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Analyzing	
  the	
  role	
  of	
  math	
  	
  
in	
  scien3fic	
  thinking	
  
Edward	
  F.	
  (Joe)	
  Redish	
  
Departme...
Outline	
  
•  Mee3ng	
  each	
  other	
  
•  The	
  structure	
  of	
  mathema3cal	
  modeling	
  
•  Analy3c	
  tools	
 ...
GeUng	
  to	
  know	
  the	
  group:	
  
Some	
  ques3ons	
  
1.  Introduc3ons:	
  Who	
  are	
  we	
  and	
  what	
  clas...
My	
  background	
  
•  Ph.D.	
  in	
  theore3cal	
  nuclear	
  physics	
  
	
  –	
  25	
  years	
  as	
  a	
  prac3cing	
...
A	
  two-­‐step	
  analy3c	
  approach	
  
•  The	
  structure	
  	
  
of	
  mathema3cal	
  modeling	
  
•  How	
  we	
  t...
Modeling	
  mathema3cal	
  modeling	
  
•  Scien3fic	
  thinking	
  is	
  all	
  about	
  epistemology	
  –	
  
deciding	
 ...
Mathema3cal	
  modeling:	
  	
  
A	
  structural	
  analysis	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
  ...
In	
  physics,	
  math	
  integrates	
  with	
  our	
  
physics	
  knowledge	
  and	
  does	
  work	
  for	
  us	
  
•  Le...
Func3onal	
  dependence	
  
•  Fick’s	
  law	
  of	
  diffusion	
  
	
  
	
  
	
  
•  The	
  Hagen-­‐Poiseuille	
  equa3on	...
Packing	
  Concepts:	
  
Equa3ons	
  as	
  a	
  conceptual	
  organizer	
  
6/24/15	
   MathBench	
  Workshop,	
  College	...
Mathema3cs	
  as	
  a	
  way	
  of	
  knowing:	
  
Epistemology	
  
•  Math	
  in	
  science	
  is	
  not	
  just	
  for	
...
Analyzing	
  Epistemology:	
  
Dissec3ng	
  its	
  role	
  in	
  learning	
  science	
  
•  Understanding	
  student	
  be...
A	
  lot	
  of	
  what	
  students	
  do	
  	
  
makes	
  more	
  sense	
  if	
  we	
  consider	
  	
  
the	
  epistemolog...
Analy3c	
  tools	
  for	
  studying	
  epistemology	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   14	
  
•...
Intro
Physics
contextEpistemological	
  resources	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   15	
  
Kno...
Intro
Biology
contextEpistemological	
  resources	
  
6/24/15	
  
MathBench	
  Workshop,	
  College	
  Park	
   16	
  
Kno...
Epistemological	
  Resources:	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   17	
  
•  These	
  groupings	
...
1. Epistemological resources:
Example from NEXUS/Physics –
Recitation: Why do bilayers form?
6/24/15	
   MathBench	
  Work...
Prompt:	
  
…explain	
  how	
  phospholipids	
  can	
  spontaneously	
  self-­‐
assemble	
  into	
  a	
  lipid	
  bilayer…...
Disciplinary	
  epistemologies	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   20	
  
•  “in	
  terms	
  of	...
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   21	
  
Intro
Physics
context
Intro
Biology
context
Physical mapp...
Epistemological	
  framing	
  
•  Depending	
  on	
  how	
  students	
  interpret	
  the	
  situa3on	
  
they	
  are	
  in...
2.	
  Epistemological	
  Framing:	
  
Example	
  from	
  Biology	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park...
Ashley’s	
  response	
  
to	
  the	
  use	
  of	
  math	
  in	
  Org	
  Bio	
  
6/8/14	
   Gordon	
  Conference	
   24	
  ...
Another	
  response	
  of	
  a	
  student	
  	
  
to	
  math	
  in	
  Org	
  Bio	
  
6/8/14	
   Gordon	
  Conference	
   2...
Ashley’s	
  dynamic	
  switch	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   26	
  
“Biological	
  authen0c...
Epistemic	
  games:	
  
A	
  poten3ally	
  useful	
  tool	
  	
  
	
  •  Epistemic	
  game:	
  A	
  structured	
  ac3vity	...
3.	
  Example	
  from	
  NEXUS/Physics:	
  
Filling	
  in	
  missing	
  epistemic	
  games.	
  	
  
6/24/15	
   MathBench	...
Many	
  students	
  were	
  seriously	
  confused	
  	
  
and	
  didn’t	
  know	
  what	
  to	
  do	
  next.	
  
6/24/15	
...
A	
  useful	
  epistemic	
  game	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   30	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   31	
  
4.	
  Example	
  from	
  Algebra-­‐Based	
  Physics	
  
s...
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   32	
  
d	
  
0	
  
p = p0 + ρgd
pceiling = p0
pfloor = p0 + ρgh
...
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   33	
  
An	
  inappropriate	
  game	
  
•  One	
  student	
  deci...
Recursive	
  plug-­‐and-­‐chug	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   34	
  
Epistemological	
  stances:	
  	
  
The	
  “go-­‐to”	
  e-­‐resource	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  ...
The figure shows the PE of two interacting atoms as a function
of their relative separation. If they have the total energy...
How	
  two	
  different	
  professors	
  explained	
  it	
  
when	
  students	
  got	
  stuck.	
  
6/24/15	
   MathBench	
 ...
Wandering	
  around	
  the	
  class	
  while	
  students	
  	
  
were	
  considering	
  the	
  problem,	
  I	
  got	
  
a	...
I	
  conjecture	
  that	
  a	
  conflict	
  between	
  	
  
the	
  epistemological	
  stances	
  of	
  instructor	
  	
  
a...
Teaching	
  physics	
  	
  
standing	
  on	
  your	
  head	
  
6/24/15	
   MathBench	
  Workshop,	
  College	
  Park	
   4...
Conclusion	
  /Discussion	
  
•  Considering	
  the	
  way	
  we	
  teach	
  math	
  and	
  how	
  
students	
  respond	
 ...
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MathBench workshop

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A description of the analytical tools developed in Physics Education Research for understanding students use of and difficulties with mathematics as used in science.

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MathBench workshop

  1. 1. Analyzing  the  role  of  math     in  scien3fic  thinking   Edward  F.  (Joe)  Redish   Department  of  Physics   University  of  Maryland   6/24/15   MathBench  Workshop,  College  Park   1  
  2. 2. Outline   •  Mee3ng  each  other   •  The  structure  of  mathema3cal  modeling   •  Analy3c  tools  for  studying  epistemology   •  Mathema3cs  as  a  way  of  knowing:     The  epistemology  of  math  in  science   •  Analyzing  epistemology:     Its  role  in  learning  science   – Epistemological  resources   – Epistemological  framing   – Epistemic  games   – Epistemological  stances   6/24/15   MathBench  Workshop,  College  Park   2  
  3. 3. GeUng  to  know  the  group:   Some  ques3ons   1.  Introduc3ons:  Who  are  we  and  what  classes   are  we  working  on?  (individual)   2.  Why  do  we  think  math  is  important  for   biology?  (Discuss  in  groups,  summarize  on   flip  charts  in  A  FEW  SENTENCES)   3.  What  are  our  goals  for  the  development  of   specific  mathema3cal  skills  in  our  classes?   (Discuss  in  groups,  summarize  on  flip  charts  –   AS  MANY  AS  POSSIBLE)   6/24/15   MathBench  Workshop,  College  Park   3  
  4. 4. My  background   •  Ph.D.  in  theore3cal  nuclear  physics    –  25  years  as  a  prac3cing  researcher   •  Switched  fields  to  Physics  Educa3on  Research        –  25  years  as  a  prac3cing  researcher   •  My  educa3on  research  has  focused  on   –  Teaching  and  learning  scien3fic  reasoning   –  Cogni3ve  modeling  of  student  thinking   –  Epistemology   –  Use  of  math  in  science   •  Past  5  years:  Building  NEXUS/Physics     –  an  introductory  physics  class  designed     to  mesh  with  and  serve  the  curriculum     of  a  bio  or  pre-­‐med  student   6/24/15   MathBench  Workshop,  College  Park   4  
  5. 5. A  two-­‐step  analy3c  approach   •  The  structure     of  mathema3cal  modeling   •  How  we  think  about  and  use   mathema3cal  modeling.   6/24/15   MathBench  Workshop,  College  Park   5  
  6. 6. Modeling  mathema3cal  modeling   •  Scien3fic  thinking  is  all  about  epistemology  –   deciding  what  we  know  and  how  we  know  it.*   •  In  physics,  mathema3cs  has  become  3ghtly  3ed   with  our  epistemology  beginning  in  ~1700.   •  As  a  result,  physics  is  a  good  place  to  start  studying   the  role  of  math  in  science.  It  plays  a  significant   role  in  all  our  professional  instruc3on,  even  in  the   introductory  classes  (not  always  in  a  good  way,  however).   •  We  don’t  just  calculate  with  math,     we  build  knowledge  with  it  and  think  with  it.   6/24/15   MathBench  Workshop,  College  Park   6  * Karsai & Kampis, BioScience 60:8 (2010) 632-638.
  7. 7. Mathema3cal  modeling:     A  structural  analysis   6/24/15   MathBench  Workshop,  College  Park   7   •  Oien  these  all  happen  at  once  –  intertwined.   (not  meant  to  imply  an  algorithmic  process)   •  In  physics  classes,  oien  the  top  element  is  stressed     and  the  remaining  elements  are  oien  shortchanged.  
  8. 8. In  physics,  math  integrates  with  our   physics  knowledge  and  does  work  for  us   •  Lets  us  carry  out  chains  of  reasoning  that  are   longer  than  we  can  easily  do  in  our  head  by   using  formal  and  logic  represented  symbolically   – Calcula3ons   – Predic3ons   – Summary  and  descrip3on  of  data   •  Our  math  codes  for  conceptual  knowledge   – Func3onal  dependence   – Packing  concepts   – Epistemology  6/24/15   MathBench  Workshop,  College  Park   8  
  9. 9. Func3onal  dependence   •  Fick’s  law  of  diffusion         •  The  Hagen-­‐Poiseuille  equa3on  for  fluid  flow   6/24/15   MathBench  Workshop,  College  Park   9   Δr2 = 6DΔt ΔP = 8µL πR4 Q
  10. 10. Packing  Concepts:   Equa3ons  as  a  conceptual  organizer   6/24/15   MathBench  Workshop,  College  Park   10    aA =  FA net mA Force  is  what   you  have  to  pay   amen3on  to  when   considering  mo3on   What  mamers  is     the  sum  of  the  forces   on  the  object   being  considered   The  total  force   is  “shared”  to     all  parts  of   the  object   These  rela3ons   are  independently   true  for  each  direc3on.   You  have  to  pick   an  object  to  pay   amen3on  to   Forces  change   an  object’s   velocity   Total  force  (shared  over     the  parts  of  the  mass)  causes   an  object’s  velocity  to  change  
  11. 11. Mathema3cs  as  a  way  of  knowing:   Epistemology   •  Math  in  science  is  not  just  for  describing     what  we  see  in  a  compact  way.   •  Math  is  epistemological  –  it’s  a  way  of   genera3ng  new  knowledge.     6/24/15   MathBench  Workshop,  College  Park   11  
  12. 12. Analyzing  Epistemology:   Dissec3ng  its  role  in  learning  science   •  Understanding  student  behavior     is  considerably  more  complex  than  figuring  out   “what  they  know  and  what  they  don’t.”   •  When  we  pay  amen3on  to  the  combina3on     of  dynamic  mental  response  and  the  role  of   epistemology,  a  lot  of  student  responses     look  different  –  and  more  complex  –     than  just  "they  don't  get  it”  or  even  “they  have   a  wrong  mental  model  (misconcep3on)”.   6/24/15   MathBench  Workshop,  College  Park   12  
  13. 13. A  lot  of  what  students  do     makes  more  sense  if  we  consider     the  epistemological  reasoning  they  use.   •  The  resources  students  bring  to  bear     in  a  classroom  are  affected  by  their   epistemological  expecta0ons              What  is  the  nature  of  the  knowledge              that  we  are  learning              and  what  do  we  have  to  do  to  learn  it?   •  Student  responses  are  complex  and  dynamic.   •  The  key  is  understanding  what  epistemological   resources  they  have  and  expect  to  use.   6/24/15   MathBench  Workshop,  College  Park   13  
  14. 14. Analy3c  tools  for  studying  epistemology   6/24/15   MathBench  Workshop,  College  Park   14   •  Epistemological  resources  (e-­‐resources)*   –  Generalized  categories  of  “How  do  we  know?”   warrants.     •  Epistemological  framing*   –  The  process  of  deciding  what  e-­‐resources     are  relevant  to  the  current  task.     (NOT  necessarily  a  conscious  process.)   •  Epistemic  games**   –  A  coherent  procedure  for  assis3ng  in  crea3ng  or   recovering  knowledge  in  par3cular  circumstances.   •  Epistemological  stances   –  A  coherent  set  of  e-­‐resources  ac3vated  together   *Bing & Redish, Phys. Rev. ST-PER 5 (2009) 020108; 8 (2012) 010105. **Tuminaro & Redish, Phys. Rev. ST-PER 3 (2007) 020101.
  15. 15. Intro Physics contextEpistemological  resources   6/24/15   MathBench  Workshop,  College  Park   15   Knowledge constructed from experience and perception (p-prims) is trustworthy Algorithmic computational steps lead to a trustable result Information from an authoritative source can be trusted A mathematical symbolic representation faithfully characterizes some feature of the physical or geometric system it is intended to represent. Mathematics and mathematical manipulations have a regularity and reliability and are consistent across different situations. Highly simplified examples can yield insight into complex mathematical representations Physical intuition (experience & perception) Calculation can be trusted By trusted authority Physical mapping to math (Thinking with math) Mathematical consistency (If the math is the same, the analogy is good.) Value of toy models
  16. 16. Intro Biology contextEpistemological  resources   6/24/15   MathBench  Workshop,  College  Park   16   Knowledge constructed from experience and perception (p-prims) is trustworthy Physical intuition (experience & perception) Information from an authoritative source can be trusted By trusted authority The historical fact of natural selection leads to strong structure- function relationships in living organisms Many distinct components of organisms need to be identified Comparison of related organisms yields insight Learning a large vocabulary is useful Categorization and classification (phylogeny) Teleology justifies mechanism There are broad principles that govern multiple situations Heuristics Living organisms are complex and require multiple related processes to maintain life Life is complex (system thinking)
  17. 17. Epistemological  Resources:   6/24/15   MathBench  Workshop,  College  Park   17   •  These  groupings  of  resources  are  labeled     as  “Intro  Bio”  and  “Intro  Physics.”   •  This  is  to  indicate  that  these  are  epistemological   resources  commonly  perceived  by  students  as   relevant  in  their  intro  classes  in  these  subjects.   •  Professionals  in  both  fields  tend  to  use  both     of  these  sets  resources  (though  with  different   distribu3ons  and  depending  on  sub-­‐field).  
  18. 18. 1. Epistemological resources: Example from NEXUS/Physics – Recitation: Why do bilayers form? 6/24/15   MathBench  Workshop,  College  Park   18   Prompt:     Which  term  wins?  
  19. 19. Prompt:   …explain  how  phospholipids  can  spontaneously  self-­‐ assemble  into  a  lipid  bilayer…why  this  par3cular  shape?   6/24/15   MathBench  Workshop,  College  Park   19   Hollis:  I  mean,  in  terms  of  like  bio  and  biochem,  the  reason  why  it  forms  a   bilayer  is  because  polar  molecules  need  to  get  from  the  outside  to  the   inside  ...  so  you  need  a  polar  environment  inside  the  cell.  But  I  don't  know   how  that  makes  sense  in  terms  of  physics.  So...   Cindy:  So  like  what  I'm  saying  is,  you  have  to  have,  like  if  it's  hydrophobic  and   interac3ng  with  water,  then  it's  going  to  create  a  posi3ve  Gibb's  free  energy,   so  it  won't  be  spontaneous.  So,  in  this  case,  you  have  the  hydrophobic  tails   interac3ng  with  whatever's  on  the  inside  of  the  cell,  which  is  mostly  water,   right?   Hollis:  Or  other  polar  molecules.   Cindy:  Yeah,  other  polar  molecules.  It's  going  to  have,  and  that's  bad  ...   That's  a  posi3ve  Gibb's  free  energy...[proceed  to  unpack  in  terms  of  posi3ve   (energe3c)  and  nega3ve  (entropic)  contribu3ons  to  the  equa3on.]   Hollis:  Yes.  See,  you  explained  it  perfectly  ...  Cause  I  was  thinking  that,  but  I   wasn't  thinking  it  in  terms  of  physics.  And  you  said  it  in  terms  of  physics,  so,   it  matched  with  bio.  
  20. 20. Disciplinary  epistemologies   6/24/15   MathBench  Workshop,  College  Park   20   •  “in  terms  of  bio,  the  reason  why  it  forms  a   bilayer  is  because  polar  molecules  need  to  get   from  the  outside  to  the  inside”       •  “  if  it’s  hydrophobic  and  interac3ng  with  water,   then  it's  going  to     create  a  posi3ve     Gibb's  free  energy,     so  it  won't  be     spontaneous  and     that’s  bad..”  
  21. 21. 6/24/15   MathBench  Workshop,  College  Park   21   Intro Physics context Intro Biology context Physical mapping to math (Thinking with math) Teleology justifies mechanismSatisfaction (smile, fist pump) Interdisciplinary coherence seeking “Interdisciplinary  coherence”  –   •  Coordinated  resources  from     intro  physics  and  biology   •  Blended  context   •  Posi0ve  affect  
  22. 22. Epistemological  framing   •  Depending  on  how  students  interpret  the  situa3on   they  are  in  and  their  learned  expecta3ons,  they   may  not  think  to  call  on  resources  they  have  and   are  competent  with.   •  This  can  take  many  forms:   –  “I’m  not  allowed  to  use  a  calculator  on  this  exam.”   –  “It’s  not  appropriate  to  include  diagrams  or  equa3ons     in  an  essay  ques3on.”   –  “This  is  a  physics  class.  He  can’t  possibly  expect  me     to  know  any  chemistry.”   •  This  can  coordinate  strongly  with  affec3ve   responses   6/24/15   MathBench  Workshop,  College  Park   22  
  23. 23. 2.  Epistemological  Framing:   Example  from  Biology   6/24/15   MathBench  Workshop,  College  Park   23   •  Biology  III:  Organismal  Biology   – A  principles-­‐based  class  that  structures     the  tradi3onal  “forced  march  through  the  phyla”     of  a  biological  diversity  class.   •  Some  of  the  principles:   – Common  ancestry  (deep  molecular  homology)   – Individual  evolved  historical  path)   (divergent  structure-­‐func3on  rela3onships)     – Constrained  by  universal  chemical  and  physical  laws.   •  Uses  Group  Ac3ve  Engagement  (GAE)  lessons   (including  math!)   “Todd the biologist”
  24. 24. Ashley’s  response   to  the  use  of  math  in  Org  Bio   6/8/14   Gordon  Conference   24   I  don’t  like  to  think  of  biology  in  terms  of  numbers     and  variables….  biology  is  supposed  to  be  tangible,  perceivable,   and  to  put  it  in  terms  of  lemers  and  variables  is  just  very   unappealing  to  me….  Come  3me  for  the  exam,  obviously  I’m  going   to  look  at  those  equa3ons  and  figure  them  out  and  memorize   them,  but  I  just  really  don’t  like  them.     I  think  of  it  as  it  would  happen  in  real  life.  Like  if  you  had  a  thick   membrane  and  tried  to  put  something  through  it,  the  thicker  it  is,   obviously  the  slower  it’s  going  to  go  through.  But  if  you  want  me  to   think  of  it  as  “this  is  x  and  that’s  D  and  this  is  t”,  I  can’t  do  it.   Discussing  the  use  of  Fick’s  Law     in  controlling  diffusion  through     a  membrane  of  different  thicknesses.  
  25. 25. Another  response  of  a  student     to  math  in  Org  Bio   6/8/14   Gordon  Conference   25                                        The  limle  one  and  the  big  one,  I  never  actually                                          fully  understood  why  that  was.  I  mean,  I                                          remember  watching  a  Bill  Nye  episode  about                                          that,  like  they  built  a  big  model  of  an  ant  and  it                                          couldn’t  even  stand.  But,  I  mean,  visually  I  knew                                          that  it  doesn’t  work  when  you  make  limle  things                                          big,  but  I  never  had  anyone  explain  to  me  that   there’s  a  mathema3cal  rela3onship  between  that,  and  that   was  really  helpful  to  just  my  general  understanding  of  the   world.  It  was,  like,  mind-­‐boggling.   The  small  wooden  horse  supported  on  dowels  stands   with  no  trouble.  When  all  dimensions  are  doubled,   however,  the  larger  dowels  break,  unable  to  support  the   weight.   Watkins & Elby, CBE-LSE. 12 (2013) 274-286
  26. 26. Ashley’s  dynamic  switch   6/24/15   MathBench  Workshop,  College  Park   26   “Biological  authen0city”  –   •  Coordinated  math  and  intui0on   •  In  a  biological  context   •  Posi0ve  affect   •  Significant  value  for   understanding  biology  
  27. 27. Epistemic  games:   A  poten3ally  useful  tool      •  Epistemic  game:  A  structured  ac3vity  usable     for  approaching  a  variety  of  knowledge  building  tasks     and  problems.  It  has  an  entry  point,  rules,  an  end  point.     –  Making  a  list   –  Compare  and  contrast   –  Cost-­‐benefit  analysis   –  Mechanism  analysis  (3me,  space,  rela3onships)   –  Recursive  plug-­‐and-­‐chug   6/24/15   MathBench  Workshop,  College  Park   27   Collins & Ferguson, Educ. Psychol. 28 (1993) 25 Bing & Redish, Phys. Rev. ST-PER 5 (2009) 020108; Bing & Redish, Phys. Rev. ST-PER 8 (2012) 010105 Tuminaro & Redish, Phys. Rev. ST-PER 3 (2007) 020101.
  28. 28. 3.  Example  from  NEXUS/Physics:   Filling  in  missing  epistemic  games.     6/24/15   MathBench  Workshop,  College  Park   28   When  a  small  organism  is  moving  through  a  fluid,     it  experiences  both  viscous  and  inerCal  drag.     The  viscous  drag  is  proporConal  to  the  speed  and  the   inerCal  drag  to  the  square  of  the  speed.  For  small   spherical  objects,  the  magnitudes  of  these  two  forces   are  given  by  the  following  equaCons:   Fv = 6πµRv Fi = CρR2 v2 For  a  given  organism  (of  radius  R)  is  there  ever   a  speed  for  which  these  two  forces  have  the   same  magnitude?  
  29. 29. Many  students  were  seriously  confused     and  didn’t  know  what  to  do  next.   6/24/15   MathBench  Workshop,  College  Park   29   •  “Should  I  see  if  I  can  find  all  the  numbers     on  the  web?”   •  “I  don’t  know  how  to  start.”   –  “Well,  it  says  ‘Do  they  ever  have  the  same  magnitude?’  How  do   you  think  you  ought  to  start?   •  “Set  them  equal?”   –  “OK.  Do  it.”   •  “I  don’t  know  what  all  these  symbols  mean.”   –  “Well  everything  except  the  velocity  are  constants  for  a  parCcular   object  in  a  parCcular  situaCon.”   •  “Oh!  So  if  I  write  it  ....  Av  =  Bv2...  Wow!  Then  it’s  easy!”  
  30. 30. A  useful  epistemic  game   6/24/15   MathBench  Workshop,  College  Park   30  
  31. 31. 6/24/15   MathBench  Workshop,  College  Park   31   4.  Example  from  Algebra-­‐Based  Physics   showing  how  e-­‐games  interact  with  framing.   •  The  following  problem  was  given  at  the  end  of   the  first  semester  of  an  introductory  class.   – EsCmate  the  difference  in  air  pressure     between  the  floor  and  the  ceiling     in  your    dorm  room.  (Note:  you  may     take  the  density  of  air  to  be  1  kg/m3.)   •  A  student  working  on  this  problem  framed     the  task  incorrectly  and  got  trapped  playing     the  wrong  game.  
  32. 32. 6/24/15   MathBench  Workshop,  College  Park   32   d   0   p = p0 + ρgd pceiling = p0 pfloor = p0 + ρgh pfloor − pceiling = ρgh ≈ 1 kg m3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 10 N kg ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3 m( ) = 30 N m2 = 30 P
  33. 33. 6/24/15   MathBench  Workshop,  College  Park   33   An  inappropriate  game   •  One  student  decided  she  needed  an  equa3on     for  pressure:  She    chose  PV  =  nRT.   •  She  decided  she  needed     the  volume  for  the  room.   •  She  decided  it  must  be  1  m3.  (?!)   •  She  maintained  that,  despite  TA’s  hint,     “I  think  you’ll  agree  with  me  this  is     an  es3ma3on  problem.”   •  She  decided  if  it  wasn’t  1  m3,  then  the  prof   probably  gave  the  value  in  a  previous  HW.  
  34. 34. Recursive  plug-­‐and-­‐chug   6/24/15   MathBench  Workshop,  College  Park   34  
  35. 35. Epistemological  stances:     The  “go-­‐to”  e-­‐resource   6/24/15   MathBench  Workshop,  College  Park   35   •  Both  students  and  faculty  may  have   developed  a  pamern  of  choosing   par3cular  combina3ons  of  e-­‐resources.   •  The  epistemological  stances  naturally   taken  by  physics  instructors  and  biology   students  may  be  drama3cally  different     –  even  in  the  common  context     of  a  physics  class.  
  36. 36. The figure shows the PE of two interacting atoms as a function of their relative separation. If they have the total energy shown by the red line, is the force between the atoms when they are at the separation marked C attractive or repulsive? C BA Total energy r Potential Energy 6/24/15   MathBench  Workshop,  College  Park   36   5.  Epistemological  stances:     An  example  from  NEXUS/Physics  
  37. 37. How  two  different  professors  explained  it   when  students  got  stuck.   6/24/15   MathBench  Workshop,  College  Park   37   •  Remember!                      (or  here)   •  At  C,  the  slope  of  the  U  graph  is  posi3ve.   •  Therefore  the  force  is  nega3ve  –     towards  smaller  r.   •  So  the  poten3al  represents     an  amrac3ve  force  when     the  atoms  are     at  separa3on  C.    F = −  ∇U F = − dU dr This figure was not actually drawn on the board by either instructor.
  38. 38. Wandering  around  the  class  while  students     were  considering  the  problem,  I  got   a  good  response  using  a  different  approach.   6/24/15   MathBench  Workshop,  College  Park   38   •  Think  about  it  as  if  it  were  a  ball  on  a  hill.   Which  way  would  it  roll?    Why?   •  What’s  the  slope  at  that  point?   •  What’s  the  force?   •  How  does  this  relate     to  the  equa3on   F = − dU dr
  39. 39. I  conjecture  that  a  conflict  between     the  epistemological  stances  of  instructor     and  student  make  things  more  difficult.   6/24/15   MathBench  Workshop,  College  Park   39   Calculation can be trusted By trusted authority Physical mapping to math (Thinking with math) Physical intuition (experience & perception) Physical mapping to math (Thinking with math) Mathematical consistency (If the math is the same, the analogy is good.) Physics  instructors   seem  more  comfortable   beginning  with  familiar   equa3ons  –  which  we     use  not  only     to  calculate   with,  but  to  code     and  remind  us     of  conceptual     knowledge.   Most  biology  students   lack  the  experience     blending  math  and     conceptual  knowledge,   so  they  are  more     comfortable   beginning  with   physical  intui3ons.  
  40. 40. Teaching  physics     standing  on  your  head   6/24/15   MathBench  Workshop,  College  Park   40   •  For  physicists,  math  is  the  “go  to”   epistemological  resource  –  the  one  ac3vated  first   and  the  one  brought  in  to  support  intui3ons     and  results  developed  in  other  ways.   •  For  biology  students,  the  math  is  decidedly   secondary.  Structure/func3on  rela3onships  tend   to  be  the  “go  to”  resource.   •  Part  of  our  goal  in  teaching  physics  to  second   year  biologists  is  to  improve  their  understanding   of  the  poten3al  value  of  mathema3cal  modeling.   This  means  teaching  it  rather  than  assuming  it.  
  41. 41. Conclusion  /Discussion   •  Considering  the  way  we  teach  math  and  how   students  respond  using  our  four  analy3c  tools   (e-­‐resources,  e-­‐framing,  e-­‐games,  &  e-­‐stances)   appears  to  help  and  give  us  insight  into   teaching  math  to  biology  students  in  a  physics   class.   •  Might  such  analyses  be  of  any  use  for  using   math  in  biology?   6/24/15   MathBench  Workshop,  College  Park   41  

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