“I’m  Confused”:  Suppor3ng  Students  Who  Struggle  

Barbara  Dougherty  University  of  Missouri  

doughertyb@missouri.edu  

Reflec%ng  on  the  Classroom  

When  you  think  about  learners  who  struggle,  what  do  you  believe  are  their.  .  .  •  Biggest  challenges?  •  Greatest  strengths?  

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Focus  of  Session    

This  session  focuses  on  ways  to  support  students  who  struggle  in  Tiers  1  and  2  to  become:  •  Confident  in  

mathema%cs  •  Successful  with  rigorous  

mathema%cs  •  Capable  of  working  with  

complex  mathema%cal  ideas  

~15%

~5%

3-Tiered Support Model

≈ 80%

≈ 15%

≈ 5%

How  do  your  students  think  about  mathema%cs?    

Structuring  Informa%on  

Compared  to  students  who  are  not  struggling,  their  brains  might  look  different!  

Skemp,  R.  (1987).  The  psychology  of  learning  mathema3cs,  The  Penguin  Press..  

Struggling  Students  in  Mathema%cs  

Students  who  struggle  with  mathema%cs  oSen  – use  procedures  that  younger,  typically  achieving  students  use;      

– make  frequent  errors  when  execu%ng  procedures;  and    – have  a  poor  understanding  of  concepts  that  are  founda%onal  to  performing  procedures  (Geary,  2004).  

Addi%onally,  they.  .  .  – do  not  see  mathema%cal  ideas  connected  – have  developed  a  high  dependency  on  help  –   exhibit  a  ‘give  up’  a[tude  

How  can  we  help  them?  

We  need  to  consider  the  type  of  mathema%cs  and  the  way  in  which  we  teach  the  mathema%cs.  

Making  Cents  

•  Take  out  some  coins.  •  Mul%ply  the  value  of  the  coins  in  cents  by  4.  •  Add  10  to  the  product.  •  Mul%ply  your  answer  by  25.  •  Add  115  to  your  answer.  •  Add  your  age  in  years.  •  Subtract  the  number  of  days  in  a  normal  year.  

Making  Cents  

•  What  do  you  no%ce  about  your  answers?    •  How  can  you  describe  what  you  no%ce  to  someone  who  is  not  present?  

 Types  of  Understandings  

Procedural  -­‐  Student  can  perform  a  computa%on  or  algorithm  by  following  a  series  of  prescribed  steps  

Conceptual  -­‐  Student  understands  the  basis  of  why  a  computa%on  or  algorithm  works.  They  can  apply  it  later  without  reteaching.  Student  can  iden%fy,  describe,  and  explain  a  big  idea  related  to  a  topic  or  a  class  of  problems  

Problem  solving  -­‐  Student  can  solve  a  problem  when  there  is  no  specific  solu%on  pathway  or  algorithm  

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Procedures  without  understanding  

   •  Students  memorize  steps  (and  are  oSen  required  to  complete  the  steps  in  a  specific  sequence)  

•  Students  prac%ce  a  large  number  of  problems  using  those  steps    

How  does  your  work  compare?  

How  did  she  do  the  work?  

I  don’t  really  get  the  way  the  way  my  5th  grade  teacher  showed  me.  I  made  it  look  like  her  work  so  I  wouldn’t  get  any  points  off.    So  I  think,  what  plus  3/8  is  1?  That’s  5/8.  I  have  1,  so  I  add  it  to  7  and  that  is  8.  16  minus  8  is  8  so  the  answer  is  8  5/8.    

Knowing how students think is important

How do you think students will solve this problem?  

How would you like students to solve this problem?

0.75  of  students  tried  to  solve  for  g  

What  would  show  strong  understanding?  

If g – 227 = 543, what does g – 230 equal? Show your THINKING. Since 230 is 3 more than 227, the difference will be less than 543. The difference is 540.

Begin  with  a    conceptual  focus  on  variable  

Problems  like:  – Zany  Z  – Perimeter  problems  like  the  hexagon  problem    Use  problems  where  the  variable  represents  a  physical  rela%onship  Use  mul%ple  representa%ons:  words,  physical  materials,  drawings/diagrams,  symbols,  graphs  CONCURRENTLY  

Use  mul%ple  contexts  

•  Use  a  number  line  to  model  generalized  rela%onships  

Think  in  generalized  terms  

•  Give  tasks  that  require  students  to  think  about  rela%onships  rather  than  algorithms  

 Joe  said,  “b  +  4  is  less  than  6  +  b.”  Do  you  think  Joe’s  statement  is  always  true,  some%mes  true,  or  never  true?  Explain  how  you  know.  

Example  based  teaching  

•  Here’s  how  you.  .  .  .  .  .  .  

•  Now  you  solve  these.  

•  I  do  • We  do  •  You  do  

Example-­‐based  teaching  requires  students  who  are  struggling  to  make  generaliza%ons  about  the  structure  of  the  class  of  problems.  

What  is  explicit  instruc%on?  

•  It  is  NOT  direct  instruc%on.    

– Direct  instruc%on  is  the  teacher  showing  students  how  to  do  something  or  giving  factual  informa%on.  

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What  is  explicit  instruc%on?  

•  Focusing  students  aoen%on  on  par%cular  structures  or  ideas  – Asking  ques%ons  so  that  students  ‘see’  the  mathema%cs    

– Providing  tasks  that  allow  students  to  explore  the  topic  

 

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What  does  it  look  like?  

•  Teacher  introduces  a  problem  that  links  to  previous  learning.  

•  Students  work  in  pairs  or  small  groups  to  solve.  •  Students  share  their  thinking  with  the  class,  cri%qued  by  others  and  teacher.  

•  Teacher  scaffolds  tasks  based  on  misconcep%ons  that  are  evident  in  thinking.  

•  Teacher  poses  ques%ons  throughout  that  focus  students  on  important  ideas  and  generaliza%ons.  

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Explicit  Instruc%on  

•  Try  to  elicit  the  informa%on  from  students  (see  Never  Say  Anything  a  Kid  Can  Say,  Mathema3cs  Teaching  in  the  Middle  School)  

•  Developing  Concepts  and  Generaliza%ons  to  Build  Algebraic  Thinking:  The  Reflec%vity,  Flexibility,  and  Generaliza%on  Approach  

 Dougherty,  Bryant,  Bryant,  Darrough,  &  Pfannens%el,  Interven3on  in  School  and  Clinic,  April  2015.  (Sage  Publica%ons)  

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Food  for  Thought  

•  Cri%cal  thinking  ques%ons  should  be  asked  in  every  class,  every  day  

•  Consistency  helps  students  understand  the  expecta%ons  and  move  toward  higher  proficiency  

Can I be excused? My brain is full.

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Closing  the  Gap  

•  Changing  the  way  tasks  are  posed  •  Crea%ng  high  expecta%ons  and  accountability  

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Curriculum  materials  

•  When  the  materials  break  skills  down  into  small  pieces,  it  requires  students  to  put  the  pieces  together  to  form  the  whole.  

Ques%oning  Techniques  

•  Factual  ques%ons  comprise  the  majority  of  ques%ons  asked  in  a  mathema%cs  class  – More  than  145  ques%ons  in  48  minute  class  period    

–  Less  than  2  seconds  for  response  

   

Dougherty  &  Foegen,  2010  

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Curriculum  materials  

•  When  students  prac%ce  only  a  skill  it  places  a  large  cogni%ve  demand  on  their  memory.  

Tradi%onal  Tasks  

       

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Reversibility question:

–  Find 2 fractions whose quotient is

–  Find another pair –  Find 3 more pairs

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Change the Task

Generalization question:

– What do you notice about the quotient when you divide a fraction greater than 1 by a fraction that is less than 1/2? Why?

– What do you notice about the quotient when you divide a fraction less than ½ by a fraction that is greater than 1? Why?

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Change the Task

Flexibility question:

–  Divide using 2 different methods. –  How are the ways you

divided them alike? –  How are they different?

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Change the Task

Flexibility question

Divide:

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Change the Task

You  try  one!  

How  could  you  change  the  ques%on  you  ask:    

Find  30%  of  $16.  

Strategies    

•  Teaching  approaches  that  focus  on  student  communica3on,  ac3ve  engagement  in  problem  solving  

Trends  from  Data  

•  Skills  are  broken  down  into  very  small  pieces  •  Heavier  focus  on  algorithms  and  procedures  •  Requirement  to  show  all  steps  •  Explana%ons  are  oSen  mathema%cal  steps  rather  than  thinking  process  

•  Vocabulary  not  consistent  •  Teacher  content  knowledge  

ShiSs  in  teaching  and  learning  Moving  away  from  .  .  .   To.  .  .  Telling/showing  how  to  do  something  

Building  from  concept  to  skill  

Teacher-­‐centric  instruc%on   Student-­‐centered  instruc%on  Problem  solving  intermioently   Problem  solving  every  day  A  focus  on  only  the  answer   A  focus  on  jus%fying  and  

explaining  Showing  the  steps   Explaining  the  reasoning  Problems  that  require  only  fast  calcula%ons  

Problems  that  require  thinking  

Pre-­‐Conference  

•  Strategies  for  Students  who  Struggle  in  Grades  3–8  

NCTM  Regional  Mee%ng,  Atlan%c  City  October  21,  2015  

Resources  

•  Recommenda%ons  are  based  on  strong  and  moderate  levels  of  evidence  resul%ng  from  comprehensive  reviews  of  current  research  literature.      

hop://ies.  ed.gov/ncee/wwc/publica%ons/prac%ceguides/