Algebra oflittlekids final
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Algebra oflittlekids final Algebra oflittlekids final Document Transcript

  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids   The  algebra  of  little  kids:       A  mathematical-­‐habits-­‐of-­‐mind  perspective  on  elementary  school1   E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   Education  Development  Center,  Inc.  (EDC)2   Asking  “When  should  algebra  be  taught?”  is  like  asking  “Is  technology  harmful  or  helpful?”  There  are  lots  of  technologies  and  lots  of  uses  of  them.  Some  are  harmful;  some  are  helpful.  Refining  the  question—asking  about  a  particular  use  of  a  particular  technology  for  a  particular  purpose  in  particular  contexts  and  at  particular  stages  in  one’s  learning—makes  the  question  researchable  and  potentially  answerable.  Similarly,  there  are  many  “algebras”—algebra  the  course,  algebra  the  discipline,  algebraic  ideas,  algebraic  language,  early  algebra,  “patterns,  functions,  and  algebra”—and  many  different  takes  on  the  learning  and  teaching  of  each  of  these.  Treating  algebra  as  an  indivisible  whole  obscures  the  options.  It’s  more  useful  to  ask  what  ideas,  logic,  techniques,  and  habits  of  mind  algebra  entails,  and  then,  about  each  of  these,  ask  when  and  to  what  extent  that  one  item  can  be  learned  with  intellectual  integrity  and  how  a  coherent  whole  can  be  woven  out  of  these  learnings.  The  answers  we  get  are  that  some  of  these  ideas  do  have  to  wait  for  eighth  or  ninth  grade,  but  that  others—even  including  aspects  of  algebraic  language—are  already  there,  early  in  the  primary  grades.  Moreover,  children  who  get  to  apply,  refine,  and  strengthen  those  ideas  and  skills  as  they  emerge  gain  the  advantage.   Any  credible  claim  about  habits  of  mind  must  surely  accord  with  features  of  mind:  children’s  cognitive  development.  For  a  charmingly  written  scientific  account  of  the  ways  that  babies  and  young  children  think,  read  The  Scientist  in  the  Crib,  by  Gopnik,  Meltzoff,  and  Kuhl  (2000).  The  habits  of  mind  approach  to  curriculum  that  we  first  described  well  over  a  decade  ago  (Cuoco,  Goldenberg  &  Mark,  1996;  Goldenberg,  1996)  and  have  continued  to  refine  (Goldenberg  &  Shteingold,  2003  and  2007;  Cuoco,  Goldenberg  &  Mark,  2009;  Mark,  et  al.,  2009)  does  accord  well  with  children’s  thinking  and  became  a  central  design  principle  behind  Think  Math!  (2008),  the  newest  NSF-­‐supported  elementary  curriculum,  developed  at  EDC.   Recognizing,  enhancing,  and  building  on  developmentally  natural  habits  of  mind  lets  us  dissect  algebra  and  sort  the  resulting  bits  and  pieces  in  a  developmentally  natural  way,  while  preserving  the  content,  concepts,  and  skills  that  schools  (and  states,  parents,  workplaces,  and  colleges)  expect.  The  fact  that  it  is  possible  to  organize  algebraic  ideas,  logic,  and  techniques  around  the  development  of  mind  makes  clear  that  we  are  truly  talking  about  thinking—habits  of  mind—rather  than  “features  of  mathematics”  or  “idiosyncrasies  of  mathematicians.”  This  article  describes  two  of  these  natural  habits  of  mind.  Two  algebraic  ideas  that  precede  arithmetic  The  common  wisdom  is  arithmetic  first,  algebra  later.  The  truth  is  not  so  simple.  Some  algebraic  ideas—ideas  about  the  properties  of  binary  operations  apart  from  the  numbers  these  operations  may  “combine”—develop  naturally  before  children  learn  arithmetic.                                                                                                                    1  An  adaptation  of  this  paper  has  been  submitted  for  publication  in  Teaching  Children  Mathematics,  NCTM.  2  This  work  was  supported  in  part  by  the  National  Science  Foundation,  grant  numbers  ESI-­‐0099093,  DRL-­‐0733015,  and  DRL-­‐0917958.  The  opinions  expressed  are  those  of  the  authors  and  not  necessarily  those  of  the  Foundation.  ©  Education  Development  Center,  Inc.     page  1  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids   In  fact,  they  must  develop  before  arithmetic  can  make  sense!  For  example,  for  many  4-­‐year-­‐olds,  even  those  who  appear  to  count  well,  seven  objects  spread  out  like  this    feel  like  “more”  than  the  same  objects  bunched  together   .  (Though  “conservation”  remains  the  familiar  name  for  this  stage  in  children’s  logic—so  we’ll  still  use  it—child  logic  is  more  nuanced  than  was  previously  thought.  It’s  known,  for  example,  that  for  small  enough  numbers  of  objects,  babies  at  eleven  months  have  not  only  stability  of  number  but  essentially  addition  as  well.  See,  e.g.,  Feigenson,  Carey,  and  Spelke,  2002.  So-­‐called  non-­‐conservers  aren’t  “enslaved  by  their  senses”  but  haven’t  yet  privileged  the  analytic  act  of  counting  over  other  ways  of  making  social  and  mathematical  sense  of  the  world.)  For  children  whose  logic  still  works  that  way,  the  claim  that    +    is  the  same  amount  as    can  hardly  make  sense.  Faced  with  the  requirement  to  assert  that  5  +  2  =  7,  “non-­‐conservers”  have  only  two  options.  Some  divorce  the  assertion  from  their  current  “common  sense”—after  all,  they  “know”  that  the  two  quantities  are  not  the  same—and  learn  “5  +  2  =  7”  as  an  arbitrary  but  learnable  fact,  the  same  way  they  learn  the  names  of  their  classmates.  For  them,  math  is  memory.  Others  find  it  hard  to  accept  what  their  logic  tells  them  is  “not  true”  and,  instead,  just  feel  like  they  “don’t  get  it.”    An  important  property  of  addition  before  addition,  itself  What  will  later  be  formalized  as  the  commutative  and  associative  laws  of  addition  begins  as  an  intuitive  sense  of  stability/invariance  of  quantity  under  rearrangement.  Piaget  (1952)  called  it  conservation  of  number;  Wirtz,  et  al.  (1964)  and  Sawyer  (2003)  called  it  the  “any  order  any  grouping  property.”  Prior  to  conservation,  while  arrangement  trumps  number,   may  not  have  a  fixed  number  associated  with  it.  Later,  the  new  conserver  may  not  yet    know  how  many  fingers    are  without  counting,  but  will  be  sure  that  the  number,  whatever  it  is,  stays  put  if  the  hands  are  moved  like  this   or  even  like  this,   .     That  algebraic  idea,  a  property  of  aggregation,  must  exist  before  the  arithmetic  fact—knowing  what  number  2  +  5  is—can  make  sense.  In  a  similar  way,  if  a  bunch  of  coins  are  hidden  and  we  ask  “how  much  money  is  there?”  children  for  whom  the  question  makes  any  sense  will  be  absolutely  certain  that  there  is  an  answer,  and  that  only  one  answer  is  correct.  They  may  be  uncertain  about  methods  of  counting,  and  may  think  that  some  methods  might  give  incorrect  answers.  The  complexities  of  communication  may  even  make  it  seem  that  they  believe  that  the  amount,  itself,  could  vary  depending  on  what  method  they  use  as  they  count  but,  in  all  likelihood,  other  means  of  questioning  would  suggest  that  they’re  sure  that  the  amount  is  stable.  In  fact,  if  they  do  believe  the  amount  can  vary,  they’re  not  cognitively  ready  for  the  question  of  what  “the  amount”  is.  There  is  no  “the  amount”  if  it  can  vary.  Some  six  year  olds,  but  not  many,  do  not  yet  conserve  number;  by  seven,  nearly  all  do.   Having  confidence  that    and    represent  the  same  quantity  is  not  the  same  as  knowing  the  commutative  property  of  addition.  The  commutative  property  is  not  about  the  arrangement  of  physical  objects  in  space,  but  about  the  behavior  of  a  particular  element  (here,  the  +  sign)  in  a  formal  syntactic  system  of  written  symbols.  In  some  contexts,  children  can  make  perfect  sense  out  of  written  symbols—even  significant  parts  of  algebraic  notation—but  most  young  children  cannot  make  sense  of  formal  operations  on  a  string  of  ©  Education  Development  Center,  Inc.     page  2  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids  symbols.  So,  at  this  stage,  commutativity  remains  largely  an  intuitively  obvious  idea  about  the  “physics  of  mathematics”:  the  nature  of  aggregation,  not  the  nature  of  symbols.  Even  so,  we,  as  educators,  can  support  the  young  child’s  logic  better  if  we  recognize  that  it  is  already  relying  on  the  underlying  ideas  that  formal  mathematics  will  later  codify.  The  fact  that  children  see  that  the  principle  applies  regardless  of  the  numbers  means  that  it  captures  the  essential  algebraic  aspect  of  the  structure  of  addition  that  commutativity  is  about.    Logical  precursors  of  the  distributive  property  of  multiplication  over  addition:    Pick  a  number.  Multiply  it  by  5;  also  multiply  it  (your  original  number)  by  2;  now  add  those  results.  You  get  the  same  answer  you’d  get  if  you  multiplied  your  original  number  by  7.  The  distributive  property,  a  general  statement  of  that  fact,  is  possibly  the  most  central  idea  in  elementary  arithmetic,  key  to  understanding  the  algorithms,  at  the  core  of  fluent  mental  calculations  (e.g.,  102  ×  27  can  be  computed  in  two  parts,  as  100  ×  27  +  2  ×  27),  and  the  logical  basis  for  many  “rules”  of  algebra  that  might  otherwise  seem  arbitrary.   This  property  relates  multiplication  and  addition,  but  children  “know  it”  long  before  they  even  meet  multiplication!  It’s  in  the  language  (and  logic)  they  use  when  they  say  that  5  (fingers,  pennies,  or  27s)  plus  2  (fingers,  pennies,  27s)  make  7  (fingers,  pennies,  27s).  These  dialogues  with  6-­‐year-­‐olds,  late  in  their  kindergarten  year,  give  a  sense  of  what  their  logic  does  and  does  not  handle.  What  distinguishes  the  questions  the  children  get  “right”  from  those  they  get  “wrong”?  What  logic  might  explain  the  particular  wrong  answers  they  get?   T   What’s  a  really  big  number?   Ne  (girl):  A  million!   T:   Suppose  I  said  “How  much  is  a  thousand  plus  a  thousand?”  What  would  you  say?   Ne:   I  have  no  idea!  (big  smile)   T:   And  suppose  I  said  “How  much  is  two  thousand  plus  three  thousand?”   Ne:   (thinks,  then  confidently)  Five  thousand!     T:  Suppose  I  said  “How  much  is  a  hundred  plus  a  hundred?”  What  would  you  say?   Gi  (girl):   A  hundred.   T:   What  about  “Two  hundred  plus  three  hundred”?     Gi:   Five  hundred.   T:   (playfully)  And  what  if  I  said  “how  much  is  a  thousand  plus  a  thousand?”  …   Gi:   A  million!     T:  Suppose  I  said  “How  much  is  a  hundred  plus  a  hundred?”  What  would  you  say?   De  (boy):  De  may  hear  “a  hundred”  as  one  word,  so  confidently  says:  Two  ahundred.     T:   And  suppose  I  said  “How  much  is  two  hundred  plus  three  hundred?”   De:   Five  hundred.     T:  Suppose  I  said  “How  much  is  a  thousand  plus  a  thousand?”  What  would  you  say?   Co  (boy):  A  thousand  two.  (Co  might  have  meant  “A  thousand,  too.”  We  don’t  know.)   T:   And  suppose  I  said  “How  much  is  two  thousand  plus  three  thousand?”   Co:   Two  three  a  thousand.  (Co  clearly  isn’t  yet  adding  naturally.)   As  soon  as  children  are  comfortable  with  the  idea  (and  language  and  knowledge)  to  answer  “what’s  three  sheep  plus  two  sheep?”  perhaps  late  in  K  or  early  in  first  grade,  they’ll  happily  apply  that  to  give  the  “correct”  answer  to  the  spoken  question  “what’s  three  eighths  plus  two  eighths?”  or  “what’s  three  hundred  plus  two  hundred?”  The  answer  is  “correct,”  but  what  they  have  in  mind  may  well  be  quite  different  from  what  we  have  in  mind  when  ©  Education  Development  Center,  Inc.     page  3  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids  we  give  the  same  answer.  We  can  see  how  different  their  ideas  are  when  we  ask  a  slightly  different  question:  “what’s  a  hundred  plus  a  hundred”  (with  no  audible  “small”  numbers  like  “two”  or  “three”).  To  this  question,  young  six-­‐year-­‐olds  may  well  repeat  “a  hundred”  or  say  something  like  “a  million.”  If,  instead,  we  ask  “what’s  an  eighth  plus  an  eighth,”  little  ones  may  just  give  a  puzzled  stare  and  not  answer  at  all;  or,  if  their  arithmetic  is  strong  enough,  they  might  possibly  count  and  answer  “sixteen”  (or,  sometimes  “nine”).     How  can  we  explain  such  different  responses  to  questions  that  adults  see  as  so  similar?  Again,  the  answer  rests  more  in  language  and  general  cognition  than  mathematics.  Kindergarteners  typically  have  hundred  and  half  as  vocabulary  items.  For  most  little  ones,  these  terms  don’t  represent  precise  or  fixed  amounts,  just  as  “a  zillion”  is  not  a  specific  fixed  amount  to  us,  but  the  children  do  know  that  “half”  means  only  part.  Most  even  know  that  halves  should  be  equal—  no  fair  if  yours  is  bigger!—though  they  might  not  know  that  they  must  be  equal  or  that  there  are  only  two  of  them.  And  they  almost  certainly  don’t  know  that  half  is  a  number.  Likewise,  they  know  that  “a  hundred”  is  big,  though  they  are  unlikely  to  know  how  big.  The  question  “what’s  a  hundred  plus  a  hundred”  is,  therefore,  more  or  less,  “what  is  a  big  amount  plus  another  big  amount?”  The  natural  response  is  “a  big  amount”  (“a  hundred”)  or  a  very  big  amount  (“a  million”),  not  “two  big  amounts”  (“two  hundred”).  But  when  fixed  specific  quantities  are  available,  children  use  them.  The  question  “what’s  two  hundred  plus  three  hundred”  is  linguistically  and  cognitively  like  “what’s  two  sheep  plus  three  sheep”—it  draws  attention  to  2  +  3,  not  to  the  nature  of  a  sheep  or  a  hundred.  Children  for  whom  2  +  3  makes  sense  answer  correctly.  Of  course,  children  for  whom  2  +  3  does  not  yet  make  sense  try  to  find  some  other  way  of  making  sense  of  the  task,  but  their  answers  don’t  reflect  addition.  (The  different  response  to  “what’s  an  eighth  plus  an  eighth”—the  puzzled  look—is  because  an  eighth  not  even  part  of  the  child’s  vocabulary,  and  thus,  with  no  meaning,  gives  the  child  less  of  a  context  for  responding.  Anna  Sfard,  2008,  suggests  that  a  child  might  well  treat  “hundred”  as  a  number,  rather  than  a  sheep,  and  still  treat  “three  hundred”  not  as  a  number,  but  as  an  expression  composed  of  two  number  words.  If  so,  our  kindergarteners  seem  to  treat  these  numbers  differently,  one  as  a  counter,  the  other  as  a  unit  or  object,  which  might  be  consistent  with  Sfard.)   Why  these  errors  are  made,  and  why  “hundred”  and  “eighth”  lead  to  different  errors,  is  a  diversion.  The  point  is  that  when  no  audible  small  numbers  like  “two”  or  “three”  are  given,  little  children  tend  to  give  wrong  answers.  But  when  we  say  how  many  eighths  or  hundreds,  and  the  numbers  are  not  too  large,  even  kindergarteners  tend  to  answer  correctly,  more  first  graders  do,  and  we  can  absolutely  count  on  it  in  second  grade.  Whatever  an  eighth  or  a  hundred  is,  the  children  are  sure  that  three  of  them  plus  two  of  them  is  five  of  them!  This  does  not  constitute  “knowing  the  distributive  property,”  but  it  does  tell  us  that  the  children  already  have  the  underlying  idea  that  the  distributive  property  will  later  encode  formally.   If  we  use  sevens  (a  fully  understood  fixed  quantity)  in  place  of  hundred  (which  may  still  be  a  nonspecific  “zillion”  for  young  children),  children  still  know  that  three  of  them  plus  two  of  them  makes  five  of  them,  but  that’s  of  little  use  if  “three  sevens”  does  not  (yet)  have  meaning.  Once  a  child  does  have  meaning  for  “three  sevens”  and  that  meaning  is  a  specific  number  (even  if  the  child  doesn’t  yet  remember  which  number),  the  child’s  long-­‐standing  logic/intuition/linguistic  knowledge  that  “three  sevens  plus  two  sevens  is  five  sevens”  becomes  arithmetically  usable.    ©  Education  Development  Center,  Inc.     page  4  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids   The  meaning  of  “three  sevens”  might  be  given  in  several  ways:  as  an  image   ,  or  a  sum,  7  +  7  +  7,  or  a  product  3  ×  7,  or  in  other  ways.  Each  way  has  something  threeish  and  something  sevenish  about  it.  Because  7  +  7  +  7  and  3  ×  7  are  both  language,  such  expressions  are  best  introduced  as  (mathematical)  descriptions  of  a  situation—for  example,  the  array  image—that  communicates  partly  without  analyzing  the  language  formally.  The  image,  of  course,  requires  some  analysis,  too—visual  rather  than  linguistic—to  see  the  three  sevens.  To  connect  “three  sevens”  with  21,  the  “normal”  name  for  that  number,  we  must  agree  that  what  makes    “seven”  is  its  seven  squares.  Then    is  21  because  of  its  21  squares,  but  it  is  also  a  picture  of  three  sevens:  a  multiplication  fact.  Similarly,  if    is  “seven,”  then    is  two  sevens.  The  picture    shows  that  three  sevens  and  two  sevens  make  five  sevens.   In  spoken  form,  “three  sevens  plus  two  sevens  make  five  sevens”  is  familiar.  The  pictures  support  the  semantics  of  the  situation,  helping  to  establish  the  role  of  sevens  and  preserve  its  numerical  meaning  rather  than  letting  it  degenerate  into  a  non-­‐numeric  object,  like  sheep.  But  the  classical  written  form—(3  ×  7)  +  (2  ×  7)  =  5  ×  7—is  quite  another  story.  Spoken  symbols  vs.  written  symbols  Knowing  that  the  finger  collections    and    can  be  described  by  the  same  number  does  not  guarantee  that  a  child  will  know  that  the  print  statements  5  +  2  and  2  +  5  refer  to  the  same  number.  The  written  language  of  mathematics  presents  challenges  that  can  be  finessed  by  spoken  language  and  by  appropriate  visual  presentations.  Perhaps  the  most  glaring  example  is  the  canonical  wrong  fourth-­‐grade  response  to   8 + 8 = ? .  No  first  grader   3 2would  ever  say  “five  sixteenths.”  It’s  uninformative—in  fact,  misleading—to  “explain”  such  errors  simply  by  claiming  that  these  expressions  are  “too  abstract”  or  that  children  “can’t  handle  symbols.”  Spoken  words  are  symbols,  too,  and  words  like  the—which  young  children  use  flawlessly—are  about  as  abstract  as  one  can  get.  It’s  worth  understanding  the  difference  between    =    and  5  +  2  =  2  +  5  to  see  why  the  challenge  of  print  for  children  may  not  be  a  mathematical  challenge.     Humans  have  evolved  to  be  quite  flexible  about  visual  order  and  orientation,  but  in  the  life  of  any  individual  human,  it  takes  some  learning.  Infants  who  have  come  to  recognize  a  bottle  when  it  is  handed  to  them  in  the  proper  orientation    do  not,  at  first,  reach  for  it  when  it  is  handed  to  them  in  some  unfamiliar  orientation  (e.g.,  with  the  nipple  visible,  but  facing  away  like  this   ).  But  very  soon  they  do  learn  to  recognize  objects  regardless  of  their  orientation.  When  you  consider  the  visual  processing  required,  this  is  quite  an  impressive  accomplishment.  Even  if  the  bottle  is    presented  in  the  same  orientation  but  at  different  distances,  very   Figure  1:    In  this  photo,  different  images  are  projected  onto  the  retina.  The  distortion  of   the  distance  from  the  tip   of  the  nipple  to  the  parts  relative  to  each  other  can  be  extreme,  and  yet  the  baby   bottle  is  the  same  as  the  recognizes  all  of  these  projections—most  of  them  never  seen   length  of  the  entire  rest  before—as  the  same  object.   of  the  bottle.  Measure  to   see  for  yourself!  ©  Education  Development  Center,  Inc.     page  5  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids  Though  this  complex  neural  computation  needs  data  (learning)  to  tune  it  up,  the  ability,  itself,  is  wired  in.  This  evolutionary  gift  is  essential  for  survival.  Otherwise,  we’d  have  been  meals  for  tigers  we  didn’t  recognize  because  they  didn’t  happen  to  be  facing  exactly  the  same  way  as  first  we  saw  them!  For  our  ancestors,  it  was  necessary  to  “see”  the  same  object  despite  different  retinal  images,  as  long  as  those  images  could  be  made  “the  same”  under  rotation,  reflection,  dilation,  or  certain  projective  transformations,  and  so  our  brains  are  adept  at  them.  (The  spatial  tests  that  some  people  find  quite  difficult  are  a  very  different  sort  of  thing.  The  “look-­‐alike”  objects  on  these  tests  require  an  analysis  that  goes  beyond  what  was  evolutionarily  useful.  Our  ancestors  didn’t  care  if  the  tiger  was  left-­‐handed!)   But  those  ancestors  didn’t  read.  The  letters  d,  b,  q,  and  p  are  the  same  shape  and  differ  only  by  rotation  or  reflection.  To  read,  children  must  learn  to  see  them  as  different  objects,  not  as  the  same  object  in  different  orientations.  Young  children’s  letter  reversals  are  not  neurological  failures  at  all—seeing  that  way  is  one  of  evolution’s  gifts—but,  just  for  this  one  purpose  of  decoding  print,  children  must  unlearn  a  principle  that  applies  to  nearly  everything  else  they  will  encounter  during  their  entire  life.  They  must  treat  print  as  an  exception  to  the  usual  rules  of  seeing.   Moreover,  w as  and  s aw—each  just  three  print-­‐squiggles  arranged  in  a  different  order—must  not  be  recognized  as  “the  same.”  Alas,  then  come  2 +5  and  5 +2,  two  perfectly  good  examples  of  print-­‐squiggles  that  are  to  be  treated  as  “the  same.”  (As  always,  the  truth  is  not  so  simple.  On  a  number  line,  numbers  represent  addresses—the  names  of  specific  points/locations  along  the  line—and  also  distances  between  addresses.  The  child  who  “enacts”  2 +5,  perhaps  by  jumping  along  a  large  number  line  on  the  floor  would  enact  5 +2  differently.)  It  is  therefore  not  surprising  that  the  notation,  in  some  contexts,  can  cause  confusions,  but  this  is  an  issue  of  notation,  not  of  concept.  Print  is  just  plain  different!   Similarly,  the  picture    lets  children  see  what  written  descriptions  like  (3  ×  7)  +  (2  ×  7)  =  (3  +  2)  ×  7  or  (3  ×  7)  +  (2  ×  7)  =  5  ×  7  typically  leave  opaque,  unless  they  are  written  as  an  abbreviated  version  of  language  the  children  themselves  are  using  to  describe  the  picture.  But  the  difficulty  is  with  the  notation—a  difficulty  with  the  manner  in  which  the  underlying  mathematical  idea  is  being  communicated—not  a  lack  of  the  idea  itself.  In  fact,  the  way  that  teachers  of  kindergarten  and  early  first  grade  teach  writing  could  help  them  teach  this  symbolic  language,  too:  children  tell  stories,  and  the  teacher  encodes  their  language  in  writing.  Here,  children  might  describe  how  a  three-­‐by-­‐seven  array  can  be  put  with  a  two-­‐by-­‐seven  array  to  make  a  five-­‐by-­‐seven  array,  and  the  teacher  can  be  writing  (3  ×  7)  +  (2  ×  7)  =  (5  ×  7)  as  the  children  speak.  Before  that  can  happen,  children  need  to  have  the  idea  that  we  can  name  the  arrays,  and  that  one  useful  name  for    is  (3  ×  7).  Imagine  that  array  to  be  on  a  card  we  hold  in  our  hands.  That  card  can  be  held  in  any  position  at  all—vertically,  slantwise,  horizontally—and  is  still  the  same  card.  It  makes  sense  to  give  it  the  same  name  no  matter  which  way  we  hold  it.  We  could  also  have  called  it  (7  ×  3),  or  even  21  (or  a  zillion  other  things,  like  “half  of  6  ×  7”  if  we  had  a  6  ×  7  array  that  we  had  already  named).  So  (3  ×  7)  =  (7  ×  3)  =…   The  visual  idea    and  the  symbols  that  describe  what  the  children  see  are  not  yet  fully  generic—not  yet  a  property  of  +  and  ×  that  can  be  used  in  syntactic  manipulations  of  ©  Education  Development  Center,  Inc.     page  6  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids  strings  of  symbols  to  generate  (a  ×  c)  +  (b  ×  c)  from  (a  +  b)  ×  c  or  vice  versa.  In  fact,  there  are  so  many  parts  to  keep  track  of  that  doing  so  is  not  trivial.  Getting  good  enough  to  recognize  and  use  this  valuable  property,  even  with  arrays  as  a  particularly  powerful  representation,  takes  time  and  practice.  But  the  underlying  idea  is  there  very  early,  as  part  of  the  child’s  cognitive  structure,  as  soon  as  the  child  can  meaningfully  make  statements  like  “two  sheep  plus  three  sheep  are  five  sheep.”  Again,  the  underlying  idea  must  be  there  before  any  practice  of  it  can  make  sense.   Written  symbols  often  present  major  challenges  that  the  spoken  symbols  do  not.  Possibly  because  of  print’s  special  status,  the  logic  that  children  apply  when  information  is  presented  in  spoken  symbols  may  not  be  applied  when  the  same  information  is  presented  in  print.  The  canonical  error  with  fractions  is  a  perfect  example:  The  spoken  question  “what’s  three  eighths  plus  two  eighths”  focuses  attention  on  “three  plus  two”  and  tends  to  evoke  the  correct  reasoning  and  get  the  correct  answer;  by  contrast,  the  written  question   3 8 + 8 = ?  doesn’t  focus  attention  only  on  the  top  numbers.  Children  for  whom  the  meaning  is   2not  already  strongly  established  tend  to  interpret  the  plus  sign  as  “add  everything  in  sight.”  In  fact,  mathematical  reading  and  writing  are  quite  different  from  prose  reading  and  writing.  For  prose,  we  proceed  in  a  line,  strictly  left  to  right.  Even  top-­‐to-­‐bottom  movement  just  accommodates  the  limited  width  of  a  page;  it  gives  no  information  that  would  not  have  been  present  if  the  writing  were  strung  out  in  one  dimension—a  line—on  a  very  wide    scroll  of  paper.  (The  real  story  is,  of  course,  more  complex.  Strict  left  to  right  reading  applies  only  at  the  very  earliest  stages,  if  at  all.  A  fluent  reader,  largely  without  conscious  awareness,  takes  in  much  more  of  the  sentence  than  a  strictly  left-­‐to-­‐right  approach  would  give.)  By  contrast,  bar  graphs,  coordinate  graphs,  histograms,  charts  and  tables,  and  the  like  are  two-­‐dimensional  records.  One  must  attend  to  horizontal  and  vertical  position  in  order  to  interpret  the  information  they  contain.  Even  symbolic  expressions  can  require  attention  to  vertical  as  well  as  horizontal  position:  32  is  not  the  same  as  32.  Moreover,  mathematical  writing  that  is  just  horizontal  are  not  to  be  read  strictly  left  to  right:  2  ×  (3  +  5),    7  +  6  ÷  2,  and  7  +  ___  =  5  +  4  all  require  attention  to  the  right  side  before  attention  to  the  left.  In  fact,  7  +  6  ÷  2  requires  both  left-­‐to-­‐right  and  right-­‐to-­‐left    analysis:  6  ÷  2  must  be  evaluated  left-­‐to-­‐right  (because   Figure  2:    Bar  graphs,  among  the  2  ÷  6  is  different),  and  yet  the  convention  about  order  of   earliest  graphs  children  make,   require  attention  to  two  operations  dictates  that  the  6  ÷  2  part  must  be  evaluated   dimensions:  which  bar  (horizontal  before  the  addition  that  is  specified  by  “7  +  .”   position)  and  the  bar’s  height.  Algebra  as  a  language  for  expressing  what  we  know  Algebraic  notation  is  used  in  two  distinct  ways:  for  describing  what  we  know,  and  for  deriving  what  we  don’t  know.  In  the  first  use,  algebra  is  a  language  for  describing  the  structure  of  a  computation,  a  numerical  pattern  we’ve  observed,  a  relationship  among  varying  quantities,  and  so  on.  Young  children  are  phenomenal  language  learners!  ©  Education  Development  Center,  Inc.     page  7  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids   Exercises  like  the  one  in  Figure  3,  but  without  the  leftmost  column,  are  familiar  enough  in  many  curricula.  Children  look  for  a  pattern  in  the  inputs  and  outputs,  figure  out  a  rule,  and  complete  the  table.  Think  Math!  often  adds  a  “pattern  indicator”  (the  first  column)  to  problems  of  this  kind.  When  Michelle,  a  second  grader  in  a  Think  Math!  classroom  finished  filling  out  this  table  before  I  had  finished  handing  out  copies  to  all  the  children,  I  asked  her  how  she  had  done  it  so  fast.    She  said  “Well,  I  saw  it  was  take-­‐away  8  because  I  looked  at  the  28  and  20,  and  then  I  saw  that  10  and  2  was  take-­‐away  8  again,  and  then  I  saw  8  and  0.”       n   10   8   28   18   17       58   57   n  –  8   2   0   20       3   4      Figure  3:  A  “pattern  indicator”  gains  meaning  from  context  when  it  accompanies  a  “find-­‐a-­‐rule”  exercise.   And  then  she  grinned  as  if  I  had  left  the  “clue”  by  accident,  pointed  to  the  left  column  and  added  “Besides,  it  says  that  right  here!”  How  did  Michelle  know?  Though  the  algebraic  language  was  there,  nobody  ever  discussed  “variables”  or  “letters  standing  for  numbers”  or  even  mentioned  that  column.  Had  Michelle  seen  just  the  table  in  Figure  4,  with  no  examples  to  infer  from,  she  most  likely  would  not  have  felt  the  symbols  “said”  anything.  But  after  she  discovered  the  pattern,  the  symbols  looked  “close  enough”  to  mean  the  same  thing,  and  so  she  then  assigned  them  that  meaning.       n   18   17       58   57   n  –  8       3   4      Figure  4:  A  “pattern  indicator”  without  a  pattern  from  which  to  infer  its  meaning  would  just  be  more  to  learn.   In  other  words,  she  did  what  little  children  excel  at:  she  learned  language  (in  this  case  “n  –  8”)  from  context.  If  algebraic  language  is  part  of  the  environment,  used  where  context  gives  it  meaning,  children  can  apply  their  natural—and  extraordinary—language-­‐learning  prowess  to  it,  and  learn  to  use  it  descriptively.  Just  as  children  learning  their  native  language  understand,  at  first,  more  than  they  can  say,  Michelle  could  not  immediately  produce  such  descriptive  language,  but  she  and  others  try  these  interesting  ways  of  writing  down  what  they  know  and,  over  time,  become  good  at  it.     Fourth  graders  learn  a  number  trick:  Think  of  a  number;  add  3;  double  that;  subtract  4;  cut  that  in  half;  subtract  your  original  number;  aha,  your  result  is  1!  They  love  it  and  want  to  do  it  to  their  parents  and  friends.  They  also  want  to  know  how  it  works,  so  we  add  pictures.  When  we  say  Think  of  a  number,  we  picture  a  bag  with  that  number  of  grapes  in  it:   .  For  add  3,  we  picture    and  double  that  becomes   .  This  act  of  doubling,  which  most  fourth  graders  find  quite  natural  and  “obvious,”  is,  again,  the  distributive  property  in  action.  While  the  expression  2(b  +  3)  does  not  make  obvious  what  the  result  is,  children  do  readily  learn  to  describe  the  picture  as  “two  bags  plus  6”  and  abbreviate  that  description  as  2b  +  6.  We  don’t  talk  about  “variables”  or  “letters  standing  for  numbers”;  we  simply  describe  what  we  know,  and  write  it  down  as  simply  as  we  can.  (See  a  detailed  description  of  the  activity  with  children  at  http://thinkmath.edc.org/index.php/Algebraic_thinking  and  see  Sawyer  (1964)  for  the  original  source  of  this  idea.)  June  Mark,  et  al.,  (2009)  describe  yet  another  way  in  which  Think  Math!  gives  students  this  algebra-­‐as-­‐description-­‐of-­‐what-­‐you-­‐know  experience.  ©  Education  Development  Center,  Inc.     page  8  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids  So  why  don’t  we  teach  algebra-­the-­course  in  grade  4?  Because  that  other  use  of  algebra—deriving  what  we  don’t  know—is  a  formal  syntactic  operation  on  a  set  of  symbols,  and  children  are  (generally)  not  able  to  divorce  symbols  from  meanings  before  roughly  age  12.  This  is  not  because  they  cannot  handle  “symbolic”  or  “abstract”  things—words  are  symbols;  pictures  are  symbols;  little  children  can  be  symbolic  and  abstract  from  babyhood—but  because  the  use  of  the  symbols  is  different.  Formal  operations  on  strings  of  algebraic  symbols—rearranging  them,  apart  from  their  semantics,  to  create  other  strings  of  symbols  that  “solve”  a  problem—are,  well,  formal  operations,  and  children  are  not,  by  and  large,  formal  operational  before  11,  and  not  reliably  so  before  about  13,  whence  the  common  need  to  wait  until  that  age  for  “algebra.”  But  only  that  part  of  algebra  that  requires  deduction  by  formal  rules  needs  to  wait  that  long.  The  part  of  algebra  that  is  expressive  of  what  we  already  know—that  is,  essentially,  a  shorthand  for  semantic  content  clearly  tied  to  a  context  we  already  understand—that  part  can  be  learned  earlier.  It  is  just  language  to  express  oneself,  and  children  are  excellent  language  learners.  They  don’t  learn  language  from  explanations  or  formal  lessons;  they  learn  it  from  use  in  context.  And,  if  is  it  learned  all  along,  as  it  becomes  developmentally  possible,  then,  when  the  child  is  in  late  middle  school,  the  transition  to  the  new  use  of  that  language  for  deductive  purposes  could,  presumably,  be  much  easier,  much  more  accessible  for  all  children,  much  less  of  a  brick  wall  of  a  million  seemingly  new  things  to  learn  all  at  once.  What  does  this  tell  us  about  elementary  school  teaching  and  learning?   Taking  advantage  of  children’s  natural  algebraic  ideas  and  honing  them  is  a  focus  on  habits  of  mind,  rather  than  on  rules  that  can  otherwise  seem  arbitrary.  The  precursors  of  commutative  and  distributive  properties  that  we  described  earlier  do  need  to  be  refined,  honed,  extended,  practiced,  codified,  and  generalized,  but  they  are  already  there  as  “natural”  logic,  the  child’s  natural  habits  of  mind  and  the  building  blocks  of  higher  mathematics.  If  children  are  to  become  competent  at  mathematics,  including  arithmetic,  those  habits  of  mind  must  take  precedence  over  rules,  formulas,  and  procedures  that  do  not  derive  from  logic  that  the  child  can  grasp.  In  fact,  children  can  grasp  a  lot  more  if  the  foundations  for  their  learning  are  grounded  in  their  logic,  which  gives  them  all  the  tools  to  understand,  not  just  memorize,  the  algorithms  for  arithmetic  with  whole  numbers  and  fractions.  But  we  all  see  the  dramatically  disappointing  results  of  “learning”  rules  without  understanding:  they  are  easy  to  mix  up  and  result  in  procedures  that  don’t  work.   Organizing  the  arithmetic  part  of  the  elementary  school  mathematics  curriculum  around  mathematical  habits  of  mind  would  not  shift  the  curriculum  dramatically  in  content,  except  to  give  more  attention  to  mental  arithmetic  than  is  usual.  Paper  and  pencil  methods  are  engineered  to  make  the  work  easy,  to  reduce  the  cognitive  load,  the  amount  of  thinking  one  needs  to  do,  of  calculation.  Judiciously  chosen  mental  arithmetic  both  exercises  and  depends  on  mathematical  ways  of  thinking  that  the  paper-­‐and-­‐pencil  algorithms  deliberately  try  to  avoid,  mathematical  ways  of  thinking  that  are  the  backbone  of  the  algebra  that  we  want  to  prepare  children  to  succeed  at.  What  would  shift  is  the  order  in  which  we  acquire  that  content.  Instead  of  being  the  preparatory  step  for  computing,  algorithms  become  the  culmination  of  understanding  how  the  computation  works,  another  case  of  describing  what  we  already  know,  and  abbreviating  that  description.  ©  Education  Development  Center,  Inc.     page  9  
  • E.  Paul  Goldenberg,  June  Mark,  and  Al  Cuoco   The  algebra  of  little  kids  References   Cuoco, A., Goldenberg, E. P., & J. Mark. “Habits of mind: an organizing principle for mathematics curriculum” J. Math. Behav. 15(4):375-402. December, 1996. Cuoco, A., Goldenberg, E. P., and J. Mark. “Organizing a curriculum around mathematical habits of mind.” Mathematics Teacher. (submitted) Education Development Center, Inc. (EDC). Think Math! comprehensive K-5 curriculum. Boston: Houghton Mifflin Harcourt. 2008. Feigenson, L., Carey, S., & Spelke, E. (2002). Infants’ discrimination of number vs. continuous extent. Cognitive Psychology, 44, 33–66. Goldenberg, E. Paul. “‘Habits of mind’ as an organizer for the curriculum” J. of Education 178(1):13-34, Boston U. 1996. (Also “‘Hábitos de pensamento’ …”Educação e Matemática, 47 March/April, & 48 May/June, 1998.) Goldenberg, E. Paul & N. Shteingold. “Mathematical Habits of Mind.” In Lester, F., et al., eds. Teaching Mathematics Through Problem Solving: prekindergarten–Grade 6. Reston, VA: NCTM. 2003. Goldenberg, E. Paul & N. Shteingold “The case of Think Math!” In Hirsch, Christian, ed., Perspectives on the design and development of school mathematics curricula. Reston, VA: NCTM. 2007. Gopnik, A., Meltzoff, A., and P. Kuhl. The scientist in the crib: what early learning tells us about the mind. New York: HarperCollins. 2000. Mark, J., Cuoco, A., and Goldenberg, E. P. “Developing mathematical habits of mind in the middle grades.” Mathematics Teaching in the Middle School. (submitted) Piaget, J. The child’s conception of number. London: Routledge and Kegan Paul. 1952. Sawyer, W. W. Vision in elementary mathematics. New York: Dover Publications. 2003 (1964). Sfard, A. Thinking as Communicating. New York, NY: Cambridge University Press. 2008. Wirtz, R., Botel, M., Beberman, M., and W. W. Sawyer. 1964. Math Workshop. Encyclopaedia Britannica Press.©  Education  Development  Center,  Inc.     page  10