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James Calleja    


OBJECTIVES OF  PROFESSIONAL  DEVELOPMENT     Ø To  explore  opportunities  in  which   collaborative  work  may  benefit  student   learning  

Ø To reflect  upon  concerns  in  introducing   and  managing  small-­‐group   collaboration  and  discussion     Ø To  understand  the  roles  of  students   and  the  teacher  within  a  collaborative   environment  

Ø To promote  effective  small-­‐group   discussion        


Teaching and  Learning  Mathematics  through  Inquiry  

LOOKING AT  STUDENT  TALK  IN  SMALL-­‐GROUP  DISCUSSIONS   Neil   Mercer   (1995)   identified   three   typical   ways   of   talking   among   students   as   they   work   in   small-­‐groups   –   disputational   talk,   cumulative   talk   and   exploratory  talk.  




Disputational talk     Involves   disagreements   and  individual  rather  than  collective   decision-­‐ making.   Exchanges   are   usually   brief   and   consist   of   assertions   or   counter-­‐assertions.   Cumulative  talk     Represents   a   building   of   ideas   based   on   each   other’s   suggestions   aimed  at  providing  a  common  consensus.  Exchanges  in  this  type  of  talk   are  usually  repetitions,  confirmations  and  elaborations   Exploratory  talk     Characterised   by   critical   but   constructive   engagement   with   each   other’s   ideas.  Challenges  are  justified  and   alternatives  suggested.   Joint   agreement  in  decision-­‐making  is  the  end  result.  

Disputational   talk,   in   which   students   simply   disagree   and   go   on   to   make   individual   decisions,   is   not   beneficial.   Cumulative   talk,   in   which   students   build   uncritically   on   what   each   other   has   said,   is   also   undesirable.   For   true   collaborative   work,   students   need   to   develop   exploratory   talk   consisting   of   critical   and   constructive   exchanges,   where   challenges   are   justified   and   alternative  ideas  are  offered.  The  most  helpful  talk  appears  to  be  that  where  the   participants   work   on   and   elaborate   each   other’s   reasoning   in   a   collaborative,   rather   than   competitive   atmosphere   (PRIMAS,   2011).   Mercer   (1995)   argues   that   in  planning  collaborative  activities  in  the  mathematics  classroom,  we  should  be   aiming  to  promote  exploratory  talk:   By   incorporating   both   conflict   and   the   open   sharing   of   ideas,   represents   the   more   'visible'   pursuit   of   rational   consensus   through   conversation.   More   than   the   other   two   types,   it   is   like   the   kind   of   talk   which   has   been   found   to   be   most   effective   for   solving   problems   through   collaborative   activity.  (p.105)    

Reference: Mercer,  N.  (1995).  The  guided  construction  of  knowledge:  Talk  amongst  teachers  and   learners.  Clevedon:  Multilingual  Matters.   PRIMAS  (2011).  Students  Working  Collaboratively:  How  can  we  foster  scientific   discussion?  PD  Module  5:  Students  working  collaboratively.  Available  online  on:   www.primas-­‐project.eu/artikel/en/1221/Professional+development+modules/view.do  

Teaching and  Learning  Mathematics  through  Inquiry  



Collaborative work  in  small  groups  has  a  positive  effect  on  both  social  skills  and   mathematics   learning.   However,   this   depends   on   the   existence   of   shared   goals   for   the   group   and   individual   accountability   for   attaining   those   goals.   Dylan   Williams   explains   that   collaborative   learning   is   effective   when   teachers   create   ‘group   goals’   –   getting   students   working   as   a   group   rather   than   working   in   a   group.   To   achieve   this,   teachers   need   to   create   two   conditions   for   students   in   their  classrooms:  collective  responsibility  and  individual  accountability.   Visit  the  ‘Videos’  section  on  our  Google  Community  and   watch  ‘Collaborative  learning  Dylan  Williams’  

Collaborative  work  is  ideal  for  developing  students’  conceptual  understanding   or  strategies  for  solving  more  challenging  and  complex  problems.     Small   group-­‐work   involves   students   working   in   collaborative   rather   than   a   competitive  atmosphere  by  sharing  and  exchanging  ideas,  and  challenging  each   other  through  critical  and  constructive  talk.   Group-­‐work  may  be  difficult  when  students  do  not  have  adequate  sharing  skills,   participation  skills,  listening  and  communication  skills.  In  this  case,  the  teacher   needs  to  provide  opportunities  for  students  to  develop  these  skills.   Visit  the  ‘Videos’  section  on  our  Google  Community  and   watch  ‘How  to  Teach  Math  as  a  Social  Activity’  


Teaching and  Learning  Mathematics  through  Inquiry  


The  list  below  is  taken  from  the  PRIMAS  PD  materials  available  online:   www.primas-­‐project.eu     Here   are   some   suggested   'ground   rules'   for   students   to   use   as   they   work   in   groups.   These  ‘ground  rules’  may  be  displayed  and  ideally  reinforced  over  time.   Maybe  you  could  also  try  to  involve  you  class  in  drawing  up  a  similar  list.       1.  


Give everyone  in  your  group   a  chance  to  speak   Listen  to  what  people  say  

"Lets take  it  in  turns  to  say  what  we  think".   "Claire,  you  haven't  said  anything  yet."     "Don't  interrupt  –  let  Sam  finish".   "I  think  Sam  means  that..."   "What  did  Sue  just  say?"    


Check that  everyone  else   listens    

"I just  made  a  deliberate  mistake  –  did  you   spot  it?  



Try to  understand  what  is   said  

"I don't  understand.  Can  you  repeat  that?"  

Build on  what  others  have   said  

"I agree  with  that  because..."  


Demand good  explanations  


Challenge what  is  said  


Treat opinions  with  respect  


Share responsibility  

"Yes and  I  also  think  that..."   "Why  do  you  say  that?"   "Go  on...  convince  me."   "That  cannot  be  right,  because..."   "This  explanation  isn't  good  enough  yet."         "That  is  an  interesting  point."   "We  all  make  mistakes!"     "Let's  make  sure  that  we  are  all  able  to   report  this  back  to  the  whole  class."   "We've  got  the  general  idea,  but  we  need  to  

10. Reach  agreement  


"Can you  show  me  what  you  mean?"  

agree on  how  we  will  present  it."    

Teaching and  Learning  Mathematics  through  Inquiry  



The  list  provide  below  is  taken  from  the  PRIMAS  PD  materials  available  online:   www.primas-­‐project.eu  

Make  the  purpose   of  the  task  clear  

Explain what  the  task  is  and  how  they  should  work  on  it.  Also,  explain   why  they  should  work  in  this  way.  ‘Don’t  rush,  take  your  time.  The   answers  are  not  the  focus  here.  It’s  the  reasons  for  those  answers  that   are  important.  You  don’t  have  to  finish,  but  you  do  have  to  be  able  to   explain  something  to  the  rest  of  the  class.’  

Keep reinforcing   the  ‘ground  rules’  

Try to  ensure  that  students  remember  the  ground  rules  that  were   discussed  at  the  beginning.  Encourage  students  to  develop  a   responsibility  for  each  other’s  understanding.  ‘I  will  pick  one  of  you  to   explain  this  to  the  whole  class  later  –  so  make  sure  all  of  you   understand  it’.  

Listen before   intervening  

When approaching  a  group,  stand  back  and  listen  to  the  discussion   before  intervening.  It  is  all  too  easy  to  interrupt  a  group  with  a   predetermined  agenda,  diverting  their  attention  from  the  ideas  they  are   discussing.  This  is  not  only  annoying  and  disruptive  (for  the  group),  it   also  prevents  students  from  concentrating.  

Join in,  don’t   judge  

Try to  join  in  as  an  equal  member  of  the  group  rather  than  as  an   authority  figure.  When  teachers  adopt  judgmental  roles,  students  tend   to  try  to  ‘guess  what’s  in  the  teacher’s  head’  rather  than  try  to  think  for   themselves:  ‘Do  you  want  us  to  say  what  we  think,  or  what  we  think  you   want  us  to  say?’  

Ask students  to   describe,  explain   and  interpret  

The purpose  of  an  intervention  is  to  increase  the  depth  of  reflective   thought.  Challenge  students  to  describe  what  they  are  doing  (quite   easy),  to  interpret  something  (‘can  you  say  what  that  means?’)  or  to   explain  something  (‘can  you  show  us  why  you  say  that?).  

Make students   do  the  thinking  

Many students  are  experts  at  making  their  teachers  do  the  work!  They   know  that  if  they  ‘play  dumb’  long  enough,  then  the  teacher  will   eventually  take  over.  Try  not  to  fall  for  this.  If  a  student  says  that  he  or   she  cannot  explain  something,  ask  another  student  in  the  group  to   explain,  or  ask  the  student  to  choose  some  part  of  the  problem  that  she   can  explain.  Don’t  let  them  off  the  hook!  When  a  student  asks  the   teacher  a  question,  don’t  answer  it  (at  least  straight  away).  Ask   someone  else  in  the  group  to  do  so.  

Don’t be  afraid   of  leaving   discussions   unresolved.  

Some teachers  like  to  resolve  discussions  before  they  leave  the  group.   When  the  teacher  leads  the  group  to  the  answer,  then  leaves,  the   discussion  has  ended.  Students  are  left  with  nothing  to  think  about,  or   they  go  on  to  a  different  problem.  It  is  often  better  to  reawaken  interest   with  a  further  interesting  question  that  builds  on  the  discussion  and   then  leave  the  group  to  discuss  it  alone.  Return  some  minutes  later  to   find  out  what  has  been  decided.  


Teaching and  Learning  Mathematics  through  Inquiry  


ü Working   in   pairs   and   in   threes   is   usually  the  most  effective.  However,   the   teacher   can   profit   from   occasionally   having   students   working   in   groups   of   four   as   it   allows   more   time   to   assess   students’   work  and  learning  process.     ü Initially   start   with   pairs   and   later   move   on   to   slightly   larger   groups   (even  within  the  same  lesson).   ü Group   dynamics   appear   to   be   more   dependent   on   students’   personalities   and  relationships  than  on  their  mathematical  competence.   ü Allowing   students   to   choose   their   group   offers   students   a   greater   responsibility  in  making  their  group  work  more  effectively.  The  teacher  then   has   ‘more   power’   in   determining   who   should   work   with   whom   when   s/he   feels  that  changes  need  to  be  done.  

ü It is   also   recommendable   to   alternate   between   student-­‐selected   groupings,   heterogeneous  (mixed-­‐ability)  and  homogeneous  groupings.      

Teaching and  Learning  Mathematics  through  Inquiry  


Profile for IBL Maths

PD: Planning for Collaboration  

Teacher guide to support in planning to integrate more collaborative work in the mathematics class

PD: Planning for Collaboration  

Teacher guide to support in planning to integrate more collaborative work in the mathematics class

Profile for iblmaths