Improving Children's Knowledge And Understanding Of Fractions, Proportions And Ratios

 

How  can  we  improve  children’s  knowledge  and   understanding  of  fractions,  proportions  and  ratios?  

  Why  research  this?   At   the   end   of   2008   all   children   at   Andersons   Bay   school   were   tested   on   the   Numeracy   Diagnostic  Test    (NumPA).       The   results   of   the   proportion   and   ratio   section   of   the   test   appeared   to   be   lower   than   the   National  Norms.     Therefore,   our   middle   team   classes   (Yr3-­‐4)   decided   to   research   the   reasons   for   these   low   scores,  and  how  we  could  improve  children's  knowledge  and  understanding  of  proportions  and   ratios.   The  NZ  National  expected  levels  of  achievement  for  numeracy  strategies  and  knowledge  are   based   on   a   Longitudinal   Study   by   Tagg,   A.   and   Thomas,   G.   (2007)     In    Fractional  knowledge,  proportions  and  ratios,  the  expected  level  for  children  at  the  end  of   Year  3  is  Stage  4-­‐5.    The  expected  level  for  children  at  the  end  of  Year  4  is  stage  5.     At  Anderson’s  Bay  School  in  2008:    At  the  end  of    Year  3:   59%  were  stage  3   41%  were  at  stage  4   At  the  end  of    Year  4:   39%  were  at  stage  4  or  below   61%  were  at  stage  5  or  above       According  to  these  results  our  school  had  59%  of  year  3  children  rated  as  ‘cause  for  concern’.     And  39%  of  Year  4  rated  as  either  ‘cause  for  concern’  or  ‘at  risk’.                      

Research     Each   teacher   carried   out   some   research   readings   into   the   teaching   of   fractions,   proportions,   ratios,  and  children’s  understanding  of  these  concepts.  

Pitkethly,  A.  &  Hunting,  R.        A  review  of  recent  research  in  the  area  of  initial   fraction  concepts.   In   this   review   it   is   suggested   that   the   basis   of   rational   number   building   is   initial   fraction   concepts,  language  and  symbols  of  fractions,  fraction  knowledge,  and  part-­‐whole  partitioning.     Children’s  informal  knowledge  is  important  in  the  development  of  fraction  concepts.   Lamon,   Susan.   J   (1949)   Teaching   fractions   and   ratios   for   understanding.   Mahwah   N.J   :   L   Erlbaum  Associates   This  reading  stated:   ‘Fraction,  ratio  and  rational  number  ideas  are  mathematically  complex  and  interconnected.  A   long   term   learning   process   is   required…current   instruction   that   gives   a   brief   introduction   through  part-­‐whole  fractions…does  not  give  the  children  the  time  they  need  to  construct  ideas   and  understandings’.     It  also  stated:     ‘Early   fraction   activities   often   involve   questions   about   pizza.   Students   answer   “how   many   slices”  of  the  pizza  were  eaten  rather  than  “how  much  pizza  was  eaten”.  The  number  of  slices  is   merely  a  counting  problem.  The  question  “how  much  of  the  pizza”  implies  “what  part  of  the   original  amount  of  pizza”  which  requires  relative  and  proportional  thinking.”     It  then  explains  the  main  reasons  why  fractions  are  hard  for  children,  with  examples:     As  a  New  Entrant,  children  learn  to  one-­‐one  count.  They  give  one  number  to  each  object.  With   fractions  the  unit  may  consist  of  more  than  one  object  but  it  can  still  be  called  one.  The  new   unit  can  also  be  divided  into  parts,  and  a  new  kind  of  number  refers  to  the  parts  of  that  whole.      The  unit  changes  in  each  new  situation      What  looks  the  same  can  be  represented  by  different  numbers      Equal  sharing  is  a  new  concept     Children  only  have  a  limited  knowledge  of  multiplication/division  before  starting  fractions     The  numbers  no  longer  have  simple  labels,  can’t  easily  be  physically  represented,  or  are  hard  to   add  or  subtract     Teachers  confuse  terminology  of  fractions  

  These  readings  were  discussed  at  our  team  meetings  and  we  came  up  with  a  plan  of  how  to  try   to  improve  children’s  understandings  and  how  to  incorporate  the  teaching  of  fractions  into  our   classroom  programme.     We  decided  to:   • Look  at  what  our  children  already  know,  in  the  form  of  a  pre-­‐test.   • Look   at   resources   available   to   help   teach   Fractions,   proportions   and   ratios   and   purchase  some  new  ones.   • Research  teaching  ideas,  online  tools,  and  children’s  online  games.   • Set   aside   one   math   session   a   week   for   teaching   fractions,   and   use   the   laptop   pod   during  this  time  to  give  children  access  to  the  online  games,  for  all  of  term  2.   • Following  this,  we  planned  to  test  all  children  again  at  the  end  of  the  term.     We  developed  our  own  pretest…   And  pre-­‐  tested  the  children…   The  results  of  the  pre-­‐test  were:   Year  3  average  score  was  23/70   This  equates  to  an  average  score  of  33%       Year  4  average  score  was  48/70   This  equates  to  an  average  score  of    69%    

Number  of  Children  

Frac7on  and  Ra7o  Test   Year  3  2009   15   10   5  

Pre  Test  

0   0  -­‐  10  

11  to  20   21  to  30   31  to  40   41  to  50   51  to  60     61  to  70   Score  Results  

 

Number  of  Children  

Frac7on  and  Ra7o  Test   Year  4  2009   14   12   10   8   6   4   2   0  

Pre  Test  

0  to  10   11  to  20   21  to  30   31  to  40   41  to  50   51  to  60   61  to  70   Score  Results  

      We   launched   into   teaching   fractions,   proportions   and   ratios   once   a   week.   For  8  weeks  the  children  played  the  fractions  games  and  explored  the  websites.   All  children  particularly  enjoyed  the  board  game  ‘Pizza  Fun’  and  the  various  websites.       To   compare   results   we   gave   the   children   the   pre-­‐test   again   as   a   post-­‐test.   Our  post-­‐test  results  showed  that  in  Year  3:   The  average  score  was  43/70  which  equated  to  a  61%  average  score     And  in  Year  4:   The  average  score  was  57/70  which  equated  to  an  81%  average  score.       Year  4’s  Pre  test:     The  average  score  was  48/70  which  equated  to  a  69%  average  score.    The  post  test  showed  an  overall   12%  increase  in  scores.  The  largest  individual  improvement  in  scores  was  an  increase  of  38%.   The  post  test  showed  an  overall  12%  increase  in  scores.  The  largest  individual  improvement  in  scores   was  an  increase  of  38%.    

Frac7on  and  Ra7o  Test   Year  3  2009   16  

Number  of  Children  

14   12   10   8   Pre  Test  

6  

Post  Test  

4   2   0  

0  -­‐  10   11  to  20  21  to  30  31  to  40  41  to  50  51  to  60    61  to  70   Score  Results    

Frac7on  and  Ra7o  Test   Year  4  2009   20  

Number  of  Children  

18   16   14   12   10  

Pre  Test  

8  

Post  Test  

6   4   2   0   0  to  10   11  to  20   21  to  30   31  to  40   41  to  50   51  to  60   61  to  70  

Score  Results    

 

      Children’s   basic   knowledge   of   fractions   appeared   to   improve   and   the   children   also   became   more  confident  in  their  understanding  of  fractions.     In   order   to   judge   whether   we   had   improved   children’s   knowledge   and   understanding   of   fractions,   proportions   and   ratios   we   then   needed   to   test   the   children   on   the   fractional   knowledge,  proportions  and  ratios  section  of  the  NumPA  test.       We   tested   all   children   the   knowledge   and   strategy   questions   on   Form   B   of   the   NumPA   test,   and   some   children   on   Form   C.      

Fractional  

Knowledge  

Results  

  The  expected  stage  for  Fractional  Knowledge  at  the  end  of  Year  3,  is  stage  4-­‐5.   In  Year  3  this  year  93.24%  scored  at  or  above  the  expected  level  of  stage  4-­‐5;  and  6.66%  below   the  expected  level.   The  expected  stage  for  Fractional  Knowledge  at  the  end  or  Year  4,  is  stage  5   In  Year  4  this  year  97.6%  scored  at  or  above  the  expected  level  of  Stage  5  and  2.44%  below  the   expected  level.    

Proportion  and  Ratios  Results    

The  expected  strategy  stage  for  Proportions  and  Ratios  at  the  end  of  Year  3  is  Stage  4-­‐5.   In  Year  3  this  year  86.58%  scored  at  or  above  the  expected  level.   The  expected  strategy  stage  for  Proportions  and  Ratios  at  the  end  of  Year  4  is  Stage  5.   In  Year  4  this  year  48.8%  scored  at  or  above  the  expected  level.                          

Summary  of  Results     The  2009  group  of  Year  3  children  showed  greatly  improved  percentages  of  children  working   at  or  above  the  expected  stages  in  both  knowledge  and  strategy  questions,  compared  to  the   2008  group  of  Year  3  children.   However,   the   2009   group   of   Year   4   children   had   mixed   results   compared   to   the   2008   group   of   Year  4  children.  The  2009  group  showed  greatly  improved  percentages  of  children  working  at   or   above   in   the   fractional   knowledge   questions,   but   there   was   a   decrease   of   percentages   of   children  working  at  or  above  in  the  strategy  questions  compared  to  the  2008  group  of  Year  4   children.  

Discussion    

To  really  track  whether  there  has  been  an  improvement  in  scores,  we  probably  needed  to  track   the  same  group  of  children  from  2008  to  2009.   If  we  compared  the  scores  of  2008  Year  3’s  to  the  2009  Year  4’s  (the  same  group  of  children),   then   there   was   an   increase   of   average   percentage   scores   in   both   knowledge   and   strategy   questions.    Limitations  of  Study     As  discussed  our  results  did  not  actually  compare  the  same  group  of  children  i.e.  The  2009  Year   4’s  were  Year  3’s  in  2008  and  the  2009  Year  3’s  were  Year  2’s  in  2008.   The   NUMPA   tests   for   this   research   were   administered   in   August,   rather   than   at   the   end   of   the   year  as  is  intended.   Proportions   and   Ratios   are   not   introduced   until   stage   4.   Children   need   to   have   sound   multiplication   and   division   knowledge   before   they   can   grasp   fractional   knowledge   and   strategies.  In  fact,  Form  A  of  the  NumPa  test  does  not  include  any  fractions,  proportions  and   ratios  questions.   The  numeracy  test  itself  is  very  limiting,  as  it  only  has  a  few  questions,  and  there  appears  to  be   a  huge  leap  between  the  strategy  questions  on  form  B.   There  may  be  marked  variations  in  how  different  teachers  assign  a  stage  to  each  answer  in  the   NUMPA  test.    

Further  Observations    

We  believe  that  it  will  be  interesting  to  see  if  most  of  the  children  retain  this  knowledge  and   understanding  in  the  longer  term.     In   reality,   when   teaching   a   balanced   math   curriculum   that   covers   all   strands,   it   is   difficult   to   implement  long  term  instruction  in  this  one  area.   We  found  that  although  the  study  was  an  interesting  exercise,  we  would  do  it  differently  next   time,  by  focusing  on  and  tracking  a  target  group  of  children  over  two  years.        

Useful  resources  used  by  classroom  teachers:   Web  

Games    

Books  

www.woodlands-­‐junior.kent.sch.uk/maths/    

Pizza  Fraction  fun       Fraction  Lotto      

Dragon  Math  1  and  2    

 

www.gameaquarium.com    

  www.coolmath.jc-­‐schools.net    

  www.coolkidsmaths-­‐games.com    

 

www.mathplayground.com       www.coolmath4kids.com       www.mathsfractions.com    

Fraction  flashcards      

Fraction  dominoes       Fraction  circles      

 

Teacher   tools-­‐   Fractions,   decimals   and  percentages  Bk1    

 

No   Nonsense   Number-­‐   Fractions   Stages  5-­‐6    

     

 

 

 

 

www.helpwithfractions.com       www.funschool.kaboose.com/funblaster      

 

 

 

 

Digistore  –  fraction  fiddle   http://www.tki.org.nz/r/digistore/protected/objects/?id=2801&vers=1.0