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                                                                                                                                                         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	
  
	
   	
  
	
   	
  
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.	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  
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	
  
	
  

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Think Board Interview, Recommendations and Reflection

  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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