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Teaching from the student's point-of-view : a developmental perspective Levitt, Lori Nadine 1988

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TEACHING  FROM THE STUDENT'S  A DEVELOPMENTAL  POINT-OF-VIEW:  PERSPECTIVE  by LORI N A D I N E LEVITT B.Ed., The University  A THESIS SUBMITTED IN THE REQUIREMENTS  of  Alberta,  1980  PARTIAL FULFILMENT O F FOR THE DEGREE  OF  MASTER O F ARTS  in THE FACULTY O F G R A D U A T E STUDIES (Educational  We  Psychology and  accept this thesis to  the  required  Special  as  Education)  conforming  standard  THE UNIVERSITY O F BRITISH C O L U M B I A 11 O c t o b e r  © Lori  Nadine  1988  Levitt,  1988  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Educational Psychology and Special Education) The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 11 October 1988  ABSTRACT  This  study  was  problems  an  which  point-of-view. describe  exploratory  are  ill-defined  The extent  those  study  to  teachers  The non-random participated  in  identify  concept the  rated  sample consisted of 27 primary  perspective  responses to  according  subsequent  of  on  to  four  in-service education.  problem  complexity,  quality  variables of  interest  questions  cognitive  the  interpret  "teacher and  classroom  the  as problem strategies  and  study  and intermediate  Participants  were  on their  student's  finder"  may  necessary  for  level teachers  introduce  asked  to  them  to  a  'The  background and experience.  associated with  included:  who  complete  accompanied each anecdotal  variables  These  point-of-view  to  faced with  courses designed to  which  process  solving.  of  and  structures  Student Anecdotes Task' and a questionnaire  Teachers'  teachers, when  perspective was also examined.  district-sponsored  developmental  the  have  teaching from a developmental  how  problems,  which  who  of  problem  developmental  problem  problem  were  finding  formulation,  teaching  included, concern for  task  and  integrative  strategies.  Additional  finding  and several  formulation,  integrative  demographic variables.  The  results  complexity strategies in  and may  ill-defined  teacher  suggested quality affect problem  thinking  were  that of  how  the  variables  point-of-view teachers  situations. discussed.  as  identify  need  ii  problem  well  and  Implications The  of  as  the  interpret  developmental  the  student's  for  teacher  education  for  clinical  interviews  and  teaching  point-of-view studies  augmented  of by  classroom observations was emphasized for future studies.  Several research questions,  related  this  to  the  cognitive  process  variables  identified  in  study  to  affect  teacher's ability to teach from a developmental perspective, were generated.  iii  the  TABLE OF CONTENTS List of Tables  vi  List of  vii  Figures  ACKNOWLEDGEMENT  viii  Chapter I. I N T R O D U C T I O N  1  A. B A C K G R O U N D O F THE PROBLEM B. STATEMENT O F THE PROBLEM 1. The Kinds of Problems Teachers Face in the Act Teaching 2. Classroom Problems as Ill-Defined Problems 3. The Teacher's Reflection-in-action 4. The Need for a Problem Finding Framework C. SIGNIFICANCE O F THE STUDY D. DEFINITION O F THE TERMS E. S U M M A R Y O F THE PROBLEM : Chapter II.  2 6 of 7 9 11 12 13 14 16  LITERATURE REVIEW A. IMPLICATIONS FOR E D U C A T I O N A L PRACTICE O F DEWEYIAN A N D PIAGETIAN THEORIES O F K N O W L E D G E 1. Dewey's Experimentalist View of Growth in Knowledge 2. Piaget's Constructivist View of Growth in Knowledge B. A FRAMEWORK FOR EXPLORING THE TEACHERS' O W N C O N S T R U C T I O N OF K N O W L E D G E 1. The Kinds of Problems Teachers Face in the Act of Teaching 2. Classroom Problems as Ill-defined Problems 3. Studies of Problem Finding C. THE TEACHER AS PROBLEM FINDER 1. The Teacher's O w n Construction of Knowledge 2. The Teacher's O w n Reflection-in-action D. S U M M A R Y  17  ;  Chapter  III.  METHODOLOGY A. DESCRIPTION O F THE SAMPLE B. PILOT STUDY C. PROCEDURES 1. Task Rationale 2. Task Development and Description 3. Task Administration 4. Rating Criteria and Examples of Teacher's Responses 5. Questionnaire: Other Variables of Interest to the Study  Chapter IV. RESULTS A. QUALITATIVE ANALYSIS 1. Problem Formulation 2. Integrative Complexity iv  .  17 18 23 29 29 30 37 40 40 41 45 47 47 48 48 49 50 54 55 67 68 71 71 77  3. Quality of Point-of-View 4. Developmental Teaching Strategies 5. Other Variables of interest to the B. THE TEACHER'S POINT-OF-VIEW  82 90 100 105  Present Study  Chapter V. DISCUSSION A. DISCUSSION O F THE RESULTS 1. Problem Formulation 2. Integrative Complexity 3. Quality of Point-of-View 4. Developmental Teaching Strategies B. LIMITATIONS O F THE STUDY C. IMPLICATIONS FOR RESEARCH O N T E A C H I N G D. DIRECTIONS FOR FUTURE RESEARCH  :.  112 112 112 114 117 119 121 128 129  REFERENCES  13.1  APPENDIX A  137  APPENDIX B  167  v  LIST  Table 1: Problem  OF  TABLES  Formulation - Scoring Categories and Examples  56  Table 2: Criteria for Rating Integrative Complexity  60  Table 3: Rating Criteria for "Quality of Point-of-View" Table 4: Developmental Teaching  .....62  Strategies  66  Table 5: Participant Ratings for Problem Formulation, Integrative Complexity, Quality of Point-of-View and Developmental Teaching Strategies by Subject Category/Grade Level  70  Table 6: Frequency Table for Problem  Formulation  72  Table 7: Frequency Table for Problem ..:  Formulation by Subject Category/Grade  76  Table 8: Frequency Table for Integrative Complexity Table 9: Frequency Table for Integrative Complexity  78 by Subject Category/Grade  Table 10: Frequency Table for Quality of  Point-of-View  Table 11: Frequency Table for Quality of  Point-of-View  Category/Grade  Level  Level 82 84  by Subject  Level  88  Table 12: Frequency Table for Developmental Teaching  Strategies  Table 13: Frequency Table for Developmental Teaching Category/Grade Level  Strategies by Subject  vi  91  94  LIST  OF  FIGURES  Figure 1: Teacher Notes for Analysis of Pretest Item  31  Figure 2: Sample Scenario  52  Figure 3: Formula for Calculating Question Quality  84  vii  ACKNOWLEDGEMENT  It  is with  their  pleasure  contributions  Words  alone  Patricia  K.  its  my  appreciation  the completion  sufficient  whose  to  of  this  to  the  following  individuals  for  and respect  I have for  Dr.  of this thesis:  express the  professionalism,  preparation  appreciation  sharing the  their  is extended to  expertise  many  final drafts  hours  gratitude  expertise  thesis  has  and  greatly  undaunted  influenced  each  encouragement step  towards  in  the  spent  Dr. Billie Housego and to  areas of providing  teacher insightful  thinking  and  suggestions  Dr. Caalen Erickson for teacher and  education,  comments  and  on  the  took  the  of this thesis.  Many thanks to time  extend  completion.  Much  for  are not  the  I  towards  Arlin,  throughout  that  and interest  the to  teachers from  Maine and Long Island, N e w York who  reveal their thoughts  about  viii  teaching.  There are those who adhere to what's etched in stone; There are those who question what's etched in stone; And then there are those who question the stone itself. -Author Unknown  To Patricia K. Arlin For teaching me how to "...question the stone itself"  ix  CHAPTER  This  study  is  an  attempt  problems which deviate point-of-view.  to  1. I N T R O D U C T I O N  describe  from  how  teachers,  their expectations,  when  identify the  What is important to the present study is the  teachers to implement strategies  faced  with  child's or  classroom adolescent's  idea that in order for  that guide students in the learning process through  helping them with questions and problems from their own points of view, need  to  be  able  to  identify  and  interpret  children's  and  adolescent's  teachers "alternate  frameworks" (Driver & Easley, 1978), "reactions to failure" (Karmiloff-Smith & Inhelder, 1974),  "faulty procedures" (Case,  1970),  "unexpected" responses  &  Fusco,  1983),  adolescent's  or  simply  1978,  (Arlin, to  1980),  1983)  "incorrect" answers  (Sinclair & Kamii,  and "mismatches" (Elkind,  understand  curriculum  tasks  from  1976; the  Brooks  child's  or  point-of-view (Elkind, 1976).  It will be argued within this study that in order to examine this view of "teaching for  thinking"  (Arlin,  1985),  understands the student's 1).  There  is  understand own on the  also  the  "how" the  researchers  need  to  explore  "how  the  teacher  thinking and her role in their learning" (Lampert, 1984, p. need  for  a  problem  teacher formulates the  finding problem  or of  formulation framework  to  instruction based on her  experience and action. This framework may inform the teacher's own reflections '-'how" a student teacher  "has  the  "has  a concept"  concept"  of  (Larsen,  the  1977),  and therefore  "child's/adolescent's  1976).  1  reveal "how"  point-of-view"  (Elkind,  2 A. BACKGROUND Researchers therefore Smith,  have  reasons  1961;  processes  of  instructional  In  the  OF THE long  maintained  and  learns  Piaget,  1972,  inquiry  which  intervention  solving of  that  reflective  for  uncertainty  inquiry  about  suggested  that  need  flexible  for  incidents coming  in  cultivated  1978).  he  are  led  thinking  a solution order  to  subject  through  the  education  continues  to  that  who  with  had  Dewey's  the  with  in  of  freshness of  of  exist. Within  to  to  take  fullness  attitudes  emphasis  influence  and  on  research  the  which  were  "reflective on  education (see Hullfish & Smith, 1961; Kitchener,  teacher  kind  of  this  and  framework,  he  there  of  was a  unexpected  on the and  favorable  to as  the  problem  interest  1983; Schon,  model  emphasized  students,  thinking,  of  S  He a  thinking"  the  describe  advantage  of  &  differences.  to  knowledge.  knowledge  ability  order  awareness  must  Hullfish  hierarchical  classroom. This, he said, depended  acquisition  to  the  an  and  identifying  needs based on these  development  problem  adult"  1933; in  associated  thinking"  place,  "miniature  primarily  identified  "reflective  take  a  (Dewey,  been be  (1933)  the  not  adult  has  assumed  to  is  the  successfully impart  in the  the  thinking".  than  Dewey  to to  teachers  "reflective  child  Interest  called  which  and questions to  the  differently  tradition,  which  processes  that  that meets the student's  Experimentalist  problem  PROBLEM  teacher's knowledge  the an  use  of  aim  development  for and  1983, 1987; Zeichner,  1983).  Recently,  in  the  developmental researchers  tradition perspective  (Glassberg  &  of  Dewey on  and  in  education  Sprinthall,  1980;  response to (Glassberg Sprinthall  & &  the  expressed need for  Sprinthall, Sprinthall,  1980), 1983a,  a  many 1983b;  3 Thies-Sprinthall, on  a  1984)  combination  have  of  begun  cognitive  to  incorporate  developmental  a developmental  theories  (i.e.,  Hunt,  1969; Loevinger, 1966; Perry, 1970; Piaget, 1972; Selman, 1980) teacher  development  successful teacher in  complex,  thus,  do  not  is viewed  flexible  development and  and programs  of  the  development  it  is then  is  neglect which  a  fallacy  consider  with the  differences  develop  and  and adults construct  construction  of  1985),  should  inform the  the  development of  classroom  their  do  if  we  what education  not  1984;  as  they  ability  to  the  engage  as  necessary  for  to  "reflective  practice"  know ought  thinking,  child,  are  be"  the  that,  (tentatively)  to  the  "we what  (Sprinthall  &  been  that  ways  to  the  and  explain  in which  researchers  adult  thought  "how"  children,  these  adolescents  and reality (Phillips, 1980). Conceptions of  exist,  the  is  adolescent  differences  do  not  respect thinking  indicate  rarely  teacher  affect  knowledge  Sinnott,  has  of  there  they  do not  thinking  problems  problem solving.  that  between  teacher's reflections.  own (Arlin,  conducive  of  this framework,  cognitive  seen  studies  think. These researchers state  suggest  that"  own  researcher"  and  teacher's  are  skills  to  views  "how"  knowledge,  (Arlin,  to  differences  "knowing  as  and  students  these  beyond  teacher  which  the  Kohlberg,  into their  education. Within  possesses  thought  may also know  go  "the  who  1961;  based  31).  problematic to  teach  we  teacher  behaviors  to  as  Sprinthall, 1983b; p.  What  of  attitudes,  ability  see  as one  levels  the  of  framework,  consider  teacher's  as a dynamic act of the  theoretical  frameworks  Rarely have the addressed,  the  1983).  Also,  been  considered  models  developmental  particularly  1981,  or  the as  in  teacher's  critical  to  own  teacher which  constraints  terms  of  on adult  own  formulation  their  subsequent  4 Instead, that  teachers  assess  problems  thus  exist  in  their  teaching  "effectiveness"  in  frameworks and  and  view tend  to  neglect the  Mackworth,  of  in  terms  predefined  examine to  focus  classroom  1965). The  on  required  to  with  within  the  these  "ill-defined"  about  the  problem defined  practical  problem  then  (Arlin,  proceeds 1983;  Lampert,  the  &  These  problems, which  also  Dillon,  1973;  identify- and interpret  1984;  curriculum and  teacher on  Csiksentimihalyi,  the  her  frames basis  1976;  its  are  solution  1983,  1987).  own  knowledge  the of  Getzels  where  tasks,  seek  Schon,  constructs  The  solution  to  problem  teacher  point-of-view".  &  problems  Cetzels  responses"  1983;  Getzels  practice.  problems. They are situations  (Arlin,  a  classroom  "well-defined"  teachers use to  student's  toward  solve  frameworks  solves  teaching  1964;  each  situations  teacher  "ill-defined"  of  "child's/adolescent's  and it  context  through  have rarely been examined.  "unexpected  nature  to  or  are, in fact, ill-defined  the  the  "good"  Cetzels,  reflection  examined  well  ability  problems  students'  identify  1982;  be  of  "discovered"  processes of  to  how  teacher's  (Dillon,  Many classroom problems faced  of  definitions  the  classroom problems as ill-defined  teachers,  continue  instructional  how &  In  she  Dillon,  has 1973;  Schon, 1983, 1987).  The  dominant  instruction (Arlin,  and  1983;  Constructivist 1958; "how"  model  Brooks, model  McNally, the  how  1977;  in  studies  they  interpret  1984; (see  of  Brooks,  Furth,  Piaget,  thought and understanding  teachers  these  problems  Fusco  &  1981;  1972,  "how"  of  1979). the  from  Grennon,  Gallagher  1978,  think  &  1983)  Reid, This  the  1981;  model  about  problems  of  child's  point-of-view  is  Piagetian  the  Inhelder attempts  child and adolescent differ  & to  or  Piaget, explain  qualitatively  5 from  each  knowledge of  other,  and  "how"  the  child  by Piaget as the process  Piaget does not go  beyond adolescent  different  from  thought  restricted  to  situations  (i.e., be  date been  Neither  adult the  teacher's Smock,  nor  model  discovery (Arlin,  as  1984;  1981).  in  Uses  formal  Piaget's  describe  of  his  their  of  mechanisms  "how" it is qualitatively are,  in  large part,  in "well-defined"  "reflexive  teacher  own  abstraction". However,  model  reasoning  concept  of  they  teacher have  Neimark,  1982;  and  1964;  1976)  Getzels  and thus  problem  reflection  been  1983;  Elkind,  conceptions  ill-defined  (1983,  model  of  to  of "reflexive  thinking to describe  1984).  use  concept  Dewey's  "reflection-in-action"  Cetzels,  construct  abstraction"  problem may,  by  not  to  thinking, but it has  used in such a fashion.  thinking  process  (Arlin,  own  a very useful  Piaget's  teacher's  adolescent  while acting on and interacting with the environment through  construction defined  analogy,  and  defined  which in  is  1981,  analogous of  1983).  to  adult  Schon's  framework  for examining perspective.  (Arlin,  Mackworth,  "how" the  teacher  to  needed  describe is  the  a  the  cognitive  processes  of  thought  in general  process  model  informed by a "developmental  &. Csiksentimihalyi, 1976;  teaching from a developmental  attempts  What is  descriptions  a problem finding  a method  reflection  situations.  Sinnott,  1987),  of  perspective"  1975-76,  1965)  1977,  of  (Arlin, 1986;  may provide this  "has the  concept"  of  6 B. STATEMENT OF THE PROBLEM  How do teachers identify the child's or adolescent's point-of-view when classroom problems which may be described as ill-defined problems?  faced with  And to what extent does the concept of "teacher as problem finder" describe those teachers who have the structures and strategies available to them for discovering "how" a particular student "has a concept" in the practical context?  In order to examine these questions,  teachers,  participating in a series of workshops  designed to introduce teachers to a developmental perspective,  were asked to select  and  respond to one  and  grade level most familiar to and/or taught by them. Each scenario, based on a  teacher's  own  problem  situation"  classroom scenario/simulation which represented the subject area  telling  of  for  a  real-life  teachers  classroom  in this  study,  problem, represented and  provided the  an  stimulus  question-asking task which accompanied each scenario. Four questions to  ask  teachers  about  the  nature  of  the  problems  might have with certain instructional tasks presented in  the  assist  present the  teachers problem  study  students  were with  also their  asked  to  problems  suggest and  to  and  "ill-defined for  the  were  designed  that  students  questions  them for solution. Teachers  possible  questions.  strategies Criteria  for  which rating  might how  identify and interpret the student's point-of-view when faced with ill-defined situations  development  was  developed  from  research  on  problem  finding,  cognitive  and reflective thinking.  There are three questions which arise from the problem statement.  These are:  1.  What kind of problem is a classroom problem?  2.  What kind of problem does a classroom problem pose for the teacher?  3.  What processes may be necessary for the teacher to begin to identify and interpret the student's point-of-view when faced with classroom problems which may be described as ill-defined problems?  7 These face  questions in  the  ill-defined  are the  act  of  problems,  result  of  teaching, and (3)  considering: (1)  (2)  the  those  the  classroom  teacher's  own  kinds  problems  of  problems  that  reflection-in-action  can  teachers  be  when  called  faced  with  classroom problems that are "ill-defined problems".  Each of these considerations the framework for the  will be  discussed  review of the  briefly in this Chapter and will form  literature related to the  problem addressed in  this study.  1. The Kinds of Problems Teachers  a. Kinds  of  Problems  Researchers (Dillon, 1982; 1965;  Wood,  problems  and  processes  of  problem what  is  Getzels, presented or  the view,  1983)  solution  1964;  or  been  presenter 1980).  problems  in  between  "discovered"  problem  along  What differentiates  the  Getzels  present  These a  situations"  and solver in the  1964;  "well-defined"  problems.  described  "discovered  problems (Getzels,  Getzels & Csiksentimihalyi, 1976;  differentiated  have  and  Yinger,  of  is  the  and  of  their  "presented  which  problem situation  well-defined  "presented"  problems  varies  (Dillon,  nature  of  in  1982;  (Mackworth, 1965)  & Csiksentimihalyi, 1976)  study  or  continuum each  Mackworth,  or  from ill-defined the  problem • of  itself, and the nature of the processes involved in identifying or formulating  problems of teachers  teaching.  have  known by the 1964;  Getzels,  "ill-defined"  situations"  discovered  teaching  Face in the Act of Teaching  instruction from the  need  to  "child's or adolescent's point-of-view".  employ a developmental  perspective  when  in this  they think about  8 This gives rise to  b.  the  The classroom  planning  difficulties where  1983; Brooks & Fusco,  lessons,  within the  children  or  "remarkably  different  observation  that  grasp  the  The  problem which present  is a classroom problem?"  lesson.  to  1983; Yinger, 1980)  anticipate  However, many  adolescents  respond  students'  to  teachers  often  experience  they  to  tasks  to  questions  in the  "unexpected",  in  that  ways  have e x p e c t e d " (Arlin,  frustration  "when  understand"  and  students  are  are  1983). The unable  is a classroom problem  to that  by teachers.  scenario  provide  are expected  result  curriculum  the teacher would  observe that teachers,  responses  lessons still  what  encountered  will  attempt  articulated  following  problem  from  concepts  is frequently  kind of  problem  Researchers (Arlin, when  question, "What  is  based  during the  basis  on  a  classroom for  teacher's  own  instruction.  It  simulations  of  telling is  one  classroom  of  an  of  several  problems  "unexpected"  used  situations in  the  study:  At the beginning of a new unit requiring an understanding of fractions, students in a grade 6 math class were asked the question, "What is a fraction?". They were asked this question after a review class on fractions. The assumption made by the teacher was that since most students are exposed to fractions as early as the first grade and build on this concept at each grade level, most students at the grade six level would already have a well-developed understanding of fractions and how to use them. These were some of the responses the teacher received from her students:  1. 2. 3. 4. 5. 6.  "The number on top is the number of pieces from the b o t t o m . " " A math problem that might be solved in steps." " A number over another number." "Tells if something is even or not". "Numbers that tell you different things like if a ruler looks just like a bunch of lines, its a bunch of fractions and you need to know them." "Equal parts."  9 In  this  instructional  problem  after  each student her  own  situation,  a review brought  frame  available  to  terms  their  of  questions the  for of  the  and the  level  presented  to  fractions.  only.  presented  this  (Arlin,  own  tended  experiences with  To  cognitive  her  students  were  presented  1986b;  a  developmental about  define  concept  They  were  (Piaget,  &  by  the  cognitive  mechanisms  question  of,  "What  1979),  Easley, 1978;  fractions. of  constrained  In  these  "fractions"  comprehending  were  situation,  processes she had  knowledge the  conceptual  problem  Driver  The students  and  with  of experiences and actions  fractions  constructing  on a literal  problem  fractions.  understanding  1987; Strauss, 1981),  most  on  lesson  Erickson,  examples,  students  her own repertoire  for  her  the  the  teacher's  by the nature  available  to  in  them  of for  organizing the information.  This  gives  rise  to  the  kind  of  problem  does  this  type  of  to  be  classroom situation pose for the teacher?"  2.  Classroom  in  the face  well-defined alternative  Problems as Ill-Defined Problems of  students'  curriculum conceptions,  "unexpected"  tasks,  the  and w h o  responses to  teacher does  who  not  go  is  what not  beyond  teachers aware the  of  students'  answers as " w r o n g " .  or  point-of-view.  Instead,  fractions  which  the  through  memorization  teacher's  definition  she  students (Brooks,  of fractions,  She does not attend  merely may  provides  simply  1984). often  layer  Although receiving  new onto the  the  students'  answers to explore the " w h y " and the " h o w " (Arlin, 1985) of their the  believe  "incorrect"  responses, sees  to the student's  information their  and  praise for their  may  framework  examples  alternative  students  students'  conceptions  memorize  "correct"  of  the  responses,  10 what  these  students  constructing  the  ready-made these  problems.  contrast,  herself  to  longer  able  their  actually  concept  solutions  "well-defined"  By  are  to  The  of  the  teacher  experience to  fractions  "masking"  (Arlin,  students'  problems  problems  students  use  who  questions  her  own  definition  students  and  the  definitions  within  1964;  problem  in  teachers,  the  a  their  1986b).  tend  to  have  finding nature  &.  the of  of  own  Teachers  seek with  (Schon, of  or  frameworks  for  who  for  immediate  look closure  for  tasks  are  instructional  actually which  the  in  1987).  the  these  from  1976), &  classroom  problem  is  like  and  situations  recognizes she her  further  conceptions  assist her  definition(s)  requires  (Getzels  to  She  received  Csiksentimihalyi, problem"  that  alternative  procedures  1983,  fractions,  she  students'  recognizes  methods  situation  Getzels  aware  and  predefined  and  (Cetzels,  is  surprise,  between  herself  is  for these teachers.  problems  sees  doing  with  from  The  and  that  her  teacher  elaboration "there  1976).  "unknown".  no  discrepancy  students.  realizes  is  students  expected  Csiksentimihalyi, is  she  the  discovery  allows  is  For  a  these  The  problem  although  there  is  "ill-defined".  No  present  teaching  that  that the  teacher extracts these c o m m o n If  perspective" 1976;  share in  exactly  features  situations.  they  situation  these (Arlin,  common  "teacher 1983;  a  (Schon, 1983).  1984;  It  one,  is through  features and maps them  reflections" . are  Brooks,  previous  Brooks,  "reflection-in-action"  (Schon,  1983,  1987),  onto  by  Fusco  Grennon,  Feldman, 1981; Sigel, 1984; Sinclair & Kamii, 1970), which  teacher's  reflection-in-action  informed &  then  the  are  a  "unexpected" "developmental 1983;  Elkind,  in turn informs  teacher  may  come  the to  11 know  the  logic  Knowledge  of  that  a child  this  logic  or  adolescent  becomes  the  brings  vehicle  to for  a  particular  the  curriculum  teacher's  task.  adaptation  of  instruction.  In  the  present  teachers  study,  believe  "ill-defined",  students'  to  and  be  "well-defined"  act as a stimulus  finding which lead toward the  This  gives  "unexpected"  rise  to  the  student's point-of-view  for  or  the  identification  question  of,  or  "unfamiliar"  "presented"  curriculum  discovery-oriented of the  "How  responses  to  what  tasks,  processes of  are  problem  "child's/adolescent's point-of-view".  do  teachers  identify  when faced with classroom problems which  and  interpret  the  may be described  as "ill-defined" problems?"  3. The Teacher's Reflection-in-action Like  traditional  problems 1933;  and  rational  Sprinthall  on-going toward  studies  &  process  the  own  something (Schon, "reflective  and  of  of  their  explanations  1983), and  Schon  as  a  something  (Lampert, 1980),  1987).  His  conversations  reflection-in-action  and  teaching of  "reflective  describes  on  in  to  1984; "thinking  notion with  ill-defined  of  thinking"  mechanism  unique or  1979)  and  about  doing  something  need  for  situations"  discovered  (i.e.,  of  (Schon, 1983). Although  Piaget,  the  "well-defined" Dewey,  "reflection-in-action"  developmental  new  based  Schon (1983, 1987) goes beyond descriptions of  thinking  1983,  logical  creativity  construction  (Yinger,  teachers  Sprinthall,  in these definitions, one's  of  may  problem  an  construction  similarities  exist  thinking  about  about  doing  thinking  while  practitioners  as  to  describe  the  situations.  It  doing  it"  engage  in  teacher's may  also  12 describe  the  through  processes of  of  development  However, not  is  may  like  Dewey,  account  qualitatively words,  which  are  "how"  describe  Schon, for  in general (Arlin,  other  concepts  of  the  the  teaching  student  teacher's  and  "has own  learning  which  a concept". construction  different there  not  development  describing  the  kind  from  child  his concept  of  1984; Neimark,  thought  1982; Sinnott,  and  his  for  theoretical  in  adolescent  Schon's  is  child's  problems".  characteristic  does  of  adult  1981, 1983); thought  which  (Arlin,  on the  Schon  the  reflection-in-action,  thought  description  framework.  of  that  may be developmental constraints  accounted  of  in  come  The process  of  when faced with classroom problems which are "ill-defined  adequately  thought  new  discovering  "reflection-in-action"  point-of-view  of  1986).  teacher's own  of  does  1984,  reflection not  thinking  and  suggest  In  in  his  how  the  \  reflective  practitioner  experience.  What  problem finding  4. The  acquires the theoretical  may  be  framework  Need for  missing to  in  framework  his  through  description  of  inform the teacher's reflections  which she filters  reflective  practice  available  to  (1975-76),  in  relationship finding the point  situation, for  her  between  ability.  view,  and  constructing  cognitive several  In view  classroom problem of  a  a Problem Finding Framework  the her  is  in the first place.  Like her students, each teacher brings to a problematic situation, her own understanding  her  of  the  cognitive  her  own  developmental cognitive  knowledge model  developmental  these findings,  as an "ill-defined"  may depend, in part,  developmental  on  the  extent  problem the  of  processes  about  the  problem  which  and thus  cognitive  the  for  she  has  situation.  finding,  process variables to  frame  Arlin  found  a  and  problem  teacher  identifies  identifies  the  student's  processes and structures  she  13 has  available to  her  "thinking-in-action"  constrained  by  available to  her for organizing the  A  finding  problem  reflections, (Schon,  may  1983,  learning,  The  the  framework,  1987). than  a  This  problem(s)  presented  own  coupled  way  of  teacher  with  reflects and  problems  SIGNIFICANCE  need  inform  the  for  on  the  teacher  her  own  conforming  to  and  that  these problems.  be  processes  who  predefined  who  the  teacher's  "reflects-in-action" of  teaching  problems,  in teaching  methods  manuals (Arlin,  1983, 1987). This framework  seeks  her  finder  own  and  who  solutions  may  identifies to  the  STUDY  often they  perspective"  are  and  is  a  to  However, upon  and  understand  development  strategies  return to  their  to  framework  to  the  of  study  of teacher education  teachers with opportunities  techniques  finding  relevant  frustration when  expected  professional  problem  particularly  development  experience  attending  provide  suggest  informs  teacher as problem  "reflection-in-action"  teachers  may  instruction.  teachers  or in-service settings problems,  and  teaching, and to the  concepts by  methods  THE  teacher's  cognitive  theories  lessons as outlined  study the  a "developmental  Observations that  articulated  of  OF  teachers and their  understand  need to  and  She, too,  and the  experience, which  describing  and religiously following  problems  1987).  information.  appreciating  to satisfy the  "ill-defined"  C.  of  (Schon, 1983,  Dillon, 1982; Getzels & Dillon, 1973; Schon,  also help her  nature  provide  rather  and solutions 1983;  the  for  that  to may  students is  programs.  are  unable  a common  programs.  Most  discuss their assist the  to  problem workshop  instructional  teachers  with  classrooms, many teachers experience  14 difficulty translating their newly acquired teaching strategies into practice.  If researchers and  seeks  her  understand the  begin to  "how"  teacher's  thinking,  own  study the solutions  may  within  the  and "why" teachers  point-of-view  they  teacher as one  into  begin  to  practical  when  own  problem(s),  then  the  they  problems  may  curriculum to the needs of the teachers. where  the  learner/teacher  is  rather  context,  they  may  If they how  questions  begin  begin to to  teach  teachers  If they ask questions  begin  on  her own problems  teachers and  to  In this view,  than  do.  teaching  when faced with "unexpected" classroom situations. teacher's  identifies  perform as they  account identify  who  where  match  to take for have  about the  teacher  education  teacher education will begin the  researcher  thinks  the  learner/teacher "ought to be". This study may provide one new means for assessing the  teacher's  point-of-view  as well  as  the  teacher's  ability to  assess the  student's  point-of-view.  D. DEFINITION OF THE TERMS This is an exploratory study adolescent's  point-of-view  of  when  the  ways in which teachers  faced  with  important for this study. They are defined  ill-defined  identify  problems.  the  Several  child's or terms  are  below:  1.  Problem Formulation refers to the extent to which the teacher identifies classroom problems as "ill-defined or "discovered" problems (Cetzels, 1964).  2.  Integrative Complexity refers to the extent to which the teacher formulates alternative explanations about how the student is thinking about a particular instructional task when faced with classroom problems which may be described as ill-defined problems (Schroeder, Driver & Steufert, 1967).  3.  Quality of Point-of-View refers to the extent to which the teacher formulates "general questions" (Mackworth, 1965) based on her hypotheses about how the student is thinking. This represents the extent to which the teacher  15 identifies the student's point-of-view when faced with ciassroom problems which may be described as ill-defined problems (Arlin, 1975-76, 1977, 1986). 4.  In  Developmental Teaching Strategies refer to the number of different strategies, which reflect a 'developmental perspective', that the teacher employs when she thinks about her teaching in ill-defined problem situations (Arlin, 1983, 1986).  the  present  adolescent's  study,  point-of-view  described  as  processes  which  represented Teaching  the  ill-defined  by  extent when  to  which  faced  problems  contribute  to  is  with  the  teacher's  represent  the  number  teacher  classroom  represented  "problem formulation" and  Strategies"  the  by  problems  "quality of  ability to  ask  "integrative of  identifies  child's  or  which  may  be  point-of-view".  "general  complexity".  different  the  strategies  The  questions"  are  "Developmental that  the  teacher  uses to assist students with the problems they identify the student as having in the first  place.  Together,  examining  the  describes  those  extent  these to  teachers  cognitive  which who  the  have  process concept  the  variables of  the  may  provide  "teacher  as  structures  and strategies  question,  "Why do  a  way  for  problem finder" available to  them  for discovering the student's point-of-view.  In  addition,  used to  teachers'  responses  provide an insight  to  the  into the  teacher's  you  teach?",  will  be  concern for problem finding. And, a  variety of demographic variables of interest to the present study will be considered. These  include: years of teaching  other  positions  adolescents,  or  reasons  experiences for  experience, requiring  participating  in  the  subject(s) taught, direct  grade  intervention  inservice  course,  Piaget's work and its implications for classroom practice (see  with and  level(s) taught, children  or  familiarity with  Appendix B).  16 £  SUMMARY OF THE PROBLEM  This study is an attempt to explore the ways in which teachers identify the child's or adolescent's  point-of-view when faced with students' "incorrect" or  responses to what the teacher's  reflection-in-action  developmental  and  insight into the contributes  teachers believe to  problem  is  thought  finding  be well-defined instructional tasks. The  to  be  framework.  extent to which the  "unexpected"  concept  be  These of the  informed frameworks  by may  a  cognitive  provide  "teacher as problem  to the identification of teachers who are able to discover the  an  finder"  student's  point-of-view in the practical instructional context.  Chapter 2 contains a review of the literature pertaining to these research  questions.  CHAPTER II.  The present study was designed to adolescent's  point-of-view  instructional finder"  when  explore, (a) how teachers identify the faced  tasks, and (b) the extent  describes those  LITERATURE REVIEW  teachers w h o  with  have the  point-of-view.  The  heavily from two  view  and  traditions  draws  Piaget's Constructivist by redefining  classroom finding  study  problem  are  traditions.  as  ill-defined  introduced  which  Finally, within  this  in an attempt to  view.  It  in  themselves  describe the type of  the  research  tend  to  Dewey's  teacher  combination  studies  of  these  understand a  related  the  to  Experimentalist  problems. To  ignore  to  strategies available  a unique  as ill-defined  responses  or  "teacher as problem  and  related traditions:  problem,  framework,  structures  represents  classroom problems an  "incorrect"  to which the concept of  them for discovering this  present  students'  child's  to  problem  Deweyian-Piagetian  as problem  finder  is discussed  "reflection-in-action" which is conceived of  in  this study as the basis for teaching from a developmental perspective.  A.  IMPLICATIONS  THEORIES OF  Researchers process  EDUCATIONAL  PRACTICE  long  (Dewey,  considered  1933; Hullfish idea that there  &  the  child  DEWEYIAN AND  PIAGETIAN  influence  the  to  are qualitative  competence  be  the  centre  of  the  Smith, 1961; Piaget, 1972, 1978, 1979)  adults think. M u c h research has focused on thinking  OF  KNOWLEDGE  have  emphasized the  FOR  of  differences between the ways in which  teachers  17  as they  interact  how  education and  have  children  these  differences  with  their  and in  students  18  daily in the  classroom. The processes of  problems  instruction  of  in ways which  thought  meet  based on theories  of growth  and  1978, 1979). These two  Piaget (1972,  relate to the  maintained that the  believed that the the  this  subject  the  need for  how  to  needs of  or reflection traditions  the  identify  students  espoused by  will  now  be  and solve  are  primarily  Dewey  (1933)  discussed as they  research problem.  Dewey (1933)  that  the  in knowledge  1. Dewey's Experimentalist View of  He  teachers use to  problem  the  as  essential to  the  intelligent  the  child reasons and learns differently  teacher  (Dewey,  had the  general habit teacher's  reactions  of  was  1933, p.  teachers who  acquire the  his/her  275).  the In  reflecting" to  teach  students,  be acquainted with the  (Dewey,  1933).  This,  "openmindedness",  and  said,  and  of  doing  with  principles  required  "wholeheartedness"  pupils  are  Dewey (1933) to  the  displayed  to  observe  "mental and  each subject of  emphasized  think well, especially  35). He saw this  approach  adult.  matter,  with. intent, to  than the  in the subject  "learn how (p.  and to  he  minds  this view,  capacity to  of  ability  "what  and knowledge, 1904,  Knowledge  problem for the student was found  for  matter"  Growth in  attitudes by  the  of  interpret  with  educational  habit"  interest  psychology  "responsibility", teacher  in  the  classroom.  In order to  provide  a way to describe " h o w "  and skills associated with "reflective process of reflective with  problem  thinking  a judgment.  solving which  was initiated The  practice", Dewey (1933) he called "reflective  after  individual's  teachers attain  the  recognition  awareness of  of  the attitudes, identified  thinking". a "true"  a problematic  behaviors  an hierarchical  He maintained problem  situation  and was  that  ended referred  19 to  as  the  "prereflective  problematic this  stage"  and therefore  process  of  inquiry  where  initiated or  the  reflective  was informed  by experience, where  those  which  have  which  come  after"  in a judgment  gone  before  (Dewey,  where the  temporarily  closed  the  process  inquiry  which  of  He  led  toward  of  reflective  of  critical  indicated  five  phases  modifies  the  situation  inquiry".  guided  by  in  some  way  a solution  that  truth  to  knowledge  "critical  the  because the  the  something  from  thinking  (Dewey,  method  to  explain  1933;  the  of  those  inquiry  ended  itself  Testing.  Intellectualization,  "Suggestion" was the  solutions. problem. which  (2)  "Intellectualization", The  acted  immediate for  (4)  Kitchener,  processes of  Dewey,  was  development  as  the  collection  a  guide  to  possible consequences of  of  better  and  more  were  by  action  (direct  action  complete  observation  of a second, better  Reasoning,  identification  the  development  the formulation  Hypothesis,  "Hypothesis" was  occurred about  tested  (3)  of  and  and  clarification  of  a supposition  of  observations.  on  the  implied  the  thought to them: (5)  of  that (1)  Hypothesis of  possible  the  specific  suppositions  Then  "Reasoning"  hypotheses and the  experimentation)  he  or  explanations. And finally, or  the  1983),  weighing  the  of  reality.  individuals use when faced with problems that are real and meaningful Suggestion,  or at least  outcome  beliefs, assumptions, and hypotheses against  inquiry  and  quality  an  that  inquiry"  problem  was  genuinely  Dewey stated  For Dewey, this process of  argued  In  "true  was  individual identified  issue.  made  each new experience "takes up  1933, p. 27).  of  analysis  process of thinking  and  continuous verification  Dewey's  "uncertainty"  which  the  further  hypotheses  might  lead  hypothesis (Dewey, 1933; Kitchener, 1983).  to  20 Dewey's  theory  of  knowledge  suggests that teachers, when  or  ideas about  a solution  verify  refute  of  fashion,  they  judgment  some  must  or  their  develop  an  decision which  emphasis  inquiry  and as  beginning  of  the  into  behavior  for  "competent" and  These  &  two  the  Sprinthall,  central  Dewey's  (1933)  incorporate 1976-77;  aim  of  Feiman,  to  teaching  collect  based  on  the  practice,  classroom setting,  must  evidence which  expectations. Then, in  gave  to  engage  led  the  an  at  hand  may  orderly  hypothesis,  question  a  "reflective  attitudes for  predicting rise  to  1983b)  to the  in  the  researchers  make or  a  which  the  processes of  to  determine  teacher".  of  the  gathering  teaching  studies  (Glassberg of  application  teachers of  they  marked  the  and  social  about  human  behavior  approach &  critical  what  behavioral  "inquiry-oriented"  approach  This  information  "effective"  teaching practice, will be briefly  (Zeichner,  Sprinthall,  and  Dewey's  and  their  1980;  teaching.  theoretical  and  reviewed.  Approach the  of  of  developmental  theoretical  "habits  the  representing  The Inquiry-Oriented  The  of  observation  1983a,  approaches,  philosophical views to  a.  eventually  purpose  cognitive  and  in the  and  ideas and  ability  qualities  of  teachers. This  applied  (Kitchener, 1983).  incorporating  methods  problem  hypothesis,  teacher's  behavioral  the  the  provides an answer to  thought  research  sciences  Sprinthall  the  reflective  perceived  1983)  on  to  original  remains open to further inquiry  Dewey's  when  faced with "uncertainty"  formulate or  reflection,  inquiry-oriented views  inquiry"  of  into  approach  knowledge teaching  1979; G o o d m a n , 1984; Joyce,  to  teaching,  based  and  "reflective  practice",  practice  (see  1972; Oja &  Bagenstos,  directly was  1975;  on to  Elliot,  Ham, 1984; Salzillo  &  21 Van  Fleet,  "active  1977;  agents"  actions on  Stratemeyer,  who  1956;  exhibited  the  Zeichner,  skills  and  desire  engage  to  develop a "disposition toward  critical inquiry"  b.  The Cognitive  Approach  in  some  researchers  of  perspective  began to  researchers designed  domains  programs  Strivers,  of  in  (i.e.,  1980;  Hunt,  1980)  of  more  problems to  1977),  the (b)  of  cognitive  needs of adults  broadly, others"  continue  teacher  education  programs  can  effects  based  possess and  (Sprinthall to  be  need  Sprinthall,  1980),  1983b;  a  can  and  &  stage  teaching.  (a)  higher  more  continual  1983a,  that,  and  Perry,  theory,  repertoire  respond  some  1966;  process of  rationale  wider  a  based on a variety  Sprinthall,  domains  for  Thies-Sprinthall,  Loevinger,  developmental  cognitive  to  ability  (1933),  teachers and their  the  their  attain the  expressed  as an active  on  of  as  what " o u g h t "  Dewey  framework  Sprinthall  higher  complexly,  more  &  of  defined  1983).  1983a,  studies  education  1980;  by  Kohlberg, 1969;  their  education  are associated with  about  the  Sprinthall,  1974;  Sprinthall,  to  developmental  into  definition  (Zeichner,  (Glassberg  of  these 1983b)  teachers, stages  of  behavioral  accurately  and  & Sprinthall, 1983b, p. 21; Walters  develop,  solving, generalization and concept formation (c)  analyze the  as defined  response  a cognitive  teacher  "function  perceive  empathetically  &  activity"  education  assumptions  (Glassberg  behaviors  and  Sprinthall,  incorporate  the  development, skills,  &  Dewey's (1933)  and  "reflective  on  Piaget, 1972; Selman,  growth,  whose  of  Dewey,  (Glassberg  In view of  &  of  developmental  1970;  form  Developmental  tradition  developmental  1984)  to  were  task for the teacher was to  to  the  Teachers  children, schools and society based on questions  be done (Zeichner, 1983). The primary  In  1981).  of  problem  (Sprinthall & Sprinthall, 1983b),  therefore  designed  particularly  to  in  promote  areas  teachers'  cognitive  22 thought  processes  thought  (Sprinthall  "we  do  not  toward &  see  development  higher,  Sprinthall,  more  abstract  1983a, 1983b).  it  as  a  fallacy  to  is, then  we  may also know what  31).  What  with the inquiry-oriented  to  is problematic  the  study  appreciating  of  teachers  and and  growth  and  achievement.  defines  the  problem  "how" the  on  based  the  on  They  pre-defined  definitions  inter-individual which  to  and  their to"  go  its  are  beyond  "knowing  differences  develop  and  of  and  they  the  is  the  of  on  what  the  teacher  identified on  and  by  student  who  context.  identify  &  approaches  teacher,  instruction  that,  (Sprinthall  behaviors  hypotheses,  both  Instead, interpret  and learning  in  and skills deemed necessary for  researchers' child,  of  teaching  neglect  to  adolescent  there  are  differences  affect  the  ways  knowledge  be"  practical  techniques  in  to  problems  stress on  flexible  (tentatively)  stress  observed  solving problems  of  know  developmental  the  within and  differences  that"  we  important  place the  theories  indicative  own  that  teacher's  Methods  "how"  is  behaviors, attitudes  intra-individual  if  and  researchers state  education ought  solution  about  the  think.  their  not  own  go  and adults construct  do  the  complex  sum, these  that  teaching  of  of  and cognitive  definitions  seeks  classroom by identifying children  to  effects  and  teachers "ought  teaching  their  conforming  researchers,  researchers,  and  In  suggest  Sprinthall, 1983b, p.  levels  in  to  which  and  explain  based  on  consider  the  adult "how"  thought these  children, adolescents,  and reality (Phillips, 1980).  23 2.  Piaget's Constructivist View of  The dominant from  the  Fusco  Reid,  The  the  child  interacting  the and  which  section  is the  Piagetian or  child  teachers examines  attempts  and  for  construct  the  deal  with  some  of  Toward  Like as  a Constructivist  Dewey important  adolescent upon  A  the  world.  concept  View  build  one and  and  of  in  (Furth, the  of  instruction  1984;  (see  Brooks,  Furth, 1981;  the from  knowledge  of  for  their  Piagetian  acting  studying  and  on  development  a framework  and  each other,  while  view of  in  thought  and  provides the  ways  classroom. This  next  theory  to  educational  apply to the research problem.  Educational  of  Practice  thinking.  out  of  of  role  Piaget  of  they  experience and  stated  experience  which  1981;  Gallagher,  will  describes the both  cognitive or  invent  child's  Dewey  1977;  that  and  actions  of  development. their  own  Piaget  focus  on  It  concepts is that  Reid,  1981)  is a means  understandings which on  or  objects,  education &  action  child  build  Gallagher  reflection-in-action  and  the  ultimately  Piaget's epistemological view  mechanisms of  Although  model  "how"  instruction  adolescents construct  Piaget, this  development.  of  a repertoire  concept  abstraction"  For  on  up  Brooks,  qualitatively  own  contributions  development  key  children  their  development problems  can reflect  represents  by which the  to  differ  Experimentalists, Piaget saw the  the  they  1978).  "reflexive  which  to  needs  which  (Piaget. of  and the  explain  Piaget's constructivist  practice and how these contributions  a.  1983;  Constructivist  to  adolescent  environment.  point  (Arlin,  problems of  & Piaget, 1958; McNally, 1977; Piaget, 1972, 1978,  model  adolesent  with the  a useful starting in  point-of-view  1981; Inhelder  of  h o w teachers think about  adolescent's  Piagetian  understanding how  or  Grennon, 1983)  Gallagher & 1979).  model in studies of  child's  &  Growth in Knowledge  of  leads  to  processes  of  24 problem the  solving  in  their  "mechanism"  of  theories  growth  of  which  reflection  from Piaget's constructivist  In  to  order  explain  action that children actions  result  knowledge  in  this  Logico-mathematical  about  one's  knowledge  also  The  includes  abstraction". individual's individuals  by  which  actions  on  those  in  the  knowledge given"  the  based  on  1981,  it p.  from  (Bruner,  29).  the In  start  other  identified  theory  of  types  knowledge.  properties  of  derived  events.  This  of  Physical  the  external  from  thinking  second  constructing  framework  two  and understanding. These  knowledge  and  and  into which  to  1973),  of  words,  concrete, By  abstract  go  beyond  where  the  reflexive  and  physical  knowledge  then  type  of  transforming  physical  the  is  knowledge  called  observables  contrast,  the  structures.  For  the  observables  or  infinitely  developmental  constructs  Contrary  to  level  an  it..."  by  the  which  Piaget, further "beyond  her  Dewey's  'replace' empirical beyond  on  abstraction". This  mental  individual  "empirical  and  process  is called "reflexive  does not  goes  knowledge physical  abstraction.  stage progression, "[r]eflexive abstraction frames  physical  organizing,  observables.  presence of  means  mechanism  the  acquire  logico-mathematical  growth  through  individuals is  is based on  on  descriptive  Piaget  knowledge  objects for  emphasis  given meaning.  process  process  information  with  Piaget's  logico-mathematical  of  becomes the  This  acquire  own  and  is  reflection.  represents  structures  and therefore  process  their  knowledge  experiences  of  it  Dewey's  construction,  knowledge  empirical data, and therefore is integrated  of  knowledge  one's  world.  own  theory  construct  physical  growth,  differentiates  mechanism  use to  represents  knowledge  own  "rules"  hierarchical  abstraction,  (Gallagher  individual  the  has  &  but Reid,  attained  25 only  puts  1984;  constraints  Biggs &  on  Collis,  her maximum  1980).  It  is this  reflexive abstraction that is the to  performance within particular "interplay"  between  basis for the following  situations  (Arlin,  empirical abstraction  implications of  and  Piaget's theory  educational practice:  Reflexive abstraction  has two  from  to  a lower  between  level  and  characteristics. First, it  a higher  level  among experiences of  this  "mirror  reflection"  (the  actions  level)  The individual  represents  to  thinking  the  the  physical  individual  a movement  about  is now able to  in  doing  is characterized by sense  from  from  that  which  infancy  being  thing  projection  provides  onward.  able  (the  a  to  For Piaget,  do  something  representational  reflect on previous actions and to  links  level).  build concepts  out  reorganization  or  of these reflections.  Secondly,  reflexive  reconstruction reflecting  of  onto  abstraction  knowledge  a higher  is  at  level,  characterized  a higher but  level.  by  a  Here,  is reconstructing  mental  the  on  individual  that  higher  is  not  level,  merely  which  is  enriched with new elements, what already existed on the lower level.  Unlike Dewey,  who  five  identified  thought, where  Piaget  which level  another leads of  more  solving  theory  at  all  to  the  thinking.  and  is  problem in  would,...,be  levels  projection  of  interested  aspects of thinking"  knowledge  then  uni-lateral  was  "[pjroblem  the operative in  a  the  based  development  regarded  as  of  fundamentally  on  cognitive the  constant  reorganization, of  the  Piaget's constructivist  spiral and  mental view  of  of so  projection forth"  structures child  and  and  (Gallagher, which  steps  of  "[g]rowth  reorganization, 1977,  characterize  adolescent  of  structures  operations  (McNally, 1977, p. 85). For Piaget then,  the  development  solving  p.  9)  one's  development  26 led to  the  expressed need on the  1981)  for  based  upon  teachers  "...all  education,  where  child  development"  (McNally,  reflect  on  their  own  of  Piaget to  matter  of  education may take their  attempt  teaching-no  a theory  who  However,  all  part of some researchers (Elkind, 1976; Feldman,  did  use  his  student's  not  theories  point-of-view  address  concept  of  the  of  what  kind-[to  p.  84).  In  1977,  teaching,  learning,  of  adult  abstraction  to  this  be] view,  instruction  into consideration while  problem  reflexive  or  cognition,  they teach. nor  discuss " h o w "  and  did  he  professionals  translate theory into action.  b.  The Teacher  Elkind  (1976)  perspective  Elkind  (1981)  education, however,  plays in  (1976)  saw  speak  neither  description  of  classroom practice  "effectiveness"  point-of-view", the  made the  child's  or  that  is,  in the  of  the  need  for  a  provides an operational the  important  role that  makes a substantial  students  A  come  further to  primary  look  as the  students'  own  contribution  reveal their  teaching see  prescription  in assisting teachers to  perspective".  in  to  adolescent's  suggestion that the  education is not rather  Psychologist developmental  definition a theory  conceptual  of  this  of  child  contribution  such a definition.  child's from  Feldman  Elkind's (1976)  development toward  and  on  perspective.  as Developmental  difficulties  perspective. contribution  of  methods  teacher's  Unlike of  with the  thinking  in to  the  take  instructional  of  the logic of  tasks  of  but  a "developmental  techniques  students:  he  research to  and specific learning principles,  provision  "the  Experimentalists,  teachers, and teachers, on  these revelations, adapt the logic of tasks to  to  child development  at classroom tasks in terms is  ability  the  whereby basis of  27 of the most important contributions child development can make is not so much particular contents and principles of learning, as a general orientation towards children. What the developmentalist has to offer the teacher is first and foremost a developmental perspective and techniques for exploring the child's own view of reality. (Elkind, 1976, p. 53)  From this view, teachers, informed about  the  child's  instruction  and  or  may  by a theory  adolescent's  begin  to  take  of  thinking  this  child development  while  thinking  into  planning  begin to  and  account  think  implementing  as they  teach  (Arlin,  1983,  1985; Brooks, 1984; Brooks, Fusco & Crennon, 1983). They may ask questions  about  that thinking  Fusco,  1983;  students they  Yinger,  often  do  in relation 1980).  display when  not  understand  be  studied  participant practical own  and  who  theories  In  of  plans and materials  They  may  also question  presented 1983;  constructivist  manipulated  defines  setting.  their  (Arlin,  1984; Strauss, 1981). In the to  to  so  and  by  teaching, and  the  new  Case,  1978,  faced with  or  1980;  Brooks &  with  Elkind,  difficulties  concepts 1976;  her  own is  (Arlin,  but  rather  problems  actively  and  engaged  1983,  becomes  1985;  "object"  an  questions in  which  Grennon,  view, the teacher is no longer an  teacher  learning  when  concepts  researchers,  interprets  doing,  with  it  (Brooks, 1984;  active in  the  constructing  her  Lampert,  1984;  Schon,  1983, 1987). They become researchers in the classroom.  c. Teachers The  as  construct  Researchers of  "teacher-as-researcher", while  not  a new  construct  (see  Hullfish  Smith, 1961; Sprinthall & Sprinthall, 1983a, 1983b; Zeichner, 1983), has recently redefined  (Lampert,  educational and  in  light  1984)  researchers of  this  within  together recognition  to to  a  project  recognize explore  designed the  intuitive  together  how  to  bring  knowledge the  teachers of  appreciation  &  been and  children, of  this  28 intuitive knowledge  Lampert  (1984)  respond  to  could be useful in educational practice (Lampert,  presents  questions  include "on-the-spot This provides begin  her  thinking"  the  raised  opportunity  how  this  affects  thinking  and her role in their  she says  is problematic  teaching  in  closer  conforming  problems  the  She  to  to  ability  interpretations on  [the  learning" (Lampert,  of the  tradition  the  work  in  of  of  of  what  of teaching, learning, instruction  learning, she  inquiry and  constructs  opportunities instruction  form  the  framework own  own  the  interpret  teacher's  to  reveal  needs to  reflection  in the  place for "think  of  and  work  to  something.  the  about  responses  teacher]  teacher one's  to  own  the  questions  understands  children's  1984, p. 1).  that  "the  project  researchers  trying  sought  to  to  consider  it  bring  understand  stress is on the teacher  researchers  how  appreciating  to  be  important  and  experimentalist  and education.  from  construction  knowledge  teachers  and education  the teacher's own  a  child's  her  for  might  teacher and her teaching is that  1984, p. 2). Again the  definitions  to  and  "how  The theories, models and research findings from fields  teachers  redefined  teacher's  depends  Experimentalist  children learn" (Lampert, and  the  with this conception  the  practice  classroom  students.  She speaks of  ask, which  continues  by  which  for suggesting an appropriate  children  What  in  clinical research" into the way a learner thinks about  lessons. and  ways  1984).  of their  the  which of  the  teacher  knowledge  teaching. own  constructive  A  thinking  can  within  framework about  be developed in order for  view  the  classroom  as  which  provides  teaching,  learning,  researchers to  face of classroom problems.  teaching,  explore  29 B. A  FRAMEWORK  FOR  EXPLORING  THE  TEACHERS'  OWN  CONSTRUCTION  OF  KNOWLEDGE  1. The  Kinds of Problems Teachers Face in the  Researchers (Cetzels,  have  1964;  "well-defined"  method been  a  for  their  and  by the  1982; Getzels,  Classroom  problems"  which  is  By  a  contrast,  situation  there  is no  "discovered"  the  the  immediate  classroom 1973;  solution  known are  situations  problems"  Csiksentimihalyi, 1965).  an  known  or  herself  each  of  which  problem faced  has  with  in  of  what  classroom situation  a  types  a continuum  varies  is  identifiable  solution. These problem  described along  or  individual  "ill-defined"  finds  and  "presented"  where  some or  The  1976)  is  (Dillon,  1964; Yinger 1980).  have traditionally  are  expected  practice  Schbn,  through  individual  or  "discovered  situation  "discovered"  where  problem  a  and  presenter/teacher and solver/students in the  problems  and  as  solvable  processes of  &  (Mackworth,  problem  been  problems where teachers are given the problems  Cetzels  described  as  which  1973;  been  corresponding  "presented"  Dillon,  "presented"  has  procedure.  described  known  &  between  "ill-defined  problem"  with or  problem and  Getzels and  "well-defined presented  differentiated  Act of Teaching  (Brooks,  1983,  treated  as  further  information  1987).  "presented  gather information  from  the  methods  to  incorporate  1984;  Dillon,  In  this  problems"  about  treated  view,  where  problematic  the student's  as "presented" or  and formulas  these  1982;  predefined  Getzels,  situation  point-of-view.  and  solving classroom  strategies  1964;  "discovery-oriented  teachers  for  "well-defined"  Getzels.  problems"  researchers  neglect  (Getzels, 1964). They  into &  their Dillon,  are  often  to  gather  neglect  to  30  2. Classroom Problems as Ill-defined Problems Many are  classroom problems  problems  their  initial  where  which  teachers  expectations  teacher's  are,  of  a wealth  of  about  particular  concept  a  situations, required,  teachers and  unidentified  that  response,  for  that  researchers providing  even  after  opportunities  and  a  task.  In  these  identify  the  explore for  the  to  child  described  as  or or  the  some  reveal  situations  "incorrect"  response  and  is  1976).  problem  still  This  for  In  are  largely provides  arriving  of  thinking  thinking  elaboration  problem  construction own  is  "ill-defined"  methods  their  from  They are  adolescent's point-of-view. own  They  way  adolescent  discovery  student's  teacher's  teachers  that  or  Csiksentimihalyi,  or  in  (Schon, 1987).  attained,  the  child's  as deviating  further  is  &  identify  " d i s c o v e r e d " problems.  "discovered"  that  solution  or  "unexpected"  "how"  Cetzels  to  learning  student's about  a  interpret  and  recognize  1964;  to  "ill-defined"  this  at  view,  knowledge about  a  by  teaching  and education.  classroom scenarios, as examples, can show how  tellings of the  to  able  "child's/adolescent's be  or  teachers  learning, instruction  Two  that  need  is, to  are  identify  information  (Cetzels,  opportunities  fact,  teaching  recognize  contains  in  point-of-view" ill-defined  when  faced  problems.  These  actual classroom problems, provide  with  teachers begin to  classroom  scenarios, the  the  which  may  teachers'  own  problems  based  framework  identify  for  on the  development  of  classroom simulations used in the present study (see Appendix A).  Scenario #1: Researchers hypotheses  have into  expressed the  way  a  the child  need is  for  teachers  thinking  about  to  formulate  something  "on-the-spot"  (Karmiloff-Smith  &  31 Inheider,  1974;  hypotheses as 1983,  1987).  Lampert, part of  This  1984),  the  gives  and  for  exploration that  rise  to  practitioners comes with  a strategy  of  in  general,  to  generate  "reflection-in-action"  "hypothesis  generation"  (see  (Schon, Arlin,  1986b).  Two 1982)  third grade teachers  discuss their experience  1986). They most  (H. Jelke & C. Reynolds, personal communication, October  begin  frequently  teacher's  with a statewide arithmetic pretest (see  their analysis  missed  by taking note  by their students.  Figure  of  the  Arlin & Levitt,  test questions  1 shows a sample  which were  item with  notations:  Figure 1: Teacher Notes for Analysis of Pretest Item  61.  Which f r a c t i o n shows the p r o b a b i l i t y a block w i t h t r i a n g l e s on i t :  a) 1/2  ////  b) 2/4  MI  C  ) 3/4  d)  4/4  n*t»n\  of p i c k i n g  the  32 The following While  "b"  teachers  information  or  2/4's  was discussed:  was  the  observed that the  correct  most  In view of this discovery, they chosen as the there  were  Figure  1).  popular  or  of  per block  From  she  this,  hypothesis  the  teachers  of  3/4's  "hypothesis that  which  then  for  an  were  herself  by  hypothesis was verified the  they  ask  Kamii's  the  In  precisely what  teachers think  the  in  "wrong wrong  was " c " or  4 blocks in  question,  "Was  3 to  4, rather  than  answers"  the  teacher.  students  to  tell  The  students  so  that of  course  were the  or refuted  the  3/4's.  With  them  that  the  a whole,  hypothesis  they with  triangles  answer  (see  2  educated guess  this  how  three  to  of  that  picture  possible  a part  them  provided  there  it  is a type  by  teachers' hypotheses, the  the  (1970)  themselves".  teachers,  these  made  in  their  a  type  of  on  a block  was  3/4's.  Had  process w o u l d  have  continued  or until  no further  information  hypothesis  generating  the  could  be  students.  teachers, through and  their  blocks,  until  gathered from  point-of-view,  children  the  generated  explaining  not  Sinclair  a part,  answer.  four  confirmed  is  asked  confirmation" there  the  adult's  the  and that there were  asked  students  These  answer for  4 blocks had triangles?" This second question  choice  and  the  "right" answer?" O n e teacher looked at the 4 blocks and noted  3 triangles  informal  mind,  from  asked themselves the question, " H o w could 3/4's be  children simply related a part to out  answer  use of comment these  the that,  situations,  that question  generating  their  and  themselves  asked  "children it  is  is (Arlin,  informal the  answer, in fact, a right answer?". With  correctly up  to  question,  answer  In  the  what  this question, the  upon  questions  to  discover  present scenario,  considered "For  drew  the  researchers  1983).  hypotheses,  strategy,  the  children's  question  is  this  teachers began  the  33 process tasks  of  identifying  and  materials  clinical method  Scenario When  the from  problems the  the  students  "child's  were  point-of-view"  became the instructional  having (Arlin,  with  the  1983).  curriculum  Here,  Piaget's  method.  #2:  teachers  introduce  that are asked to  concepts  to  students,  elicit the student's framework  know  about...?" or  "What  been  discussed as  a way  points-of-view  new  for  is...?" for  This  strategy,  teachers  certain concepts before  to  the  most  common  or point-of-view which  identify  attempting  will  questions  are: "What  do  you has  be  called  "framing",  childrens'  naive  frameworks  to  teach these  or  concepts (Arlin,  1986b; Erickson, 1987; Strauss, 1981).  A  seventh  grade  teaching strategy  teacher  (see  by framing  Arlin  &  a particular  Levitt,  1986)  question  on  modified the  this  basis of  the  developmental science unit  she was beginning on the solar system. She asked her students,  Why  do you think we always see the  same side of the m o o n from  the  earth?  These are some of the responses she received from 1.  "Because when we rotate  2.  "Because we that's  move  at  her students:  we are always in the same position every the  same  time  and  speed  as  the  moon  why".  3.  "Because the m o o n is moving with us".  4.  " O n e side is facing space and we get stuck with just one side".  night". does  and  34 The  teacher  was  amazed  expressed  among  she found  that  their  frameworks.  basic  to  find  42 students. half  of  that  there  were  14  U p o n further examination  them  could  be  Nearly  80% of  put  into  the  one  students'  different  of  points-of-view  her students' responses,  general  category  responses could  to be  describe placed  in  one of five categories.  This  teacher  students'  (N.  points-of-view  the  same relative  her  conception  youngsters  of  the  believe  during  the  "how"  and  "why"  earth-moon  day  motion" phases  most  "rotation"  in  strategies  which  made  classroom  situation,  hypotheses  ongoing which  in  up  the  the  identifies  problems,  researchers  need  ask  students  given concept?  in  order  the  new  to  concepts of  teacher  "theories  put in  faced with  as,  side  of  students'  the  examples  "tilt",  To  in  these  (out  of  knowledge  of  about  and  the  experiences "relative  seasons, eclipses and symbolic 1983,  the  In  situation, which the  face  questions  about  form"  1987).  determine  "What  is  that  earth  questions  in  her  describe  "orientation",  problem  responses  to  conceptions"  asking  question,  moon  think  teacher's  (Schon,  point-of-view  the  of  This  an ill-defined by  the  "I  "explicit,  points-of-view.  consider elicit  into  interpreted  She goes on  day, night,  action"  problems  child's  that  "alternate  about  the  to  have  talk  students'  to  moon)".  order  to  believe  other  types  students'  the  the  the  the  her  teacher  their  to  1984)  framework  influenced  teacher,  from  goes  May  object)".  popular  students  relation  Here  about  responses the  directly  youngsters  (a fixed  most  her  selected  moon.  time  during  of  communication, these  moon  (even  the  her  of  all the  the  relationship,  and  think  second  that  subsequently  personal  as, "I  position  sight)  she  Darling,  how  the this  verified elicited  extent  of do they  to  ill-defined teachers have  a  35 In  both  of  curriculum student  these  task  available of  to  these  terms  the  concepts  for  their  questions  could not  students  on  own  tended  level  and  the  define  only.  the  the  The students cognitive  provided  and  could  an  were  structures  unexpected  constrained available  situation  to  for  by  the  descriptions  of  the  problems provide an insight identify  Faced  the student's  with  students'  expectations,  the  conceptions",  and  explore  them  the  not  for  the  the  "why"  strategies into the  teachers cognitive  to  the  both  them  in  physical problems  and  therefore  teacher's  own  presented  information.  attempted  assist  had  the  problem  point for further  use  she  presented  the  then  for  to  of  understand  of  each  concepts. In  terms  organizing  teachers who  to  This  identify  intervention.  students  with  their  processes teachers may use to  begin  point-of-view:  responses  teacher who and  does  which  who  is  not  go  "how"  (Arlin,  deviate not  the  various of  students'  "incorrect"  their  In  other  words,  Rather,  she  to  student's  or  point-of-view.  from  students'  of  (Brooks, 1984).  ways  the  1985)  answers as " w r o n g " framework  in  aware  beyond  student's the  in  nature  the students' constraints in order to find a starting  to  the  a  frame  processes  concepts taught  comprehending  They  own  describe what the concept was in a way that matched the  definitions.  These  about  with  situations,  her  developmental  concepts  were  presented  problem  actions,  knowledge  to  were  presented  cognitive  her  with  literal  students  experiences  concepts. They  a  the  these  the  constructing  the  To  of  and  experiences  of  situations,  teachers.  repertoire  examples, the  characteristics  and  their  her  her  of  and  by  brought  understanding  instructional  responses, will  her  "alternative answers interpret  she does not merely  own  provides  to the  attend new  36 information their  and examples of the concepts which the students may simply layer  alternative  situations", for  students  doing  concepts.  so,  who  are  the  see  the  the  teacher  student  solution  memorization^ the  use  this  their method  are having with the  of  framework  tend  to  the  problem  as merely  expected. Correct  answers  surprise, or  and  "child's  are  and  questions  own  understanding  her  students,  presentation generates Steufert,  no  (Schon,  and of  of the  the  alternative 1967)  and  in  useful  1983,  1987).  the  concepts,  "thoughtful their  for  the  problem  receive praise  constructing  the  nature to  the  of  both  the  teacher,  "alternative  1976),  allows  the  herself  for  assisting her recognizes  the  understanding  she actually  the  materials.  From  explanations  about  the  students'  thinking  students  (Arlin,  (Brooks, the  "how"  She  based on  sees  based  "incorrect  1983).  1984)  teacher  the  the  with  received from  or  hypotheses,  on  to  experience  their  her  this  own  asks  these  hypotheses, the  her  expected  from  students  upon  the  students, "how"  students'  Driver  repertoire  answers" could be the  of  problems  between  (Schroeder,  her  methods  recognition,  then  nature  and  initially  tasks  informal  conceptions"  discrepancy she  answer that  predefined  students  She  the  evidence  herself  " u n e x p e c t e d " . situations,  understandings  makes  Here,  students'  discovery of  curriculum  questions"  answers.  of  (Elkind,  these  longer  experiences and actions, about answers"  aware  point-of-view"  recognizes that  procedures  is  the  are for  who  teacher  the  identify  requiring  By  values  for  curriculum task as known  of teaching (Paley, 1986).  who  "well-defined and who  own  she is doing a g o o d job  the  these  (Getzels, 1964; Getzels & Csiksentimihalyi, 1976). These teachers  originally  contrast,  In  teacher's definitions  "masking"  who  students  through  memorize  only  Teachers  problem the and to  conceptions  onto  she & of  "correct through they  problems  got as  37 unknown  to  both  "discovered"  or  herself  and  "ill-defined  to  the  problem"  students.  as  She  a situation  sees  the  requiring  solution  further  to  this  discovery  and  elaboration (Getzels, 1964; Getzels & Csiksentimihalyi, 1976).  In  the  present  "well-defined" situations" finding  the  (Arlin,  teacher.  1975-76).  understanding  of  which  Studies of  The  need  responses  informs  to  what  The  problems  concept  of  problems  the  as  teacher's own  represent  are, in  problem  teachers  fact,  ill-defined  basis for  problem  contribute  to  can  of  be  problem  problems  construction  to  "ill-defined  the  finding  believe  by  the  providing  knowledge.  How,  a  then,  classroom problems as ill-defined problems?  Problem  for  Ill-defined  classroom  d o teachers identify  3.  students'  curriculum tasks have been redefined to  for  framework  study,  Finding  studies  of  problem  finding  in  educational  settings  (Dillon,  Getzels,  1964; Getzels & Dillon, 1973; Mackworth, 1965) and in studies of  general  (Arlin,  studies  exist  1984, (Arlin,  1986;  Sinnott,  1975-76,  1977,  1981, 1986;  1983) Dillon,  has  been  1982;  suggested.  Getzels  &  1982;  adults Few  in  such  Csiksentimihalyi,  1976; Yinger, 1980).  Getzels  (1964)  that we  stated  [researchers]  that, know  "it  method within  of  problem  of  solution  this  model  the  between  processes involved from  the point  in  [problem  formulation]  of view of either learning or  his observation, Getzels (1964) proposed a three  solving: the and  the  least about  teaching" (p. 247). In view of model  is  formulation  acquisition "presented"  of  of  the  problem,  a solution. and  He  "discovered"  the made  development the  problems  step of  a  differentiation in  terms  of  38 "what  is  known  and  problem-situation" operations  involving  solutions  as  creative  problem  types degrees  problem  typifies  required  to  types  even the Cetzels, finding  the  situation  the  given  highest  (1965),  difference  form  at  between  an  into  of  discovery  about  He defined  as "the  detection  "thought". specific outcome  solving The  "experiment"  answer  problem  to  finding  one was  of  The  first  formula  or  to  use  solutions,  finding.  Cetzels,  problem  within  problem  is  merely  the  known problem  and  perhaps 242). For problem  the  problem" many  qualitative information  selection and use of  new  method  of  a  57). By contrast,  solving was  discovery  of  the  an  program  of  problem "the  (p.  he  by  57). Methods also differed. the  require type  However,  proposed  finding  solving as "the  while  and  creativity.  as  well-defined "the  solutions.  which  problem,  of  mental  "discovered"  a  problem  need for  (p.  and  given  "principle  the  methods  types  solution.  programs" (p.  the  programs"  successful outcome  acceptable of  was  of  and  242). The remainder of  time  problem  problem  problem  problem  in  the  discovered" (Cetzels, 1964, p.  and  from an existing set of  "presented"  for  given  requires  same  solving  existing and expected future  the  of reflection than of  the  routine,  problems,  eight  the be  existing program  of  the  solver]  unknown  individual,  where  itself, must  problem  processing framework.  finding  of  as  the  methods  "creativity"  where  thinking",  may be more a matter  Mackworth  Between  and  problems,  involving  answer" (Cetzels, 1964, p.  "creative  described  formulated  and  data  presenter  He  continuum  "innovation"  essential question the  already  a  the  241).  processes.  proposed  find the  require  p.  [by  operations  mental  of  "plug  to  or  described the  he  unknown  1964,  known  he  method  is  (Cetzels,  By contrast,  varying  what  general  defined  comparing  The  method  finding  discovery  57).  The  an  of  was one  successful  questions  from  39 many ill-defined problems" (p. 57).  Yinger  (1980)  planning  adapted  (Clark  &  these  Yinger,  models  1977;  of  Yinger,  problem 1980)  planning processes of teachers in laboratory  in  finding order  to  to  studies  examine  of  teacher  the  cognitive  school settings. He saw problem  finding  as the first step in the teacher's planning process where the outcome was an initial problem-conception, generate  ideas. In  as "the  (p.  among  teacher  the  to  which  be  the  knowledge  used  In  process solved,  this  the  components:  (1)  by (2)  which the  has  the  learned  of  teaching  goals  to  specific  groups  teacher  becomes  knowledge perceive  include of  is  attempt teaching,  problematic to  apply  they  with notions  neglect  teacher comes to  to  these of  indicate  to  finding  her  aware  experience  implicit  and  and  (4)  of  finding this  problem to  is  identify classroom problems as ill-defined  -  the  finding of  that  ways  in the  teacher's  notions  teaching  that  the  and  the  explicit  of  materials  is and can be  occurs in  studies  ability  problem  (3)  the  information finding  a  situation^  memory,  an  confronting  of  her  and  represented  dilemma  necessarily in the act of  "how"  problem  process  problem  from  problem  conceptions problem  and  the  students,  classroom. For Yinger (1980),  defined  finding  planning  teacher can draw which  process was  idea that requires further planning  include concrete materials and any source of in the  problem-finding  problem  to the interactions with students and not  What  the  the  teacher's  the  for  of  instructional  model,  methods  teaching  purpose  teacher planning, Yinger (1980)  and  conceptions effective  of  four  teacher  overall  a potential  84).  interaction  which  terms  discovery of  elaboration"  needs  and the  planning  prior  instruction.  is that while teachers  acquired  and  and "how"  problems. Although  they their the these  40 researchers  describe  neglect  discuss the  to  and  skills  the  Deweyian-Piagetian  within  of  what  a  problem  student's an insight  into  point-of-view  are  of  faced  in  this  "How  with  processes  development  have  A description  question,  when  basic  for  framework  conceived  the  the  a  study  problem which  ignore  educational is  of  through  acquired. A n d , they  traditions finding  point-of-view.  be  mechanisms of finding  problem  reflection-in-action  may  description  as  the  of  the  do  teachers identify  of  classroom  problems  is  the  teaching  teacher as problem the  implications What  basis for  they  these processes  the  practice.  finding,  finder  that  needed type  of  from  the  may  provide  child's or adolescent's  which  may  be  defined  as  ill-defined problems?"  C. THE TEACHER AS PROBLEM  1. The Teacher's Own Arlin  (1975-76),  Mackworth Several  cognitive  finding  ability.  the  intellect  information In this  problem situations  work  Knowledge of  a cognitive  process variables were  Getzels  and  developmental identified  Csiksentimihalyi model  as potential  of  model"  Steufert's  (1956),  of  Inhelder  and  Driver  and  Piaget's (1958)  objects, they  "stimuli" For  the  or Arlin  (1967)  Mackworth  (1975-76,  processes of  1977,  problem  subjects  (1965)  referred  1986),  these  finding,  where  problem  levels  of of  operational schemes.  are provided with an opportunity  what  of  and  finding.  Guilford's "structure  formal  process model, it was proposed that when  situations". initiated  Schroeder,  (1976)  problem  predictors  from  processing, and  face  the  of  These included: selected variables drawn  cognitive  the  on  proposed  with "problem-rich" in  Construction  building  (1965),  FINDER  to to  are  ask questions as  "ill-defined" a  presented  subject's  "ill-defined problem failure  to  41 employ &  a mainline  Steufert,  designated mainline  process, such as the  1967),  resulted  as "output".  processes  of  (Schroeder  et  ill-defined  problems  question'  represented  as,  "the  al.,  various  in the  production  By contrast,  tended  (Arlin, the  of  information  of questions posed and the types of outputs."  Findings  relationship  (Arlin,  between  cognitive  1975-76,  problem  and  a  and  1965).  For  be  1986)  the  of  Arlin,  inferred  set  the  integration from  the  described  'general  this  from  many  process  the  types  correlate with these  supported  proposed  was  employing  questions'  She  could  of level  operations which  1977,  finding  high  'general  finding.  that  Driver  level question, which evidence  many  problem  (Schroeder,  gave  Mackworth,  of  information  a lower  who  generate  1975-76;  organizations  of  differentiation  to  outcome  of  subjects  discrimination,  1967)  integration  the of  hypothesized  cognitive  process  variables.  2.  The Teacher's Own  Recently,  Schon  repertoire  of  includes  Reflection-in-action  (1983,  1987)  the  whole and  and  as  actions"  of  her  action. well  as  Fusco, 1983; Yinger, 1980)  the  is interesting  that  "practitioner's  describe the which  that  examples, images, understandings  understanding  It  proposed  For the  Piaget  to  "ground"  Schon chooses to  repertoire".  these  solutions  b e c o m e the  and  insofar  teacher,  previous  child's "reflexive  leads to  experience  "the  actions..." as  it  is  also  (1983,  chose  term  new  (Arlin & Levitt,  which  her  for  (Brooks  &  interactions. actions  "repertoire"  abstraction" and the adolescent's "reflective  concept development  a  understandings  experiences and  the  up  138) to  problems  analyzing the  previous  built  p.  images,  instructional for  has  accessible  "examples,  call these  (1979)  practitioner  to  abstraction"  1986; Gallagher & Reid,  1981).  42 Like  the  child  repertoire  adolescent,  building.  teacher's 1987),  or  It  is  too  situation.  If  construct  her  builds  the  use  to  responses  to  construct  present that  teacher,  through  on  the  to  which  is  they  new.  the  beyond  situation  share  in  is  "knows-in-action" and  learning,  faced  practical  and  with  the  of  1983,  she  and  Schon's  the  practice  actions,  student's  of  that  will  education.  processes that the  context.  Piaget's conception  in  instruction,  cognitive  when  type  (Schon,  actions  experiences  the  exactly  common.  reflection-in-action,  new  of  a similar  teacher  "unexpected" description  development  to  of  include  inform the adult's experiences and actions.  teaching  features  go  in  she  in  "reflection-in-action"  experiences  teaching,  knowledge  engaged  of  these  nature  tasks  As  of  upon  of  at the new  may  the structures which  No  reflects  is  process  a repertoire  curriculum  "reflection-in-action"  the  (adult)  constructed.  concepts  "Reflection-in-action" hints may  is  up  teacher  own  teacher  through  "knowing-in-actioh"  she  the  situation, This  is  making illustrated  like  It  is  extracts  a in  previous these  these  through  Schon's  and  "unique"  common  moment-to-moment  one  yet  there  are  that  the  situations  features  and maps  subtle  adjustments  (1983,  1987)  them  to  that  definition  of  "reflection-in-action":  There is some puzzling, or troubling, or interesting phenomena with which the individual is trying to deal. As he tries to make sense of it, he also reflects on the understandings which have been implicit in his action, understandings which he surfaces, criticizes, restructures, and employs in further action. (Schon, 1983, p. 130)  Schon situation  (1983, as  1987) one  of  refers  to  "problem  chooses and names the things  the  practitioner's  setting".  When  way a  of  understanding  practitioner  she will notice. Then, through  sets  a  a  unique  problem,  complementary  acts  she of  43 naming states  and  framing,  that  their  she  individuals  backgrounds.  selects  frame  Furthermore,  a teacher's available theories  as  an  rules she  in is  her  store  confronted  terms, the situation  In order to a  kind  of  devising" she  problem, of  is  the  theories  not  is, one  35).  A  a surprise  unexpected  phenomena, experiments  between  to of  prior  makes something.  unique  and  of  uncertain  solved  knowledge.  In  other  "in  reappreciation" levels of  the  book"  by  words,  (Schon,  Schon  depending the  In  on  categories situation  application  the  1987).  them.  treat the  the  unique  of  specific  the  situation  problem  testing  (Schon,  situation, the teacher must  in  the  situations,  strategies  that  rethink  leads  by ways  put  her  to  finding  thought  framing  takes  some the  place  understanding  and  It  1983,  development  of  p.  132).  through  as a "continuous  to the  130),  that  her  in  finally,  test  new  a  situation  process of  ways  of-  action,  (c)  invents  takes  where place.  conversation  cognitive  The  in  a  concept  action  function  an  with a  "conversation"  the  (a)  (Schon, 1987, p.  appreciation,  described  own  responds  Schon describes his  stages of  also  strategies  phenomena  p.  so by  a process where,  and  is a "reflective  1983,  Piaget  her  each  present  is interesting  as "[spiraling]  of  knowing-in-action  problem,  with  Her reflection-in-action (Schon,  do  and operations, (b) she then  restructuring of  her  her new understandings  situation"  conversations  cognitive  ways  outside  teacher follows  sense (Schon, 1983).  "reflective  falls  5). Specifically, the  or  teacher  metaphorical  case  be  with the  situation  transformation  integration  different  to  inventing,  (Schon, 1987, p.  on-the-spot  organizes  is "ill-defined".  "improvisation,  of  in  unique  that go beyond available rules, facts, theories to  and  and techniques, she is unable to  that  deal competently  experiences  attention,  situations  because a  professional  with  for  problematic  of  instrumental  things  of  and his  structurations".  44 However, own  Schbn's  thinking  (Yinger, a  practice  the  to  before  like  teacher  finding  more  'expert'  on-going instruction  What  Schon's  1983;  like  1978)  to  and  beyond  thinking while  the  teacher  constructs  it.  Through  a research scientist behavior  is  this  work  suggests (her  for  own  the  present  theories  study,  about  1976;  Feldman,  1981),  which  in  is  that a child or adolescent brings to  information  student's  the  teacher gains about  the  operations  themselves. Schon's  description  into the  problem describes situation  that  In  the  system  this  view,  performance  if  the  of  teacher's  knowing-in-action),  practice  may  describe  "how"  finding the  framework  steps  which  informs  a teacher goes through  may describe the is able to  identify the student's  in  is  teacher's know  curriculum  professionals their  own  the task.  will be about  the  translate  knowledge  practice setting. In addition, a  making  sense out  finder",  that is, the  classroom problems point-of-view.  "how"  to  teacher's reflection-in-action  "teacher as problem  identify  a particular  construct  in the  the  come  framework  of  teachers  the  informs  teacher may  student's  child's or adolescent's point-of-view  teacher  begin to  the  turn,  logic or framework  the  smooth  childrens'  (Schon, 1983, 1987), then  about  sees  an expert  35-36).  of  by a "developmental perspective" (Arlin, 1983; Brooks, Fusco and Crennon, Elkind,  theory  1983).  situation  Schon  model  her  something  (Schon,  own  process,  into the  "reflection-in-action"  The  it  her  m o d e l e d " (1987, p.  about  doing  doing  "trying to  reflection-in-action  thinking  about  something  solve  integrate  go  (Schon, 1987).  "knowledge-in-action" informed  to  teacher  doing  framework  whose  is able to  the  Piaget,  about  attempting  acting the  allows  1984;  thinking  problem  practitioner than  (Lampert,  1980)  Within  description  as ill-defined  of  the  extent  problems  and  which  ill-defined to and  which thus  45 D.  SUMMARY  The  nature  of  the  the  child's  or  adolescent's point-of-view  may  be  which  the  the  "teacher  instructional about  phenomena  and  the  as  and strategies  hypotheses  to  study  is to  described as ill-defined  structures in  present  formal  on  research  the  setting,  testing  of  it will  explore  teachers  discovering the  The  of  confirmation  the  emphasis  hypotheses of  teachers  from  the who  student's study  is on a data  natural versus experimental  By asking the curriculum  teacher what  task  presented  methods  kind in  the  reveal the teacher's framework adolescent  "has  explore the use this Sinclair  a  have  &  teacher's methods  to  generate  description  students'  1970).  responses  and  The  about  description formulations  student of and  thinking  the  steps  solutions  1977;  interpreting  the  base as o p p o s e d  she while  Strauss, the  goes  for  teachers'  1986b).  researcher  In  the  by  selection  attempting  has the  to  It  opportunity  "how" is  the  own  the  thoughts  the  teacher  arrive  identify  at and  to  child  or  possible  to  answers and  to  then  "wrong"  formulate  assisting  to  is having with a  teacher's thinking  to  short,  through  1981).  student's  asked  strategies  reveal the (Arlin,  is  a student  for understanding  understanding  teacher  suggest  "problems". These questions and  of  of  hypotheses. This is supported  she thinks  simulation,  as a guide to  Kamii,  problem  (Larsen,  the  for classroom study (Shulman, 1965).  or point-of-view  concept"  information  of  to  point-of-view  is  the  which  extent  research which suggests that research on teaching should be based on the of  identify  classroom problems  those  purpose  questions.  and  addition,  for  major  development  ways in which  describes  them  The  the  faced with  In  finder"  available to  these  when  problems.  problem  context.  explore  (Arlin, 1983;  questions, students about  with  their  interpret  their  teaching  is asked to particular  predict  give a problem  classroom  46 problems  from  the  child's  or  adolescent's point-of-view.  These procedures  may  be  described as the set of questions which inform the present research.  In  order  with be  to  explore  how  classroom problems provided  for  the  teachers which  terms  are  identify  the  ill-defined  associated  with  student's problems,  each  of  Chapter 3 will describe these procedures and methods.  point-of-view operational  the  four  when  faced  definitions  questions  will  posed.  CHAPTER III. METHODOLOGY  In  order  when  to  faced  set of  four  describe with  how  teachers  ill-defined  questions  which  identify  problem  the  situations,  child's  or  teachers'  accompany a selection  of  adolescent's  responses to  point-of-view the  following  classroom scenarios, will  be  explored (see Appendix A):  1.  What kinds of problems d o you think assignment? Why do you think so?  2.  What questions worked through  3.  What questions would you, as their teacher, ask these students about the problems that you think they are having with the assignment? What responses would you expect to get? W h y do you think so?  4.  H o w would you proceed to help these students having? What strategies would you use? Why?  In  addition,  variables of  Chapter  the  concern  a description  of  and procedures as they relate to  The  present  a suburban to  introduce  Feldman,  having  for  problem  with  finding  the  and  problems  several  with  the  as  they  they  are  demographic  the  sample  and  a discussion of  the  methods  the research questions.  OF THE SAMPLE  study  district-sponsored  are  to the present study will be considered.  3 provides  A. DESCRIPTION  students  do you think these students were asking themselves the assignment? Why do you think so?  teacher's  interest  these  consisted of  a total  of  27  experienced  teachers,  participating  in  in-service courses in a suburban/urban coastal area of Maine and in  area on  Long  participants  to  Island,  New  York.  a developmental  1981), and to strategies  for  "teaching  47  The  in-service  perspective for  courses were  (Arlin,  thinking"  1983;  (Arlin,  designed  Elkind,  1985)  from  1976; "the  48 child's  point-of-view"  categories  based  in-service Primary  This method on  the  on  course  Grades  (Elkind,  demographic  (see (K-3)  of  Appendix and  grouping  quality  of  provides additional  1976).  12  Participants  information  B,  collected  Questions  participants  2-7):  responses  demographic  (see  information  at  15  represented  takes into account the  teachers'  represented the  effects  grade  beginning  participants  the  section  two  of  the  represented  the  Intermediate of  on  familiarity 'Task  which may be of  level  Grades  (4-8).  and experience  Administration')  interest  to the  and  present  study (see Appendix B).  B. PILOT STUDY An  initial  set  of  12  anecdotes  pre-service teachers in order relevance  of  the  examination to  the  own  interpretation  'The  administered  to validate the  and  grade  levels  responses given  on  8 tasks chosen for  tasks eliminated  teacher's  C.  the  final set of  The four a  of  subject  was  telling  from the of  a  clarity  consenting  of the  in-service  tasks and to each  task.  set were problem  present  study  those which and/or  Based  the  on  were  an  made  (see Appendix A).  were  which  and  examine  initial tasks, modifications  use in the  classroom  26  associated with  these  final  to  not  tended  based on to  pose  problems for the teachers.  PROCEDURES Student  Anecdotes  teachers identify the problems description  which of  Development  child's  may the  and  Task'  be  task  is  (Levitt  &  Arlin,  1986)  was  or adolescent's point-of-view described  as  divided  into  Description,  (3)  Task  ill-defined four  when  problems  sections:  Administration,  used  (1) and  explore  "how"  faced with classroom  (see Task (4)  to  Appendix  A).  Rationale,  (2)  Rating  Criteria  The Task and  49 Examples  of  variables  Teachers'  of  teacher's  concern  variety of  1. Task  interest  to  the  for  present  problem  for  the  based  development  on  the  often  the  deviate  from  teachers  "unexpected"  problems  that, in order problems  to  and  researchers situations  their  from  identify  include  Appendix  to  ask  based on  a  discussion  of  B). These include,  B,  Question #1),  they  might  have  the  context  of  exploration.  by their  routine  other (a)  and  expectations.  knowledge  student's  There  It  the  (b)  a  questions  which  elicit  the  view  to  the  of  In  the  them  a way for  "how"  present  educational  provide to  and the  of  and in  deal  the  with  proposed  interpret  the  classroom  responses about  clinical  interview  practice were  used to  exploring  processes and  'Student  identify  these  of the teacher's responses from  study,  identifying  classroom. The  to  therefore  tasks  teachers'  instruction  as they  about  & Arlin, curriculum  a need  was  teachers think curriculum  "why"  is  to  classroom  (reflect-in-action)  point-of-view.  (Levitt  responses  teachers during  with  available of  students'  have  a constructivist  teachers in order  Anecdotes Task'  students  point-of-view.  from  Student  ways in which  and to explore the own  'The  original  own  the  the  questions  need  teacher's  method  will  (see Appendix  (see  that  them  teacher's  construct  of  premise  presented to  within  study  finding  tasks and materials  the  (5)  Rationale  was  "how"  Section  demographic variables (see Appendix B, Questions #2-7).  The rationale 1986)  Responses.  the  how  a  child  Anecdotes  elicit responses  "has  Task'  questions  a  structures concept"  provides  this  50 2.  Task Development and  a. Task A  Development  variety  different  of  instructional  grades  anecdotes to discovery from A).  subject  teachers, (see  Therefore,  the  might  development situations  worksheets areas  and  classroom  were  used  to  discussion construct  serve as the stimuli or "ill-defined" problems for  in  teachers  the  and  a teacher's own  more  Description  Figure  telling  is supported  implying  construct  about  scenarios  normally  an activity  represent  in her own  simulations  in  by findings  in the  their  uncertainty,  of  verbal or written responses to  classrooms. literature  and  that,  control  in  uncertainty,  order  the  to  tasks  is  instructions designed abstract of  record  and to  for  Feldman,  "conversation"  four  reflect  stimulus  "teaching  1976;  accompanied  pertaining  thinking" The  (Arlin,  1985)  major  participants  in  of  the  a  the  to  method  "[i]n  general,  given  of  task  ...stimulus  domain, 1967,  that  p.  produce 186).  In  the  of  study  points-of-view,  sheet  scenario. of  based on  a "developmental  to  containing The  stimulating  these find  questions out  points-of-view,  and  task performance.  students'  literature  from  function  taken  situations  This  Steufert,  answer  method  developed from  processes of  and control" is represented by students'  standardized  Piagetian clinical  and were  with  a  questions  the  1981).  by  student  scenario was  curriculum tasks which reflect their  teachers' interpretations  from  classroom (see Appendix  which deviate from the teacher's own expectations of student  In  initiating  problematic  responses" (Schroeder, Driver &  present study, this "conflict;  "real-life"  Sample Scenario) Each  encounter  conflict,  relevant  2 for  scenarios  each  specific  questions  thinking  of  were  about  a constructivist  an view  perspective" (Elkind, was  "how"  to they  conduct "have  a the  51 . c o n c e p t " of the child's  The  clinical  interview  participants situation is  an  found  in  "permit  by  findings  "natural  to  accompany  respond  that  inquiry  learning  influenced  which  to  1975-76, 1986; Mackworth,  teachers to  teaching  questions  opportunity  (Arlin,  supported  point-of-view.  the  by the  valid  (Shulman,  literature  on  1980, p.  Task  A  total  problem  the  or  the  "ill-defined" of  of  and  own  terms  about  form  of  as the  eliciting  teachers'  1965),  finding  scenarios  that  there  give  problem responses  thinking is  are  a need  all aspects of  the  questions  questionnaire  to the  is also  is designed  to  formulations.  Description of  anecdote of  accounts  66). The  elicit the teacher's own questions and problem  b.  discrepant  express themselves in their  process" (Hunt,  of  1965). This method  most  situations"  a  each  8  student  anecdotes  is classified into  student(s)  who  one  of  completed  make two or  up  main  'The  Student  categories  responded to  the  Anecdotes  representing particular  Task'.  Each  grade  level  the  curriculum  task  or  material: Category I: Primary (Grades K-3) Category II:  The  tasks  Physical  Intermediate  are  further  (Grades 4-8)  classified  Sciences/Mathematics  (English/Reading;  Social  participants  Studies). were  into  the  (Science; This asked  following  Math),  method to  select  of  two  and  (2)  Subject  Humanities/Social  organization  from  which  one  task  their  background and experience (see section on 'Task  Categories:  for  provided  a  completion  Administration').  (1)  Studies  framework based  on  52  Figure 2: Sample Scenario  PRIMARY (Grades K-3) - MATH (C. Reynolds, personal communication, October  1984)  The following worksheet on finding the standard number was completed by a grade 3 student  at  Please examine page.  the  beginning of  the worksheet  the  school  year  and was  carefully and answer the  marked  by the  questions on the  teacher. following  53 (Sample Scenario cont...2) Name: ANSWER SHEET - (PRIMARY/Math)  Imagining  that  you  are  questions  as thoroughly  the  teacher  as you  in  this  situation,  can: (Feel free  to  please answer  use  the  back  the  of  following  the  page  if  necessary) 1.  What  kind(s)  of  problem(s)  do you think  this  grade 3 student  is having with  this assignment? W h y do you think so?  What  question(s)  do  you  think  this  student  was  asking  him/herself  while  working through the assignment? Why do you think so?  3.  What  question(s)  problem(s)  that  would you  you,  think  as he/she  the is  teacher,  ask  having  with  this  student  the  about  assignment?  the What  responses would you expect to get? W h y do you think so?  4.  How would  you  proceed to  help  this  student  having? What strategies would you use? Why?  with  the  problem(s)  he/she is  54  3. Task Administration At  the  beginning  complete  of  their in-service  course,  consenting  participants were  asked  to  'The Student Anecdotes Task' (Levitt & Arlin, 1986). Two grade levels were  represented:  15 participants represented  the  Primary Grades (K-3) and 12 participants  represented the Intermediate Grades (4-8).  Participants were from  the  general  instructed to select one task and its corresponding response  Subject  age  Category  and  group and subject  familiar and comfortable  Grade  Level  area that  group  he/she  with. This selection  most  tend to  perform better when their experience  1972;  Sinnott,  1975;  & Guttman,  1978),  best  often  procedure was  adults  Sinnott  which  is taken  and  on  subject  it is assumed that familiarity with age/grade  matter represented  context  for  processes  the  that  in the  teachers are  to  most  chosen  reveal  often  the  activated  and  used  1972;  of the  findings  the most that  (Piaget,  that  adults  Sinnott, 1975).  student(s) and the  knowledge when  on  suggestions  task would provide the experiences,  or was  into account  should be tested using materials most familiar to them (Piaget, In this view,  represented  taught,  based  sheet  they  most appropriate and are  cognitive  faced  with  ill-defined problem situations.  Participants  were  given  the  the  time  provided at the beginning of the course or in a place of their own choosing  over  a period of  two  weeks,  returned  the  investigator.  to  option  of  completing  at which time they were In  addition,  their anecdotes  collected  participants  were  during  by course  leaders and  asked  complete  to  a  questionnaire requesting their responses to several variables which may be of interest to the present study (see  Appendix B).  55 4. Rating Criteria and The  standardized  specific the was  rated  contains four of  answer  directions  teacher's  to  according  sections  of  include:  point-of-view, for  sheets  four  thinking  examples  "concern  Examples of Teacher's  to  the  developed  criteria  teachers' (1)  problem 1976)  questions  accompany  each  from  classroom  the  (see Sample Scenario, Figure #2).  and (4)  Csiksentimihalyi,  which  Responses  specified  formulation,  developmental  finding  in  a  relate  (2)  general of  to  the  the  integrative  context  demographic  review  following  teaching strategies.  the  variety  Each of  the  responses as they  problem  and  in  literature  scenario  contain  to  four  questions  discussion. rating  complexity,  her  (3)  work"  variables  of  Each  criteria.  In addition, the  of  probe  The  quality  teacher's  (Cetzels  interest  to  & the  present study are also presented.  #1.  Problem Formulation: What kind(s) of problem(s) do you having with this curriculum task? Why do you think so?  Teachers'  responses to  problems  and  continuum known problem 1976;  processes of  from  situation  formulation  question  were  solution  &  is unknown (Arlin, Dillon,  by the  1986a,  in  1973).  have been derived to  scored  identified  "presented" problems to  and what  Getzels  this  according  by  Getzels,  Modifications the  the  Getzels (1964).  presenter/teacher  reflect  to  " d i s c o v e r e d " problems  press;  think  and the  1964; of  nature  eight  student  is  types  of  These represent  in terms  of what  solver/student  Getzels &  these  this  in  a is the  Csiksentimihalyi,  categories  of  problem  of the task (see Table 1):  56  Table 1: Problem Formulation - Scoring Categories and Examples  Type 1:  The teacher interprets the student's problem as known and thinks that there is a standard method for solving it available to the student and to herself and therefore guarantees a solution in an identifiable number of steps. Example: "They are having trouble with place value (10's)".  Type 2:  The teacher interprets the student's problem as known but indicates that no standard method for solving it is known to the student, although it is known to herself. Example: "The student is having problems with the concept of place value and expanded notation. The child is not ready to abstractly apply the concept of place value".  Type 3:  The teacher interprets the student's problem as known, however indicates that no standard method for solving it is known by the student or herself. Example: "The students seem to be having a problem with the definition of fraction. In some cases, it looks as if they don't understand that a fraction can be more than just 1/2 etc. Perhaps the teacher's methods (examples) were not clear."  Type 4:  The teacher states that the student's problem exists but indicates that it remains to be identified or discovered by the student, although she herself knows the nature of the problem. Example: "Students don't know what to do with unlike denominators -- they have never been exposed to this before, [therefore] novel -- Problem establishing a relationship between (1/2 + 1/4) = (4/8 + 2/8) = because it's too abstract -haven't dealt with it at the concrete level..too complex."  Type 5:  The teacher states that the student's problem exists but indicates that it remains to be identified or discovered by the student and by herself. Example: "It doesn't seem that the students have grasped the concept of fractions with similar denominators because none of the students got the correct answer. The problem is probably that the students don't know how to find a common denominator (or maybe they don't know what a c o m m o n denominator is!) Another problem might be with fraction addition itself."  57 Type 6:  The teacher states that the student's problem exists but indicates that it remains to be identified or discovered and that there is a standard method for solving the problem known to the student and to herself, once the problem is discovered. Example: "The children appear to be having a problem transferring what they already know about numerals and their values to the new knowledge about how values change according to the place of the numeral.,.1 feel that at this point the students have not received enough practice in finding groups of tens in a set and then labelling the set according to the fens and ones."  Type 7:  The teacher states that the student's problem exists but indicates that it remains to be identified or discovered, and that no standard method for solving it is known to the student although known to herself. Example: "These children (most of them) would be in the concrete operational stage. These children would have difficulties relating the concrete with the abstract. If the teacher showed them addition with pie fragments these kids would have no problem answering the questions. However, when doing problems from the textbook the kids might have difficulty".  Type 8:  The teacher states that the student's problem exists but indicates that no standard method for solving it is known to the student or to herself. Example: "The questioner is asking her [Roberta] to give information she cannot answer from the paragraph. He/she is asking Roberta questions that require prior knowledge inappropriate for a 7 year old to be expected to have."  Type  1 problems  teacher's formula to  view or the  that  itself,  8,  must  (Problem  in  "the be  Type  "need  method  to  task. This teacher  information  through  "presented"  students  known  a curriculum  new  represent  an  attempt  principles  of  to  or  "well-defined"  only  find  to  the  obtain  solution,  problem  the  given  [correct] answer"  may suggest that the the and  discovered" (Getzels, 1964, 8) requires  plug  problems  finding  p.  desired perhaps 242).  data  results.  The  (Cetzels, 1964).  reflect  into  the  (Cetzels, 1964, p.  students  even  which  be presented  In  problem  the  highest  essential form  In view  of  of  the given 242) with  types  2  question discovery  findings  that  58 problem  finding  functions  effectively  within  discovered  or  ill-defined  problem  situations (Arlin, 1975-76, 1977, 1986; Cetzels & Csiksentimihalyi, 1976), teachers w h o view  classroom  strategies  problems  necessary for  as  ill-defined  identifying  may be problem finders in their  #2.  may  classroom problems  have  the  structures  and  as ill-defined  problems.  They  own classrooms.  Integrative Complexity: What question(s) do you think this student him/herself while working through the task? Why do you think so?  The to  problems  teacher's formulate  they  work  deviates  response to alternative  through  from  the  recognize the task,  the  this  question  explanations  instructional  discrepancy between the she  expects  who  has  generate  alternative  For what  question  student?  (Arlin,  structure  the  the  explanations for  teacher  uses  to  which she produces alternate  Criteria Four  Used  to  categories  of  teaching  students  for  which  where  the  student's  and learning, the  undertake,  she  the  use to and  1970).  generate  her  interpretations  represent  of  this  view,  teacher  response  is  operations  the the  situations,  point-of-view the  will  question:  right answer for level  of  related  to  the  for  must  methods  the  (organizations) of the  general  used by the teacher to  In  response  respond to  the  student's  answer, in fact, the  attempts  ask themselves as  these differences by asking herself  Complexity:  conceptual structure  to  identifying  Kamii,  set  to  the task. In these ill-defined problem  Rate Integrative a  extent  questions students  situations  is this student's wrong  1983; Sinclair &  the  processes she would  the  potential  the  In  teacher's expectations  methods  teacher  about  tasks.  students actually take to respond to the  reflects  was asking  the  conceptual extent  to  level  of  situation.  inferring  generate her response to  the  question  #3:  59 1.  The  Low  Integration  Index  consists  of  "a  hierarchical  and procedures" (Schroeder, Driver & Steufert, 2.  3.  The  Moderately  Index  is  absolutism accompanied by a lack of  The  Moderately  High  patterns  high  by  a movement  consistency and  rigidity.  characterized  the  of complex  of  interactions  moderately  high,  conceptual  structure.  complexity  of  integrative  high  the  levels In  of  teacher's  scale which  integrative to  potential  to  (Schroeder et al., 1967, p.  complexity.  order  points-of-view).  gives the person the  are scored along a 4-point levels  several  by  away  the  The High Integration Index  Responses  23).  characterized  is  rules  complex rules for comparing and relating (i.e., a person can observe behavior from  Index  established  of  his own  Integration  1967, p.  of  provision  effects of  to  Integration  from  more  4.  Low  set  make  responses,  Each  represent  major  complexity) adequate the  point  (low,  represents  have  alternate  23).  a continuum  inferences  raters  generate  moderately a  modal  about  been  from  the  low low,  level  of  integrative  directed  to  ask  themselves:  Can it be inferred that the teacher's response was generated by a conceptual structure which failed to produce alternative interpretations (organizations) of the event?  The following the  descriptions  of the  levels of  nature of the task (adapted from  integration  have been modified  Schroeder et al., 1967, p.  187-189):  to  reflect  60  Table 2: Criteria for  Scale Value Jh Low Integration o o  o o o o  Rating Integrative  Index  The teacher views the student's response as unpleasant or as a flaw or weakness in the student, The teacher seeks fast and unambiguous closure or resolution to the student's problem and reacts in such a way as to engage internally consistent processes that reduce incongruity or dissonance, The teacher offers the student a specific guide or rule to correct the problem and thus reduce conflict, The teacher implies that an absolute solution can be found to the student's problem. The teacher states that effects are compartmentalized, are all one way or all another way. The teacher presents only one side of the problem (her own side) while ignoring differences and similarities with the student's views. Example: " W h y isn't it worth '1'? Why isn't I circled the 1st one on the left."  Scale Value 2\ M e d i u m Low Integration o o o o o o  Complexity  it correct  the  one  is on the  Index  The teacher lists similarities and differences between her own views and those of the student without considering relationships, The teacher specifies at least two different hypotheses about the nature of the student's problem(s). The teacher uses "either-or" types of responses expressing a possible conditional rule about two ways of categorizing, The teacher makes probability statements about the occurrence of the student's problems. The teacher reacts against absolutism in general (the possession of more than one view without the rejection of a particular view), The teacher avoids dependency on external imposition; that is, the availability of alternatives (this dees not include opposition to a particular categorization). Example: "Where is or are continents/oceans relative to M E / U S A / N person works from homebase outward or familiar to unknown."  Scale Value _3j Medium o o  left,  High Integration  America? A  Index  The teacher integrates the student's and her own conflicting interpretations of the instructional task so as to preserve and not "ward off" the conflict. The teacher generates various meanings of alternate perceptions, such as various meanings of the perception of conflicting views about how the student  61  o  o o  has the concept. The teacher's response provides evidence that the completion implies the ability to take the student's perspective or point-of-view into account and to relate different perceptions of different people, The teacher's response implies that one's behavior is affected by the way another behaves, as in a give-and-take strategy game The teacher views social relationships as anchored in mutual responsibility (as o p p o s e d to fixed beliefs or rules), in which each person can "place himself in the other person's shoes" (relate alternate schema). Example: "Possibly repeating the teacher's question and trying hard remember the exact words stated by the teacher in her presentation. With concrete examples, memorization is all the students had to use."  to no  Scale Value A\ High Integration Index o o o o  o  The teacher's response indicates that conflicting alternatives lead to new organizations and information, The teacher uses alternatives through exploratory action in order to obtain new information from the student, The teacher generates functional relations between alternatives, The teacher considers relationships among similarities and differences between the sides of a problem or question, a development of relationships between alternate reasons as to why these differences and similarities exist, The teacher produces responses reflecting more "connectedness" between alternatives by theorizing as to why these reasons exist. Example: "I think Roberta is trying very hard to draw on prior knowledge and she is trying to concretize her answers to the questions' and draw on her experiences."  #3.  Quality of Point-of-View: What question(s) would you, as the teacher, ask the student(s) about the problem(s) you think they are having with the task? What responses would you expect to get? Why to you think so?  The  kinds  of  questions  understood  in  terms  those situations  of  teachers the  pose  in  ill-defined  developmental  capacities  problem that  each  situations  may  be  teacher  brings  to  (Arlin, 1977):  Questions which are sensitive to a student's cognitive level are questions which support the student's thinking, which challenge him, and which encourage him to construct his knowledge about the content of the question. (Arlin, 1986b, p. 17)  62 The  quality  their  of  the  students  are  questions  teachers  having with  use to  instructional  elicit  information  tasks may  they discover the child's or adolescent's point-of-view  Criteria  Used  Rating  to  Rate  "Quality  of  for  question  #3  categories  six intellectual (Mackworth,  Within  products  (Arlin,  are  based on  to  which  Guilford's  (1956)  1977).  a modification  categories. These are used to  identify  all other  of  the  'general  Category 2: classes  Category 5:  transformations  Category 3: relations  Category 6:  implications  represent  the  framework way  in Table  in  (Arlin,  which  1975-76,  1977,  information  3 are modified  to  reflect  is  questions'  questions posed:  Category 4: systems  finding  problems  extent  Category 1: units  Category definitions Arlin,  describe the  the  Point-of-View":  1965) posed as distinguished from  a problem  categories  about  1986),  structured the  nature  each by of  of the  this  the  six  teacher. study  (in  1975-76).  Table 3: Rating Criteria for 'Quality of  Category  Point-of-view'  Definition  The teacher: 1. units  attributes the student's problem to lack of information; gives reasons why the student doesn't something Example: "What does the 1 stand for in 16? ' 1 0 ' . "  basic know  2. classes  attributes the student's problem to different classes of learning problems; gives slow-learning problems as the reason why the student doesn't know something  63 Example: "...Is this a city or a country? Is it near Alabama or far away? Can a city be inside a state?...She doesn't have enough information or experiences to understand." 3.  relations  attributes the student's problems to observable connections between ways in which the student knows the information Example: "How can she understand what belongs together and why?"  4.  systems  attributes the students' problems to different ways in which information is organized within a system; conducts a task analysis Example: "1 wouldn't have asked her any of those questions past the 3rd one. I would have proceeded differently. I would have asked: Is Alabama in the United States? What is the United States? H o w is Wisconsin like Alabama? H o w is the place Caddie lives like the place you live? H o w is it different?...! would never have introduced "country" and " t o w n " in my questions. I think it confused her. 1 would have asked further questions about how countries are alike and different. Once introduced, I might have asked her what the differences between a town and country were".  5.  transformations  attributes the student's problem to changes in the way the student knows the information; identifies the student's point-of-view Example: "It depends greatly upon what has been previously introduced to the children. If this is a new topic of study I don't believe I would ask many questions except maybe to spark interest in the upcoming subject of place value. I might ask, " W h y didn't you circle aN of the wheels?", "If there are 16 wheels and I ask you to circle both numerals in this number, why haven't all of them been circled?" I would expect them to explain to me what they know about the values of J _ and _6. Then this could lead to a discussion of J _ . standing for 1 group instead of J _ object."  6.  implications  questions the nature of the child's problem and predicts the student's point-of-view based on a developmental analysis Example: "First I would ask why they think the answer they gave is correct. From their frame of reference, I'm sure their answer makes sense to them; their answer merely mirrors their reality of the matter."  64 Each category  represents  the  is, the way in which the and  processed  Guilford, types  and  1956).  of  (Arlin,  problems, questions  questions  represent of  organization  the  (1)  a  represent  students.  (Arlin,  response  indicates  that  information  while  those  abstraction  (Arlin,  1977).  Calculation of Question  the  indicate  the  degree  Regarding of  ability  298).  the  to  implications  teachers who  ask  units  implications  stimulus  which  requiring  problems  of  questions  questions  a high  problem  of  tend  reflect  =  the units of  level  of  situation  in  in  the  abstraction  in  the  use  higher  concrete level  of  questions  by  Question Quality:  quality  represents  a  weighted  average  of  the  number  of  2) + 3(cat 3) + 4(cat 4) + 5(cat 5) + 6(cat  6)  category (Arlin, 1975-76, 1977):  Quality  in  apparent  to a  are  recognition  readily  degree  which  present  the  two  organize  information,  reflect  ill-defined  1986a;  elicit  teachers  only  questions  the  (Arlin,  abstraction  that  is organized  are used to  classroom of  is processed,  question  is abstracted that is not  who  ask  a  in  organize the  Information reflecting  it  the  organization  organization  By contrast,  of  ways  to  level  once  categories  with  teacher's  1977, p.  the  faced  such a way that a "general quality stimuli"  form  when  1977).  low  the  298)  (2)  (Arlin,  in  question #3,  to  p.  and  and the  of  takes  a person takes from  produced  case  1977,  teachers'  attributes  the  that information  information  finally  information:  information ill-defined  In  form  1(cat) + 2(cat  total number  of questions asked by the  teacher  65 A  higher  order  approach the  (Arlin,  finding.  In  1975-76,  the  represents the  #4.  (transformations  (5)  "general question" (Mackworth,  question  ill-defined  category  1977)  present  teacher's  and  study, ability  to  implications  1965) more  hence,  an  or  to  of  identify the  is  closely than  indicate  outcome  (6))  a  high  a lower  level  transformations  student's  defined  of  or  point-of-view  to  order  problem  implications  in the  face  of  problems.  Developmental Teaching Strategies: H o w w o u l d y o u proceed to help this student with the probiem(s) he/she is having? What strategies would you use? Why?  Researchers  have  suggested  ongoing  response from the  may  be  related  the  instructional  to  in  the  step  In  place,  point-of-view.  The  teacher's  developmental  analysis of  generating  the  and  need  framing  extent by  to  of  strategies  the  present  may  the to  represent  the  study,  the  with the  teacher  student's  point-of-view  in  the  further  response  1984)  strategies  teachers  that  of  that  process  given  the  which  indicate  finding  the  she identified  analysis  may  is  (Brooks,  elicit  step  strategies  problem  to  suggests as a next  problems  of  is able  questions"  a developmental discussion  no  the  teacher  thematic  continue  until  which  asking "thoughtful  in assisting students  first  acknowledges  the  student  kinds  process.  suggest as a next having  the  that  (Arlin,  them  student's reflect  the of  in  a  teacher  hypothesis  1985;  Brooks,  1984). Criteria The  Used  rater's  defined discussion  to  Rate  "Developmental  task was to  as  establish the  developmental  of  procedures.  teaching These  "developmental  perspective"  the  necessary for  techniques  Teaching  (Arlin,  number  of  strategies  strategies, 1985,  exploring  Strategies"  that  delineated  1986b; and  different  Elkind,  revealing  strategies which  emerged from  research  1976), their  in  provide  students'  each  could  teacher's  focusing  on  teachers own  be  views  a  with of  66 reality. the  These teachers  tend  minute-by-minute  point-of-view'"  to  "consider  interactions  curriculum  with  children  and in  instruction  particular  (Arlin, 1983, p. 103). The use of the following  next step in the  instructional  process once the student's  in  general,  from  the  and  'child's  strategies provides  point-of-view  the  is discovered:  Table 4: Developmental Teaching Strategies  1.  Hypothesis 1983,  Generation  1987). The teacher formulates  learner  is  in  educational 2.  The  asks  (Arlin,  1974, Lampert,  in  order  hypotheses  to  find  a  1984;  Schon,  about  where  the  starting  point  for  1986b; Erickson, to  "What  write  down  1987; Lampert, or  share  is . . . ? " (Arlin,. 1986, p.  in  19)  1984; Strauss, 1981). discussion "What  at the  beginning  they of  or  a lesson.  'concreteness'  object  or  and/or  (b)  between  principle  experience to  (a)  (Arlin, to  encourage  concepts and their continuum  (Arlin,  principle'  to  the  concept  include  development  memorization  show  the  students  to  build  what  up  a  a concept  repertoire  "looks  of  like"  connections  experience. 1986b).  careful  and to  (Arlin,  1985). The teacher uses a concrete, physical  students  Examples  rote  inhelder,  "on-the-spot"  process  students  about...?" or  during  4.  learning  &  intervention.  teacher  know  3.  the  Framing/Frameworks The  (Karmiloff-Smith  selection  promote  1986b,  important in the experience to  The  p.  teacher of  different  student 19).  helps  repertoire  the  examples  understanding  She  assist in their  extends  'concreteness to  rather  students building.  encourage than  notice  simple what  is  67  5. Questionnaire: Other Variables of Interest to the Study At  the  beginning  of  the  questionnaire  which  teaching  educational  insight  and into  the  "Why  addition,  a variety  the of  present the  Chapter  course,  background  practice, concern  study.  of  participants  information  (see Appendix for  problem  d o you teach?" (adapted  experience  repertoire  requested  teacher's  question,  inservice  demographic  from  variables  4  will  Cetzels  may  gleaned from  and  that  provide  an  indepth  research questions and findings.  the  discussion  areas order  presented related  to  &  were  the  a  children, additional asked  the In  information  to  have  of  results  within  as  interest  represents their  in present instructional  of  with  Csiksentimihalyi, 1976).  questionnaire  may  to  provide  participants  provide  participants  and which may influence their thinking  In  finding,  The information knowledge  B)  in  were  they  some  teaching  situations.  relate  the  CHAPTER IV.  This chapter will of  the  Arlin,  four  provide an exploratory  questions  1986). The qualitative  obtained tables  from  and  anticipated by  question,  "Why  variables  subject  may  do  with  for  additional  formulation  of  level.  to  an  rated  high  on  of  of  raw  the  frequency  the  group  analysis of  a variety to  the  of  the  similarities  of  the  child's  the  demographic  present  into  discussion will  &  data  responses and  a qualitative of  (Levitt  the  of  each  study.  all variables and a participant  concept  questions raised as a result  group  interest  insight  the  Anecdotes Task'  discussion  addition,  examination  further  have  a  overall  of  responses to  an examination  and  In  participants'  'Student  consist of  information  provide  teachers  the  pertaining  a participant  how  in  variable,  teach?" and  all variables may to  each  correlations  you  provide  respect  them  category/grade  comparison between on  to  discussion of  analysis will  participants  responses  low  posed  RESULTS  and  A  rated  differences  point-of-view.  The  be presented with  further elaboration in Chapter 5.  Teachers were  presented with  classroom scenario in which task  based  received  on  students  assumptions  instruction  accompanied  several anecdotal tasks and were  the  from  of  previous  their  teacher.  scenarios  was  participants in this study in order to teachers, available  the to  developmental  experience them  for  and  thinking  perspective.  provided  An  "incorrect"  teaching Each  designed  to  elicit their  actions about  and  the  teaching  examination  68  of  and of  responses to learning  the  four  conduct  a  points-of-view, cognitive and their  asked to  or  select one a curriculum after  having  questions  which  "conversation" that is, to  elicit  developmental  teaching  for  responses  from  processes  thinking may  with  from  provide  a an  69 interesting  insight  into  "incorrect"  response  the  to  research  what  h o w do teachers identify  they  the  believe  5 provides  questions  of  identification subject  the  ratings  the  'Student  number  from  category/grade  and  allotted  be  faced  with  a well-defined  the  present  labelled as follows: will  provide  ratings  oh  chapter,  to  the  to  #27.  from  27  participants  Task'.  Each  Participants  which  their  on  each  participant  are  grouped  scenarios  all  each  example  used  to  [see Appendix A, Scenario #00;  the . reader tasks  student's  instructional  were  is  task,  with  a  as well  relatively  as  the  efficient  classroom  method  scenario  four an  to  The  the eight  in Appendix A.  the  results  will  5, Subject #00]  of  the  the  assigned  selected.  describe  see Table  of  according  classroom scenarios of the 'Student Anecdotes Task' can be found  Within  a  Participants' Ratings  Anecdotes  #1  level  to  When  child's or adolescent's point-of-view?  Identification of Student Anecdotes Table  question:  locating  be This  participant's  participant  chose  for  completion.  Interrater Reliability A  random  using  selection of  the  agreement  categorical between  variable  being  reponse  was  strategies,  but  10 of  the  sets of  descriptions  the  coded.  two The  more  at  one  rater  outline  raters  of  Both  identified  in  ranged  number  issue.  responses were c o d e d by a second rater Chapter  from  66%  developmental  raters  additional  agreed  3. to  90%  strategies  on  strategies  The  the not  rater. The highest percent of agreement was on integrative  initial  percent  depending present  presence identified  by  on in  of  of the each  certain the  complexity and quality  first of  70 Table 5: Participant Ratings for Problem Formulation (PF), Integrative Complexity Quality of Point-of-View (QP) and Developmental Teaching Strategies (DT) by Subject Category/Grade Level Primary Math/Science (n = 7) Subject 2 3 4 5 6 7  1  PF (1-8) 1 2 1 2 2 6 1  •  IC (1-4)  QP (0-6)  DT (1-4)  - 1 1 1 1 1 3 1  1 2.7 1 2 1 5 1.4  nil 3 3 1,3 3 1,3 2,3,4,  Primary Hum/Social (n = 8)  PF  IC  QP  DT  Subject 8 9 10 11 12 13 14 15  1 2 2 1 2 8 1 2  1 1 2 1 1 • 4 1 2  0 1.5 2 1 2.5 4 1 1.2  nil 1,2,3 3 3,4 3 1,2,3,4 3 3  Interm. Math/Science (n = 9)  PF  IC  QP  DT  1 1 '2 1 1 6 4 1 1  1 1 1 1 1 3 2 1 1  1 1 1 1 1 6 3.5 •1 1  3 3 3 3 3 2,3 3 3 3  Interm. Hum/Social (n = 3)  PF  IC  QP  DT  Subject 26 27  1 2 1  1 2 1  1 1 1  nil 1 3  Subject 17 18 19 20 21 22 23 24  16  25  (IC),  71 point-of-view. one  Initial  category  removed  disagreements employed  disagreements  were  except  resolved  and the  nature  with for  the  through  of  the  respect  to  problem  responses  discussion.  responses, the  of  formulation  one  Given  subject.  the  inter-rater  All  types  reliability  were  of  of  only the  categories  was more  than  acceptable.  A.  QUALITATIVE  ANALYSIS  The results will be presented and discussed in the following 1.  Problem Formulation  2.  Integrative  Complexity  3.  Quality of  Point-of-view  4.  Developmental Teaching Strategies  In addition, the variables will  teacher's concern for  be  examined as other  problem finding variables of  four sections:  and a variety  interest  to  the  of  demographic  present  study  (see  Appendix F3).  1. Problem Formulation The  first  extent  question  the  teacher  ill-defined  problem  is having  with  to  posed  these  by  (1964)  Getzels  differentiation  situation:  the  will eight in  between  the  "What  curriculum  questions of  teachers  identifies  this  modifications  to  kind(s)  examined  types  of  of do in  problems  description  presented  designed to  classroom  task? Why  be  his  was  or  of  examine  problem  as  problem(s) you  think  terms and  do  and to  well-defined  you  think  this  what or  an  student  so?" Participants' responses of  their  processes of  problem  well-defined  a  how  concordance solution  formulation,  problems  and  that  with  delineated is,  discovered  the or  72 ill-defined problems and their corresponding processes of solution.  Responses  were  nature  the  of  student  and  formulation,  rated  according  student's herself.  to  problem  Table  6  the  and  provides  degree  its  solution a  *Highest to  identified  as  distribution  the  known of  of  (level the  teacher or  the  8 7 6 5 4 3 2 1  01 00 02 00 01 00 09 14  03.7 00.0 07.4 00.0 03.7 00.0 33.3 51.9  student's  Of  by  the the  problem  formulation)  Problem Formulation (PF)  Percent  1).  for  problem  Frequency (n = 27)  responses  viewed  unknown  ratings  PF Type *  Lowest Levels of  majority  formulation  which  (see Chapter 3 for a description of the types of  Tabie 6: Frequency Table for  The  to  Problem Formulation  were  the  problem  total  concentrated number  as a type  of  at  the  lowest  participants  level  (n = 27),  1 problem. The following  of 14  problem or  51.9%  is an example  of a level 1 response:  They [students] don't know the function of numerators and denominators. They don't see the necessity for including fractions to be added (or subtracted) in the same class. They also have not demonstrated that they are evaluating their answers for reasonableness by using estimating or ranging (forecasting that the answer is known to fall between two extremes and those extremes only), [see Appendix A, Scenario #6; see Table 5, Subject #19]  73 Responses teacher  views  students' is  rated the  to  problems tend the to  of  reflect  intimate  the  of  formulation  as well-defined, teacher  and  students  (1964),  the  a "minimum  of  scale  that  is,  therefore  once  innovation  or  identify the  method  problems  the  to  solve  of  student's  types  problem  2  through  8  and  solve of  problems  indicated that they had identified method  took  the  solution  has  creativeness". As evident  in  problem  these  taken of  or  that  the  as requiring  (Mackworth,  1965).  teachers gave level 2 responses. Nine, or  instruction  of  formulation  tend  or use the  teacher  further  views  the  discovery and  8 represent increasing  and subsequent - problem solving; the processes required  ill-defined  task as well as the  the  arrive at the correct answer.  suggest  as ill-defined,  that  types  are rated at level 1 of  of solving the problem to  problem finding  majority  further  nature  instruction  elaboration (Getzels & Csiksentimihalyi, 1976). Types 2 through levels of  the  a ready-made  additional  processes used  suggest  that their students " n e e d only plug i n " additional information  contrast,  nature  problem  by the  and  Cetzels  the  problems  known  teacher  to  to  1  above example, teachers w h o  teacher's own  By  are  the  According  level students'  problems  available  place.  at  the  the student. The following  solve the problem, but  method  of  solution  levels  33.3% of  problem that the student  required to  place, the  Of  would  2 the  through  8,  participants  was having with  the  suggested that  until  remain  unknown  is an example of a level 2 response:  From my own experience in 3rd grade "difficult" at this age (but less so than relationships; differences between states, be beyond even the 3rd grader--I think that, [see Appendix A, Scenario #4; see  to  I know that spatial concepts are time, I think). Directionality; size countries and continents can all they're pretty impossible before Table 5, Subject #15]  to  74 These responses may reflect a  teacher  to  begin  to  the  lowest  identify  a  level of  creative thought  student's  problem  processes used by  from  the  student's  own  point-of-view.  Teachers  in  the  present  study  did  not  provide  one teacher responded at level 4 of problem  responses at  levels  3 or  5.  Only  formulation:  They [the students] are dealing with a new situation in the abstract notational form apparently without prior experience in exploring the situation in the concrete situation, [see Appendix A, Scenario #6; see Table 5, Subject #22]  The  remaining  indicates thought the  that  responses were a minority  of  at  teachers  processes associated with  students'  problems  as  levels  6  (2  provided  problem  ill-defined.  teachers)  8  responses which  finding  The  and  is  an  teacher).  reflect  at levels high  following  (1  the  enough  example  of  to a  This  use  of  identify level  6  response:  The students appear to presented. Answers given lesson with perhaps no really lack the experience #7; see Table 5, Subject  From  this  response it  can  are having a problem the  students'  the  introduction  students'  problems  be  inferred  understanding  problems of  lack the conceptual understanding of material are partially correct but after only a brief concrete examples demonstrated, the students to grasp this topic, [see Appendix A, Scenario #21]  remains to  concrete will  become  that the  certain be  known  concepts, but  identified.  examples,  a to  teacher  that the  However, once  standard both  is aware that  the  of  student  and  7.  students  actual nature  discovered  method  There were no responses which reflected a problem of type  the  solution to  the  of  through to  the  teacher.  75 According to Getzels (1964), type 8 problems can be described as the highest of  discovery. The  Since,  ill-defined  (Mackworth,  use of  this  problems  1965),  and  problem  provide a  creativity  has been  teacher  response  reflects  cognitive with  identify  and  ill-defined  tasks  interpret  problem  from  a  stimulus  for  between  level  of  identifying  "incorrect"  that  is,  as  processes of  problem  finding.  processes  of  problem  finding  finding  and  problem the  literature  problem  the  developmental  students'  situations,  requires  established in  this  processes necessary for  instructional  the  relationship  associated with whose  type  nature  of  responses  and elaboration. Only one teacher provided a type  (Arlin,  formulation her  perspective.  problems  form  which  1975-76), may  have  students'  She  to  variables  may  the  problems  be  instructional require  the  further  able  to  tasks  as  discovery  8 response:  The questioner is asking her [Roberta] to give her information she cannot answer from the paragraph. He/she is asking Roberta questions that require prior knowledge inappropriate for a 7 year old to be expected to have, [see Appendix A, Scenario #3; see Table 5, Subject #13]  This teacher's response reflects with the evident  her awareness that the  student  instructional task. However, no immediate attempt to in  contributing  her  response. Furthermore,  to  the  problem,  and  the  teacher  indicates  that  sees the  the  is having a  provide remediation is instructional  methods  process as  used  were  matched to this particular student's developmental level or point-of-view. seems to identify  leave the situation  students'  situations,  may  be  "incorrect" using  in her own classroom.  open to  processes of  instructional  problem  finding  tasks  as  in these  is able  ill-defined types  not  This teacher  further discovery. The teacher who  responses to  problem  of  to  problem situations  76 Frequency  of  Problem  Table  7 provides  each  teacher's  category/grade subject  adequate them  level  use  most  and  for of  by Subject  a distribution selection  category  opportunities  Formulation  the  ratings  of  the  familiar  to  or  of  students  grade  teachers the  of  level to  (Piaget, 1972; Sinnott,  of  problem  anecdote  and  which  taught  respond  experiences  Category/Grade  to  by  the  actions,  and  formulation represented  teacher.  was viewed situations  Level  so  cognitive  1978; Sinnott & Cuttmann,  the  Familiarity  as likely as  according  to  to  to  subject with  the  increase  the  reveal  the  most  processes available  to  1978).  Table 7: Frequency Table for Problem Formulation (PF) by Subject Category/Grade Level  PF Type *  Prim. Ma/Sc (n = 7)  Prim. Hum/Soc (n = 8)  Int. Ma/Sc (n = 9)  8  --  1(2.5)**  --  1(14.3) —  —  —  --  --  1(11.1) — 1(11.1)  3(42.8) 3(42.8)  4(57.1) 3(37.5)  1(11.1) 6(66.7)  1(33.3) 2(66.6)  7/  6 5 4 o J 2  1  Int. Hum/Soc (n = 3)  •  --  --  * Highest to Lowest Levels of Problem Formulation ' * N u m b e r of responses(percentage of n)  As  indicated  all subject the  subject  7, the  category/grade  teachers  when  in Table  in  the  attempting to  majority  level  present  groups.  study  of  Although  level  these  use a minimum  identify the student's  category/grade  responses are rated  groups  of  results creative  at  levels 1 and  suggest that thinking  or  most  lntermediate(Humanities/Social  in of  discovery  problem, some teachers from each of  (except  2  the  Studies))  77 engage  in  processes associated with  and highest primarily  (level  high  8) levels of problem  or low on problem  level seem to contribute in ill-defined problem  2.  problem  finding  formulation.  formulation.  at both  N o one group  Neither  to the ways in which  the lowest  subject  tends  category  teachers identify  (level 2) to score nor grade  students'  problems  situations.  Integrative Complexity  The  second question  student's  posed to teachers  point-of-view  him/herself responses  while to  working  this  modifications  was, "What  will  of the levels of  answers them  to to  research  the  on  questions  fOr solution  (Arlin,  the  integrative  a  task?  Why in  do  h o w they this  you  terms  complexity  of  student  think  their  (conceptual  identify  the  was asking  s o ? " Teachers'  concordance  structure)  with  delineated  (1967).  developmental  they  to explore  d o you think  be examined  by Schroeder, Driver and Steufert  According  question(s)  through  question  in order  perspective,  ask themselves  1983; Sinclair &  about  Kamii,  students  the  1970).  It  provide  problems  correct  presented  to  is the teacher's job to  identify  the problems and questions students have with instructional tasks. She needs  to  able  be  thinking  to  about  "correct" distribution  formulate  on-the-spot  about  something, that is, h o w the student's  answer for the question(s) of  hypotheses  the  (conceptual levels).  ratings  for  the student the  teacher's  the  way the  "incorrect"  student  is  answer may be the  asks him/herself. Table 8 provides a levels  of  integrative  complexity  78 Table 8: Frequency Table for  Integrative Complexity  (IC)  IC Levels*  Frequency (n = 27)  Percent  4 3 2 1  01 03 03 20  03.7 11.1 11.1 74.1  *4 = High Integrative Complexity; 3 = Moderately High Integrative 2 = Moderately Low Integrative Complexity; 1 = Low Integrative Complexity  The  large  (level to  1).  majority  Of  been  the  responses  From these  generate  have  of  27  integrative  responses it  alternative asking  participants  in The  about  as they  the  the  lowest  can be inferred  explanations  themselves  complexity.  reflect  study,  following  the  were 20 is  of  integrative  complexity  that these teachers do  kinds  of  working  or an  level  Complexity;  questions  through  the  74.1% responded example  of  a  at  the  not  tend  students  may  instructional this  response  low  task.  level  rated  at  of  scale  value 1:  I think some [students] didn't ask themselves anything; they said the first thing that came to mind. I feel this way because many answers ' seem rote while others don't make sense, [see Appendix A, Scenario #5; see Table 5, Subject #17]  These  teachers  responses. own  in  the  information  Instead,  teachers  to  use lower  answers  understanding  thinking this  Their  tend  of  same  the  do  not  reflect  problem  situation.  levels  They  level  do  conceptual  an  situation  in such a way as to providing  of  ability  processes to  to  take  as it differs  not  take the  from  discriminate, students'  1 responses tend  to  into how  generate  their  consideration  their  the  students  differentiate  points-of-view  attribute  are  or  integrate  into  account.  students'  "incorrect"  79 answers  to  through  weaknesses in  the  "incorrect" present  provision  of  students  which  appropriate  are thought  information.  may  about  not  how  have their  the  necessary  students  cognitive  construct  to  They  responses as " w r o n g " . This suggests that the  study  alternatives  the  their  be  quickly  interpret  majority  the  of  students'  teachers in  processes  own  remedied  for  knowlege  the  formulating  about  certain  curriculum tasks.  Scale  value  inferred some were the  2  from  reflects this  predictions thinking same  a  moderately  level about  of  teacher  the  similarities  while working  task.  Of  low  the  through  27  response  the  participants,  rated at this moderately low level of  level  and  of that  conceptual  structure.  they  the  differences  have  between  It  ability  how  the  can to  make  students  task and how they themselves think only  3  or  11.1% of  the  be  about  responses  were  conceptual structure. An example is:  What shape do 1 give the world? What goes in the middle? What's next to it (or me)? H o w does everything "fit"? [see Appendix A, Scenario #4; see Table 5, Subject #15]  A  moderately high  level of  responses. An example of  conceptual structure  was reflected in 3 or  11.1% of  the  a level 3 response is:  I think the students were simply asking themselves how many wheels they had because they were asked to count them. I do not believe the children understood that the numeral 16 could be broken up into tens and ones, [see Appendix A, Scenario #1; see Table 5, Subject #6]  This  teacher  the  concept  students of  the  appears of  place  are thinking student's  to  work  question value.  to on  further  whether  She  leaves  exploration  her resulting  or  not  her  the  initial  students hypothesis  and discovery and  prediction  that,  "...the  actually  understand  about  how  the  bases further analysis students  were  simply  80 asking themselves how  many wheels  off"  keeps the  the  problem,  formulating  had...".  problem  Rather than  situation  open  attempting  for  further  alternative conceptions about h o w the students might  while working students'  she  they  through  the  interpretations  task. She integrates  conflict  with  her  and indicates that there is a need to  own  her own  have been  of  information  "ward  discovery  understanding of  interpretations  collect further  to  the  task  about the  by  thinking how at  the hand  students'  problems.  The  one  (level  4),  group. that  participant,  whose response reflected a high  chose to  This  she  complete  teacher  had  of  taught  15  a scenario from  years,  revealed  English as well  students in grades 3 to  as  6. The following  in  level of  integrative  complexity  the  Primary(Humanities/Social Studies)  her  questionnaire  programs  for  the  (see  gifted  Appendix  B),  talented  to  and  is her response to question #2:  I think Roberta is trying very hard to draw on prior knowledge and she is trying to concretize her answers to the question and draw on her experiences, [see Appendix A, Scenario #3; see Table 5, Subject #13]  It  can  be  description working one  inferred about  through  which  through  the  she  response that thinks  the  teacher of  the  this  student  response itself the  freedom  discrimination,  developmental framework  requires formal  reasoning ability;  *  teacher might  can to  be  is  providing  have  been  a  theoretical  thinking  alternative  and  explanations  integration,  about the student's own  (Piaget,  while  described as an hypothesis;  formulate  differentiation  hypotheses from this information  a cognitive  hypotheses  this  task. The  processes  generate further  Within  how  allows  the  from  1972), the  a necessary but  not  ability  and  to  thinking.  to  formulate  sufficient  condition  81 for  problem  finding  in which the teacher ability)  (Arlin,  1975-76, 1977, 1986). Therefore, in addition  teacher organizes information  abstracts  or  selects  may contribute,  the student's point-of-view  to  from  these  alternatives  high  situations  level  of  problem the  It  is  a high (Arlin,  interesting  formulation  level  of  1975-76,  integration 1977,  (Arlin,  1986)  to  note  (see Table  that  5). The  she may also have the high levels of the student's point-of-view  Frequency  Integrative  Ratings the  generate the  teacher  is  1984).  of  majority  of  alternative  a  might  teachers  identify  participating or  have  thinking  been of  the  in  formulate  levels  addition, a the  problem  problem finding (Arlin,  if  scored high  the  teacher  processes of  on  identifies  problem finding,  and abstraction required to  Category/Grade  all subject  explanations  distribution  reasoning  identify  in her own classroom.  by Subject  teachers' responses from  students  provides  Complexity  if  (Subject #13)  the  In  required  suggestion is that  integration  and interpret  that  this  student's problem as ill-defined, thus initiating  of  the  hypotheses in ill-defined  finding is to result in the 'general question', the outcome of 1975-76).  way  in an ill-defined problem situation.  requires  integration  (formal  the way in which the teacher begins to  Within a problem finding framework, the ability to formulate problem  the  (conceptual level), the way in which  information  in part,  to  of  categories and the  present  hypotheses  while  Level  working  integrative  category and grade level selected by each teacher.  grade  study in  did  order  through  to  the  complexity  by  levels indicate not  tend  predict task. the  to how  Table  9  subject  82 The  distribution  category/grade Neither  of  teachers  level  subject  has  nor  within  a  each  significant  grade seem to  formulates alternative  group  effect  contribute  explanations about the  does  not  suggest  level  of  integrative  on to  the  extent  to  that  subject  complexity.  which  the  teacher  questions students ask themselves about  particular curriculum tasks.  Table 9: Frequency Table for Integrative Complexity (IC) by Subject Category/Grade Level  IC Type*  Prim. Ma/Sc (n = 7)  Prim. Hum/Soc (n = 8)  Int. Ma/Sc (n = 9)  Int. Hum/Soc <n = 3)  4 3 2 1  —1(14.3) —6(85.7)  1(12.5)** —2(25.0) 5(62.5)  1(11.1) 1(22.2) 7(77.7)  1(33.3) 2(66.6)  *4 = High Integrative Complexity; 3 = Moderately High 2 = Moderately Low Integrative Complexity; 1 = Low Integrative * * Number of responses(percentage of n)  3.  Quality of  In  order  to  teacher task?  provide  ask the student(s)  on-going  1984)  Brooks, has  responses  exploration  when  about  responses would  1983;  (Brooks,  direct  participants  question was posed to (Arlin,  a  point-of-view  problems,  What  Complexity;  Point-of-View  child's/adolescent's ill-defined  Integrative Complexity  faced  were the  you  asked,  been which  in  expect  which  discussed as reveal  with  how  teachers  classroom  "What  the  to  get?  on the the an  Why  essential  of  are the  they are having with  the  you  you,  the  as  do  would  which  think  s o ? " This  developmental perspective  formulation  students'  identify  problems  question(s)  problem(s) you think  teachers drawing 1984)  of  "thoughtful  instructional  points-of-view.  literature  questions",  tool  for  eliciting  The  purpose  of  83 eliciting these responses is to refute  the  hypotheses  they  provide a means by which the teachers can verify  formulate  about  the  nature  of  the  problems  they  or  think  students are having with particular tasks.  Research suggests that the tend  to  approach what  calls,  the  1977,  1986), the  from  many  level of the  'general  to  problem, extent  "mainline"  ill-defined  problem  (Arlin,  (1965),  Within  a  in  his  problem  processes required  problems  include,  a  which  these  situations  information finding  to  formulate  1975-76; Schroeder, Driver &  high  level  of  formal  processes are  may influence  the  what  extent  to  many  Steufert,  questions  and  for  a  In  high  addition,  " s o l v i n g " the  (Arlin,  teacher for  which  1975-76,  general  1967).  thought  framework  (Arlin,  is appropriate  the  situations  processing  discrimination  operational  available to  problem  framework  differerentiation,  select from available information,  requires  to  questions posed in ill-defined  Mackworth  question'.  integration  ability  kinds of  1984).  use in  she is able to  The  ill-defined  formulate  the  'general question'.  The  kinds  of  questions  examined in terms intellectual  of  products  descriptions  of  teachers  their  in  ill-defined  as  finding.  delineated  Each  by  question  Arlin  posed  according to the  six categories (see Chapter 2) and the  by  was  was  each  teacher  calculated  intellectual participant  by  products  problem  concordance with a modification  categories  problem  ask  tallied.  obtaining category  The the  quality  of  "weighted  over  the  the  total  by  of  number  (Arlin, 1975-76). This formula is illustrated  the  1977)  teacher  number of  the of  will  be  Guilford's (1956)  (1975-76,  questions  average"  of  situations  posed  in  was  six her  rated  questions asked by  each  questions questions  teacher  according asked  in Figure 3 below:  by  to the  84  Figure 3: Formula for Calculating Question  1(cat)  +  2(cat)  +  3(cat)  total number  The  following  table  represent quality  provides  of  a  +  of  4(cat)  +  distribution  of  the  01 01 01 01 04 18 01  03.7 03.7 03.7 03.7 14.8 66.6 03.7  were  reflect  low  10, the units  of  a  4 = systems;  majority  questions.  levels  integration)  example  questions  which  Point-of-View (QP)  6 5 4 3 2 1 0  problems  following  of  Percent  Table  discrimination,  of  Frequency (n = 27)  As  which  categories  QP Category*  5 = transformations; posed  their  6(cat)  point-of-view:  *6 = implications; 0 = no questions  in  +  questions asked  Table 10: Frequency Table for Quality  indicated  5(cat)  Quality  of  and category  of  18 or  3 = relations;  questions 66.6% of  both  information  abstraction  (formal  one  (units)  teachers the  2 = classes;  asked students  teachers  posed  organization reasoning)  response  1=units;  questions  (differentiation,  (Arlin,  suggests  about  that  1977). the  The  teacher  85 thinks  the  information  student  is  having  required to  a  problem  provide the  with  "correct"  the  task  due  to  a  lack  attempt  to  identify  the  student's  point-of-view  teacher's response. He merely indicates, through lack  knowledge  of  fractions. The  on a literal interpretation  Four or  14.8% of  can be inferred slow  learning  the  from  and  of the  information  can  direct  he  be  the  Don't  inferred  from  observation, that the  elicits  from  the  students  this  students is based  problem.  teachers responded with questions in the their  basic  answer:  Back up. What do [you] use fractions for? - [Student response] know, [see Appendix A, Scenario #6; see Table 5, Subject #16]  No  of  classes category.  responses that they attribute their students' problem(s)  need  for  specific  remedial  assistance. The  following  is  It to an  example of a Category 2 response:  I would ask her [Roberta] to tell me what a town is, what a state is and what a country is. I would expect to get answers like - "Where you live". " A place to live, but different from where I live." "Where different people live". I feel she would give these answers because of the responses she's given to specific town, state, and country questions, [see Appendix A, Scenario #3; see Table 5, Subject #10]  Only one of  the  relations  category  between  the  27 participants (3).  students'  This  in the  teacher  responses to  present study  suggests the  task  that and  asked questions reflecting there  how  the  may  be  student  a  the  relationship  understands  concepts relevant to the task:  The Problem Solving strategy to be focused o n : What do you know how to do with fractions? What is different about the new situation? H o w can you change the unfamiliar to the familiar? [see Appendix A, Scenario #6; see Table 5, Subject #22]  the  86 O n e participant  posed questions reflecting  the  systems category  (4):  I wouldn't have asked her any of those questions [past] the 3rd one. I would have proceeded differently. I would have asked: Is Alabama in the United States? What is the United States? H o w is Wisconsin like Alabama? H o w is the place Caddie lives like the place you live? H o w is it different? D o you know the name that is given to the different parts of the United States where people live like Alabama, Wisconsin, Calif., N.Y., etc.? D o you know the name given to places where people live like Birmingham? I would never have introduced "country" and " t o w n " in my questions. I think it confused her. I would have asked further questions about how countries are alike and different. O n c e introduced, I might have - asked her what the difference between a town, and country were, [see Appendix A, Scenario #3; see Table 5, Subject #13]  This  teacher  conducts  organizing the task.  The  One  point-of-view.  question'  analysis which  available to  the  teacher  in order to verify  teacher  integration  information  information  questioning  a task  posed The  required is evident  processes of if  the  that student collects  or refute  questions  reveals  which  of  for  may  she  thinks  generating  indicate  the  the  student  a response to need  for  her hypotheses about the student's  reflects  discrimination,  process  how  problem  in her response (Arlin,  the  ability  to  differentiation finding  is to  identify and  the  result  in  is the  further problem.  the  student's  high  level  the  of  'general  1975-76):  It depends greatly upon what has been previously introduced to the children. If this is a new topic of study I don't believe I .would ask many questions except maybe to spark interest in the upcoming subject of place value. I might ask, " W h y didn't you circle all of the wheels?", "If there are 16 wheels and I ask you to circle both numerals in this number, why haven't all of them been circled?" I would expect them to explain to me what they know about the values of J _ and _6. Then this could lead to a discussion of J _ standing for 1 group instead of _1_ object, [see Appendix A, Scenario #1; see Table 5, Subject #6]  87 This  teacher  goes  integrating  the  and  reflect  then  from  the  beyond  hypothesis on  stating into  this  students'  the  question  in  the  form  of  an  her teaching. She may think about  thinking  while  points-of-view.  This  formulating teacher  questions  may  teach  hypothesis  her own  to  elicit  from  a  to  thinking responses  developmental  perspective.  At  the  highest  following  level  of  question  quality,  implications  (6),  one  teacher  posed  the  question:  First 1 would ask why they [students] think the answer they gave is correct. From their frame of reference, I'm sure their answer makes sense to them; their answer merely mirrors their reality of the matter, [see Appendix A, Scenario #7; see Table 5, Subject #21]  Research on problem finding a high  level of  information apparent  in in  processes  (Arlin, 1977)  organization, that is, the such a way  the  stimuli"  available  to  that (p.  her  ask  298).  for  (category  reflect  higher  levels  1977,  1986;  see  the  information problem questions  situations. from  of  other  reflecting from  Table  to  be  The  many  5)  or  abstraction  ability to  tends  In  quality  on  5,  her  than  formulate  #6  and  questions organize  that  teacher  theories  may of  according to  to  suggestion  is  the that  problems"  use  incoming readily  have  the  teaching  and  the situation  at  (Arlin, 1975-76) that, the ability  to  (category  to  questions #21).  of  teachers may  require  is not  6) in  questions  other  Within  a  be  abstract  tends  categories problem  hypotheses based on the selection of  related  ill-defined  asking  this  own  these reflections  implications  Subjects  is abstracted  words,  is further enhanced by findings  transformations  framework,  teacher may be able to  a "general  learning and selecting information hand. This notion  indicates that implications  thought  finding  appropriate  in  ill-defined  who  generate  "many  facile  in  use  their  (Arlin,  of  general formal  88 operational thought  (Arlin,  #21  ratings  receive  high  point-of-view.  Teachers  about  doing  something  doing  it  available  (Schon, to  them  on  who  1983, for  1975-76).  It  is interesting  integrative  complexity  think  about  their  (Lampert,  1984)  and  1987)  may  have  "reflecting-in-action"  own think  the on  of  Point-of-View  by Subject  their  Category/Grade  The following table provides a distribution  note as  that  well  thinking about  own  Subjects #6  as  on  (Piaget,  doing  cognitive  learning. They may have the ability to teach from  Quality  to  quality  of  1972),  think  something  while  developmental theories  and  of  processes  teaching  and  a developmental perspective.  Level  of the quality of  questions teachers asked  by subject category and grade level:  Table 11: Frequency Table for Quality of Point-of-View (QP) by Subject Category/Grade Level  QP Type*  Prim. Ma/Sc (n = 7)  Prim. Hum/Soc (n = 8)  Int. Ma/Sc (n = 9)  6 5 4 3 2 1 0  —1(14.3) ——2(28.6) 4(57.1) —-  —-  1(11.1)  1(12.5) —2(25.0) 4(57.1) 1(12.5)  *6 = implications; 5 = transformations; 0 = no questions posed  The  majority  as units  of  responses within  questions. N o one group  1(11.1) 7(77.7)  4 = systems;  each subject appears to  Int. Hum/Soc (n = 3)  3 = relations;  category  and  3(100.0)  2 = classes;  grade  1=units;  level were  have a significant number  of  rated  high  or  89 low  ratings.  contribute  From  this  the  extent  to  from the student's  The  question  observation, to  neither  which  subject  teachers  ask  nor  questions  grade  level  seem  to  which  elicit  responses  point-of-view.  quality  score  for  the  entire  sample  has  been  calculated  below  (see  Figure 3 for the Question Quality Formula):  1(49)  +  2(15)  +  3(5)  +  4(8)  +  5(0)  +  6(1)  78 (total number of questions asked)  The question quality score for the  In  the  present  teachers notion may  study,  participating that  the  an overall in the  students'  quickly  remediated  problem  finding  framework  of  teachers  many  ill-defined  Their  responses  necessary  for  in  do  curriculum  task.  appropriate  for  the  teachers  the  present  in  once  the  (Arlin,  reflect  do to the  alternatives  Hypotheses situation study  not  may not  to  to  not elicit  1.48  score  a lack  1977,  of  1986),  generate ongoing  selection  this  reflect  the  suggests  general  students  (Arlin,  the  which  is provided. Within  differentiation  the  that  information  responses from  the  formulated  indicates  basic  many  how  have the  1.48  information  discrimination,  on  of  pose questions which  appropriate  about  based are  due  1975-76,  study  =  quality  study tend are  in order  not  generating  present  this  problems  question  problems  be  majority  27 participants  of  are  that  questions their and  other  processes associated with  from  integration  thinking  In  the  students.  information  1977).  a  about  a  deemed words, problem  90 finding  and  problem 1983,  subsequent  situations.  1987).  They  formulation',  necessary  but  and  may  1987)  not on  therefore  sufficient own  to  "reflect-in-action"  theories  for  of  thinking  and  the  in  the  for  use  practice  'quality  of  teacher to  teaching,  (Arlin,  them  in  ill-defined  setting  discovery associated with high  complexity'  conditions  teach for  available  processes of  'integrative  her  to  solving  not  In this view, the  'problem  1983,  problem  learning,  1985) from  (Schon, levels of  point-of-view'  may  "reflect-in-action" instruction  and  a developmental  The developmental teaching strategies that teachers use to  be  (Schon,  education, perspective.  assist students  with  their  would  you  problems may provide further insight into this question.  4. Developmental Teaching Strategies Teachers proceed What will  were to  help  strategies be  asked  to  provide  this  student  would  you  examined  in  with use?  terms  of  responses the  the  problem(s)  Why?" their  to  Teachers'  concordance  question,  that you  "How  think  responses to with  the  he/she is having? this  final  question  developmental  teaching  strategies delinated by Arlin (1983, 1986b).  One  of  the  teaching for Piagetian  key  thinking  and of  student  thinking  Teaching  in  (Arlin,  instructional about  strategies which  a constructivist  1985, 1986b)  neo-Piagetian  knowledge is  elements  research strategies  a  particular  are based on  the beginning of teaching from  view  involves  for  of  thinking  knowledge  educational  which concept  provide  teaching  the  implications  of  practice an  (Arlin,  point-of-view:  insight  1983,  a 'developmental  the child's  about  and of  (Arlin,  1983)  and  into  "how"  the  1986b;  Brooks,  perspective' represent  1984). only  91 They represent first steps in coming to understand the thinking demands of tasks in the light of the logic of the students. The use of these strategies depends on teachers thinking about teaching and teaching for thinking. The use of these strategies involves an awareness of the types of thinking which many curriculum tasks require. (Arlin, 1986b, p. 20)  Developmental providing  a  problems  she  may in  means  influence the  teaching by  identifies the  classroom.  strategies which her  the  12  for  teacher  students  way in which Table  account  can  as having  the test  with  her  provides  a  distribution  the  Percent  1 2 3 4 Nil  05 04 23 03 03  18.5 14.8 85.2 11.1 11.1  2 = framing; 3 = concreteness principle; teaching strategy not stated (See Chapter  entire  that  objects  of  assisting task.  experiences alone:  concrete them  Twelve  with  teachers  the  tasks. They problems  of  different  procedures:  Frequency (n = 27)  provision  about  instructional number  by  Developmental Teaching Strategies (DT)  According to Table 12, of the  instructional  theories  DT Strategy*  * 1 = hypothesis generation; continuum; 6 = developmental for a full description)  for  of  point-of-view  instructional  she assists her students with  Table 12: Frequency Table for  strategy  own  certain  strategies that emerge in each teacher's discussion of  the  student's  sample of  the  or  27 teachers, 23 or  experiences to problem(s)  suggested  the  use  they of  85.2%  students  was  were  having  concrete  4 = examples 3, Table 4,  indicated  an  effective with  materials  the and/or  92 [Concreteness Principle, Subject #10]: I would show her a map of the United States and explain what towns, states, and a country is. She at 7, w o u l d most likely not completely understand all the concepts involved. I would use a paper wall map, and a puzzle map to help her understand states and their relationship to each other. These would also help to show Roberta country and town relationships. Children need a variety of approaches and by being able to look at, and feel with her hands, this would give her the variety she needs, [see Appendix A, Scenario #3; see Table 5]  In  only four  instances was the  concreteness' principle  not  included in the  teacher's  description: Three  teachers  did  not  mention  at  least  one  of  the  developmental  strategies  in  their responses:  [Subject #8]:  I don't know,  [see Appendix A, Scenario #4;  see Table  5]  [Subject #1]: Take the 16 wheels and break them up into two groups 10 and 6. Try to explain why the 1 is worth 10 when you have the total. O r try 10 + 1, 10 + 2, etc. [see Appendix A, Scenario #1, see Table 5]  [Subject #25]: It would depend on the grade level, ability level, and interest and attitude level of the students involved, [see Appendix A , Scenario #8; see Table 5]  Only  one  teacher,  suggest the  who  use of the  mentioned  the  strategy  'concreteness principle'  of  'hypothesis  generation',  did  not  in his response:  [Subject #26]: It would appear that I could begin to evaluate strategies used to create the confusion and begin again, [see Appendix A, Scenario #8; see Table 5]  The  remaining  continuum', 18.5%  of  strategies  were the  of  'hypothesis  much less frequently  teachers'  generation',  mentioned.  responses indicated  Of  that they  'framing' the  and  the  27 participants,  would  formulate  'examples only  5 or  "on-the-spot"  93 hypotheses  about  assistance.  Only  the 4  way the  or  14.8% of  conversation with the about  the  student's  that they would  student  about  the  task  thinking.  And,  only  3  use different  of  point-of-view  variables tended responses identify  did  how  unlikely  to  or  not the  reflect  the  students  that these  the  use of  were  teachers would  concrete  combination principle'  of  and/or  participants  encourage concept majority  of  problem  or  their  own  strategies. The following  complexity low  and  on  of  it  all  Their  or discovery  Therefore,  theories  a 'developmental  tended  well-defined.  finding  formulate  experiences  development.  scored consistently as known  a  indicated  participants  integrative  12 reveals that teachers tended  manipulatives  different  the  problems.  of  of  information  their  Further  Table  further  about  on these theories from  examination  conduct  of  formulation,  providing  would  18.5%  processes of  while  they  collect  problems  thinking  that  task  to  5). Teachers who  students'  the  order  observations that the  learning and reflect  use  in  examples in order to  (see Table  identify  about  indicated  receive consistently low ratings on problem  quality  be  teachers  student  These results may be related to to  the  was thinking  to  would  teaching  and  perspective'.  to  exclusively  either or  examples illustrate  mention  as  one  of  the a  the 'concreteness  as it is combined with one or more of the remaining three  strategies:  [Concreteness Principle, Framing, Hypothesis Generation, Subject #9]; Roberta's responses to my questions would tell me which direction to go in next. I would continue with questions until I narrowed down enough facts to satisfy my suspicions as already stated. Continued dialogue is the most important strategy l could use...That would be another place to start with her; to point out the similarities we all have with each other. Map work is in order... [see Appendix A, Scenario #3; see Table 5]  [Concreteness Principle, Examples Continuum, Framing; Subject #7]: I believe this student could be helped by going back to the initial steps in developing an understanding of the place value system. I would start  94 with tens and ones and would progress through the developmental stages and use manipulations, then pictorial and finally symbolic representation of the numbers 0-19. 1 would question the student's understanding of the null or empty set and instruction on this concept if necessary...  Teachers other  who  strategies  3 variables than teachers who  Distribution The  combined  of  following  subject  who  not  Teaching  table  a distribution  provides  level. The  mentioned  one  to  receive  higher  ratings  on  the  Strategies  number or  tend  mentioned the 'concreteness principle' alone.  Developmental  category/grade  category  did  of  of  more  of  developmental  teachers within the  teaching  strategies  each subject/grade  strategies  in  their  by level  discussion  of  procedures is documented.  Table 13: Frequency Table for Developmental Teaching Strategies (DT) by Subject Category/Grade Level  Strategy*  Prim. Ma/Sc (n = 7)  Prim. Hum/Soc (n = 8)  Int. Ma/Sc (n = 9)  Int. Hum/Soc (n = 3)  1 2 3 4 nil  2(28.6)** 1(14.3) 6(85.7) 1(14.3) 1(14.3)  2(25.0) 2(25.0) 7(87.5) 2(25.0) 1(12.5)  —1(11.1) 9(100.0)  1(33.3) 1(33.3)  —-  1(33.3)  *1 = hypothesis testing; 2 = frames; 3 = concreteness (See Chapter 3, Table 4 for a full description) **Responses(Percentage by n)  Although  there  level on the to  note  does not  appear to  principle;  be a significant  effect  4 = examples  of  subject  number of developmental teaching strategies mentioned, it  that  teachers  in  the  intermediate  groups  tended  to  continuum  category/grade is  interesting  mention  the  95 'concreteness'  principle  subject  and/or the  A  matter  comparison  or  and  moderately  complexity number  of  discovering tasks,  but  their  problems.  observation:  rather  grade level  and  quality  how  the  The  following  received high ratings  responses to rated  developmental and  #4  uses  received  on  indicates teaching  two  question #4  high  questions  teacher  to  the  nature  of  the  by teachers  rated  low  formulation,  integrative  taught.  point-of-view  problems  Subject  of  teachers  of  different the  exclusively. This may be related  examination  low  and  almost  that  students the  ratings  it  may  strategies  will on  to  be  are  about  be  all  not  that  have  strategy/ies  examples  low  problem  the  types  important  certain  or for  instructional  assist the  students  with  used  examine  this  Subject  #13  variables  to while  (see Table 5):  [Subject #4]: Place value work with the counting of real objects. This child should count out 1000 objects to get a " f e e l " for how much 1000 is. I would backtrack to an area of place value success (maybe 1's and ten's). Then work through several standard numbers with counters, occasionally leaving out a place (ex: 1000, 20, 4) and discussing "what happens". Child needs several experiences of this type until concept is "real", [see Appendix A, Scenario #2; see Table- 5]  [Subject #13]: I would have had maps and globes next to me to help her. I would have started with a map of Alabama and picked out Birmingham. Then I'd take out a US map and w e ' d find [Alabama] and Birmingham on the map. Then w e ' d look at a world map and find the U.S. Depending on how she reacted we could talk about how long it would take to get to Wisconsin from Birmingham walking, by car, by plane, etc. W e could look at pictures, etc. [see Appendix A, Scenario #3; see Table 5]  Both of these teachers mention they  would  of their  assist the  responses, the  students  more with  than one strategy their  major question for  problems.  in their  However,  consideration  on  discussions of close  becomes, " H o w  how  examination might  these  96 teachers use these  strategies  different  which  strategies  can  teacher's  be  practice  in  provide  an insight into these questions.  his discussion of  that the student's and  that  this  knowledge,  to  the  make  this  concept  of  real  "real"  opportunities  for  with  concept  of  place  value  and  teacher  does  not  suggest  the  student  continues of  understand  the  this  indicated  length.  If  say, why more than  in  a child isn't  in  knowing  with providing  By contrast, proceed to  type".  It  the  Subject #13 assist the  in  a  for  words,  Although  repertoire with  steps seems  strategies  in  " u n e x p e c t e d " ways even after  value  through "I  me!". the  student  What  student  comes  to  that  the  a problem +  can  to  provide  the  between  the  objects,  this  provide  understand  will  teaching  with  suggests  event  student  demonstrate this  take  that  exposed to  predetermined  have  a concept  is for that  assumption  in  is that  correct the  the  "several come  to  methods  is  worksheets  such  on  would  real the  Based  he  connections counting  will  place value  intervention.  he  of  strategies  has established  of the concept of  specific  many  discussion  that Subject #4  the  alternative  "How  teaching  following  appropriate  experiences  comment:  knowing  a "correct"  up  need  understand  o.k.?  student.  his/her  place  his final  how  build  the  outline  This teacher's  of  can  this  interested in  respond  concept  further  to  other  developmental  inferred  is  to  the  in  procedures?" The  objects"  proceeds  student  experiences  as  is in his/her understanding  teacher  to  of  strategies, it can be  "counting  the  discussion  problem  context?"  defined  emerge  From  each  in the  of  this  5 examples, Subject #4  responses  concept(s)  is  rather  associated  response.  indicates  student  that she  with her  has some  problems,  but  ideas about does not  how  explicitly  she state  would what  97 that  problem  is.  For  concrete  materials,  with  the  student  how  the  prepared  teacher,  examples which  student to  this  is  thinking  on  how  globes]...".  She  seems  about  student's  she  leave  discovering and framing them the  case of  Subject #4,  This teacher identifies and  elaboration;  may  have the  she  she  sees  These  student's this  formulate to  those  complexity  and  well-defined may  not  reacted,...[to selection  conducts  context  suggest  who quality  point-of-view. strategies, but  conversation  in  open  that her  with to  about she  definition  for  is  comment, maps  her teaching rather  and by  than, as in her teaching.  problem  as one which  type  problem  as an ill-defined  problem. She  on  own  teaching  to  of  reflect  her  situation  at  requires further discovery  theories  of  and  her own repertoire, those strategies  hand  (see  Appendix  that  there  of  be  a relationship  and the ability to one's own  received of  may  low  formulate  knowledge.  A,  Scenario  ratings  point-of-view  tended  on to  on  Instead, may not  this these  knowledge teachers  and  may  use them when  use  provide  between  the  #3).  problem describe  it verbal  of to  processes  and reflect on strategies  U p o n examination  teaching methods. They may have knowledge reflect  Using  information  instruction  strategies of  a  indicates  instruction  initial  of  further  teacher  during  unknown.  a developmental perspective in her own classroom.  construction  responses,  This  and select from  the  associated with problem finding based on the  task.  within the  identifies  as appropriate  inferences  teacher collecting  point-of-view  her  processes required  She may teach from  in  remains  using predefined strategies as a framework  the  learning and therefore  problem  the  interest the  [Roberta]  to  student's  discussion,  suggests an  consider the  "Depending  and  the  of  formulation, methods  descriptions  integrative  indicative  development, identify  teachers'  the of  engaged in the act of teaching.  but  of they  student's appropriate  98 Themes which  arose in these teachers' discussions of strategies included:  (a) presenting  new information  to  students,  [Subject #19]: If the explications kids gave above [to question #4] didn't lead to constructing a method which resulted in the traditional answer, I'd present new information using concrete materials...[see Appendix A, Scenario #6; see Table 5]  (b) covering steps missed in previous lessons and/or (c) starting from  the  beginning,  [Subject #5]: I would teach place value strategies -- ones, tens, hundreds ~ I would use manipulatives and place value cards to teach the concept, e.g. 95 is not the same as 905. The student needs to see the difference in concrete examples before he/she moves to abstract (i.e. expanded notation), [see Appendix A, Scenario #2; see Table 5]  (d) guiding  students  to the  "correct"  answer,  [Subject #17]: I would start from scratch to teach the concept. I would use visual examples and hands-on activities to explain. This way I hopefully would cover the steps they missed in previous lessons, [see Appendix A, Scenario #5; see Table 5]  (e)  reversing the  process by attempting  asking the students  to  questions about their  "correct"  the  students'  problem(s)  before  problem(s),  [Subject #18]: I would have students (the moon) rotate in place with reference to teacher (as the sun) noting that their back was turned at one point. Then rotate always looking at one spot on the floor (move around it) noting they always faced the spot (earth) even though they were at some time back to the sun...Students will be helped with visual demonstrations and actual participation when teaching movement concepts such as this, [see Appendix A, Scenario #7; see Table 5]  Teachers w h o can  be  variables  used  received low effectively  indicated  that  scores on  while they  those "knew  all variables teachers how"  to  "knew  who use  that"  received the  a particular high  strategies  ratings to  strategy on  assist  all the  99 students from use  the  teacher  their own points-of-view. Teachers' explanations about  strategy  to  constructs  developmental instructional  assist  her  own  perspective,  situations.  their  Only  students  knowledge  and  suggest  three  provides  about how  additional  teaching they  and  might  responses indicated  insight learning  apply  the  how they  use  this of  a  into  would how  a  on  a  based  knowledge  to  developmental  perspective:  [Subject #13]:  As discussed in comparison with Subject #4.  [Subject #6]: The children need practice finding sets of ten within a group of objects and then labeling the set as to h o w many groups of ten they found and h o w many were left over. Therefore, I would begin with manipulative objects and have each child "find t e n " in each set and then tell me how many groups of ten he/she found and how many were left over. W e would record these numerals on the board and then " g u e s s " how many objects total were in that set. Ex: _1_ ten and _3_ ones is 13. [See Appendix A, Scenario #1; See Table 5]  [Subject #21]: Having the students take part in a demonstration of this relationship - discussion on the topic - have the students draw diagrams of the process of the Earth & M o o n . This would give the students an experiential base to work with. Any questions could be approached in the discussion. The students would do the diagram as an activity to use what they have just learned. [See Appendix A, Scenario #7; See Table 5]  Teachers w h o  use strategies that assist students from  actively  engaged  become  active  questions in the  in  constructing  participants  who  practical setting  their  own  define  theories and  rather than  the students' points-of-view of  interpret  conforming  strategies. They become researchers in the classroom.  teaching their to  and own  predefined  are  learning. They problems methods  and and  100  5. Other Variables of Interest to the Present Study Participants  in  information  regarding,  question,  "Why  questionnaire  a. Concern in  their  (1976)  the  you  Problem  studies  of  study  concern  teach?",  and  reproduce  finding  that  their  in  selected  predefined  problems  which  have  you  teach?",  In  the will  concern  for  modified  and  Question  #1).  The  education  may  be  problem(s)  expression  of  solutions. They  contrast,  by  the  variables.  The  approach  by  teacher's to  finding  terms  finding  or  of  &  own  theory  the  way  in  ways.  with  a  things  These  work  view  their  the  individuals  uncertainty  their  work  conscious  in  the  Csiksentimihalyi of  teaching,  which  teachers  with  as  an  need  to  purpose "Why  of do  with  the  teacher's  of  her  work  as  Appendix  B,  context (1976)  out  uncertainty  question,  concordance  simply  by seeking  as a central  responses to  they are having with instructional  felt  approach  discovery  discovery  Cetzels  or  problems,  problems. They approach  or  participants' in  who work  unknown  problem study,  their  to  Csiksentimihalyi  ill-defined  known  tend  individuals  and  or  base their  problem  related  discovered  to  examined  delineated  of  which  for  present be  requested  represented  Cetzels  on  new theories which apply to  work.  artists,  problems  things  the  face  as an  ready-made  concern  young  pleasing  unknown  their  which  demographic  in  expression  overall  of  reality  By  an  a variety  finding,  of  theories.  with  problem  aspects  pre-formulated  overall  in  the  work  adopt  formulate  (b)  questionnaire  Finding  purposely  of  for  a  in Appendix B.  problem  view  completed  their  evidence  who  to  (a)  can be found  provide  want  do  for  individuals  present  (see  learning, assist their  tasks.  9  instruction  and  students  with  101 Of  the  27  participants  which reflect  in  the  present  study,  a concern for discovery in their  only  3  teachers  provided  responses  teaching:  [Subject #13]: I have always found children to be fascinating. I'm never bored because I've never taught the same lesson twice the same way because the cast keeps changing. It's mentally stimulating and I grow and learn each year, [see Table 5 for cognitive process ratings] [Subject #6]: I enjoy finding different ways of helping children to grow and learn and achieve some success. I am rewarded by the enthusiasm that children have about learning, [see Table 5] [Subject #21]: I love working with children and am confident provide child centred activities to foster their acquisition of [see Table 5]  These teachers suggest that they process of  teaching itself.  ratings  problem  on  addition,  the  By contrast,  the  of  in  the  teaching rewards  is interesting  formulation,  strategies  discovery which  It  receive rewards from  they  integrative  used  to  to  note  their  goes along which teaching from  remainder terms  which  of  come  this decreasing concern for  of the from  teachers' personal the  problem  teaching  students  reflected  a developmental  obtained  itself.  that is, from  and quality of  responses move benefits  work,  that these teachers received  complexity  assist  their  that I can knowledge,  The  the  sense  high In of  perspective.  increasingly from  point-of-view.  the  toward  teaching  following  a view  rather  examples  illustrate  finding:  [Subject #11]: Because it's a joy to teach when you feel like you or are making a difference in the learning or the way each child about him/herself and others, [see Table 5]  than  have feels  [Subject #15]: It is the professional career I chose/backed into. I am continuing into my 24th year--beyond retirement age-because (1) it makes me feel "useful to society", (2) I still love children, (3) I still need the responding love of children, (4) I'm still working towards improving, perfecting my craft, [see Table 5]  102 [Subject #25]: I enjoy working with young people, and wish to help them in ways similar to those that my teachers used to help me. I enjoy contact with young people both in and out of the classroom, contact which I think helps to keep me young and fresh in outlook, [see Table 5] [Subject #18]:  These  I enjoy working with young people, [see Table 5]  responses range  from  the  satisfying  effects of one's teaching on students to come by  directly  these  the  and  that  descriptions of the  students. This is consistent with  on  problem  their  tendency  formulation, to  use  direct  that  they  seeking satisfaction from  themselves  the  feel  process of  observing  lower  ratings  complexity  teaching  will  from  methods  the  discovery which  and  to  helping their  benefit  the  satisfying feelings which  the  integrative  problems. These teachers seek rewards from  knowledge  come  the  teachers  point-of-view with their  from  feelings  received  quality  of  assist students students  students  comes from  acquire  rather  than  teaching  from  a developmental perspective.  Teachers whose theories as  a  central  awareness such the  b.  that  problems. practical  information Appendix the  of  teaching their  work  meaningful  yet  They  identify  may  reflect tend  unresolved the  an to  overall share  problems  student's  concern a  for  problem  questioning  finding  attitude,  exist  and  a desire  to  problem  and  seek  solution  its  an  identify in  context.  Demographic  Participants  of  purpose  of  Variables  were in  asked to areas  respond to  related  to  B, Questions 2 through  knowledge  of  teaching  a questionnaire  children,  teaching  and  7). This demographic  and learning that  which  requested  educational  information  participants  may  background  practice  (see  represents some bring  to  bear  on  103 present  instructional  completed to  this  and  situations.  returned  portion  of  the  the  Of  the  27  questionnaire.  entire  participants The  in  following  population. The following  the  present  discussion will  study, be  21  limited  is a summary and discussion  of participants' responses: 1.  Years of teaching experience ranged from 0-5 years =  4  6-10 years =  3 participants 6 participants  16-20 years =  6 participants  21-25 years =  2 participants  The  =  6 participants  median  completed  24 years:  participants  11-15 years =  unknown  2 years to  years the  of  teaching  questionnaire  experience  was  indicated  approximately  by  14  the  years  21  of  teachers  experience.  who The  mean number of years experience was 11.9 years. 2.  Subjects within  taught  one  varied  grade  level  among (i.e.,  participants. primary  Some  taught  grades) while  a variety  others  taught  of  subjects  one  subject  or several subjects at one or many grade levels. The  main subjects taught  included:  spelling,  being taught  or  than any of the Subjects More  taught  by teachers  reading,  language  had at some time  in the and  humanities/social studies  social  been taught  studies. by  more  category  Social  studies  of  21 teachers  the  was  other subjects. within  participants  within  the  Math/Science Category  this  group  indicated  that  included they  math  and  had taught  science.  math  than  science. Additional  subjects  included:  physical  education,  health,  art,  religious  studies,  special  education  vocational  classes,  education.  gifted  Two  and  talented  participants  classes,  mentioned  career  their  education  substitute  and  teaching  experience. 3.  Grade  levels  ranged from  taught  by  kindergarten  Kindergarten through  8 =  Grades 9 through  12  4.  Other  Level = positions  1  21  =  participants  filled  out  the  questionnaire  15  participants  17  participants  5  participants  participant  or  experiences  requiring  adolescents have been summarized to (a) extra-curricular  who  to college classes:  grade 3 =  Grades 4 through  College  the  school activities  direct  intervention  with  represent the following  (student  children  or  categories:  graduation; plays;  out-trips)  (b) coaching (field hockey; basketball; gymnastics) (c) volunteer  work  (aide  cooperative volunteer  in  a religious  education  (f)  community  service;  work)  (d) counsellor (camp counsellor; guidance (e)  program;  counsellor)  parent piano  teacher  (g) supervisor/leader (4-H leader; Bible school; playgound leader; student  aide)  (h) research technician Only  5.  one  participant  of  the  21 w h o  answered the  questionnaire  absence of  additional experiences with children or adolescents.  Reasons for  participating in the inservice course revealed two  (a)  an  Examples  interest include:  and "  desire gain  to  expand  knowledge  one's and  that  the  main themes:  knowledge  insight  indicated  will  about make  teaching. learning  105 more (b)  6.  productive required  to  did not  Familiarity  with  generally  expressed  as  Piagetian  theory  college,  be  required,  Piaget's  in  for  educational  major  (project  previous  of  ratings  "...4  day  the  seminar  what  they  had  theories  exposure  each  Piagetian Piaget,  in  was  exposed  and  in  activities Erickson,  their  college  classroom therefore  for  as  stage,  data  and on  to  learned.  developmental  "...have many books and articles  identify  a the  may  from  discussion  and  Cesell  /  Piaget / can  see and work  be  two  described  participants  there  are differences  with  the  same  same  scenario Subject  classroom (see  #6  of  each  of  the  variables  child's/adolescent's point-of-view  how  while  at  it".  practice  been  "...much  on  principal  had  Piaget's  children  conducted  students  provided  which  elicited  faced  completed  behavior,  classroom  of  of  revealed:  studied  for  recall  knowledge  by  3  " system requires  participants  little  up  for  with  POINT-OF-VIEW  teachers  of  had  and  [inservice]  "...signed  implications  Most  practice -  levels of  sections  responses  its  "...need  teacher",  now".  problems  description  but  general knowledge";  B. THE TEACHER'S  classroom  and  some  recorded  class differently  how  had  results)";  see evidence  include:  recommended";  "minimal".  implications  explore  not  work  who  and  Examples  know what course was about";  Participants  attained  low  inservice.  although  interpreted  when  attend  will  observed  of  students"; " help become a better  credits,  education  The  for the  in  between  the the  problem.  Appendix  received  as ill-defined  A,  high  when  problems.  study  will  A  selected faced  to with  comparison  provide  a  concise  ways in which  teachers  think  Both  selected  and  teachers  Scenario #1). ratings  for  Subject problem  #1  received  formulation,  106 integrative were  complexity and quality  based  reflected  the  responses to  on  pre-defined  use of  of  point-of-view.  teaching  developmental  strategies  teaching  Strategies described by Subject while  strategies,  those  used  (see Table  each of the four questions posed in Scenario #1  by  Subject  5) The  #1 #6  teachers'  are as follows:  Question #1: What kinds of problems do you think these students this activity? W h y do you think so?  are having with  Subject #1:  They are having trouble with place value (10's).  Subject #6:  The children appear to be having a problem transferring what they already know about numerals and their values to the new knowledge about how the values change according to the place of the numeral. They understand fully what the numeral one stands for but they have not understood what the numeral one stands for when it is in the tens place. They do not understand that 1 can be representing an entire group (one ten). They have learned h o w to count and record the number of objects in a group, but they have not yet learned h o w to think about the set in terms of subsets, i feel that at this point the students have not received enough practice in finding the groups of tens in a set and then labeling the set according to the fens & ones.  Question #2: What questions d o you think these children were asking themselves as they worked through the assignment? W h y do you think so?  Subject #1:  Why isn't it worth " 1 " ? Why isn't it correct the left, I circled the 1st one on the left.  - the  one  Subject #6:  l think the students were simply asking -themselves how many wheels they had because they were asked to count them. I do not believe the children understood that the numeral sixteen could be broken up into tens & ones.  Question #3: What questions would you, as the teacher, ask these children the problems that you think they are having with this assignment? responses would you expect to get? Why?  is  on  about What  107  Subject #1:  What does the  1 stand for  in 16? " 1 0 "  Subject #6:  It depends greatly upon what has been previously introduced to the children. If this is a new topic of study I don't believe I would ask many questions except maybe to spark interest in the upcoming subject of place value. I might ask, " W h y didn't you circle al[ the wheels"?, "If there are 16 wheels & I ask you to circle both numerals in this number, why haven't all of them been circled?" I would expect them to explain to me what they know about the values of J _ & _6. Then this could lead to a discussion of J _ standing for 1 groups instead of J _ object.  Question #4: H o w would you proceed to help these children they are having? What strategies would you use? Why?  with  the  problem(s)  Subject #1:  Take the 16 wheels and break them up into two groups - 10 and 6. Try to explain why the 1 is worth 10 when you have the total. Or try 10 + 1, 10 + 2, etc.  Subject #6:  The children need practice finding sets of ten within a group of objects and then labeling the set as to how many groups of ten they found and how many were left over. Therefore, I would begin with manipulative objects and have each child "find t e n " in each set and then tell me h o w many groups of ten he/she found & h o w many were left over. W e would record these numerals on the board & then "guess" how many objects total were in that set. Ex. J _ ten and _3_ ones is 13.  These  two  described brought  teachers as  her  an  she  had  students  ill-defined  own  understanding the  were  presented or  repertoire students'  available understand  to the  her  with  discovered of  concept  problem, each teacher identified  her  "place value".  and interpreted  To  and  and the  constructing of  classroom  problem.  experiences  points-of-view for  a  problem  this  situation,  actions,  cognitive own  her  may  each  own  be  teacher  frame  for  developmental processes  knowledge  Faced  which  with  the situation  the  about  how  the  same classroom  differently:  108 Regarding with  question  of  students  might  therefore  does  "why"  or  have  #1  indicates  not  "how"  of  between  her  expected  from  that  stating  their  the  nature  transferring  about  the  concept  For  Subject #6 place  concept  and  the  definitions  understand  the  they  problem:  the  "The  already know the  numeral..." This teacher identifies  values  and interprets  value,  to  actually  and  explore  the  not  attend  to  problems  is aware the  are  of  the  discrepancy  definitions  received. This  appear  numerals  value  on her response.  the  children  elaborating to  teacher  on,  be  she  rather  having  and their values to  according  the  place  she  by  change  the  students'  concept  about  be aware that  recognizes  place  she  the  does  that  trouble  knows  of  and  the  She  of  teacher  elaborate  indicates  value.  the  not  her,  "having  answers  responses as " w r o n g "  of  are  appear to  "incorrect"  of  of  how  about  points-of-view.  do  what  students  indicates that this  students'  By contrast,  students  the  knowledge  or  students  the  responses. She does not  students'  definitions  the  the  the  conceptions  own  statement  conceptions  beyond  frameworks  alternate  problem  go  that  problems. She does not  alternate  well-defined.  students'  notices  students'  interprets  students'  known  new  the  and the  This teacher  than  Subject  place value (10's)". This direct  exact nature  the  #1,  to  the  place  classroom problem  a the  of  the  as ill-defined,  that is, as a problem situation which requires further discovery and elaboration.  Subject  #1's  alternative Instead, teacher  to  explanations she  asking  that  question  about  attributes  indicates  appropriate were  response  their this  information. themselves  This as  how  #2  the  "incorrect" problem is they  is  evident were  indicates  students  might  responses easily in  that  to  she  be a  remediated  the  working  questions through  does  thinking lack  about  of  through  not  formulate the  task.  knowledge.  This  the  she  thinks  the  task:  provision the "Why  of  students isn't  it  109 correct does  - the not  one  is on  appear  thinking  about  makes  predictions  to  the  experimentation  the  left,  differentiate  task  and  about  "incorrect"  answers could be the  about  "solving" the  problem  based her  the  on  formulates "correct"  solution  of  that  she  students' before  problems.  attempting  In  to  the  appear to  teacher  students'  originally of  problem  expected.  the  1 stand  Although  this  teachers'  expected  acceptable to  question  for may  answer  of  in  the  teacher  how  the  students'  She then  or refutation  students'  initially  this problem by asking a question which does  ways  1975)  "place value (10's)"  "10"  by  task.  She  based  on  the  than  finding students'  of  be  getting  closer  to  attempts  thinking  to  hypotheses,  might  these  "solve"  differ  from  hypotheses  into  her responses. As indicated  problem  identified  as solved.  the the  is designed to to  responses that  She sees  discovery of nature  (see Question #1),  suggests  might  in the next step of the process.  requiring  different  of  students  formulating  integrates  16?" She expects  elicit  been  same  how the  the  with  words,  as merely  She  how  (Arlin,  the  have  rather  an hypothesis about  require verification  identifies  the  solving  it  same situation.  might  thinking  with  teacher  other  find  left". This teacher  answers for the students when she says,  this  hypotheses, about  in the  this  problems as one  "What  are  problem  provide  the  about  the  does not  #3,  students  on  students  thinks  associated  her own thinking for their verification  question  the  believe....". These hypotheses about  thinking  Subject #1  how  not  alternative  own  1st one  herself  the  1965)  By contrast, Subject #6  thinking  she  how  (Mackworth,  think....! do  between  how  problems.  "I  I circled the  of  the  the  get  "10"  from  other  her. The processes of problem solving initiated  students'  answers by the  solve  information:  as the  the  the  answer  and proceeds to  elicit this same  in  response.  students, may  not  identification  this be of  110 the  students'  (Mackworth,  problem 1965).  in  well-defined  Subject #1  with problem finding points-of-view  as  does  are  not  indicative  appear  that may be necessary for  ill-defined  problem  to  of  have  situations.  By  about  she  students  task. in  With  order  these  questions,  verify  or  to  information  about  how  the  value:  might  ask,  "Why  them #6  "...I to  explain to  may have the  for identifying  A  key  to  uses to the  the  identifying  through  that  required  to  pre-defined 1986). include: would  of  place  circle  aH the  they  know  about  the  #6  by  the  students  methods  assist the  you  teacher  This  strategies  "frames"  didn't  a student  value  assist them  Subject  thinking  #6  formulates  the students' are  hypotheses  be  This  which  the  Subject  students'  responses  and  about  to  the  thinking  from  collect  concept  further of  wheels"...?...! w o u l d  values of J _ and  6...".  her  place expect Subject  points-of-view.  how  memorization.  knows  processes associated  ongoing  may  has a concept  students with their  total...".  concept  what  her  response  processes associated with problem finding which may be necessary  assist the  the  students  the students'  problem  have  me  refute  elicits  of  and interpreting  contrast,  h o w she thinks  type  the  identifying  questions based on her hypotheses about the  this  she  with  and  "the  provides  knows  two  that  problems. based on  their  new  the  the  is  their  worth  10  and  have  them  when  examples  the  of  strategies Her  "good" in  her  and  on  teaching (Paley, response. These  description sense of  of  information  may base her teaching  definitions  you  conceptions  a problem  with  one  students with  alternative  students  reflects the  strategies  assists the  information  provide  principle".  problems  1  Subject #1  developmental  concreteness  the  layer onto  chosen will  their  students with  "...why  may simply  strategies  mentions and  merely  has  through  problems. Subject #1  explaining  teacher  is  of  how  she  discovery which  111 goes  along  with  identifying  with  her  "conversation" manipulation as  she  of  students  student's  as  they  point-of-view.  attempt  to  "'find  objects. She keeps in touch with each student's  teaches.  learning  the  while  This in  teacher  the  may  practice  formulate  setting.  She  her  own  may  She  conducts  ten"  through  frame  of  theories  teach  from  a the  reference  of  teaching  and  a  developmental  perspective.  The  major  questions  difference posed to  between them  in  Subjects #1 the  and #6  Student  be  constrained  and  may also be  her  for  the  constrained  organizing  information  by  and  appropriate  nature  of  by the  abstracting  to  the  same questions  on  available  for  her  when  her  faced  with  constructing  of  presented  developmental  repertoire  of  own  level.  knowledge  classroom  problems  which  present  study was to  may  to  the  comprehends  the  be  for  and  solution  actions,  Subject #6  She may of  her  processes available  experience  at hand. By contrast,  an abstract  of  her responses indicates that she  problem  cognitive  situation  comprehend the to  the  her  interpretations  Anecdotes Task. Subject #1  the questions on a literal level only. The nature may  is in their  appears  to the to  have the processes  student's  described  point-of-view as  ill-defined  problems.  The  purpose  discussed formulation  of  in  the  this of  questions  teacher education the limitations  chapter.  Further and  the  generate  discussion of implications  and teacher thinking  will  be  the of  questions results  these  presented  based on as they  questions  for  the  apply  results to  research  in Chapter 5. In  the on  addition,  of the study and directions for future research will be discussed.  CHAPTER V. DISCUSSION  This  study  problems  was  designed  which  adolescent's  may  to  be  explore  described  point-of-view.  The  how  as  extent  teachers,  ill-defined  to  which  may have the  for  of  reflecting  was  also  on their  examined.  thinking,  problem  purpose of the the  extent  own  to  own These  finding  the  implications of teacher  results  formulated  with  identify  the  of  and  limitations  child's  teacher  as  practice  research  setting  on  teacher  education. The  which  the  tested. study  major  questions, related  Chapter as  or  problem  a developmental perspective in  further of  from  from  classroom  and strategies necessary  perspective on  teach from  delineated and  structures  provide a framework  which teachers might  of  were  and a developmental  be  concept  faced  teaching and learning in the  questions  study was to  classrooms, could  discussion  theories  problems,  the  finder describes those teachers w h o  when  they  5 will  their  provide  contribute  to  to  a the  and generation of future research questions on teacher education and  thinking.  A. DISCUSSION  OF THE RESULTS  1. Problem Formulation Classroom that  students  different 1983)  often  were  redefined  respond  to  as  and on  ill-defined  instructional  from what the teacher would  well-defined which  problems  tasks  represented  situations an  (Cetzels,  ill-defined  in  ways  based that  on  observations  are  "remarkably  have e x p e c t e d " (Arlin, 1983; Brooks &  observations that teachers often  problem  problems  1964).  problem  112  treat  ill-defined  Faced  situation,  with  problem a  participants  situations as  classroom were  Fusco,  scenario asked  to  113 respond to having  the  with  examined  question, "What  this  along  "ill-defined"  instructional  problem  majority  (level 1) of  of  situations  or  each  level  problem  finding.  problem  type.  discovery  stimulus  One  which  they  students' they  think this  so?"  "well-defined" in  what  student  Responses  and  is  were  " d i s c o v e r e d " or  was  classroom situation.  3  the  problem  finding  problems  "incorrect"  may have the  By  lowest a  known  by  Findings  the  indicated  the  problem  the  and  as ill-defined interpret  responses cognitive  levels  students'  indicative  2  through  with of  identified  8  (Arlin,  be  discovery  level  the  students'  to  teaching from  nor  formulation.  acts  as  1977,  teachers  and  of  category  1975-76,  If  4  form  situation  related  points-of-view.  a  the  Neither subject  to which the  might  further  processes which  "minimum  discovery associated with  problem  process  processes necessary for  perspective. These may include  a  level of problem  the extent  to  of  teachers  ill-defined  problems  as open  reflected  Nine, or 33.3% of the teachers  level  These  finding  the  problems as  contrast,  responses  significantly to  framework,  students'  response . associated  provided finding.  the  problems  further discovery and elaboration.  initiates  identify  1964).  provided  teachers  with  well-defined  problem formulation.  teacher  Only  problem  students'  you  think  varied  in the  Mackworth, 1965). Within this framework, the  you  which  (Cetzels,  2 responses, reflecting  associated  which  do  do  These teachers identified  grade level appeared to contribute  a  of  associated with  creativeness"  problems as open to  Within  Why  solver/student  represented increasing levels of provided  problem(s)  of the 27 participants, 16 or 59.3%, responded at the lowest level  Processes  innovation  of  "presented" or  problem formulation.  well-defined. of  task?  a continuum  presenter/teacher and the that the  kind(s)  the  a  1986;  identified extent  teachers  to  identify  elaboration,  then  a developmental  are associated with problem  finding:  114 integrative  complexity  and quality  of point-of-view  as well as developmental  teaching  strategies.  2.  Integrative Complexity  In  view  they  of  research  ask themselves  which  suggests  (Arlin,  as  suggestion  was  students'  they  worked  that  thinking  students  1983; Sinclair &  respond to the question, "What themselves  that  as it differs  Kamii,  who from  this  task?  formulate their  that  the  processes  integration  with  classroom  scenario)  question'  (Mackworth,  led  to  the  respect  to  are  the  that  if  or  the  problem  which  might  be related to the extent  problem  " u n e x p e c t e d " classroom problems  might  also  were  you  the  students'  problems  and questions  (1975-76)  comment  that,  how  the  and  is  a  problem to  in these  asking  s o ? " The about  situations.  the  finding teacher  high  identifies This  finding level  situation a  of (i.e.,  result  in  (Arlin,  1975-76). This  organizes  or ill-defined  teacher  to  be using processes associated  and  'general  information  problem  to which he/she identifies the students'  affect  asked  think  explanations  ill-defined  finding  the  way in  do  discrimination  of  faced with  It  stimuli  questions  solving. Studies of problem  the outcome  when  ill-defined.  differentiation,  required  1965),  suggestion  of  the  these students were  Why  o w n , might  answer  teachers  alternative  with problem finding and its subsequent problem indicate  1970),  questions d o y o u think  through  teachers  correctly  situations,  problems as  interprets  is suggested  the  by Arlin's  [t]he structural properties of differentiation, discrimination and integration characteristic of this process [of information processing] are analogous to the problem situation [i.e., ill-defined problem situation, and] the opportunity to raise questions [i.e., quality of point-of-view]... (Arlin, 1975-76, p. 101)  In addition to of integrative from  the formulation  Within  a cognitive  formal  reasoning  (Arlin,  some type  developmental framework,  level (Arlin,  1984).  her  repertoire  teacher  to  the  reflects  depend  this the  at  this  not  extent  problem  actions,  finding  the  in  which  that  teacher  indicate  that  the  reflect  the  indicate for  finding  an  that  a  she  finding select  sees  framework,  "good  the  formal  problem  that  ill-defined  is  101).  teacher is able to  information  a problem it  1975-76, p.  selection occurs at  condition  the  uses  to  and  (Arlin,  process of  thought,  hand. Within  information  strategy"  sufficient  level of  experiences and  situation  on  on  of  of  Studies of  reasoning is a necessary but  appropriate  may  according to  1975-76, 1977, 1986). At this  from  explanations, which requires a high level  complexity (conceptual level), problem finding also involves " a selection  available stimuli  operational  of alternative  as  how  problem  the  situation  problem  finder"  (Arlin, 1975-76).  Results  from  74.1%,  provided  (level  1).  students curriculum own the  the  present  responses which  These in  the  teachers  did  scenarios  might  tasks. They did  thinking teachers  about  reflected  similarities  and  own  the  thinking  the  responded  responses  indicated  study  an  and  to  provide  have  and  both  in  ability formulate  the levels  to  thinking  differentiate  students' 2  and  interpret  student to  lowest  and  integrate  questions  majority  level  alternative  been  appear to  ability  differences teachers'  not  task at  not  large  of  explanations as they or  the  discriminate  students'  these  based directly  one teacher's response (level 8) reflected this integration  on of  how  the  through  the  between or  Level  in  these  Level 2 terms  3  conceptions  their  11.1% of  complexity.  questions  thinking.  alternative  Three,  or.  complexity  about  worked  integrative  teacher  teachers, 20  integrative  points-of-view. 3 of  of  of  responses into  their  alternatives.  Only  alternatives (the  teachers  116 thinking from  and the  reflection  student's  thinking)  and the  formulation  of  hypotheses which  on these alternatives. This teacher theorized as to  "how"  come  and  "why"  these similarities and differences exist:  [Subject #13]: I think Roberta knowledge and she is trying to and draw on her experiences.  This  teacher  ill-defined problem and  (see  Table  situations  interprets  elaboration about  5,  seek  the  She  associated  from  the  problem  alternative  It  finding  responses  engage  are indicative  available  nature  explanations  may  with  the  Formulation).  "incorrect"  complexity which  abstracting  interpreted  Problem  initiate  problems.  processes  the  and  students'  may  their  integrative in  identified  is trying very hard to draw concretize her answers to the  problem information  of  follows  that  that  if  as  further  discovery  requiring how  (i.e.,  which  the  students  related  finding.  to  and  thinking levels  of  She may also engage  formal she  identifies  are  high  as  ill-defined  who  problem  solving  then,  problem  teacher  processes  of  student's  process, the  about  in  this  on prior questions  reasoning  selects  level)  as appropriate  by to  situation.  The majority  of teachers in the  present study do not  associated with integrative  complexity  questions  and  students  them  solution.  for  significantly to  problems  Neither  subject  level of integrative  appear to  at levels high enough to  may  have  category complexity.  with nor  instructional  grade  level  be using processes  begin to tasks  identify  presented  appeared to  the to  contribute  117  3. Quality of Point-of-View The  ability  discussed 1984).  to  ask questions which  as a key method  In  situations  the  present  were  study,  associated  finding.  Question quality  of  questions  the  students  about  Ratings  "represent  the  102). In the questions Neither  ways  on  category  point-of-view  Teachers  who  of  the  who  implications  of  with  tended  (1956)  the  grade  to  "What  identification  (Arlin,  levels.  1977) (i.e.,  a cognitive  These  which  low  ratings low  high  problem  "weighted average"  expect  you  to  products  ask  get?  the  Why?".  categories  which  1975-76,  p.  6) and implications  (category  7)  of  point-of-view.  the to  student's  contribute  findings  are  problem quality  scores at the classes  ratings  (2)  significantly  to  on  consistent  and  process model of problem finding  By contrast, formulation  research  questions  require  (i.e.,  or  level and 4,  transformations  with  and  Eighteen,  units (1)  problem  abstraction  formulation  scores.  level.  scores at the  implications  complexity)  on  question  quality  suggests that  integrative  of  (Arlin,  appeared  received scores the  received question  outcome  quality.  receive  consistently  problem  is structured"  (category  level  ill-defined  the  you  intellectual output  (Brooks,  in  the  has been  perspective  questions would  responses would  consistently  received  (6)  posed  question',  asked,  vis-a-vis question  teachers  complexity  organization  Within  questions  'general  students  a developmental  the teachers received question quality  integrative  finding  of  informational  nor  received  complexity  14.8%  teachers  the  Guilford's  in which  quality of  or  kinds  response from  scores were calculated according to  associated  subject  66.6% of  with  from  present study, transformations  were  integrative  the  problems? What  based  ongoing  teaching  teachers posed when  their  were  for  elicit  on  (5)  .and and  problem  a high  formal  those  level  reasoning).  (Arlin, 1975-76, 1986), question  118 quality  scores  below  the  categories  of  tranformations  and  implications  are  not  indicative of the  'general question'. Only 2 teachers posed questions at this level of  problem finding.  It is interesting that Subject #13, w h o  ratings on problem formulation score  at  the  differentiation  systems  problems.  questions  which  goes  of  the  "maximum  She  are  However,  beyond  "why"  level.  associated with  student's  thinking.  (4)  and integrative  the  thinking  about  students'  questions  collect  does  her  own  responses,  questions posed in ill-defined  terms  developmental  finding  the  has been referred  creative  production"  formulation.  (Arlin,  Further findings  finding  skill of  finding  (Arlin,  necessary  to  but  posing  as a "critical p.  questions to  about  differences  and  about- the  which  and  poses  students'  indicate  that  she  and  the  discover  the  "how"  section  for  a  discussion  of  problem situations can be understood  in  situation)  teacher  brings  to  that  situation.  Problem  process that links Piagetian operations  297).  the  This  problem  indicate that "operational level is related to the  problem  1986)  condition  indicate for  links  formal  1977, p. that  problem  reasoning  to  to  1977,  sufficient  these  quality  discrimination  explanations  information  'Limitations'  general questions" (Arlin,  1975-76, not  capacities the  1977,  about  pose  performance" as it applies to this  the  alternative  thinking  (see  reflect  further  not  The types of of  of  hypotheses  to  teacher  complexity, received a question  formulation  formulates  designed  this  Her  received the highest possible  298).  formal finding  Studies of  reasoning and  that  problem  may  be  a  transformations  or implications questions do not appear below this level of reasoning.  The question quality score calculated for the that  the  majority  of  teachers  in  the  entire  present  sample was 1.48 which suggests  study  tend  to  formulate  questions  119 which than  are designed to questions  appears  from  elicit  which  specific pre-defined  elicit  ongoing  these results that the  of  problem  formulation,  intregrative  to  him/her  for  ill-defined  use  in  identifies and interprets  responses from the  students  rather  responses from  the  students'  extent  the  teacher has the processes  to  complexity problem  the student's  which and  quality  situations  of  may  points-of-view.  point-of-view  affect  how  It  available  each  teacher  point-of-view.  4. Developmental Teaching Strategies Teachers were asked to with the  discuss the  problems that they were  strategies they would having with the  instructional  study, developmental teaching strategies accounted for providing  a means  problems  she  Teaching  strategies  by  which  identified  the  her  based  teacher  students  on  a  could  as  the  present  materials  as  'hypothesis 'examples Three  the  85.2%  major  generation', continuum'  teachers  developmental responses not  study,  having  developmental  remedial 14.8%  not  mentioned  strategies  that  the  from  the  "framing"  number  of  in  represented  provision  Chapter  of  by the  tasks.  only  the  use  of  mentioned  the  strategy  and  11.1%  the  represent 3.  of  the  any  by  each  appropriate  of the  strategies).  of  Examination  mentioned  concrete  mentioned  the way in which teachers described their direct  about  instructional  a description  which  strategies  point-of-view  theories  certain  present  (Arlin, 1983).  18.5%  strategies  students  tasks. In the  own  mentioned  3, Table 4 for  described  her  with  assist the  student's  perspective  teachers  strategy.  mention  be related to  ranged  the  (see Chapter  teaching  indicated  appear to  descriptions  did  of  the  test  beginning of teaching from the student's point-of-view  In  use to  the  of  four  teachers'  teacher  did  use. Rather,  information  with  or  120 without  student  questions  while  teachers high  who  to  interacting  with  received  ratings,  Consideration may  the  students. low  problem  processes  provide  an  teacher engages in when  thinking  at  discovering the These  ratings  students'  descriptions  and  those  formulation,  who  integrative  are  problems  and  consistent  received  with  consistently  complexity  and  quality  (see Table 5).  of  Research on  attempts  consistently  respectively, on  of point-of-view  adulthood  participation  postformal  which  postformal thinking  insight  into  constructing  thought  involves  hypothetico-deductive  associated the  her own  relativity found  of  in  adolescence  reflection-in-action  knowledge of  argues for  of  in  reflection  kind  in adulthood  a  thought  with  a structural  thought  adolescence  teaching  as (Arlin,  and  that  the  and learning.  model of  opposed 1984).  adult  to  the  Within  this  model, Arlin (1984) suggests that,  ...[teachers] may not only exhibit competencies and skills in one area that are typical of one set of operations, but also exhibit quite a different set of competencies and skills in another. The choice of competencies and skills is dependent on the ...[teacher's] perception, selection, and/or construction of a particular problem space or spaces and the priority given them. The behaviors observed and the sequences of actions selected are, in this view, the direct result of the choice of the [teacher]. (Arlin, 1984, p. 261)  In  the  present  formulation, by  the  study,  integrative  complexity  experiences and  identifying  the  teachers  student's  actions  who and and  point-of-view  reflecting  on their  applying  this  knowledge  directly  received  high  ratings  all  were quality  the in  rated of  cognitive ill-defined  on  variables,  new  in  levels may  of  be  processes available problem  situations.  problem  constrained to  them  for  They may  be  previous teaching success and  situations.  suggest  low  point-of-view  own observations or experiences of to  at  their  By  contrast,  teachers  responses that  they  who select  121 from  their  modify their  repertoire  and integrate  o w n thinking  (Yinger,  1980) to  1987),  then  they  of  experiences  and  actions,  into the new situation.  If  previous  information  teachers go beyond  that  thinking  (Lampert,  1984; Piaget, 1978) and thinking about doing  thinking  about  doing  may "reflect-in-action"  something  while  doing  it  they about  something  (Schon, 1983,  (Schon, 1983, 1987) o n their  o w n theories  of teaching and learning in their o w n classrooms.  B. LIMITATIONS OF THE STUDY Several  methodological  classroom  simulations  issues  including  as a measure  the  generalizability  of teacher  thinking  of  results,  the  use of  and the conclusions  which  can be drawn from the results require careful consideration.  The  present  based  on  several  the  a  total  was exploratory  development  research  development of  study  traditions.  and problem of  of  only.  Its  a framework  These  intention for  traditions  was to  teacher are  thinking  reflective  finding. The sample for the present  27 experienced  teachers  participating  generate  in  questions  which  combines  thinking,  cognitive  study  was made up  district-sponsored  in-service  courses designed to introduce teachers to a developmental perspective on education. Participants  were  not  randomly  selected, therefore  and to other teacher education programs must the  size  number teachers  generalizations  participants  selected  Studies)  group.  variables  of  within  scenarios  Therefore,  problem  the  each which effect  formulation,  subject  subject  integrative  nature  category/grade  represented of  other  teachers  be considered with caution. Although  of the sample was adequate for the exploratory of  to  the  level  varied.  Only  3  lntermediate(Humanities/Social  category/grade  complexity  of the study, the  level  and quality  of  on  the  main  point-of-view  122 as well as the  developmental teaching strategies could not  However, in view of the exploratory close  examination  appear  to  The  use  problems  when  of  and  stimulating  the  that  neither  teacher's  classroom  subject  ability  problems  to  which  nor  identify may  through  grade the  be  level  student's  described  by  elaborations clinical  within  employed.  students  teachers'  instructional  tasks  based on  reponded  associated  and. provided thinking a  to  the  method.  thinking  gives  a teacher's  an instructional  as  about  rise  own  task  about  the  Although  for  sheet  The the  the  to  several  telling  of  in ways  in  and the  points-of-view.  which  form use  finding  teachers  students'  answer  teachers.  problem  of  of  was  used  questioning  a  interview the the  observations  setting,  exploratory task and  was  factors  to  which  nature developed  limitations  must  as  the  study  of on  the  be  carefully  cost,  basis  study  scenario elicit  was  a  of  subsequent  present  paper-and-pencil  such  its  Each  of responses and explanations that are elicited  an and  with  opportunities  standardized  from  Although  further use:  with  Each scenario was  processes  achieve the quality  administration  have  stimulate  the teacher's original expectations. The scenarios represented a way  own  accompanied  following  with  where  thought  their  method  to  students  concerns.  solving,  Piagetian  faced  questions  deviated from  and  significantly  tables,  classroom simulations/scenarios to  classroom situation  reveal  frequency  of the study, it can be inferred  problems.  methodological  problem  the  contribute  point-of-view ill-defined  of  nature  be empirically established.  was  responses  based task  to  on  the  does  not  by Piaget's clinical time,  economy  influenced  the  of  research,  sound  considered  prior  of  method  to  the its  123 Discrepancies response  sheet  responses to low  levels  indicated  the  question #4. his/her  in  consistency  of  responses  were  identified  in  two  questions #1 of  and #2  problem use  of  consistently low quality  to  of  the  the  score.  questions Subjects  integrative  principle  "Why  do  the  characteristic of systems and transformations  #17  method  teach?"  normally  of  #20  provided reflected  Neither  of  teacher  intervention  problem  responses  finding  Responses  to  the  for in  reflecting of  question  a low  #3,  are  questions:  [Subject #17]: They have no idea what the numerator and denominator are not  Question #2: What problems they are expect? Why?  4  be indicative  Question #1: What kinds of problems do you think having with this assignment? Why do you think so?  [Subject #20]: Knowing C o m m o n Denominator. answers.  and  a concern for you  subjects'  through  complexity.  as a  main variables would  However,  1  (see Appendix A) which  teacher indicated  question,  ratings on the  cases.  and  'concreteness' neither  to  Scenario #6  formulation  Furthermore,  responses  question  the  these  students  a fraction is. The significance understood.  of  which number to add, try to figure out Lowest I know by looking at their work and their  questions would you ask these students about having with this task? What responses would  the you  [Subject #17]: 1. I'd ask " H o w did you get that answer?" or "What makes you think that?" 2. I'd probably get some "justification" for the answer although it might be misdirected. Or, I might get an answer that applies to another concept. Perhaps the student just confused terminology. [Subject #20]: W o u l d you please tell me how you got your answer? All kinds. Because they're each making sense of the problem in their own ways (although there may be some "stabs in the dark").  These  teachers'  indicative  of  the  responses kinds  of  to  question  questions  #3,  posed  in  as  compared  ill-defined  to  problem  question situations.  #1 In  are the  124 present result  study, in the  these  situations  interpret  the  responses.  questions  Questionnaire  changing  to  This  implications  responses to  well-defined  period allotted  on  is  a  (see Appendix  for  appropriate  or  B) regarding  teachers  provided  practice:  on  Therefore,  received  task,  interest  the  teachers.'  compulsory, to  and  the  their  to  with  completely may  and  not  question #1  are  two  tend  week  to time  task may have affected  response  There  do  teachers  the  to  received by  be  these  Question  #7  unfamiliar,  but  a problem  of  responses  the  of  Piagetian theory  the  and  its  that's subjects  having  their  elaborations who  do  not  appear  responses. The in  possibility  for  received  each question, consistently  while  high  responses as did many with consistently  completing of  hear are some variables which  given by the teachers.  of the  order  question #3  both  addition,  his familiarity  responses  participants  all variables elaborated  on  that  In  "Almost  responses and  teachers elaborated  in  little  1984).  responses alone. All  ratings  #20's  assigned."  anticipated  Some  only.  completion  Subject  readings  upon practice (Arlin,  the  finding  Their responses to  suggests  level  the  in  educational  quickly with the  which  literal  suggested  effect  provided  problem  consistently high level ratings  question #1.  problems  participants  for  verbalizing  of  on all questions, these teachers' responses to  correspond with their of  processes  'general question'. Unlike the  some participants  indicative  require  the  teachers might  to  be  amount task,  affected  by  others  ratings low  whether  on  scores. or  not  of  time  given to  complete  the  the  fact  that  in-service  was  providing  have affected  the  answers that the  nature  researchers want of  the  responses  125 On  two  occasions, teachers  seemed to  interpret  the  question  differently  than  was  intended:  Question #2: What problems they are expect? Why?  questions would you ask these students about the having with this task? What responses would you  [Subject #24]: If my assumptions of memorizing a rule is correct, these students would want to be spoonfed another rule to apply to unlike fraction addition, [see Appendix A, Scenario #6] [Subject #19]: I would suspect that the child w h o said, "I don't would have a question on how to compute the answer but the may have no questions at all. [see Appendix A, Scenario #6]  These  teachers  themselves, the  task?  interesting  might  such What to  be  providing  as, "What  questions  responses  note  that  Subjects #24  and #19  subject  area  and  related  to  responded  the to  suggest the points  to  would  from  grade  Scenario  the  they  level or #6  for  and  group,  to  direct  the  #20  the  questions  students Why from  ask their  do  you  the  they  think  previous  Intermediate(Math). #3  itself.  evidence  design of clinical  of  this  The  Although problems,  particular  interviews  the  It  is  discussion, and from  the  might  be  other  teachers  present  findings  scenario. This  complemented  about  so?"  problem two  asked  teacher  discussion, all chose Scenario #6  question  without  to  expect?  present  need to examine the need  would  Subjects #17  scenario  the  responses  know" others  observation  by  classroom  teachers'  immediate  observations.  Clinical  interviews  which  probe  for  reactions to  the  scenarios may  provide  which  to  arise  the  More  seem  importantly,  the  due  to  opportunity  information  on  a standardized way  flexibility for  based  the  of  time  interviewer  given to  for  controlling  to  complete  elicit  variations the  task.  continuous responses  126 from  each  question the  teacher  posed as well  interviewer  elicit  the  rather  will  is  able  teacher's  than  on  help  to  determine  how  the  as how  he/she is thinking  to  as  ask  framework.  simulations  many  Ratings  and  about  questions  based on  paper-and-pencil  teacher the  deemed  is  interpreting  task.  In this  necessary  in  the  situation, order  to  Piagetian clinical interview  methods  responses tend  in  to  result  more  reliable and valid responses.  In addition, the knowledge learning  extent  must  this  knowledge,  be established through  suggestion is that teachers w h o associated  with  problem  reflecting-in-action which and  teachers  reflect  strategies  they  this  may  may best  on have  and then  finding  their  construction  his/her own theories uses  this  knowledge  on  In the  of teaching in  the  classroom  present study,  the  all variables may have the processes  available  to  them  for  organizing  and  practice setting. The extent  "knowledge-in-action"  may depend  the  to  them  role  of  be established through  of and  knowledge within the  available  redefine  on his/her own  classroom observation.  score high  on their own  perspective. This for  teacher reflects  (experiences and actions), formulates  based on  setting  to which the  for  the  teaching  teacher  from  on a  as problem  clinical interview  techniques  the  to  structures  developmental finder.  Evidence  augmented  by  careful classroom observation.  It  is  interesting  often mentioned principle, while be  note  that  the  by teachers at the other  concentrated  size and the  to  kind  of  developmental  intermediate  primary  inclusion of  senior  groups  strategy  concreteness principle, tended  (see Chapter 3, Table 13). A higher  level teachers  most  grade levels was the 'concreteness'  strategies, combined with the  in the  teaching  may  provide  further  insight  to  sample into  the  127 possibility that the to  which  the  processes  maximum  levels.  more  subject  the  the more  focus of  It  by  been  in  that  the  taught  studies  of  postformal  state  that  one  Biggs  1980),  provide  given  Collis,  to  levels  of  used  as  than  problem  the  questions  than  explanation  is  of  to  used  at  the  grade  level  and  physical sciences and  their the  mathematics,  teachers at  that  integrative  quality  this  teacher  on  the  basis of  previous  to  ask  questions  which through  or  of  an  of  other  questions these might  the  interview.  in  complexity. her  be  based  into  formulated how  student's  by  However, reflected  the level  her and  student of  lower  of  in  own  possible  intervention.  This  teacher quality  systems verbal  of  the  hypotheses  her  was thinking This  would  category.  thinking.  selected  thought.  ratings  were  the  on  the  strategies  the  the  level  highest  of  261; in  four  discussion  hypotheses have  All  maximum  discrepancies  found at the  questions . posed  might  a  (Arlin,  1984, p.  the  be  adults on  (Arlin,  reflect  can  point-of-view  hypotheses about matched  that  in  limits  some  implications.  student's  of  sets  5). This teacher was rated  that  integration  into  responses  strategies  teacher's  finding  performance"  processes  teaching  the  attained  insight  other  and  the  problem  minimum  an  transformations  this  the  been determined  being  higher  or  has  of  by  (see Table  elicit  that  on  use  suggested  suggest  used  the  formulation  level  suggests  rather  of  developmental  be  This  is  Subject #13  normally  at  may  example  would  extent  fact,  teaching strategies than  thinking  say about  performance  the  in  the  teach humanities/social studies courses.  "[t]he  An  are,  the  to  use direct  has little to  results.  that  is related  performance...[i]t &  grade level may affect  teacher  suggested  likely the teacher will  notion  1984)  teaching and the  used  matter  grade levels w h o  The  has  the  Another questions in  order  could  have  128 Despite  these  limitations,  teaching emerged from  of  thinking  tend  to  neglect problem  the  teachers  consider  finding  framework how  based on  for to  ON  of  teaching  based on  behaviors,  to  are  teach the  framework  growth  teachers  in  and  the  his/her  ability  to  questions  for  future  cognitive think  studies  teaching  of  of  research  on  models  of  developmental  attitudes to  a  that  think.  are  What  developmental  teacher  researchers  thinking.  have  the  teaching  assumed to these  perspective  own  may  thinking  be  from  be  researchers  a  and  a  constructivist  opportunity  processes which  and teach for  1983a, 1983b)  Within  and learning. The teacher's  developmental  about  and  students  implications  for  cognitive  1980; Sprinthall & Sprinthall  skills  their  knowledge,  think about  actions,  and  TEACHING  (see Glassberg & Sprinthall,  be  necessary  RESEARCH  teachers and their  adult  implications  the findings. These will now be discussed.  C. IMPLICATIONS FOR Studies  several  to  examine  experiences  and  associated  with  a  developmental  perspective (Arlin, 1983, 1985; Brooks, 1984; Brooks, Fusco & Grennon, 1984; Elkind, 1976; own an  Feldman,  1981)  may  "reflection-in-action". interesting  development  metaphor of  teacher  provide  researchers  The  concept  of  for  further  research  education  with  "teacher  programs  an  insight  as problem  on for  teacher both  into  the  finder"  thinking  in-service  teacher's  may and  and  provide for  the  pre-service  teachers.  If  researchers  problems context, they  do.  and then If  begin  to  questions they  study and  may begin  researchers  the who to  teacher seeks  understand  ask questions  about  as  her  one own  "how" the  who  identifies  solutions and  "why"  teacher's  own  within  his/her the  teachers thinking,  own  practical  perform then  as they  129 may  begin  to  they  begin  to  identify  the  process.  match teacher education observe  nature  of  teachers the  in  curriculum  the  problems  act  teachers  In this view, teacher education will  on where  of  to  the  needs of  teaching, then  face  as they  begin where  the  they  engage  the  teachers. If  may in  begin  the  teaching  teacher is rather  researchers and teacher educators think the teacher "ought  to  to  than  b e " . What  the present study may offer the researcher and teacher educator is a 'developmental perspective' of  reality  ill-defined  and when  The intent of the  she  is  for  faced  identifying with  FOR  interpreting  classroom  problems  the  which  teacher's may  be  own • view defined  as  the  cognitive  FUTURE RESEARCH  present study was to processes proposed  classroom situations, identify  generate questions based on an  to  and interpret  with instructional tasks. As a result of the to  and  problems.  D. DIRECTIONS  of  techniques  stimulate  further  research  on,  describe the  how  teachers,  when  exploration faced  with  problems and questions students  have  findings, several questions were  "Teaching  from  the  student's  generated  point-of-view:  A  developmental perspective". These questions are:  1.  To what extent do teachers identify students' " u n e x p e c t e d " reponses to what teachers believe to be well-defined instructional tasks as ill-defined problems?  When faced problems, a.  with  classroom situations  which  may  be  described  as  ill-defined  To what extent do teachers formulate alternative explanations about their students understand a particular concept in ways which differ h o w the teachers themselves understand the same concept?  how from  and To what extent do teachers select information from their repertoire of experiences and actions and formulate hypotheses based on these  130 alternatives in point-of-view?  order  the  process  of  identifying  the  student's  To what extent do teachers formulate questions based on their hypotheses about h o w the students are thinking? D o they elicit ongoing responses from the students? D o they verify or refute their hypotheses about h o w the students "have the concept"?  c.  To what extent do teachers formulate their own theories of teaching and learning in order to begin to assist students with their problems and questions from the students' own points-of-view?  from  these questions that a problem finding  teacher's "reflection-in-action" making  sense  problem finder  1.  begin  b.  It follows  in  to  of  an  and which describes the  ill-defined  problem  situation  framework  which informs  steps a teacher goes may  describe  the  the  through  teacher  as  in her own classroom:  To what extent does the contribute to the teacher's problems?  W h e n faced with ill-defined problem,  a  cognitive ability to  classroom  process variable of problem formulation identify classroom problems as ill-defined  problem  which  may  be  described  as  an  a.  What cognitive process variables (integrative complexity, formal reasoning level, quality of point-of-view vis-a-vis question quality, developmental teaching strategies), singly or in combination, best predict the teacher's ability to identify the student's point-of-view when faced with classroom problems which may be described as ill-defined problems?  b.  To what extent do the cognitive process variables (integrative complex complexity, formal reasoning level, quality of point-of-view vis-a-vis question quality, developmental teaching stragies) describe those teachers who have the structures and strategies available to them for "reflecting-in-action" on their own theories of teaching and learning in their own classrooms?  REFERENCES Arlin,  P. K. (1975-76). A cognitive Horizons, 54, 99-106.  process  Arlin,  P. K. (1977). Piagetian Psychology, 13, 297-298.  Arlin,  P. K. (1983). 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Alternative Education, 34, 3-9.  paradigms  planning.  and  of  The  Elementary  field-based  teacher  Implications  School  experience  education.  Journal  for  journal,  in  teacher  of  Teacher  APPENDIX A THE STUDENT  A N E C D O T E S TASK  (Levitt & Arlin, 1986)  137  138 SCENARIO #1: PRIMARY (Grades K-3) • MATH (adapted by P. K. Arlin, May 1986 from Kamii, C , 1985, pp. 60-61) Near the end of the school year, first graders were  given the following activity to  do in their math lesson:  The  children were  asked to  take  each given a piece  out  their  of  paper,  16  tinker  crayons. The children were  wheels. Rather, they were  not  toy  told  wheels  and  that there  were  were  16  asked to take each wheel and draw an outline of it on  their paper. After they had completed their outlining, they were asked to count up the  number  paper  as  of  wheels  shown  in  that  they  Drawing  A  had  drawn  below.  and  Then  the  to  write  children  the  number  on  their  asked  by  their  were  teacher to circle the number of wheels that the 6 showed in the number "16". As the  teacher  Drawing now  gave  these  B shows the  circle the  number  their papers. 32-out-of-32  instructions, she underlined completed of  wheels  activity.  Then the  that the  the  "6"  teacher  1 stood for  in the  number  asked the  in the  "16".  children  number  "16"  children completed the task as shown in Drawing C.  to on  139 Name: ANSWER  SHEET - (PRIMARY/Math)  Imagining  that  you  are  questions  as thoroughly  the  teacher  as you  in  this  can: (Feel  situation free  to  please  use  the  answer back  the  of  following  the  page  if  necessary) 1.  What  kinds  of  problems  do  you  think  these  children  are  having  with  this  as  they  about  the  assignment? Why do you think so?  2.  What worked  3.  What  questions  do  you  through  the  assignment? W h y do you think so?  question(s)  would  think  you,  these  as  the  children  teacher,  were  ask  problems that you think they are having with this would  4.  How  you expect to  would  you  asking  these  themselves  children  assignment? What  responses  get? Why?  proceed  to  having? What strategies would  help  these  children  you use? Why?  with  the  problems  they  are  140 SCENARIO #2: PRIMARY (Grades K-3) • MATH (C. Reynolds, personal communication, October 1984) The  following worksheet  3 student  on finding the standard number was completed  at the beginning  Please examine  of  the worksheet  the school  year  by a grade  and was marked by the  carefully and answer  the questions on the  page.  Directions: Wrlta eh* s t a n d a r d nuabar. 1)  3000 + 200 + 20 + 6 - ?)n2 2. fS  2)  6000 + 500 + 2 -  3)  2000 + 0 + 0 + 9  4)  S00 • 30 + 7 •  5)  7000 + 900 • 90 • 1 -  6)  1000 • 200 • 80 + 0 -  7)  3000 • 200 • 10 • 7 -  S)  9000 + 100 + 8 -  9)  1000 + 700 + 60 -  6  10)  3000, + 300 + 70 • 8 -  11)  200 + 10 • 7 «  12)  8000 + 3 -  R  747  1  J^&LJJL  33  2- (  7 t 7  000 •*• 100 + 30 + 7 « HJ37 19) 700 + 20 •« 3 20)  5000 • 400 + 50 + 8 - fiJrfS %  756  21)  2000 + 600 • 10 • 9 -  H 6 3 Y  22)  8000 + 300 • 60 • 2 -  23)  500 • 30 • 1 •  W  800 • 70 + 7 -  16)  1000  400 • 8 -  15 )  700 + 50 + 6 -  16)  4000 + 60 • 3 -  17)  7000 • 700 + 50 • 6 -  18)  200 • 90 + 5 -  1 6 %Y  7-?56  2-3 5 25 )  72-5  800 + 60 • 6 -  f? <? £  2r£ ff.^  § % \  37  7  \ <\ £  Z  teacher. following  141 Name: ANSWER SHEET - (PRIMARY/Math)  Imagining  that  you  are  questions  as thoroughly  the  teacher  as you  in  this  situation,  can: (Feel free  to  please answer  use the  back  the  of  following  the  page  if  necessary) 1.  What  kind(s)  of  problem(s)  do  you think  this  grade 3 student  is having with  this assignment? Why do you think so?  2.  What  question(s)  do  you  think  this  student  was  asking  him/herself  while  working through the assignment? Why do you think so?  3.  What  question(s)  problem(s)  that  would you  you,  think  as he/she  the is  teacher,  ask  having  with  this  student  the  about  . assignment?  the What  responses w o u l d you expect to get? W h y do you think so?  4.  How  would  you  proceed to  help  this  student  having? What strategies w o u l d you use? Why?  with  the  problem(s)  he/she is  142 SCENARIO #3: PRIMARY (Grades K-3) • READING/SOCIAL (P. Arlin, personal communication, April 1986) A  bright  about  7 year old from  the main character in the book  completed  Prior the  Alabama named  to  STUDIES  Roberta was interviewed  called Caddie  by her teacher  Woodlawn which  she had just  reading.  the interview,  Roberta was asked to  reread  the introductory  paragraph  of  book:  In 1864 Caddie W o o d l a w n was eleven, and as wild a little tomboy as ever ran the w o o d of western Wisconsin. She was the despair of her mother and of her elder sister Clara. But her father watched her with a little shine of pride in his eyes, and her brothers accepted her as one of themselves without a question. Indeed, T o m , w h o was two years older and Warren, w h o was two years younger than Caddie, needed Caddie to link them together into an inseparable trio. Together they got in and out of more scrapes and adventures than any one of them could have imagined alone. A n d in those pioneer days Wisconsin offered plenty of opportunities for adventure to three wide-eyed, red-headed youngsters. Roberta lived  was then  while  following  being  asked  the  allowed  to  interview took  following reread  questions  the text  about  as many  where times  Caddie  Woodlawn  as she wanted. The  place—  Interviewer:  Where did Caddie W o o d l a w n live?  Roberta:  In Wisconsin.  Interviewer:  Is that in another  country  Roberta:  In America—is that  right?  Interviewer:  Why d o you think it is in America?  Roberta:  I know Wisconsin is a place in America.  Interviewer:  Is it a country?  Roberta:  Yes—is it?  Interviewer:  Is it a town?  Roberta:  No, it's like where we live in Alabama but  Interviewer:  Is it like where y o u live?  or is it in America?  different.  143 Roberta:  Yes, sort of but not all countries are alike—they have some thing different.  If you are b o m in Wisconsin you aren't an  Alabamian. Interviewer:  Is Wisconsin in the city or in the country?  Roberta:  It is in the country because there is a farm in the picture and Caddie must live there.  Before  continuing,  below:  What do Roberta's  Woodlawn  lived?  please  answer answers  the reveal  following about  question  in  the  her understanding  space of where  provided Caddie  144 To  get  a more  Woodlawn drawn then  explicit  lived.  a circle  The  on  asked to  description,  final  a blank  draw  in  Roberta  product sheet  is of  Alabama.  was  shown paper  asked  below.  to  to  illustrate  Initially,  the  where  Caddie  interviewer  represent  Birmingham.  As can be seen from  her drawing  had  Roberta  was  on the  next  page, she drew a circle around Birmingham and labeled it 'AL'. When  asked to  draw  the  United  States,  Roberta  drew  an  even  Wisconsin  in  the  larger  circle and  labeled it 'US'. On  request,  marked  it  Roberta  drew  a small circle for  "WC". Following this, she was asked to  on her map.  She selected  Birmingham to  jersey and drew  New  show its location. She continued  York within little circles inside Alabama. When  think of  United  another  States  and  place to  put  a little circle in Alabama outside placing California, Texas, and  she was asked if New  New  York was a  city or a state, she said that is was a city. Using the to  the  map  space provided below, Roberta drew: Why  do  please answer the following you  think  Roberta  drew  question with respect  that  map  this  way?  145 Name: ANSWER  SHEET - (PRIMARY/Reading/Social Studies)  Imagining  that  questions  as  you  are  the  teacher  as  possible:  thoroughly  in  this  (Feel  situation, free  to  please  use  the  answer back  the  of  the  following page  if  necessary) 1.  What Why  What  kind(s)  of  problem(s)  do  you  think  Roberta is having with  this lesson?  do you think so?  question(s)  do  you  think  Roberta was  asking  herself  as she  formulated  her responses? Why do you think so?  3.  What that  question(s) you  expect to  4.  How  think  would you, as the teacher, ask Roberta about she  get? Why  would  you  is  having  with  this  lesson?  What  the  responses  problem(s) would  you  d o you think so?  proceed to  help  Roberta with the  What strategies would you use? Why?  problem(s)  she is having?  146 SCENARIO #4: PRIMARY (Grades K-3) • SOCIAL STUDIES (A. Goran & B. Davis, personal communication, December 1986) Students in Social Studies classes in a school district in Maine were given a blank sheet of of the on  the  paper  and were asked to  maps submitted answer  sheet  draw  a map of the world.  by various students. provided,  indicate  the  Please examine grade  think each map came from and why you think so:  MAP A  level  of  the the  Below are samples maps carefully and student  that you  147  MAP B  -v  148 MAP  C  151 Name: ANSWER  Please  SHEET • (PRIMARY/Social Studies)  indicate  the  grade  level  of  the  child  you  think  each  map  came  from  and  why you think so:  Map  A  Map  B  Map  C  Map  D  Map E  Imagining  that  you  are  the  questions  regarding  the  map  each: (Feel free to 1.  What  kinds  use the of  teacher  in  drawing  task  back of the  problems  do  this  you  assignment? Why do you think so?  situation, and  the  please  grade  answer  level(s)  you  the  following  assigned  to  page if necessary) think  these  students  are  having  with  this  152 What worked  What  questions  do  you  through  the  assignment? Why do you think so?  question(s)  would  think  you,  problems that you think they would  How  you expect to  would  you  these  as  the  children  teacher,  were  asking  ask  are having with the  these  themselves  children  assignment? What  as  they  about  the  responses  get? Why do you think so?  proceed  to  help  having? What strategies would you  these  children  use? Why?  with the  problems  they  are  153 Name: SCENARIO #5: INTERMEDIATE (Grades 4-8) - MATH (B. Brown & L. Stofan, personal communication, February 1984) Students  in  grades 3 and  IS A FRACTION? indicate whether student  Please read through you think  the  and why you think  so.  (a) Part of an object  (b)  6 gave the  following the  question,  WHAT  responses carefully, and beside each one,  that is broken in pieces  things  like if a ruler looks like just a bunch  lines, its a bunch of fractions and you need to  know  them  number  (d) The number on top  (e)  the  response was given by a grade 3 or by a grade 6  Numbers that tell you different  (c) Half of a whole  responses to  is the  number of  pieces from the  Part of a whole thing (such as a pizza)  (f) A piece of a " h o l e "  (g)  Part of  something  (h)  Part of  a whole  thing  bottom  of  154 (i) Two numbers or more divided into two  (j) A whole  or more  number divided into separate pieces of  parts  numbers  (k) A math problem that you might solve in steps  (I) A number over another  number  (m) Tells if something is even or  not  (n)  Numbers which tell you how many are colored and how many are  (o)  Part of  not  something  (p) Something you use to  (q)  Equal parts  (r)  Something which  count how many halves  makes a sentence shorter  three are colored, its easier to  say 3/4  --  instead  of  saying four  parts  and  155 Name: ANSWER  SHEET - (INTERMEDIATE/Math)  Imagining you  that you  think  following  were  are the teacher in this situation given  questions  by  grade  3  as thoroughly  and  grade  and with regard to 6  students,  as possible: (Feel free  to  the answers  please  use  the  answer back  the  of  the  with  the  asking themselves as  they  page if necessary) 1.  What  kinds  of  concept of  2.  What  3.  What  you  do  you  think  these  that  would you  you,  think  as  they  responses would you expect to  4.  think  these  students  are  having  students  were  their responses? W h y do you think so?  question(s)  problems  do  "fraction"? W h y do you think so?  questions  formulated  problems  H o w would  you  proceed to  the  teacher,  are  having  ask with  these  students  fractions?  about  the  What  kind  of  problems  they  are  get? W h y do you think so?  help  these  students  having? What strategies would you use? Why?  with  the  156  SCENARIO #6: INTERMEDIATE (Grades 4-8) • MATH (N. Carroll, personal communication, May Students  in  fractions  having  taught The  a  grade unlike  equivalent  present  math  4  denominators.  fractions  assignment  class  and the  was  were  given  Prior addition  preceded with  to of  1986)  a worksheet  this  assignment,  fractions  a review  of  with the  on  the  addition  students similar  had  been  denominators.  addition  of  fractions  with like denominators.  The students were given the following problem to solve:  h + k =  These were their answers:  1. I don't know. 2. h + k = 2  3. h + h - k + h = 2 4 4. \ + k - 1 - 1 4-2  =  0 2  6  k + h = 1+ 1 -  4  f  2  =  3  8. h + k = 2+ 1  ^  = 6  1  1  3  h + k - 2 2  5. 4 + k = I  6.  7. h + h - 2 + 1 4-3  9.  2  2  10. h + k = 2 + 2 6  of  =» 4_  6  6  11. h + k = 2 + 1 4-3  = 2_ 1  12. 1{ + Jj; = 1 + 1 2x4  = 2_ 8  157 Name: ANSWER  SHEET - (INTERMEDIATE/Math)  Imagining  that  you  are  questions  as thoroughly  the  teacher  as you  in  can:  this  (Feel  situation, free  to  please  use  the  answer back  the  of  following  the  page  if  necessary) 1.  What  kinds  of  problems  do  you  think  these  students  are  having  with  this  as  they  about  the  assignment? Why do you think so?  2.  What worked  3.  What  questions  do  you  through  the  assignment? Why do you think so?  question(s)  problems would  4.  How  would  think  you,  that you think they  you expect to  would  you  these  as  the  children  teacher,  were  asking  ask  are having with the  these  themselves  children  assignment? What  responses  get? W h y do you think so?  proceed  to  having? What strategies would  help  these  children  you use? Why?  with  the  problems  they  are  158  SCENARIO #7: INTERMEDIATE (Grades 4-8) (N. In  a grade  posed  the  relationship  Why moon  7 general following  question  you  think  from the  The following  "One  science class of  between the  do  to  average to  her  students  earth and the  we  always  earth (no  1983)  average  giving  a  ability, brief  the  teacher  lesson on  the  moon:  see  the  matter where  same we  side is facing space and we get  it is daytime  high  after  responses were given by different  "Because when  - SCIENCE  Darling, personal communication, April  side  are on  (near the  side)  of  the  earth)?  students in the  class:  stuck with just one side."  it is on the  other side."  "Because that side always faces the s u n . "  "I  think we  and the  always see the  moon  rotates,  same side of  by the time  it  the  moon  gets dark  out  because when we it  is back where  rotate it was  before."  "Because  the  moon  rotates  one way  and we  rotate  the  other  way  so  when  we see it it is always the same side."  "We  always see the same side because the sun never shines on one side."  "Because that's  we  move  at  the  same  time  why."  "Because the m o o n  is moving with us."  and  speed  as  the  moon  does  and  159 Name: ANSWER  SHEET - (INTERMEDIATE/Science)  Imagining  that  you  are  questions  as thoroughly  the  teacher  as you  in  this  can: (Feel  situation, free  to  please  use  the  answer back  the  of  following  the  page  if  necessary) 1.  What  kinds  of  problems  do  you  think  these  students  are  having  with  this  as  they  about  the  assignment? Why do you think so?  2.  What worked  3.  What  questions  do  you  through  the  assignment? W h y do you think so?  question(s)  would  think  you,  problems that you think they w o u l d you expect to  4.  How  would  you  these  as  the  students  teacher,  were  asking themselves  ask  are having with the  these  students  assignment? What  responses  get? W h y do you think so?  proceed  to  having? What strategies would  help  these  students  you use? Why?  with  the  problems  they  are  160 SCENARIO #8: INTERMEDIATE (Grades 4-8) - SOCIAL STUDIES (A. Goran & B. Davis, personal communication, December 1986) Students sheet  of  of the on  in Social Studies classes in a school district in Maine  the  paper  and were asked to  maps submitted answer  sheet  draw  a map of the world.  by various students. provided,  indicate  the  Please examine grade  think each map came from and why you think so:  MAP A  were given a blank  level  of  the the  Below are samples maps carefully student  and  that you  162  MAP  C  165 Name: ANSWER  Please  •  SHEET - (INTERMEDIATE/Social Studies)  indicate  the  grade  level  of  the  child  you  think  each  map  came  from  and  why you think so:  Map  A  Map  B  Map  C  Map  D  Map E  Imagining  that  you  are  the  questions  regarding  the  map  each: (Feel free to use the 1.  What  kinds  of  teacher  in  drawing  task  back of the  problems  do  this  you  assignment? Why do you think so?  situation, and  the  please  grade  answer  level(s)  you  the  following  assigned  to  page if necessary) think  these  students  are  having  with  this  166 2.  What  questions  worked through  3.  What  question(s)  do  you  would  would you expect to  How  would  you  these  students  were  asking themselves  as  they  about  the  the assignment? W h y d o you think so?  problems that you think  4.  think  you, they  as  the  teacher,  ask  are having with the  the  students  assignment? What responses  get? Why?  proceed to  help  these  students  having? What strategies would you use? Why?  with  the  problems they  are  APPENDIX B OTHER VARIABLES O F INTEREST:  Concern for  Problem  Finding  Demographic Variables  168 NAME:  Please  answer  the  following  questions  regarding  your  background  and  experience(s)  as a teacher:  1.  Why d o you teach?  2.  H o w many years of teaching experience do you have?  3.  What subject(s)  4.  What  5.  List any other  have you  age group(s)  interaction  have you  positions  or  taught?  experiences that you  have  had that required  direct  with children and/or adolescents.  6.  W h y are you participating  7.  How  familiar  practice?  taught?  are  you  in this inservice?  with  Piaget's  work  and  its  implications  for  classroom  


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