Page 1


James Calleja    


Plan for  the  Session        

Thinking about  Student  Agency  &  Responsibility  





What do  you  expect  to  get  from  today’s  session?   Aims  of  the  session  

¼ h  

Working on  a   Task  

Teachers work  on  the  ‘Mathematics  around  a   Giant  chair’  task  

½ h  

Sharing reflections  about  your  experiences   working  on  this  task  

¼ h  

Watch and  discuss  a  video  of  Keith  using  this  task   with  his  Year  7  class  

¾ h  

Follow-­‐up Reflection     Lesson  Video  

A Teacher’s   Narrative  

Read and  reflect  on  an  excerpt  from  a  teacher’s   reflections  as  she  implements  changes  to  adopt   more  inquiry-­‐based  pedagogies.  

½ h  

Which issues  do  you  find  most  striking?  Why?   Shifting   Responsibility   to  Students:   The  Case  of  two   Teachers  

What actions  might  the  teacher  take  to  shift  more   instructional  responsibilities  onto  the  students?  

The Didactical   Contract  

What is  it?  Why  may  it  be  important  to  discuss,   share  and  establish  with  your  students?  

Which ones  would  you  ‘risk’?  How  would  you  do   that?  With  whom,  when  and  why?  

¾ h  

½ h  


Aims of  the  session   For  today’s  session  we  will  have  the  following  aims:   o To  explore  opportunities  where  the  teacher  may  shift  instructional   responsibilities  to  the  students   o To  reflect  upon  concerns  in  giving  students  more  responsibilities  over  their   learning   o To  understand  the  responsibilities  and  the  roles  of  students  and  the  teacher   within  a  collaborative  classroom  community   o To  create  an  effective  classroom  culture  based  on  habits,  rules,  expectations,   behaviours,  actions,  interactions,  beliefs  and  values  which  the  teacher  and   the  students  establish,  understand  and  share  


Teaching and  Learning  Mathematics  through  Inquiry  


30 min  

  This  is  how  you  will  work  on  the  ‘Mathematics  around  a  Giant  chair’  task:     • The  task  is  presented  to  you  

2  minutes  

• You have  some  time  to  work  individually  generating  your  own   questions  about  the  displayed  photo  

3  minutes  

• It is  now  time  for  you  to  work  in  groups  of  three  or  four  to   investigate  a  question  of  your  choice.   15  minutes   • You  will  present  and  share  your  ideas  with  the  whole  group    

10  minutes  



15 min  

As a  whole  group  you  are  asked  to  reflect  on  the  following  questions:   1. Comment  on  your  experience  working  on  this  task.   2. Comment  on  the  characteristics  of  the  ‘Mathematics  around  a  Giant  chair’  task.   3. For  which  year  group,  do  you  think,  would  this  task  be  suitable?  Why?   4. What  do  you  anticipate  would  be  the  challenges  for  students  working  on  this  task?   How  would  you  try  to  address  these?   5. Would  you  consider  doing  this  task  with  one  of  your  classes  next  year?  Why?   6. Would  you  present  the  task  in  a  similar  way  as  it  was  presented  to  you  or  would  you   do  it  differently?  Why?  How?    

Teaching and  Learning  Mathematics  through  Inquiry  


The Giant Red Chair

! Student!Name:!________________________________! Students!I!am!working!with:!________________________________________________________________! ! Mathematical9related!questions:!(individually)!

Nehmen Sie Platz! !

! !

Question that we are going to answer is… !



Teaching and  Learning  Mathematics  through  Inquiry  


10 min  

You will   now   watch   a   video   of   a   teacher   (Keith)   using   the   ‘Mathematics   around   a   Giant  chair’  task  with  his  Year  7  (form  one)  class  of  students.   Note   how   the   teacher   structures   the   lesson,   the   mathematical   ideas   valued,   the   difficulties  that  students  encounter  and  how  the  teacher  deals  with  these  issues.   For   the   follow-­‐up   discussion,   you   are   encouraged   to   write   down   some   notes/points   you  might  see  as  important.   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________      


20 min  

What are  your  comments  about  the  lesson?   Would  you  structure  the  lesson  as  the  teacher  did  or  would  you  do  it  differently?   Who  generates  the  mathematical  ideas  that  get  discussed?  Who  evaluates  and/or   responds  to  these  ideas?   How  deeply  do  students  get  to  explain  their  ideas?   How  does  the  teacher  respond  to  students’  struggles?   To  what  extent,  do  you  think,  the  teacher  stimulated  student  agency  and  shifted   responsibility  onto  the  students?  


Teaching and  Learning  Mathematics  through  Inquiry  



30 min  

The   following   extract   is   an   account   by   Brea,   a   teacher   of   mathematics,   who   highlights  changes  in  her  thinking  and  practice  as  she  shifts  to  implement  more   inquiry-­‐based  pedagogies.   As   you   read   underline,   highlight   or   take   notes   on   aspects   that   strike   you   in   one   way  or  another.     These  issues  you  highlight  will  guide  our  follow-­‐up  discussion.   For   most   of   my   teaching   career,   I   felt   my   job   was   to   simplify   mathematics.   As   a   student  of  mathematics,  I  was  led  to  believe  that  math  should  be  simple,  then  when  I   started   teaching,   I   thought   that’s   what   my   job   was   –   to   make   math   simple   into   little   bits   so   the   students   could   consume   it   and   regurgitate   it.   So   I   aimed   to   cover   the   curriculum  in  consumable  bits  that  could  easily  be  delivered  and  tested.   I   planned   to   teach,   not   prepared   to   teach.   Units   were   outcomes   based,   laid   out   in   a   day-­‐by-­‐day  orderly  manner.  I  delivered  lessons  with  notes  already  compiled  with  set   examples,  complete  with  what  pages  and  questions  to  do  in  the  text.  I  did  not  realize   how  important  the  intellectual  part  of  this  job  is  and  how  I  very  easily  could  get  or  did   get  wrapped  up  in  the  skills  and  techniques  of  what  to  do  in  a  classroom.   Discourse   has   always   been   fundamental   in   my   classroom,   even   when   I   wasn’t   really   working  in  inquiry.  Conversation  and  dialogue  has  been  the  basis  of  my  class.  So  the   notion   of   relationship   and   conversations   with   kids   was   always   there,   but   I   never   stepped  outside  of  my  preplanned  boundary.  In  my  classes,  I  would  think  kids  were   asking  ‘good’  questions,  but  I  now  realize  they  were  for  clarification  or  procedural.  We   never  critically  entered  a  topic,  looked  at  the  bloodlines  or  cared  for  it  in  a  way  that   honored   it.   If   a   student   asked   a   question   that   seemed   off   topic   or   confusing   to   me,   I   would   seldom   really   listen,   often   dismissing   it.   Even   though   we   might   discuss   more   than  one  way  to  a  solution  of  an  assigned  problem,  there  was  still  a  solution,  that  is,   the   problem   was   treated   as   closed.   The   focus   of   my   efforts   then   would   be   on   the   students   and   building   relationships,   having   conversations   etc.   At   times   I   felt   I   was   doing  a  good  job  because  I  was  liked  and  I  liked  the  kids  also.  Each  semester  brought   newness   in   the   form   of   students   but   the   topics   were   set,   flattened   and   I   wondered   how  much  longer  I  could  do  this.   Since   starting   to   engage   in   the   inquiry   kind   of   work   that   we   are   doing   in   my   classroom,   mathematics   has   become   beautiful   again.   I   want   my   students   to   understand  that  mathematics  is  not  simple,  that  it  is  complex  and  complicated,  that  it   does  exist  in  the  world,  that  it  is  a  ‘living  discipline’,  that  it  has  bloodlines.  I  want  them   to   understand   that   there   are   patterns,   but   there   are   also   no   answers,   there   is   no   certainty.  When  they  enter  into  the  field,  they  are  contributing  in  some  way  to  it,  but  it   is   not   meant   to   be   simple   and   easy.   I   am   finding   that   it   is   the   structure   of   the   mathematics  and  the  patterns  and  the  connections  that  seems  to  keep  coming  up  as   an  entry  point  for  me  to  be  able  to  start  to  look  at  something  to  do  with  the  kids.  It  is  


Teaching and  Learning  Mathematics  through  Inquiry  

through its   structure,   patterns   and   connectedness   I   can   see   many   possibilities.   Where   does   this   come   from?   Why   do   we   still   talk   about   it?   How   does   it   live   and   contribute   to   the  world  today?   There   are   times   I   see   clearly   the   mathematical   connections   either   through   the   structure   of   math,   its   beauty,   complexity   or   imagery.   The   world   has   opened   up   and   through   discourse   math   presents   itself   as   complicated,   uncertain,   and   unfinished.   It   is   no  longer,  as  tends  to  be  in  the  math  classroom,  certain,  linear,  and  algorithmic.  I  have   begun  to  see  more  connections  within  topics  and  in  interdisciplinary  ways.  The  more   we  enter  into  a  topic,  the  more  exciting  it  becomes,  it  all  seems  new  to  me  again,  it  is   exciting   and   alive.   But   there   are   challenges.   For   example:   How   do   I   open   topics   in   a   generous  way?  Do  I  look  for  the  topic  in  the  world  or  see  the  world  through  the  topic?   Discourse  continues  to  be  fundamental  in  my  classroom.  Topics  always  open  up  with   conversation.   Students   are   always   in   partner   or   groups   talking.   They   are   always   writing  and  sharing  in  some  way,  so  that  their  work  is  always  public  in  some  form.  But   unlike  the  pre-­‐inquiry  classroom,  the  world  has  now  opened  up  in  the  discourse  and   conversations   are   rich   and   complicated,   answers   are   uncertain,   the   work   constantly   unfinished.  I  now  want  students  to  question,  and  wonder,  and  ask  why.  I  want  them  to   make   connections   and   to   see   things   as   interconnected.   I   am   also   now   deeply   trying   to   listen   to   their   inquiries.   There   are   now   portals   in   my   lessons   that   call   me   to   really   listen,  become  attuned  to  what  students  are  wondering  about.   They   are   wondering   about   math   and   are   inquiring   into   topics   that   come   up   in   class.   For  example,  we  were  talking  about  the  names  of  polynomial  functions  with  a  degree   of   one   to   five.   A   student   asked,   ‘‘what   about   6,   7   etc.?’’   He   was   assigned   the   task   of   finding  out  about  this  for  us  all.  The  next  class  he  said  he  could  not  look  it  up  last  night   but  four  others  responded  that  they  had.   What   I   have   noticed   of   late   is   the   openness   of   my   students   to   think   and   go   places   they   have   not   before.   As   I   open   a   topic,   I   never   know   where   it   will   go.   More   often   than   not,   we   end   up   in   territory   way   beyond   the   ‘curriculum’  for  that  grade.  For  example,   the   grade   10’s,   in   a   conversation   about   the   sine   and   cosine   of   supplementary   angles,   ended  up  describing  the  unit  circle.  In  an  assignment  in  which  they  researched  the  life   of  a  mathematician,  they  then  wrote  about  how  they  could  come  to  understand  who   they  were  and  who  they  could  become.  It  seemed  natural  to  discuss  these  things.  Yet  I   know  if  I  had  tried  this  before,  I  would  not  have  had  the  open  reception  or  the  effort   they   put   into   their   writing.   I   am   constantly   amazed   at   their   thoughtfulness;   at   times   they  seem  so  much  smarter  than  I.  I  truly  feel  privileged  to  be  in  the  face  of  the  young.       The  extract  is  taken  from  Chapman  and  Heater  (2010,  p.  450-­‐451)   Chapman,  O.,  &  Heater,  B.  (2010).  Understanding  change  through  a  high  school  mathematics  teacher’s   journey  to  inquiry-­‐based  teaching.  Journal  of  Mathematics  Teacher  Education,  13(6),  445-­‐458.  

Teaching and  Learning  Mathematics  through  Inquiry  



40 min  

In most  classrooms,  it  seems  that  the  teacher  carries  much  of  the  responsibilities  for   student   learning.   And   rightly   so,   some   might   claim.   However,   teachers   seem   to   undertake   full   responsibility   for   whatever   goes   on   in   the   classroom,   with   students   ‘passively’  waiting  for  things  to  be  done  –  by  the  teacher,  for  the  students  –  because   that’s  the  way  it  is  and  that’s  the  way  it  should  be!  Let’s  consider  some  key  decisions,   actions  and  expectations  traditionally  undertaken  by  teachers.   Maria   and   Helen   are   secondary   school   teachers   and   have   taught   mathematics   for   over  five  years.  Both  Maria  and  Helen  feel  that  they  work  under  constant  pressures   and  constraints  mainly  related  to  the  mathematical  content  that  they  are  required  to   teach,   the   time   factor   and   high-­‐stakes   examinations.   Moreover,   they   feel   that   they   also  carry  much  of  the  responsibility  for  student  learning.   The   statements   that   follow   are   taken   from   a   conversation   that   Maria   and   Helen   had   regarding  their  classroom  practices,  their  role  as  teachers  and  that  of  their  students  as   learners  of  mathematics.   Issue  


Who decides  on  the   I  decide  which  exercises  and   work  that  students   problems  my  students  should   do?   do.  

I provide  a  list  of  exercises  but   then  allow  my  students  to   select  which  problems  or   questions  to  do.  

How much  work   should  I  expect   students  to  do?  

I expect  and  make  sure  that   all  students  do  all  the  work   that  I  assign.  

I allow  some  degree  of   freedom  with  the  amount  of   work  students  do.  

Who corrects  the   students’  work?  

I always  collect  and  correct   students’  work  on  class  tasks   and  homeworks.  

My students  usually  get  to   correct  their  own  work  and   only  get  to  hand  it  in  when   they  cannot  sort  out  problems   on  their  own.  

What if  students   have  issues  that   they  cannot  solve?  

I make  sure  that,  by  the  end  of   the  lesson,  I  sort  out  students’   unresolved  mathematical   issue.  

I prefer  to  leave  my  students   to  struggle  with  their   unresolved  mathematical   issues.  

What about  notes   taking?  

I make  sure  to  give  students   my  own  set  of  notes  about   each  topic.  

I expect  my  students  to  write   their  own  notes  about  the   topic  being  done  in  class.  

How do  I  set   I  always  choose  and  decide   students  to  work  in   with  whom  they  get  to  work.   groups?  



Teaching and  Learning  Mathematics  through  Inquiry  

I provide  students  with  the   opportunity  to  choose  and   decide  with  whom  to  work.  


30 min  

When   students   are   given   the   task   of   deciding   how   much   they   need   to   do   to   understand   the   mathematics   involved,   an   important   shift   takes   place:   away   from  doing  something  because  they  have  been  told  to  do  so,  and  towards  doing   something  because  they  recognize  the  value  of  making  progress.     Ollerton  (2006,  p.  198)     References:        Ollerton,  M.  (2006).  Getting  the  Buggers  to  Add  Up  (2nd  Edition).  London:  Continuum.  

Pair work   Using  the  above  quote  as  our  guiding  principle,  think  about  and  identify  how   you  may   negotiate  new  ways  of  working  with  the  students  in  your  classroom.   Some  questions  to  help  you  focus:   How  would  you  communicate  the  classroom  rules?   Which  rules  do  you  value  most?  Why?   How  do  you  expect  your  students  to  work  in  your  class?   What  habits  and  behaviours  would  you  like  to  instill  in  the  students?   How  do  you  intend  to  get  students  to  accept  more  responsibility  for  their   learning?  

___________________________________________________________________________________________________ ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________  

Teaching and  Learning  Mathematics  through  Inquiry  



10 min  

Ø Briefly describe  your  experience  during  today’s  session.   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Ø What  did  you  feel  un/comfortable  doing  during  the  session?   Comfortable:  ___________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Uncomfortable:  ________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Ø I  used  to  think...  but  now  I  know…   I  used  to  think  __________________________________________________________________________________   ___________________________________________________________________________________________________   Now  I  know  ____________________________________________________________________________________   ___________________________________________________________________________________________________   Ø What  will  you  take  with  you  and  try  to  implement  in  your  class?   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Ø Any  other  comments/suggestions  that  you  would  like  to  add.   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Thank  you  for  your  participation  and  reflections.  


Teaching and  Learning  Mathematics  through  Inquiry  

Profile for IBL Maths

PD: Thinking about Shifting Responsibility  

Session guide to support teachers in shifting more responsibility in an inquiry-based learning environment

PD: Thinking about Shifting Responsibility  

Session guide to support teachers in shifting more responsibility in an inquiry-based learning environment

Profile for iblmaths