Supporting  students  in  “productive  struggle”:  What  can  teachers  do?  

Megan  Franke  UCLA  

The Issues •  Students  need  the  opportunity  to  work  and  wrestle  with  mathema;cs    (Hiebert  &  Grouws,  2007)  

•  expending  effort  to  make  sense  of  mathema;cs  and  figure  out  something  that  is  not  immediately  apparent    

•  Common  Core  Standards  for  Mathema;cal  Prac;ce  call  for  students  to  “construct  viable  arguments  and  cri;que  the  reasoning  of  others”  including  jus;fying  their  conclusions,  communica;ng  them  to  others,  listening  to  the  arguments  of  others,  responding  to  the  arguments  of  others,  deciding  whether  they  make  sense,  and  asking  useful  ques;ons  to  clarify  or  improve  the  arguments    

•  How  can  teachers  produc;vely  engage  students  in  wrestling  with  mathema;cs?  

Maggie  had  3  packages  of  cupcakes.    There  were4  cupcakes  in  each  package.    She  ate  5  cupcakes.    How  many  cupcakes  are  leW?  

Adam  has  21  cents.  Gumdrops  cost  3  cents.    How  many  gumdrops  can  Adam  buy?    

Student  explanations  in  mathematics  classrooms  

•  Related  to  student  achievement  •  Our  research  •  Others:  Gillies,  Webb  &  Mastergeorge,  Safard  &  Kieran,  Nathan  &  Knuth  •  Not  just  any  explaining  –  details  ma_er    

•  Teachers  can  support  students  to  explain  •  O’Connor  &  Michaels,  Gillies,  Mercer  •  Ini;al  ques;on:  how  did  you  get  that?  not  sufficient  

•  98%  of  the  ;me  teachers  asked  

•  Probing  ques;ons  (sequences),  leading  ques;ons,  bundled  ques;ons  

Multiple  Dimensions  of  Student  Communication  

•  Explaining  their  own  thinking  

•  Engaging  with  others’  thinking  

•  You  engage  with  others’  ideas  

•  Others  engage  with  your  ideas  

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Engaging with each others’ ideas

Referencing/acknowledging  somebody’s  ideas  in  general  without  details  •  I  used  Nomi’s  strategy    •  “Does  everyone  agree  with  Sam?”    S  says  no.  •  Students  point  to  strategy  like  their  own    

Addressing  the  student’s  idea  with  detail  •  repea;ng  what  was  already  said  or  wri_en  on  the  board  •  sharing  someone  else’s  strategy  for  them  •  disagreeing  while  referencing  detail  of  what  they  disagree  with    

Addressing  the  detail  of  someone’s  idea  and  contributes  to  someone  else’s  ideas  •  S  disagreeing  with  what  was  shared  and  shares  an  alterna;ve  •  responds  to  a  S’s  shared  idea  and  adds  to  it  •   construct  strategy  together    

•  Explain  your  thinking  

•  Engage  with  others’  ideas  to  a  high  degree  

•  Have  others  engage  with  your  idea  to  a  high  degree  

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Relationship  between  Student  Participation  and  Achievement    

   

Par%al  correla%on  with  achievement  

Provided  fully-­‐detailed  explana;ons  of  how  to  solve  the  problem  

.30*  

Highest  level  at  which  you  engaged  with  other  students’  ideas  

.44*  

Highest  level  at  which  other  students  engaged  with  your  ideas  

.41*  

If  Seily  has  Iive-­‐thirds  liters  of  soda,  what  would  that  look  like?  Draw  and  label  all  parts    

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Some Observations about the Teacher Practices

•  Teachers  made  a  variety  of  ini;al  moves  to  help  students  a_end  to  each  other’s  ideas  

•  These  moves  all  concerning  having  students  compare  their  ideas  to  other  students’  ideas  

•  Do  you  understand  what  Adrian  did?  •  Do  you  agree?  •  Is  your  approach  the  same  or  different?  •  Point  to  the  solu;on  that  is  like  your  own  •  Can  you  explain  what  he  did?  

•  More  important  than  the  ini;al  move  for  how  students  engaged  with  others’  ideas  was  how  the  teacher  followed  up  to  press  students  to  make  their  thinking  explicit  

Supporting engagement

T  directed  students  to  discuss  answers  and  strategies  with  each  other  (norm  moves)    

•  You  are  not  talking  to  me,  points  to  the  class,  you  have  to  convince  them  

•  Emmy  is  not  convinced  yet  so  lets  see  if  we  can  get  some  more  ideas  

•  Carlos  are  you  paying  a_en;on,  you  want  to  see  if  you  agree  with  her    

Supporting engagement

T  asks  students  to  elaborate  on  answers  and  strategies  (to  make  it  possible  for  others  to  a_end  to  its  detail)    

•  asks  specfic  ques;ons  that  require  students  to  a_end  to  the  detail  

•  asking  students  to  discuss  specific,  centrally  important  steps  embedded  in  their  strategies    

Supporting engagement

T    posi;ons  students  to  par;cipate  in  another’s  thinking    

•  Selects  strategies  that  are  different  to  be  shared  with  the  whole  class  

•  Have  more  than  one  set  of  work  shared  and  explained  on  the  board  for  each  problem  

•  Asks  students  to  offer  sugges;ons  to  each  others  strategies  

•  Did  you  do  it  the  same  way  –  how  is  it  the  same?  •  I  love  what  you  did…..    Look  at  this  Lily,  because  Cole  is  at  the  same  place  where  you  are.”      

•  Cris  maybe  it  will  help  if  you  listen  to  someone  else  and  then  try  again  

Disrupting existing cultural practice

•  Exis;ng  prac;ce,  expected,  experienced  •  Dinner  ;me  conversa;on  

•  Teachers,  students  and  families  all  have  ways  they  are  used  to  doing  school  math  •  Classrooms,  schools,  districts  

•  Requires  renego;a;on  rather  than  telling    

Count by 1, start 180, count to 230 Count by 7/8 Count by .004 start at 53.280 Count by 10 start 66, count to 266 Count by .99, start at 1 Count by 2, start at 0 Count by 11, start at -77

Create  a  space  to    begin…  

One  of  these  things  is  not  like  the  others…  

Listen…  replace  your  talk  with  another  move  

Create  visible  reminders  

Consider  what  gets  displayed  

Tasks  with  mul;ple  entry  points   What do you notice?

• 6 x 2 = 3 x 4 • 12 x .5 = .25 x 24 • 416 x 2 = 4 x 208

True or False 26 + 18 - 18 = 25 17 - 9 + 8 = 17 7 + 6 = 8 + 5 67 + 83 = 66 + 82

Transform  the  Textbook  pages  

Ask:    Circle  all  the  number  sentences  with  an  answer  less  than  5.  

Draw  on  non-­‐school  tasks..  

Productive struggle for each student

•  Challenge  our  assump;ons  about  student  par;cipa;on  

•  Find  the  produc;ve  –  what  students  have  communicated  

•  Make  explicit  what  it  means  to  par;cipate  (norms)  

•  Support  it  (follow  up)  

•  Watch  for  how  the  status  quo  limits  opportuni;es…find  ways  to  challenge  it