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James Calleja    

©2015


Plan for  the  Session        

Thinking about  Collaboration  

Task

Topic

Time

Introduction

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

¼ h  

Working on  a   Task  

Collaborative work  on  the  ‘Human  Tower’   assessment  task  

½ h  

Reflecting about  your  experience  working  in  a   small-­‐group    

½ h  

Follow-­‐up Reflection   Lesson  Video:   Students’  Talk  

Teacher Concerns     Watching  and   Analyzing  a   Lesson  

Watch and  discuss  a  video  showing  Year  7   students  discussing  the  ‘Human  Tower’  task  

½ h  

Dylan Williams  Video:  Group  goals   Obstacles  and  concerns  with  collaborative  work     When  should  I  engage  my  students  in  a   collaborative  activity?  

½ h  

Setting the  ‘ground  rules  for  discussion’   Watching  and  analyzing  a  discussion  lesson   The  teacher’s  role  during  small-­‐group  work  

¾ h  

What do  we   Reflections  and  follow-­‐up  discussion  about  the   learn  from  this   classroom  setting,  the  phases  of  the  lesson,  the   lesson?   teacher’s  role  and  student  learning  

½ h  

     

Aims of  the  session   For  today’s  session  we  will  have  the  following  aims:   o To  explore  opportunities  in  which  collaborative  work  may  benefit   student  learning   o To  reflect  upon  concerns  in  introducing  and  managing  small-­‐group   collaboration  and  discussion   o To  understand  the  roles  of  students  and  the  teacher  within  a   collaborative  environment   o To  reflect  critically  on  a  small-­‐group  collaboration  based  lesson   o To  experience  features  and  aspects  that  promote  effective  small-­‐group   discussion  

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Teaching and  Learning  Mathematics  through  Inquiry  


WORKING ON  A  TASK  IN  SMALL-­‐GROUPS  

20 min  

  Work   on   the   ‘Human   Tower’   task   (see   pages   4   and   5),   looking   at   ways   in   which   students  might  try  to  solve  the  problem  posed.   You  will:   •

First reflect  on  the  problem  individually    

Then work  as  a  group  (of  4)  to  solve  the  problem    

(12 minutes)  

Finally present  your  solutions  to  the  whole  group    

(5 minutes)  

(3 minutes)  

   

REFLECTING ON  YOUR  EXPERIENCE  WORKING  IN  A  SMALL-­‐GROUP  

20 min  

Some of  the  questions  below  are  adapted  from  the  PRIMAS  PD  materials.   PD  Module  5:  Students  working  collaboratively  available  online:   www.primas-­‐project.eu   As  a  whole  group  you  are  asked  to  reflect  on  the  following  questions:   1. Was   it   helpful   for   you   to   first   have   some   time   to   think   about   the   problem   before   you   discussed  it  in  your  group?   2. How  would  you  describe  your  role  in  the  group?     Did  someone  take  over?     Was  someone  more  of  a  passenger?     Were  you  given  the  opportunity  to  contribute  your  ideas  to  the  group?     Did  you  consider  the  alternative  views  of  everyone  in  the  group?   3. Did  each  member  assume  a  different  role?  Why  do  you  think  this  happened?     What  role  did  you  assume  while  working  on  the  task?  Did  anyone  decide  about  this?   4. Did  you  feel  uncomfortable  or  threatened?  If  so,  why?   5. Did  the  discussion  stay  ‘on  task’  or  were  you  ‘wandering’  at  times?   6. How  do  you  see  this  collaborative  task  in  your  classroom?      

Teaching and  Learning  Mathematics  through  Inquiry  

3


Imagine your  group  of  friends  is  asked  to  stand  on  each  other’s  shoulders  to  build  a   human  tower  like  the  one  similar  to  the  one  in  the  picture  below.    

How  high  would  your  tower  be?             The  challenge  for  your  group  is  to  design  the  best  structure  for  building  your  tower.   Your  tower  needs  to  be  made  up  of  at  least  three  to  four  people.   Think  about  every  detail.   Show  your  design  and  describe  how  high  this  tower  would  be.  

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Teaching and  Learning  Mathematics  through  Inquiry  


The photo  shows  a  ‘CASTELL’.   A  Castell  is  a  human  tower  built  traditionally  in   festivals  by  people  in  Catalonia  (Spain).     Explain  how  you  would  estimate  the  following:   a) The  number  of  people  involved  in  building   the  castell  shown.  

b) The height  of  the  human  tower  in  the  picture.  

    __________________________________________________________________________

Fill in  Group  Members’  Names  

Teaching and  Learning  Mathematics  through  Inquiry  

5


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.   ü

ü

ü

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  

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Teaching and  Learning  Mathematics  through  Inquiry  


WATCHING A  LESSON  VIDEO  

10 min  

You will  now  watch  a  video  of  a  teacher,  Joanne,  using  the  ‘Human  Tower’  task  with   her  Year  7  class.   Pay   particular   attention   to   how   the   students   work   in   their   groups.   You   are   asked   to   determine  whether  students’  talk  is  more  disputational,  cumulative  or  exploratory.   You  may  want  to  write  down  some  notes/points  that  you  might  consider  important.   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________      

REFLECTING ON  STUDENTS’  TALK  

15 min  

What are  your  comments  on  the  lesson?   Who  generates  the  mathematical  ideas  that  get  discussed?     Who  evaluates  and/or  responds  to  these  ideas?   In  your  opinion,  do  the  discussions  help  or  hinder  learning?     Can  you  identify  groups  of  students  in  the  video  in  which  this  occurred?   Would   you   say   that   there   is   evidence   of   disputational,   cumulative   and/or   exploratory   talk  within  the  groups?      

VIDEO: DEVELOPING  GROUP  GOALS  

10 min  

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.     How  can  students  become  collectively  responsible  and  individually  accountable?  

Teaching and  Learning  Mathematics  through  Inquiry  

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TEACHER CONCERNS  WITH  IMPLEMENTING  COLLABORATIVE  WORK  

15 min    

While working  in  pairs:    

Think about  potential  obstacles  and  concerns  that  may  hinder  teachers   from  engaging  students  with  collaborative  work.  

Pick out  one  particular  concern  that  a  teacher  may  have.  Think  about  how   you  would  respond  to  that.  

Share your  choice  together  with  your  response.  

     USING  COLLABORATIVE  WORK     As  a  whole  group,  think  and  share  your  ideas  about  this  question:    

15 min  

For which  lessons  may  collaborative  work  be  a  suitable   pedagogy  to  use  in  the  mathematics  class?          

TEACHING STUDENTS  HOW  TO  PARTICIPATE  IN  DISCUSSION  

30 min  

Have  a  look  at  this  scenario!   Stephan   has   been   teaching   mathematics   for   twelve   years.   He   is   very   enthusiastic   about  teaching  and  likes  to  try  out  new  ideas  in  his  class.   Stephan  has  learned  that  his   students  always  seem  to  lack  conceptual  understanding.  They  seem  to  find  it  hard  to   tackle  the  challenging  problems  that  he  usually  presents  them  with.  This  year  Stephan   is  willing  to  address  this  by  using  collaborative  work  with  his  class.  He  thinks  that  it   would  be  a  good  idea  to  establish  some  ‘ground  rules  for  discussion’  with  his  class.   What  ‘ground  rules  for  discussion’  would  you  suggest?   How  would  it  be  best  for  Stephan  to  establish  these  rules?   How  could  Stephan  use  these  rules  to  engage  students  in  valuable  discussion?    

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Teaching and  Learning  Mathematics  through  Inquiry  


Now let’s  have  a  look  at  Stephan’s  list  of  ground  rules…   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  could  be  displayed  and  reinforced  over  time.   Maybe  you  could  involve  your  class  in  drawing  up  a  similar  list.     1.  

Give everyone  in  your  group   a  chance  to  speak  

"Lets take  it  in  turns  to  say  what  we  think".   "Claire,  you  haven't  said  anything  yet."    

2.

Listen to  what  people  say  

"Don't interrupt  –  let  Sam  finish".   "I  think  Sam  means  that..."  

3.

Check that  everyone  else   listens    

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

4.

Try to  understand  what  is   said  

"I don't  understand.  Can  you  repeat  that?"   "Can  you  show  me  what  you  mean?"  

5.

Build on  what  others  have   said  

"I agree  with  that  because..."   "Yes  and  I  also  think  that..."  

6.

Demand good  explanations  

"Why do  you  say  that?"   "Go  on...  convince  me."  

7.

Challenge what  is  said  

"That cannot  be  right,  because..."   "This  explanation  isn't  good  enough  yet."        

8.

Treat opinions  with  respect  

"That is  an  interesting  point."   "We  all  make  mistakes!"    

9.

Share responsibility  

"Let's make  sure  that  we  are  all  able  to   report  this  back  to  the  whole  class."  

10. Reach  agreement  

"We've got  the  general  idea,  but  we  need  to   agree  on  how  we  will  present  it."    

 

Teaching and  Learning  Mathematics  through  Inquiry  

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WATCHING A  DISCUSSION  LESSON  BY  STEPHAN  

15 min  

This 15-­‐minute   video   captures   a   40-­‐minute   lesson   in   which   my   Year   7   students   (11   year-­‐olds)   work   on   an   inquiry-­‐based   task,   dealing   with   properties   of   triangles.   Before   you  watch  the  video,  I  would  like  to  give  you  some  useful  information  about  my  class.     The  26  girls  taking  part  are  the  highest  set  class.  They  are,  by  and  large,  considered  to   be   average   to   high   ability   students.   This   is   not   the   first   experience   that   my   students   had   working   in   small   groups.   Throughout   the   year,   I   provided   students   with   numerous  opportunities  of  learning  mathematics  within  a  collaborative  setting.   The  topic  was  ‘new’  to  the  students  –  in  the  sense  that  I  planned  this  task  to  serve  as  a   topic   starter.   Students’   prior   knowledge   relied   within   what   they   had   done   and   recalled  from  the  previous  years,  that  is,  while  they  were  in  primary.  Indeed,  I  knew   my  students  well  enough  to  believe  that  they  would,  at  least,  be  able  to  remember  the   names  of  particular  triangles  and  identify  some  basic  properties.   This  discussion  lesson  highlights  three  main  phases:   1.

Task Presentation  –  I  first  assess  students’  prior  knowledge  of  triangles  and  then   explain  the  purpose  of  the  task  –  classifying  triangles  in  a  two-­‐way  table.  

2.

Small-­‐Group Discussion   –   Students   work   in   groups   of   4   to   5   (I   prefer   a   heterogeneous   group   composition).   I   like   to   define   my   role   as   a   ‘guide   by   the   side’  as  students  work  on  the  task.  

3.

Students Presentation   –   Students   communicate   their   work   to   the   whole-­‐class   justifying  their  conclusions.  

Enjoy  watching  the  video!  

  This  video  is  also  available  on  YouTube.  

Stephan   Teacher  of  Mathematics   St  Clare  College,  Malta    

Follow the  link  below:   http://www.youtube.com/watch?v=dT5NLZ2GQQo&feature=youtu.be      

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Teaching and  Learning  Mathematics  through  Inquiry  


ANALYZING STEPHAN’S  DISCUSSION  LESSON  

30 min  

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

THE TEACHER’S  ROLE  DURING  SMALL-­‐GROUP  DISCUSSION  

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  

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QUESTIONS FOR  ANALYZING  STEPHAN’S  DISCUSSION  LESSON  

Think and  talk  about  the  following:   •

How does  the  teacher  introduce  the  lesson?    

Does he  emphasise  specific  ground  rules  for  discussion  during  the  lesson?  Why,  do   you  think,  he  did/didn’t  do  that?  

What is   the   teacher’s   role   during   each   the   three   phases   of   the   lesson?   Does   his   role  change?  How?  

How does  the  teacher  assist  students  during  the  small-­‐group  activity?  

To what   extent,   do   you   think,   was   setting   students   to   work   in   heterogeneous   groups  of  4  to  5  students  beneficial?  

In your  opinion,  what  works  and  what  would  you  improve?  

         

ADDITIONAL MATERIAL  

There is  a  very  good  video  that  demonstrates  how  teachers  may  set  up  a  collaborative   learning  community  in  their  classroom.   Click  the  link  below.  It  is  worth  watching!   http://www.youtube.com/watch?v=kZxNldBEU6o    

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Teaching and  Learning  Mathematics  through  Inquiry  


SESSION EVALUATION  

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  

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Profile for IBL Maths

PD: Thinking about Collaboration  

Session guide to support teachers in planning for and using more collaborative work in their mathematics class

PD: Thinking about Collaboration  

Session guide to support teachers in planning for and using more collaborative work in their mathematics class

Profile for iblmaths
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