Top Banner
GOOD TO GREAT Number Sense
38

Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Jul 20, 2020

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

GOOD  TO  GREAT  

Number  Sense  

Page 2: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Mas-cate  and  Ruminate  

1.  When  two  even  numbers  are  mul-plied,  the  product  will  always  be:  

A.  Even  B.  Odd  C.  Unknown  

2.  When  two  odd  numbers  are  mul-plied,  the  product  will  always  be:  

A.  Even  B.  Odd  C.  Unknown  

3.  When  an  even  number  and  an  odd  number  are  mul-plied,  the  product  will  always  be:  

A.  Even  B.  Odd  C.  Unknown  

Page 3: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Why  Number  Sense?  

•  Number  sense  in  kindergarten  and  1st  grade  predict  math  achievement  through  3rd  and  5th  grade  

•  Kindergarten-­‐entry  number  knowledge  is  a  beMer  predictor  of  overall  academic  achievement  in  3rd  and  5th  grade  than  early  reading  skills  or  aMen-on  

•  An  accurate  mental  number  line  in  3rd  grade  predicts  math  achievement  on  statewide  tests  

•  5th  grade  frac-ons  and  division  mastery  predicts  math  achievement  in  high  school  

Sources:  Jordan  et  al.  2009;  Geary,  2011;  Duncan  et  al.,  2007;  Claessens  et  al.,  2009;  Booth  and  Siegler,  2006;  Siegler  et  al.,  2012  

Page 4: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

What  if  our  students’  performance  in  number  sense  equaled  geometry?  

Page 5: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

What  Is  Number  Sense?  

•  Coun-ng  – Cardinality  – Es-ma-on  

•  Number  Knowledge  – Ordinality  – Magnitude  

•  Number  Opera-ons  – Number  combina-ons  – Math  facts  

Page 6: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

How  many  kiMens?  

Page 7: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Mental  Number  Line  

Children  “o]en  fail  to  appreciate  an  integral  aspect  of  the  number  system—that  the  whole  numbers  are  evenly  spaced  along  the  number  line.  Instead,  it  seems  that  in  young  children’s  mental  representa-on  of  magnitude,  small  values  are  far  more  different  from  each  other  than  larger  values;  for  example,  1  and  2  are  more  different  than  8  and  9.”  

Source:  Ashcra]  and  Moore,  2012    

Page 8: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

0   10  5  1   2   3   4   6   7   8   9  

0   10  

Number  Line  Es-ma-on  0-­‐10  

Page 9: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

0   100  50  10   20   30   40   60   70   80   90  

0   100  

Number  Line  Es-ma-on  0-­‐100  

Page 10: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

0   1,000  

Number  Line  Es-ma-on  0-­‐1,000  

0   1,000  500  100   200   300   400   600   700   800   900  

Page 11: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Number  Line  Accuracy  

•  About  one-­‐half  of  kindergarten  students  have  an  accurate  0-­‐10  mental  number  line  

•  About  one-­‐half  of  first  grade  students  have  an  accurate  0-­‐100  mental  number  line  

•  About  one-­‐half  of  fourth  grade  students  have  an  accurate  0-­‐1,000  mental  number  line  

Sources:    Siegler  &  Booth,  2004;  Siegler  &  Opfer,  2003;  Bertelef  et  al.,  2010  

Page 12: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

0   10  

8 + 2 = 10

0   10  

0   10  

8  

2  

Page 13: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

0   100  

10 x 3 = 30

0   100  

0   100  

10  

10  

0   10   20   30  

Page 14: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Frac-ons  

“Learning  about  frac-ons  requires  children  to  recognize  that  many  proper-es  of  whole  numbers  are  not  true  of  numbers  in  general  and  also  to  recognize  that  the  one  property  that  unites  all  real  numbers  is  that  they  possess  magnitudes  that  can  be  ordered  on  number  lines.”  

Source:  Siegler  et  al.  2013  

Page 15: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

0   1  

0   1  ½  ⅓   ⅔  ¼   ¾  ⅕   ⅘  ⅗  ⅖  ⅙   ⅚  ⅛   ⅜   ⅝   ⅞  

What  Frac-on?  

Page 16: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Understanding  Number  

“There  are  paMerns  within  number  combina-ons  and  rela-onships  between  them.  For  example,  paMerns  in  combina-ons  of  6  include  3  +  3  =  6;  therefore  4  (which  is  1  more  than  3)  +  2  (which  is  1  less  than  three)  =  6,  5  +  1  =  6,  so  1  +  5  =  6,  and  so  forth.  Children  with  an  intui-ve  grasp  of  number  paMerns  can  readily  derive  answers  from  known  combina-ons  to  solve  unknown  ones.  This  ability,  in  turn,  helps  them  master  or  become  fluent  with  number  combina-ons.”  

Source:  Jordan  et  al.,  2006  

Page 17: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Number  PaMerns  1  

Page 18: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Number  PaMerns  2  

Page 19: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Number  Combina-ons  

Page 20: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

How  many  kiMens  again?  

Page 21: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Addi-on  Strategies  

Single  Digit  •  Coun-ng  all  (with  or  without  fingers  or  manipula-ves,  

verbal  or  nonverbal)    •  Coun-ng  on  (from  the  larger  number  if  commuta-vity  is  

understood)  •  Retrieval  from  memory  •  Derived  facts  (memory  and  number  paMerns)  

Mul-ple  Digit  •  Decomposi-on  (making  10s,  friendly  equa-ons)    •  Standard  algorithm  

Source:  Geary,  1994  

Page 22: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Addi-on  Accuracy  

99%  of  1st  graders  used  more  than  one  strategy  and  62%  used  at  least  3  strategies  •  Coun-ng  all  (50%  error  rate)  •  Coun-ng  on  (17%  error  rate)  •  Decomposi-on  (8%  error  rate)  •  Retrieval  from  memory  (4%  error  rate)  

Source:  Ashcra],  1992  

Page 23: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Friendly  Addi-on  Equa-ons  

0   20  10  2   4   6   8   12   14   16   18  

20  +  3  =  23  17  +  6  =    23  

Page 24: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Standard  Par-al  Sums  

437  +86  523  

1437  +86  400  110  13  

523  

Page 25: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Addi-on  Regrouping  Errors  

Source:  Geary,  1994  

Page 26: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Finger  Coun-ng  

•  Strong  posi-ve  correla-on  with  solving  addi-on  problems  in  kindergarten  

•  The  correla-on  decreases  over  K-­‐2  •  Small  but  significant  nega%ve  correla-on  with  solving  addi-on  problems  at  the  end  of  second  grade  

•  Children  from  low-­‐income  families  use  their  fingers  to  count  less  than  others  in  kindergarten  but  more  in  second  grade  

Source:  Jordan  et  al.,  2008  

Page 27: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Subtrac-on  Strategies  

•  Coun-ng  up  (from  subtrahend  to  minuend)  •  Coun-ng  down  (usually  reserved  for  problems  where  the  minuend  is  much  larger  than  the  subtrahend,  e.g.  27-­‐3)    

•  Addi-on  reference  (fact  families)  •  Decomposi-on  (10s,  friendly  equa-ons)    •  Retrieval  from  memory  

Source:  Geary,  1994  

Page 28: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Friendly  Subtrac-on  Equa-ons  

0   20  10  2   4   6   8   12   14   16   18  

20  –  11  =  9  17  –    8  =     9  

Page 29: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Subtrac-on  Regrouping  Errors  

Source:  Geary,  1994  

Page 30: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Mul-plica-on  Strategies    

Single  Digit  •  Count  equal  groups  (arrays,  repeated  addi-on)  •  Rules  (zero  property,  iden-ty  property)  •  Derive  from  known  facts  (especially  from  doubles)  •  Retrieve  from  memory  

Mul-ple  Digit  •  Decomposi-on  (distribu-ve  property)  •  Place  value  (par-al  products)  •  Standard  algorithm  (long  mul-plica-on)  

Source:  Geary,  1994;  Lampert,  1986  

Page 31: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Mul-plica-on  Accuracy  

92%  of  3rd  graders  used  more  than  one  strategy  on  basic  mul-plica-on  facts  (0-­‐9)  and  65%  used  at  least  3  strategies  •  Wri-ng  the  problem  (39%  correct)  •  Sets  of  tally  marks  (55%  correct)  •  Repeated  addi-on  (59%  correct)  •  Retrieval  from  memory  (78%  correct)  

Source:  Siegler,  1988  

Page 32: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Retrieval  Errors    

•  Near  misses:  10%  larger  or  smaller  •  Opera-on  confusions:  add  instead  of  mul-ply  

•  Table  errors:  correct  answer  to  another  mul-plica-on  problem  

•  Table-­‐related  errors:  another  mul-ple  of  one  of  the  operands  

Source:  Geary,  1994;  Lampert,  1986  

Page 33: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Friendly  Mul-plica-on  Equa-ons  

(20  +  4)  ×  6  =  

24  ×    6  =     144  

(20  ×  6  )  +  (4  ×  6  )  =  

120  +  24  =  144  

Page 34: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Standard  Par-al  Products  

47  ×  32  94  

1,410  1,504  

147  ×  32  1,200  210  80  14  

1,504  

2

Page 35: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Mul-plica-on  Regrouping  Errors  

Source:  Lampert,  1986  

Page 36: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Division  Strategies  

•  Trial  and  error  (mul-ply  the  divisor  by  a  succession  of  numbers)  

•  Grouping  (for  12÷4,  make  4  groups  and  add  one  to  each  group  un-l  there  are  12)  

•  Repeated  addi-on  of  the  divisor  •  Mul-plica-on  reference  (fact  families)  •  Retrieval  from  memory  (but  not  much)  

Source:  Geary,  1994;  Robinson  et  al.,  2006  

Page 37: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Problem  Size  Effect  

•  Math  facts  with  larger  addends  or  mul-pliers  are  more  difficult  to  retrieve  from  memory  

•  Math  facts  with  larger  addends  or  mul-pliers  appear  less  frequently  in  math  textbooks  

•  Problem  difficulty  or  fewer  associa-ons  with  larger  math  facts  stored  in  memory?  

Sources:  Hamann  &  Ashcra],  1986;  Ashcra]  &  Christy,  1991;  Dehaene  &  Mehler,  1992  

Page 38: Number Sense’ - nesacenter.org...2015/10/07  · Mas-cate’and’Ruminate’ 1. When’two’ even’numbers’ are’mul-plied,’ the’product will’always’be:’ A. Even’

Number  Sense  is  Conceptual    

•  Early  understanding  of  number  rela-ons  and  opera-ons  provides  support  for  learning  complex  calcula-on  procedures  

•  Children  who  understand  concepts  tend  to  solve  problems  more  effec-vely  across  mathema-cal  domains  and  age  ranges  

•  Conceptual  knowledge  of  frac-ons  predicts  gains  in  procedural  knowledge  of  frac-ons  

•  Conceptual  knowledge  of  inversion  supports  but  does  not  guarantee  development  of  strong  calcula-on  skills  

Sources:  Jordan  et  al.,  2009;  RiMle-­‐Johnson  &  Siegler,  1998;  &  Siegler  et  al.,  2013;  Gilmore  &  Bryant,  2006