Module  2:  This  is  a  pedagogy,  not  a  program

Module  2  focuses  on  the  teaching  and  learning  cycle  and  effective  pedagogy  and  practices  in  mathematics.  

The  effective  teaching  of  mathematics  results  in  students  developing  efficient  numeracy  strategies  that  they  can  apply  within  mathematics,  across  KLAs  and  beyond  the  classroom.

Overview  of  Module

This  Module  has  a  focus  on  professional  readings.  It  is  important  that  teachers  make  informed,  evidence-­‐based  decisions  on  how  to  teach  and  what  strategies  to  use  in  the  classroom.  

Overview  of  Module

Embedding  the  teaching  practices  explored  in  this  module  will  allow  teachers  to  see  that  this  is  

‘just  the  way  we  teach.’

It  is  not  a  numeracy  program  to  be  implemented  in  isolation  from  mathematics  lessons.  This  is  quality  teaching  for  all  students  in  all  classrooms.  

Overview  of  Module

Teaching  students  efficient  numeracy  skills  forms  one  aspect  of  explicit  mathematics  teaching.  

These  skills  can  also  be  explored  through,  and  developed  in  other  KLAs.  

This  is  a  pedagogy,  not  a  program

This  is  a  pedagogy,  not  a  program

purposeful  questioning

practicing  formative  assessment

differentiating  tasks

ongoing  assessment

making  connections

communicating  efficient  strategies

Examples  of  effective  teaching  strategies

Effective  teaching  strategies  are the  pedagogy  that  form  how  we  teach,  they  are  also  underpinned  by  the  teaching  and  learning  cycle.

This  is  a  pedagogy,  not  a  program

Strategies  that  are  used  in  a  Year  1  classroom  are  transferable  to  a  Year  5  classroom,  the  strategies  remain  the  same.  What  changes  is  the  content  and  students’  needs.

• The  aim  is  for  effective  teaching  strategies  to  be  utilised  in  all  mathematics  classrooms  from  Kindergarten  to  Year  8  and  beyond  to  assist  students  in  becoming  numerate.

• Developing  numeracy  skills  mainly  occurs  in  mathematics  lessons.

• Strategies  that  teachers  use  should  be  embedded  within  classroom  practice  and  not  seen  as  extra  to  what  is  currently  happening.  

This  is  a  pedagogy,  not  a  program

• Some  numeracy  strategies  can  be  practised  and  repeated  throughout  the  day  (generally  in  primary  classrooms  where  the  teacher  has  the  same  students  all  day,  secondary  teachers  could  possibly  use  the  start  of  each  lesson  to  reinforce).  

• Any  short,  focused,  frequent  (SFF)  activities  used  throughout  the  day  need  to  be  aligned  to  the  concept  students  are  currently  learning  and  be  based  on  the  needs  of  the  students  being  targeted.

This  is  a  pedagogy,  not  a  program

• Use  the  Participant's  Activity  Journal  to  complete  this  task• Choose  either  Option  A  or  Option  B  for  this  reading  task• Print  out  the  two  appropriate  readings• Once  read,  complete  the  questions  in  the  Activity  Journal• If  completing  the  course  with  another  teacher  or  a  small  team,  discuss  the  reflections  on  the  readings  as  a  group

Activity  1

The  teaching  and  learning  cycle:  Part  1

The  teaching  and  learning  cycle  represents  the  four  stages  that  occur  in  the  design  and  delivery  of  classroom  tasks  that  incorporate  an  outcomes-­‐based  approach.

The  cycle  has  no  start  or  end  point,  with  each  step  informing  the  next  through  ongoing  evaluation.  It  is  the  process  of  gathering  data  and  reflection  that  dictates  where  in  the  cycle  you  need  to  be  operating.

The  teaching  and  learning  cycle

Students  differ  in  many  ways  fundamental  to  learning:  prior  knowledge,  ability,  conceptions  of  learning,  learning  styles  and  strategies,  interest,  motivation,  self-­‐efficacy  beliefs  and  emotion,  as  well  in  socio-­‐environmental  terms  such  as  linguistic,  cultural  and  social  background.

[Our]  challenge  is  to  manage  such  differences,  …within  a  shared  education  and  culture.

The  Nature  of  Learning:  Using  Research  to  Inspire  Practice  ©  OECD  2010

Teaching  is  a  balancing  act

‘Prior  knowledge  is  one  of  the  most  important  resources  on  which  to  build  current  learning  as  well  as  one  of  the  most  marked  individual  difference  among  

learners.’

The  Nature  of  Learning:  Using  Research  to  Inspire  Practice  ©  OECD  2010

Prior  knowledge

Using  assessment for  learning  teachers  can  learn  more  about  their  students  and  their  prior  knowledge.Teachers  should  be  able  to  answer  these  questions  in  preparation  for  planning:• What  do  my  students  already  know  and  what  can  they  do?  • How  do  I  know  this?  What  is  my  evidence?• Which  students  require  extension  in  particular  areas?  • Which  students  require  support  in  particular  areas?

Where  are  my  students  now?

The  assessment  data  collected  informsteachers  of  the  numeracy  skills  and  strategies  students  need  to  develop.

As  teachers  develop  a  unit  of  learning,  they  use  the  mathematics  syllabus and  a  scope  and  sequence  to  determine  allocated  outcomes  and  content.  

It  is  important  to  make  connections  between  subtrands  to  assist  in  students’  conceptual  understanding.

What  do  I  want  my  students  to  learn?

•Use  the  Participant's  Activity  Journal  to  complete  this  task  along  with  an  example  lesson  plan  

• Print  out  and  read  the  Capacity  Building  Series:Differentiating  Mathematics  Instruction  reading

•Once  read,  annotate  a  current  lesson  plan  focusing  on  how  it  could  be  differentiated:• e.g.  open  or  parallel  tasks,  content,  process  or  product

• If  completing  the  course  with  another  teacher  or  a  small  team,  discuss  reflections  on  the  reading  and  the  lesson  plan  as  a  group

Activity  2

The  teaching  and  learning  cycle:  Part  2

Students  will  ‘get  there’  through  planned  quality  learning  experiences.  Teachers  differentiate practice  in  response  to  students’  learning  and  support  needs  (identified  in  your  initial  assessments).Teachers  modify  lessons  to  cater  for  students’  needs.Content  -­‐ Change  the  range  of  numbers  Process  -­‐ Use  flexible  grouping  (ability,  interest,  mixed)  Product  -­‐ Have  students  record  in  different  ways  

How  will  my  students  get  there?

Teachers  differentiate  learning  in  ways  that  are  responsive  to  students’  needs.  

While  units  may  be  planned  collaboratively  and  shared,  teachers  will  need  to  differentiate    units  to  cater  for  particular  students.

Differentiation  will  be  covered  further  in  Module  5.

Developing  units  of  learning

By  monitoring  and  tracking  through  ongoing  assessment,  teachers  can  collect  evidence  of  student  learning,  moment-­‐by-­‐moment.  This  informs  teachers  about  what  students  currently  understand  and  where  to  next  in  the  learning.

How  do  I  know  when  my  students  get  there?

Effective  and  informative  assessment  and  reporting:• has  clear,  direct  links  with  outcomes• values  teacher  judgement• recognises  individual  achievement  and  progress• conveys  meaningful  and  useful  information.

• Use  your  Participant's  Activity  Journal  to  complete  this  task• Print  out  the  Effective  Pedagogy  in  Mathematics  reading• Once  read,  complete  the  reflection  task  in  the  Activity  Journal• If  completing  the  course  with  another  teacher  or  a  small  team,  discuss  reflections  on  the  reading  as  a  group  and  share  ideas  around  the  ten  principles  of  effective  pedagogy

Activity  3

Effective  pedagogy  in  mathematics

Quality  pedagogical  approaches  to  

teaching  mathematics are  the  building  blocks  for  

numeracy.  

‘The  principles  of  practice  identified  by  Askew1 and  Clarke2  as  being  

central  to  effective  teaching  of  numeracy  were;  making  connections,  challenging  all  pupils,  teaching  for  

conceptual  understanding,  facilitating  purposeful  discussion,  maintaining  a  focus  on  mathematics,  and  possessing  and  

instilling  positive  attitudes  towards  mathematics.’

Muir,  Tracey.  (2008)  Principles  of  practice  and  teacher  actions:  influences  on  effective  teaching  of  numeracy.  Mathematics  Education  Research  Journal,  Vol.  20,  No.  3,  78-­‐101

1Askew,  M.,  Brown,  M.,  Rhodes,  V.,  Johnson,  D.,  &  Wiliam,  D.  (1997).  Effective  teachers  of  numeracy.  London:  School  of  Education,  King's  College.  2Clarke,  D.,  Cheeseman,  J.,  Gervasoni,  A.,  Gronn,  D.,  Horne,  D.,  McDonough,  A.  et  al.  (2002).  Early  numeracy  research  project  final  report.  Melbourne,  Australia:  Australian  Catholic  University.  

making  connections-­‐ connecting  different  areas  of  mathematics  e.g.  area  and  multiplication

challenge  all  pupils-­‐ setting  high  but  realistic  expectations  e.g.  ‘your  next  step  in  learning  is  counting  by  tens  off  the  decade’

teach  for  conceptual  understanding-­‐ awareness  of  conceptual  connections  between  areas  e.g.  to  understand  

how  to  add  and  subtract,  students  must  have  a  strong  concept  of  ones,  tens,  hundreds

purposeful  discussion-­‐ use  a  range  of  question  types  to  probe  and  challenge  students’  thinking  

e.g.  How  do  you  know  when  you  have  all  the  combinations?Muir,  Tracey.  (2008)  Principles  of  practice  and  teacher  actions:   influences  on  effective  teaching  of  numeracy.  Mathematics  Education  Research  Journal,  Vol.  20,  No.  3,  78-­‐101

‘…teachers  must  see  mathematics  as  a  set  of  conceptual  relationships  between  numbers  and  number  symbols  rather  than  as  numbers  that  are  manipulated  by  rules.  

The  questions  we  ask,  the  tasks  we  design,  and  the  discussions  we  prompt  can  refocus  students  on  discovering  such  relationships  themselves.’

Teaching  number  sense,  Sharon  Griffin,  Educational  Leadership  Vol.  61  (2004)

Habits  of  mind

The  ten  principles  of  effective  mathematics  pedagogy  in  the  Anthony  and  Walshaw  reading  should  not  be  taken  in  isolation  but  interpreted  as  part  of  a  complex  web  of  factors  that  can  affect  student  learning.  

The  following  presentation  discusses  a  few  of  the  key  principles  in  more  detail.

Effective  pedagogy

Effective  teachers  provide  opportunities  for  students  to  work  both  independently  and  collaboratively.

…teachers  invite  students  to  explain  their  solutions  to  others;  they  also  encourage  students  to  listen  to  and  respect  one  another,  accept  and  evaluate  different  viewpoints,  and  engage  in  an  exchange  of  thinking  and  perspectives.

Key  principle  2:  Arranging  for  learning

There  are  times  when  teachers  group  students  by  ability  so  students  can  work  alongside  peers  at  the  same  level  of  understanding.  

Allowing  teachers  to  work  with  one  specific  group  at  a  time  while  the  other  groups  work  independently.

At  other  times,  mixed  ability  groups  are  important.  Students  can  share  efficient  strategies  and  different  ways  of  working  to  find  varied  solutions  to  tasks.  

At  times,  when  explicit  teaching  occurs,  the  whole  class  may  work  together  or  as  individuals.

In  the  mathematics  classroom:  Flexibility  in  the  learning  environment

Effective  teachers  support  students  in  creating  connections  between  different  ways  of  solving  problems,  between  mathematical  representations  and  topics,  and  between  mathematics  and  everyday  experiences.

The  ability  to  make  connections  between  apparently  separate  mathematical  ideas  is  crucial  for  conceptual  understanding  and  numeracy  development.

Key  principle  5:  Making  connections

For  students  to  be  able  to:  • develop  the  formula  for  the  volume  of  prisms  by  considering  the  numbers  and  volume  of  ‘layers’  (base  area  x  height)  Volume  Stage  4,  NSW  mathematics  K-­‐10  syllabus

We  need  to  first  make  the  connections  for  students  between  the  concepts (links  to  Aspects  1,  3,  5  and  7  on  the  Numeracy  continuum)

Key  principle  5:  Making  connectionsIn  the  mathematics  classroom

Skip  counting Repeated  addition Multiplication Area Volume

Students'  numeracy  and  underlying  mathematical  understanding  will  be  enhanced  through  engagement  with  a  variety  of  applications  of  mathematics  to  real-­‐world  situations  and  problems  in  other  learning  areas.  

NSW  mathematics  K-­‐10  syllabus

Key  principle  5:  Making  connectionsOutside  the  mathematics  classroom

Key  principle  5:  Making  connections

The  Improving  Mathematics  Education  in  Schools  (TIMES)  Project  2009-­‐2011

Example  of  linking  Multiplication  and  Division  and  Ratios  and  Rates  in  the  syllabus,  aligned  to  Aspect  5  Multiplication  and  division  as  operations  and  Aspect  3  Number  properties  of  the  numeracy  continuum,  to  real  life  problems  involving  money.

Effective  teachers  use  a  range  of  assessment  practices  to  make  students’  thinking  visible  and  to  support  students’  learning.

Moment-­‐by-­‐moment  assessment  of  student  progress  helps  them  [teachers]  decide  what  questions  to  ask,  when  to  intervene,  and  how  to  respond  to  questions.

Key  principle  6:  Assessment  for  learning

Example  of  questions  to  ask  during  an  activity  as  part  of  formative  assessment  (example  Make  100  activity  from  Counting  On  activities,  NSW  DET,  2004)

Key  principle  6:  Assessment  for  learningIn  the  mathematics  classroom

Questioning1. After  two  rolls  the  teacher  asks  students  to  name  all  possible  scores.  (In  the  sample  game  the  possible  scores  are  5,  23,  32,  and  50.)  Note:  If  the  first  two  numbers  rolled  are  the  same,    there  are  only  three  possible  scores,  e.g.  rolls  of  3,  3  produce  possible  scores  of  6,  33,  60.

2.  When  there  are  two  rolls  to  go,  ask  students  to  stand  if  it  is  still  possible  for  them  to  score  exactly  100.  Ask  their  scores  and  check  as  a  whole  class  activity.

3.  When  there  are  two  rolls  to  go,  ask  students  to  announce  their  scores  and  say  what  they  would  like  the  next  two  rolls  to  be,  e.g.  a  student  with  a  score  of  88  could  score  exactly  100  with  2  sixes.  The  same  result  could  be  achieved  with  a  1  and  a  2  (12).

Effective  teachers  develop  and  use  sound  knowledge  as  a  basis  for  initiating  learning  and  responding  to  the  mathematical  needs  of  all  students.  They  understand:• the  big  ideas  of  mathematics• mathematics  as  a  coherent  and  connected  system• how  to  extend  and  challenge  student  ideas• how  to  adapt  and  modify  their  routines  to  fit  student  

needs

Key  principle  10:  Teacher  knowledge

The  numeracy  continuum  K-­‐10  needs  to  be  viewed  not  as  separate  aspects  to  teach,  but  as  interrelated    concepts  that  build  on  each  other  to  assist  students  in  developing  efficient  numeracy  strategies.  

Understanding  the  big  picture  of  how  being  able  to  identify  and  count  numbers  leads  to  understanding  patterns  in  mathematics  is  vital.  

Teachers  also  need  to  understand  the  links  between  additive  and  multiplicative  thinking,  flexibility  with  numbers  and  the  commutative  property  of  addition  and  multiplication.

Key  principle  10:  Teacher  knowledgeIn  the  mathematics  classroom

End  of  Module  2:  This  is  a  pedagogy,  not  a  program

For  accreditation  please  ensure  all  activities  for  this  module  have  been  completed.

All  activities  are  to  be  completed  in  the  Participant  Activity  Journal,  the  annotated  lesson  from  Activity  2  also  needs  to  be  attached  for  this  Module.

This  is  Module  2  of  5,  all  five  modules  must  be  completed  to  receive  the  12  hours  of  registered  professional  learning.  

Once  all  five  modules  are  completed,  the  principal/delegate  will  sign  off  on  the  deliverables  and  the  participant’s  completion  in  the MyPL@Edu  event  that  was  scheduled  by  the  school.