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Mathematics as a Creative Endeavor

Dana Ernst
September 25, 2014

Mathematics as a Creative Endeavor

Is Mathematics Communication? How do we communicate with Mathematics? Is it only about numbers? Join Dana Ernst, Assistant Professor of Mathematics, Terry Blows, Professor of Mathematics, and Nancy Barron, Professor of English and Director of the Interdisciplinary Writing Program, who will answer these questions and more in two stimulating presentations.

This talk was given at the Fall 2014 Liberal Studies Town Hall on Thursday, September 25.

Dana Ernst

September 25, 2014
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  1. Is  Mathema*cs  Communica*on?  
     
    Mathema*cs  as  a  Crea*ve  Endeavor  
    1:00-­‐1:50  
    Thursday  September  25th  
     
    Terence  Blows  &  Dana  Ernst  

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  2. To  build  a  house  …  
    …  you  start  with  a  plan  and  
    a  box  of  tools  

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  3. To  do  mathemaCcs…  
    …  you  start  with  a  plan  and  
    a  box  of  tools  

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  4. Mathema*cs  
     
    For  most  people  mathema&cs  means  the  subject    of  that  name  done  
    in  schools.  To  those  in  the  discipline  this  is  not  mathemaCcs  just  the  
    building  of  skills  in  order  to  do  mathemaCcs.    
     
    At  NAU  we  offer  two  skills  courses:  MAT  100  (all  majors)  and  MAT  108  
    (STEM  majors).  These  cover  content  seen  in  high  school  algebra  
    classes.    
     
    We  start  doing  mathemaCcs  in  Liberal  Studies  MathemaCcs  
    FoundaCon  courses  such  as  MAT  114.    
     
    While  some  skills  are  sCll  being  developed  in  courses  such  as  MAT  
    119,  MAT  125,  MAT  136  etc.  these  courses  do  involve  some  
    mathemaCcs.  

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  5. Liberal  Studies  
     
    The  Liberal  Studies  Program  at  Northern  Arizona  University  currently  
    has  five  defined  essen&al  skills:    
     
     
     
     
     
     
     
    The  remedial  courses  MAT  100  and  MAT  108  involve  the  building  of  
    skills  and  involve  li\le  in  the  way  of  essenCal  skills.  But  beyond  these  
    effec&ve  wri&ng  is  an  important  component  in  many  of  our  classes.  
    As  much  so  as  cri&cal  thinking  and  quan&ta&ve  reasoning.  
    •  CriCcal  thinking  
    •  EffecCve  wriCng  
    •  EffecCve  oral  communicaCon  
    •  ScienCfic  inquiry  
    •  QuanCtaCve  reasoning  

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  6. Cri*cal  Reasoning  
     
    Before  we  get  into  the  subject  of  mathemaCcs  as  a  creaCve  
    endeavor  here  are  a  couple  of  examples  of  problems  from  MAT  114  
    that  involve  criCcal  reasoning.  
     
    In  the  first  we  have  shown  students  how  to  find  the  median  and  
    quarCles  of  a  data  set  and  how  to  draw  a  box  plot.  
     
    In  the  second  we  have  introduced  students  to  formulas  from  
    personal  finance    involving    compound  interest,  inflaCon,  savings  
    and  loans.    They  learn  which  numbers    go  where  in  the  formula  and  
    how  to  carry  out  quite  complex  calculaCons  on  their  calculators  .  
    (We  also  have  them  build  an  amorCzaCon  table  using  Excel.)  

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  7. Example  
     
    In  the  first  half  of  2011  civil  unrest  led  to  the  toppling  of  certain  
    regimes  in  north  Africa.  Part  of  the  displeasure  of  the  people  came  
    from  the  fact  that  these  countries  had  a  very  small  elite  of  wealthy  
    ciCzens  while  most  were  poor  -­‐  indeed  over  half  the  populaCon  
    lived  in  poverty.    Which  of  the  following  box  plots,  represenCng  
    annual  income,  best  represents  this  situaCon?    
    LOW   HIGH  

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  8. Example  
     
    Suppose  you  invest  a  certain  amount  of  money  each  month  at  a  
    fixed  rate  of  interest  to  give  you    a  nest  egg  a  number  of  years  in  the  
    future.  
     
    True/False  
     
    (i)      If  you  doubled  your  monthly  investment  then  the  value  of  your  
    nest  egg  would  also  double.  
     
    (ii)    If  your  broker  could  find  you  an  investment  with  twice  the  
    return  then  the  value  of  your  nest  egg  would  also  double.  
       
    (iii)  If  you  put  money  for  twice  as  long    then  the  value  of  the  nest  
    egg  would  also  double.  
     

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  9. Crea*ve  Endeavor  
     
    The  above  examples,  although  different  from  what  has  been  taught  
    do  not  represent  anything  creaCve.  
     
    Merriam-­‐Webster  On-­‐line  Dic*onary  
     
    1cre·∙a·∙*ve  
     
    adjec&ve  \krē-­‐ˈā-­‐Cv,  ˈkrē-­‐ˌ\    
     
    •  having  or  showing  an  ability  to  make  new  things  or  think  of  new  
    ideas  
    •  using  the  ability  to  make  or  think  of  new  things  :  involving  the  
    process  by  which  new  ideas,  stories,  etc.,  are  created  
    •  done  in  an  unusual  and  oken  dishonest  way  

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  10. “I  believe  there  is,  in  mathemaCcs,  in  contrast  to  the  
    experimental  disciplines,  a  character  which  is  nearer  
    to  that  of  free  creaCve  art.”    -­‐  Hermann  Weyl  
    “MathemaCcs,  rightly  viewed,  possesses  not  only  
    truth,  but  supreme  beauty.”  -­‐  Bertrand  Russell  

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  11. There  is  evidence  that  the  relaCon  of  arCsCc  beauty  
    and  mathemaCcal  beauty  is  more  than  an  analogy.  
    Zeki  et  al.  recently  published  a  paper  that  suggests  
    the  same  part  of  the  brain  responds  to  both.  
     
    “The  experience  of  mathemaCcal  beauty  and  its  
    neural  correlates”,  FronCers  in  Human  Neuroscience  
    (2014).  

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  12. Three  strangers  meet  at  a  taxi  stand  and  decide  to  share  a  cab  to  
    cut  down  the  cost.  Each  has  a  different  desCnaCon  but  all  are  
    heading  in  more-­‐or-­‐less  the  same  direcCon.  How  much  should  
    each  contribute  to  the  total  fare?  
    Burger  &  Starbird’s  Taxi  Problem  
    What  is  the  most  
    common  answer  
    to  this  problem?  

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  13. Problem  1  
    An  ant  is  crawling  along  the  edges  of  a  unit  cube.  What  
    is  the  maximum  distance  it  can  cover  starCng  from  a  
    corner  so  that  it  does  not  cover  any  edge  twice?  
    Problems  to  work  

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  14. Problem  2  
    A  mouse  eats  his  way  through  a  3  ×  3  ×  3  cube  of  cheese  
    by  tunneling  through  all  of  the  27  1  ×  1  ×  1  sub-­‐cubes.  If  
    he  starts  at  one  corner  and  always  moves  to  an  uneaten  
    sub  cube,  can  he  finish  at  the  center  of  the  cube?  

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  15. Problem  3  
    Take  15  poker  chips  and  divide  them  into  any  number  of  piles  
    with  any  number  of  chips  in  each  pile.  Arrange  piles  in  
    adjacent  columns.  Take  the  top  chip  off  every  column  and  
    make  a  new  column  to  the  lek.  Repeat  forever.    
     
    What  happens?    
     
    Make  conjectures  about  what  happens  when  we  change  the  
    number  of  chips.  

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  16. Problem  4  
    Show  that  in  any  group  of  6  students  there  are  3  students  
    who  know  each  other  or  3  students  who  do  not  know  each  
    other.  

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