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James Calleja

Plan for  the  Session

Introduction

Topic Introducing  the  aims  of  this  series  of  the  four   summer  workshops   What  do  you  expect  to  get  from  today’s  session?

Time

½ h

Aims of  the  session   Working  on  a   Mathematical   Teachers  work  on  the  ‘Four  Fours’  task.   Task

¾ h

Teacher views  on  their  experience  working  on  the   ‘Four  Fours’  task

½ h

Video Discussion

Teachers reflect  and  discuss  the  roles  of  the  teacher   and  that  of  students  during  the  ‘Four  Fours’  task.

½ h

Teachers are  presented  with  a  range  of  tasks.     Which  tasks  are  more  likely  to  promote  inquiry?     What  characteristics  should  they  have?

½ h

Teachers work  in  groups  to  plan  a  task  for  inquiry.   Teacher  present  and  share  their  work  with  the   whole  group.

¾ h

Aims of  the  session   For  today’s  session  we  will  have  the  following  aims:   o To  understand  the  role  of  tasks  in  planning  to  teach  mathematics   through  inquiry   o To  experience  mathematical  inquiry  by  working  on  a  task     o To  explore  tasks  that  provide  opportunities  for  students  to  engage  in   mathematical  inquiry   o To  reflect  critically  on  an  ‘inquiry’  lesson   o To  experience  features  and  aspects  of  mathematical  inquiry

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Teaching and  Learning  Mathematics  through  Inquiry

45 min

You are  provided  with  two  sets  of  numbers:

Four 4’s

&

4 4 4 4

four  basic  operations

+ −

× ÷

Part  1  –  On  your  own

10 min

Use all  four  4’s  and  any  of  the  four  basic  operations  provided  to  write   equations  that  give  answers  from  0  to  10.     You  may  want  to  use  other  operations  than  the  ones  given!

Part 2  –  In  Pairs

15 min

Work with  a  partner  and  investigate  whether  it  is  possible  to  come  up   with  more  than  one  equation  for  each  answer  0  to  10.

Part 3  –  In  Groups  of  three

15 min

Use all  four  4’s  and  any  operation  to  write  equations  that  give   answers  from  11  to  20.  Explore  possibilities  for  other  ways  of   writing  equations  for  each  number.       This  is  how  you  will  work  on  this  task:   • The  task  is  presented  to  you

3  minutes

• You have  some  time  to  work  individually  on  the  problem

7  minutes

• You will  be  asked  to  work  with  a  partner

10  minutes

• It is  now  time  for  you  to  work  in  a  small-­‐group     10  minutes   • You  will  present  and  share  your  findings  to  the  whole  group

15  minutes

Teaching and  Learning  Mathematics  through  Inquiry

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SPACE FOR  WORKING

4

Teaching and  Learning  Mathematics  through  Inquiry

DISCUSSION POINTS

30 min

As a  whole  group  you  are  asked  to  reflect  on  the  following  questions:   1. What  opportunities  does  the  task  provide  for  students  to  struggle  with   mathematical  ideas?   2. How  do  you  see  students  engaging  with  important  mathematical  ideas?   3. What  could  the  mathematical  goals  for  a  lesson  using  this  task  be?  How  would   you  plan  a  lesson  using  this  task?     4. How  do  you  see  this  task  integrated  within  a  unit  of  study?

WATCHING  A  LESSON  VIDEO

10 min

You will  now  watch  a  video  of  a  teacher  (David)  using  the  ‘Four  Fours’  task  with  his   form  two  (year  8)  levels  7-­‐8  class.   Note   how   the   teacher   structures   the   lesson,   the   mathematical   ideas   valued,   the   difficulties  that  students  encounter  and  how  the  teacher  deals  with  these  issues.   For   the   follow-­‐up   discussion,   you   are   encouraged   to   write   down   some   notes/points   you  might  see  as  important.   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________

20 min

What are  your  comments  about  the  lesson?   Would  you  structure  the  lesson  as  the  teacher  did  or  would  you  do  it  differently?   Who  generates  the  mathematical  ideas  that  get  discussed?  Who  evaluates  and/or   responds  to  these  ideas?   How  deeply  do  students  get  to  explain  their  ideas?   How  does  the  teacher  respond  to  students’  struggles?   To  what  extent,  do  you  think,  students  engaged  in  inquiry?

Teaching and  Learning  Mathematics  through  Inquiry

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LOOKING AT  MATHEMATICAL  TASKS  FOR  INQUIRY

30 min

You are  presented  with  a  set  of  tasks  –  also  available  on  the  teacher  booklet.     These   tasks   are   taken   from   the   work   of   Malcolm   Swan   and   two   websites   –   Inquiry   Maths  and  Bowland  Maths.       COLLABORATIVE  LEARNING  TASKS   Malcolm   Swan   created   a   framework   with   five   ‘types’   of   activities   that   encourage   distinct  ways  of  thinking  and  learning.  These  are:   1. Evaluating  mathematical  statements  –  ask  students  whether  statements  are   always,  sometimes  or  never  true,  and  developing  proofs

2. Classifying  mathematical  objects  –  ask  students  to  devise  or  apply  a  classification                             Worksheet  1

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Teaching and  Learning  Mathematics  through  Inquiry

Worksheet 2

How c an  you  justify  each  of  (a),  (b),  (c)  as  the  odd  one  out?

3. Interpreting multiple  representations  –  draw  links  and  develop  mental  images   for  concepts

4. Creating  and  solving  problems  –  ask  students  to  create  problems  for  the  class

Teaching and  Learning  Mathematics  through  Inquiry

7

5. Analyzing reasoning  and  solutions  –  diagnose  errors  and  comparing  solutions   Cut  up  the  following  cards.     Rearrange  them  to  form  two  proofs.   The  first  should  prove  that:     If  n  is  an  odd  number,  then  n2  is  an  odd   number   The  second  should  prove  that:     If  n2  is  an  odd  number,  then  n  is  an  odd   number.   You  may  need  to  use  all  the  cards.

INQUIRY  MATHS  PROMPTS     From  http://www.inquirymaths.co.uk   On  the  website  pages:   Inquiry  maths  is  a  model  of  teaching  that  encourages  students  to  regulate  their   own  activity  while  exploring  a  mathematical  statement  (called  a  prompt).  Inquiries   can  involve  a  class  on  diverse  paths  of  exploration  or  in  listening  to  a  teacher's   exposition.  In  inquiry  maths,  students  take  responsibility  for  directing  the  lesson   with  the  teacher  acting  as  the  arbiter  of  legitimate  mathematical  activity.   Prompts  are  mathematical  statements,  equations  or  diagrams  stripped  back  to  the   bare  minimum,  while  simultaneously  loaded  with  the  potential  for  exploration.  In   short,  a  prompt  should  have  “less  to  it  and  more  in  it”.   Inquiry  is  not  about  discovering  a  pre-­‐determined  outcome;  rather,  it  is  a  joint   mathematical  exploration  initiated  by  the  student  and  supported  by  knowledgeable   others,  be  they  peers  or  adults.

PROMPT  1:     A  NUMBER  PROMPT     Why  is  one  statement  correct  when  the  other   one  is  not?

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Teaching and  Learning  Mathematics  through  Inquiry

PROMPT Â 2: Â  AN Â ALGEBRA Â PROMPT Â  Â  Encourage Â students Â to Â come Â up Â with Â the Â questions Â on Â the Â following Â prompt! Â

Â Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

đ?&#x2019;&#x161; â&#x2C6;&#x2019; đ?&#x2019;&#x2122; = đ?&#x;&#x2019; Â

Â PROMPT Â 3: Â  A Â GEOMETRY Â PROMPT Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Class Â posing/answering Â some Â questions Â in Â response Â to Â the Â prompt: Â  â&#x2021;&#x2019;

What Â is Â different Â and Â the Â same Â about Â the Â rectangles? Â  Â

â&#x2021;&#x2019;

How Â many Â rectangles Â are Â possible Â with Â the Â same Â area? Â

â&#x2021;&#x2019;

Which Â has Â the Â longest Â perimeter? Â ... Â the Â shortest? Â

â&#x2021;&#x2019;

Is Â there Â a Â rectangle Â with Â an Â area Â equal Â to Â the Â length Â of Â its Â perimeter? Â

Â

Â Â  BOWLAND Â MATHS Â TASKS Â  Â  From Â http://www.bowlandmaths.org.uk Â  On Â the Â website Â pages: Â

Â

Bowland Â Maths Â aims Â to Â make Â maths Â engaging Â and Â relevant Â to Â pupils Â aged Â 11-Â­â&#x20AC;?14, Â  with Â a Â focus Â on Â developing Â thinking, Â reasoning Â and Â problem-Â­â&#x20AC;?solving Â skills. Â In Â these Â  materials, Â the Â maths Â emerges Â naturally Â as Â pupils Â tackle Â problems Â set Â in Â a Â rich Â  mixture Â of Â real-Â­â&#x20AC;?life Â and Â fantasy Â situations. Â

Teaching Â and Â Learning Â Mathematics Â through Â Inquiry Â

Â

9 Â

Three  Unstructured  Problems

ORGANISING A  TABLE  TENNIS  TOURNAMENT

You have  the  job  of  organising  a  table  tennis  league.       • 7  players  will  take  part   • All  matches  are  singles.   • Every  player  has  to  play  each  of  the  other  players  once.     • There  are  four  tables  at  the  club.   • Games  will  take  up  to  half  an  hour.     • The  first  match  will  start  at  1.00pm.     Plan  how  to  organise  the  league,  so  that  the  tournament  will  take  the  shortest  possible   time.  Put  all  the  information  on  a  poster  so  that  the  players  can  easily  understand   what  to  do.

PROBLEM 2:

DESIGNING A  BOX  FOR  18  SWEETS

You work  for  a  design  company  and  have  been  asked  to   design  a  box  that  will  hold  18  sweets.     Each  sweet  is  2  cm  in  diameter  and  1  cm  thick.     The  box  must  be  made  from  a  single  sheet  of  A4  card   with  as  little  cutting  as  possible.     Compare  two  possible  designs  for  the  box  and  say  which  is  best  and  why.     Make  your  box.

PROBLEM  3:

CALCULATING BODY  MASS  INDEX

This calculator  shown  is  used  on  websites  to   help  an  adult  decide  if  he  or  she  is  overweight.     What  values  of  the  BMI  indicate  whether  an   adult  is  underweight,  overweight,  obese,  or   very  obese?       Investigate  how  the  calculator  works  out  the   BMI  from  the  height  and  weight.           Note  for  pupils:     If  you  put  your  own  details  into  this  calculator,  don’t  take  the  results  too  seriously!     It  is  designed  for  adults  who  have  stopped  growing  and  will  give  misleading  results  for   children  or  teenagers!

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Teaching and  Learning  Mathematics  through  Inquiry

30 min

What are  the  essential  differences  between  these  tasks  and  those  commonly  found   in  textbooks?   Why  are  these  tasks  more  likely  to  promote  inquiry?   What  characterizes  tasks  that  promote  inquiry?   What  pedagogical  issues  do  you  believe  will  arise  when  teachers  use  these  tasks?

45 min

Choose a  task  (from  the  ones  provided  above)  that  you  feel  would  be  appropriate  to   use  with  one  of  your  classes.   In  groups,  discuss  how  you  will:   ⇒

Organise the  classroom  and  the  resources  needed

Introduce the  problem  to  your  students

Explain to  students  how  you  want  them  to  work  together

Challenge/assist student  that  find  the  problem  straightforward/difficult

Help students  share  and  learn  from  alternative  problem-­‐solving  strategies

Conclude the  lesson

Teaching and  Learning  Mathematics  through  Inquiry

11

SESSION EVALUATION

10 min

Ø Briefly describe  your  experience  during  today’s  session.   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Ø What  did  you  feel  un/comfortable  doing  during  the  session?   Comfortable:  ___________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Uncomfortable:  ________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Ø I  used  to  think...  but  now  I  know…   I  used  to  think  __________________________________________________________________________________   ___________________________________________________________________________________________________   Now  I  know  ____________________________________________________________________________________   ___________________________________________________________________________________________________   Ø What  will  you  take  with  you  and  try  to  implement  in  your  class?   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Ø Any  other  comments/suggestions  that  you  would  like  to  add.   ___________________________________________________________________________________________________   ___________________________________________________________________________________________________   Thank  you  for  your  participation  and  reflections.

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Teaching and  Learning  Mathematics  through  Inquiry