An  Empirical  Study  of  the  Process  of  Crafting  and  Using  Definitions     By     Angela  Jean  Little       A  dissertation  submitted  in  partial  satisfaction  of  the     requirements  for  the  degree  of     Doctor  of  Philosophy     in   Science  And  Mathematics  Education     in  the     Graduate  Division     of  the     University  of  California,  Berkeley           Committee  in  charge:     Professor  Andrea  A.  diSessa,  Chair   Professor  Alan  H.  Schoenfeld   Professor  Eve  E.  Sweetser       Fall  2013 An  Empirical  Study  of  the  Process  of  Crafting  and  Using  Definitions     Copyright,  2013     By     Angela  Jean  Little Abstract     An  Empirical  Study  of  the  Process  of  Crafting  and  Using  Definitions     by     Angela  Jean  Little     Doctor  of  Philosophy  in  The  Graduate  Group  in  Science  and  Mathematics  Education     University  of  California,  Berkeley     Professor  Andrea  diSessa,  Chair       In  this  dissertation  I  analyze  the  process  of  crafting  definitions  whose  purpose  is   classification.    The  context  I  examine  is  undergraduate  upper-­‐division  physical   science  majors  defining  and  naming  sub-­‐categories  of  a  physical  phenomenon  in  the   context  of  a  design  task  over  an  extended  period  of  time.    The  goal  of  the  design  task   is  one  of  classification:  help  people  better  identify  the  phenomenon  out  in  the  world.         I  first  develop  an  analytic  framework  for  the  process  of  improving  a  definition.    This   framework  involves  an  interplay  between  four  main  elements:    the  current  state  of   the  definition,  criteria  for  what  makes  a  good  definition,  examples,  and  definitionally   unarticulated  knowledge  (DUK).    By  DUK,  I  mean  implicit  judgments  that  definers   make  about  categorization  that  have  not  been  incorporated  into  the  definition,  per   se.    I  show  how  criteria  and  the  practices  associated  with  meeting  them  guide  the   refinement  work.         Once  participants  craft  their  definition  a  question  emerges:    to  what  extent  does  it   serve  them  as  a  tool  when  they  make  sense  of  new  candidate  examples?    I  zoom  in   on  moments  where  new  candidate  examples  are  introduced  by  participants  or  the   facilitator  and  describe  the  extent  to  which  students  rely  on  their  own  crafted   definition.    The  consideration  of  a  new  candidate  example  can  sometimes  lead  to   participants  naming  a  new  subcategory  and  I  analyze  this  process  as  well.       The  detailed  empirical  analysis  is  bookended  by  a  set  of  workshop  design  principles   that  made  the  analysis  possible  and  some  educational  implications  of  the  work.    I  lay   out  a  set  of  design  principles  for  creating  activities  wherein  people  engage   productively  in  collaboratively  defining.    I  conclude  by  exploring  the  educational   implications  for  college  physics  teaching  as  well  as  more  general  instances  where   one  is  interested  in  crafting  a  definition  for  the  purpose  of  classification 1 Acknowledgements     I  have  to  thank  my  parents  first  and  foremost.    Their  encouragement  and  emotional   support  all  throughout  my  K-­‐12,  college,  and  graduate  schooling  were  central  to  my   finishing  this  Ph.D.    Others  that  played  particularly  pivotal  mentorship  roles  along  the  way   were  my  high  school  physics  teacher,  Michael  Lampert,  and  my  undergraduate  honors   thesis  advisor,  Stamatis  Vokos.    Without  Michael  suggesting  that  I  major  in  physics  and   without  the  incredible  community  and  support  that  I  found  in  SPU’s  physics  department,   including  Stamatis’  wonderful  mentorship  and  support  in  particular,  I  would  not  be  where  I   am  today.    I  also  am  deeply  grateful  for  all  of  the  graduate  and  undergraduate  students   involved  in  The  Compass  Project,  a  program  I  co-­‐founded  during  my  time  as  a  Ph.D.  student   in  the  physics  department  at  Berkeley.    I  discovered  a  calling  in  Compass  and  developed   friendships  that  supported  me  when  I  thought  I  would  leave  graduate  school.    Compass   gives  me  hope  for  positive  change  in  the  world.       Before  I  thank  them  for  their  intellectual  support  of  my  research,  I  must  first  thank  my   committee  members  for  their  mentorship  during  a  difficult  decision  I  had  to  make  about   switching  from  my  physics  Ph.D.  program  to  an  interdisciplinary  program.    Eve  Sweetser   was  first  in  this:    I  took  her  endlessly  interesting  class  The  Mind  and  Language  during  a   semester  when  I  was  beginning  to  understand  that  I  had  more  interdisciplinary  interests   but  was  still  a  physics  graduate  student.    We  discussed  the  possibility  of  my  being   interdisciplinary  between  the  linguistics  and  physics  departments.    It  turned  out  that  an   interdisciplinary  graduate  group  already  existed  in  STEM  education,  however.    Around  the   same  time,  I  happened  to  be  hired  to  co-­‐design  and  co-­‐teach  a  course  with  Alan  Schoenfeld   and  Dor  Abrahamson.    I  may  not  recall  the  exact  specifics  (was  it  Alan  or  Dor  who  first   suggested  I  consider  SESAME?),  but  Alan  was  instrumental  in  supporting  me  to  switch  over   and  mentoring  my  early  education  research.    He  introduced  me  to  Andy  diSessa  who   empathized  with  my  physics  to  STEM  education  transition  and  turned  into  my  primary   advisor  as  I  finished.         Andy  was  also  incredibly  supportive  when  I  had  difficult  injuries  from  a  bicycle  accident,   helping  me  get  back  on  my  feet  by  meeting  with  me  weekly  for  a  couple  of  months  when  I’d   finally  healed.    Thank  you  to  all  of  my  aunts  and  uncles  and  grandmother  (rest  in  peace,   Lucy)  who  banded  together  and  supported  me  financially  when  I  would  have  gone  into   debt  from  the  dental  bills  associated  with  the  accident.    It  helped  me  stay  focused  on  my   dissertation.             Intellectually,  I  am  thankful  for  a  committee  that  has  always  treated  me  like  a  colleague.    I   really  can’t  tell  you  how  much  I  am  appreciative  of  that.    Every  step  of  the  way,  their   feedback  has  felt  much  less  like  harsh  judgment  and  much  more  like  thoughtful  experts   who  want  the  best  for  my  burgeoning  work.    I  think  I  will  forever  more  have  Andy’s  voice   in  my  head  when  I’m  editing  papers  that  make  general  sweeping  statements  (including  my   own):  “Do  you  really  mean  EVERYTHING  works  like  that?  I  bet  I  can  find  a  counter  example!”   I’ve  learned  so  much  about  all  aspects  of  writing  and  researching  from  Andy.    From  Alan,  I   have  to  say  that  foremost  I  think  about  his  incredible  skill  of  top-­‐level  organization.    I’ve  so     i often  found  myself  in  his  office  tossing  out  a  set  of  mushy  ideas  and  then  getting  to  watch   as  he  organizes  it  into  a  coherent  flow  of  ideas  that  make  so  much  more  sense.    It’s  such  an   incredibly  useful  skill  that  has  helped  me  improve  my  work  a  ton.    From  Eve,  I  picked  up   my  fascination  with  and  close  attention  to  language.    She  showed  me  how  much  comes  out   of  seemingly  subtle  differences  in  language.         In  addition  to  my  committee,  SESAME  and  EMST  have  an  incredible  group  of  graduate   students,  many  of  whom  I  count  as  friends  as  well  as  colleagues.    A  huge  thank  you  to  the   Patterns  and  Functions  research  groups  where  I  received  incredible  feedback  on  my  work   and  found  a  warm  and  supportive  community  that  cared  so  deeply  about  students  and   STEM  education.    Thank  you  to  each  and  every  member.    In  particular,  I  don’t  think  I  would   have  survived  my  qualifying  exam  studying  without  Kim,  who  made  it  extra  special  with   her  lobster  cookies.    Hillary:    I’m  so  glad  we  could  talk  about  work  in  the  context  of  life   goals  more  generally  and  share  strategies  for  holistic  living.    Lauren  and  Colleen,  what   better  officemates  could  a  lady  ask  for?    Thank  you  to  people  who  provided  critical  last   minute  feedback  on  aspects  of  my  dissertation:    Gina,  Lama,  Lauren,  and  Colleen.    Thank   you  to  Kate  Capps,  also,  our  SESAME  administrative  support  person  who  listened,  cared   deeply  about  us,  shepherded  us  through  all  of  the  bureaucratic  requirements.      Although   she  isn’t  “SESAME,”  I  should  also  thank  Anne  Takizawa,  our  administrative  support  person   for  graduate  students  in  the  physics  department.    She  was  one  of  my  biggest  supporters   during  my  three  years  in  the  physics  department  and  I  was  so  glad  I  had  her  as  a  listening   ear.    Plus,  she  made  me  applesauce  when  I  had  my  bike  accident  and  couldn’t  eat  solid   foods.     There  were  other  important  supports  to  my  dissertation:    my  counselor  Lisa,  my  friend   Denia  who,  what  can  I  say,  we  “ran  the  race  together,”  my  partner  Mel  who  drove  me  to  the   coffee  shop  on  cold  mornings  when  I  lacked  motivation  to  write  and  helped  me  with   annoying  Microsoft  Word  format  issues  near  the  end.    My  friend  Jess,  who  listened  to  me   whenever  I  needed  it  that  final  semester.    I  am  incredibly  thankful  for  two  coffee  shops,   their  warm  staff  members,  and  tasty  coffee:    The  Coffee  Studio  in  Chicago,  IL,  and  Local  123   in  Berkeley,  CA.    Thank  you  for  being  a  positive  space  where  I  looked  forward  to  writing   and  for  all  the  cappuccinos  and  “physics-­‐ccinos”  that  got  me  through  this.           Well,  I’m  starting  to  tear  up  now  because  you  see  how  my  finishing  my  Ph.D.  was  due  to  the   support  of  so  many  incredible  people  along  the  way.    I  mean,  seriously.  It’s  ridiculous.     Truly,  I  couldn’t  have  done  it  without  them.    I  hope  to  live  a  life  of  supporting  others   because  they  have  supported  me.         I  would  like  to  thank  some  additional  people  that  did  not  fit  neatly  into  any  of  the  previous   paragraphs.    Kathleen  Metz  for  all  of  the  wonderful  discussions  about  qualitative  research   methodology  both  in  her  course  and  as  a  member  of  my  qualifying  exam  committee.    Also,  I   learned  a  great  deal  and  had  way  too  much  fun  in  Coye  Cheshire’s  quantitative  methods   course.    Best  course  ever!    Marjorie  Shapiro  and  Bob  Jacobsen  for  their  support  of  my   educational  interdisciplinary  interests  in  the  physics  department.    Nicci  Nunes  for  hiring   me  to  teach  for  Cal  Teach.    It  was  an  incredible  learning  experience  and  influenced  my   switch  to  SESAME.    Nicci  herself  provided  so  much  wonderful  mentorship.    Also  Dor     ii Abrahamson  who  I  co-­‐taught  the  Cal  Teach  course  with  for  three  years  and  learned  a  great   deal  from.    And  Nilofar,  for  all  of  the  support  and  friendship  during  graduate  school,   particularly  during  my  qualifying  exam  preparation  (yay  for  de-­‐stressing  with  David   Attenborough  documentaries  about  birds).    And  Kristin,  for  all  the  fun  swimmuritos  and  le   med  dinners  and  encouraging  notes  along  the  way.    And  all  of  my  wonderful  Cal  Teach,   CS301,  physics,  and  Compass  students  who  inspired  me  and  who  I  learned  a  great  deal   from.    And  Mari  and  Eric  who  co-­‐thought  with  me  about  my  dissertation  title.         Finally,  I  thank  all  of  my  dear  friends  and  family.    I’m  so  thankful  for  you  all  for  many   reasons,  but,  pertaining  to  this  dissertation,  your  support  really  got  me  through  graduate   school.                                                                         iii Table  of  Contents   Chapter  1:  Preface  and  Literature  Review  ......................................................................  1   Personal  Preface:  Background  in  Designing  Curriculum  on  Crafting  Definitions  ......  1   Literature  Review  .............................................................................................................................  1   Existing  Research  On  Defining  ...................................................................................................................  1   Purposes  of  Crafting  Definitions  ...............................................................................................................  4   Research  Questions  ........................................................................................................................................  4   Related  Research:  Influences  on  Classification  and  the  Context  Dependent  Nature  of   Knowledge  ..........................................................................................................................................................  4   Related  Research:  The  Process  of  Crafting  Representations  ........................................................  5   Dissertation  Overview  .....................................................................................................................  6   Chapter  2  –  Workshop  Design  and  Data  Collection  Methodology  ...........................  8   Summary  of  Chapter  .........................................................................................................................  8   Summary  of  Workshops  and  Description  of  Threshold  .......................................................  8   What  is  Threshold?    What  contributes  to  people’s  experience  with  the  phenomenon?  ..  8   Overview  of  Workshops  ...............................................................................................................................  9   Design  Goals  and  Design  Choices  ..............................................................................................  10   Design  Variable  Choices  ............................................................................................................................  11   Workshop  and  Interviewer  Script  ............................................................................................  16   Introductory  Script  For  Both  Small  Group  and  Individual  Participants:  .............................  17   Script  for  Interaction  After  Participants  Craft  Definition  ...........................................................  20   Artifacts  Collected  ..........................................................................................................................  22   Room  Context  and  Video  Set-­‐up  ................................................................................................  22   Transcript  Convention  .................................................................................................................  23   Conclusion  ........................................................................................................................................  23   Chapter  3  –  Three  Episodes  Of  Definitional  Refinement:  Identification  of                         Criteria  and  Practices  Associated  with  Meeting  Them.............................................  24   Introduction  .....................................................................................................................................  24   Analysis  Methods:    Development  of  Analytic  Focus  and  Creation  of  Analytic                   Framework  for  Definitional  Refinement  ................................................................................  24   Development  of  Initial  Framework  ......................................................................................................  24   Development  of  an  Improved  Analytic  Framework  ......................................................................  25   Adding  A  New  Framework  Element  .....................................................................................................  26   Final  Framework  For  The  Process  of  Definitional  Refinement  ................................................  27   Better  Criteria  Identification  ...................................................................................................................  28   Identification  of    ‘Practices  Associated  with  Meeting  Criteria’  .................................................  28   Data  and  Analysis  ...........................................................................................................................  28   Overview  ..........................................................................................................................................................  28   Definitional  Refinement:  Episode  One  ....................................................................................  29   Definitional  Refinement:  Episode  Two  ...................................................................................  34   Definitional  Refinement:  Episode  Three  ................................................................................  39   Conclusion  ........................................................................................................................................  44   Chapter  4  –  A  Cross-­‐Case  Analysis  of  How  Participants  Make  Sense  of  New           candidate  examples  ..............................................................................................................  47   Introduction  .....................................................................................................................................  47   Methodological  Details  about  DUK  ...........................................................................................  48   Identifying  Use  of  Crafted  Definition  in  Transcript  Data  ............................................................  48     iv Identifying  Types  of  Definitionally  Unarticulated  Knowledge  (DUK)  ...................................  48   Implied  DUK  ....................................................................................................................................................  49   Outside  DUK  ....................................................................................................................................................  51   Section  One:  Cross-­‐Case  Comparison  Of  Stoplight  Example  ............................................  51   Focal  Example:  The  Stoplight  ..................................................................................................................  51   Presentation  of  the  Data  and  Analysis  ................................................................................................  51   Group  A  Decides  To  Name  A  New  Subcategory:  Data  and  Analysis  .......................................  52   Student  Q  Accepts  A  Version  Of  The  Stoplight  Example:  Data  And  Analysis  .....................  55   Student  G  Rejects  The  Stoplight  Example:  Data  and  Analysis  ...................................................  60   Comparison  Analysis  Across  Group  A,  Student  Q,  and  Student  G  ...........................................  65   Conclusion  to  Section  One  ........................................................................................................................  67   Section  Two:  How  Do  Participants  Name  and  Populate  A  New  Subcategory?  ...........  69   Episode  One:  Group  A  Names  and  Populates  “Systems”  of  Threshold  .................................  69   Episode  Two:    Student  G  Names  and  Populates  The  Subcategory  of  Gradual     Thresholds  .......................................................................................................................................................  74   Section  Two  Conclusion  .............................................................................................................................  76   Chapter  Four  Conclusion  .............................................................................................................  77   Chapter  5  –  Conclusion  ........................................................................................................  78   Overall  Results  ................................................................................................................................  78   Educational  Implications  and  Future  Work  ..........................................................................  78   Crafting  Definitions  As  Supporting  Students  In  Making  Conceptual  and  Mathematical   Connections  Between  Different  Areas  of  Physics  ...........................................................................  79   Epistemological  and  Affective  Affordances  of  Undergraduates’  Crafting  Definitions  ....  82   Future  Research  Direction:    The  Importance  of  Engaging  in  Crafting  Definitions  To     Uncover  Tacit  Influences  On  Classification  And  The  Role  of  Particular  Kinds  of     Examples  ..........................................................................................................................................................  87   Conclusion  ........................................................................................................................................  89   References  ...............................................................................................................................  91                 v Chapter  1:  Preface  and  Literature  Review   Personal  Preface:  Background  in  Designing  Curriculum  on  Crafting  Definitions   This  dissertation  is  about  the  process  of  crafting  and  using  definitions.    Although  I   focus  on  particular  phenomena  and  contexts  that  are  physics  related,  my  goal  is  to   build  a  general  framework  for  the  process.    I  will  also  examine  the  benefits  of   educators  involving  their  students  in  crafting  definitions  when  I  address  the   educational  implications  of  my  work.         Defining  is  an  incredibly  rich  process  that  plays  out  in  a  wide  range  of   contexts.    In  graduate  school,  my  teaching  has  frequently  involved  supporting  my   students  in  crafting  conceptually  central  definitions.    I  developed  and  taught   curriculum  where  freshman  physics  students  worked  to  define  mechanical  wave  in   the  context  of  addressing  a  research  question  about  what  earthquakes  can  tell  us   about  the  interior  of  the  earth.    I  co-­‐led  a  discussion  where  a  class  of  freshman  in   physics  had  to  come  to  consensus  on  the  definition  of  physics  model  to  support  their   work  in  using  models  in  pursuing  their  own  research  questions.       Defining  happened  in  more  than  just  the  physics  classes  I  developed.    I  facilitated   workshops  for  instructors  who  had  to  co-­‐design  curriculum  together.    They  thus   needed  a  set  of  agreed-­‐upon  principles  to  appeal  to  during  design  decisions.    The   process  of  articulating  these  principles  was  a  definitional  conversation:  we  asked   everyone  to  share  out  examples  of  educational  experiences  that  had  been  positive   and  negative.    We  then  drew  out  themes  and  used  this  as  a  starting  point  to  refine  a   set  of  articulated  principles  for  what  we  wanted  to  achieve  in  the  classroom.    You   could  call  it  a  definition  for  ideal  classroom.    I’ve  recently  led  workshops  on  defining   proudness  where  participants  share  out  examples  of  things  they’ve  been  proud  of   and  develop  a  set  of  principles  to  create  educational  experiences  where  their   students  can  be  proud  of  their  work.    As  an  educational  researcher  working  on  the   process  of  defining,  I  have  even  taken  on  the  meta-­‐task  of  defining  defining,  even.     The  process  of  crafting  a  definition  is  an  interesting  and  challenging  design  task.   My  background  in  designing  and  facilitating  definitional  discussions  heavily  informs   the  research  described  here.    In  addition  to  building  an  analytic  framework  for  the   process  of  crafting  definitions,  I  will  speak  to  some  design  principles  for  building   productive  definitional  conversations.     Literature  Review   Existing  Research  On  Defining   The  process  of  crafting  definitions  is  an  important  yet  understudied  part  of   professional  practice.    STEM  classrooms  that  take  an  inquiry  approach  and  attempt   to  bring  students  into  authentic  scientific  or  mathematical  practice  often  engage  in   crafting  definitions.    Researchers  have  examined  classroom  conversations  where   students  define  terms  like  alive  (Warren  and  Roseberry  2011;  Bang,  et  al.  2013)  and   whether  orcas  are  whales  or  dolphins  (Engle  and  Conant,  2002  ).    In  these  cases,   definitional  conversations  underlie  framework  building  for  topics  like  productive   disciplinary  engagement  and  how  desettling  can  support  the  inclusion  of   epistemological  commitments  of  non-­‐dominant  communities.    As  for  research     1 examining  the  process  of  crafting  definitions,  however,  there  is  surprisingly  little  in   science  education,  though  there  is  some  in  mathematics  education.       One  of  the  earliest  studies  to  describe  the  process  of  crafting  definitions  in   mathematics  is  Fawcett’s  book  (Fawcett,  1938)  on  teaching  a  high  school  geometry   course  that  took  a  different  approach  from  the  more  traditional  textbook  problems   of  the  time.    Instead,  Fawcett  supported  his  students  in  the  practices  of  professional   mathematicians,  which  included  crafting  and  defending  definitions.    Schoenfeld,  in   his  book  Learning  To  Think  Mathematically,  gives  a  thorough  summary  (Schoenfeld,   1992):     Fawcett's  goals  were  that  students  develop  a  good  understanding  of  the  subject   matter  of  geometry,  the  right  epistemological  sense  about  the  mathematics,   and  a  sense  of  the  applicability  of  the  reasoning  procedures  that  they  had   learned  in  geometry  to  situations  outside  of  the  mathematics  classroom.    In   order  for  this  to  happen,  he  believed,  (1)  the  students  had  to  engage  in  doing   mathematics  in  a  way  consistent  with  his  mathematical  epistemology,  (2)  the   connections  between  mathematical  reasoning  in  the  formal  context  of  the   classroom  and  mathematical  reasoning  outside  of  it  would  have  to  be  made   explicit,  and  (3)  the  students  would  need  to  reflect  both  on  their  doing  of   mathematics  and  on  the  connections  between  the  reasoning  in  both  contexts...     Fawcett  pointed  out  that  definitions  have  consequences:  in  his  school,  for   example,  there  was  an  award  for  the  "best  teacher."    Many  students  favored  the   librarian  -­‐-­‐  but  was  the  librarian  a  teacher?    Or,  he  used  sports  as  an  analogy.     In  baseball,  for  example,  there  might  be  varying  definitions  of  "foul  ball"  (is  a   fly  ball  that  hits  the  foul  pole  fair  or  foul?)  -­‐-­‐  but  once  one  sets  the  rules,  the   game  can  be  played  with  consistency.    After  such  discussions,  Fawcett  notes   "[n]o  difficulty  was  met  in  leading  the  pupils  to  realize  that  these  rules  were   nothing  more  than  agreements  which  a  group  of  interested  people  had  made   and  that  they  implied  certain  conclusions"  (p.33).    In  the  mathematical  domain,   he  had  his  students  debate  the  nature  and  usefulness  of  various  definitions.     Rather  than  provide  the  definition  of  "adjacent  angle,"  for  example,  he  asked   the  class  to  propose  and  defend  various  definitions.           Fawcett’s  work  was  some  of  the  earliest  to  suggest  that  it  was  important  to   engage  students  in  doing  what  mathematicians  do  and  also  show  that  it  was   possible  to  do  so.   Contemporary  research  that  focuses  on  crafting  definitions  generally  falls   into  two  camps:  existence  proofs  that  students  can  productively  engage  in  crafting   definitions  for  important  terms  over  weeks-­‐long  curriculum  (Atkins  and  Salter,   2010;  Swinyard,  2011)  and  more  general  framework  building  (Nachlieli  and  Sfard,   2003;  Zandieh  and  Rasmussen  2010).    These  two  more  general  frameworks  focus   on  opposite  ends  of  the  spectrum  with  respect  to  the  length  of  time  that  people   engage  in  defining.    Nachlieli  and  Sfard’s  work  on  “the  activity  of  defining”  (AoD)  is   focused  on  shorter  clarifying  moves.    They  define  AoD  as  “the  activity  that  aims  at   keeping  communication  effective  by  clarifying  the  use  of  words”  and  note  some   important  practices  associated  with  this  that  they  name  “meta-­‐discursive”  practices.       2