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Is  it  true?  Always?  Supporting  Reasoning-­‐and-­‐Proof  Focused  Collaboration  among  Teachers  

2013  NCTM  Annual  Conference  Denver,  Colorado  

Nicole  Miller  Rigelman  *  Portland  State  University  rigelman@pdx.edu  

 

http://goo.gl/ys4Qd    

In  this  session  we  will:  *  Unpack  what  it  is  meant  by  reasoning-­‐and-­‐proving  and  consider  what  it  looks  like  to  prompt  such  thinking.  *  Explore  tools  to  support  teacher  collaboration  with  selecting  tasks,  planning,  and  observing/examining  classroom  practice  and  student  thinking  with  an  eye  on  supporting  students  with  developing  convincing  arguments.  

Session  Overview  

*  Growing  consensus  in  the  community  that  it  should  be  “a  natural,  ongoing  part  of  classroom  discussions,  no  matter  what  topic  is  being  studied,” (NCTM,  2000,  p.  342).      

What  gets  in  the  way?  *  It  is  difficult  for  teachers  and  student.  *  We  have  limited  conceptions  of  what  counts  as  “reasoning  and  proving.”  

Why  focus  on  reasoning  and  proof  in  your  PLC?  

Instructional  programs  [preK-­‐12]  should  enable  students  to  develop  and  evaluate  mathematical  arguments  and  proofs.  

 -­‐  NCTM,  2000,  p.  56  

 [In  grades  3-­‐5]  mathematical  reasoning  develops  in  classrooms  where  students  are  encouraged  to  put  forth  their  own  ideas  for  examination.  Teachers  and  students  should  be  open  to  questions,  reactions,  and  elaborations  from  others  in  the  classroom.    Students  need  to  explain  and  justify  their  thinking  and  learn  how  to  detect  fallacies  and  critique  others  thinking.    

-­‐  NCTM,  2000,  p.  188  

Reasoning  and  Proof  

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…both  plausible  and  flawed  arguments  that  are  offered  by  students  create  an  opportunity  for  discussion.    As  students  move  through  the  grades,  they  should  compare  their  ideas  with  others’  ideas,  which    may  cause  them  to  modify,  consolidate,  or  strengthen  their  arguments  or  reasoning.    Classrooms  in  which  students  are  encouraged  to  present  their  thinking,  and  in  which  everyone  contributes  by  evaluating  one  another’s  thinking,  provide  rich  environments  for  learning  mathematical  reasoning.  

     -­‐  NCTM,  2000,  p.  58  

Reasoning  and  Proof  

In  the  domain  of  Number  and  Operation…  *  Computation  strategy.  Purposeful  manipulations  that  may  be  chosen  for  specific  problems,  may  not  have  a  fixed  order,  and  may  be  aimed  at  converting  one  problem  into  another.  *  Computation  algorithm.  A  set  of  predefined  steps  applicable  to  a  class  of  problems  that  gives  the  correct  result  in  every  case  when  the  steps  are  carried  out  correctly.    

-­‐  K-­‐5  Progressions,  Number  and  Operations  in  Base  Ten,  2011,  p.  3    

Reasoning  and  Proof  

1.  Make sense of problems and persevere in solving them 2.  Reason abstractly and quantitatively 3.  Construct viable arguments and critique the reasoning of others 4.  Model with mathematics 5.  Use appropriate tools strategically 6.  Attend to precision 7.  Look for and make use of structure 8.  Look for and express regularity in repeated reasoning

- Common Core State Standards for Mathematics, 2010, pp. 6-7

Standards  for  Mathematical  Practice   Mathematics  Task  Framework  

                                               Cognitive  Demand                                        Design                                      Set-­‐Up                          Implementation                    Student                                                      Learning  

 Stein  et  al,  1998  

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Mathematical  task  as  represented  in    

curricular/  instructional  materials  

Mathematical  task  as  set  up  by  the  teacher  in  the  classroom  

*Task  features  *Cognitive  demands  

Mathematical  task  as  implemented  by  students  in  the  

classroom  *Enactment  of  task  

features  *Cognitive  processing  

Students’  Learning  outcomes  

Relationships  among  various  task-­‐related  variables  and  students’  learning  outcomes.  Henningsen  and  Stein,  1997,  p.  528;  Stein  and  Smith,  1998,  p.  270  

Factors    influencing  setup  *Teachers’  goals  

*Teachers’  knowledge    of  subject  matter  

*Teachers’  knowledge    of  students  

Factors    influencing  students’  

implementation  *Classroom  norms  

*Teachers’  instructional    dispositions  

*Students’  learning    dispositions  

Mathematics  Task  Framework  

* Jamie’s  family  visited  their  grandmother  who  lives  634  miles  from  their  house.  On  the  first  day  they  drove  319  miles.  How  many  miles  did  they  have  left  to  drive  the  second  day?    

From  Investigations  in  Number,  Data,  and  Space,    Russell  &  Economopoulos,  2008    

Visiting  Grandma  

Task  Analysis  Guide  

Memorization   Procedures  with  Connections  •  involve  either  reproducing  previously  learned  facts,  rules,  

formulas,  or  definitions  or  committing  facts,  rules,  formulas  or  definitions  to  memory.  

•  cannot  be  solved  using  procedures  because  a  procedure  does  not  exist  or  because  a  time  frame  in  which  the  task  is  being  completed  is  too  short  to  use  a  procedure.  

•  are  not  ambiguous.    Such  tasks  involve  exact  reproduction  of  previously  seen  material,  and  what  is  to  be  reproduced  is  clearly  and  directly  stated.  

•  focus  students’  attention  on  the  use  of  procedures  for  the  purpose  of  developing  deeper  levels  of  understanding  of  mathematical  concepts  and  ideas.  

•  suggest  explicitly  or  implicitly  pathways  to  follow  that  are  broad  general  procedures  that  have  close  connections  to  underlying  concepts.  

•  usually  are  represented  in  multiple  ways,  such  as  visual  diagrams,  manipulatives,  symbols,  and  problem  situations.    Making  connections  among  multiple  representations  helps  develop  meaning.  

•  require  some  degree  of  cognitive  effort.    Although  general  procedures  may  be  followed,  they  cannot  be  followed  mindlessly.    Students  need  to  engage  with  conceptual  ideas  that  underlie  the  procedures  to  complete  the  task  successfully  and  that  develop  understanding.  

Procedures  without  Connections   Doing  Mathematics  

•  are  algorithmic.    Use  of  the  procedure  either  is  specifically  called  for  or  is  evident  from  prior  instruction,  experience,  or  placement  of  the  task.  

•  require  limited  cognitive  demand  for  successful  completion.    Little  ambiguity  exists  about  what  needs  to  be  done  and  how  to  do  it.  

•  have  no  connection  to  the  concepts  or  meaning  that  underlie  the  procedure  being  used.  

•  are  focused  on  producing  correct  answers  instead  of  on  developing  mathematical  understanding.  

•  require  no  explanations  or  explanations  that  focus  sole  on  describing  the  procedure  that  was  used.  

•  require  complex  and  non-­‐algorithmic  thinking  -­‐-­‐  a  predictable,  well-­‐rehearsed  approach  or  pathway  is  not  explicitly  suggested  by  the  task,  task  instructions,  or  a  worked-­‐out  example.  

•  require  students  to  explore  and  understand  the  nature  of  mathematical  concepts,  processes,  or  relationships.  

•  demand  self-­‐monitoring  or  self-­‐regulation  of  one’s  own  cognitive  processes.  

•  require  students  to  access  relevant  knowledge  and  experiences  and  make  appropriate  use  them  in  working  through  the  task.  

•  require  students  to  analyze  the  task  and  actively  examine  task  constraints  that  may  limit  possible  solution  strategies  and  solutions.  

Henningsen  and  Stein,  1997,  p.  528;  Stein  and  Smith,  1998,  p.  270  

 

Task  Analysis  Guide  

Procedures  with  Connections  •  focus  students’  attention  on  the  use  of  procedures  for  the  purpose  

of  developing  deeper  levels  of  understanding  of  mathematical  concepts  and  ideas.  

•  suggest  explicitly  or  implicitly  pathways  to  follow  that  are  broad  general  procedures  that  have  close  connections  to  underlying  concepts.  

•  usually  are  represented  in  multiple  ways,  such  as  visual  diagrams,  manipulatives,  symbols,  and  problem  situations.    Making  connections  among  multiple  representations  helps  develop  meaning.  

•  require  some  degree  of  cognitive  effort.    Although  general  procedures  may  be  followed,  they  cannot  be  followed  mindlessly.    Students  need  to  engage  with  conceptual  ideas  that  underlie  the  procedures  to  complete  the  task  successfully  and  that  develop  understanding.  

Henningsen  and  Stein,  1997,  p.  528;  Stein  and  Smith,  1998,  p.  270  

 

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0.    Setting  Goals  and  Selecting  the  task  

1.  Anticipating  (e.g.,  Fernandez  &  Yoshida,  2004;  Schoenfeld,  1998)  

2.  Monitoring  (e.g.,  Hodge  &  Cobb,  2003;  Nelson,  2001;  Shifter,  2001)  

3.  Selecting  (Lampert,  2001;  Stigler  &  Hiebert,  1999)  

4.  Sequencing  (Schoenfeld,  1998)    

5.  Connecting  (e.g.,  Ball,  2001;  Brendehur  &  Frykholm,  2000)  

The  Five+  Practices  

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Orchestrating Productive Mathematical Discourse Chart for Monitoring, Selecting, Sequencing, and Connecting Student Thinking

Strategy Work of Specific Students Sequence Compare

A.

B.

C.

D.

Adapted from Smith and Stein (2011).

Orchestrating Productive Mathematical Discourse Chart for Monitoring, Selecting, Sequencing, and Connecting Student Thinking

Strategy Work of Specific Students Sequence Compare

A.

3 A and B, A and C

B. Number line connect to add up

1

C. Add up to nearest hundred

2 C and B

D.

5 D and F

E. Base Ten Pieces connect to take away showing regrouping**

6 delay if

not enough

time

F. Base Ten Pieces or number line connect to take away too many (320) and add 1 back

4 Ask students for connections

they see

Adapted from Smith and Stein (2011).

** By the end of the unit, the target is: 4.NBT4 Fluently add and subtract multi-digit numbers using the standard algorithm. Since this is the first lesson on multi-digit subtraction, this is not the main emphasis.

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-­‐  Boaler  &  Humphreys,  p.  37,  2005  

! !

!

Examining    Student  Thinking  

&    Teacher  Moves  

*  Type  1  –  Answering,  Stating,  or  Sharing    

*  Type  2  –  Explaining    

*  Type  3  –  Questioning  or  Challenging  

*  Type  4  –  Relating,  Predicting,  or  Conjecturing  

*  Type  5  –  Justifying  or  Generalizing    -­‐  Weaver,  Dick,  &  Rigelman,  2005  

Discourse  Types  

*  Type  1  –  Answering,  Stating,  or  Sharing  A  student  gives  a  short  right  or  wrong  answer  to  a  direct  question  or  makes  a  simple  statement  or  shares  work  that  does  not  involve  an  explanation  of  how  or  why.  *  Type  2  –  Explaining    *  Type  3  –  Questioning  or  Challenging  *  Type  4  –  Relating,  Predicting,  or  Conjecturing  *  Type  5  –  Justifying  or  Generalizing    

-­‐  Weaver,  Dick,  &  Rigelman,  2005  

Discourse  Types  

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*  Type  1  –  Answering,  Stating,  or  Sharing    *  Type  2  –  Explaining  A  student  explains  a  mathematical  idea  or  procedure  by  describing  how  or  what  he  or  she  did  but  does  not  explain  why.    *  Type  3  –  Questioning  or  Challenging  *  Type  4  –  Relating,  Predicting,  or  Conjecturing  *  Type  5  –  Justifying  or  Generalizing    

-­‐  Weaver,  Dick,  &  Rigelman,  2005  

Discourse  Types  

*  Type  1  –  Answering,  Stating,  or  Sharing    *  Type  2  –  Explaining  *  Type  3  –  Questioning  or  Challenging  A  student  asks  a  question  to  clarify  his  or  her  understanding  of  a  mathematical  idea  or  procedure  or  makes  a  statement  or  asks  a  question  in  a  way  that  challenges  the  validity  of  an  idea  or  procedure.    *  Type  4  –  Relating,  Predicting,  or  Conjecturing  *  Type  5  –  Justifying  or  Generalizing    

-­‐  Weaver,  Dick,  &  Rigelman,  2005  

Discourse  Types  

*  Type  1  –  Answering,  Stating,  or  Sharing    *  Type  2  –  Explaining    *  Type  3  –  Questioning  or  Challenging  *  Type  4  –  Relating,  Predicting,  or  Conjecturing  A  student  makes  a  statement  indicating  that  he  or  she  has  made  a  connection  or  sees  a  relationship  to  some  prior  knowledge  or  experience  or  makes  a  prediction  or  a  conjecture  based  on  an  understanding  of  the  mathematics  behind  the  problem.    *  Type  5  –  Justifying  or  Generalizing    

-­‐  Weaver,  Dick,  &  Rigelman,  2005  

Discourse  Types  

*  Type  1  –  Answering,  Stating,  or  Sharing    *  Type  2  –  Explaining    *  Type  3  –  Questioning  or  Challenging  *  Type  4  –  Relating,  Predicting,  or  Conjecturing  *  Type  5  –  Justifying  or  Generalizing  A  student  provides  a  justification  for  the  validity  of  a  mathematical  idea  or  procedure  or  makes  a  statement  that  is  evidence  of  a  shift  from  a  specific  example  to  the  general  case.    

-­‐  Weaver,  Dick,  &  Rigelman,  2005  

 

Discourse  Types  

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For more information about the Student Discourse Observation Protocol see, Weaver, D. & Dick, T. (September 2006). Assessing the Quantity and Quality of Student Discourse in Mathematics Classrooms, Year 1 Results. Paper presented at Math Science Partnership Evaluation Summit II, Minneapolis, MN. Available at: http://ormath.mspnet.org/index.cfm/14122.

Discourse Analysis Tool Discourse Types Predicted Actual Type 1 – Answering, Stating, or Sharing A student gives a short right or wrong answer to a direct question or makes a simple statement or shares work that does not involve an explanation of how or why.

Type 2 – Explaining A student explains a mathematical idea or procedure by describing how or what he or she did but does not explain why.

Type 3 – Questioning or Challenging A student asks a question to clarify his or her understanding of a mathematical idea or procedure or makes a statement or asks a question in a way that challenges the validity of an idea or procedure.

Type 4 – Relating, Predicting, or Conjecturing A student makes a statement indicating that he or she has made a connection or sees a relationship to some prior knowledge or experience or makes a prediction or a conjecture based on an understanding of the mathematics behind the problem.

Type 5 – Justifying or Generalizing A student provides a justification for the validity of a mathematical idea or procedure or makes a statement that is evidence of a shift from a specific example to the general case.

Implications  

As  you  reflect  on  what  you  heard  in  this  session,  what  are  take  aways…    *  For  your  classroom?  

*  For  your  professional  learning  community?  

*  For  your  work  supporting  teachers  in  either  setting?  

Questions?  

Nicole  Miller  Rigelman  *  Portland  State  University  rigelman@pdx.edu  

 

http://goo.gl/ys4Qd