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Mathematics as a Creative Endeavor

77d59004fef10003e155461c4c47e037?s=47 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  
  2. To  build  a  house  …   …  you  start  with

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

     and   a  box  of  tools  
  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.  
  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  
  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.)  
  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  
  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.    
  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  
  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  
  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).  
  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?  
  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  
  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?  
  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.  
  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.