Unpacking the relationship between classroom practice and student learning in

mathematics: Examining the power of student explanations  

Megan Franke, Mathematics Classroom Practice

Study Group UCLA

 

Session  Overview  •  Understanding  the  rela.onship  between  classroom  prac.ce  and  student  outcomes  

•  Prior  research  on  students’  explana.ons  and  teachers’  support  of  those  explana.ons  

•  Engaging  students  in  each  other’s  mathema.cal  ideas  

•  Findings  related  to  student  par.cipa.on,  teaching  and  student  learning  

Results  of  a  large-­‐scale  intervention  study  

•  Recruited  volunteer  teachers  at  19  schools  in  low-­‐performing,  urban  school  district  

•  On-­‐site  professional  development  focused  on  algebraic  reasoning    

 

•  Thinking  Mathema-cally:  Integra-ng  Arithme-c  and  Algebra  in  Elementary  School    

•  Equal  sign,  Rela.onal  thinking  •  Orchestrate  conversa.ons  

Jacobs, V., Franke, M.., Carpenter, T., Levi, L. & Battey, D. (2007). Exploring the impact of large scale professional development focused on children’s algebraic reasoning. Journal for Research in Mathematics Education 38 (3), pp. 258-288.

The schools and district

Overview of Classrooms: Mean Achievement

01020304050607080

Low (2 classes) Medium (1class)

High (3 classes)

Ach

Collecting  data  around  interactions  

Student communication  

•  Explaining  to  other  students  is  posi.vely  related  to  achievement  outcomes,  even  when  controlling  for  prior  achievement    (Brown  &  Palincsar,  1989;  Fuchs,  Fuchs,  HamleG,  Phillips,  Karns,  &  Dutka,  1997;  King,  1992;  NaNv,  1994;  Peterson,  Janicki,  &  Swing,  1981;  Saxe,  Gearhart,  Note,  &  Paduano,  1993;  Slavin,  1987;  Webb,  1991;  Yackel,  Cobb,  Wood,  Wheatley,  &  Merkel,  1990).    

• When  describing  their  thinking,  students  must  be  precise  and  explicit  in  their  talk,  especially  providing  enough  detail  and  making  referents  clear  so  that  the  teacher  and  fellow  classmates  can  understand  their  ideas  (Nathan  &  Knuth,  2003;  Sfard  &  Kieran,  2001).  

Potential Benefits of Explaining Your Own Thinking

• Transform  what  you  know  into  an  explana.on  that  is  relevant,  coherent,  complete,  and  understandable  to  others  

• Bring  concepts/details  together  in  ways  that  you  hadn’t  thought  of  previously  

• Recognize  misconcep.ons,  contradic.ons,  incompleteness  in  your  idea  

• Develop  a  sense  of  yourself  as  someone  who  can  do  mathema.cs  and  communicate  mathema.cally  

8

Coding Student Participation  • Accuracy  of  answer  given  •  Correct  •  Incorrect  •  No  answer    

• Nature  of  explana.on  given  •  Correct  and  complete  •  Ambiguous  or  incomplete  •  Incorrect  

•  Further  elabora.on  aPer  teacher’s  ques.ons  

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Types of Student Explaining

•  Gives  correct/complete  explana.ons  

     10  +  10  –  10  =  5  +  □      

     Five?  ‘Cause  10  plus  10  equals  20,  huh?    And  then  it              says  minus  10  equals  5  plus  blank.    So  it  go\a  be  10,  so              5  plus  5  equals  10.  And  that’s  how  I  got  it.  

•  Gives  incorrect  or  incomplete  explana.ons  

     50  +  50  =  50  +  □    +  25  

   [50].    It’s  just  like  50  plus  50.    They  are  kind  of  partners              because  they  are  the  same.          8  +  2  =  7  +  3  (True  or  False?)  

   [True]  because  there’s  a  2  and  a  3  and  a  7  and  an  8.                They’re  like  an  order.    

Table 3. Correlations between Student Participation and Achievement Scores

Highest level of student participation on a problema

Achievement Scoreb

Gives explanation .69*

Correct and complete .73*

Ambiguous, incomplete, or incorrect -.01

Gives no explanation -.69*

aPercent of problems in which a student displayed this behavior. Problems discussed during pairshare and whole-class interaction are included. bPercent of problems correct. Note: Number of students = 35.

*p <.05

Profiles of Students’ Contributions

0%10%20%30%40%50%60%70%80%90%

100%

LowClasses

MediumClass

HighClasses

CorrectexplanationsIncorrectexplanationCorrect answeronlyIncorrect answeronly

Moving  toward  understanding  the  details  of  practice    in  relation  to  student  outcomes  

 •  Teachers’  support  explaining  (Lampert,  2001),  Revoicing  (Forman  et  al.,  1998;  O’Connor  &  Michaels,  1993,  1996;  Strom  et  al.,  2001)    Press  (Kazemi  &  S-pek,  )  Teachers’  ques.oning  (Wood,  1998)  

Filtering  approach    (Sherin,  2002)  

•  Teachers’  prac.ce  supports  students’  produc.ve  explana.ons  (Gillies,  2004;  Rosja-­‐Drummond  and  Mercer,  2003)    

•  And  while  evidence  shows  these  prac.ces  are  not  likely  in  many  classrooms,  they  are  even  less  likely  in  classrooms  of  low-­‐income  students  of  color  (Anyon,  1981,  Ladson-­‐Billings,  1997;  Lubienski,  2002;  Means  &  Knapp,  1991).    

 

Teachers’ Supporting of Students to Share their Thinking

•  98%  of  segments:  Teachers  asked  the  target  students  to  explain  their  thinking  

•  91%  of  segments:  Teachers  requested  an  explana.on  at  the  outset  of  the  segment,  or  aPer  an  answer  was  given  

•  76%  of  segments:  Teachers  asked  the  student  to  elaborate  further  on  their  explana.on  

•  Frequent  reminders  about  listening  to  explana.ons:    •  “Give  [name]  a  chance  [to  explain]”  •  “I  like  the  way  [Student]  is  paying  close  a\en.on  to  what  [Students]  are        about  to  share”  

•  “Let’s  understand  [Student’s]  thinking.”  

Whether teachers elicited student thinking beyond initial explanations and how the engagement ended

0%10%20%30%40%50%60%70%80%90%

100%

Low Medium High

Yes: correctexplanationYes: incorrectexplanationNo: correctexplanationNo: incorrectexplanationNo: correct answer

No: incorrectanswer

Example: General Question

Problem:  375  =  __  +  (3  x  10)    •  Student:  345  •  Teacher:    I’m  just  a  li\le  unsure  of  how  you  came  up  with  345.    Can  you  show  me  what  you  did?    

Example:  Speci?ic  Question  

• Problem:  100  +  __  =  100  +  50   Student:  The  50  will  go  right  there  because  it  has  to  be  the  same  number.  

Teacher:  What  has  to  be  the  same  number?    

Probing Sequence

• Used  when  teacher  was  unclear  about  a  student’s  explana.on  

 • Used  to  highlight,  clarify  or  make  explicit  por.ons  of  a  student’s  strategy  

 • Used  when  teacher  is  trying  to  help  a  student  understand  a  problem  

Engaging  with  each  others’  ideas  

•  New  study:  •  K-­‐5  teachers  •  Mul.-­‐age  school  •  School  describes  itself  as  a  learning  environment  that  values  diversity,  encourages  crea.vity  and  innova.on,  supports  disciplined  inquiry,  involves  families  and  their  communi.es,  and  makes  a  commitment  to  mee.ng  the  needs  of  the  whole  child.  

•  All  teachers  par.cipated  (12  who  taught  mathema.cs)  •  25-­‐35  students  per  classroom  

•   Classroom  observa.ons  •  Spent  the  year  in  classrooms  approximately  once  a  week  •  Video  and  audiotaped  2-­‐3  days  in  March,  April  each  class  •  Collected  student  work  •  Researcher  designed  assessment  and  standardized  test    

Collecting  observation  data  •  One  sta.onary  video  camera  with  two  flat  microphones  captured  the  ongoing  flow  of  the  class  and  the  interac.on  of  up  to  two  groups  of  students.    

•  Four  Flip  video  cameras  captured  the  interac.on  of  the  remaining  students.    •  one  Flip  video  camera  was  sta.onary  and  the  other  three  were  operated  by  research  team  members.    

•  Distributed  six  digital  audio  recorders  to  pick  up  sound  not  captured  by  Flip  

•  Created  a  single  movie  for  each  classroom  observa.on  by  combining  all  of  the  video  and  sound  sources  

•  Movie  analyzed  using  Studiocode  so  that  we  could  code  the  details  within  the  context  

The above figure illustrates the way the codes are applied to a video timeline. Instances of codes are represented by rows. In addition, codes have labels as can be seen in the table in the upper right hand corner.

Relationship between Student Participation and Achievement

     

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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*  

•  Explain  your  thinking  

•  Engage  with  others’  ideas  to  a  high  degree  

•  Have  others  engage  with  your  idea  to  a  high  degree  

26

Teacher Support of Students’ Engagement with Each Other’s Ideas

 

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Why the invitation was not enough  •  Student  had  no  readily  available  response  or  a  response  that  provided  any  detail,  and  so  the  teacher  had  to  find  ways  to  work  with  the  student  to  elaborate  and  extend  their  engagement  with  the  other  student’s  idea    

•  Student  did  not  discuss  the  mathema.cal  idea  in  what  had  been  shared  or  did  not  address  the  par.cular  mathema.cal  idea  that  the  teacher  wanted  to  address    

•  Students  did  not  know  how  to  take  up  the  teacher’s  invita.on  

Teacher  support  for  engaging  in  other’s  ideas  

•  Student  did  not  have  much  of  a  detailed  response  

Jack ate 6 peanut butter sandwiches. He ate 1/6 of a sandwich and decided he didn't want more. How much does Jack have left?

Ms.  A:    Okay.  Who  can  explain  what  Yadira  did?    Who  can  explain,    Cole,  who  can  explain  what  Yadira  did  here?    Cole,  can    you  come  and  explain?    invita.on  

Cole:    She  took  these  things  (poin.ng  to  6  rectangles)  and  then    she  did  this  (mo.oning  over  the  lines  dividing  one  of  the    rectangle  into  sixths)  so  that  she  can  throw  away  this    (poin.ng  to  the  shaded  part).  

Ms.  A:    But  what  did  she…  what  are  these?  (poin.ng  to  the  5    wholes  in  Yadira’s  picture)  probe  

Cole:    Wholes.    Sandwiches.  Ms.  A:    Those  are  the  sandwiches.    Ok.  Those  are  the  whole  

 sandwiches.  Yes.  And?  probe  Cole:    And  then  she  did  that  (mo.oning  to  the  lines  dividing  one  

 whole  into  sixths  again)  so  you  can  see  that  she  colored  in    one,  and  that's  the  one  he  ate.  

Ms.  A:    So  how  many  does  he  have  leP?  Cole:    Umm.  5  wholes  and  5  sixths  sandwiches.  Ms.  A:    Do  you  agree  with  Yadira  and  Cole?  (to  class)  

Teacher  support  for  engaging  in  other’s  ideas  

•  Student  needed  support  to  get  to  the  mathema.cal  ideas  

The students were in the middle of a conversation about 0/3. The question arose as the students were counting backwards by 1/3 from 4. Ben stated that he thought 0/3 was a whole and should follow 1/3 in counting backwards.

Ms. J: Sara has her hand raised. Sara do you want to add something? Invitation

Sara: I don’t agree with, I wanted to actually kind of come up and, Ben said that 0/3 is a whole. Ben it is not. It is not a whole.

Ms. J: Can you give him any evidence of that? Probe

Sara: It is not a whole because none of the, like like, for 3/3 [she is drawing a picture and all of it is shaded in and walks over to Ben’s picture and shows that none of it is shaded in] None of it is shaded in, in this one.

Ms. J: Well Sara what is this, 3 parts of what [pointing to her picture] Lets make it a real thing, it is easier to talk about scaffold

Sara: 3 part of, pieces of chocolate, a brownie [students in the class are also calling out different things it could be]

Ms. J: So this is chocolate, a, brownie, a tray of brownies, alright brownies. So you are saying that these 3 pieces, this is a whole, a whole brownie that has been what probe

Sara: cut Ms. J: into? Sara: 3 Pieces Ms. J: Ben do you have something to say? invitation Ben: yeah Ms. J: come on up.

Teacher  support  for  engaging  in  other’s  ideas  

•  Student  did  not  know  how  to  take  up  the  teacher’s  invita.on  

If Seily has five-thirds liters of soda, what would that look like? Draw and label all parts. Two students have written their solutions on the board

Ms J: Carlos. come on up and explain Daniel’s because you said that yours was more like Daniel’s (she had asked earlier for students to point to the strategy on the board that was like theirs). You said yours was a little more like Daniels. (As Carlos is walking to the board with his paper) Can you explain that one? invitation

Carlos: (walking slowly and pauses) No.

Ms J: You can’t explain this picture (pointing to Daniel’s picture)? Yours is very much like it. positioning

Carlos: (looks at the drawing for about 6 sec) I understand that (points to one part of Daniel’s picture) but not the lines.

Ms J: Oh, Can you ignore the lines and explain the picture? scaffolding

Carlos: Yes

Ms J: Okay

Carlos: What Daniel did, right here (points to his picture) is 5/3 which is one liter (pointing to Daniel’s picture) and 2/3 of a liter (pointing to the picture).

Supporting teachers to engage students in mathematics

•  A  number  of  researchers  have  not  only  engaged  in  significant  research  in  this  area  but  they  have  also  begun  to  support  teachers  •  Mercer  and  colleagues:  rules  for  par.cipa.on  •  Gillies  and  colleagues:  communica.on  skills  •  O’Connor  and  Michaels:  Talk  moves  

•  Consistent  with    •  Seymour  &  Lehrer  

•  Hueris.c  moves  vs.  specialized  version  of  the  move  •  Kazemi  &  S.pek  

•  Norms  for  press  

•  Cengiz,  Kline  &  Grant  (2011)    •  Combina.on  of  instruc.onal  ac.ons  

•  Lampert  

Teacher Practice

Student Achievement

0.198 (0.082)

Teacher Practice

Student Achievement

Student Participation 0.323

(0.104) 0.297

(0.092)