immediate goals. The identification of these episodes through a taxonomy would help in distinguishing routine student occlusions from those with greater po- tential for being more consciously processed and for producing criti- cal moments. Distinctions might also be made and ratios formed of the satisfactorily or unsatisfactorily resolved student occlusions relative to the total. A neutral category might need to be included as well. A comparison of different quantitative measures is needed to see which best characterizes teachers reported experiences or their lessons. Teaching Mathematics The primary argument used to support the need for this study of critical moments grew out of recommendations for teaching mathematics proposed recently by the National Council of Teachers of Mathematics. The Council urged that students be given more problem solving experiences and that teachers focus more on developing concepts, using concrete materials, and solving real world problems rather than limiting instruction to computational skills. The concern of the investigator 230 is that in carrying out these recommendations, teachers mayhave to cope with more unpredictable student performance (n: student occlusions Of a somewhat different nature. Either could cause teachers to resist implementing the recommended style. Whatever verbal support teachers have given for providing problem solving experiences and encouraging understanding of mathematics, they have made little progress in changing how they teach mathematics. If teachers do find it difficult to respond satisfactorily to unpredictable student performance and critical moments, then every effort should be made to find ways to help them. In this section the findings and implications for teaching mathematics will be discussed, after which recommendations for training teachers and preparing cur- ricular materials will be offered. Findings and Implications Findings which had particular significance for the teaching of mathematics dealt with l. The nature of student occlusions which produced critical moments; 2. Causes of teachers' critical moments; 3. The role of activities with regard to student performance and teachers' cognitive processing and planning; and 4. The role of teacher antecedents--mathematical goals, conceptions of learning, and instructional style--in teachers' responses to critical moments. Student Occlusions in Critical Moments Student occlusions did not have equal potential to cause teachers to consciously process these events and to experience critical moments. The distinctive features of the student occlusions and teacher elective 231 actions associated with critical moments were due primarily to one or more of the following: (a) the mathematical content of a student re- sponse or contribution in comparison with the task and questions of an activity, (b) the response of a student in comparison to the norm Of responses to similar questions or in comparison to the plans and expectations of the teacher, and (c) recurring incidents of the same dif- ficulty or insight from the same student or over the same questions. Recurring incidents were presented as distinctive until the teacher man- aged hasatisfactorily resolve the problem or set a goal which encom— passed student repetitions. For the most part the student errors were patterns indicative of misconceptions or an inability to respond to a particular question or task. Student insights were suggestions of new rules or variations in the tasks--usually more complex ones. The implication is that teachers' critical moments come close to matching student occlusions which have greater potential for flexible teacher actions and constructive use of the information. Two are pos- sible advantages to finding teachers conscious of and interested in responding to distinctive student difficulties and insights. One is that teachers may fail to respond or may respond inappropriately to critical moments because they lack sufficient knowledge, not because they fail to note the potential for action. The second advantage is that if teachers are provided with apprOpriate information, they might improve their responses to critical moments and the student occlusions associated with them. The question, then, is what do teachers need in the way of assistance? The answer must be found in the causes of teach- ers' critical moments. 232 Causes of Critical Moments The primary causes of teachers' critical moments were their limited knowledge of mathematics and, to a lesser extent, their limited repetoire of pedagogical options. These deficiencies sometimes in- terfered with teachers' interpreting student occlusions, searching for elective actions,and evaluating the effectiveness of their actions. Since the teachers did not receive help in these areas, they COuld not have been more effective in COping with their critical moments. Teachers did not anticipate the types of student difficulties or insights which might occur nor did they plan any Besponses to them. The teachers in this study were usually quite surprised by the students' performance at critical moments. Whether this is due to a failure of teachers' planning practices or their knowledge is not totally clear, but there was evidence to suggest that the teachers in this study were incapable of anticipating many of the more distinctive student occlu- sions. They simply did not understand the mathematics well enough and had not taught the same unit enough times to be able to draw upon their past experiences. Prescriptions given to teachers for actions in specific situa- tions for specific content would suggest an approach similar to the diagnostic-prescriptive programs in mathematics. The difference is not in theory, but in practice--the way diagnoses are to be made, prescrip- tions implemented, and for what purpose. The diagnostic-prescriptive materials have been designed primarily for testing and interviewing students on an individual basis with the emphasis on helping students who are having difficulties achieving mastery. The recommendations in this 233 section have to do with total class instruction which introduces stu- dents tO new mathematical ideas and tasks through activities. Teachers must respond to both student difficulties and insights immediately. Much of the information which has been learned from the diagnostic clinicians (e.g., Ashlock, 1979), however, would undoubtedly be of some use to the teacher engaged in total class instruction. Similarly, information about student understanding of mathematics could be useful in identifying the types of occlusions teachers might encounter (e.g., Behr & Post, 1971; Pulos, Stage, Karplus & Karplus, 1981; Erlwanger, 1973; and Moser, 1981). Research on the effectiveness of specific ac- tions in particular settings, however, is still greatly needed. The prob- lem with past research efforts has been that attention was focused on general pedagogical moves without sufficient regard to the details of the instructional setting and the mathematical content. Teachers' experiences with critical moments seem to indicate that they do not need general information as much as they need task-and topic specific mathematical knowledge and situation-SPECITIC P9d890- gical advice. Teachers need to be made aware of the student diffi- culties and insights that could arise, what they might indicate, and possible courses of action. In order to be able to prescribe specific actions, however, more research is needed to find what Whitfield (1971) termed the "temporal best" decisions. He was concerned with the mechanisms teachers might use to make their choices as well as the information available for them to use. If experiences of teachers in this study are any indication, efforts to train teachers in more gen- eral instructional strategies might be less effective. 234 Teachers did not have a clear understanding of abstractions. Terms such as concept are often used indiscriminately to describe what- ever ideas were being taught. Similarly, without the specific mathe- matical knowledge these strategies require, it is doubtful that teach- ers could be successful in implementing them. Teaching strategies, at least as they have been characterized by Cooney, Kansky, and Retzer (1975), also do not provide pedagogical techniques for using the strate- gies with an entire class or even with the different types of materials teachers are expected to use. Herscovics. (1980) may come to similar conclusions in training teachers to conduct clinical interviews. In- stead, teachers may need to learn their general tactics for responding to pedagogical situations or to mathematics through examples as Resnick (1981) has indicated. Teachers, like students, develop their heuris- tics by performing in a more algorithmic or prescriptive manner. Shul- man (1977) reminds us, "One of the inexorable blessings of human cogni- tion is the process of invention and construction. We do not merely imitate; we always construct" (p.271) Role of Activities for Students and Teachers The importance of activities to student performance and to teach- ers' cognitive processing and planning must not be overlooked. In this study different frequencies and distributions of student occlusions could be noted across different activities and teachers. Some of the more salient features of the activities which seemed to influence the amount and types of unpredictable student performance were (a) the task and student's familiarity with it, (b) the teacher's questioning routine, instructional strategy and the type of learning encouraged, and (c) the instructional style and goals of the teacher. 235 The activity is also an important determinant of the problem space in which teachers are attempting to process the student occlu- sions including the more distinctive ones associated with critical mo- ments. Researchers from the South Bay study (Joyce et al., 1977) came to similar conclusions about the importance of the selection of the materials and instructional flow of an activity in determing the boundaries of the teachers' mental processing of unpredictable events. Once an activity is underway and a teacher has established a routine for questioning the students about the task, most of the in-flight de- cisions or choices of elective actions are made within the same frame. When the mathematical knowledge required exceeds that being used in an ongoing activity, teachers experience difficulties. Similarly, as long as teachers are able to rely on pedagogical routines which they have acquired as habit they are able to select and use these tech- niques without much conscious effort. However, when the cognitive de- mands for interpreting a student remark or for searching for an elec- tive action exceed the available frame, teachers are not able to pro- cess them. Instead, they have to draw upon knowledge they may not be as familiar with or may not even have. The difference is essentially the same as that between cognitive processing within the short-term memory or having to search the long-term memory where information is not as readily accessible. An activity is also the product Of a teacher's planning, albeit not always a very complete product. Teachers plan and implement ac- tivities as research on teacher planning has shown (Yinger, 1977; Clark & Yinger, 1979; Shulman, 1981) which this study has supported. 236 Role of Teacher Antecedents in Their Responses to Critical Moments The diversity with which teachers in this study responded to their critical moments has already been discussed. The range was from simple discomfort and little or no flexibility to discomfort with more flexibility and constructive use of critical moments. The question which has yet to be answered is whether teachers can respond dif-' ferently to critical moments--even with assistance-~or if their re- sponses are a reflection of more stable characteristics. The less flexible teacher taught with goals for instrumental understanding and in a manner similar to what Easley (1978) de- scribed in his depiction of the typical mathematics teacher. The more flexible teacher was the one who sought relational understanding and who taught in a manner encouraged by the National Council of Teachers of Mathematics (1980). To what extent these differences can be al- tered by different curricular materials or by different training prac- tices has yet to be learned. Teachers' goals are another indication of their responses to their critical moments. Because of the interrelatedness of teachers' goals, conceptions of how students learn mathematics, and attributions of student difficulties, it may be possible to predict how teachers re- spond to distinctive student occlusions and the constructive use to which they put the critical moments using one or a combination Of factors. 237 Directions for Training Teachers and Development of Curricular Materials In their recommendations for the 'BOs, the National Council of Teachers of Mathematics encourages teachers to provide problem-solving experiences for their students and calls for the improvement Of teacher training and curricular materials. Because instruction based on these recommendations tends to stimulate unpredictable student per- formance, findings on teachers' critical moments offer some directions for improving teacher training and the preparation Of curricular ma- terials. Critical Moments as Training,Sites Critical moments provide a natural research and training site. Because critical moments are particularly salient events, they provide Opportunities for teachers to investigate probable causes of the student difficulties and insights and elective actions which might be tried. For teachers to be trained with their own critical moments in a personalized manner has disadvantages due to time and personal con- straints. There is also the possibility that sufficient or desired ex- amples might not occur. Consequently, there is a need to consider training by simulation. A library of critical moments, gathered from videotapes and transcripts of actual lessons, could be used in simula- tion experiences for training teachers or conducting research. Respond- ing to critical moments in the context of the activity being taught would provide a more realistic teaching situation than the descriptive sketches written by Bishop and Whitfield (1972). Critical moments would provide case studies as already used in the study of medicine and DUS] MESS. 238 In a critical review of research in mathematics education con- ducted in the United States, Kilpatrick (1981) calls for more research conducted in the classroom. Integrating the findings of research conducted during actual instruction with research conducted on stu- dents' understanding of mathematics outside the classroom is one way of lending more credibility to both. For example, in their study of students' knowledge of rational numbers, Behr and Post (1981) found that visual-perceptual distractors affect students' under- standing of fractions. One Of Martha's critical moments vividly il- lustrates this same difficulty. Student difficulties or insights which occur could also point to needed areas for research on student understanding and the effects of teachers' actions. Preparation of Curricular Materials Alerting teachers to the potential student difficulties and insights would not necessarily diminish the distinctiveness of these incidents in teachers' minds. Instead, the advanced information could alert them to signals that otherwise might be overlooked or misinterpreted. Similarly, prescriptions or recommendations for action would be useful in cases in which teachers were not so sure Of how they might respond. Teachers could also benefit from sug- gestions for follow-up activities to be used with individual stu- dents or to use when they encountered difficulties with their activities. Supplying teachers with this information along with the acti- vities to be taught would serve five functions. First, it would assist 239 the teachers in what they claimed wasa difficult task--planning intro- ductory activities--particularly when more than just basic facts and skills are to be learned. Since teachers process unpredictable student occlusions in terms of the mathematical task and questioning routine of an activity, the prepared activity would help to define the para- meters. Thirdly, activities could be designed to illustrate different kinds of instruction such as direct instruction of skills or'guided. discovery with problem solving. This leads to the fourth function of the activity, the impact of the task and teaching method on the type and number of student occlusions. If, for example, instruction is to encourage student conjecture, explanation, and experimentation, the activity must be taught so as to stimulate this type of student behavior. Lastly, for information about probable student occlusions and prescriptions for elective actions to be cred— ible, it needs to be gathered in learning environments which are simi- lar to what teachers will create. Prepared activities would help stabilize these learning environments. The author's recommendation, then, calls for the development of curricular materials packaged in activities for which teachers' actions can be tested for their effectiveness. An exemplary unit of this type has been developed by Fitzgerald and Shroyer (1979). Addition- al units of the same type are in the process of being developed by the Middle Grades Mathematics Project (Lappan, 1980). These units consist of a carefully sequenced set of activities designed to teach mathematical concepts and relationships in a problem solving mode. Teachers are 240 given general information about the content and how it is to be taught as well as scripts for the activities. The scripts provide teachers with specific information about what to say and do, how students might respond, and what actions might be taken. Scripts are the algorithms from which it is hoped that teachers will achieve successful teaching experiences Hawkins (1966) believes are necessary: The good teaching I have observed, teaching by teachers who are accustomed to major success, owes little to modern theories of learning and cognition and much to apprenticeship, on-the- job inquiry, discussion, trial--ceaseless trial--within a com- mon-sense psychological framework; a framework that is not all unSOphisticated, however, and is able to accept individual in- sights from psychological sources but without jargon or dogma: keeping the practice dry (p. 5). The availability of such curricular materials-even with the ad- ditional information about student performances and possible percep- tions for responding--may still not be sufficient. There is ample evidence that curricular materials are not always implemented as in- tended. Smith and Sendelbach (1979), for example, found that teachers misinterpret, alter, and ignore suggestions in the teachers' guides for science programs. They selectively choose and plan according to their goals and conceptions. The extent to which teachers' personal- ized versions of curricular materials influence the type of student performance, however, has yet to be documented. Similarly, the effect of the recommended way of preparing curricular materials on teachers' responses to critical moments also has to be investigated. Instructional units of the type recommended and just described could also be used in preservjce teacher education. _This would provide 241 trainees with examples of teachable units which illustrate different topics, grade levels, materials, and instructional styles. TOO Often teachers are left to discover what constitutes an activity when they begin teaching. Case studies of units Of activities could help to overcome this deficiency. Summary This discussion has focused on the findings, implications, and directions for research and development about critical moments. The important findings about teacher thought and teaching had to do with (a) the different types of critical moments and teachers' cognitive processing; (b) the distinctive and observable characteristics, Of student occlusions; (c) interrelatedness of teachers' activities, actions, goals, conceptions, and attributions; and (d) the variation in teaching and student occlusions. Implications were indicated (a) the role of an activity in de- termining student performance and teachers' mental problem space, (b) the role of student performance in determining teachers' actions and level of processing, and (c) the variation of teacher responses to critical moments. Methodological concerns and recommendations had to do with the (a) criteria for identifying critical moments, (b) the classification of student and teacher behavior, and (c) the method and effect of stimulated recall on teachers. The possibility of investigator bias was also indicated. 242 The important findings with regard to the teaching of mathematics dealth with (a) nature of student occlusions in critical moments, (b) causes of critical moments, (c) role of activities, and (d) role of teacher antecedents in their responses to critical moments. Directions for teacher training had to do with using critical mo- ments as training sites. Recommendations for improving the prepara— tion of curricular materials were also given. CHAPTER VIII SUMMARY AND CONCLUSIONS The purpose of this study was to investigate how' teachers cope with unpredictable student performance which is in- evitable in the problem-solving mode of teaching. Problem solving has long been a desired goal of instruction in mathematics, and the recommendations of the National Council of Teachers of Mathematics for the 19805 declared it a numbercxe priority in the teaching of mathematics. Admittedly this is not an easy task for teachers to undertake, and it makes their instruction more difficult. Address- ing the teaching of problem solving with curricular changes and training programs leaves out the vital role of the teachers, their thoughts and actions, and their interaction with the behavior Of their students. Several studies have established the failure of curricu— lar projects and training programs over the last.twenty years to bring about any change in the teaching of elementary school mathematics. If this is, indeed, because the teacher as vital element has been ignored (Shulman, 1975), the need to explore teacher behavior and thought processes is apparent. The initial chapter raised questions about four areas: the nature of the instructional environment dealing with how teachers organ- ized units, their teaching styles, their goals, student perfor- mance and teachers' elective actions. The second question intro- 243 244 duced the concept of critical moments and teacher reaction to it. The third proposed creating models of how teachers mentally process student performance during critical moments. Emanating from these three areas of inquiry, the fourth set of questions proposed to consider the implications for further research, for developing ap- prOpriate teacher training and preparation of curricular materials. This study attempts to understand more about the mental processing and behavior of teachers in relatiOn to unexpected student perfor- mance; it studies critical moments--a source of difficulties for teachers. A review of the literature ‘reported several studies on inter- active teacher thought and decision making. After a brief descrip— tion of the research and an extensive analysis of the findings, the methods of stimulated recall, coding, and data collection were detailed as a preamble to this study. Prior research conducted over the last decade indicates that teachers do experience difficulties in coping with unexpected student performance, but the use of student difficulties and insights as a strategic research site had not been explored. Similarly, only general relationships between content taught, teacher behavior and teacher thought have been examined. This study proposed to explore in depth the rela- tionship between student performance, teacher thought and behavior, and the subject--mathematics--being taught at critical moments. The teachers selected for study were three experienced teachers who had had some affiliation with the teaching and philosophy of the Unified Science and Mathematics Program in Elementary School, a 245 curriculum project committed to solving real problems. The teachers were teaching upper elementary grades to predominantly fifth and sixth graders and a few fourth grade students. Case studies were conducted using the process-tracing technique and stimulated recall to gather teachers' thoughts and feelings while teaching mathematics to supplement the observable behaviors. The procedure was similar to the one used by Conners (1978). Each teacher taught a six- or seven-day unit on rational numbers, fractions, or decimals. These units were introduced so that teachers were engaged in interactive teaching with their whole class for at least a portion of each day's lesson. The focus was on studying the incidents of unexpected student difficulties or insights which caused the teachers some cognitive or emotional difficulties. Behavioral data were broken down into activities according to the content, task, and mode of instruction. Potential teacher difficulties were identified from student insights and difficulties--the unexpected student contribution. Both student occlusions and teachers' elective actions were categorized and then compared with the teachers' thoughts and feelings to identify the critical moments-~those occasions for which teachers experienced cognitive difficulty and emotional discomfort. As this was both a descriptive and theoretical investigation of critical moments, descriptions from both observable and self-report data were used to answer the questions posed in Chapter 1. Teachers in this study, as shown in Chapter IV, bore many simi- larities to the characteristics of more effective teachers identified 246 in the process/product studies of Evertson et a1. (1980) and of Good and Grouws (1979). Teachers also exhibited variation in the content emphasized, instructional materials, strategies, and style. These overt variations were manifestations of different instructional goals of the teachers and of their conceptions of the teaching and learning of mathematics. They were also consequences of the differences in students' ages, past learning experiences, and capabilities. Using a thumbnail description, the teachers could be character- ized as follows. One was a typical teacher using traditional methods-~exemplification and questioning. With an emphasis on paper- and-pencil tasks and symbolic reasoning, he was encouraging instrumental understanding. At the other end of the con- tinuum was a teacher who was atypical, but close to the NCTM recom- mendations. Wanting to provide students with a solid base, she gave them concrete manipulatives for concept development and rela- tional understanding. The third teacher was a combination of both. She used concrete tasks to introduce the concepts and then shifted to symbolic skills. She also provided alternative symbolic approaches. Her approach was an attempt to bridge the gap between concrete experiences and textbook problems. The impact of the instructional environments was evident in the variations in frequency, proportion, and density of student difficulties and insights. Unsolicited student contributions were more prevalent with the less traditional instructional approach, and student difficulties outnumbered student insights. Teachers' elective actions for student difficulties were similar when classified 247 according to the broad categories of exploiting, alleviating, and avoiding moves. .Their techniques for alleviating student dif-' ficulties varied. Teachers responded differently to student insight; both exploits and avoids were more common. One teacher avoided every instance of student insight. A distinction had to be made between critical moments of a short- term nature and those based on pervasive problems. Critical moments arose over specific incidents of student difficulty or insight, while what was eventually labelled critical discrepancies reflected a more general pattern of student performance. There were four types of conditions that produced cognitive difficulties and emotional discomfort for teachers. The resulting critical moments were labelled as follows: (Type A) student difficulties, (B) student insights, (C) instructional pace, and (D) unanticipated success. All but the last type were further subdivided according to the particular difficulties teachers reported. Of the three Type A teaching difficulties, two were critical moments which occurred when teachers were trying to correct student mis- conceptions or to elicit desired responses to teacher-initiated questions. The third occurred when teachers had trouble diagnosing student difficul- ties with understanding a task. This constituted a critical discrepancy. Difficulties teachers had with student insights were in carrying out their exploits and in having to avoid student suggestions. Pacing dilemmas were produced by trying to meet the divergent needs of two groups--students having difficulty and students having to wait while assistance was given to others. This difficulty occurred in relation 248 to student occlusions and to a recurring pattern of giving individual assists during a group activity. Critical moments occurred in relation to student occlusions that were distinctive: student errors indicative of misconception; student performance distinct from the norm; unsolicited student contribu- tions--insightful or inappropriate--about mathematics not covered in class. Typically, elective actions associated with critical moments called for the use of mathematics not routinely used within the activity which placed a greater cognitive demand on the teacher. Elective actions varied in accordance with the teacher's under- standing of math, their conceptions of learning and their goals for instruction. Likewise, their negative emotional reactions which varied in intensity and type were found to be partially a function of their limited understanding of math which affected not only their choice of actions, but their understanding of their students difficulties and insights. At times, teachers altered their plans and activities which showed that critical moments had an effect; however, when these critical moments were no longer thought about, they did not influence the planning process. The nature and function of teacher mental processing of unex- pected student performance was examined using research on cognitive processing. Using the theoretical writing of Skemp (1979) to interpret much of what was found from examining critical moments, four cognitive processes were identified and described in the processing of student occlusions: interpreting, goal setting, searchipg for elective 249 actions, and evaluating the effectiveness of the actions. Typically, it was the more distinctive student occlusions and elective actions which teachers processed at a more conscious level--a necessary con- dition for more adaptive behavior. Action was initiated to set a goal to act, then find, execute, and test the results. Teachers failed to achieve their goals by reason of default--being unable to think of something else to try--or by interference from antigoals-- situations to be avoided. When teachers encountered difficulties in achieving their goals, they could either continue to search for another elective action or change their goals to secondary goals that could be satisfied more easily. A goal had to be satisfied; if not the initial one, then a secondary one. A relationship was found among teachers' affective reactions, the movement towards or away from goals, and teachers' perceived abilities to bring about change. Teacher attributions for student difficulties were found to reflect their instructional goals and conceptions of learning more than their expectations for students. Critical discrepancies were threats to teachers' more enduring goals for the activities. A switch to secondary goals could provide relief from the discomfort of critical situations, but this was not easily done. Instead, teachers would endure the discomfort as these discrepancies faded in and out of consciousness--at least for a while. Awareness of the situation recurred as long as the condition remained. Escape came from changing the routine of the activity or ending it. 250 The model of cognitive process used by Peterson and Clark (1978) to analyze interactive teacher thought was more appropriate for critical discrepancies than critical moments. Different models were proposed for teachers' cognitive processing of unpredictable student performance-—one for student occlusions and critical moments and another for pervasive patterns of student performance and critical discrepancies. Factors found to influence teacher thought and behavior included the more stable factors of mathematical and pedagogical knowledge, conceptions of learning, principles of teaching, expectations, and instructional goals. The less stable factors identified were the event itself and the intermediate consequences of the stable factors, such as the teacher's perceived ability to effect change. Of particu- lar importance was mathematicalknowledge which influences all four cognitive processes, and pedagogical knowledge which influences goal setting and searching for elective actions. A second point of interest was the complexity of the goal setting process when teachers' initial goals were being threatened. Teaching was made difficult by critical moments because teachers could not achieve their goals--momentary goals for responding to stu- dent occlusions or activity goals. The primary cause of teachers' difficulties was their limited knowledge of mathematics. In some cases, teachers'actions were constrained by their repetoire of pedago- gical techniques or fears of consequences. Teacher thought, as revealed in this study, was similar to the findings of other researchers investigating teacher decisions and thought. Implications of particular importance include (a) the role 251 of the activity in influencing unpredictable student performance and teachers' problem space for processing the same, (b) the influence of student performance on teachers' cognitive processing and actions, and (c) teachers' varied responses to critical moments. Teachers' flexi- bility over and constructive use of critical moments was consistent with their choice of activities. In the critique of methodology for studying teachers' critical moments, recommendations were made about the coding of student and teacher behavior and the techniques to obtain teacher thought with and without the use of stimulated recall. Implications for the teaching of mathematics had to do with the nature of student occlusions as a determining factor of critical mo- ments, the importance of teachers' knowledge of mathematics, and some- what less their pedagogical routines as causes of critical moments. For teachers to cope more effectively with critical moments, they need task-and topic-specific information about the types of stu- dent occlusions to expect and prescriptions for elective actions. Recommendations were given for the use of critical moments in training teachers and for preparing curricular materials by activities and units. Directions for future research and development included the fol- lowing: 1. Investigate the effects of an activity-~task, question- ing, routine--on the number and type of unpredictable . student performance and on the teachers' instructional approach and response to critical moments. 2. Investigate the consistency of the relationships be- tween teachers' goals, conceptions of learning, at- tributions, and emotional responses to critical mo- ments. 252 Refine and validate taxonomies for identifying poten- tial critical moments from observable characteristics. Compare methods for coding student occlusions and teacher elective actions, for gathering teachers' thoughts, and for identifying teachers' critical mo- ments and discrepancies. Compile cases of critical moments to illustrate dif- ferent types of student difficulties and insights, teacher elective actions for different mathematical tasks, and types of instruction for use in training by simulation. Prepare instructional units of the type recommended and use to identify types of unpredicted student difficulties or insights by tasks and topics and determine effective actions for use by teachers. Investigate the effects of training teachers with simulated critical moments and with improved pre- paration of curricular materials. APPENDIX m1 1 253 Table A.1 Martha's Fraction Unit: Time in Minutes and Content by Activity Content Activity Time Ippig Task Domain 1 14 Concept name+model I/a 2 5 Concept modelename 1/6 (3) (5) (test) 4 4 Concept name-rmodelb 1/3 5 3 Concept name+model 1/a 6 20.5 Concept model+name b/ag] 7 3.5 Concept name+model b/aSl (8/9) (14) (test/game) 10 3.0 Relationship compare b/aSl like denom. 11 17 Concept model+name b/aSI (12) (6.5) (test) 13 5.5 Relationship compare/order b/asl _ like denom. 14 9 Relationship compare like numerators 15 38.5 Relationship compare b/aSl 16 9.5 Concept model+name b/a21 17 5.5 Concept name+modelb 6/4' 18 7.5 Concept model+name b/aSI 19 14 Relationship find equivalent b/aSl 20 5.5 Operation add b/aSl like denom. 21 24.5 Operation add b/a51 22 27.5 Operation add b/aSI 23 4.5 Concept define all types (24) (11) (Operation) (add) (b/ail) 25 11 Operation add b/aSl a rods = Cuisenaire rods b no unit rod specified Materials rodsa pictures rods rods pictures pictures symbols pictures symbols symbols rods rods rods rods rods/symbols rods/symbols rods rods words/symbols (rods) rods/symbols Day Table A.2 Activity 1 1 2 (3) (3) 4 5 6 7 (8) 9 10 11 12 13 (14) 15 (16) 17 (18) Zelda's Fraction Unit: Content by Activity Time 10 4.5 (9) (7) 12.5 14 15 12.5 14 19 18.5 11 19 (21) 5.5 (45) 17 (37) 254 TOpic Concept Concept (work period) (work period) Concept Concept Relationship Relationship (work period) Relationship Operation Operation Relationship Relationship (work period) Relationship (work period) Relationship (work period) 9111321 Task name model name Time in Minutes and Domain 1/a 1/a (report activity 3) name model name equivalents generate set of equivalents compare name add model add find simplest equivalent find simplest name find missing value generate set equivalent b/aSI I/a b/a£1 b/ail like denom. unlike denom. b/aSI b/ail b/aSI b/aSJ Materials picture rods rods rods/symbols rods rods/symbols symbols symbols symbols symbols 255 Table A.3 Ralph's Decimal Unit: Time in Minutes and Content by Activity Content Day Activity Time Topic Task Domain* Materials 1 1 23.5 Concept translate deci- decimal symbols mal to fraction (2) (22) (work period) 2 3 6.5 Concept (report answers from activity 1) 4 21.5 Concept translate deci- decimalk symbols mal to fraction orlO (5) (18) 3 6 15.5 Concept translate frac- IOk symbols tion to decimal (7) (30) (work period) 4 8 19.5 Relationship order and equate 10k symbols 9 8. 5 Concept translate frac- 15 10k symbols (10) (20.5) (work period) 5 11 25.5 Operation add/sub. frac— decimal symbols tion explanation- (12) (20) (work period) 6 13 19 Operation add/subtract decimal symbols (14) (27) *Fraction domains are in terms of denominator: 10k orthOk. 256 Table A.4 Distribution of Instructional Time in By Content and Teacher Minutes Content Teacher Martha Zelda Ralph Concgpts 94 (40)* 41 (25) 75.5 (54) model/name unit (l/a) 26 27 proper (b/a j_l) 48.5 14 imprOper (b/a > 1) 15 define/translate all fractions 4.5 denom = 10k 67 denom f lOk 8.5 Relationships 70 (30) 88 (53) 19.5 (14) compare/order rods/pictures 38.5 19 symbol like denom 8.5 like num. 9 equate rods/picture 14 15 19.5 generate set 12.5 find missing value 14.5 Operations 68.5 (30) 37.5 (22) 44.5 (32) add (subt) like denom 5.5 8 44.4 rods/symbols unlike denom rods 52 9 symbols 11 29.5 232.5 166.5 139.5 *(percent of instructional time) 257 .coNNusepmcw mo muscwe Log mcowmzNuuo mo conga: :N umczmmme NN xuchmoe N._ N N_ NN N N. N N NN _._ F a NN N. o a N_ N._ o N NF N._ N NN NN o N.F NN _N _N N.N N N ON o._ o N_ Na N. N OF NN N. P N ,_ _._ N c_ N, N. N N N, N o.F N N_ _, N._ N N N_ a. o N N N. N N o_ N.N o e_ NF N. o N N 0.? o N_ N N. o NN N, a e._ N o_ e_ 0.0 o o N_ N. N N N N._ a N_ N m. N N __ N N. o o N o._ o N ON a. _ N a N. o N N N._ o N N N. o N N a. o N a N. _ OF N 0.0 o o N N N._ F N a e. o N N N. o N N N. 0 NF _ N.F 0 NF N N. a N N _ Noam NLNNNCN .ecao .Nmfl NNNN NemamcH .amae .Nwfl Nuwm Nemamea .CNTN .NMH .Nmm -cao acaeaum ->_uu< -eao Nemesum ->Nuu< -eao Nemesum ->eoua £N_NN Ne_aN agate: .NuN>Npu< use Locumme an mconsNuuo pawnspm mo *Nwwmcmo vcm Nocmnvwcm m.< argue REFERENCES REFERENCES Ashlock, R. B. Research and development related to learning about numerals for whole number51 implications for classroom/re- source room/clinic. In M. E. Hynes (Ed.), Topics related to diagnosis in mathematics for classroom teachers. Seminole County, FL:7_(no publisher given), 1979. Barnard, C. I., The function of the executive. Cambridge, MA: Har- vard University Press, 1938. Behr, M. J., & Post, T. R. The effect of visual perceptual distrac- tors on children's logical-mathematical thinking in rational- number situations. Paper presented at the International Group for the Psychology of Mathematics Education, Grenoble, France, 1981. Bishop, A. J. Simulating pedagogical decision making. Visual Educa- tion, November, 1970. Bishop, A. J., & Whitfield, R. C. Situations in teaching. London: McGraw-Hill, 1972. Bloom, 8. S. The thought processes of students in discussion. In S. J. French (Ed.), Accent on teaching: Experiments in general education. New York: Harper, 1954. Brophy, J. E. Teachers' cognitive activities and overt behaviors (Occasional Paper No. 39). East Lansing, MI: Institute for Research on Teaching, Michigan State University, 1980. Brophy, J. E., & Good, T. L. Teacher-student relationships: Causes and consequences. New York: Holt, Rinehart, & Winston, 1974. Bussis, A., Chittenden, E. A., & Amarel, M. prond surface curricu- lpm, Boulder: Westview Press, 1976. Clark, C. M., & Joyce, B. R. Teacher decision making and teaching ef- fectiveness. Paper presented to American Educational Research Association Conference, Washington, D. C., 1975. Clark, C. M., & Peterson, P. L. Teacher stimulated recall of inter- active decisions. Paper presented to American Educational Re- search Association Conference, San Francisco, 1976. 258 259 Clark, C. M., & Yinger, R. J. Teachers' thinking. In P. L. Peter- son & H. J. Walberg (Eds.), Research on teaching: Concepts, findings, and implications. Berkeley, CA: McCutchan Publish- ing Co., 1979. Clarkson, G. P. E. Portfolio selection: A simulation of trust invest- ment. Englewood Cliffs, NJ: Prentice Hall, 1962. \/ Conners, R. D. An analysis of teacher thought processesgpbeliefs, and principles during instruction. Unpublished doctoral dissertation, University of Alberta, 1978. Coopen.H. M. Pygmalion grows up: A model for teacher expectation, communication, and performance influence. Review of Educational Research, Summer 1979, 49, 3, 389-410. Cooper, H. M., & Burger, J. M. How teachers explain students' academic performance: A categorization of free response attributions. American Educational Research Journal, 1980, 11, 95-109. Cooney, T. J., Kansky, R., & Retzer, K. A. Protocol materials in mathematics education: Selection of concepts. Bloomington, IN: Indiana University School of Education, National Center for the Development of Training Materials in Teacher Education, Report No. 7, February, 1975. de Groot, A. 0. Thought and choice in chess. The Haguec Mouton, 1965. Doyle, W. Classroom tasks and student abilities. In P. L. Peterson & H. J. Walberg, Research on teaching: Concepts, findings, and implications. Berkeley, CA: McCutchan Publishing Co., 1979. Easley, J. A portrayal of traditional teachers of mathematics in American schools. Report #4, Committee on Culture and Cognition. Champaign, IL: University of Illinois, 1978. Eicholz, R. E., & O'Daffer, P. G. Elementary school mathematics, books #5 and 6. Palo Alto, CA: Addison Wesley Publishing Co., 1964. Elstein, A. S., Shulman, L. S., & Sprafka, S. A. Medicalgproblem solving: An analysis of clinical reasoning. Cambridge, MA: Harvard University Press, 1978. Erlwanger, S. H. Benny's conceptions of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1973, 1, 7-26. Evertson, C. M., Anderson, C. W., Anderson, L. M., & Brophy, J. E. Relationships between classroom behaviors and student outcomes in junior high mathematics and English classes. American Educational Research Journal, Spring, 1980, 11, 1, 43-60. 260 Fey, J. T. Mathematics teaching today: Perspectives from three na- tional surveys for the elementary grades. What are the needs in precollege science, mathematics, and social science education? Views from the field. Washington, D. C.: National Science Foun- dation, 1980. Fitzgerald, W. M., 8 Shroyer, J. C. Mouse and elephant unit, revised edition (Produced under NSF grant, SED #77-18545). East Lansing, MI: Department of Mathematics, Michigan State University, 1979. Good, T. L. Teaching mathematics in elementary schools. Educational Horizons, Summer 1979, 178-182. Good, T. L., & Grouws, D. A. The Missouri mathematics effectiveness prOJect: An experimental study in fourth grade classrooms. Journal of Educational Psychology, 1979, 71, 3, 355-362. Good, T. L., & Grouws, D. A. Project for increasing mathematics ef- fectiveness. Teachers' manual, project PRIME; no further data given. Hawkins, 0. Learning the unteachable. In L. S. Shulman & E. R. Keislar (Eds.), Learningpby discovery: A critical appraisal. Chicago: Rand McNally, 1966. Herscovics, Nh,& Bergeron, J. C. The training of teachers in the use of clinical methods. In Equipe de Recherche Pedagogique (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education. Berkeley, CA: 1980. Heymann, H. W. Subjektive unterriahtstheorien von mathematik lehrern als gegenstand fachbezogener lshr-lem-forschung. Germany: University of Bielefeld, Institute for Study of Mathematics, 1980. Jefferies, R., Polson, P. G., Razran, L., & Atwood, M. E. Mission- aries and cannibals. Cognitive Psychology, October 1977, 9, 4 412-440. Joyce, 8., & McNair, K. Teaching styles at South Bay School: The South Bay study, part I (Research Series No. 57). East Lansing, MI: Institute for Research on Teaching, Michigan State Univer- sity, 1979. Joyce, B. R., Morine, G., & McNair, K. Thought and action in the classroom: The South Bay study. Report prepared for the Insti- tute for Research on Teaching, Michigan State University. Palo Alto, CA: 1977. Kagan, N. I. Interpersonal process recall: A method of influencing human interaction. East Lansing, MI: Michigan State University, 1975. 261 Kagan, N., Krathwohl, D. R., 8 Miller, R. Stimulated recall in ther- apy using videotape--a case study. Journal of Counseling Psy: chology, 1963, 19, 237-243. Kilpatrick, J. Research on teaching mathematics to the elementary school pupil. Proceedings of the Research-on-Teaching Mathema- tics Conference, May 1-4, 1977. Conference Series No. 3. East Lansing: Institute for Research on Teaching, Michigan State University, 1977. Kilpatrick, J. Research on mathematics learning and thinking in the United States. Equipe de Recherche Pedagogique (Ed.), Psychology of Mathematics Education. Grenoble, France: 1981. Kounin, J. S. Discipline and group management in classrooms. New York: Holt, Rinehart 8 Winston, 1970. Kounin, J. S., 8 Doyle, P. H. Degree of continuity of a lesson's signal system and the task involvement of children. Journal of Educational Psychology, 1975, p], 159-164. Kounin, J. S., 8 Gump, P. V. Signal systems of lesson settings and the task-related behavior of preschool schildren. Journal of Educational Psychology, 1974, pp, 554-562. Lappan, G. Middle grades mathematics project (Produced under NSF grant, SED #80-18025). East Lansing, MI: Department of Mathe- matics, Michigan State University, 1980. Marland, P. W. A study of teachers' interactive thoughts. Unpub- lished doctoral dissertation, University of Alberta, Edmonton, Alberta, Canada; Fall 1977. Marx, R. W., 8 Peterson, P. L. The nature of teacher decision making. Paper presented to American Educational Research Association Conference, Washington, D. C., 1975. McNair, K., 8 Joyce, B. Thought and action, a frozen section: The South Bay study. Educational Research Quarterly, 1978-79, p, 16-25. McNair, K., 8 Joyce, B. Teacher thoughts while teaching: The South ’I Bay study, part II. (Research Series No. 58.) East Lansing, MI: 3 Institute for Research on Teaching, Michigan State University, 1979. Merton, R. K. Notes on problem finding in sociology. Introduction in R. K. Merton, L. Brown, 8 L. S. Cottrell (Eds.), SOCiology today; Problems and prospects. New York: Basic Books, 1959. Miller, G. A., Galanter, E., 8 Pribram, K. Plans and the structure of behavior. New York: Henry Holt 8 Co., 1960. 262 Morine, G., 8 Vallance, E. A study of teacher and and pupil percep- tions of classroom interactions. Beginning Teacher Evaluation Study, Technical Report 75-11-6, Special Study B. San Francisco: Far West Lab for Education, R 8 D, November 1975. Morine-Dershimer, G. How teachers "see" their pupils. Educational Researchpguarterly, 1978-79, 3, 43-52. Morine-Dershimer, G. Teacher plans and classroom reality: The South Bay study, part IV. Research series No. 60. East Lansing: Institute for Research on Teaching, Michigan State University, July 1979. Moser, J. M. The emergence of algorithmic problem solving behavior. In Equipe de Recherche Pedagogique (Ed.). Proceedings of the Fifth Conference of the International Group, Psychology of Mathe- matics Education. Grenoble, France: 1981 National Conference on Studies of Teaching. Panel Seminars, National Institute of Education, Washington, D. C., Decmeber, 1974. National Council of Teachers of Mathematics. An agenda for action. Recommendations for school mathematics of the 19805. Reston, VA: NCTM, Inc., 1980. Newell, A. You can't play 20 questions with nature and win: Projec- tive comments on the papers of this symposium. In Chase, W. G. (Ed.), Visual information processing. New York: John Wiley, 1973. Peterson, P. L. Direct instruction reconsidered. In P. L. Peterson 8 H. J. Walberg (Eds.), Research on teaching: Concepts, findings, and implications. Berkeley: McCutchan 1979. Peterson, P. L., 8 Clark, C. M. Teachers' reports of their cognitive processes during teaching. American Educational Research Journal, Fall 1978, 15, 4, 555-565. KjPeterson, P. L., Marx, R. W., 8 Clark, C. M. Teacher planning, teacher behavior, and student achievement. American Educaional Research Journal, 1978, lp, 417-432. Polya, G. How to solve it: A new aspect of mathematical method, 2nd edition. New York: Doubleday Anchor Books, 1957. Pulos, S., Stage, E. K., Karplus, F., 8 Karplus, R. Spatial visuali- zation and proportional reasonin of early adolescents. In Equipe de Recherche Pedagogique (Ed.). Proceedings of the Fifth Conference of the International GrOpp, Psychology of Mathematics Education. Grenoble, France: 1981. Radford, J. Reflection on introspection. American Psychologist, 1974, 245-250. 263 Resnick, L. B., 8 Ford, W. W. The psychology of mathematics for in- struction. Hillsdale, NJ: Lawrence Erlbaum, Associates, 1981. Rosenshine, B. V. Content, time, and direct instruction. In P. L. Peterson 8 H. J. Walberg (Eds.), Research on teaching: Concepts, findings, and implications. Berkeley, CA: McCutchan, 1979. Schwab, J. J. Science, curriculum, and liberal education: Selected essay , I. Westbury 8 N. J. Wilkof (Eds.). Chicago: University of Chicago Press, 1978. Shavelson, R. J. Teachers' decision making. In N. L. Gage (Ed.), Psychology of teaching methods, 75th Yearbook of National Society for the Study of Education (Part I). Chicago: University of Chicago Press, 1976. Shavelson, R. J. What is the basic teaching skill? Journal of Teacher Education, Summer 1973, 24, Shulman, L. S. The psychology of school subjects: A premature obituary? Journal of Research in Science Teaching, 1974, ll, 4, 319-339. Shulman, L. S. Strategic sites for research on teaching. Paper pre- sented to American Educational Research Association, Toronto, 1978. Shulman, L. S. Recent develOpments in the study of teaching. In B. R. Tabachnick, T. S. Popkewitz, 8 B. B. Szekely (Eds.), Studying teaching and learning: Trends in Soviet and American research. New York: Prager Publishers, 1981. Shulman, L. S. .Research on teaching mathematics to the elementary school pupil. Proceedings of the Research-on-Teaching Mathema- tics Conference, May 1-4, 1977. Conference Series No. 3. East LanSing: Institute for Research on Teaching, Michigan State University, 1977. Shulman, L. S. Teaching as clinical information processing. Panel #6, National Institute of Education Conference on Studies in Teaching, Washington, D. C., 1975. Shulman, L. S., 8 Elstein, A. S. Studies of problem solving, judgment, and decision making. In F. N. Kerlinger (Ed.), Review of research in education, 3. Itasca, IL: F. E. Peacock, 1975. Shulman, L. S., Loupe, M.J., 8 Piper, R. M. Studies of the inquiry process: Inquiry patterns of students in teacher training pro- grams. East Lansing: Final Report, Project No 5-0597, Office of Education, Michigan State University, 1968. 264 Simon, H. A. Models of man. New York: John Wiley and Sons, 1957. Simon, H. A. The sciences of the artificial. Cambridge, MA: MIT Press, 1969. Skemp, R. R. Intelligence, learning and action: A foundation for theory and practice in education. Chichester, England: John Wiley 8 Sons, 1979. Skemp, R. R. What is a good environment for the intelligent learning of mathematics? 00 schools provide it? Can they? In Equipe de Recherche Pedagogique (Ed.), Proceedings of the Fifth Conference of the International Group, Psychology of Mathematics Education. Grenoble, France: 1981. Smith, L. M. An evolving logic of participant observation, educational ethnography, and other case studies. In L. S. Shulman (Ed.), Review of research in education, 6. Itasca, IL: E E. Peacock, 1978. Smith, E. L., 8 Sandelbach, N. B. Teacher intentions for science instruction and their antecedents in program materials. Paper presented to American Educational Research Association, San Francisco, 1979. Snow, R. E. Theory construction for research on teaching. In R.M.W. Travers (Ed.), Second handbook of research on teaching. Chicago: Rand McNally, 1973. Soar, R. S., 8 Soar, R. M. Emotional climate and management. In P. L. Peterson 8 H. J. Walberg (eds.), Research on teaching: Concepts, findings, and implications. Berkeley: McCutchan, 1979. Stake, R. E., 8 Easley, J. (Eds.). Case studies in science education. Urbana, IL: University of Illinois, 1978. Stallings, J. A., 8 Kaskowitz, D. H. Follow through classroom obser- vation evaluation, 1972-1973. Menlo Park, CA: SRI, 1974. Sutcliffe, J., 8 Whitfield, R. Decision-making in the classroom: An initial report. Cambridge, England: SSRC/BERA Seminar on Class- room Interaction, November 1975. Suydam, M. N. Review of recent research related to the concepts of fractions and of ratio. In E. Cohors-Fresenborg 8 I. Wachsmuth (Eds.), Proceedipgs of the Second International Conferepce fpr the Psychology of Mathematics Education, Fachbereich Mathematik, Uhiversitat Osnabruck, 1978. 265 ' math- dam, M. N., 8 Osborne, A. The status of pre-college sc1ence,. Suy ematics, and social science education: 1955-1975. Vol. II. . Mathematics education. Columbus, OH: The Ohio State UniverSity CEnter for Science and Mathematics Education, 1977. Tuckwell, N. B. Content analysis of stimulated recall protocols. Oc- casional paper series, research report #80-2-2. Edmonton, Alber- ta: Centre for Research on Teaching, UniverSity of Alberta, 1980. Walker, D. F., 8 Schaffarzick, J. Comparing instruction. Review of Educational Research, 1974, 44, 83-111. Weiner, B. The role of affect in rational (attributional) approaches to human motivation. Educational Researcher July-August, T980, 9, 7, 4-11. Weiner, B. A theory of motivation for some classroom experiences. Journal of Educational Psychology, 1979, 11, 1, 3-25. Weiss, I. Report of the 1977 national survey of science, mathematics and social studies education. Research Triangle Park, NC: Research Triangle Institute, 1978. Whitfield, R. C. Teaching as decision making for and in the science room. In P. L. Gardner (Ed.), The structure of science education. Melborne, Australia: Longman, 1971. Zahorik, J. A. Teachers' planning models. Paper presented to Ameri- can Educational Research Association, Washington, D. C., 1975. a. I .. i I ... as. I! .I . .. . K, 1 . .1. Iii. .. i . .u... . .i . i. ..I .il . u.. are... .a i n. i .... 91:4. 1|, All-10.. all. .. .. . - . I i a! . . .‘ ,. . . \.. is ...o. «u-.. i»: ll, .-.o I i. . . . . 1 ., . "II)![I]![lililliillis - u— _ A _