What	  do	  children	  know	  about	  numbers?	  Background	  	  The	  purpose	  of	  this	  report	  is	  to	  present	  ...
Teaching	  Mathematics	  in	  Early	  Years	                                                                    	      EDU...
Teaching	  Mathematics	  in	  Early	  Years	                                                                          	   ...
Teaching	  Mathematics	  in	  Early	  Years	                                                                    	      EDU...
Teaching	  Mathematics	  in	  Early	  Years	                                                      	     EDUC8502	  	  Reco...
Teaching	  Mathematics	  in	  Early	  Years	                                                               	     EDUC8502	...
Teaching	  Mathematics	  in	  Early	  Years	                                                                         	    ...
Teaching	  Mathematics	  in	  Early	  Years	                                                                         	    ...
Teaching	  Mathematics	  in	  Early	  Years	                                                                          	   ...
Teaching	  Mathematics	  in	  Early	  Years	                                                                      	       ...
Teaching	  Mathematics	  in	  Early	  Years	                                                                              ...
Teaching	  Mathematics	  in	  Early	  Years	                                                                    	     EDUC...
Teaching	  Mathematics	  in	  Early	  Years	                                                                	      EDUC850...
Teaching	  Mathematics	  in	  Early	  Years	                                                                     	      ED...
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Think Board Interview, Recommendations and Reflection


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Numeracy Interview with Year 2 student using think boards top identify student strengths and weaknesses and recommend activities.

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Think Board Interview, Recommendations and Reflection

  1. 1.     SEMESTER  1   2011       EDUC8502                                                                                 TEACHING  MATHEMATICS                                   IN  EARLY  YEARS       Assignment  2     What  do  children  know  about  numbers?       Due:  Friday  May  13th  2011   Sharon  McCleary   19113469      Unit  Co-­ordinator:  Associate  Professor  Christine  Howitt    Tutor:  Ms.  Clair  Kipling          
  2. 2. What  do  children  know  about  numbers?  Background    The  purpose  of  this  report  is  to  present  findings  and  recommendations  arising  from  an  interview  with  Gabi,  a  Year  2  female  student,  conducted  on  1st  April  2011  at  Hollywood  Primary  School  in  Perth,  Western  Australia.  The   duration   of   the   interview   was   approximately   40   minutes.     It   focussed   on  determining  what  the  child  knew  about  numbers,  using  Think  Boards  as  a  strategy  to  promote  communication  about  her  number  knowledge  and  connections  across  various  modes  of  representation.  The   Think   Board   is   a   recording   format   that   allows   the   student   to   express   their  understanding   of   a   concept   in   various   ways   (i.e.   using   stories,   symbols,   pictures   and  real-­‐life   representations).     It   gives   valuable   insight   into   the   connections   the   student  has   formed   between   enactive   (concrete   objects),   iconic   (pictures,   diagrams)   and  symbolic   (words,   symbols)   representations   of   mathematical   concepts   (Frid,   2004),  shedding   light   on   the   individual   student’s   process   of   mathematical   meaning-­‐making  and   providing   a   useful   method   of   identifying   future   learning   areas   within   the   child’s  zone  of  proximal  development  (Krause,  2010).  Introduction    The  interview  commenced  with  a  relaxed  discussion  about  numbers,  aimed  at  making  the   student   feel   comfortable,   developing   rapport,   and   determining   her   general  disposition  towards  numbers.      The   book   “10”   by   Vladimir   Radunsky   was   read,   and   the   child   was   introduced   to   the  pre-­‐made   Think   Boards.     She   was   asked   to   choose   3   numbers   within   set   ranges   to  represent   in   different   ways   on   the   Think   Boards,   and   guided   through   each   of   the  sections.     She   displayed   genuine   excitement   about   numbers   and   was   eager   to  participate,  initially  asking  if  she  could  use  larger  numbers  outside  the  given  ranges.  
  3. 3. Teaching  Mathematics  in  Early  Years     EDUC8502    Gabi   selected   ‘15’   (recommended   range   of   11-­‐19)   as   her   first   number,   and   80   as   her  second  (recommended  range  50-­‐100).    After  completing  the  second  Think  Board  she  realised   she   would   have   difficulty   representing   larger   numbers   and   chose   ‘2’   as   her  third   number.     Attempts   to   persuade   her   to   choose   a   higher   number   resulted   in   her  choice   of   ‘20’   as   her   third   number.     She   was   clearly   outside   her   comfort   zone   when  larger   numbers   were   suggested   and   it   was   not   appropriate   to   challenge   her   further  that  particular  day.  The   resources   provided   for   the   child   to   use   included   various   sets   of   counters,  environmentally   available   natural   materials   such   as   leaves   and   stones,   coloured  pencils,  stickers,  stamps,  and  lead  pencils.  The  three  Think  Boards  are  included  for  reference  in  Appendix  A,  B  and  C  respectively,  along  with  photographs  of  the  ‘real’  items  used  on  each  Think  Board.  Student  Profile:  Analysis  Think  Board  One:  ‘15’  Gabi  was  able  to  represent  the  number  ‘15’  correctly  in  symbolic  form,  as  can  be  seen  in   the   “symbol”   section   of   Think   Board   One   in   Appendix   A,   where   she   wrote   ‘15’   using  the  correct  pencil  grip  and  number-­‐writing  formation.    When  asked  to  represent  ‘15’  using  real  objects,  she  hesitated,  asking  “15  of  anything?”    This  demonstrated  an  understanding  of  ‘number’  as  an  idea  that  describes  things  in  a  group,  independent  of  what  is  being  counted  or  labelled  (Demant,  2008).  She   proceeded   to   collect   and   count   15   leaves   from   the   surrounding   gardens,   initially  counting   in   1’s,   then   collecting   groups   of   two   and   skip   counting   (“7,9,11”),   before  collecting  a  group  of  four  and  counting-­‐on  to  arrive  at  15.  She   then   verified   there   were   15   leaves   using   rational   counting   (Cathcart,   2011):  making   a   one-­‐to-­‐one   correspondence   between   each   leaf   and   the   sequential   number  name  as  she  placed  it  on  the  Think  Board.    This  revealed  a  solid  understanding  of  the  principles   of   counting   identified   by   Gelman   and   Gallistel   (1978),   namely   the   Stable  Order   Principle,   One-­‐to-­‐one  correspondence,  Cardinal  principal,  abstraction  principle  and  the  order-­‐irrelevance  principle  (Compton,  2007).  Sharon  McCleary   3    
  4. 4. Teaching  Mathematics  in  Early  Years     EDUC8502    Gabi  initially  had  difficulty  representing  ’15’  pictorially,  and  was  unable  to  respond  to  prompts   requesting   her   to   think   about   instances   of   this   number   in   everyday   life.    However,   after   further   explanation   (i.e.     ‘3’   could   be   represented   by   three   little   pigs   or  a  triangle),  she  produced  an  example  relating  to  the  ‘real’  section  on  her  Think  Board,  drawing  three  flowers  with  five  petals  each  (See  Picture  Section  of  Think  Board  One).    She   counted   each   petal   individually,   then   stated   “5+5+5   equals   15”.     This   shows   she  successfully   decomposes   and   recomposes   numbers,   and   has   an   internal   concept   of  multiplication   as   repeated   addition   of   equivalent   groups,   consistent   with   the   second  level   of   conceptual   development   for   multiplication   representations   given   by   Thomas  (Thomas,  1997).  The   Story   section   of   Think   Board   One   indicates   she   has   sound   knowledge   of   the  standard   classroom   number   practise   of   creating   and   representing   word   problems  using   conventional   symbols   (i.e.   14+1=15).     It   also   shows   she   was   building   meaningful  connections   for   the   context   of   this   particular   Think   Board   as   she   engaged   with   the  activity,   as   her   symbolic   (story)   and   enactive   (real)   representations   related   to   the  same  theme  (i.e.  garden).  During   this   part   of   the   interview,   Gabi   demonstrated   good   early   number   sense,   a   solid  understanding  of  counting  and  the  beginnings  of  calculation.  Think  Board  Two  ‘80’:  Think  Board  Two  (Appendix  B)  shows  ‘80’  represented  in  a  non-­‐standard  form  in  the  Picture  section:  seven  longs,  nine  units  and  one  separated  unit;    the  place  value  chart  was   suggested.     Gabi   was   unable   to   create   an   equivalent   representation   of   ‘80’   when  requested.    She  did  not  recognise  ‘9+1’  could  be  traded  for  a  ‘10’  and  represented  by  an  additional  long,  displaying  confusion  even  when  this  was  explicitly  demonstrated  and  stated.     This   indicates   she   has   not   fully   abstracted   the   concept   of   a   unit   of   ten   (Gray,  1999);  she  is  operating  within  the  extended  stage  of  structural  development  described  by   Thomas   (2002),   using   the   sub-­‐system   of   units   to   form   her   understanding   of   the  base-­‐10  system.  This   developing   understanding   of   the   base-­‐10   system   is   also   apparent   in   the   Real  section  of  Think  Board  Two  where  she  has  used  seven  bananas  to  represent  seven  tens  and  ten  random  fruit  counters  for  the  remaining  ten.    Sharon  McCleary   4    
  5. 5. Teaching  Mathematics  in  Early  Years     EDUC8502    Both   sections   indicate   she   can   partition   the   decade   and   represent   the   number  accurately,   but   reveal   a   limited   understanding   of   grouping   and   place   value   concepts.    They  also  indicate  strain  on  her  working  memory  since  she  finds  it  difficult  to  consider  the   discrete   parts   and   the   whole   number   simultaneously   in   part-­‐part-­‐whole  relationships   (Gray,   2000).     This   may   result   from   repeated   classroom   experiences   of  partitioning   ten,   and   shows   she   has   not   conceptualised   groups   of   ten   as   a   unit,   or  visualised  the  pattern  of  tens  making  up  100.  Gabi   did   not   provide   authentic   real-­‐world   connections   in   the   Story   section   of   Think  Board  Two,  indicating  her  limited  awareness  of  real-­‐world  contexts  for  this  number.  Think  Board  Three  ‘20’:  Gabi  initially  chose  ‘2’  for  this  Think  Board,  stating  “I’ll  pick  an  easier  number,  ‘2’.    It’s  my   Birthday!”.     This   indicates   her   awareness   that   the   previous   representations   had  been   difficult,   and   shows   she   is   capable   of   building   authentic   real-­‐world   connections  for   numbers   with   which   she   is   familiar   and   comfortable.     She   proceeded   to   use   the  birthday  connection  with  the  number  ‘20’.  Examination  of  the  Picture  section  of  Think  Board  Three  shows  that  she  drew  twenty  cupcakes   to   represent   the   number,   linking   this   drawing   to   her   Story   section   by  showing  the  17  cupcakes  separated  from  the  “3  new  cupcakes”.      The   Real   section   of   this   Think   Board   reinforces   this   link   by   representing   the   ‘17’   using  bananas   and   differentiating   the   ‘3’   using   bunches   of   grapes.     This   shows   her  understanding  of  part-­‐part-­‐whole  relationships,  however,  as  can  be  seen  in  the  Story  section   of   the   Think   Board,   she   represents   her   number   sentence   incorrectly   as  “19+1=20”,  again  indicating  some  confusion  with  part-­‐part-­‐whole  relationships.    When  asked   to   write   the   number   sentence   corresponding   to   her   story,   she   produced   the  “17+3=20”,  as  shown  on  Think  Board  Three.  Again,  she  did  not  use  a  place  value  chart.    This  demonstrates  it  is  not  a  natural  part  of  her   expressive   repertoire;   she   thinks   of   multidigit   numbers   in   terms   of   units   and   is  operating  within  the  first  layer  of  the  number  system  (Geist,  2009).      (842  words)    Sharon  McCleary   5    
  6. 6. Teaching  Mathematics  in  Early  Years     EDUC8502    Recommendations:  The  main  areas  Gabi  requires  support  in  are:  • Developing   the   underlying   conceptualisations   involved   in   grouping   in   tens   and   place  value  operations.  • Consolidating  her  number  sense    for  multidigit  numbers,  initially  up  to  100.  These   areas   have   been   identified   using   observations   from   the   interview   and  prioritised  using  the  WA  Curriculum  Framework  (WA  Curriculum  Council,  2005),  First  Steps   Documents   (Willis,   2004)   and   The   Australian   Curriculum,   Mathematics  (Australian  Curriculum,  Assessment  and  Reporting  Authority  [ACARA],  2010).  They   represent   the   foundation   for   developing   understanding   of   our   numeration  system   and   higher-­‐level   concepts   of   number,   including   estimation   and   computation  (Cathcart,  2011).  WA  Curriculum  Framework:  (Curriculum  Council,  2005)  Gabi   has   predominantly   achieved   Level   2   of   the   WA   Mathematics   Curriculum  Framework  (Curriculum  Council,  2005):  “Understand   Numbers   (N6.a.2):   Reads,   writes,   says   and   counts   with   whole   numbers  beyond  100,  using  them  to  compare  collection  sizes  and  describe  order.”  Understand   Operations   (N7.2):   Understands   the   meaning   and   connections   between  counting,   number   partitions,   addition   and   subtraction;   uses   this   understanding   to  represent  situations  involving  all  four  basic  operations.  Calculate  (N8.2):  Counts,  partitions  and  regroups  in  order  to  add  and  subtract  one-­‐and  two-­‐digit  numbers,  drawing  mostly  on  mental  strategies  for  one-­‐digit  numbers  and  a  calculator   if   numbers   are   beyond   the   student’s   present   scope.”   (WA   Curriculum  Framework  Progress  Maps  Mathematics  Outcomes  Overview:  Number,  2009)  In   her   Think   Board   representations   (See   Think   Boards   Two   and   Three),   Gabi  partitioned   the   last   decade,   demonstrating   she   thinks   of   numbers   as   part-­‐part-­‐whole  relations.    There  was  little  evidence  of  her  understanding  the  regrouping:  she  seemed  to  create  the  seven  tens  from  procedural  knowledge  as  she  was  unable  to  explain  the  Sharon  McCleary   6    
  7. 7. Teaching  Mathematics  in  Early  Years     EDUC8502    base-­‐10   grouping   concepts   behind   the   procedure,   demonstrating   a   lack   of   relational  understanding  (Cathcart,  2011).  She  also  revealed  limited  number  sense  for  larger  numbers,  ‘20’  and    ‘80’,  relying  on  counting   in   units   (rather   than   grouping   tens)   to   represent   these   numbers,   indicating  she  has  not  fully  internalised  the  concept  of  grouping  in  tens  to  facilitate  more  efficient  counting.     Therefore   activities   emphasising   counting,   grouping,   place   value   and  number  patterns  up  to  100  should  be  introduced.  The  Australian  Curriculum:  (ACARA,  2010)  The   Australian   Curriculum   Year   Two   elaboration   requires   students   to   “Recognise,  model,   represent   and   order   numbers   to   at   least   1000”,   and   “Group,   partition   and  rearrange  collections  up  to  1000  in  hundreds,  tens  and  ones  to  facilitate  more  efficient  counting.”  (Australian  Curriculum,  Assessment  and  Reporting  Authority  [ACARA],  The  Australian  Curriculum,  Mathematics,  2010).    It   would   be   difficult   for   Gabi   to   build   number   sense   for   numbers   up   to   1000   as  required   by   The   Australian   Curriculum   since   she   has   not   yet   consolidated   grouping,  place  value  and  number  patterns  for  numbers  under  100.  Grouping   by   tens   is   fundamental   to   the   place   value   system,   and   a   thorough  understanding  of  place  value  is  necessary  for  the  development  of  higher-­‐order  number  sense  and  operations  (Reys,  1989).    Therefore,  Gabi  would  benefit  from  consolidation  of  the  Year  1  outcome  “Count  collections  to  100  by  partitioning  numbers  using  place  value.”   (ACARA,   The   Australian   Curriculum   Mathematics,   2010).     This   is   the   earliest  curriculum   outcome   which   she   is   not   confidently   able   to   demonstrate,   and   it   has  therefore   been   prioritised   in   order   to   minimise   misconceptions   and   build   a   solid  foundation  for  future  work.  First  Steps  in  Mathematics  Documents:  (Willis,  2004)  During   the   interview,   Gabi   displayed   several   characteristics   typical   of   the   First   Steps  Quantifying   Phase   (Willis,   2004),   automatically   selecting   counting   as   a   strategy,   skip  counting   leaves   when   constructing   Think   Board   One   and   realising   it   would   give   the  same   result   as   counting   by   ones.     She   was   able   to   write   number   sentences   matching  Sharon  McCleary   7    
  8. 8. Teaching  Mathematics  in  Early  Years     EDUC8502    the   semantic   structure   for   each   of   the   Think   Boards,   producing   small   number   addition  problems.  Each   Think   Board   demonstrated   her   tendency   to   think   about   number   in   terms   of   part-­‐part-­‐whole   relations   (e.g.   19+1=20   Think   Board   Three),   typical   of   a   child   in   the  Quantifying  Phase  (Willis,  2004).      A  key  element  of  this  phase  is  conservation  of  number,  which  Gabi  demonstrated  when  re-­‐arranging   counters   without   having   to   re-­‐count   them.     This   indicates   she   is  developmentally   able   to   deal   with   abstract   symbolic   activities   and   can   mentally  manipulate  numbers  represented  by    symbols  with  a  real  understanding  of  what  she  is  doing   (Charlesworth,   2007).     She   demonstrated   this   confidently   for   smaller   numbers  (i.e.   ‘15’   Think   Board   One),   but   did   not   display   an   understanding   of   the   place   value  symbols   used   to   represent   larger   numbers   and   would   benefit   from   more   concrete  experiences  constructing  systems  of  10’s.    This  would  consolidate  her  understanding  of   the   Base-­‐10   patterns   and   place   value   representations   up   to   100,   and   eventually  translate  to  larger  numbers.  Recommended  Activities:  The  following  two  activities  have  been  designed  to  give  exposure  to  these  outcomes:  Activity  1:  Build  a  100’s  Chart  using  Tens-­Frames.  First  Steps  in  Mathematics  -­  Number:  (Willis,  2010):    Understand  Whole  and  Decimal  Numbers  Key  Understandings  (Willis,  2010,  pg  52):  “KU5     There  are  patterns  in  the  way  we  write  whole  numbers  that  help  us  remember  their  order.”  Reason  About  Number  Patterns  (Willis,  2010,  pg  242)  “KU  5  Our  numeration  system  has  a  lot  of  specially  built-­‐in  patterns  that  make  working  with  numbers  easier.”  Materials:   Lead  Pencil,  coloured  pencils,  paper,  die,  two  different  coloured  counters,  10  ‘tens  frames’,  a  100’s  chart  cut  into  strips  of  10  (i.e.  1-­‐20,  11-­‐20,  21-­‐30,  etc).  Sharon  McCleary   8    
  9. 9. Teaching  Mathematics  in  Early  Years     EDUC8502    1.   Roll   the   die   and   use   coloured   counters   to   fill   in   the   tens   frame.     Alternate   the   colour  of  counters  used  for  each  roll  of  the  die.      2.   When   the   first   tens   frame   is   completely   full,   write   the   corresponding   number   sentence  using  the  coloured  counters  to  assist.  4.   Trade  the  completed  tens  frame  for  the  first  row  of  the  100’s  chart,  and  colour   the  numbers  corresponding  to  the  coloured  counters.  5.   Continue  this  process  until  all  10  tens  frames  have  been  completely  filled,  and   the  entire  100’s  chart  has  been  generated.  This   activity   capitalises   on   Gabi’s   ability   to   partition   ten   (shown   on   Think   Boards   Two  and   Three),   integrating   visualisation   to   assist   recognition   of   the   part-­‐part-­‐whole  relationships   within   the   tens   frame   (McIntosh,   1997),   but   extending   her   thinking   to  the   next   level   of   counting,   where   ten   units   are   grouped   together   and   ‘ten’   becomes   the  iterable  unit  (Jones,  1994).  Studies   show   that   imagery   is   used   extensively   in   the   construction   of   mathematical  meaning,   with   Presmeg   (1986)   identifying   five   main   types   of   visual   imagery:   concrete,  pattern   (relationships),   memory,   kinaesthetic   (involving   muscular   activity)   and  dynamic  (Thomas,  2002).  This  activity  utilises  concrete,  pattern,  kinaesthetic,  and  memory  imagery  to  reinforce  connections  between  verbal,  imagistic  and  formal  notation  systems  of  representation  (Goldin,   1987).     It   strengthens   the   connections   between   the   concrete   counters   and   the  conventional  symbolic  representation  by  writing  the  corresponding  number  sentence,  providing  explicit  links  which  encourage  mathematical  learning.        It  also  introduces  the  idea  of  trading  ten  units  for  a  single  entity  of  ten,    allowing  the  student   to   use   their   previous   constructs   of   the   system   of   1’s   to   develop   an  understanding   of   the   Base-­‐10   system   and   the   patterns   within   the   100’s   chart.     This  concept   of   grouping   is   a   crucial   part   of   the   numeration   system   and   understanding  base-­‐10  and  place  value.  Sharon  McCleary   9    
  10. 10. Teaching  Mathematics  in  Early  Years     EDUC8502    The  activity  gradually  builds  multi-­‐digit  number  sense  by  reinforcing  the  structure  and  order   of   the   100’s   chart.     This   is   an   essential   pre-­‐requisite   for   understanding   larger  numbers  up  to  1000.    Activity   Two:   Jelly-­Bean   Party   Bag   Game   (Estimation,   Counting   and   Grouping  using  Place  Value  Mats)  First  Steps  in  Mathematics  –  Number:  (Willis,  2010):    Understand   Whole   and   Decimal   Numbers   Key   Understandings   (Willis,   2010,  pg12&60):  “KU1  We  can  count  a  collection  to  find  out  how  many  are  in  it.  KU6  Place  value  helps  us  to  think  of  the  same  whole  number  in  different  ways  and  this  can  be  useful.”  Materials:  Large  bag  of  painted  beans  (e.g.97),  party  bags,  place  value  charts,  die.  1.   Ask  student  to  estimate  how  many  jelly  beans  are  in  the  large  bag.  2.   Roll   the   die,   explaining   the   place   value   chart   by   representing   single-­‐digit   numbers  as  individual  beans  in  the  1’s  column.  2.   Explain  once  there  are  10  jelly-­‐beans  they  can  be  put  into  a  party  bag  in  the  10’s   column:  this  will  help  us  count  faster.  3.   When   the   bag   is   empty,   ask   student   to   count   using   the   party   bags,   and   write   the   number  in  the  place  value  chart,  comparing  it  with  their  estimate.  4.   Ask   how   many   jelly-­‐beans   in   each   column   to   encourage   partitioning   of   this   number  and  demonstrate  the  difference  between  face  value  and  complete  value.   (i.e.90+7=97).  This  activity  balances  challenge  and  success,  providing  a  meaningful,  real-­‐life  context  for  counting.      Sharon  McCleary   10    
  11. 11. Teaching  Mathematics  in  Early  Years     EDUC8502    It   encourages   estimation,   which   is   an   effective   way   of   developing   number   sense   (Reys,  1989),  and  allows  direct  links  from  the  enactive  processes  of  counting  and  grouping  to  the   written/symbolic   place   value   representations   of   number.     Failure   to   understand  place  value  systems  often  stems  from  an  inability  to  differentiate  between  face  value  and   complete   value,   since   the   same   number   can   represent   several   values   (Varelas,  1997).     Focusing   on   the   semiotics   aspects   of   the   written   place   value   system   during   the  activity   helps   students   differentiate   between   face   value   and   complete   value,   and  assists  conceptual  understanding  of  place  value,  which  is  crucial  for  developing  higher-­‐level  number  concepts.  Rubin   and   Russell   (1992)   state   counting,   grouping,   estimating   and   notating   are  essential  in  developing  representations  of  the  number  system  (Thomas,  1994).  The   activity   utilises   a   game   format   for   problem   solving   to   promote   automaticity   and  consolidate   key   concepts   by   encourage   justification   of   mathematical   ideas   in   a   social  context,   thereby   improving   mathematical   fluency   (Geist,   2009).     Fluency,   Reasoning  and   Problem   Solving   are   Proficiency   Strands   within   the   Content   Structure   of   The  Australian  Curriculum  (ACARA,  2010).                                  (1378  Words)  Critical  Reflection:      The   interview   process   made   me   realise   children’s   mathematical   thinking   is   highly  personal   and   very   different   to   adult   thinking   (McIntosh,   1997;   Sfard,   2005);   their  number  concepts  are  limited  by  their  developmental  stage  and  real-­‐world  experiences,  their   own   ability   to   make   mathematical   sense   out   of   these   experiences,   and   to  communicate  these  understandings  effectively.  Gabi’s   interview   responses   indicated   her   conscious   exposure   to   numbers   was   mostly  confined   to   the   classroom,   since   she   had   difficulty   providing   authentic   examples   for  the  Think  Board  representations.    I  was  surprised  how  difficult  it  was  for  Gabi  to  make  real-­‐world  connections  and  realised  the  links  between  classroom  mathematics  and  the  real-­‐world  need  to  be  regularly  and  explicitly  made  to  become  meaningful  for  children.  The  Think  Boards  challenged  her  to  consider  number  concepts  using  different  modes  of   representation,   and   enabled   her   to   explore   connections   between   these  representations.     I   observed   Gabi   actively   trying   to   make   meaning   in   true  Sharon  McCleary   11    
  12. 12. Teaching  Mathematics  in  Early  Years     EDUC8502    constructivist   fashion   as   she   engaged   in   the   activities   -­‐   similar   to   Wilkerson-­‐Jerde   &  Wilensky’s   (2011)   description   of   mathematical   learning   as   the   process   of   building   a  network   of   mathematical   resources   by   establishing   relationships   between   different  components  and  properties  of  mathematical  ideas.  I   realised   children   are   only   able   to   reveal   their   knowledge   if   they   are   given   the  opportunity   to   do   so,   and   classroom   activities   need   to   be   open-­‐ended   to   allow   them   to  demonstrate   and   explore   their   own   mathematical   thinking   without   placing   limitations  on   it.     The   teacher’s   role   is   to   provide   opportunities   for   deep   understanding   of  concepts  and  make  clear  links  between  the  concept  and  the  conventional  mathematical  symbols,   allowing   semiotic   meaning   making   without   stifling   their   inherent  mathematical   thought   processes.     In   order   to   achieve   a   deep   understanding   of  mathematical   concepts   and   achieve   autonomous   learning,   children   must   be   allowed   to  reinvent  mathematical  concepts  in  their  own  minds  (Kamii,  1984).  During   the   interview,   communication   was   pivotal   in   encouraging   further   learning.    Talking   about   the   Real   and   Story   sections   of   the   Think   Boards   made   Gabi’s   thinking  visible   (Whitin,   2000)   and   generated   further   learning   opportunities;   it   motivated  learning   in   the   child,   and   teaching   in   the   adult   (Sfard,   2005).     This   was   a   clear  demonstration   of   Gabi   constructing   knowledge   in   a   social   context   (Vygotsky,   1978),  where   communication   clarified   and   consolidated   her   thinking.     It   also   demonstrated  how   meaning   arises   from   the   tension   (Radford,   2011)   between   the   child’s   inner  understanding   of   mathematical   ideas   and   their   functioning   in   a   shared   sociocultural  world  of  semiotic  systems  (Fried,  2011).  Given   that   very   young   children   are   still   developing   knowledge   of   mathematical  language   and   conventions,   and   the   limitations   of   interpreting   their   external  representations,   it   is   important   to   observe   them   on   multiple   occasions,   through  various   representational   modes,   using   active   listening,   and   interpreting   gesture,  pictures  and  symbols  to  determine  their  mathematical  understandings  and  assist  them  towards  achieving  autonomous  learning.                                              (446  words)  Sharon  McCleary   12    
  13. 13. Teaching  Mathematics  in  Early  Years     EDUC8502    References:      Cathcart,  W.,  Pothier,  Y.,  Vance,  J.  &  Bezuk,  N.,  (2011),  Learning     Mathematics  in  Elementary  and  Middle  Schools:  A  Learner-­‐Centred     Approach,  Pearson  Education,  Boston,  MA.  Charlesworth,  R.  &  Lind,  K.,  (207),    Math  &  Science  For  Young  Children,     Fifth  Edition,  Thomson  Delmar  Learning,  New  York.  Compton,  A.,  Fielding,  H.  &  Scott,  M.,  (2007),  Supporting  Numeracy:  A  Guide     for  School  Support  Staff,  Paul  Chapman  Publishing,  London.  Conklin,  M.,  (2010),  It  Makes  Sense!  Using  Ten-­‐Frames  to  Build  Number     Sense,    Math  Solutions,  CA.  Curriculum  Organiser  Outcomes  Overview  -­‐  Mathematics,  (2009),    Curriculum  Council       Of  Western  Australia,  WA.    Retrieved  from  http://www.curriculum.wa.edu.au  Demant,  D.,  (2008),  A  Story  of  Natural  Numbers,  Black  Dog  Books,  Fitzroy,     Victoria.  Frid,  S.,  (2004),  Cross  Section,  Volume  15  No.  3,  pgs  1-­‐7.  Fried,  M.,  (2011),  Signs  for  you  and  signs  for  me:  the  double  aspect  of     semiotic  perspectives,  Educational  Studies  in  Mathematics,     doi:10.1007/s10649-­‐011-­‐9319-­‐0.  Geist,  E.,  (2009),  Children  Are  Born  Mathematicians:  Supporting     Mathematical  Development,  Birth  to  Age  8,  Pearson  Education,  New     Jersey.  Gray,  E.,  Pitta,  D.  &  Tall,  D.,  (1999),  Objects,  Actions  and  Images:  A  Perspective  on  Early     Number  Development,  Mathematics  Education  Research  Centre,  Coventry  UK.  Jones,  G.,  Thornton,  C.,  &  Putt,  I.,  (1994),  A  Model  for  Nurturing  and     Assessing  Multidigit  Number  Sense  Among  First  Grade  Children,     Educational  Studies  in  Mathematics,  27,  117-­‐143.  Kamii,  C.,  (1984),  Autonomy  as  the  aim  of  childhood  education:  A  Piagetian     Approach,  Galesburg,  IL.    Sharon  McCleary   13    
  14. 14. Teaching  Mathematics  in  Early  Years     EDUC8502    Krause,  K.,  Bochner,  S.,  Duchesne,  S.  &  McMaugh,  A.,  (2010),  Educational     Psychology  for  Learning  and  Teaching,  3rd  edition,  Cengage  Learning     Australia,  pp262-­‐287.  Maslow,  A.  (1968),  Toward  a  Psychology  of  Being,  (2nd  edition),  Van     Nostrand  Reinhold,  New  York,  pp  3-­‐67.  McIntosh,  A.,  Reys,  B.  &  Reys,  R.,  (1997),  Number  Sense:  Simple  Effective     Number  Sense  Experiences,  Dale  Seymour  Publications,  New  Jersey.  McIntosh,  A.,  De  Nardi,  E.  &  Swan,  P.,  (2007),  Think  Mathematically!  How  to     Teach  Mental  Maths  in  the  Primary  Classroom,  Pearson  Education     Australia,  Australia.  Nataraj,  M.  &  Thomas,  M.,  (2009),  Developing  Understanding  of  Number     System  Structure  from  the  History  of  Mathematics,  Mathematics     Education  Research  Journal,  Vol.  21,  No.  2,  96-­‐115.  Radford,  L.,  Schubring,  G.  &  Seeger,  F.,  (2011),  Signifying  and  meaning-­‐   making  in  mathematical  thinking,  teaching  and  learning,  doi:     10/1007/s10649-­‐011-­‐9322-­‐5.  Radunsky,  V.,  (2002),  10  (ten),  Viking,  New  York.  Reys,  R.,  Suydam,  M.  &  Lindquist,  M.,  (1989),  Helping  Children  Learn     Mathematics,  Second  Edition,  Prentice  Hall,  New  Jersey.  Sfard,  A.  &  Lavie,  I.,  (2005),  Why  Cannot  Children  See  as  the  Same  What     Grown-­‐Ups  Cannot  See  as  Different?-­‐Early  Numerical  Thinking     Revisited,  Cognition  and  Instruction,  23(2),  237-­‐309.  The  Australian  Curriculum-­‐Mathematics,  Version  1.1,  (2010),  Australian  Curriculum,       Assessment  and  Reporting  Authority  [ACARA],  Retrieved  from:       http://www.australiancurriculum.edu.au  Thomas,  N.  &  Mulligan,  J.,  (1994),  Researching  Mathematical     Understanding  Through  Children’s  Representations  of  Number.       Retrieved  from  http://www.aare.edu.au/94pap/thomn94173.txt  Sharon  McCleary   14    
  15. 15. Teaching  Mathematics  in  Early  Years     EDUC8502    Thomas,  N.,  Mulligan,  J.  &  Goldin,  G.,  (2002),  Children’s  representation  and     structural  development  of  the  counting  sequence  1-­‐100,    Journal  of     Mathematical  Behaviour,  21,  117-­‐133.  Varelas,  M.  &  Becker,  J.,  (1997),  Children’s  Developing  Understanding  of     Place  Value:  Semiotic  Aspects,  Cognition  and  Instruction,  15(2),  265-­‐   286.  Vygotsky,  L.,  (1978),  Mind  in  Society,  Harvard  University  Press,  Cambridge,     MA.  Whitin,  P.  &  Whitin,  D.,  (2000),  Math  is  Language  Too:  Talking  and  Writing     in  the  Mathematics  Classroom,  National  Council  of  Teachers  of     English,  USA.  Wilkerson-­‐Jerde,  M.  &  Wilensky,  U.,  (2011),  How  do  mathematicians  learn     math?:  resources  and  acts  for  constructing  and  understanding     mathematics,  doi:  10.1007/s10649-­‐011-­‐9306-­‐5.  Willis,  S.,  Devlin,  W.,  Jacob,  L.,  Powell,  B.,  Tomazos,  D.  &  Treacy,  K.,  (2004),     First  Steps  in  Mathematics:  Number  (Book  1  &  2),  Rigby,  Australia.      Sharon  McCleary   15